Signature Redacted Signature of Author

A SEARCH FOR ASTRONOMICAL GRAVITATIONAL RADIATION

WITH AN INTERFEROMETRIC BROAD BAND ANTENNA

DANIEL DEWEY B.S., Massachusetts Institute of Technology (1979)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN PHYSICS

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY February, 1986

@ Massachusetts Institute of Technology 1986

Signature redacted Signature of Author ...... ,w . - ... epartment of Physics Signature redacted January 17, 1986 Certified by . v ...... Rainer Weiss Thesis Supervisor redacted AcceptedAcce by...... Signature tedby.. . . . g .- n . \. .. /. , -. .x . . George Koster Chairman, Department Committee

FEB 1 4 198&

Archive'1 A SEARCH FOR ASTRONOMICAL GRAVITATIONAL RADIATION

WITH AN INTERFEROMETRIC BROAD BAND ANTENNA

DANIEL DEWEY

Submitted to the Department of Physics on January 17, 1986 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Physics

ABSTRACT This thesis describes and implements a data analysis scheme designed to detect short, burst-like gravity waves in the output of the interferometric antennas. This represents a first attempt at integrating the astrophysical predictions of expected waveshapes with a data analysis scheme for their detection. An experimental program was carried out involving the construction of a prototype gravitational radiation antenna, the diagnosis of some of its noise sources, the study of expected astronomical sources of radiation, and the collection and analysis of data from the instrument. Construction included the implementation of a system to damp the motions of the test masses, the design of a servo system to hold the interferometer to a dark fringe and the assembly of a microcomputer-A/D system to record data and instrument parameters. Diagnosis of noise sources identified noise due to scattered light as a problem, and methods to suppress this noise through external phase modulation of the input laser beam were devised and implemented. The results of the analysis are encouraging from an instrument performance perspective; the noise obeys Gaussian statistics to signal-to-noise ratios of 5.5, and the number of events in the non-Gaussian tail is not excessive, ~ 500 per day. As a detector of gravity waves, the sensitivity of the prototype is very low due to its size, remaining noise sources, and low light power. Detected burst events with amplitudes of ho ~ 5 x 10- 4 correspond to the signal expected from an optimisti- cally large source at the distance of Proxima Centauri, the closest star beyond our Sun.

Thesis Supervisor: Dr. Rainer Weiss Title: Professor of Physics

-2- Scientific Acknowledgments

This work is greatly indebted to Prof. Rainer Weiss for both its scientific content and flavor. A project of this size is the work of many people and this work owes much to each of them. In June of 1985, the MIT gravity group consisted of Rai Weiss, Paul Linsay, Peter Saulson, Andrew Jeffries, Jeff Livas and Richard Benford, with past member David Shoemaker. Specifics of the division of labor are given at the end of the Introduction. Thanks to Steve Meyer for introducing noise into my work. Paul Linsay introduced and addicted me to servo systems. It was a pleasure to share the intensive week of data taking with Jeff Livas and the egg timer. Thanks also to Andrew Jeffries for several days and nights of helpful organization during this period-lots of orange juice. Interactions with the MPQ Garching group has been a source of pleasure and scientific advance. The task of putting together this thesis was made infinitely easier and more satisfying through pretty plot programs and TEX, both due to Ed Cheng. Thanks to David Shoemaker for a crash course in TEX. Nearly as essential were the XEROX machine and two wonderful Rapidograph pens. Thanks to Ed Cheng, David Shoemaker and Claude Canizares for proof reading this thesis. All errors, however, are the author's.

This work supported through NSF grant PHY 8109581.

-3- Personal Acknowledgments

Graduate studies and research can be hazardous to both body and mind. To all who have added joy, from the passing smiling stranger to the closest of friends, I acknowledge my debt and offer my thanks. In particular, the early years of graduate study were positively fun thanks to Luz Martinez-Miranda, Steve Evangelides, Mehran Kardar, Bruce McClain, and Mary-Lou Powers. The warm, friendly atmosphere of the Ashdown House Coffee Hours, under the care of Prof. and Mrs. Hulsizer, contributed immensely to the enjoyment of these years. Very special thanks to Chris for many shared years of love. Thanks to all in "the lab" for the sense of community and the knowledge that they're always there. Special thanks to Richard Benford for providing much of this lab spirit and overlooking gross incompetence in the machine shop. Thanks to Kathreen Gimbrere, a.k.a. "the neighborhood pest", for several fun evenings when they were needed most. Lyman Page aided in the writing of this thesis by making periodic visits to the author's office, instilling humor, confidence and determination. The "extended family" provided by Steve Meyer, Sharon Salveter, and Ed Cheng provided essential and appreciated support during the writing of this thesis. Special thanks to Ed for an altogether not unpleasurable living arrangement, including habitually "yummy" meals and leftovers. Tony Patera provided his hard-boiled egg recipe and many crazy ideas that have pleasantly punctuated the past ten years. Thanks to Mehran for the five thousand six hundred and sixteen sweaty games of squash, and as many locker room conversations, which kept the body in shape and saved the mind from many an abyss. A smiling wink to David Shoemaker, who introduced me to homemade pasta, cloth napkins, dBs and BDs, the fun of working together, carburetors, the "slap" bass, black jeans, the National Audio-Radio Handbook, Italy, the END DO, empty letters, and most recently TEX. Thank you, David. The love and faith of my family has sustained me for the past 28 years. A heartfelt thanks to Dan, Jean, Tim, Bill, John and Kate.

-4- TO MY GRANDPARENTS

Daniel and Georgia Gray Dewey

William and Jean Long Moyles

- 5- Table of Contents

1 Introduction 10

2 Theory of Gravity Waves 18 2.1 Introduction 18 2.2 Propogation and Polarization of Gravity Waves 18 2.3 Generation of Gravity Waves 20 2.4 Detection of Gravity Waves 21 2.4.1 Resonant ("Bar") Antenna Operation 21 2.4.2 Free-mass ("Interferometric") Antenna Operation 22 2.4.3 Comparison of Antenna Sensitivities to Burst Sources 23

3 Astrophysical Sources 24 3.1 Introduction 24 3.2 System of Units 24 3.3 Detectability of Burst Events 25 3.4 Particle-Blackhole Events 26 3.5 Stellar Collapse Events 28 3.6 Chirp Events 29 3.6.1 Chirp Source Waveforms and Detectability 30

4 The Instrument 33 4.1 Introduction 33 4.2 Mechanical 33 4.3 A Walk through the Optical Components 36 4.3.1 Laser 36 4.3.2 External Phase Modulator 36 4.3.3 Fiber 37 4.3.4 Central Mass 38 4.3.5 Delay Lines 38 4.3.6 Photodetectors 40 4.4 Electronic Systems 40 4.4.1 Mass Damping System 40 4.4.2 Fringe Servo System 41 4.4.3 Monitoring and Data taking 41

5 Noise Sources and the Prototype Noise Performance 42 5.1 Noise Sources 42 5.2 Prototype Noise Performance 42 5.2.1 Shot Noise 44 5.2.2 Electronic Noise 45 -6- 5.2.3 Laser Amplitude Noise 45 5.2.4 Seismic and Acoustic Noise 46 5.2.5 Scattered Light Noise 47 5.2.6 Laser Frequency Noise 50 5.2.7 Thermal Noise 51 5.2.8 Beam Jitter 53 5.2.9 E/M Fields 54

6 Data Analysis Scheme 55 6.1 Introduction 55 6.2 The Matched Filter 55 6.2.1 Signal to Noise Ratio (SNR) 56 6.2.2 Pulse Height Distribution (PHD) 57 6.3 Practical Considerations 60 6.3.1 Sample Rate and the White Assumption 60 6.3.2 The Effect of Mismatched Templates 60 6.3.3 Multiple Detections 61 6.4 A Set of Templates 62 6.4.1 The N 1 , No, NHC Templates 62 6.4.2 Choosing a Subset of the Templates 63 6.4.3 Implementation 64

7 The Data Taking Run and its Analysis 66 7.1 The Run 66 7.2 Analyzing the Data 66 7.2.1 Introduction and Overview 66 7.2.2 The HKP and PHD files 68 7.2.3 Data Synthesis 84 7.2.4 Windows and the Final Data Set 84 7.2.5 Final PHDs and a List of Events 87 7.3 Examination of the Events 94 7.3.1 Template "102" Events 94 7.3.2 Other Events 97

8 Discussion of the Results 106 8.1 An Instrument Performance Perspective 106 8.2 An Astrophysical Perspective 106 8.3 Conclusions 108

References 111

-7- Appendices

A Quadrupole Radiation from Masses in a Circular Orbit 116

B Thermal Limit for a Bar Antenna 117

C Interferometer Response 118

D Comparison of Bar and Interferometer Response to GW Bursts 120 D.1 The Canonical Form of a Burst of Gravitational Radiation 120 D.2 Interferometer Response 120 D.3 Bar Response 121 D.4 Comparing Bar and Interferometer 122 D.5 Comparison for an Arbitrary Wave Shape 123 D.6 Discussion 123

E Mass Damping System 130 E.1 Overview 130 E.1.1 The Sensing Capacitor Plates 133 E.1.2 The HV Drive Plates 133 E.2 Capacitive Displacement Transducer 136 E.2.1 The Capacitance Bridge 138 E.2.2 The Mixer 140 E.2.3 Transfer Functions from x to VCMO and VCMON 141 E.3 Equations of Motion 144 E.3.1 Exact Damping Equations and the Root-Locus 146 E.4 Damping System Noise 148 E.5 Parameter Trade-offs 151

F The RF Modulation/Demodulation Scheme and its Noise Sources 152 F.1 Overview 152 F.2 The Modulation/Demodulation Scheme 152 F.3 Noise Terms 155 F.3.1 Shot Noise 155 F.3.2 Thermal and Mixer Noise 156 F.3.3 RF Amplitude Noise 157 F.3.4 The Total Noise vs. b Plots 157

G Driving the Internal Pockels Cells 160

H The Loop System 163 H.1 Overview 163 H.2 Linear Analysis of the Loop System 167

-8- H.2.1 Pockels Loop 167 H.2.2 Mass Loop 169 H.2.3 Transfer Functions of Interest -Signal and Noise in the Loops 176 H.3 Nonlinear Operation of the Loop System 182 H.3.1 The Fringe Circuit 182 H.3.2 Motions that Cause Loss of Lock 182

I Attenuation of Scattered Light Noise with Phase Modulation 188 I.1 The Problem 188 1.2 Single Tone Modulation 189 1.3 Gaussian Noise Modulation 192 1.4 Pseudo-random Digital Modulation 194 1.5 Experimental Details and Results 196

J Derivations Related to the Matched Filter 201 J.1 The Matched Filter Gives the Optimum SNR 201 J.2 The SNR Formula for Digitized Signals 202 J.3 Calculating SNR/SNRoptimum for Mismatched Templates 203 J.3.1 Mismatch of the Component Pulse Shape 203 J.3.2 Mismatch in the Number of Component Pulses 204

K The Data Taking System 205 K.1 A/D Systems 205 K.2 Clock and Timing Board 205 K.3 The Data Format 208

L Assigning ho Values to the Events 211

-9- 1. Introduction

The prediction of gravitational radiation, "gravity waves", was made by Ein- stein in 1916 and has remained, until very recently, an unverified consequence of the theory of general relativity. One result of the theory, which suggests the existence of gravity waves, is the finite speed of propagation of the gravitational field, limited to the speed of light. This means that if the Sun were to vanish, not only would we see the sun for an additional eight minutes, the time required by the last emitted rays of light to reach us, but the gravitational pull of the sun that holds us in orbit would remain during those eight minutes as well. The "knowledge" that the sun has disappeared propagates both optically and gravitationally at the speed of light. Thus, anytime matter in the universe moves, the change in its gravitational field is propagated throughout space and gives rise to gravity waves. As the discussion above suggests, the detection of gravity waves is strongly linked to astrophysical phenomena. There are two reasons for this. First, gravity waves that could be generated in the laboratory are too weak to be detected by present day detectors; the equivalent of the Hertz radio wave (i.e., light wave) experiment is not feasible for gravity waves. Second, many of the unknown and exciting areas of astrophysics involve phenomena that may be better understood through their gravity wave emissions. These include events like the collapse of stars during supernovae, the encounters of matter with black holes, and the spiraling to coalescence of two compact objects (neutron stars, black holes) in a decaying binary system. The field of gravity wave detection had its beginnings in the 1960's with the work of Weber. The end of that decade saw the publication by Weber (1969) of "Evidence for discovery of gravitational radiation". Though Weber's results were not confirmed, the reported coincidences between separated detectors lead to interest in the field and an increase in experimental efforts to detect gravity waves In particular, many "Weber bars" were built. These "bar" antennas operate by detecting the oscillations a passing gravity wave sets up in a large homogeneous cylinder of material, typically aluminum. The bars are carefully isolated from laboratory disturbances. In addition, because a bar vibrates simply due to its warmth, the bars are, now, cooled to close to absolute zero to reduce these thermal vibrations. In the early 1970's a new approach to the detection of gravity waves began to look promising: the laser interferometric antenna. In this antenna a laser beam is used to measure the very small changes in the separation of "free" masses that are predicted to be the result of a passing gravity wave. Here, "very small" means 10- 7 of a meter, one ten-millionth the size of an atom! That it might be possible to measure these very small changes in length with a laser interferometer was demonstrated by Blum and Weiss in 1967. Applying the technique to the detection

- 10 - of gravity waves was put forth by Moss, Miller and Forward in 1971 and by Weiss in 1972. Two important properties of the interferometric antenna design are its broad band nature and the scaling of its sensitivity with the separation of the masses (explicitly the optical path length). The broad band nature of the interferometric antenna design allows it to be used in a search for many types of gravity wave emissions: periodic signals, bursts, and "chirp" waveforms. The data analysis scheme employed determines the type of signal actually being search for. The scaling with size promises to aid in overcoming many potential noise sources in the instrument. The mid 1970's found room temperature bars nearing limits set by thermal noise and transducer technology. Even so impressive searches had been carried out at levels far more sensitive than Weber's. In particular, the extensive search made by the Munich/Frascati teams, see Kafka and Shnuipp 1978, represents 580 days of coincidence data, yielding no gravity wave events. With the rise in experiments, a corresponding growth in theoretical papers appeared. The null results of the searches, though disappointing, were not surprising in view of the expected strength and rate of predicted events. The one test for gravity waves that has had positive results, Taylor and Weis- berg 1982, demonstrates the emission of gravity waves from a system of orbiting compact objects. In this system, two 1.4 solar mass objects perform a pas de deux, orbiting each other with a period of a little less than eight hours. Due to their orbital motion, gravity waves are emitted and carry energy from the system. The masses slowly fall together and the orbital period decreases, largo to allegro. Mea- surements of the change in this orbital period agree with those predicted for gravity wave emission. Though these gravity waves are weak and at too low a frequency (one cycle per eight hours) to be detected, in 108 years the tempo becomes presto with each cycle taking only one one-hundredth of a second. It is during this end stage, as the bodies near coalescence, that the gravity waves emitted by the system will be observable from the earth. In the late 1970's to the present the emphasis has been on the cooled bars on the one hand, and interferometers on the other. With the cooled bar at Stanford, Boughn et al. 1982 have set even lower limits on possible sources with 74 days of data. With the exceptions of searches by Forward 1978, for pulses, and Hereld 1984, for periodic signals from the "millisecond pulsar", the interferometer group's efforts have concentrated on understanding noise sources in the instruments, typically presented as a spectrum of the antenna's operating noise level. The ultimate goal of this prototyping work is the design and construction of large scale antennas having sensitivities able to detect predicted sources at a rate that does not try an experimenter's patience. Presently several groups are engaged in prototype interferometer research: Cal-

- 11 - 100 1000 10000 2 3 4 5 6 789; 2 3 4 5 6 789

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The dashed lines are the receiver noise required for a detection (SNR = 1) of a burst source located at the Galactic center (R ~ lOkpc), plotted for several values of the emitted gravitational wave energy, given in units of solar mass equivalents (from equation 3.5).

- 12 - Tech, MIT, the University of Glasgow, and the Max-Planck-Institute fur Quan- tenoptik at Garching. A comparison of the present-day performance of several of these interferometric antennas is given graphically in Figure 1.1; the noise level of the instruments, in strain/diz, is plotted as a function of frequency. The MIT and MPQ antennas are of similar optical design, employing delay lines. The CalTech instrument uses a Fabry-Perot optical system. The factor of more than two orders of magnitude worse performance of the MIT instrument is due to the combination of it's shorter length (factor of 20), imperfect contrast and lower power levels (factor of 5), and noise sources above shot and electronic noises in the region below 20 kHz (as much as a factor of 20 at 1 kHz). It is useful to compare the performance of the interferometer and bar antennas. To put this in perspective, an interferometer having the MPQ Garching sensitivity, hN ; 2 x 10-19 strain/v/iii, is compared with the Stanford bar. To also set the astrophysical scale the response of these antennas is calculated for a source located at the Galactic center that emits 10-2 solar masses of energy. (This source is optimistic but not absurd). The response of the antennas depends on the details of the emitted gravity wave shape and the frequency at which it is emitted. Figures 1.2 and 1.3, each for a different wave shape, show the signal-to-noise (SNR) ratio expected for each antenna as a function of the frequency of the burst. The broad band nature of the interferometer response, falling smoothly with frequency, is a chief selling point. Note that the bar response depends on how well defined in frequency the waveform is, and has a clear resonance for very sinusoidal pulses, Figure 1.2. However, the bar response to a less structured pulse, Figure 1.3, shows it to be a broad band detector as well, with a sharp low frequency cut-off. That the SNRs are comparable for the Stanford bar and the MPQ antenna indicates that the interferometric antennas are on the verge of leaving the prototype stage. In fact, the field is now in a period of transition, with experimenters looking towards the future: the development of second generation cooled bars and construction of large-scale interferometric antennas. In this spirit, much of the work going on in the laboratories is aimed not so much at improving the operating sensitivities of the existing prototypes, but rather examining and evaluating the consequences of scaling these antennas to larger, more sensitive designs. This thesis seeks to contribute to the search for astronomical gravitational radiation through the development of a data analysis scheme designed to detect short, burst-like gravity waves in the output of the interferometric antennas. This represents a first attempt at integrating the astrophysical predictions of expected wave shapes with a data analysis scheme for their detection. As discussed further in the introduction to Section 6, this data analysis has two goals: the detection of gravity wave events and the characterization of the interferometer burst-noise performance. Though the low sensitivity of the MIT antenna precludes a contribution to the former, a clear demonstration of the latter is made. In the design of the data analysis scheme presented here, there was much give

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The source, located at the Galactic center, has the wave shape shown. A total gravity wave energy of 10-2 ME) is assumed to be emitted. The interferometer noise is 2 x 10-19 strain/V Hz, The bar noise is 20 m'K.

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The source, located at the Galactic center, has the wave shape shown. A total gravity wave energy of 10 - 2 ME) is assumed to be emitted. The interferometer noise is 2 x 10-1g strain/VlHz, The bar noise is 20 mOK.

- 15- and take between the astrophysics and the data analysis. This has lead to some problems of organization. The concept of the matched filter, which pervades much of the thesis, is not explicitly described until Section 6. Section 2 presents a terse summary of the formulas relevant to the theory of gravity waves. In Section 3, many of the predicted astrophysical sources of gravitational bursts are examined: their waveforms are used to guide the template selection in the matched filter scheme and estimates of the antenna sensitivities needed to detect the events are made. Sections 4 and 5 present details of the MIT instrument's operation and its noise sources. Section 7 describes the application of the matched filter scheme of Section 6 to data from the MIT prototype, and Section 8 summarizes the results of this work. The appendices are largely concerned with details of the electronic systems associated with the prototype. Appendix D, however, contains an explicit calculation of the response of a bar antenna to various burst waveforms and has allowed the comparisons presented in Figures 1.2 and 1.3. Though the description by Weiss 1972 was complete and today's implementation differs only slightly, some of the material and many of the details in this thesis are original. In particular, the bar comparison, the data analysis scheme, much of the electronics including the loop system, and the invention of digital phase modulation are due to the author.

This thesis is the first from the MIT prototype; a summary of the efforts at MIT during the last decade and a half is appropriate. These efforts began in the late sixties/early seventies with work carried out by Rai Weiss and, then graduate student, Kingston Owens. The assembly of tubing and the Model 165 laser as well as the 1972 description were the outcome of this period. The project was not considered appropriate for doctoral research. Funding flagged, and other interests beckoned for several years. In the late seventies work was resumed by Peter Kramer, then a post-doc, and the unit assembled on the present-day granite table and initial mass-damping designs developed. Finally, in 1980 the project gained momentum with Paul Linsay and David Shoemaker carrying out the work. In addition to the work of Rai Weiss (which cannot be overly stressed), the lion's share of the recent work on the prototype is the result of many people. Most of the logistics and work in the initial assembly and operation were performed by Richard Benford, Paul Linsay, Jeff Livas, and David Shoemaker. The need, design and construction of an external phase modulator was due to Rai Weiss. Jeff Livas took on the difficult and often thankless job of maintaining a clean optical system and high fiber throughput. The monitoring program for the interferometer masses was the brainchild of David Shoemaker. The quick-look data taking and analysis programs were a joint effort with Jeff Livas and David Shoemaker. With David Shoemaker, the "fast" A/D system was built. The second and third rounds of improvements in the suspension design and mass damping were carried out by

- 16 - Jeff Livas, Peter Saulson and Rai Weiss. Paul Linsay contributed the program that allowed continuous data to be taken to tape. Andrew Jeffries contributed the M-Timer board, research on clocks, and a working, understood, backed-up VAX system. The actual work during the data taking runs was shared equally with Jeff Livas.

- 17 - 2. Theory of Gravity Waves

2.1 Introduction This section presents some of the results of the theory of general relativity relevant to the propagation, generation and detection of gravitational radiation. Further details of GR theory are available in Landau and Lifshitz 1975, Weinberg 1972, and Misner, Thorne, and Wheeler (MTW) 1973. The discussion below follows Landau and Lifshitz 1975; equations from their work are indicated and numbered as such, e.g., (L+L 110.9).

2.2 Propagation and Polarization of Gravity Waves In general relativity the properties of space-time are determined by the distribution of mass and energy in space-time; the geometry of space-time varies throughout space-time. In particular, measurements of the distance (interval) between two points (events) is locally determined by the symmetric space-time metric, gik:

ds2 = g i dxidxk (L+L 82.1)

Euclidian geometry is the result of the Galilean metric:

1 0 0 0 (0) 0 -1 0 0 gik 0 0 _1 0 (L+LLL 82.2)22 0 0 0 -1

Changes in the mass distribution give rise to changes in gik and, as in E/M theory, the speed of propagation of these changes is limited to the speed of light, c. Thus, the possibility (necessity) of gravity waves. In the weak-field low source velocity limit, gravity waves can be described as a perturbation of the locally Galilean metric:

gik - giO) + hik (L+L 107.1) The hik can take on plane wave solutions, obeying the wave equation:

(82 1 a2 hk = 0 (L+L 107.9) Xx2 c at2 / 1 Specializing to the geometry shown in Figure 2.1 where plane waves travel in the x, direction, hik is given explicitly by:

- 18 - 0 asf.EV.P

Figure 2.1 Plane Wave Geometry

0 0 0 0 0 0 0 hik = 0 (2.1) 0 h 0 h2 2 2 3 0 h32 h33

with

h22= -h 33 = h+ ; h2 3 = h32 =hx .

The reduction of the four possible entries to two occurs because of the symmetry of gik and the gauge choice that hik be traceless. The remaining two degrees of freedom can be viewed as two polarization states of the plane wave, given by:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2.2) += 0 0 1 0 ex 0 0 0 1) 0 0 0 -11 (0 0 1 0) and thus

hik = h+e+ + hxex (2.3)

The labels "+" and "x" are derived from the effects of the polarization states on matter, shown graphically in Figure 2.2. As in E/M theory, at any instant the field can be considered in terms of its decomposed values h+ hx, or a single magnitude polarized at an arbitrary angle. For gravity waves the relationship between these is given by (MTW 1973 section 37.2):

hil = h+e+ + hxex = heo (2.4)

-19 - j~xi 2 -7/

N X2a N xa lb

/ I

r41-7l YN -

Figure 2.2 Polarization States of hij

where

h = [h2 + h2 -tan-(hx 2 h+

2.3 Generation of Gravity Waves As in E/M theory where accelerating charges give rise to radiation, so in general relativity: the acceleration of masses gives rise to gravitational radiation. Unlike in E/M, where the charge distribution can have a dipole moment with a nonvanishing second time derivative, a gravitational source, consisting of a complete and isolated system, has a dipole moment with a vanishing second time derivative because of the conservation of momentum. Radiation is emitted, however, through changes in the quadrupole moment of the system, defined as:

Dap = J j(3xaxp - r 2 6ap)dV (L+L 99.8)

For plane wave propagation along the x1 axis the h waveforms are given by:

-2G" -2G h2 3 c4R23 h22ha = c4R (Di22 -i33) (L+L 110.9)

An example of the application of these formulas is given in Appendix A where the radiation from two point masses in a gravitationally bound circular orbit is calculated. This calculation is relevant to the "chirp" source discussed in Section 3.6.

- 20 - In estimating the possible strength of gravity wave signals, the energy carried by the wave is of importance. In particular the energy flux in the x1 direction is given by:

dE c3 . 1. 2 [hi +-( 22 - h 33) ] . (L+L 107.12) dArea dt = 167rG4[

2.4 Detection of Gravity Waves The effect of a gravity wave on a mass element, in the proper reference frame of the detector center of mass, is an acceleration (Thorne 1982 section 4.1, MTW 1973 section 37.2). Explicitly,

j= hijxi (2.5)

The acceleration field due to each polarization state of hij is graphically displayed in Figure 2.2. Equation 2.5 above is the starting point for the analysis of gravity wave detectors; the two types of laboratory-scale antennas currently being developed are described below.

2.4.1 Resonant ("Bar") Antenna Operation The bar antenna detects a gravity wave through its excitation of the normal mode of oscillation of an elastic cylinder. A simple model for a bar antenna is the mass spring arrangement shown in Figure 2.3. The change in the separation of the masses x from the nominal L is given by:

mi+mw 2x = Fg(t) = m L h(t) (2.6) 2 where w is the resonant frequency of the system and Fg is the gravity wave force on the system. The fundamental noise source in such an antenna is the white thermal noise force due to the finite Q of the normal mode (see Appendix B ). In practice the noise and signal forces are converted, through the bar resonance, to displacements which are measured in the presence of additional displacement noise terms. Because of the high Q of the bar, this transducer noise is (ideally) dominated by the bar thermal noise near the bar resonance; away from resonance, however, the transducer noise becomes limiting. The frequency range in which there is useful signal to noise is presently limited to less than 10 Hz. When used as a detector of burst gravitational radiation, changes in the state of oscillation of the normal mode of the bar are monitored. Each change represents an energy input to (or output from) the bar and through Boltzmann's constant

- 21 - Figure 2.3 Model of Bar Antenna Figure 2.4 Free-Mass Antenna Arrangement this energy is expressed as a temperature; likewise the rms noise of this energy measurement is expressed as a noise temperature, Td.

2.4.2 Free-mass ("Interferometric") Antenna Operation If test masses in space are free of all but the gravitational force, then equation 2.5 above can be integrated, yielding:

xi hijxi (2.7)

The effect of a gravitational wave is to produce a "strain" field in space similar to the acceleration fields shown in Figure 2.2. In principle a detector could be made by measuring the distance between two free test masses; in practice (primarily because of noise reduction due to the symmetry) an arrangement of three masses is used, shown in Figure 2.4. Using a laser interferometer the difference between the lengths of the two arms is measured. For an optimally polarized wave, propagating perpendicular to the plane of the antenna, the change in path length difference is given by:

dl(t) = dL 1(t) - dL2 (t) = L h(t) (2.8)

(A detailed calculation of the interferometric antenna response is presented in Ap- pendix C.) The fundamental noise source in this free-mass antenna is the white displacement noise due to laser shot noise; this and other noise sources are discussed further in Section 5.

- 22 - 2.4.3 Comparison of Antenna Sensitivities to Burst Sources The two antenna types, though based on the same fundamental physics, have different responses and (at present) different fundamental noise terms. Because of these differences, work is required to accurately compare their relative sensitivities to expected burst sources. Weiss 1972 has presented a bar-interferometer comparison for the case where each antenna is thermal noise limited and a single square pulse is incident; this analysis shows the interferometer superior to a bar. For realistic cases the interferometer is shot noise limited, and the bar antenna has a fixed rms energy noise. Appendix D presents a comparison of bar and interferometer responses for canonical and real (i.e. predicted) waveforms. The result of this comparison shows that the Stanford bar and the MPQ Garching interferometer have comparable sensitivities, see Figures 1.2 and 1.3. It is instructive to look at general scaling laws. Let a canonical gravity wave have amplitude ho, frequency fg, and a total duration NHC/2fg. Then the energy carried by the burst scales as (from equation L+L 107.12):

EGW Oc 2 dt oc f 9h! (2.9) g

The SNR for interferometric detection scales as (from equation 6.1):

SNRinterf oC h2 dt oc NC'1/ 2 ho (2.10) J 1/2 fg

The SNR for a bar detection, with fbar = fg, scales as (from equation D.4):

SNRbar oc Jhjdt oc NHCf2ho (2.11) fg Finally, these lead to:

SNRbar ar cx NHC . f3 (2.12) SNR interf

- 23 - 3. Astrophysical Sources

3.1 Introduction This section presents a survey of predicted "burst" sources of gravitational radiation. These are transient sources expected to produce detectable radiation during a finite period of time, typically less than one second, but up to one minute for the "chirp" sources. The results of theoretical calculations can provide important information about the sources, aiding in their detection; among these are the wave shape, the wave amplitude, and the rate of events. The wave shape is of prime importance to the detection scheme (see Section 6) and the discussions below focus on this. The amplitude, along with the the wave shape, allows the intrinsic detectability of the source to be evaluated. Finally, questions of rate are secondary to the actual detection process, though important for the likelihood and statistics of a detection. Note that many of the references in the sections below were chosen primarily for their published h(t) waveforms and are a small fraction of the literature on source physics. Good overviews can be found in the edited collections Smarr 1979 and Deruelle and Piran 1983 and in Linsay, Saulson, and Weiss 1983.

3.2 System of Units Many of the papers on source physics use units of length and time such that c and G have the numerical value 1 and the unit of mass is the solar mass. This set of units is summarized in Table 3.1. In this "G=c=1" system of units the distance to the Galactic center, 10 kpc, is equal to 2 x 1017 "1.5 km"s; the distance to the center of the Virgo culster is a factor of 2000 times farther away.

Table 3.1 System of Units

Choose G=c=1 and measure mass in solar masses, then the fundamental units are:

Length in 1.5 km Mass in 2 x 10" kg Time in 5 x 10-6 sec

Length, time and angular momentum can be given in mass units:

Length = (G/c 2 ) x M Time = (G/c3 ) x M LZ = (G/c) x .t x M

- 24 - 3.3 Detectability of Burst Events A measure of the "detectability" of a source, independent of its distance and questions of rate, can be given by determining the receiver noise hN required to detect the source with unity signal to noise ratio SNR if it were located at the Galactic center. This measure allows a realistic comparison of source strengths to be made, and indicates the instrumental sensitivities required to begin doing burst source astrophysics in the Galaxy. Note that these numbers do lead to optimistic receiver noises. In practice a recognizable event will require an SNR > 6, and the source will most likely be located as far away as Virgo, requiring receiver noises four decades smaller. These noise levels are the goals of the large antenna projects. The detectability as defined above can be calculated from the published h(t) waveforms using the fundamental equation of the data analysis scheme, equation 6.1 in Section 6.2.1:

2 h2(t) dt SNR - fh 2(d (3.1) N Setting SNR=1 and solving for the required receiver noise gives:

hNSNR=1,GC - 2 h2(t)dt (3.2)

=2 h[xR(t)]2dt where the quantity h x R/p is the usual result of theoretical calculations. The integral can be approximated by adding the contributions from each component half-cycle of the waveform, see Figure 3.1. Assuming these are roughly sinusoidal yields:

hNSNR=1,GC h x R), At, X 1.1 X 10- 2 0 X A X V/M strain/vHz (3.3) where h x R/p and At are in "G=c=1" units and the value of RGC has been combined with the conversion from "G=c=1" units to strain//Hz. Note that the VK/Y dependence comes from the M dependence of the time axis. Typical values of this detectability are presented in the discussions below. Even without exact waveforms, however, an approximate determination of the detectability can be made from estimates of the energy and frequency of the emitted gravitational radiation. (Section 5.2 of Abramovici 1985 presents similar ar- guments.) If an arbitrary source emits EMc2 of energy in a near sinusoidal pulse

- 25 - txR

Figure 3.1 A Theoretical Waveform Viewed as Several Half-cycles then, using equation (107.12) of Landau and Lifshitz 1975, the integral required by equation 3.1 can be approximated:

EMC 2 = 47rR 2 - C 2 dt (3.4a) 167rGc3

2 ~ 47rR C 2 h2dt (3.4b) 167rG J Solving equation 3.4b for the integral and using it in equation 3.1 to solve for the required receiver noise gives:

hNSNR=1,GC = 101s strain/v/Hz x -M 1 (3.5) Msolar fkHz Thus, one percent of a solar mass radiated at 1 kHz requires a 10-19/N/riH strain sensitivity for detection. Note, that because of the 1/fkHz term in equation 3.5 the energy radiated by a system is not the sole factor in determining its detectability by an interferometric antenna.

3.4 Particle-Black Hole Events There are many published h(t) waveforms for the gravitational radiation emitted by a test mass encountering a black hole. The parameters of the encounter (Detweiler 1979) include the masses of the particle and black hole (it and M), the angular momentum of the particle about the black hole Lz, the spin parameter of the black hole a, and the energy of the particle at infinity. The types of encounters are conveniently grouped into six categories defined by the parameters a and Lz: the black hole is described by the Schwarzschild (a = 0) or the Kerr (a 0 0) metric; the particle trajectory is a radial infall (L. = 0), a plunge (0 < |LzI < 4.0), or a scattering (JLzI > 4.0).

- 26 - a) b)

C) d)

e) f)

Figure 3.2 Burst Waveforms and Templates for their Detection a) Particle-Schwarzscild black hole radial infall, Detweiler 1979. SNR/SNRoptimum = 0.81 b) Rotating stellar collapse, Stark and Piran 1985. SNR/SNRoptimum = 0.83 c) Particle-Kerr black hole scattering, Kojima and Nakamura 1984. SNR/SNRoptimum = 0.77 d) Particle-Schwarzscild black hole scattering, Oohara and Nakamura 1984. SNR/SNRoptimum = 0.75 e) Damped ellipsoidal stellar collapse, Saenz and Shapiro 1981. SNR/SNRoptimum = 0.81 f) Cold rotating stellar collapse, Saenz and Shapiro 1978. SNR/SNRoptimum = 0.84

- 27 - The simplest case of Schwarzschild radial infall (Davis et al. 1972, Detweiler 1979, Ferrari and Ruffini 1981) is shown in Figure 3.2a. Three regions of the waveform are identified: a precursor, a sharp pulse, and a ringing tail. The precursor is expected on the basis of the quadrupole formula; the sharp pulse comes from the actual encounter; and the ringing tail is the result of excitation of the black hole quasi-normal modes. The time scale of the waveform is set by the black hole mass-the frequency of the ringing tail is z 10 kHz/M (M in solar masses). The important feature for the detection of this event is the main pulse, as shown by the template location in Figure 3.2a. The detectability (see Section 3.3) of this source is ; 3 x 10-20 x A x VMK strain/v'Hz, masses in solar masses. If the test particle has non-zero orbital angular momentum L., as much as 50 times more energy can be released in gravitational radiation (Detweiler and Szedenits 1979). The waveform for L. = 3.9 in Figure 1c of Detweiler and Szedenits, which is similar to the stellar collapse waveform in Figure 3.2b, shows that this energy is radiated by the orbital motion of the particle as it spirals through one orbit into the black hole. For the case of L. > 4.0, the waveforms take on a sinusoid burst character (Oohara 1984, Oohara and Nakamura 1984); Figures 3.2c and 3.2d are representative. Because of the high frequency at which this radiation is emitted, the detectability of the waveform is typically increased by only a factor of two or three over the radial infall case. Waveforms for particle-Kerr black hole events (Kojima and Nakamura, 1983, 1984, 1984) have shapes and detectabilities similar to the Schwarzschild cases; there is a somewhat greater variety of wave shapes due to the variation of the black hole quasi-normal mode frequency with Kerr parameter and the possibility of co- and counter- rotating encounters.

3.5 Stellar Collapse Events Calculations of waveforms expected from stellar collapse events have been carried out inspite of the greater complexities and unknowns involved. Because of lower efficiencies these sources are typically ten times less detectable than the particle- blackhole events of comparable mass. Some of these waveforms (Cunningham et al 1980, Stark and Piran 1985) have the shape and radiate at the frequencies of particle-black hole events, Figure 3.2b. The 1.7 kHz many cycle burst waveform of Saenz and Shapiro 1981 is very intriguing because it is so clean, Figure 3.2e. The waveforms of Muller 1982 are dominated by single large pulses of ~ 1 ms duration; with efficiencies of 10-6, the detectabilities are 10-". The waveforms "catalogued" by Saenz and Shapiro 1978 are all very pulse-like with efficiencies ranging from 10-15 to 10-3; the waveform in Figure 3.2f has a high detectability, 2 x 10-20, due to a comparatively high efficiency, 10-, at a low frequency, 300 Hz.

- 28 - 3.6 Chirp Events A very appealing source of gravitational radiation, both to theorists and experimenters, is the gravitational decay of a binary system. In particular, if the components are compact objects, neutron stars or black holes, the orbital frequencies can become very high before any of several effects destroy the assumption of two point masses. The calcualations are straight forward, involving Newtonian dy- namics and the energy and angular momentum losses due to gravitational radiation. Because of the change in frequency with time these sources are referred to as chirp sources. The classic work on radiation from Keplerian orbits is Peters and Mathews 1963. The enhancement in gravity wave emission due to an eccentricity of the orbit leads to the interesting result (Peters 1964) that the orbit will circularize as it decays; chirp sources radiating at frequencies of interest to present antennas will have essentially sinusoidal waveforms. The response of a bar antenna to a chirp source (as well as the term "chirp") is presented in Forward and Berman 1967 even though "no neutron star has yet been identified, much less a neutron star binary system". The picture has changed with the work of Taylor and Weisberg 1982. At least one such system does exist, the decay of the orbit is as predicted, and it will chirp in 108 years. In an important and informative work, Clark and Eardley 1977 calculate the evolution of a neutron star binary system, including mass transfer. Their results show that the chirp can proceed to radiation frequencies as high as 1 kHz before mass transfer or "immediate tidal disruption" limits or destroys the system. They also calculate the response of a bar antenna to the system. The most uncertain aspect of this source is the likey rate of events. An estimate has been made (Clark 1979, Clark et al. 1979) of the galactic rate, 3 x 104 per year; this might lead to a rate as large as one per year for sources within 15 Mpc. More speculative is the hypothosis of Bond and Carr 1983 "that a large fraction of the 'missing mass' may be in binary black hole remnants of population III stars with mass M exceeding 102 Msoiar ." Located in the Galactic halo, to 60 kpc, these sources might be expected to produce a chirp every ten years; the chirp proper would be confined to frequencies below 30 Hz but with a detectability (SNR = 1 at 60 kpc) of 2 x 10-17 using the few cycles from 10 Hz to coalescence, one second later.

- 29 - Figure 3.3 Chirp Waveforms for m, = m 2 = 1.4MO and mi = m2 = 4.6M®

3.6.1 Chirp Source Waveforms and Detectability Using the nonrelativistic equation for the decay of a binary orbit due to gravitational wave emission (Taylor and Weisberg 1982), the gravitational radiation frequency as a function of the time to collapse is:

WGW = 2 Worbital = 2 5 G5 F(mlM2 t ]-3/8 (3.6) 5 c5 Fmim)]/(36

with mim2 F (ml, M2) = I2)/ (Mi + m2)1/3

Figure 3.4a shows the time remaining until collapse as a function of the present GW frequency of the system. Making use of the quadrupole formula for gravitational radiation and assuming a circular orbit, the amplitude of the emitted waves is given in Appendix A:

4 hamp = 1 5G")1/ 4 F3/4(M, M2) t~1/ (3.7) 3 R ( li / (i1 2 Note that the chirp waveforms are a one parameter family: the amplitude and frequency of the chirp are functions of the one parameter F(Mi, M2 ). Waveforms are shown in Figure 3.3 for mi = m 2 = 1.4M 0 and mi = m 2 = 4.6M® as the waves chirp from 300 Hz to 1 kHz.

- 30 - _ a 3 4 567 3 4 5 6799

tilt i ilitllt -rdl~liti

4 .)

L) n

a.)

(1) CU- (U ...... CL- ...... 0-

4.U--r o ~ (S) U .) ...... CD l

1 2 3 4 5 6 7 89 3 456789 10 100 300 2 3 4 5 6 7U 3 4 5 6 7Be

CD-0 N

L ~(

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a 3 4 5 6 789i 3 4 5 6789 10 100 1000 ON Frequency (Hz) Figure 3.4 A Chirp Source of two 1.4M0 Objects a) The time to collapse for a given GW emission frequency. b) The receiver noise needed for an SNR=1 detection with the source located at the Galactic center.

- 31 - Assumming a matched filter detection, the detectability can be calculated from equation 3.2:

hNSNR=1,GC 2 h2(t)dt

1 (5G5l 4 R( c ) 1 /4 F 3 / 4 j t-1/ 2 dt (3.8)

1 20G 5 )1/4 F3/4 T1/4 RGC c 1 1 Even though h decreases at lower frequencies, the increase in time spent at the low frequencies leads to a net improvement in SNR as the starting frequency of observation is decreased. The optimistic receiver noise needed to detect the chirp is plotted in Figure 3.4b as a function of the starting frequency of observation. The required receiver noise of 3 x 1019 strain/ /Hz for observations starting at a frequency of 500 Hz are already at hand. However Clark's 1979 estimate of the Galactic rate of these events, 3 x 10- 4 per year, indicates that thcy will rarely be observed at these levels.

- 32 - 4. The Instrument

4.1 Introduction The design of the instrument is essentially that proposed by Weiss (Weiss 1972). There are several descriptions of similar, operating instruments in the literature (Forward 1978, Billing et al. 1979). The description here is of the MIT prototype during the June 8, 1985, data runs. See also the description in Livas et al. 1985. As shown in the block diagram, Figure 4.1, the antenna consists of three masses located at the corners of a right isosceles triangle. Through a suspension system these masses behave as essentially free test masses at frequencies of interest. The suspension also provides isolation from seismic and acoustic noise sources. Relative motions of the masses are measured by a Michelson interferometric optical system which uses multi-pass delay lines to achieve greater sensitivity. Finally, there are several electronics systems which damp the mass motions in the vacuum, hold the interferometer output to a dark fringe, and monitor and collect data from the instrument. The components of the antenna are described in detail below, grouped roughly into mechanical, optical, and electronic and computer systems.

4.2 Mechanical The antenna consists of equipment mounted on two granite tables; the division of equipment is indicated in Figure 4.1. The laser table consists of the laser, an external phase modulator, a diagnostic Fabry-Perot, and the fiber input optics. This equipment is separated from the interferometer proper to reduce sources of noise on the interferometer table, in particular noise due to the water cooling of the laser. An optical fiber serves to couple the laser light from the laser table to the interferometer input. A vertical cross section through the arm of the interferometer containing end- mass one (EM1) and the central-mass (CM) is shown in Figure 4.2. The optical components are inside a vacuum system held to a pressure of 10-6 mm Hg by ion pumps. Bellows in the arms of the interferometer and translation stages under the end-mass tubes allow adjustment of the optical cavity length; independently evacuated compensation bellows offset the force of air pressure tending to collapse the arms. The entire table can be floated on pneumatic cylinders to provide isolation against ground motions. The masses (EM1, 2: 8.5 x 14 x 24 cm, 8 kg; CM: 18 x 18 x 18 cm, 15 kg) are each suspended from a single fine wire, ; 3 x 10-4 m diameter, approximately one meter long. This gives rise to a pendulation period of 2 seconds and a torsional period of 90 seconds. The suspension points of the wires can be adjusted in X, Y and Z by translation stages at the tops of the tubes. A set of three glass plates, each with several deposited copper electrodes, is associated with each mass. These

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-- ; e & nQ9 - n 'v- I -i kj Lj V~c'~Qd Y-oS5SQ -cIOr Li allow the motions of the mass to be sensed and appropriate feedback applied to damp the pendulum and torsional oscillations of the masses.

4.3 A Walk through the Optical Components The following description of the interferometer optics follows roughly the order along the path of the laser beam. 4.3.1 Laser A Spectra Physics model 2020-05/2560 laser was used for the data taken on June 8, 1985. Prior to this date the Spectra Physics model 165 was used. Both lasers are argon ion lasers and were operated at 514.5 nm. Single mode operation is achieved with an intercavity etalon. The lasers were typically run at a power output of 150 to 250 mW. This is the maximum available from the model 165 and less than maximum for the model 2020. The power limit of 250 mW was determined by apparent thermally related blooming effects in the external phase modulator crystals (below). The laser output beam size has a 1/e 2 (intensity) diameter of 1.3 mm. The model 2020 required modification to make it acceptable for use with the interferometer. The problem that lead to modifications was a broadband excess of amplitude noise in the frequency range above a few MHz. The excess RF noise showed modulation at the line frequency (60 Hz) and it was discovered that the filament AC was responsible for the modulation. When a DC supply is used to power the filament the noise is no longer line modulated and, if the DC supply has the correct polarity, the noise level is shot noise limited above 4 MHz. (This same problem and proposed solution had been recognized and described by the group at MPQ; Riidiger et al. 1982, p 17.)

4.3.2 External Phase Modulator A dominant noise term in laser interferometric antennas is noise due to scattered light in the system. A variety of schemes have been developed to diagnose and combat this noise source (Schilling et al. 1981, Weiss 1982, Schnuipp et al. 1985, Appendix I of this thesis). All of these schemes use external phase modulation of the input laser beam in the 1-100 MHz range to achieve noise reduction. Scattered light noise and these schemes are discussed further in Section 5.2.5 and Appendix I. The laser light is passed through a phase modulator after leaving the laser. The phase modulator is made of two LiTaO 3 crystals optically in series and electrically in parallel. Focussing lenses are used to match the laser beam size to that of the crystals (1 x 1 x 25 mm each). Spaced at twice their focal length (25 cm each), the lenses focus the beam to a waist diameter of 0.13 mm. The two crystals give a total phase shift of 7r for 43 V applied. An RF amplifier drives the crystals and a termination load through a 50 ohm coaxial cable. The crystal holder was designed

- 36 - to have low stray capacitance and inductance; system roll-off occurs at 250 MHz primarilly due to the capacitance of the crystals. The phase modulator suffers from two major problems. First, in addition to the desired phase modulation, amplitude modulation and beam motion are generated by the modulator. These can lead to RF amplitude noise on the interferometer output which is above the shot noise level. Second, an apparently power dependent and permanent blooming occurs; the wave shape of the output beam is severly distorted. This produces poor coupling to the fiber, resulting in an overall reduction in the optical throughput. After passing through the modulator, the beam can be directed by a movable mirror into a swept Fabry-Perot to display the spectrum of the light. This allows verification of the single mode operation of the laser and provides a means of measuring the amount of phase modulation applied in the single frequency and white noise modulation schemes.

4.3.3 Fiber One of the important additions to the antenna design since the 1972 description is the use of a single mode optical fiber at the input of the interferometer. The fiber solves two important problems associated with the input beam: beam jitter which gives rise to noise terms (identified and suppressed with a mode cleaner in Riidiger et al. 1981), and wavefront distortion which increases the effective shot noise limit by lowering the interferometer contrast. The fiber has the important property that it is single mode (in fact double mode, coresponding to the two possible polarization states), and thus the only variables in the output beam affected by changes of the fiber input beam are the intensity and polarization of the light. With the addition of a polarizer at the fiber output, all changes at the fiber input result in low-frequency amplitude noise at the output; the interferometer is relatively insensitive to this noise source (Section 5.2.3). The actual beam shape out of the fiber is essentially Gaussian and well matched to the optical delay lines. The contrast of the instrument can exceed 95% when correctly aligned, indicating the cleanness of the wavefronts. Given the 1/e2 (intensity) half-angle 0 for the fiber output beam, input and output matching optics for the fiber can be easily designed. Input matching is achieved with a single lens following a A/2 plate used to maximize the throughput of the desired polarization; output matching to the cavity system is performed by a zoom lens allowing for continous variation of the Gaussian beam parameters: waist size and position. In operation an overall fiber system transmission of greater than 40% is possible.

- 37 - P isPSe-?C-,CCL-NOfCsn)[0je- -~~- rr;~ 9-y-, ro~\Cy\t

I r .

F4 M cis-s

Figure 4.3 Delay Line Geometry

4.3.4 Central Mass The central mass has mounted on it a beam splitter, two optical delay line mirrors, and two Pockels cells (with some electronics, Appendix G ). The input laser beam is split at the beam splitter, and the outgoing beams are sent into each of the delay line cavities through holes in the mirrors. The beams returning from the cavities pass through the Pockels cells and are then recombined at the beam splitter, generating the two interferometer output beams.

The beam splitter produces a nominal 50/50 split; its losses amount to 20%, i.e., it is a 40/40 beam splitter. The Pockels cells are AD*P 45 degree y-cut cells (3 x 5 x 50 mm each) with AR coated input faces; transmission losses are typically 10 percent. The cells produce an optical phase shift proportional to the voltage applied across them; operated together, 150 volts gives a phase shift of 7r at 514.5 nm. They are an important element in the antenna, serving to interrogate the interferometer fringe and provide feedback to hold a dark-null-fringe. Further details of the Pockels cells operation can be found in Appendices F, G, and H.

4.3.5 Delay Lines Through the use of multipass optical delay lines in each arm of the interferometer, Figure 4.3, the sensitivity of the interferometer to mirror motions is increased: the change in optical path length is Nb times the actual mass motion; where Nb is the number of passes in the delay line. (Nb originated as the "number of bounces"

- 38 - made by the light beam, counting the entrance/exit hole as a bounce.) Many noise sources in the interferometer, in particular the fundamental shot noise, are constant in terms of the optical phase shift of the interferometer (Section 5); thus, the contributions of these sources to the equivalent mass motion noise are reduced by a factor of Nb. The delay lines are the circular spot pattern, re-entrant cavities described in Herriott et al. 1964, and Herriott and Schulte 1965. A light beam entering the cavity through a hole in the input/output mirror on the central mass bounces back and forth along the interferometer arm many times. Successive reflections occur at a fixed radius from the mirror centers and are equally spaced in angle. With a judicious choice of the cavity parameters, the beam will exit from the entrance hole after a finite number of passes in the delay line. Concave dielectric mirrors, with a 10 cm diameter and a 1.31 m radius of curvature, are mounted on the central and end-masses to define the cavities. They are spaced by 1.46 m to operate at Nb = 56; the number of spots on each mirror is Nb/2 = 28. The circular spot pattern has a radius of 3.8 cm from the mirror center. Using the theory of Gaussian beams (Yariv 1967), the spot size (1/e 2 intensity radius) on the mirrors is 0.5 mm for the matched condition-spot sizes are constant and the beam waist is located midway between the mirrors. Because the cavities are re-entrant-the input and output beams intersect at the virtual surface of the input mirror-the optical path length is first order insensitive to rotations and transverse translations of the mirrors. Thus, the optical system provides some rejection of all but the desired beam- axis motions. There are many considerations which limit the number of bounces in the delay line system. Most fundamental is the requirement that the cavity storage time, Nb x Lcavity/c, be less than the period of the gravity waves of interest. This avoids the filtering effects inherent in the finite time averaging of the gravity wave field performed by the delay lines. For the 1.46 m cavity of the prototype this limit is ~ 2000 passes for a 20 kHz bandwidth. Secondly, the reflection losses in the cavity increase exponentially with the number of reflections-reducing the output light intensity and increasing the shot noise limit. For a given reflectivity R of the mirrors, the number of bounces that maximizes the signal to shot noise ratio is 2/(1 - R), and for this number of bounces the power out of the interferometer is reduced by a factor of 1/e 2 . The prototype mirrors have reflectivities of 0.994; thus, reflection losses limit Nb to less than 330 passes. Finally, there is the simple physical constraint that the spots next to the coupling hole must not fall through the hole. With the present circular pattern, the condition that the spots do not overlap leads to a maximum number given by: Nb_.. = 2 x 27rrcircie/2rspot For the values above, Nb is limited to less than 480 passes, requiring an input/output

- 39 - hole of 1 mm diameter. In operation a hole size of 2.5 mm is used, limiting Nb to less than 270 passes.

4.3.6 Photodetectors The light out of each interferometer port is incident on a photodetector, an E.G.&G. model SGD444A photodiode with 1 cm2 area. It is operated in a photo- conductive mode with an applied bias voltage of 220 V and a load resistor of 150 ohms. The RF signal across the resistor is amplified by a preamp (using an AT17a transistor) and sent to the input of the RF mixer circuitry. The DC voltage across the load resistor is used to measure the average light intensity at each port. Cali- bration of the RF gain and the equivalent load impedence is done by illuminating the diode with laser light and plotting the mixer output noise versus the square root of the photocurrent; the parameters of a linear fit to the plot determine the product Rioad x Gsystem and the no-light noise floor.

4.4 Electronic Systems

4.4.1 Mass Damping System Because the masses are suspended in a vacuum, the damping force of air is not present. Without this damping the various modes of the mass suspension, in particular the pendulation modes, are driven by ground noise and execute large amplitude motions. To damp these modes, without introducing the corresponding noise terms (Callen and Welton 1951), a system to sense the mass motion and apply damping forces is provided for each mass. The operation of the system is described briefly below and in great detail in Linsay and Shoemaker 1982 and Appendix E. The motions of the six degrees of freedom of each mass are sensed with RF capacitive displacement transducers: changes in the spacing between the mass and a set of plates changes the capacitance of these plates. Each plate is an element in an RF bridge circuit and the resulting unbalance of the bridge provides a measure of the mass motion. The electronics associated with the bridge circuit produces signals proportional to the velocity and position of the mass in the frequency range of interest. Forces can be applied to the mass by another set of plates nested with the sensing plates. These drive plates are held at a high DC bias potential and, through the electrostatic force, are able to push and pull the mass about its equilibrium position. In the matrix box the sensor signals are amplified and combined to produce signals to the drive plates to damp the various modes. In addition there is an input which will apply forces to the mass along the beam axis; this is used by the loop system to hold the interferometer to a null fringe.

- 40 - 4.4.2 Fringe Servo System The motions of the masses, even though they are damped, are still large enough to produce rms phase shifts on the order of tens of fringes, primarily at low frequencies. However, the shot noise limit coresponds to measuring fringe phase to less than 10-8 of a fringe. Besides the simple problem of dynamic range, which could be solved by suitable filtering, the multiple fringe excursions lead to a varying gain and sign of the output signal. In order to make the measurement feedback is applied to hold the interferometer at a dark fringe (where the signal to shot noise ratio is a maximum). This null servo technique also provides rejection of some important noise sources (see Section 5). The feedback system consists of two servo loops. The interferometer fringe phase is measured by the photodetector/mixer system (Appendix F), and, after suitable filtering, feedback signals are applied through the electrostatic mass damping system of end-mass one and through the Pockels cells on the central mass. The mass feedback provides large-amplitude corrections at low frequencies; the Pockels feedback allows small-amplitude changes to very high frequencies. The output of the interferometer can be taken from any one of several signals in the servo system; the output of the mixer is typically used. The signals and spectra measured at this point must be corrected for the action of the loops. The details of the servo loops are discussed in Appendix H.

4.4.3 Monitoring and Data Taking Because of temperature changes the masses drift in position; these motions can be large enough to exceed the dynamic range of the mass damping system sensors. Large trucks and large bumps occasionally conspire to produce ground motions that give the masses a kick, sending them into wild swinging. In order to monitor the positions of the masses, an 11/23 computer with A/D systems and an X-Y display is used to display in real time the motions of the masses derived from the damping system signals. The monitoring program also allows quick-look spectra to be taken, displayed, and stored. For the data runs, a timing board and tape drive are added to the system, allowing continuous, timed, and phased data to be taken. A description of this system is provided in Appendix K.

- 41 - 5. Noise Sources and the Prototype Noise Performance

5.1 Noise Sources The output of the interferometer is sensitive to a variety of signals in addition to gravity waves; a number of these noise sources are listed in Table 5.1. A description and evaluation of the importance of most of these noise terms is presented in Weiss 1972. Two noise sources that were not immediately recognized are beam jitter and scattered light noise; these are described in Riidiger et al. 1981 and Schilling et al. 1981, respectively. As Table 5.1 indicates, the noise terms can be classified according to their origin and mode of coupling to the interferometer output. It is also interesting to ask how the hN noise limit of the interferometer would scale with changes in larm (the arm length), Nb (the number of passes in the delay lines) and Po (the light power being detected) when the given source is the limiting noise term. Noises which are constant in mass motion, optical phase shift, or equivalent light intensity lead to the simple dependences:

hN OC dXmass X la- oc doptical X la x N-1 (5.1)

oc dPo x am x N-1 x P-1 The hN scaling indices for the various noise sources are given in Table 5.1. From the ubiquitous inverse dependence on larm in Table 5.1, it is clear why long baseline antennas are being pursued. Much of the research on interferometric antennas has involved identifing and reducing these sources of noise. The two physically fundamental noise terms are the shot and radiation pressure noises. At present power levels, the shot noise is dominant; a white shot noise limited output at as high a power level as possible is the Holy Grail of interferometric antenna research.

5.2 Prototype Noise Performance A spectrum of the MIT prototype interferometer output noise, typical of the instrument's performance during the June data run, is shown in Figure 5.1. The spectrum is far from the ideal white shot noise limited case; it has a roughly 1/f2 fall off and is shot and electronic noise limited above 10 kHz (not shown). The string modes of the wire suspension are evident as well as a number of unidentified resonances. Because of these and the limited frequency resolution of the spectrum, the exact level and structure of the underlying broadband noise is not known. For the purpose of the search for gravity wave signals in the antenna output, the spectrum is taken as given; that the search sensitivity is limited by photon shot

- 42 - Table 5.1 The Scaling of hN for several Noise Sources

hN OC larm arm X NbINb X Po'PO

Ii INb 'PO

Optical Shot Noise -1 -1 -1/2 Laser Amplitude Noise -1 -1 0 Laser Frequency Noise -1 -1 0

Opto-Mechanical Radiation Pressure -1 1/2 1/2 (Quantum Limit) Scattered Light -1 -1 0 Index Fluctuations -1/2 -1/2 0 Beam Jitter -1 -1 0 Radiometer Effect -1 1 1

Mechanical Thermal -1 0 0 Seismic -1 0 0 Acoustic -1 0 0 Gravity Gradient -1 0 0 Cosmic Rays -1 0 0 E/M Fields -1 0 0

Electronic Sensor Noise -1 -1 -1 Feedback Signals Optical -1 -1 0 Mass -1 0 0

-43- 100 1000 10000 7 2 3 4 5 6 8 9 2 3 4 5 6 7 8 9 _IIII I I II I . I , I I , I , I , i l l I I_ I I_ I CU- - ru M, ...... -CD Go- ...... 0 -0

...... -. 4 ..... 4I . . 4 4...... - .h6

N Cu- -Cu

CD- - C E -0

'-4 ...... 4..4 ...... 0 Cu - ...... -4 -CD _P- 0 CD- E ...... ~T....T1 Lit -CS

CD- -wu

V__4

-CD

CU -

2 3 4 56799' 4 5 4 5967 89 100 1000 10000 Frequency (Hz) Figure 5.1 Representative Spectrum with 1/f2 Trend and Shot and Electronic Noises

noise, laser frequency noise, or noise from a fan is irrelevant. (However, if the source can be identified a subtraction or veto scheme can often be employed to reduce the effective noise level.) For purposes of improving the performance of the instrument it is useful to examine the possible contributions to the spectrum from suspected noise sources. Because of hurried improvements to the suspension design and a deadline imposed by renovation of the laboratory space, little time was available for a detailed characterization of the spectrum. What is known or suspected about the various noise sources is presented below.

5.2.1 Shot Noise The shot noise limit for the instrument is calculated in Appendix F. Using equation F.12, the shot noise limit for the parameters typical of the June 8 run (Po = 11.5 mW, K = 0.73, 6 = 0.90) is 2.4 x 10-1 7 m/vfEz. This level is shown in Figure 5.1 as the lower dotted level. (Note that with perfect contrast, K=1, the limit would be 1.1 x 10-1 7 m/Vz.)

- 44 - 100 1000 10000 2 3 4 5 6 7 8 9 2 3 4 i 6 7 8 9 , I Ii I .. 1.1 I .. 1 I I i e I I I

cu ...... ------...... r ...... ru

...... : ...... f CD) W-...... Lw

N cu- ......

...... Wj ...... CD ...... : ...... i ...... ! ...... I ...... CS)

C -n -4 4P --4 0 CUwR iMatIT

......

Cu: GOD1...... 3.....4.... 56799' 2.. 3 ..4.... 5679

100 1000 10000 Frequency (Hz) Figure 5.2 Upper Limits to Low Frequency Laser Amplitude Noise

5.2.2 Electronic Noise The label "electronic" is here applied to all noise sources that enter the instrument through the sensors and transducers used to operate the interferometer. The sum of all the noises described in Appendix H gives 2.9 x 10-1 7 m//iHz, valid for frequencies above 1 kHz (rising to 3.6 x 10-1 7 m/N/III at 250 Hz), about equally due to mixer noise and loop noises. The sum of the shot noise and these electronic noises gives a limit of 3.8 x 10-1 7 m/V/IIz; this agrees well with the measured high frequency (> 10 kHz) performance, and is shown as the solid horizontal line in Figure 5.1.

5.2.3 Laser Amplitude Noise Amplitude noise on the laser beam interferes with the measurement process in two basically different ways in two distinct frequency regemes. High frequency amplitude noise (at the RF interrogation frequency, Appendix F) has the effect of an increased shot noise level. Low frequency amplitude noise (f < 20 kHz, i.e., in the measurement band) is largely attenuated through the null servo scheme, only making its appearance through the error signal excursions around fringe null caused by the mass motions and finite loop gains.

- 45 - The presence of high frequency amplitude noise is easily detected: the laser beam (or the output beam from the fiber, etc.) is shown on the photodetector and the noise at the RF interrogation frequency (5.38 MHz) is compared with shot noise. At these high frequencies the only source of this noise would be the laser itself; however, schemes to attenuate scattered light noise (see below) require phase modulation of the laser beam at high frequencies (1 to 100 MHz depending on the scheme) and can produce amplitude noise terms as well. The data was taken using digital phase modulation (Appendix I). This modulation scheme has the advantage that the signal applied to the phase modulating crystals is periodic and thus made up of a discrete set of lines. If these are chosen to avoid crystal resonances (and the RF interrogation frequency) the amount of amplitude noise produced is below shot noise. High frequency amplitude noise is not expected to contribute to the noise spectrum. Low frequency amplitude noise can be produced by a variety of sources: the laser, vibrating components in optical system, the fiber system, etc. Amplitude noise is characterized by the ratio of noise amplitude to DC signal, dV/V. The expected feedthrough is proportional to the error signal, in this case the voltage at the w mixer output V,:

1 dV noise VPR Nb 27r/A V( amP (5.2)

Note that because of the V, dependence, the gain (including the sign!) of the feedthrough of amplitude noise is time dependent, determined by the mass motions; this can lead to anharmonic terms in the noise spectrum. A limit on the low frequency amplitude noise contribution to the noise spectrum can be set by measuring dV/V of the beam as it enters the interferometer. If the major excursions of V, are assumed (rightly) to be at low (< 150 Hz) frequencies, then the requirement that the interferometer remain locked leads to the requirement that IV. 1 < 0.7V (with Gp = 1) to keep the pockels cells in range. Using this value of V,, and measured dV/V values in the equation above gives the limits shown in Figure 5.2. The lowest level shown represents a dV/V ; 6 x 10-6 level. Additionally, no change in the spectrum is seen when the laser is run in "current" or "power" regulation modes even though there is a > 10 dB change in the measured amplitude noise level.

5.2.4 Seismic and Acoustic Noise Both of these noise sources can cause motions of any of the components of the interferometer; the most important of these is (suspected to be) motion of the tube tops from which the masses are suspended. This motion is communicated through the wire suspension to the masses with an expected attenuation of:

- 46 - 5k~UJ~ ~ S C.O, t{lrJ\t

Figure 5.3 The Effect of a Scattered Beam on the Main Beam's Phase

Xmass 2 20.5 (5.3) Xtube top fHz s or ~ (0.5/87)2 (; -90 dB) if the ideal 1/f 2 dependence continues to the first string mode at 87 Hz. An accelerometer on the tube top measures broad band motions of order 10-3 /fkhZ m/VN/z. Even with a leeway of 20 dB this is unable to explain the (supposed) broad band noise; however, the resonances could be driven by this noise level as the acoustic sensitivity below suggests. Note that the transmission through the gradient of the electric bias field due to the damping scheme, equation E.12, is comparable to the suspension transmission. An indication of the sensitivity of the instrument to .acoustic fields can be obtained by increasing the ambient sound level with a loud noise source (e.g., a WBCN rock-and-roll radio station broadcast) and noting changes in the spectrum. With the present wire suspension the acoustic sensitivity is confined principly to the string modes and a forest of resonances in the 600 Hz to 2.0 kHz range. A study of these resonances has not been made.

5.2.5 Scattered Light Noise Scattered light noise refers to the signal produced at the output of the interferometer by stray, coherent beams which interfere with the main beams. This noise source was first brought to light through the work of Schilling et al. 1981 (refered to here as MPQ81); their picture of the effect of the scattered beam on the interferometer phase is shown in Figure 5.3. The effect on the measured optical path length is given (approximately) by:

dxoptica) = Omain = A Escat sin scat W (5.4) 27r 27r Emain

- 47 - ES

Figure 5.4 Time Series of Scattered Light Noise

Because of the dependence on Escat/Emain, a scattered beam of minuscule intensity relative to the main beam, i.e., 10-12, can produce noise terms devastatingly above shot noise. In order to produce a noise signal, the scatterer phase with respect to the main beams, oscat, must change with time. This can occur either because the location of the scatterer changes in time or the frequency of the laser fluctuates. The sin dscat dependence combined with Oscat excursions larger than ir lead to the interesting property of scattered light noise that its amplitude is independent of Oscat. Rather, Oscat determines the frequency content of the noise, through frequency up-conversion. The time series shown in Figure 5.4 make this clear; the FM looking waveform of sin Omain has a white spectrum out to a highest frequency given roughly by:

d~scat Wmax~ dt max (5.5)

Scattered light beams have a variety of origins in the interferometer. One source is the delay line itself, as noted in MPQ81. Referring to Figure 5.5, as the main beam exits the delay line some of it can be scattered back into the delay line and make a further round trip. This extra round trip (or trips!) leads to a phase difference between the main and scattered beams of:

2ir = - Nb [1 + dlc(t)] (5.6) Oscat(t) A where 1 is the nominal arm length and dle(t) is the common mode change in the arm length. (The loop system holds the differential motion of the arms to a null;

- 48 - ..~~... ~J...t..

' ...~~......

Figure 5.5 A Mechanism for Scattered Light Noise- Scattering at the Delay Line Hole however, common or breathing mode motion of the arms driven by ground noise can take place.) Though these common mode motions are at low frequencies (0.5 Hz, 10 Hz, 87 Hz, etc.) they can have large amplitudes and, through frequency up-conversion, can lead to scattered light noise frequencies in the kilohertz range. Other sources of scattered beams include reflections from any of the optical elements (beams scattered before entering the delay line have the 0scat dependence above), scattering from adjacent or overlapping spots in the delay line, and scattering from the walls of the vacuum tubes. There are direct techniques to combat the effects of scattered light noise, con- centrating on the Escat/Emain term or the scat term. The amplitude of the scatterer can be reduced; in the prototype, introduction of blackened rings around the delay line entry/exit holes reduced the noise level and its frequency extent. The variations in kscat can be decreased through frequency stabilization and reduction of the common mode mass motions; these methods have been successfully applied by the MPQ group (Shoemaker et al. 1985). Another set of schemes is based on the (often large) path length difference between the main beams and the scattered beam(s). By deliberately applying phase modulation to the laser beam, 4 scat can be manipulated and the noise power spread out in frequency or moved to frequencies out of the measurement range. In the original modulation scheme (MPQ81), a sine wave modulation is used resulting in the elimination of the effects of some scatterers, but not all. To attenuate scatterers occuring at a variety of path lengths a scheme using band limited white Gaus- sian modulation (Weiss 1982) and ones employing digital modulation (Appendix

- 49 - 100 1000 10000 2 3 4 5 6 7 8 9. 2 3 4 5 6 7 8 9 I . g . I ... 1. . 1 . I I . I ~ I I . i

...... C,)

...... : ...... : ...... : ...... -- 4 (U- ...... j .... I .... C ..I ...... I ...... j ...... I ......

...... -CDu

...... N ...... I ...... 3 ..... I...... C.... I ...1...1-4-4.4,

...... -a- C4...... ; ...... - .0 ......

CU ...... r C)

...... 0 1 ...... CD ...... C

...... i... i i. . - - - 4 ...... - 4 .. .i .04- I () ......

CU ......

...... ' 1 ' .- 4

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 100 1000 10000 Frequency (Hz) Figure 5.6 Spectra with/without Digital Phase Modulation

I, Schnuipp et al. 1985) have been developed. Appendix I describes these in more detail. The contribution of scattered light noise to the interferometer noise spectrum can be assessed by comparing spectra with and without applied phase modulation. Figure 5.6 shows such a comparison; digital phase modulation has been used. Signs of scattered light noise are present above the resonance at 800 Hz. From the analysis and demonstration in Appendix I, digital phase modulation provides greater than 20 dB suppression of (most) scatterers having path length differences greater than 6 meters. The observed suppression of at most 12 dB suggests another noise floor or short scatterers. The Gaussian modulation scheme will work for scatterers as short as one meter-the two methods result in the same suppression where comparison is possible, strengthening the "another noise source" argument. (The clump of resonances from one to two kilohertz shows up clearly.)

5.2.6 Laser Frequency Noise Laser frequency noise, in concert with a static path length difference between the interfering beams, produces an equivalent displacement noise given by:

- 50 - 100 1000 10000 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 I ~ IL II II

IJ CU- .e...... I~ IC 4 -...... {) )......

S-..- - ......

Frequ.n.y ...... Hz)...... F Cf Laser Fr.quen .y N ..s .C ntri ..ti.

Cu E ~10 00 00

dxopticai = dLopticai (5.7)

Billing et al. 1979 report a measurement of the frequency fluctuations of an un- modified argon ion laser. Using their values and our measured limit of optical path length difference for the prototype (56 x 0.2 mm) gives the curve plotted in Figure 5.7. The du/v value corresponding to the lowest level shown is i0' 3 . Though highly inconclusive this suggests that frequency noise may be important in the < 1 kHz range. Further suspicion is thrown on frequency noise by a significant change in the shape of the noise spectrum from data taken with the model 165 and model 2020 lasers, shown in Figure 5.8.

5.2.7 Thermal Noise Thermal noise sources in interferometric antennae have been described (Weiss 1972 and Shoemaker et al. 1985, for example). The displacement spectrum of a thermally driven resonance is divided into three frequency regions: below resonance (spectrum flat), resonance (spectrum peaks), and above resonance (spectrum falls as

- 51 - 100 1000 10000 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 ......

N .. -...... -.-. CU*

OD4 5 ...... 3 4 .... . 7 ...... 0 4 ......

CU- ...... Z...... r 0 ...... L ...... -4 CD E U-...... ca) CD0 .. ..-...... T...... W ...... ;.

...... U -...... 7 ......

(U ...... T-lI

2 3 4 56 7 89 2 3 4 56 89 10 0 1000 10000 Frequency (Hz) Figure 5.8 Comparing Spectra taken with "Old" and "New" Lasers

1/f 2). Because the resonance peak is the largest feature it is sensible to concentrate on it. For a typically high Q (> 100) resonance in the kilohertz range the bandwidth of the (low resolution) spectrum is comparable to or greater than the line width. Thus, the height of the resonance peak in the spectrum is determined by the rms amplitude of the resonance and not particularly by its Q. The rms amplitude (an example of equipartition) is given by:

kT Xrms = 2 (5.8)

Choosing a spectral bandwidth of order 7 Hz (i.e., a 2000 point, 0-10 kHz, Hanning apodized spectrum), the thermal peaks would be expected at an equivalent displacement noise level of:

xnoise = 3.8 x 10-15 1 1 m/v/Hz (5.9)

This line is plotted in Figure 5.9 for several values of the resonant mass. From

- 52 - 100 1000 10000 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 I I t

CU - -Co C,) C -4 -CD W

N CU -

CU- 0 U, ...... 4...... 4...... 7... 9 0 G) 0 --4 L) W -wu (U-

-CD

cu - -Cu

100 1000 10000 Frequency (Hz) Figure 5.9 Loci of Thermal Noise Resonances and Low-f Floors these lines it is reasonable to expect that thermally driven resonances are important in the spectrum. Note that because of inefficient coupling of the resonance motion to interferometer phase changes, the measured peak heights will fall below the levels predicted. (Detailed identification of resonances has not been carried out.) Contributions to the noise spectrum from either the below resonance or above resonance terms are unlikely. The below resonance displacement noise is given by:

4kT Xnoise = - 2.6 x 10~16 1 (5.10) WOmQ f3mkg m/Hz

Some of the levels expected from this equation are also shown in Figure 5.9; note that Q = 1 and so these are upper limits to realistic levels by probably an order of magnitude.

5.2.8 Beam Jitter A measurement of the interferometer sensitivity to changes in the input beam angle gives a maximum value of ; 10-' m/radian for near vertical beam deflections (produced about a point 0.5 m from the beam splitter). To produce the observed

- 53 - noise levels near 1 kHz of 10-15 m/ViiHz, an angular input noise of 10' radian/ Hz is required; or alternatively a relative vertical motion of the central mass with respect to the laser beam of ~ 5 x 10-10 m/V Hz is necessary. The motions of the table at 1 kHz are measured to be ~ 10-13 m/N/rHz suggesting that, except for mechanical resonances in the input optics, broad band beam jitter is not a problem.

5.2.9 E/M Fields The 1/f2 trend of the noise spectrum is exactly what would be predicted by assuming that a random white voltage noise is present on the mass-driving plates. Using the known voltage-to-force coefficient for the capacitor plates (see Appendix E), the required voltage noise is ~ 7 x 10- V/ /Hiz. This is one thousand times the high voltage amplifier noise level.

- 54 - 6. Data Analysis Scheme

6.1 Introduction Work on data analysis schemes for gravity wave antennas to date has been confined principally to analysis schemes for resonant (bar) detectors. Analysis schemes for the potentially more informative broadband detectors (laser interferometric) have received little attention with two exceptions: a search by Forward 1978 for burst events in the output of his interferometer using his ear and brain for the event detection, and a search for periodic gravitational radiation from the millisecond pulsar, PSR 1937+214, with the Caltech instrument (Hereld 1984). This section presents a scheme using matched filters to detect burst sources in the output of interferometric broadband antennas. The principle motivation behind developing a data analysis scheme is to detect and analyze the radiation from expected sources. Because of the infancy of the gravity wave field, the problem of detection is of prime interest. This, however, is not the only reason for developing analysis schemes. In addition to providing a means for source detection, a data analysis scheme provides a concrete framework in which to analyze the sensitivity of an antenna to a given source. In particular, the signal-to-noise ratio, SNR, for various waveforms can be evaluated. This allows a meaningful comparison between different antenna types to be made, in particular between bars and interferometric antennas (Appendix D). Finally, a search for realistic events with an antenna, even if the events are known to be infrequent and likely undetectable at operating sensitivities, has value in uncovering and setting limits on noise sources. At present, the performance of broadband detectors is given in terms of the power spectrum of the output noise, ideally shot noise limited. This spectrum is sufficient to evaluate the detectability of periodic sources, however it gives only an incomplete picture of the antenna's noise characteristics when used to detect weak and infrequent burst type events. The following sections present a summary of the matched filter concept, a discussion of the practical considerations of implementation, and a canonical set of filters to detect many of the predicted source waveforms.

6.2 The Matched Filter The matched filter addresses the problem of detecting a signal with known shape in the presence of additive, white, Gaussian noise. This is indeed the type of noise expected at the output of a shot noise limited interferometric antenna. The signal at the output of such an antenna is proportional to h(t), and many calculations of the expected wave shapes have been made (Section 3). The assumption of known wave shape is discussed further in Section 6.3.2 below. The terms white and Gaussian are independent: "white" implies an average lack of correlation from sample to sample, i.e., a flat power spectrum; "Gaussian" - 55 - /..A..VV..WM.j..... \NV\JVJVtVJN/\NVNV\f Figure 6.1 Examples of White Noise (left) and Gaussian Noise (right) describes the probability distribution of the time series. A Gaussian signal has the important property that when it is filtered by a linear filter the output signal is also Gaussian. Figure 6.1 shows examples of white and Gaussian noise. The solution of the above detection problem grew out of radar research (North 1943) and is now a standard method (Helstrom 1969, VanTrees 1968). The optimum signal to noise ratio, SNR, for detection is achieved by cross correlating the data with a template of the searched for signal. (Appendix J.1 presents a justification of this.) This is equivalent to filtering the data with a filter whose impulse response is the time reversed signal, thus the term matched filter.

6.2.1 Signal to Noise Ratio The fundamental equation of the matched filter expresses the SNR for detection in terms of the characteristics of the signal and noise. Given a waveform h(t) in the presence of additive, white, Gaussian noise with a single-sided power spectral density of h (in strain squared per Hertz), the resulting SNR, when optimally detected, is given by:

2 SNR-I2f h (t)dt (6.1) N

(A derivation of this formula is presented in Appendix J.2. Note that the integral in equation 6.1 can be viewed as the "energy" of the signal; the term is avoided in this discussion because of possible confusion with the gravitational wave energy in Section 3.) This SNR is an amplitude SNR, proportional to h; it gives

- 56 - the ratio of the peak filter output when detecting a signal to the rms filter output in the presence of noise only. The filter can be analyzed and implemented in either the frequency or time domain. Here, the time domain is used because of conceptual and computational (Section 6.4.3) advantages.

6.2.2 Pulse Height Distribution In operation, the input time series, consisting of noise and, possibly, signal, is assumed to have a Gaussian distribution, given by:

p(x) = e-(x/o)2/2 (6.2) a-v/2-7 where

p(x) = the probability density of x - = the rms value of the series SNR = x/-

If there are no signals present at the input, the filtered time series will also have a Gaussian distribution. If, however, a signal with a large enough amplitude is present at the input, non-Gaussian "tail events" will show up in the output distribution near the coresponding SNR value. Because of the filtering action of the template, the effect of a signal on the output distribution is greater than on the input. A demonstration of the matched filter operation is presented in Figure 6.2. In this figure and elsewhere pulse height distributions (PHDs) are plotted as log N vs. SNR 2 . From equation 6.2 this results in a straight line for a Gaussian distribution; the expected Gaussian distribution is shown as a dotted line. In Figure 6.2a an input Gaussian, white time series is filtered yielding a Gaussian, non-white time series. In Figure 6.2b a pulse, with characteristics similar to the filtering template, has been added to the data and the resulting non-Gaussian tail events are apparent in the output PHD.

- 57 - 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 WWL i IlIUM I 1111111 IWIUI ~ MMIII LJIMMI I JJUMI I LWUIJ ... ?HD ......

.(S)-

1- I ...... !...... !...... I ...... m CDw M L ...... i ...... L ...... L 0 a) -Te rr e L

z ...... CS) ...... t.tfl...... z

cI' 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 00 SNR**2 SNR**2

700 720 740 760 780 800 820 840 860 880 700 720 740 760 780 800 820 840 860 880 1 1 1 111 111 1.i~a 11I..m..1.1. . 1 .11 6 .. 6 1 1

U) W) - cr- CL(U \WV\/VYP&Af\ ICE

700 720 740 760 780 800 820 840 860 880 700 720 740 760 780 800 820 840 860 880 Time (samples) Time (samples)

o A(.S% on~y 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 1. . .I I L. . - UIUU{Wi llii ilU&U IIUUJIl1IIIU UIIIIUU4IIIiIIUW -

.,S V\ o t c4ak kQ ,c \ . il t......

(S) Q- LA _

M c G)

L a L CS) M M L 6y i ap *+ .0 ...... E (S) ...... G) z 0 z-. _ir1 -ct

-4

1Tmmnrinnm S", I ,t"" 7I" 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 SNR**2 SNR**2

700 720 740 760 780 800 820 840 860 880 700 720 740 760 780 800 820 840 860 880 e a a i t s a i i s a i a liala,.i I S I.111 a ,.1... em I. a a I I I I I a I I I I I a j I I a a I I a i a I i I a I a

(n,

N, cr- (U- O0

E. (E

700 720 740 760 780 800 820 840 860 880 700 720 740 760 780 800 820 840 860 880 Time (samples) Time (samples)

J c 6, 2 6>n o Nots5 + PkJ's. 6.3 Practical Considerations

6.3.1 Sample Rate and the White Assumption The data from a gravity wave antenna will typically be taken, stored, and filtered digitally. Ideally the input signal will be bandlimited at the Nyquist frequeny, one half the sample rate, to avoid aliasing of higher frequency noise into the data. In this case the SNR for a given signal is independent of the chosen sample rate. The sample rate is chosen to keep the number of points in the, now discrete time, templates to a minimum while retaining a reasonable approximation to the wave shape. The assumption of a near white noise spectrum is justified, over a decade or more in frequency, by the performance of present-day interferometers. However, increasing ground noise and decreasing isolation of supports lead to a steep rise in the noise spectrum at low frequencies. Filters to remove this low frequency noise must be included in the system, a combination of analog filtering of the antenna output and digital filtering of the sampled data. The scheme will work in the presence of non-white noise with the rms value out of each template a function of the input spectrum.

6.3.2 The Effect of Mismatched Templates At first glance a drawback of the matched filter scheme is the large number of possible templates-do all of these have to be implemented? In fact, a template can vary substantially in appearence from the actual signal and still obtain a near optimum SNR, provided gross characteristics are similar. Many of the waveforms of likely sources can be viewed as composite: the waveform is made up of a series of component pulses-each a half cycle of a sine wave- delineated by the waveform's zero crossings. A template mismatch can be examined in terms of the effects of a mismatch in the shape, amplitude, spacing, and number of these component pulses. A mismatch in the shape of a component pulse leads to only slight reductions in the SNR for detection. For example, using a three level approximation to a sine wave (Whitney et al. 1976) results in a SNR reduction of 4% for detecting a sinusoidal pulse, and a loss of 6% percent for detecting a triangular wave shape (Appendix J.3.1). From the integral term in equation 6.1, the relative value of each of the component pulses to the overall SNR is seen to depend on the product of the square of the amplitude of the pulse with the length of the pulse. Thus, only the largest pulses need to be considered for inclusion in the template; precursor and ringing tail pulses, even with amplitudes as large as one third to one half of the largest pulses, can be ignored. Likewise, slight differences in the relative amplitudes of the

- 60 - 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 S...... I...... I

S CD -. 7. % E0 Z Nc

W. ..I...... * iii I ii i i i i ii i

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 F/F-Opti mum

Figure 6.3 Effect of Frequency Mismatch of Template and Signal component pulses of the signal from those of the template result in a small reduction of the SNR. Finally, the spacing and number of component pulses, NRC, are the most critical parameters of a template. The spacing of the pulses determines the frequency of the filter; the number of pulses, the bandwidth. The effect of a frequency error is shown in Figure 6.3; it can be seen that the effective bandwidth of a template decreases with increasing NRC. The reduction in SNR for a mismatch in the number of component pulses is derived in Appendix J.3.2 and discussed further in Section 6.4.2.

6.3.3 Multiple Detection As the example in Figure 6.2 shows, the presence of a single burst event in the data typically leads to more than one "event" in the output template pulse height distribution (PHD). In fact, the expected distribution of events added to the PHD by a matched burst is given by the PHD of the template's ambiguity function-this is just the output of the template for a matched input, i.e., the auto-correlation of the template (appropriately named because of its role in determining the timing of events, see Theriault 1975).

- 61 - FigureCO 6. FomaC annclBrttsheDgtlTmtt NI 3 No=jI N HCL

DL L-Loyl ~ tQ

Figure 6.4 From a Canonical Burst to the Digital Template

In addition to the event multiplication due to the ambiguity function, there is event multiplication due to detection of the same event by several templates. Note that in this regard, the scheme described below is designed to detect any interesting event, not to provide an orthogonal classification of the events. The removal of multiple detections is straight forward and is described and demonstrated in Section 7.2.5.

6.4 A Set of Templates From the waveforms presented in Section 3, it is reasonable to choose simple sine bursts as templates. Each of these is parametrized by two parameters: the number of half-cycles in the burst NHC, and the burst frequency. Because the implementation of the matched-filter is digital, the filter frequency is given normalized by the Nyquist frequency (one half the sampling rate). The set of 22 filters described below covers a range of bursts having NHC = 1 to 8 and fburst/fNyquist = 0.08 to 0.55. The signal-to-noise ratio achieved for bursts in this range is typically within 85% of optimum; applied to "real" waveforms, see Figure 3.2, the SNR is typically within 75% of optimum. Because of the simplicity of the filters they can be easily and efficiently implemented on a digital computer.

6.4.1 The N1 ,NoNHC Templates In order to reduce computation time, the templates are 'clipped' to take on only the values +1, 0, and -1 . An example of a sine burst, the clipped approximation, and the resulting digital template are shown in Figure 6.4.

The digital template is described by three integers : N 1 , No, and NHC. The number of sample times per half-cycle during which the template has the value +1

- 62 - 0 10 20 30 40 50 60 Time (Digital Samples)

Figure 6.5 The Set of Templates and their N 1,NQ,NHC Codes

(or -1) is given by N1 . Likewise N0 gives the number of digital periods spent at the 0 value. Finally NRC specifies the total number of half-cycles in the burst. The similarity of all half-cycles aids in high-speed implementation but restricts the template frequencies obtainable. A template's frequency is given by:

ftempiate (6.3) fNyquist N1 + N0 63

Note that the three degrees of freedom of the template, N1 , N0 , NRC, are reduced to two by requiring N0 ~ N1/3 to obtain the best match to a sine wave.

6.4.2 Choosing a Subset of the Templates The number of templates required to cover a given region of NRC-frequency space is reduced by two considerations. First, the filters have a finite bandwidth and thus a finite set can cover a large frequency range; the frequency spacing of the filters is chosen to ensure detection of the bursts with tolerable reduction in SNR. Because the bandwidth is inversely proportional to NRC, more filters are required to cover the same frequency range for larger values of NRC. Second, detecting a

- 63 - burst which has some number of half-cycles by a template with a different number results in a small reduction of the SNR if the numbers are close. The change in SNR is given by (Appendix J.3.2):

SNR NHCsmaier (64) SNROPt NHCiarger

The chosen set of templates is shown in Figure 6.5. Note that template 101 is a simple impulse and thus its output is the input series. Templates 301 and 601 have response to DC and the 102, 104 templates have broad responses at the Nyquist frequency. The SNR response of the remaining templates with NHC=2,4,6 is shown in Figure 6.6. This response is the frequency response of the digital filter weighted by Vf/ to keep the detectability constant. For signals with fburst/fNyquist = 0.08 to 0.55, the response is within 85% of the peak (with a few exceptions in the NHC=4 and 6 sets). Likewise the exclusion of templates with NHC=3, 5, 7 or 8 does not seriously reduce the SNR for sources with these characteristics; the additional reductions are 87, 91, 93, and 87 percent respectively.

6.4.3 Implementation Simultaneous implementation of all templates is easily accomplished with only additions and subtractions. By making use of past values of the template outputs it is easy to generate essentially one template output value per addition: 101(I) is the new input data point from this is generated 301(I) = 201(1-1) + 101(I)

312(I) = 301(I) - 301(1-4) 314(I) = 312(I) + 312(1-8) 316(I) = 312(I) + 314(1-8) As presently implemented, computing the new values of the 21 (not counting 101) templates requires 34 addition operations per input point. For comparison, the computer time required to pre-high-pass the data and, after template filtering, form the 22 pulse height distributions is about 12 times that required to do the actual filtering. A large improvement in computation speed could be made by using an analog high-pass filter; the integer A/D data could then be directly template filtered with all operations in integer arithmetic.

- 64 - 0 0 0 1 0 2 0 3 0 4 0 5 0 6 1 1 1 f I II i I I I I III I I I I1I1a1I I1I1I1ItIuIII111111

~8~~~ 3; ~ .~

x 00 0 E z En z U) -,

-CD

00 0.1 0 2 0 3 0 4 0.5 0 6 NH C F/F-Nyqui st

0 0 0 1 0 2 0 3 0 4 0 5 0 6 11111 111 1 111111 1 11 11 11]1 111 1 11 11 I~ -

1639 6a4 4;'1 3'9 .20 4

-, x 0 CS) NHC: E a, z Ur) (S) z aS) LO '-D a,

0.0 0 1 0 2 0 3 0.4 0.5 0 6 F/F-Nyquist

0 0 0.1 0 2 0 3 0 4 0 5 0 6

1 1 1 1 1 1t t~ 1 ihh l th hth 1 11111 11111 1 1 1 1 1111111111 1 t l-

13( 4;L6 4.6 16 316 16 206 CS) CD x 0 E CD z a, CD z

11 111111 1 " " 0 0 0 1 0 2 0 3 0 4 0 5 0 6 F/F-Nyquist

Figure 6.6 SNR Response of the Digital Templates

- 65 - 7. The Data Taking Run and its Analysis

7.1 The Run The goal of the data run was to obtain data enabling searches for burst and periodic sources to be carried out. In order to obtain a long "time baseline", data were taken with the prototype on six of the seven consecutive nights, June 3 (Monday) through June 9 (Sunday). A total of 15 hours of data was recorded, approximately equally divided between single channel 10 kHz recordings and three channel 3 kHz recordings. (The data taking system is described in Appendix K.) Of the 15 hours, six were taken during the nights of June 3,4,5; one hour each were taken on June 7 and 9; and seven hours on the night of June 8. On June 6 the change in lasers (Section 4.3.1) was made. Because of the susceptability of the instrument to loss of lock due to ground disturbances of passing vehicles, the data were taken primarilly during the hours of 2 a.m. to 6 a.m. In addition to making the data collection an arduous task for the experimenters, this severely limits the total efficiency. In order to collect a good night's data, Vassar street and the East Parking Garage were closed for the nights of June 8 and June 9. This allowed data to be collected from 10:30 p.m. to 8:30 a.m. on the night of June 8. After initial adjustments and verification of the instrument's operation, data could be collected with little fussing by the experimenters. The data collection duty cycle of 50 to 70 percent is determined primarily by the time required to rewind, log, and load a new tape.

7.2 Analyzing the Data

7.2.1 Introduction and Overview There are two goals of this search for burst sources in the interferometer output. First, from the point of view of the noise performance of the instrument, the search can provide information not available from the short term averaged spectra typically used for characterizing the interferometer. Secondly, the results of the search can detect or provide limits on possible astrophysical sources. The analysis is performed on the 10 kHz data from the single night of June 8. This set is made up of 11 tapes of - 15 minutes each; a list of the tapes and their time coverage is given in detail in Table 7.1 and pictorially Figure 7.4. The analysis, from the data tapes to a final list of candidate events, proceeds in several steps, schematically summarized in Figure 7.1. Each tape is first processed to produce a disk file, suffix HKP, containing the housekeeping information from the tape and some summary information derived from the data on the tape. The tapes are processed a second time to produce pulse height distribution (PHD) disk

- 66 - - Remove housekeeping data - Calculate max, min, rms and average in each chunk of data

11 .HKP files

11 Tapes from June 8 to 9. 39 .PHD files

- High-pass - Template filter - PHD bin - Save high SNR events

Instrument parameters Time information .HKP files List of Events Pulse height distributions .PHD files Template rms values Events - ho rms and threshold selection - Windows - Multiple detections removed

Figure 7.1 Schematic of the Data Analysis Proceedure

- 67 - files which contain pulse height distributions and lists of selected (high SNR) events for each of the 22 templates. The information in these disk files is then synthesized to produce a final list of events, calibrated in time and ho amplitude, and an overall pulse height distribution. The tapes are used one final time to retrieve and examine the waveforms of the statistically interesting events. It is necessary to introduce some terminology used with the data. As described in Appendix K, the fundamental unit of data-taking is the "chunk". It consists of 32768 words (16 bits each); of which 32608 are data samples and the remaining 160 are made up of two sets of housekeeping samples. For the 10 kHz data, one chunk represents 1.63 seconds of data; there are ~ 550 chunks per tape.

7.2.2 The HKP and PHD files The generation of the HKP files is straight forward: for each chunk of input data the two housekeeping sets are copied to an output block of data. In addition, the data in each chunk (and each channel for multi-channel data) is processed to calculate its rms, average, maximum and minimum values within that chunk. When the interferometer goes out of lock the maximum and the minimum values will saturate at the A/D limits and, thus, the HKP file can be used to determine these unlocked times (for an example see the lower traces in Figures 7.5 to 7.15). This process takes less than an hour per tape. The generation of the PHD files involves four sequential operations on the data: the data is high-pass filtered; the data is filtered by all 22 templates (Section 6.4); the resulting 22 output time series are binned to produce 22 pulse height distributions; and the events with the largest signal to noise ratio for each template are saved. Thus, the PHD file contains 22 PHDs and 22 lists of events. A spectrum of the interferometer signal as recorded on tape is shown in Figure 7.2. The non-white nature of the data is clear; in particular, the sharp rise below 1 kHz is apparent. Because the templates do not have sharp roll-offs towards DC, it is necessary to digitally high-pass filter the data. The high-pass filter used is a simple finite impulse response filter of length 61 points and apodized with a Kaiser window (see Bozic 1979). Its roll-off below 1 kHz is close to four poles. The spectrum of the high-passed data is shown in Figure 7.3 along with the locations of many of the templates. The output of the high-pass filter is then filtered by the 22 templates as described in Section 6.4.3. Each point of each output series is then binned, according to its amplitude (separate bins for positive and negative SNR values), into pulse height bins corresponding to each template. The first chunk of data is used to estimate the scaling of the pulse height bins-chosen to cover an SNR range of 0 to 6. Events with SNR above 6 are binned in the 6 bin. In addition to filling the PHD bins, in which each event loses its identity, events with large SNR values are "saved": their SNR value and location in the

-68 - -30 dB

0) -60 AB cx

Figure 7.2 Spectrum of the Data as Recorded on Tape

() j B

-J 3134

_ I_I ,_ A _ILL 1_ _ -tj - 6 0 4B

0 5kH,

Figure 7.3 Spectrum of the High-pass Filtered Data

- 69 - tape are recorded in a list of events. There are two approaches to saving these "tail" events: a fixed SNR threshold is chosen and all events above this vlaue are saved or a number of events to be saved is picked and the SNR threshold, the "event threshold", is adjusted to retain this number of events. The latter scheme was chosen for this analysis; the 128 events with the largest SNR in each template are saved. The smallest SNR of these saved events gives the event threshold for the template. This is a dynamic quantity: as data is analyzed this threshold (set initially at 3.5) increases as events with larger SNR are found. For the PHD files used in the analysis, the SNR threshold is typically between 4 and 5. Thus, all events with SNR greater than this value are saved-their location on tape and their SNR value recorded. For several reasons a tape is analyzed in several sections; typically, four PHD files are produced from each tape (each covering 128 chunks and skipping the first 30 chunks). One important reason for this sectioning is the occurrence of occasional losses of lock producing worthless data. (See Appendix H.) For example, tape JUN8TT was analyzed in four non-contiguous sections as indicated in Figure 7.13; the lower traces show the saturating of the A/D output during unlock. A more practical reason is the time required to process the data: one tape requires approximately 30 hours of VAX 730 CPU time. By dividing a tape into 4 sections they can each be done in one overnight session. Finally, there are occasional periods of time during which the instrument is excited-presumably due to acoustic or seismic stimulation. These periods claim many events of the 128 allotted, and can push the SNR event threshold to unacceptably high levels. (In hindsight, some elements of this strategy could be improved; this is discussed further in Section 8.3.) The PHD and HKP files represent a compression of the amount of data by a factor of about 60; even so it is difficult to "view" all the data. A quick and informative overview of the results of the PHDs can be had through the event distribution plots, Figures 7.5 to 7.15. In these plots the saved events from each of the templates are plotted in time with a height proportional to the square of their SNR; they are clipped at an SNR of 8 and, thus, won't protrude into the next template. The curves at the bottom of the plot show the maximum and minimum values of the data within a chunk (from HKP) and clearly indicate regions where the interferometer has lost lock. Regions of dotted data indicate areas excluded from further analysis, as discussed in Section 7.2.4. For comparison, the last of these plots is from a tape of shot noise data: the interferometer was bypassed and light shone directly on the photodetector with the signal chain the same as for data taking.

- 70 - Table 7.1 List of the Windowed Data Files and their Parameters

Window Time of Window Parameters Housekeep file PHD file Start. End Start End P mW VPR (chunks) (EDT) 0 JUN8AA.HKP J8A1.PHD 71 -> 134 22:37:42.43 22:39:26.77 12.2 61.6 J8A2.PHD 182 -> 277 22:40:43.40 22:43:19.92 12.2 61.9 J8A278.PHD 278 -> 392 22:43:19.92 22:46:27.42 12.1 61.1 J8A3.PHD 415 -> 554 22:47:03.29 22:50:51.54 12.1 61.0 JUN8CC.HKP J8C1.PHD 30 -> 125 23:23:15.56 23:25:52.07 11.4 57.5 J8C2.PHD 200 -> 300 23:27:52.72 23:30:37.39 11.3 57.2 310 -> 333 23:30:52.07 23:31:31.20 11.2 57.1 J8C3.PHD 394 -> 554 23:33:09.02 23:37:31.52 11.4 57.2 JUN8EE.HKP J8E1.PHD 30 157 0:10:59.65 0:14:28.34 11.0 55.2 J8E2.PHD 158 285 0:14:28.34 0:17:57.04 10.9 55.1 J8E3.PHD 286 319 0:17:57.04 0:18:52.47 10.9 55.1 323 395 0:18:57.36 0:20:56.38 10.9 54.8 406 413 0:21:12.68 0:21:25.73 10.9 55.3 J8E4.PHD 414 541 0:21:25.73 0:24:54.42 10.8 54.7 JUN8GG.HKP J8G1.PHD 40 -> 157 1:49:50.87 1:53:03.26 11.7 58.2

JUN8II.HKP J811.PHD 30 84 2:46:33.12 2:48:02.79 11.4 56.9 90 157 2:48:10.94 2:50:01.81 11.5 57.1 J812.PHD 158 285 2:50:01.81 2:53:30.50 11.5 57.0 J813.PHD 286 413 2:53:30.50 2:56:59.19 11.5 57.0 J814.PHD 414 541 2:56:59.19 3:00:27.88 11.5 57.4 JUN8KK.HKP J8K1.PHD 22 96 3:36:37.46 3:38:39.74 11.6 58.1 J8K97.PHD 97 171 3:38:39.74 3:40:42.02 11.6 58.1 J8K3.PHD 172 271 3:40:42.02 3:43:25.06 11.6 58.0 J8K4.PHD 272 340 3:43:25.06 3:45:17.56 11.6 57.9 J8K5.PHD 341 436 3:45:17.56 3:47:54.08 11.6 57.9 J8K6.PHD 437 552 3:47:54.08 3:51:03.21 11.6 58.0 JUN8MM.HKP J8M2.PHD 158 -> 223 4:32:11.71 4:33:59.31 11 .5 57.4 230 -> 285 4:34:09.10 4:35:40.40 11 .5 57.5 J8M3.PHD 286 -> 413 4:35:40.40 4:39:09.09 11 .6 57.5 J8M4.PHD 414 -> 541 4:39:09.09 4:42:37.78 11 .5 57.4 JUN800.HKP J801.PHD 30 -> 123 5:14:04.02 5:16:37.27 11 .7 58.1 J802.PHD 250 -> 399 5:20:02.71 5:24:07.27 11 .6 58.0 J803.PHD 430 -> 557 5:24:56.18 5:28:24.87 11 .7 58.2 JUN8RR.HKP

JUN8TT.HKP J8T1.PHD 30 123 7:16:46.52 7:19:19.78 11.4 56.8 J8T2.PHD 141 262 7:19:47.50 7:23:06.40 11.4 56.7 J8T3.PHD 288 362 7:23:47.16 7:25:49.44 11.4 56.5 J8T4.PHD 424 472 7:27:28.90 7:28:48.79 11.3 56.5 481 526 7:29:01.83 7:30:16.83 11.4 56.7 JUN8VV.HKP J8V1.PHD 323 -> 425 8:07:37.69 8:10:25.62 11.3 56.2

- 71 -2 0 2 4 6 8 I I I I ... I1... I1 . I , I .. 1 . - t I I I I I I I I A C E G I K M 0 T V

-1 ~n

WjW

I I S i I I -2 0 2 4 6 8 Time, June 9, EDT (hours)

Figure 7.4 Temporal Coverage of the Windowed Data Files 0 100 200 300 400 500 600 i I II I I J8AI.PHD I J8A2.PHD I J8A278.PHD I J8A3.PHD ...... 736 ...... a n ...... Kit. tIis Ijli.1.kt1.1S1il111 ilft I 626 JJ. .Li. lmii. l J.1...... i. . I . I.I ..... t . J...... J.. . ..a L...l..I. ..IL...J..L LK...... i... .A.I .. . 1JJ1.1 LAI.1 .1...... N .. (U 426 A... 1.1. IM .1.1 ... 11.. I i... .11...1.11.0. J. A L. 1i.1J J I.... -i. 31f.... i. J.. .. 1... - 1 . 416 .1Lj . I .Al .. I.it Nl . I IAi~l h.I . i A .. ! l. ... J....Aftil ;I i .Id... .t 11.JRi. J.. Nil 1.1 .... Oili It...lILIAl 316 A imiillum.1 .JMiLi I .I ...1. 1.JM AJifti .tim..1JL IlII Jd tI. li.twi t.il.2 4!3.110 Ib ... sIWA. t 2 R.olmima j ma8 ...... i ..... 1.1111.1 ru s.131f11141 in 216 a .1i.t.. a m i.| . l a Ii aid1 .31.1I IJ..I...... I ...... it .t.. in 206 A.I JbIoAml.. a .faun hillu MI l issala ill i: Ill vanal JIL 1. 3 J. 11 ill i. 1.1. II LI mili 834 .Iis ma. ININN..u .U.. l. it lal gaii .. h .1.3.. j.uJUa m.i.. ... ti...... idi .t. 1111.1.AL.. 1im ...... It hi.LOI.t m a) 624 J . a11..1..d. a IL..... I ... 1 .ihM .... II l.u.i .... lot I .l. u. ..n I. .. I As.to ... Ills.. J.1.011 I NII .I~.t .91 I .I _0 0 424 ulit .il. Nii J. 1..L.f10i4li.iIl . 111. II.. ht 131 US .J ... .. J. A. 1. m.. L.A. 1..U .. A. It I I 10 .11 1.1. A 111.1. 1 JI I A -(S) 314 .UM. .u! Idm iit .i.l s.01 1 .1. i- 11 .1111 .. J. di ljl 1.. .L .110... Jo.0.d.Oki.iil Ik..J.. 1. 1J.D.Oli 11l 214 11 .111 1e .1111I' n ll i 111 11L J flm.l J. awnli III 1.I1.ljait "JI .1a. ... .1..i~ l. .Llj ..It . I..I...... 1.11L! . *-p CD 0 -4 204 im..iJiakiiuausA i ili mewit.1 1 miL.ilitsuidt 1IM111.61 llIi bi m a wn 1si.l.. . ti .I.* .I.fl.AL.1i.J.1....itll1it..LJOIlbf '-4 inikmi.iuutGum.iiii. mii uwG ma .LLII Ji ll~l. ilihilt l li. til B si .. CIO 104 E i A I i .uai .. it i f L . . . iN: a 832 .oei1..I J 1l.111 ill 2- 522 .1. . lI. . .im. I 1 3. 4l m..i, l I.. 1. J. A...... J.96

312 .lb N.111 1 . J:: Ji.a."" ." - il . .11 JEti.i..atai..i . in 1111...LJMI.Jl.MJ..1JAMlN.I.J~I~uII.t 202 KIMJ~ hiil LICk tka i i . IN021tsn mt 11 o I I IN.M Ai1.1-11 I ISt. I I11.111A iI 11111 . muamIN.JJ~. W. A. "lebIta 102 .ilii ....I. O. iiitI LJ M.1. A1 .ft .011 . I. . J. I I. I .1111.1 . 11 . 1111 .11 . 11 0. A 601 2.ialn 411if i I It i milka it. 11111N. a ...111.11.1 .: 301 .IO..ll kkililliti Jf. I f limi nIibl a t..ls.t a..l 10 101 N|J. Ill.. .11111,1.1. I...5IMI.1. W1111 l ik MAX, M IN fromI - .-.-- .HKP~ l -1file: I. -'

' i ' ' ' ' ' ' ' ' ' 1 ' ' ' ' ' ' ' 1 ' ' ' ' '1 ' ' ' 1 ' ' ' ' ' ' ' ' ' '1 ' ' ' ' ' ' ' ' 1 ' ' ' ' '11 ' ' 1 1 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.5 Event Distribution Plot from JUN8AA.HKP 0 100 200 300 400 500 600 . I I I I - t1 I i i I I I i i i i i lI II I I I I II Ii I 11i i i i i I I II I J8C1.PHD I J8C2.PHD I J8C3.PHD

736 . Jil .lIill . 1111 .... hilliiLI:::.:...... L...... JJIL~It.E 11.4 1i.. i.. 1.ll 111.....VI..11.111 .:.: ...... 11i...... 1 .. . 626 .I...... 1. . .1 is Il .. i ..... t--.. ...: ...... 1.Li ..... J... . Jt...... i... I t .... I ..I . I... A l . .U. isI .. I ... J..... -m 426 .IJ..I...... I...L ...... L...... J .. lj . . l.. I I ... r.. . .l.. .ii. .. .. 416 1. . 1.1 .LI ...L... LI i ... MI.. : . :..i .... LL .L .L I..L .11.1.1.1 J.1.. . 1...IjL A..L.. JI. L... I .IL.1...... ii.. 316 1J.li.J.IH . .Ai.. .l11.11..: .:.:1.. . I..IJ.. i.L . .L...... lt. J. . A1 I1....J.L.Ii.ol.. i ml mil ah. lI .j l .. W\ - 216 ...... jilt 0.1A. ... ii.J.II 111j.hIimanjIdI. AilU..111 ...... i. .. .01 1AJ... U1.111j .J. .6 I I ili t 1111.11HILI ID- 206 834 fiI .L J. I...I I... LLI. ...1. .. I I .lnd.Ik 7cR 0 624 .111. 11.1. 1 .... Jill. I.L N..A. Jli.t:::. ....~~.8.. 1 1 111111bi .... I AJAi .1.1. AM.A I... 0 - 424 . J ... .1. ... .l....L I..U . J...... 314 ... I. .. 111. ... L 1... . 11.1.11...... J.ill.] 1..11 - d ia 11 .11 . I. INll~u~~I ..K 214 ...... : . ::.:...... 204 ...... i.: ...... L...... :...... 1. 104 . I I..... I I .litIJL itt II~llit. JI i I I .. I l.. f .lIA .. 832 Mi. f. om. .If1. H - fl .' I. j..... I... IL.. I .. I...... 1. It ll . .8.I.Nhtb a Al11.1e..KlIAi l 1) - L0 522 I.J A II. t.11 J....A..MLIM..::.:. .... 111....i. .. JAL. It.:... L.1 lill.il..lli.idili X. l 81ldl.l. 114111a.1-1 312 N.]...... L..Iu .. 11.1.1 1.. J.0.. 1. 1.1.. I. .. .AIt J.11. In 16.1..1 fiJ . I" .. .A . .1 AIII. II. .l . 1LJ.. 11011Ol ... I II.. ItI ...... : . . : ...... J...... 1. ....I ..I .. . 202 ..10 l l li l.I .lI J..ll . It IN. L . .1. 11M~t .. 102 :...... J.1 ..... 1.i .. t ...... CD) 601 '::::IIJ.Ul...L11.1. 1... 116.. ...tL. ILI :: . 1.. . 301 ..A. 1.1.11 i .11 .... 11.] [ A . . I:. . I II... 8 ~ J.... 11..1 . 1 . 1 L1. It .1. 1.11. ...L. I.I.. I .'II ..... 1.01j... : . . ::.: .1~...L 1. .1~. .. A..A . i..AtI 1.1.9 I .1 l looki. Ilk.I. t. JIMI luk IJ.111dil. jowl 1.1 i.. MAX, MIN from .HKP f ile: Ji ,I- - -06. .A--A --. A- -k- __ -L - --- __ -. I 0 ___ ------I,

I~ I I I I I I II I II I I 'I I II 1I II II I II 1 1 1I II I I II I II II II 1 1 1 1 1 1 1 1 1 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.6 Event Distribution Plot from JUN8CC.HKP 0 100 200 300 400 500 600

I J8E1.PHD I J8E2.PHD I J8E3.PHD I J8E4.PHD

626 ...... I.... ii ...... l...... L..t. .. .. 42 6 ...... I. u...... I.L ..m...I u ...... J...u.l...I...... I ...... 1...... ii... L... .

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202 ., l.Ia..m 1 ii,. .. .itiam md. It. U1U1311.m11i.lleamIlli..iuui.N.li. fl.l..1I L.l ... i. lI I .iilli l:zl.i..iiiiuiii..i.bi.iu.il.i...m.

102 . 4 .Lau. ii.i a . ..lM..uiLt.utu 1 s.M.wL iLfi I.mL lbiii iii.i,.iiditb..hii..u iii . 131Ii .ml.G i1 Iii :z--titlia i Miii.i ~ 3.31 JMll 3. jiiiilimit 601 .1 1... i 1.. Li.s. Jii i .1. I.. L . II .1... .il l I LI I I. t .11 11.L...... IL ...... L ui i . 11...... IL . .mi..... 1060 11 J.1 util...tli.a ..Is0 .I ai ....11.aii .. i..tt it mia 11.13si lhlt i l... .t iel a. i1 J L..lIi . .sl ..l ....L...... ll..l....h ... ii 1...... :l.- ilJ I J.1L I.i.1 i. . m . Iiim.. Lst. . di.. .I. I 3.1t...... m... i. MAX, MIN from .HKP file: ~

0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.7 Event Distribution Plot from JUN8EE.HKP 0 100 200 300 400 500 600 Ill1 1illI I 11111111111111111111111 I J801.PHD

736 .:.L.11.J.I..IIU. i.tIA It UEJ..It 6 2 6 . .. ..&t.. 1 ...... 1. A. .... I 416 ...... I..... 3 1 6 .. A .. . L ...... I...... I 21 6 ...... C)3.:..ii..I..L.. UI...... Il. I... o>: 6 .-i. b..I l .. 1.. I.l...... I.. . .I.. 24 ...... m.....u. .. u. . U 3104 14 .-- .-.ui .J I.L.t etII I.A. . o it,ILI I jii... 111...... i Utt a 2 14.. .. u...... 204 ::EU.I 1nuI.. i . 1.N. ... .sI..... ItI.. it 104 1u .utnl.i.i ... i t 1. 1 i . 1... E3..... L.. J. .1.l.1 .N1 1... A

3 12 ::1LI..t.. .8..,1t.i I.. 1.s If..,.. ILLt 20 2 :- i t111* ....IJ .tu l...... A L 102 :-Ju .t . nst .. ntt ... . 6 01 -: ... IL. .. J.1. I. .. I. .. N 3 0 1 . ne .s ..... JJ1.A . 1. 1 I..#.. . 10 1 . , A.11..111 ..... LL AJ 1 .. m ll...I.. i. MAX, MIN from .HKP file:

' ' 1' ' ' ' ' '1' '' 1'' ' ' ' ' I I I ii I i I 'i ' ' ' ' ' '' ' II 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.8 Event Distribution Plot from JUN86G.HKP 0 100 200 300 400 500 600

i i i ii I1 11111 |1 1 11 1 | 1111 1 1 1 111111 | 11 111 _ I J8I1.PHD I J8I2.PHD I J813.PHD I J814.PHD ...... 736 - "1. -.it...I...i. *...... s...... ij...... Ljit.. . ll.m...... t .u.t.. .J i x. 5. II. I...I.. Ihig I. o il....i1.1u ... J. lt .1. . J .1.111. l. 11 Jlm..1. LII.... 1idN 1 11. A ... I . . i. l.

216 .1. 1.... i ..*. l. i. L.Al .i.i .. 1... . 1.1i...... i.JJJ iL. i t . t . .ml.ui i .i.. ..n.li...... Jui11.i11. mii .i1...it Lm. 1..1 : 26 4....11n. .i.u..i...... I.. . ..i.. I...... L.1111.11.1. A .A.I..1 ... . I..J.u ...... t .s . .L.IL .. ...

2061112 4 .1. O. .11 i .11.11 il i uI I I. .l. 1i..iI ...I u. ii1Al. I.IL. lill il . A t mi ull . M.bil Olad g . i..i.ti tua i . a .hiiium O liA liL 1 .. 8LI. 4 16)1j 4 li isi I II... U:: I I.i i t iU. I, iligAi. .1.11 . 1 . .1 .. IL igliu .1 ...I lliS ., ll.1.111 i A 1111 .1 . Illi. ... A.IiKi1 -tiiiiA 11113J 1. .11111111.LA .3I11113 83 4 .. .i it. i i1....m .it..Nilm .sbt nli.s J0 1 .. [i 1. . I.j . 1. 1. .N.l . 1i.11u1 . 1 1.. . i. mi.. lD. J. 1.l.il .1i L .iu..ik il A .11 aI. i Eml mu1.... . 3 11 . tii . J. I . . : . 1. Wl li l t.i.lIN. -1 AA.1 ... J. 1ii.II .11. 111.1m-6 111uli. J1.1 1i hl i . It mI.u I I ul.I uit .g iujisiiju. .A.11.13 ik dit.11...1.JA.1 .NOIt1. 1 1 i i.1 J u.01 162 4.. 1 .... lit .... I. . .A.. d.I .. i. . imJi.IA i.. i. . ... LL ..... It ...... lilt ... A . .. i. ....t i... ibIh...m.. i..Jit. l. I.i.dJ. IJiJ.l ll 11 . It . 424 .. 12046~ mitt~ls..l..wII ... J. I... i.l A....101 ....i ~II...i ii~Iii...t... 1iulha .l..1. i...in1 .... .1111a. i.u.nita~ . Iautl haii Ill.hiilL... i .. . a u. .uul.. u.... .uu bnsailui. J.. 1. .3 .. 11.n m..J.m..bI... M iiti .imjtuak i .1.111. I kI Ammiisi .... ti 11..a U. . a . .1..i . i . .- 114 m. 1. si i s. . ... iii .M.u II. I.t. II.i . UJ.l. uti.I1.1. 1..6iil.i1i.I i l il t .. II.il.i .. i ih tiMi ili .lii I. . k i 5. IN..i k.i #it lo IhNJh .t I. -4 204 . ..i 1. 1. 11 1Inj ....- 1.1. J11 11... NJ.111I U.llubitilfil i i i .uJlllb i KNOW It 1IL A Ji h -l . 6 .ilils 11 ka~a Yn~ 11--1. 11 ..4 Ii ,it I..u ..l j f. 11se. A...li. Ill. 11.1... 111.111.l~ ttlt AVt~ li aa .l..111.11 .blmifte"u.N1.11 Ill. lli t~.t .1.0110114I. JHib .MA et11I s -8 3 2 ... u.. w t I. it I. =La..t .t. .l. 1 1. 0. h.Alf..I I..1. J. 1. J.1.Ill. AD.... iUJ.Utd N ...I... .I~hIA I se s. l. n 111.11 N Ilt1It . 1111 M 1111111. OIL .11 ill....~ ) .... i ..ui i .i..l.. 1 II 1.1.1 i 11. A..I .1. W .LL il. J. .. . i ..i. I .I.. I Ini. 1.0 11.. U i i. I. ~ 2 11 Ilu~ S. JJI. uJ . A1*ti t ..ln .i... JILIAIIIILIJ .It~i 1111 J.1s J. fili.1.I.u .111... I W b.0 IJ ... u 1111 .. Ilt . L11 . J.1111. INN 1.111 . No tI Ae 20 2 11m.ll :u.t ".I .. I.. -O..i ill u.1 . I tIIIIII L .. it i as. i Ain i. ~iti sus m 11.8 1 .uh j . U0IN I 1h . 111111. .kdlli k Jn a u il Bi ll LA

60 1 - I 5 ._ 2 = ll .It 11JI JDA.I..I0 _1. i...... 1.1.L 1A tA .1 a. .tiln..D.o..t... .. 1111. ..mi. 1..k -I ....oil Ito. ....1.1b.~m .1~tj.1a~l id~ i.ti~l~~ei 1assai 10i Manca~ . . . II ~t . DO1. -ll it.. It I ttlJ.nl. . H .tl [O t~hNIIIs~un .s. ji iil ~W . li.. 11 a 1 .1611ta ee n MAX , MIN f rom .HKP file : ~

0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.9 Event Distribution Plot from JUN8II.HKP 0 100 200 300 400 500 600

lillIIi I i ii iiiiiiii 111 1 1 11111111111111 1111111 1 i i I 111111111111111ii i _ I J8K1.PHi J8K97.PD J8K3.PHD I J8K4.PHD J8K5.PHD I J8K6.PHD

736s.sni mim ai ihll m i i ii AE sa.lim i. .ft..umsiI.. iii.. a illiiimlua.mi ,,hBibI. iu.NIkhIft. JJLsuli i . li. ~ .L .....d.. 11I.IN.ii. i.... 416.. u ...... L . i. uu...... tm la .. i . ....a . .i. ....J.L...... i... J.. . . I ... L......

626 .1111 . JI . -111 . l. ...ll.NS I tIIII. 11 ILL J . J. l. .1.1 aLI ..I.I.l. JJ.. . i IA.I . 111At. h.. J. 1. 1B.aL 1. 1 86 J- I J .. i 0.1 ..1 t.1.1 I L i l t . litit.. I...... it-O.M...... 3 16ai.,,imim. ,ll l ki da Nd. ui IAi .. i.i.. Jtoll. i.... Jilil iJi i.i.lli.mlmutii.miiiim met iii. I Jit Ii.KII . J . .i.f. I..I 1..u Im . .t i .... . 2 6(U.26 .i.. u. .. i. I iioi um.... im.I.. it.1 ... t li L .. J..1 I .. L. i ii...u i 1tii . J. J i.. 11. Ifti j .. tI ... I I .. im .u.AJ . I J .. 1 A .l J ... L .Q

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4u2 0 6 da. ui.kitt lab .Mit b.nhi.t iitiinilim ii u lp i.. I . l . .ii.a 4i3 1.. hum .AIit. ioiiawi -1li ii iuiii. .lilmum .iimik1.0 ..i ..nl in 8804 3 2 iuin I. o iu.ii. ii n m i u ui . .. uawmgifgLD..iuI ,b . U11.1. 1. L UJIIt suiiniiuinutiui ai.iu a1u1 i iti. 1 . .1.I..fI. LJ .ai....u .i. 01 624 1. 1m.11-611..a aLI. uaLm IL... a ai.J. ... J1.i... .. 1. . 110 ..i LEE i. iklk0- .11.i .i.A ...i.ll*. .. i. . i..I.. .. I J...u.1. uuIN . lL 0i 5 22 . j...... Lj. i Il l.. u ... .u. 1.-.1.u..1.im 1 N.11 111 i. .. u. i 1 1 11.. .1. 1i. 1. . i .. I i1 .ilNM..i.i L.u-1 i...i . .. tu.i. I. .ii. M.1 J.1 11... . it- 312.lt JI.dli.j.ll li. H tadii a .. mai.iuiwomb.it l . 10 .111. 1 A 1. 0 toll O Ni. i d.ii hui.i tmuii.M IAitIi .M.I.MIM.li . 1i.d I i j.l N.. . Old 1111111 Ji.11.11L6 it 6 202mi.mmi.. ii I~ii atiui sh i..i.u.itiui.iiiM. O EiI ii lilihi liarI1.mi tui.iiimL uimila m .di.Ikm.Mialmdli ju.i.u.iiaiitaill aii.u t ui 10211.111 u ir n lain *maai mijiiinit auadai. 2a 4imh . .l4i n ll miuiha lii.ii hint kiitinai. j.,l. I i i J.l fit wi lu L.u i UM i l 60 l.i.i . .u. JLJgHHLI.III I.Mi Kit u.m..m.utdj.hi i.i.iliji itijE J.i .Jii.Mt ijliji.ih.1t 1... i i mh Ji~aiibih.i.Lt it l.ii.i.....il.i j.ii.itili bus.u. .1 ii 301 i. i dul .. i.ling M.i t .illM t..i M .t lh i . li iti. 000t .MiiLui limNijM. li a. I MA a it ili ii t i ]iN . a..... l i -i iltiihii.lN.i ER l ju . nid ul.iuii s liii iohi.mhi.i.hgij.umihui huhl iH.i bilL u.arn..iJ.i aasUuita iajuulgesjanim. . i.ltalii iii. Il.. sUnmInIsJI.IuI.I.L 3.1, m SMAX, MIN from .HKP file:--

I* I I I ii I I i I iOi i 1 1 1 1 1 1 ii 0 il ii W. I I il I11 ... i I I i 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.10 Event Distribution Plot from JUN8KK.HKP 0 100 200 300 400 500 600 i Ii ||1111111i Ii | 1 111111i i | III I I II II I J8M2 .PHD I J8M3.PHD IJ8M4.PHD

626 ..... 0 i...... : ...... j. bI. ..I..n i...mL...ili.uiJ.I..I.... .L...... I..L...... hK 42 6 .1.. I....i ...... I.. .1...... INIONA 1.... . 111-...... i.1. I1. . .i.l111.

316 -i-LJ...I.iJ...... I.: ...... I N 1.1.1 . II .h...IiI.. JiL.U.11 JI . L IL.h.U. .. .. I . .II.. iiliU.L .iiA. ... n 204 IJ.[....I .. . .L. .I j.. . I...I ..... 11 i.jA L. 1.i .. .1...... I.u . ... I..t ...... I...... I ...... i.Il.L1W t in- 104 . .... i iiil. i I J..Lit hill n. hi..Iiuihl2i . lii. .1....Iu..u.. i Ill.. .. .j. .... f ... 3162 .111.i2J0.. .J...... 1 ...... L...... la ...... I ..th. I.l ltt.11 ~ . ut.. iJ .ILA .di..A~ l.. JL.11 J .. iiJUA.11.1. . . .111.I.. 109.i.. Ji.l.IU ultbk . L.1Il.I. .tk..lbiilt . . 0 22 ...... L. ...li ii. ..I.t J. ..I ii i ... Jrniu . 1. 1 1.1...... i. ... fti Lu . 9- - li.l. ll jii i a) 601.21 I. .I...1 . .. jl..1 i ...... 11....1i. .ml .. ii.L. .... J.I . i.. . .l. 1 i. L. J I.. . Ii. .I .d 4.-) 3.4.ill 1 . 11 1 I I. :' ... . 111.it . 11. 111. At lll 11 ... 9111.1. Ill..1110. s ll i . .1 .. ! .1d .. t li G i t . 0 M. .24...... fl...... L . U.J... L 21.4. .Il ll l ....1 .. ..>.11.1 ...... t Lf .J.. ji ... l. .. - . .111 1. L...J.01 . J .I L I . I it IS . a-4 .20 4 1:'.lii . .. ~lIl~ .. Jii.ll 11 ll.L . .l I ...ill nII ..M ..tt~ Bi~ i 1--

52 .1 ..l.It .101 . l. J. . 11.MI. . ll.1.1..b. 11.1. i t 1-11...l 1...L... .l l . ... u . . 1... .11.1.11 .1. .11l.Ii . J1ti.til - . .1 $2. .. .1 l 811ls.h i ..ta s Jill Li . .. 12. 111u1. i~ 11. . . .1 .

.0. .1.11.1 . 1.1 . .J 1 .. U. . . . I A .tlll.lit..lt . l. til .L..1.1 . 1J AU11.Jit .C AMlL

MAX, MIN from .HKP file:-

A - A I - -- ,a - -A-- - --. A- A -, - -jRAA -& p - -. . -- t, A,,A

I I I 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.11 Event Distribution Plot from JUN8MM.HKP 0 100 200 300 400 500 600 . III ! I I I I I I I I i i i I I I I I I , i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 11 1 I I I J801.PHD I J802.PHD I J803.PHD

736 .ANIMAL. J..lin llE.AL. lild 1&1.1 J... J.im..I .1...1.m lit .1.1. J iJI.. lii ii j.mill.illim.mi siiamtmz 626 JJI.M bi. Jil .... Am1.0.1... I .1...... LLJ. i 1.1. ....L ... JibmI. ... 1 ...... 1i.... 11.I1 i u.1. i .... I.. . J .. .. iILi at R) (U 426 ... L....i...... 1..... JA.... J..iLj .. i.... i i....L.. .. l .1.L iti .ill. JI . J."l.J. J. .. . it ...... I illt. I 416 . 11. i bImIJ .... k.... . li ...... JI.1..L 11. J.ii. 11...Ii .B U. 1.i miitt .. l... Ill .I.. i i. . 11. . I i. l.N.. .uUI.miE.. 316 1111.1, f .. iJ i. i.L.ini.l li itii . lt 1i111li iLi ll. Il IM.uft.L.JU II . I ....L 216 ... . A ...... A .. t..i. I Al. I I illAIN. ii.hi I J. 11. iiiit il. i1.1 i. Ji If) 206 .l. . .Is liii. InIAJ11...1.. Il. i. .uum... J .... |i.b .omiladbuseni I IJ01JR1. M.I..L l. tilla 834 it . . wili 1. it luimitil .Iu a IM ... .. L.J. JR. iNiiJ..t i.i.i lL miii4111A.1 .it . i. .. bI. 1.1. . .. Ii11Id. J. 1 Ill. -In a) 624 .us. Smi I.... I..Jil...... ll...... 1 .. I...... IIA.I ..J.. .. I ..iim..im..mf.m1t... .11. I. .1dt..... ii.t I U 4Jm..L... Iti.thm. Imb l.. .. i i. : 0 424 i X. Jl...... I.....i. 11 . . .J J... 1. . ItJ.t. I. I.... I..... 11.1J. L .I.I. .1 1 . -... 1111M.J. 1l.. i .. m.... 1ll. L . I 314 0 i1mi emA .J.J.l illu JitI d.

-4 214 ..IdJ11.1.ti....1. JUL.. J..d. .ii dI iLi i. It 01.1111u. 1.mlait.if M .l.: 1. 1. J i . 111 1lu11Ima11 1 161im1 IIiri. .ii11 a.I~b ..Wi .it..1-tt .li. iAt: 0J~ I0 0j0 204 J1111 1 .OMI. -... i.. III.It I.. I.I. Ikl.mmil 2Ii . Iit.m.ii.ilt..l.S li.It .IdLI LLI lit imiji.Li. it. 11i .11.. 6 .1t- Go 104 1i. J.i ..L.lii Ji..iii. It~ lt N a i~i .. tI L I Il d. .. 11.11 it 832 .MJt illiL.. . ill 1111...... a; RIIR I. t oo..I. I..JIL. J..ldo J .II .. i1.. 111 .11J.11.. llli i I J...1 I I . 1.. 1 1. 522 11N.Ii.iW...... JtA I. AJ..AI I I. I... I..... J..-J Of11111,1. J.It 1.1.It 312 i f..oim .... f.e...... t. .111I.A.A111.1. J i 1. in1...IlS uI I. lJ .t If) JIU.1. 110.1 t JI..I. 1I. JI Ill ei 1. . it.0.1.: .1j lit I .fo il l "I .1111 1 1.I1 011..I 202 Ji mi IflI Ji. I J. 1 III 1. .1i A -I11 102 it i mill 91. J.i. l~J IIIJ. itIII. 11.JIN i I1MuN HIililE..do.21Kt 601 iL. 111111.... L.1... It.Ill I.. .I. JI..11 ~tJill ~~~~.JA .J.. JINDIJtLALJIL 1.It .AILIIA 1. . 11i.~ .. Ili tolNu".. 11. l t.t 301 11Wlail ...... I...1 .1IL11ni .1111 ItIII .1.s 1itit L...... 1oul ..1.1111.01i .1111.111..01 l .A.JID..1:16 .. 11 ..l 11111. 1 101 it .link IJ.... I...I1J. 1.1. 1|.I t IIL LI 11L11. It~iti a|ll IJ I sIIJI .L.I. 1.111111.1.N.IKIJAI 181. 1 inJim ..I lt .11 MAX, MIN from .HKP file :

.. A -dhb------_- -- - - &- - --_- A A, -- -go A_ - -. a%------

0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.12 Event Distribution Plot from JUN800.HKP 0 100 200 300 400 500 600 LLi hLi I I I I I I I I Ill LI I I I I I i i i i i I I I tt Ii I I II I i i L i i i 1 S 1 _i_I 1 1 I J8T1.PHD I J8T2.PHD I J8T3.PHD I J8T4.PHD

736 dI bin iiImii iustBIL t ...... 1.i IlimbI. M.....LM c ...... L...... :...... l. .. J1. ...Il 0. 626 J.. l. ub~.lhi.. g i. J.. i.I. .... ILL ... I...... J.Li .. J.... Ill - ') 426 ...... L...... L. .111.1 1 11. JA.IL ... i...11 I A.U A..IJ. ... 13. -1.1... 1 1.... .1...... 1 ...... - S 416 .1 ii .u.tmnii I. h.mtiuj. .. 1i ...... 316 l. il.u1uu tminl.liJa eI.u. .111. J.. . . I J. . ... i...... lit.[.:-iJktiiM. 1. .unt ost ...... 0 216 .. . it .it .U ... AmnUl .3. I.li ii. 1.t I . I .. 1 1. 1.1 . J. 206 .JIu i fl t Alli iNIi. .L .U...UIlJ i 3 $..I am..UJ.1 1 1.0211.11: $A1111. Si i.i 11 I isI ...... 1...... A. .. Ia.:-.I ...... 11.. 834 ni .t 11.111.1m. 1.m11116a1 J.Iit u . 1. 1 muakil.lt. J.. ...amI...o l CA i..~lalh.ui.IJ .m ms.iau.a..it 624 .I 110- Uft.ili iii.1.1oli. IIRi -iItU .1AIII. .. .. LL... . 424 .1.1..iIahit.1 .111 ...i IIIIII.huaIihsii . ....iukth.e.Li 0. . im ii. i . 11....1W.It . 1 Li . ii .I UU..11.. 111.1 .. . .I A . i . li 3. .. I I I ...).ll....|.( .11.. .~ . U 314 ...... 214 .01001. It .IMNJu.l.U.ilim A .. h. 1j.. 1111i.. [* I l. il.LJ.ia ll: JmII31i. IaIAll *1I ...... inWohi lhaawat 204 mtiliiibi. iiktmiiiOtl.L I f ilel.il l.. .I.J.3...... I....ill. U' ltsto simi~at 00 - 104 sdiINiNMi*lIIiii.II...JILUI1II Il Sit l.MidlLI I 832 .Ind.. 11111 i. idifu. aw I. it .. a.. las J. 3.I ifti .llal lut ..11 ...... I L.. a) ddH. n a. ...1..S. J.6111 I I .. L It. 11 522 ...1.... I.L in.a j.. voilk ...I ... I- 312 .SAJ. li .J..Mlllil.giM litt. .91-111 s Illi.Kt l .. 118 illlsIt I JLUL Jlh#.MiL N31Jlit ..1...... :...... L. 202 liawi~l.I. NJI (LIOahit to I .oil. . .. M ii j~ nLml l: MAJOIthibmeiittU.0. lib.111li1 111 MILMUJ In ...... :...... 102 MiliMi tliWNItLijditcht ..LII...I..:zI.J..... 601 .1.1.... I.Aul .I Jill a Jll l l. I Iu1N I 1ima a awlu ...Lu~...i.~.i..l. 301 .. IllmlgI ilk Ism I 210al.NLMI.Ml W111i...k .laim.11-4. 0.9il = 11t 1a11 $ in ...... L . 101 NJL111911.1tLMUIlIj. J. .11 It 2111.12. at I witilul ai .1...... ::. ...,...... L... MAX, MIN from .HKP f Ile:

I S- .LL~&A. iL

I V I V -W 'o ...... ' I r 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.13 Event Distribution Plot from JUN8TT.HKP 0 100 200 300 500 600 I i i i ii i II I I I tI i i i I II I I I I I I I I 1 1 1 1 1 1 1 1 1 1 1 1i 11ii i .I I J8V1.PHD 736 626 J.. All.. I.A.111. . Ill IL J.a 1.11. 426 -JJR .1 . J.D I.. ...U.. ..l AM. JI1VJML . 416 ..1 . .1..11..iI.ti . tg.lh m.. 316 11JJ11111. I .llbi.iImim.dh. Atil 0~ 216 Ii~i B tit .81 l.IJJ.l.s. 206 hIMIltibiam mik.al M1 N I .. i1iti. a00 834 t ioo 1111111E1 an1..L. .*. ll I.. &.LR 11 624 424 .t~i.J.l. . 1.1 .11. .18 ) Mk. Go- 314 JI". I liii .iltil.iARMui 11011. 214 1NILL .1 5 ilkId.l. 1 L6.1mhii.i 204 L V) - 104 .IN .111m11.101btii..IItWA)A.

832 it.lld11.0.1. LIM.Lin.i t. U. 522 .AIL I. J..wJ n ]IllkU. dillJl 1101 0 - 312 ht I.t 9 Jill NOLUN a t J~A iftb. 202 .M Jil.a Ii JUlhIMl . .Umlijah it 102 mblilhl f.1i Ji l. l10.hIl1111 ti.U 601 ua iM 1.u.tu m i uk.u. Mh~uuif luibi 301 L.bitnbu.uijil lillul. EN.illS lii. 101 ielL~inl.lI J.dlidati~h@UuJ i l MAX, M N from .HKP file:

~.ifttL2~ A iz-7rI7777z=n-mfi AMr~ - S III I I I II IIIII I1 I III I I I I I I III I I I I II 0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.14 Event Distribution Plot from JUN8VV.HKP 0 100 200 300 400 500 600

I SHOT1.PHD I SHOT2.PHD

736 111.1 11 J. hlu IUI.ll..1131 u..I.L .U.. Ji*.1.. JlI3Il.L .1. I b II J.1.. .L 626 .1.91 L .. 1.miJi. .. . l . W.i. I.L 1.1.iiu L Ait . m .. 1. 11 J .1IJU

2 14 6 .a. slum. a. Imw *3. a... l. uiih~ l. .n J~iuiiii. 1 .u.i. ll .1..*11 ..i 206 s. i .11. lt liu . lmaii.ulii.WiI.IL giiI litu I.liulmlahi.Ulnt -o-a.' 3 64 i .L.lau s..li1.IIU.auts t .jii.11.11tu.itIIali.iu.lunii I...lJ u.I

o 424 .Ita..mw1a1 illnuW..I au1i. J.I .1. . i uJ iu 3u*.i uI. t si .. l 31 4.1111..*111 imm I. Aiut Ia h.ui. ia i i 11.111 .1.lia I I.. .m1au u .

162j1i4 . 21nhuaii.t .11.111L11J.Jiil bn 11it lul.N Itud i h i tm hi 1 u1at A.lu. -.

00 104 .111UU1Ihl II N I .i. 1.6 INX il il .iINi 1ii1d.Mil 0.hill 1ilui i 832 I AiNU IE*1 1 .. .d.i s...... t .E *l15* 111li. llI IMa. I. J il 522 Ji . IllI SUirnl i uM s l . 1.1h IL Ii. 0 11 M.. .0.11 -11lii.lf 111. tl W.i I. U 3~~1 4 .i i iuim mI 1.11 N hinaA pi t t.I I i. N it .III o.11t3 i a I hiItu . Sli ms I a) CD 321014 1 2 kiunssistm.Iis. li sii . iJN Ai. . n sma liniin u win. E.U. i t. II.. itm I hhisaiNl.mea alef lii s.ajiissliatsicma. i. .iiuii Itl I u. 20 2 *hiimhtu 31uh~lui hu. ImilkI~te f t .a i u.uktiiuWs.u ludaum l aiu lii NA 102 ~lilustiLal ammuI.m ui.lbuskliimtmu.IlNfluJli.mIUI i.3.1. i.liblt 60 1 .It l I flhUiftI. i t ii Iliial.i J nI . itg. liti. IL. I U1131 j 1.l .. 1J 30 1 .1. aIsmnm t I . 33 11 111111I JiMJ.uJ.I .. 1 limitit*iil at . liii t I. flu l LI 8 MAX MIN from .HKP f11e : -

0 100 200 300 400 500 600 Time (chunks, 1.63 sec/chunk)

Figure 7.15 Event Distribution Plot from SHOT.HKP 7.2.3 Data Synthesis The next step in the data analysis is combining the results of the many PHD files. A list of the HKP and PHD files used in the data analysis is given in Table 7.1. This table also gives the values of the power (Po) and volts-per-radian (VPR) derived from the housekeeping data in each section of data. The contrast, K, of the interferometer remained constant at ~ 73%. The relative constancy of these values demonstrates the stability of the instrument (i.e., laser power, fiber throughput, interferometer alignment) over the course of the night. An important question is the stationarity of the instrument's noise performance in time. A very coarse measure of the average noise properties of the instrument is given by the rms values of the pulse height distributions for the templates. This rms noise represents an average over the time of one PHD file (~ 3.5 minutes); it is insensitive to occasional bursts but will reflect a change in the fundamental noise levels. The average rms value out of each template is given in Table 7.2, and, with the exception of the JUN8RR data tape, these values remained constant during the night. (In fact, JUN8RR was taken without digital phase modulation and has a different spectrum and higher noise levels, see Figure 5.6.) The rms value of each template PHD, computed in A/D units, has been converted to the equivalent amplitude, ho, of a pulse having the same shape as the particular template. Appendix L describes the conversion process in detail and, in Figure L.2, compares the template ho,rms values with the measured interferometer noise spectrum. Because the template event threshold is chosen to keep 128 events above it, the event threshold reflects the number of "tail" events in the distribution. A comparison of these template event threshold levels is more sensitive to the tail of the distributions and thus the occasional large events. Typically the event threshold is at SNR ~~4 - 5 (Again JUN8RR is abnormal, with thresholds in the SNR ~ 5 - 7 range.) There is far more variation from PHD file to PHD file in the threshold levels than in the rms values. This suggests that the noise terms responsible for the tail events are not stationary on 3.5 minute time scales; this can be seen by the structure shown in the event distribution plots of Figures 7.5 to 7.14. Note the uniformity of the shot noise data, Figure 7.15.

7.2.4 Windows and the Final Data Set The goals of the data analysis are understanding the noise properties of the instrument and identifying candidate gravity wave events. The non-stationarity of the tail events, discussed above, combined with the known presence of resonances and acoustic and seismic disturbances leads to a noise picture of an essentially stationary character with infrequent large disturbances. These can be isolated (though not forgotten) from the analysis to reveal more accurately the statistics of the nearly stationary noise terms. With this philosophy in mind, in combining the data from the files some data have been left out of further analysis. The complete tape JUN8RR, because of its - 84 - Table 7.2 RMS Noise Levels of the Templates

Template fkHz ho,rms Template fkHz ho,rms Template fkHz ho,rms x10- 15 x10- 1 5 x10- 1 5

101 - 46.5 104 10 11.3 206 5 4.3 301 - 28.2 204 5 7.0 216 3.3 7.7 601 - 35.4 214 3.3 8.8 316 2.5 11.3 102 10 16.5 314 2.5 15.5 416 2.0 19.2 202 5 13.8 424 1.7 29.2 426 1.7 27.3 312 2.5 24.3 624 1.25 20.5 626 1.25 14.6 522 1.4 32.8 834 0.9 11.8 736 1.0 8.3 832 0.9 17.5 completely different noise level and character is left out. The PHD file J8M1 is likewise excluded because its event threshold in many templates has been pushed into the SNR=5-7 range by the single large event near chunk 110. Finally nine other regions in the data files have been excluded as given in Table 7.3 and shown dotted in Figures 7.5 to 7.15. The resulting list of data to be synthesized is given in Table 7.1 along with the associated times of the windows. The coverage in time of the data is given graphically in Figure 7.4.

- 85 - Table 7.3 List of Regions Windowed from the Analysis

File Removed region Explanation (Number of chunks)

J8C1.PHD 126-139 Extended Period of Excitation (14) J8C2.PHD 188-199 Extended Period of Excitation (12) 301-309 A Giant Glitch at ~ 307 (9) J8E3.PHD 320-322 A Giant Glitch at ~ 321 (3) 396-405 Extended Period of Excitation (10) J8G1.PHD 30-39 A Few Giant Glitches at ~ 38 (10) J811.PHD 85-89 A Giant Glitch at ~ 88 (5) J8M1.PHD The whole file A Giant Glitch at ~ 110 raised (128) the event threshold J8M2.PHD 224-229 A Giant Glitch at ~ 228 (6) J8T4.PHD 473-480 Several Giant Glitches (8)

Total Number of Excluded Chunks : 77 + 128

Fraction of Total Un-windowed Data : 0.020, 0.033

- 86 - X Template SNR JUN8M. HKP J8M2.PHD 158-> 223 230-> 285 201.063 102 15.8 + 167.529 102 11.7 195.023 102 11.7 401.063 102 1 3.3 195.023 102 15.1 & 167.529 102 13.4 201.ea- 414 - e.e 201063-404- 11. 167.529 104 10.8 195.023 104 11.8 --1W.03 -104 12. 195.023 104 11.0 167.529 104 10.1 Figure 7.16 Multiple Detection of the Event at X=201.063

7.2.5 Final PHDs and a List of Events The pulse height distribution for all events from all of the windowed data is given in Figure 7.17. The dotted line gives the expected Gaussian distribution. The event threshold for most templates is between 4 and 5 and the event counts above SNR2 = 20 (SNR=4.5) agree well with theory. The existence of tail events is clear. Examination of the events that make up this PHD leads to the conclusion that multiple detections (Section 6.3.3) are leading to a density of tail events that does not reflect the actual event rate. For example, Figure 7.16 lists the events in J8M2.PHD with SNR>10. The single "event" at chunk location X = 201.063 (because a chunk has 32,698 points, specifying the X value to 0.001 represents a 30 point, 1.5 msec, range) has been detected twice by template 102 and three times by template 104. The other templates will see the event at smaller SNR values but at more locations due to their increased length. Removal of multiple detection proceeds in a straight forward way as suggested in Figure 7.16. The event, among all templates, with the largest SNR in the PHD file is selected as a "special event". All events within 60 points are then eliminated from the event list as duplicates. Then, among the remaining events the one with the largest SNR is selected as special and the process is repeated. The result of applying this removal of multiple detections is shown in the PHD of Figure 7.18. In spite of the reduction of multiple detections, there are still a large number of tail events. By examining a list of events, most were found to have been detected by template 102; the PHD of only the template 102 events is shown in Figure 7.19. These events are examined in detail in Section 7.3.1 below. The PHD of the remaining templates (with the exclusion of templates 101, 301 and 601 which, in any case, saw no events above SNR 2 = 32) is shown in Figure 7.20; a list of the events from these templates, with SNR>5.9, is given in Table 7.4. This table and those regions windowed from the analysis, Table 7.3, are the end result of the analysis-they contain all the statistically interesting events that occured during the data taking.

- 87 - Pulse Height Distribution 0 50 100 150 200

I I I I I I I I I I i t i i i i i I I i i C- f-u' I-

0- - O -0'h - .61

.4 - -wu C Cu - -r-4 .. ; ; m -4 L - -mU

0D- L *r 4~ .. -...... - . .... CS

- OD

5...... *. . Cu-

0D- - i'

- CID

Cu-

JTWrTTrrFV pr-r-FUT- T-(-T--f-T-1-fT1-irr-r[T-1-rrrr--r-r-J-t-i-t-t-t- 0 50 100 150 200. SNR**2

Figure 7.17 PHD from All Templates-No Multiple Detection Removal

- 88 - Pulse Height Distribution

0 50 100 150 200 I-I -4-i-I-. CD- -0O

-fu cu- 4 4 3 I 4 I I*I -0

C- -06 1) "- 0'u CD- C =OD (&3 ...... L A0 0- L C-(u- .A. .~~~ t t z Ca-C- 0 --

cu- -- b -4 - : : : 0D-

40 - -m

- - r , i 01-1 I I I 11111 150I'I i A1 I I I I I I I I I 1 I I 0 50 100 150 200 SNR**2

Figure 7.18 PHD from All Templates (Multiple Detection of Events Removed)

- 89 - Pulse Height Distribution

0 50 100 150 200 l i i i 111111 I I I I1I11I1 iI -CD

CD c- 4------+ - .------'- '- .- '- - d-b- -...... ;-L-4-J...... 4C

c CD -. 4 -S m L cu-

Cu CuC

z- -4 ------.. -...... CD-- -CD

4A.

'-4

- --,,-. C- r . ru

0 50 100 150 200 SNR**2

Figure 7.19 PHD of the Template "102" Events Only

- 90 - Pulse Height Distribution

0 50 100 150 200 cui LiI IIi.1 J 11 J I I I i i I- -TI

r4.

-. . . .o .. .. .4 ...... -- 4

C -- 4 .. -i .. +.. . .. cn"-4 L (U- cu t Cu - I.ttv ...... -... V-4 L 4D- ru

rU- --4 - TI) .------r- . --. - r -w -0~ 4q.-

.ii.... CU- ...... -- 4- -w

CU-

ff -r i I I I I I I II1I11I1II1I1 F1IV I 2II''' 0 50 100 150 200 SNR**2

Figure 7.20 PHD from All Templates Excluding the "102" Events

- 91 - Table 7.4 List of Events with SNR > 5.9 (excluding template 102 events) <- indicates isolated events

-15 PHD File: Location Time Template SNR h x10 0 X Chunk Word E.D.T.

J8A1.PHD 94.888 94 28969 22:38:21.38 736 6.7 52.71 <- J8A3.PHD 540.967 540 31535 22:50:28.66 216 6.0 44.13 540.964 540 31440 22:50:28.66 216 5.9 43.43 J8C1.PHD 109.123 109 4014 23:25:24.57 214 7.1 62.98 109.121 109 3943 23:25:24.57 214 6.9 60.72 J8C2.PHD 331.144 331 4701 23:31:26.55 736 15.2 121.85 <- J8E1 .PHD 38.358 38 11689 0:11:13.28 736 23.5 189.98 <- 115.013 115 421 0:13:18.26 316 7.0 76.54 115.008 114 32864 0:13:18.25 104 6.0 66.02 48.015 47 33082 0:11:29.02 104 5.9 65.37 <- J8E4.PHD 523.414 523 13492 0:24:24.11 314 6.2 92.30 523.412 523 13425 0:24:24.11 206 6.1 26.47 523.407 523 13275 0:24:24.10 316 6.1 66.78 J8G1.PHD 48.925 48 30165 1:50:05.43 204 6.3 45.91 <- J812.PHD 247.381 247 12431 2:52:27.54 316 6.6 74.04 247.384 247 12519 2:52:27.54 214 6.3 54.70 247.402 247 13124 2:52:27.57 216 6.2 47.71 247.390 247 12712 2:52:27.55 316 6.0 67.38 247.386 247 12590 2:52:27.55 416 5.9 113.37 J814. PHD 455.814 455 26540 2:58:07.37 204 6.3 44.22 455.817 455 26631 2:58:07.37 314 6.0 93.09 J8K1 .PHD 44.566 44 18451 3:37:14.26 316 6.7 75.68 44.587 44 19150 3:37:14.29 214 6.4 56.34 44.573 44 18689 3:37:14.27 216 6.3 49.64 44.557 44 18152 3:37:14.25 214 5.9 52.00 J8K3.PHD 216.621 216 20255 3:41:54.78 206 6.7 29.35 216.632 216 20621 3:41:54.80 316 6.6 73.75 216.636 216 20751 3:41:54.80 316 6.3 70.96 216.640 216 20884 3:41:54.81 214 6.1 53.52 174.031 174 995 3:40:45.34 624 6.0 122.85 <- 216.626 216 20424 3:41:54.79 214 6.0 52.54 216.645 216 21033 3:41:54.82 202 6.0 81.88 216.629 216 20522 3:41:54.79 206 6.0 26.20 216.624 216 20350 3:41:54.78 316 5.9 66.48 J8K5.PHD 436.974 436 31775 3:47:54.04 206 8.5 36.93 436.969 436 31610 3:47:54.04 206 6.6 28.62 J8K6.PHD 483.636 483 20742 3:49:10.12 736 12.2 102.82 <- 509.000 508 32612 3:49:51.47 206 9.1 39.29 497.429 497 13985 3:49:32.61 736 8.0 68.00 <- 509.014 509 447 3:49:51.50 214 6.9 60.79

- 92 - Table 7.4 (continued)

-15 PHD File: Location Time Template SNR h x1O 0 X Chunk Word E.D.T. J8M2.PHD 185.517 185 16872 4:32:56.58 104 7.6 87.69 <- 182.491 182 16006 4:32:51.64 316 7.3 82.71 182.505 182 16466 4:32:51.67 316 6.1 69.10 182.495 182 16130 4:32:51.65 316 6.1 68.56 J8M3.PHD 372.431 372 14054 4:38:01.32 216 6.9 52.15 <- J8M4.PHD 434.369 434 12028 4:39:42.31 316 6.8 77.51 434.376 434 12270 4:39:42.32 206 5.9 27.44 J801 .PHD 57.586 .57 19111 5:14:49.00 216 7.8 60.82 57.592 57 19300 5:14:49.01 216 6.9 54.26 121.276 121 8995 5:16:32.84 206 6.9 31.43 121.281 121 9147 5:16:32.85 204 6.6 47.39 121.284 121 9256 5:16:32.85 206 6.6 30.08 55.402 55 13119 5:14:45.44 426 6.5 169.27 <- 121.286 121 9334 5:16:32.86 204 6.5 46.19 57.584 57 19040 5:14:49.00 216 6.5 50.59 121.299 121 9749 5:16:32.88 214 6.3 56.68 121.292 121 9525 5:16:32.87 206 6.3 28.71 121.273 121 8891 5:16:32.83 206 6.0 27.59 121.273 121 8889 5:16:32.83 206 6.2 28.27 121.289 121 9417 5:16:32.86 206 6.1 28.08 57.589 57 19209 5:14:49.00 216 6.0 47.38 J8T1.PHD 80.462 80 15081 7:18:08.80 832 6.2 106.79 <- J8T2.PHD 197.860 197 28051 7:21:20.20 214 10.7 91.63 197.863 197 28144 7:21:20.21 206 7.5 31.39 197.874 197 28488 7:21:20.22 214 7.3 62.40 197.858 197 27982 7:21:20.20 214' 6.6 56.26 197.867 197 28270 7:21:20.21 316 6.2 68.50

- 93 - 7.3 Examination of the Events

7.3.1 Template "102" Events Having identified the non-statistical events in the data it is useful to examine their time series. The uniqueness of the template 102 events is made clear from their time series. Figure 7.20a shows an SNR=11.1 event from file J8E1, plotted at the output of the digital high-pass filter, i.e., at the input to the template filters. The spike-like nature of the event explains why the 102 template was most sensitive to it. Figure 7.20b shows, for the same event, the unfiltered data as recorded on tape. The single time sample (50 As) character of the event is very suspicious in view of the 6 kHz roll-off frequency of the anti-aliasing filter at the A/D input. The amplitudes of the 102 pulses do not appear to cluster around particular values suggesting single bit errors in the recorded data are not the problem. Likewise, the values before high-pass filtering, i.e., out of the A/D, do not show "quantization" to significant-bit boundaries, ruling out successive approximation errors in the A/D. The cause of these spikes has been determined by plotting the distribution in time of the template 102 events modulo one chunk of data, Figure 21. The events cluster near the beginnig and middle of one data taking cycle-this is when the buffers are written to tape! The tape drive and A/D are mounted in adjacent cabinets. Presumably, the tape drive motors, etc., produce pulses that are picked up by the A/D. The magnitude of the pulses are ~ 100 A/D units, or 0.5 V equivalent at the actual A/D input.

- 94 - JUN8EE 1.675 1 680 1 685 1 690 mmmli mlimmijimi ml...iii am... mlaa I , m ml M CU 0 L

D1 . . 0 0 1 1. f (U I a) L

0 'a

E a:_ %D -I I CS)t.

1.675 1.680 1 685 1,690 Time (104 samples)

JUNBEE 1 675 1.680 1,685 1.690

Dn In

cr- C

Cr ru (U CD

.... II' ' ' 'I ' ''t ' 'I 'If 'I 1.6750 1 6800 1.6850 1.690 Time (104 samples)

Figure 7.20 Time Series of a Template 102 Event a) The pulse is seen after digital high-pass filtering. b) The pulse is seen before filtering, as recorded on tape. (An offset of 30 points is due to the length of the filter.)

- 95 - 0.0 0.2 0.4 0.6 0.8 1.0 I I I I I I I I KU-

CO 0 0 C FLL (au 0 L

0 0 Ca) E 4- o

-Q

-RD UU UHL-] I I L I I I I I I I I 0.0 0.2 0.4 0.6 0.8 1. 0 Time (Fraction of a chunk)

Figure 7.21 Distribution of "102" events modulo one chunk 7.3.2 Other Events Many of the events are associated with a local excitation of the instrument; as the list of events, Table 7.4, makes clear by the often close temporal association of events. Particularly blatant examples are given in files J812, J8K3, and J801. Events that are isolated (12 of them) are indicated in Table 7.4 by an arrow. The rest of the events can be assigned to 15 regions during which the interferometer is assummed to be excited. That this division into isolated/non-isolated events may be meaningful is supported by the comparison in Table 7.5 of the distributions of event templates for the two classes of events. The predominance of the 204/6, 214/6, and 316 templates for the non-isolated events may reflect the presence of excitable resonances. Figure 7.22 shows the spectrum of the data around chunk 401 of JUN8EE, a region windowed from the analysis, compared with the standard spectrum (from JUN8KK, chunk 60). Figures 7.23 to 7.29 show some of the detected pulses and the PHDs of their corresponding templates. The high SNR template 736 pulses are very exciting. The results of the analysis are summarized in Table 7.6.

Table 7.5 Distribution of the SNR>5.9 Events among the Templates

Tmpl Isolated Non-isolated 101 301 601 102 202 312 522 832 / 104 // 204 / 214 314/ 424 624 / 834 206 216 / 316 416 426 / 626 736 /////

-97- 3 4 5 6 7 8 9,1 2 3 4 5 6 7 8 9

ICU- ......

...... 001- ...... * ...... *...... i4.4.i ...... i ...... i ...... i ...... i .....I ...

......

...... N (U_ i ...... ro

...... :-!." -f ...... OD - ...... CD ...... L ...... -01 ...... _46

CU_ ...... 0 In _rU

...... (S) CM7 ...... 0 -4 CD %D ...... i ...... i ...... i ...... E i i i i ;.4.4.444 ...... e ...... Ln ...... 0) ...... ICU- ...... _rU

...... i...... i ...... i ...... i ..... i .... i ...i...i ... .4.4.444444 ...... i ...... i ...... i ...... i CD - ...... -4 ...... L -CD ......

ICU- ......

2 I I 1 1 6T-7 7rTr"r 8 97 2 3 4 5 6 7 8 9

100 1000 10000 Frequency (Hz)

Figure 7.22 Spectrum from a Region of Excitation

Table 7.6 Summary of Non-statistical Occurences

Excluded from analysis: 3 Regions of Excitation 5 Giant Glitches 2 Several Giant Glitches

Template 102 events: ;zz 120 Single Spikes Synched with Tape Drive ;z: 1 per minute

From the event list: 12 Isolated Eventsi ho > 5 x 10-14 15 Regions of Excitement

37 Total Number of Non-statistical, Non-102 Events in 1.7 hours, ;z 1 every 2.7 minutes

-98- Pulse Height Distribution 0 50 100 150 200

L tr

L 0.. .50.100150 .00 -o

Ow-** . JUN8M...M ...... 167 1 68 1.68 1 69 1.9 0

.. ... W

CS)

a CD

E (S) ICu aCUD

)

1.675 1.680 1.685 1.690 1.695 1. 700 Time (104 samples) Figure 7.23 PHD and a Waveform from Template 104

The time series of the 104 event with the highest SNR, 7.6, is shown. The high frequency nature of the template is seen in the alternation in sign from one sample to the next. The open circles are the digital samples. This is one of the "isolated" events.

- 99 - Pulse Height Distribution 0 50 100 15 0 200

C CD

I I I I I I I I I I I I I j i I II I I I I L z

-1

0 50 100 150 200 SNR**2

JUN800 I 895 1 966 1 965 1.916 1 915 1 926 1.925 i...... ,.,,i, ,, , , , , , , , , , ,. .,, .. .. r\,)-\ --44

D CD

- I Ea- CU- 4J CS_

E -ru '4- U _S I. U

II I IIIIIII IIIII I I I 1 TI TI II T 1 895 1 900 1 905 1.910 1 91 5 1.920 1- 925 Time (104 samples) Figure 7.24 PHD and a Waveform from Template 216

The time series of the 216 event with the highest SNR, 7.8, is shown. The event was strongly detected by the 216 template due to its maintenance of the correct phase over the six half-cycles. This event was labled "non-isolated" and the time series shows lots of activity at 216-like frequencies.

- 100 - Pulse Height Distribution 0 50 100 150 200

......

CU I.-

CS) L C- -,U -

e-u

1 0 50 100 150 200 0 SNR**2

JUN800 .295 1 300 1 305 1.310 1.315 1 320 1.325

,I sI I ,I I I I 1 ,I, , , , , , , , , , , , I ,,. I, , 1, ,, , , ,,,I

CU C

a) - o

CU -0 -j -0 E 01:1- CE Ia 'fl--1

1 295 1 300 1 305 1 310 1.315 1 320 1 325 Time (104 samples) Figure 7.25 PHD and a Waveform from Template 426

The time series of the 426 event with the highest SNR, 6.5, is shown. The burst clearly stands out from the background colored noise of the instrument. This is one of the "isolated" events.

- 101 - Pulse Height Distribution 0 50 100 150 200

-ofu

. . -D :

0 50 100 150 200 SNR**2

JUN8KK< 1 385 1 390 1 395 1.48e 1.405 1 410

I D I

-- T -n)

0 Cu

-) - I CUIS) -P)

E T

1) - w -wI

1.385 1.390 1 395 1 400 1 405 1 410 Time 104 samples) Figure 7.26 PHD and a Waveform from Template 736

Template 736 detected several "isolated" and high SNR bursts. The burst shown here has an SNR of 8.0. Because the 736 template is so long, higher frequencies mask the 736 component- that there is something present is clear. This is one of the "isolated" events.

- 102 - JUN8KK 2 060 2.065 2 070 2.075 2.080 2 085

-U

i~~~~ I IIit 1 1 1 1 1 1 11 Iti e III I I II II I II Lii

FTier Wavefsames)

-D V0- T0m AVomls

C~CD

2 060 2 065 2 070 2 075 2 080 2,085 29 Time (104 samples)

This burst, detected by the 736 template, has an SNR of 12.2. The top waveform is after digital high-pass filtering, as seen by the template filters. The lower waveform shows, with a 30 point offset, the same data as recorded on tape after the analog 200 Hz high-pass filter. This is one of the "isolated" events.

- 103 - JUNBEE 1 155 1 160 1 165 1 170 1.175 1 180 1 185 ,l, , ,IIJ iI IIa I ,t ,, ,I,, ,, , , , I,, , , , ,i , , , i , 1

7-(S CS)

L11I I(S) (-

'a I31 0:a:

La:

1 155 1 160 1 165) 1 170e) 1 175 1 180 1 185 Time (104 samples)

JUNBEE 1 155 1 160 1 165 1 170 1 175 1 180 1 185

(S)

('J) Ca: Ea F r .nr Tim(1S)a Ths bbm. pl s a: CD CD

1 155 1 1600 1.1650 1 170 1 175 1.180 1 185

Time (104 samplIes)

Figure 7.28 Yet Another Waveform from Template 736

This burst, detected by the 736 template, has an SNR of 23.5! The waveforms above are as described in Figure 7.27. The cause of these bursts is unknown. This is the most exciting of the "isolated" events.

- 104 - JUN81I1 2 170 2 175 2 180 2 185 2 190 2195 11 1 , 1 ,,1 , I,,,i tl Illl~l li111 l~iii i

CD - ...... *0 ......

.4-)

a) _CD

I 2III I II I I I I I 9 I I 2 170 2 175 2 180 2 185 2.190 2,195 Time (10' samples)

JUN8MM 2.375 2 380 2.385 2.390 2.395 2.400 2.405 , J,,, ,L4JL J,

CD -

C ......

Mu -D Cu 7 - I

(S) C:

2.375 2 380 2.385 2.390 2.395 2 400 2 405 Time (10' samples)

Figure 7.29 Two Glitches Windowed from the Analysis

The upper glitch is from JUN8II, chunk 87. The lower is from JUN8MM, chunk 226. The origin of such huge and localized disturbances is unknown. Both waveforms are without high-pass filtering-directly from the tape.

- 105 - 8. Discussion of the Results

8.1 An Instrument Performance Perspective If the noise properties of the instrument are viewed in their own right, independent of its operation as a gravity wave detector, the conclusions are optimistic. In particular, the constancy of the rms levels over the course of a night and the adherence to Gaussian statistics out to an SNR of 5.5 are very encouraging. Critical evaluation of the instrument burst noise properties is contained in the tail distribution. As listed in Table 7.6, the number of non-statistical occurences needed to produce the observed tail events (ignoring the template 102 events) is only 37 for the 1.7 hours of observation, ~ 500 per day. It is difficult to decide if this number is high or low, but that it is finite and the events are well defined leads to a clear and complete set of candidate events. Some comparison can be made: Boughn et al. 1982, in reporting results from the Stanford bar, have tail event rates on the order of 10 per day. Amaldi 1980 presents data indicating a 400 tail events/day. The usefulness of this data analysis scheme is demonstrated by the detection of the template "102" events, presumably caused by interference from the tape drive. The existence of these spikes, at a rate of about one per minute, would not have been detected through the conventional spectral analysis.

8.2 An Astrophysical Perspective The 12 isolated events could be the result of gravity waves; in particular the five template 736 (~ 1 kHz) events are interesting. In order to set the results of the data analysis in an astrophysical context, it is informative to estimate the distance at which the source would have to be located to produce events of the recorded strength. Using equation 3.4b of Section 3.3 and the approximate integral of h for the canonical events,

h2 dt 1 h2 NC (8.2) 2 2f the expected ho amplitude of a burst can be calculated as:

ho = 4.5 x 10- 17 (8.2) R/RGC Msoiar NRHC fkHz The ho values of the detected events are > 5 x 10-14. Other values needed in equation 8.2 are f ~ 1 kHz, NHC=6 and EM might be 0.01 solar masses in optimistic cases. Solving for R yields:

- 106 - R < 3.7 x 10-5 x RGC = 0.37 pc

Proxima Centauri, the second closest star to the earth, is located at a distance of 1.3pc. With the observed event rate of several per hour, solar masses per day must be consumed in a volume of space containing a few stars! These pulses are unlikely to have astrophysical origins. For comparison with the Stanford bar, events the size of those detected here would deposit temperature equivalent energies of 40 million degrees into the bar. The bar has an rms noise of order 20 m*K and their tail events are those above 250 m0 K. Very large events are those above 10 'K and occur at a rate on the order of once per week. Thus, a coincidence experiment with the MIT prototype would appear fruitless. However, the temperature equivalent of a pulse scales with h 2 and thus the MPQ Garching antenna's factor of 2000 lower strain noise at 1 kHz reduces the 40 million degree equivalent event to a ten degree equivalent event, and coincidence experiments appear feasible.

- 107 - 8.3 Conclusions The matched filter technique for the detection of bursts in the interferometer data looks promising; it is certainly a useful tool for single-antenna noise diagnosis. In the case of multiple antenna systems, it could serve as a method of data compression before applying the final coincidence criteria. More importantly, because of the inevitable differences in the waveforms received by separated antennas, this analysis scheme may succeed where direct cross-correlation schemes do not. The implementation of the analysis scheme described here represents a first design. Many possible improvements in the processing algorithm are suggested. The need to perform digital high-pass filtering, and the required CPU time, can be eliminated by using analog filters at an appropriate high-pass frequency. The template filtering can be done in real time through improved coding, custom hardware or even analog filters. (The similarity of the analysis scheme to the operation of the human ear is clear; Forward 1978 used a very efficient processor.) More efficient and clever ways of handling unlocked times, glitches, regions of excitation, and multiple events could be developed; much of this could be done in real time, eliminating either an increase in the SNR threshold or the recording of unnecessary events. Due to the lack of sensitivity of the MIT prototype, little astrophysics can be learned from the data. However, a sensitivity of 10-19 strain/Hz has been achieved in the Garching prototype antenna (Shoemaker 1985) in the 1-2 kHz range-this allows events as described above to be detected to 1.5 kpc- 1/6 of the distance to the Galactic center. The important goal of searching for gravitational radiation from the Galaxy, especially its center, is less than a factor of ten away from interferometer and bar technologies. Coincidence searches at these sensitivities will inaugurate Galactic gravitational wave astronomy. The expected event rate is low-once per decade is optimistic. Thus, it is sensible to consider looking farther in order to include a larger region of potential sources. The graph of Figure 8.1 seeks to suggest the next step. Here the total mass within a given radius of the earth is plotted as a function of that radius. The data is from Bachall and Piran 1983 (for the Galactic distribution of mass) and Allen 1976. The next step then, is an increase in the observable stellar mass, and presumably the event rate as well, by a factor of one thousand by extending observations to include the Virgo cluster. This will require a further factor of two thousand in antenna amplitude sensitivity or a factor of 4 million in the effective bar noise temperature. The ultimate goal of the large antenna projects is to acheive this sensitivity. There are a variety of pressures and concerns directing future research efforts. From the above, the construction of large systems is clearly scientifically of the highest priority. There are, however, issues of technology and engineering that must be explored in scaling an instrument by a factor of one hundred in size and several

- 108 - 3 4 1 10' 102 10 10 1 106 2 4 I 2 4 66 P- 4 1 1 1 1 1 1 2 4 68 2 4 68

11 Aeage -- 4 :density ...... C...... in ...... v-4 (S)

......

V r g:0 ......

0 ...... :...... *......

CS) i ...i-4-i ...... i ...... M ...... E -4 ...... ; ...i.4 ...... ; ...... ;..i.; ...... % ...... -P 0 cu L.: ...... z ...... -- I

......

...... :..:.i ...... r

U ...... : ......

-4

...... :--

684 2 4 681 2 4 68 1 ' 4 6 S'1 2 4 68 2 4 68 1 1 10 10 3 104 10 5 106 Distance (kpc) Figure 8.1 Cumulative Mass Distribution

The total mass within a given radius of the earth is plotted as a function of that radius. The dashed line, for distances beyond the Virgo cluster, is derived from the average density of galactic material in the universe, 2 x 1022 kg/M 3 . Closure density is a factor of 50 times greater.

- 109 - decades in sensitivity. Often these argue for step-wise progress and specific research on side issues. Finally, questions of finances and politics influence the choice of designs and time tables. Though the route to the construction and operation of large gravity wave observatories is uncertain, it is clear that they will eventually contribute to our understanding of many interesting astronomical phenomena.

- 110 -- References

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- 112 - Linsay P, Saulson P and Weiss R 1983 "A Study of a Long Baseline Gravitational Wave Antenna System", Prepared for the National Science Foundation under NSF Grant PHY-8109581 to the Mass. Inst. of Tech. Linsay P S and Shoemaker D H 1982 "Low-noise RF Capacitance Bridge Trans- ducer", Rev. Sci. Instrum. 53, 1014. Lipa J A 1980 "Gravitation Experiments at Stanford" in V124 Lecture Notes in Physics: GravitationalRadiation, Collapsed Objects and Exact Solutions, Ed. C Edwards, Perth Jan. 1979 Springer-Verlag. Livas J, Benford R, Dewey D, Jeffries A, Linsay P, Saulson P, Shoemaker D, and Weiss R 1985 "The MIT Prototype Gravitational Wave Detector", in Proceedings of the Fourth Marcel Grossman Meeting on General Relativity edited by R. Ruffini, North-Holland, Amsterdam. Michelson P F and Taber R C 1981 "Sensitivity analysis of a resonant mass gravitational wave antenna with resonant transducer" J. Appl. Phys. 52, 4313. Misner W, Thorne K S, and Wheeler J A 1973 Gravitation, W. H. Freeman and Company, San Francisco. Moss G E, Miller L R, and Forward R L 1971 "Photon-Noise-Limited Laser Trans- ducer for Gravitational Antenna" Appl. Optics 10, 2495. Muller E 1982 "Gravitational Radiation from Collapsing Rotating Stellar Cores" Astron. and Astrophys. 114, 53. North D 0 1943 "An Analysis of the Factors Which Determine Signal/Noise Dis- crimination in Pulsed-Carrier Systems", reprinted in Proc. of the IEEE 51, 1016 (1963). Oohara K 1984 "Excitation of the Free Oscillation of a Schwarzschild Black Hole by the Gravitational Waves from a Scattered Test Particle" Progress of The- oretical Physics 71, 738. Oohara K and Nakamura T 1984 "Gravitational Waves from a Particle Scattered by a Schwarzschild Black Hole" Progress of Theoretical Physics 71, 91. Peters P C 1964 "Gravitational Radiation and the Motion of Two Point Masses", Phys. Rev. 136, B1224. Peters P C and Mathews J 1963 "Gravitational Radiation from Point Masses in a Keplerian Orbit", Phys. Rev. 131, 435. Riidiger A, Schilling R, Schnu'pp L, Winkler W, Billing H and Maischberger K 1981 "A mode selector to suppress fluctuations in laser beam geometry", Optica Acta 28, 641. Riidiger A, Schilling R, Schnu'pp L, Winkler W, Billing H and Maischberger K 1982 "Gravitational Wave Detection by Laser Interferometry" Max-Planck-Institut fur Quantenoptik report MPQ 68.

- 113 - Saenz R and Shapiro S 1978 "Gravitational Radiation from Stellar Core Collapse: Ellipsoidal Models" ApJ. 221, 286. Saenz R and Shapiro S 1981 "Gravitational Radiation from Stellar Core Collapse: III. Damped Ellipsoidal Oscillations" ApJ. 244, 1033. Smarr L L 1979 editor of the volume Sources of GravitationalRadiation, Cambridge University Press. Schilling R, Schnuipp L, Winkler W, Billing H, Maischberger K, and Rudiger A 1981 "A method to blot out scattered light effects and its application to a gravitational wave detector" J. Phys. E: Sci. Instrum. 14, 65. Schnuipp L, Winkler W, Maischberger K, Rnidiger A, and Schilling R 1985 "Re- duction of noise due to scattered light in gravitatonal wave antennae by modulating the phase of the laser light", J. Phys. E: Sci. Instrum. 18, 482. Shoemaker D H, Winkler W, Maischberger K, Riidiger A, Schilling R, Schnuipp L 1985 "Progress with the Garching 30-meter prototype for a gravitational wave detector" in Proceedings of the Fourth Marcel Grossman Meeting on General Relativity edited by R. Ruffini, North-Holland, Amsterdam. Shoemaker D H 1985 Private Communication. Stark R F and Piran T 1985 "Gravitational-Wave Emission from Rotating Gravi- tational Collapse" Phys. Rev. Lett. 55, 891. Taylor J H and Weisberg J M 1982 "A New Test of General Relativity: Gravita- tional Radiation and the Binary Pulsar PSR 1913 + 16" ApJ 253, 908. Theriault K B 1975 "Optimum Arrival-time Estimation in Exploration Seismol- ogy", MIT E.E. Thesis, Supervisor: A.B. Baggeroer. Thorne K S 1982 "The Theory of Gravitational Radiation", in Rayonnement Grav- itationnel, GravitationalRadiation, edited by N. Deruelle and T. Piran, North- Holland 1983. VanTrees,Jr. H L 1968 Detection, Estimation and Modulation Theory, pt.1 Wiley and Sons (New York). Weber J 1960 "Detection and Generation of Gravitational Waves" Phys. Rev. 117, 306. Weber J 1969 "Evidence for Discovery of Gravitational Radiation" Phys. Rev. Lett. 22, 1320. Weiss R 1972 "Electromagnetically Coupled Broadband Gravitational Antenna, Quarterly Progress Report", Res. Lab. Electron. MIT 105, 54. Weiss R 1982 "Noise due to Scattered Light in Interferometric Gravitational An- tennas and its Reduction by Random Phase Modulation of the Input Beam" Private communication in appendix II of a document to the NSF. Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons.

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- 115 - Appendix A Quadrupole Radiation from Masses in a Circular Orbit

Two masses orbit in the x-y plane, viewed from z = +oo: At the instant shown the positions, velocities, and accelerations of the masses Y are: j L

xi M2 a; -min X2 - M 1 + m 2 Ml +M2 rc~a ,' I . = Gm1m 2 . -Mi . a(mi Y2 Y1 + M 2 ) M2 L--2

-Gm 2 I. Gm, 2 a X2 a2

The orbital frequency is:

2 G(mi + M2 ) Worbital a3 The quadrupole moments are given by:

Dag = mi(3xixP - boprF) in particular,

Dxy =3mix1 y1 + 3m2x 2y 2 Dxx - Dyy 3 2 2) 3 2 2 2 3i1(x- y1)+ -m 2 2 2 2 (x2 -y) Differentiating and substituting the values at the above pictured moment:

Dxy = 0 DXX - Dyy = 6 Gmim 2 Iac=4 2 a The quadrupole formula then gives:

2 5 3 _ 1 4G Mim2 1 4G / mim2 2/3 R 4 3 R c (Mi + M2 )1/ Worbital This is the amplitude of the sinusoidal radiation emitted at the frequency 2 Worbital. This calculation was made at a point in time when hxx is maximum and hxy was 0; at a general time, hxx and hxy will be 900 out of phase.

- 116 - Appendix B Thermal Limit for a Bar Antenna

This can be easily calculated in the framework of the matched filter. In the ideal case, the gravitationally induced force signal,

1 Fg = - mbar lbar h(t) (B.1 = 2.6) 2 is to be detected in the presence of the white thermal noise force background:

F2herm = 4kTbar Mbar(bar B.2) Q Assuming an h(t) waveform composed of several, i.e., NHC, half cycles at a frequency WGW and with amplitude ho, the force waveform is approximately:

1 Fg(t) ~ mbar lbar WGW h0 x (B.3)

Now, using the matched filter formula, equation 6.1, the optimum SNR for detecting this "burst force" is given by:

2 f:F2(t)dt SNRoptimum = F2 (B3.4) therim

Setting SNR=1 and solving for h2 x NHC gives:

h xNHC = 16 kTbar Wbar 1 (B.5) 7r Q mbar lbar WGW

In practice, because of displacement transducer noise and the bar resonance,

WGW KoWbar. If values comparable to those of the Stanford bar are used:

Tbar = 4.3 *K mbar = 2.4 x 10 Kg3 lbar = 1.2 m

Q 5 x 106 Wbar = 27r 840 sec-1 k = 1.38 x 10-23 J/ K then, for NHC=6, an ho of 10-20 can be detected. (The present performance is ho ; 10-18; this shows the effect of the narrow banding required due to the dominance of the transducer noise away from the bar resonance.)

- 117 - Appendix C Interferometer Response

The response of the interferometer to a source at an arbitrary orientation can be calculated using equation 2.7 for the motion of a free test mass, rewritten in vector-matrix form, retaining only the space coordinates:

dX = [-]X (C.1) 2 with

0 0 h 1 0 h+ 2 =2g 0 hx The geometry of the antenna is shown in Figure C.1. The vectors are arbitrary in 3-D. The interferometer measures the difference between the lengths (measured along an arm) of the two arms:

dl = d1j - d12 (C.2) This difference is independent of vcm and is given by:

dl [-]'h h (C.3) dl Hal2 a2 -[-a2 (It has been assumed, correctly for the small prototype, that variations of [ ] across the antenna are small; in particular the gravitational wavelength is much longer than the optical path length.) If the arms have equal (for h=0) lengths L then:

dl hh = a -[-]a, - - (C.4) Li 2 a2 [a22 With [}] conveniently expressed in the source coordinate system, the antenna unit vectors can be evaluated in the source system using the geometry shown in Figure C.2. The results of the rotations are:

- sin 0 sin # al,rot = cos # (C.5) -cosO sin#J

sin 0 cosq# a2,rot = sin (C.6) cos 0 cos J

- 118 - Substituting these and [h] into equation C.4 gives:

dl 1+cos2O - = h+ cos 2q ( - hx sin 2 cos 0 (C.7) L 2 2 ) The maximum response occurs for 0 = 0:

dl- dl =- h+ = - -hx (4 = 7r/4) (C.8) L (0 0) ; L

EMi

M %

Cernter 4- naC~0 sS

Figure C.1 Definition of mass vectors

X3 S o VIC C P- Atv--nrc Ac'rr C oa U Y \ Pv Civ rS X r AL~ nro,- eP

/ A

Figure C.2 Spherical geometry for antenna response calculation

- 119 - Appendix D Comparison of Bar and Interferometer Response to GW Bursts

In order to compare the response of an interferometric antenna with that of a bar, a canonical "burst" gravitational waveform as well as models for the response of each antenna must be specified. These ingredients are described below.

D.1 The Canonical Form of a Burst of Gravitational Radiation The gravitational radiation is assumed to be that from a burst source. Although there are many calculated and proposed h(t) waveforms in the literature, most of them can be well approximated by a simple sinusoidal burst. The parameters of this burst waveform are the frequency wg, the length of the burst measured in half-cycles NHC, and the amplitude ho of the waveform:

+ W3NHC=LI

The delta functions shown in the h waveform are used for analytic simplicity. They are required by the slope discontinuities and keep hafter burst - hbefore burst = 0. In "real" waveforms, "wings" or "tails" are present (dotted lines above); approximating these by the delta functions introduces negligible errors.

D.2 Interferometer Response The interferometer is characterized by one parameter: its noise level, assumed white and Gaussian, in strain per root Hertz, hN. Given the h(t) from a source the expected amplitude signal-to-noise ratio for optimum matched-filter detection is given by:

SNR = 2fh2(t)dt (D.1) N Most of the contribution to the SNR comes from the few half-cycles of large amplitude and the SNR is insensitive to slight changes in the wave shape, i.e., details of wings and tails. For the canonical burst:

- 120 - SNR h NHC7r (D.2) hN Wg

Note, that if the wave shape is maintained but its frequency increased, it is necessary for ho to grow as V/_ to maintain a constant SNR.

D.3 Bar Response The response of a bar antenna is modeled by using a harmonic oscillator equivalent (Amaldi 1980). The equation of motion of the equivalent harmonic oscillator is given by:

maR + mawax = Fg(t) = -mala ii(t) (D.3) 2 with

ma = Mbar/2 4 la = 4 Lbar

Wa = 27r fbar resonance

Note that a damping term has not been included because the duration of the burst is assumed to be much shorter than the ringdown time of the bar. It is straight forward to show that the energy deposited in the harmonic oscillator by a burst is given by:

E = kTburst = Fg (t) sin wat dt 2 + (] Fg(t) cos Wat dt) J 2ma2 - M a "Fourier component of h-double-dot at Wa" 12 8 (D.4) In particular the energy deposited in the bar by one of the canonical burst waveforms can be calculated. The force term is:

1 Fg(t) = - mala h(t) 2 (D.5) = - mala Who - - 2 ia gI 1 X\9~

- 121 - Performing the required integrations gives:

Tburst = maaa NHC 2 w h- C 2 (-,NHC) (D.6) where 3kW COS Wa NHC-r C( WaNHC) sinx siny -4 (N C x y NHCr si w 2 with

x NHC7rNRC (1 --Wa)) 2 Wg (D.7)

y NHC7r +Wa) = ( W

upper of 1 st and 3 rd choices for NHC odd

upper of 2 nd choice for NHC=1,2,5,6,...

Note that C(1, NHC) = 1 and thus the tuned, i.e., wg = Wa, response of the bar is a simple expression. Because the energy density of the incoming wave is proportional to W2h , C 2 (...) gives the change in cross section of the bar as the gravity wave frequency is detuned. Using the values for the Stanford bar (Lipa 1980, Michelson and Taber 1981, Boughn et al. 1982):

3 ma = 2.4 x 10 kg la = 1.2 m

Wa = 27r 841 sec- Td 20 m*K

Choosing NHC=3 and wg = wa, the 20 m*K rms level corresponds to ho = 10-18. A 10 *K event interpreted as a several half-cycle burst has ho = 2 x 10-17.

D.4 Comparing Bar and Interferometer To get an idea of the relative sensitivity of the bar and interferometer to a given GW burst, the temperature-equivalent energy deposited in the bar, Tburst, is plotted vs. the frequency of the GW burst. The amplitude ho of the burst at each frequency is chosen so that the interferometer SNR is constant. For the canonical burst this requirement leads to:

ho = SNR hN Wg (D.8) NHC 7r

- 122 - Substituting this into (D.6) gives the equivalent bar temperature:

Tburst, const. = mal2 NHC 2w2 (SNR 2h2 C) C2 ( a, NHC) (D.9) brt os.-32k aa9NNHCWr Wg interf. noise

This is plotted in Figures D.1-D.3 for several values of NHC (2,4,6) with h(t) shown inset. The bar parameters are those of the Stanford bar and the interferometer noise level is that reported by the MPQ Garching group, ho = 2 x 10-1 9 strain/ z (Shoemaker et al. 1985). As is apparent from equation D.9: Tburst oc SNR2 . To compare SNR values, the amplitude SNR for the bar is used:

SNRbar = Tburst (D.=_) rd Thus, an event that deposits 0.5 *K in the bar has the same SNR as the SNR=5 events in the interferometer. Whether the curve is above or below the 0.5 'K line indicates which instrument is more sensitive to the burst.

D.5 Comparison for an Arbitrary Wave Shape For arbitrary wave shapes the comparison can still be made using numerical integration and differentiation to evaluate SNR and Tburst. A frequency is assigned to a wave shape on the basis of the period of the largest few half-cycles. This frequency is varied by changing the scale of the time axis. Again the amplitude of the waveform at each frequency is adjusted so that the interferometer matched-filter detection gives SNR=5. -The results for a few wave shapes are shown in Figures D.4-D.6 (with h(t) and h(t) shown inset).

D.6 Discussion Comparison of the "canonical burst" results with the "real" waveform results shows that the canonical burst gives a good ballpark estimate of the real response. More interesting are the actual values of the curves. In the frequency range below 500 Hz and above 2 kHz, the MPQ instrument is more sensitive than the Stanford bar. Even in the 500 Hz to 2 kHz range the interferometer could serve as a useful veto for the greater than several degree events. The reality of the > 10 'K events could be very certainly determined. The geometry of the detectors and location and polarization properties of the source have been ignored; it would be interesting to plot the relative response to polarization averaged Galactic center sources vs. Galactic center location for the antennas. All in all, a bar-interferometer coincidence experiment looks promising.

- 123 - 100 1000 10000 a 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

CD ...... : ...... -CD %D...... : ...... :

......

CU ...... -ru

CS) ...... (!D --- 0 M ...... CD CD r 0) ...... Qj L CU...... -ru

CD ...... e ... %D...... L ...... m ru ...... z ...... L...... t..; .. -ru ...... M ...... t ...... CD

...... ! ...... : ......

ru ......

0 ...... 7 ...... (Lj CS) CD ...... %D ...... ; ..... CA

......

0) L (U- ...... W C -- 4 CD Ui 0 ...... t...... : ...... CD ...... r* - T T*: --Z-- -Z- CD CD CSD : : : : : : : : ...... ; ...... Z ...... ; .....

CU ...... : ......

1 3 14 15 61 71 81 91 . 3 41 5 6 7 8 9 100 1000 10000 GW Frequency (Hz) Figure D.1 Bar Response for constant Interferometer SNR = 5.0 Canonical Burst with NHC = 2 Interf. Noise is 2-OOE-19 h/rHz R 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

-124- 100 1000 10000 3 4 5 6 7 8 9 3 4 5 6 7 8 9

OD ...... : ..... OD %D ...... ; ..... CIS ......

......

CU ...... L...... ru CS) (S) ...... : ...... I V-4 CD ...... CD CS) %D ...... I ...... 1 ......

...... ;: .....

...... L CU ...... M

CD ...... -- 4 z ...... z...... : .. : ...... CD ...... r ...... (D CS)

%D...... L 0 ...... : ...... m

CU ...... ro

...... T--4 ...... : ...... I.- 1z CD- ...... -CD -4 ......

...... _0 ...... ; ...... GU

ru ...... ra

-4 0 CS) ...... 0- ...... r ...... r m ...... 4. 4.- 1...... CD ...... z ......

...... *......

0) L CU ...... 0) L.-.; ...... -tu C 1ji 0 ...... t ...... CD ...... t ..... CO CD ......

......

ru ...... Y ...... -tu

4 5 6 7 8 9 5 6 7 8 9 100 1000 10000 GW Frequency (Hz) Figure D.2 Bar Response for constant Interferometer SNR = 5.0 Canonical Burst with NHC = 4 Interf. Noise is 2.OOE-19 h/rHz R 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

- 125- 100 1000 10000 3 4 5 6 7 8 9 3 4 5 6 7 8 9

CD ......

...... r

...... U CS) CS) ...... ; ..... -4 CD ...... 0 CD

...... : ......

L CU ...... T ......

CD ...... -- I ...... CD ...... 0 (!D ...... ; ..... L 0 ...... m

CU ...... r ......

V--4 ...... : ..... CD- ...... -0 -- 4 %D...... T ......

...... _0 ...... : ...... Qj -0 -4 ...... ; ...... 0) ...... 0 ...... * CD CS) CD ...... I ...... *......

0) L ...... -- 4 (!D ...... m ...... CD CS) ...... f ...... -0 ......

Cd -4-4-.4-4.4-444.; ...... ru

41 5 6 7 8 9 P- 3 4 5 6 7 8 9 100 1000 10000 GW Frequency (Hz) F 1 gure D. 3 Bar Response for constant Interferometer SNR = 5.0 Canonical Burst with NHC = 6 Interf. Noise is 2.OOE-19 h/rHz R 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

- 126- 100 1000 10000 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

...... -CD ......

...... :

ru ...... : ru

...... C .... 1...... w ...... T..r ...... t...... 7 ......

W ......

L fu ...... C ...... ru 0 ) (D 0 ...... r...... : ...... L ......

CU_ ...... -IV

...... i ...... i ...... CD_ .. -CD ...... 7 ......

...... (D

W C) G ...... CS) CD ...... T.Y.- ...... -CD W ......

...... : ......

L ru ...... (D z ...... ; ..... _rU CS W CS) ...... CS CS) %D......

V ...... : ......

CU ......

9 3 45 6 7 8 9 -7 8

100 1000 10000 OW Frequency (Hz) Figure D.4 Bar response for constant Interferometer SNR = 5.0 Particle - Schwarzschild B H, J=O, Detweller, Fig.3 Interf. noise is 2.OOE-19 h/rHz A 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

- 127- 100 1000 10000 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 I I I - I I I A I LA I I - I I

...... : ...... - CD ...... -V%

...... %.. . .-...... 4 ......

ru4 ...... - ...... CD ...... : ......

......

(j) ......

L CU ...... -ru

M CD ...... -4 E m:]D ...... i ...... i ...... i ..... i ...... CD ...... : ...... 0% L ...... *...... : .....

...... t ...... _fV

...... i-.-.i ... i_.4.4. -4.444.1444 ...... i ...... !...... C ...... CD- ...... -W

...... t...... 2..... f......

-- 4 Cl ...... 0) 0 ...... CD- ...... o ......

>1 ...... I.- ...... 0) L CU ...... _rU

...... :,44.44444-i ...... i ...... i .... i .... i ... ;_4..

CD ...... t ...... CD

w ......

.V ...... i ......

CU ...... -IV

2 3 4 5 r 3 4 5 6

100 1000 10000 GW Frequency (Hz) F 1 gure D. 5 Bar response for constant Interferometer SNR = 5.0 Particle-Kerr BH, Kojimo + Nakamura, Fig.3(c), Lz=-4.9 Interf. noise is 2.OOE-19 h/rHz A 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

- 128- 100 1000 10000 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

...... M ...... CO ......

...... r ......

CU ...... -ru

CD ...... : ..... (S) -- 4 CD ...... -CD ......

......

L CU ...... -ru 0) aj (S) ...... CD ...... M L %D ......

ICU ...... -ru

...... j.j ...... I ...... CO ...... :.:*......

It ...... : ......

-4 Cd ...... V...... ru 0) 0 ...... CS) M ...... *......

L ...... a) (U ]z -4 Ui M ...... -44444- ...... -GD ...... ;..; ......

......

cu ...... TU

2 3 4 5 6 7 13 9 a 3 4 5 6 7 8 9

100 1000 10000 GW Frequency (Hz) Figure D.6 Bar response for constant Interferometer SNR = 5.0 Damped Ellip. Stellar Collapse, Saenz Shapiro 1981, Fig.6 Interf. noise is 2.OOE-19 h/rHz A 0.5 K Event has SNRbar = 5.0, Td = 20.0 mK

-129- Appendix E Mass Damping System

This description is of the system in early 1983; the introduction of wire sus- pensions in May 1985 required further modifications not described here.

E.1 Overview The mass damping system is designed to damp motions of the suspended mass in six degrees of freedom: three translations and three rotations. The mass is suspended in a cage whose 3 faces have sensing and driving plates evaporated on them (copper on glass), see Figure E.1. The two sensing plates on a face can thus sense common mode motion (a "pendulum" mode for the X and Y plates and reffered to as such for all plates) and differential mode motion (such as that produced by "rocking" rotations of the mass (X and Z axes) and torsional rotations sensed by the Y axis-all refered to as rocking modes). Coresponding drive plates, biased at a nominal high voltage (HV), can be used to push (smaller HV) or pull the mass and exert forces and torques. The sensing and driving plates of one face are closely orthogonal to those of the other faces-this allows modes to be damped in a 3 x 2 scheme where each plate (face) senses and damps two modes with no information from the other faces. Much of the electronics and design is devoted to the capacitive displacement transducer units (CDTs); the hard work was done by Linsay and Shoemaker 1982. These units, one for each sensing plate, through an RF bridge circuit measure the capacity of the sensing plate to the (nearly) grounded mass, and produce a voltage output proportional to the relative velocity of mass and plate. The sum and difference of the velocities from the plates on one face generate the PMO ("pendulum monitor output") and RMO ("rocking ...") signals: the PMO signal containing no sensitivity to rocking mode motion, and vice versa. These signals are then amplified (gains Gp and GR) and applied, in common mode (P) or differential mode (R), to the HV driving plates; this summing and differencing to generate the feedback signals takes place in the "matrix box". In this way the fundamental requirement of damping is achieved: forces proportional to velocity (with correct sign!) are applied. Much of the work involved in the system goes into choosing the parameters of the CDTs such that the output is in fact close to proportional to velocity in the frequency range to be damped. Deviations from a strictly velocity-proportional output give rise to changes in the mode's natural frequency in addition to a change in Q. The exact performance is best calculated through the technique of the root-locus. However, for initial design and intuitive understanding simple approximations work very well. These are discussed in the CDT section, E.2. The units used in this section are cgs with occasional mms, pf, and "lab volts ".

- 130 - e~d

Drive plote Xt'- D

X+E S Y-X YjJx X- te

\1- IA

En 'sS

EcA.0 Co0 PjjeS

Figure E.1 An End Mass and its Cage of Evaporated Plates

- 131 - IV) v Aj~

+ + G

F- C-c la~s\Je ADl

y AV~ RTFs'Z

D~r am /srj\y !&x5tQ\fof on e- \t , tw jec J5 ce#ts o r E.1.1 The Sensing Capacitor Plates Motions of the mass are sensed through changes in the capacity of six "sensing" plates distributed around the mass as shown in Figure E.1. The capacity of each of these plates as a function of the mass position is given by:

Cdet(x) Coffset + Cpiate (E.1) Xmm with capacities in pf and xmm in millimeters. Coffset includes stray and cable (hard- line) capacity. Cpiate is the active capacity at a 1 mm mass-plate spacing and has units of "pf - mm". (The 1 mm value is used because, in operation, this is the typical mass-plate seperation.) The numbers of interest for the sensing plates are:

Cdet(xo) ~~ 140 pf (E.2) and:

dCdet Cpiate dx =-- x2 ~ 430 pf/cm (E.3) where the values above are typical for a 1mm spacing.

E.1.2 The HV Drive Plates In order to apply forces to the mass another set of six plates is used, the HV driving plates. For these plates the useful constant is:

a d(Force in cgs units applied to the mass) (E.4) d(Vpiate in "lab" volts) The plate operates about an HV bias voltage, VO, and the static force is:

Fs - 1 VOA 1 V0,labA (E.5) 2 cgs 87r x 2 (300)2 87r X and thus:

dFcgs _ 1 2VoA (E.6) dVo,lab (300)2 87r x 2

The effective A is determined from the measured Cpiate:

1 1 A 2 Cp = .4 (E.7) 0.9 47r xcm

- 133 - thus,

= Cpiate Acm2 = 47r xcm 0.9 CPf with CPf (E.8) XmM so:

dFcgs 1 2VO 0.9 xcm Gpiate dVo,lab (300)2 87r x2m Xmm (E.9) 1 VO Cpiate 105 xcm Xmm

Typical values are: xcm 0.1 cm, Cpiate a 45 pf mm, and Vo a 400 V; these give an a e 1.8 Fcgs/Viab.

Also of interest is the constant:

dFcgs d [1 V2A 1 (E.10) dxplate,cm dx (300)2 87r x2

Evaluating the derivative:

dFcgS s = 10_ 5V0 Cpiate dxplate XO xmm (E. 11) = 7200 for the typical values above.

This force gradient gives rise to a coupling of plate motion to mass motion. The magnitude of this coupling for an end mass is given by:

Ma = F

2 Ms Xmass = 2 x 7200 xplate

For an end mass, M = 8000 g, so: (E.12)

Xmass 2 Xplate s2 where the factor of two comes in because there are two HV driving plates. Thus, to achieve a mass motion of < 10-15 cm/Viz at 1 kHz, the plate motion must be < 2 x 10-8 cm/ Hz.

- 134 - 2.0

1.5 (..ooY... .. 1.e

Ln 0.5 V - -j- U e.0

- - ->.

-1.

.0 0.' B.s I I .. 2. X(

12.

11.

10.

b. 0 - ----

5. 0 0 0 r\OX OAYC/ a., C

3. rA- ...--v

0.. 0.0 1.0 2.0 3.0 4.0 5.0 r.0 7.0 8.0

'V (x100 vo\ts )

Figure E.3 The Potential Well in which the Mass Sits a) above shows the potential well. b) gives the locations of the local maximum and local minimum as a function of the HV bias voltage. Note their disappearance above a critical HV value.

- 135 - Finally, it is instructive to plot the potential well that the mass sits in when HV is applied:

Mg V 2 A I U(x) 21 (x - x) -- (E.13) : (x2 8 ) 8irx

This is plotted in Figure E.3a for typical end mass x-axis values:

M = 8000 g V = 400 V 1 xo = 1 mm x. = 1.14 mm A = 134 cm 2

Notice that the mass can get "stuck" to the plate if it gets over the barrier at x - 0.4 mm. To avoid this, teflon bumpers protrude by 0.5 mm from the masses surfaces. In addition, for a given x., increasing V causes the equilibrium point xO to move toward the plate and the location of the barrier to move farther away from the plate. At a critical value of V these meet and the minimum goes away altogether. The locations of the minimum xO and the barrier xb are determined by:

dU 0 3 2 2 x _ xx + A 0 (E.14) dx 87r Mg

This has 3 roots of which one for x. < 0 is unphysical. The other two are xO and xb. Their locations versus V are plotted in Figure E.3b for the typical end mass parameters.

E.2 Capacitive Displacement Transducer A near-schematic of a CDT is shown in Figure E.4. An RF oscillator, through a resonant step-up transformer, drives the bridge with VP' ; 40 VPP. Motion of the mass causes a change in Cdet which unbalances the bridge causing a net RF output from the bridge. After amplification this signal is demodulated and the resulting DC error signal is fedback to rebalance the bridge by changing the capacitance of the varactor, C,. The signals VCMO ("capacitive mixer output") and VCMON ("capacitor monitor", the voltage on the varactor) are the transducer outputs. The relationship between VCMO (and VCMON) and the mass motion is more easily explored when a simplified diagram for the system is used. To this end the operation of the capacitive bridge and the mixer are analyzed.

- 136 - PP PP e-so nort Step- v p I I RF osc.

GB GB vcro

RF RE Amp.,

c~3 c Ct-I +-v, c ciaC SAL I cv C~c Axv7

~'VCO

S e-nsltND Corc-LOc BcI~ Boc tMixe.r B~k

r,,U,,_ E.9 CI - o~citive- Dis~1oxzuvmxxnt -F(-Ckf)-5AVh~ve C e- C E.2.1 The Capacitance Bridge The bridge circuit can be viewed as having two inputs, x and Vv, and an output V , the differential RF voltage at the bridge output. Thus, the operation of the bridge is given by the constants which specify the change in the bridge output VfjP as a function of changes in x0 ut and Vv. These are calculated below. A change in x, i.e., Cdet, shows up as a change in VP ; this is calculated as:

1 1 = PP Cet V outPP B 1 1 (E. 15) Cdet C 3 CD Differentiating, we have:

dV - VP 1 dCdet (E.16) dx + g Cdet + CB dx

For maximum sensitivity CB is chosen to be ~ Cdet.

To calculate the change in V. with Vv we note that near balance we must have: Cdet ~ CBAL + (Cs in series with Cv) (E.17)

Thus, analogously we have (with a sign change):

dVP V_ P 1 d(CBAL + (Cs series Cv)) (E.18) VV + Cdet + CB dVv

The last derivative can be calculated:

d(CBAL + (Cs series Cv)) d CS) 1 dCv (E.19) dVV dVV dVV 1+ C. (1 + ) and thus:

dVPP 1 dCV V P (E.20) VV 1+ Cdc (1 +_Y)2 dVV

These two equations, E.16 and E.20, characterize the bridge.

- 138 - Typical bridge parameters are:

VPP a40 VPP CB 120 pf dCdet Cdet a 140 pf; dxdx e 430 pf/cm C ~ 47 pf C ~ 321 pf @ V, = +5 V, for a CV1666 varactor dC~ d~V -23.2 pf/V @ Vv = +5 V dVv d(C. series Cv) 0.38 pf/V dVv

The series capacitor C. considerably reduces the "gain" of the varactor; however, it is needed to keep the RF voltage on the varactor less than the bias voltage (+5 V) to avoid non-linearities. This voltage is:

1 1 0 RF varactor voltage - V P Cdet CV CB e + C (E.21) ; 2.4 VPP for V P = 40 VPP

When the bridge is locked, the conversion from Vv to an equivalent distance of mass motion is given by:

dx 1 1 dCV dVv (dCdet) (1 + Cv)2 dVv (E.22) 8.8 ptm/volt

Thus, the bridge dynamic range is limited to 15 Am by the varactor dynamic range.

- 139 - E.2.2 The Mixer The output of the bridge is an RF voltage having a peak-to-peak amplitude of Vo. This signal is amplified and demodulated yielding a DC voltage proportional to V . Ideally the mixer performs:

VIF = sign(VLO) -VRF ; where sign(x) = { > (E.23)

Pictorially, for signals in phase with the LO:

RF Mixer IF -- LO

Thus, an RF input of VP becomes a "rectified sine wave" of amplitude VP/2. The average, i.e. DC, component of the output is:

vDCout V IN [sinOdO 2 r 2 ~(E.24) =VP

If this relationship is assumed for non-ideal mixers as well, then the mixer losses are accounted for by a reduced effective gain of the preceeding RF amplifier stage.

- 140 - E.2.3 Transfer Functions from x to VCMO and VCMON Using the bridge equations (E.16 and E.20) and the mixer relationship (E.24), we can draw the system of Figure E.4 as:

kv +

1+ sRC. T

AYj A

where we have:

dV _ 1 VP 1 dCdet dx 7r 1+ -AL CB + Cdet dx 1 V P 1 1 dCv (E.25) b = ACFB.! V B 7r 1 + AiC-d C33 CB + Cdet (1 + Gv)2 dVv a = equivalent gain of RF section The mixer factor of 1/7r has been included in dV/dx and dV/dVv (i.e., b). Thus, these voltages are equivalent DC demodulated values. Typically:

a = 800 b = 0.05

fRC = 0.16 Hz, (RC = 1 sec) dV = 9.7 V/cm dx dV = 8.6 x 10-3 dVv ACFB = 5.8

- 141 - Of interest are the transfer functions from x to the outputs VCMO and VCMON. These are given by:

VCMO dV 1+sRC =a - x dx 1+ab+sRC (E.26) VCMON _ dV a ACFB x dx 1+ab+sRC

The DC loop gain, ab a 40, and the RC corner frequency, fRC, define 3 frequency regions of interest. In each region the transfer functions can be approximated:

f < fRc(0.16 Hz) fRC < f < ab fRC (6.4 Hz) ab fRC < f

VCM0 dV 1 dV RC s x dx b dx b a dvdx

VCMN ;dV (1/ dV dV ab dV x dx dV, dx sRC dVV

The exact transfer functions and the approximations are plotted in Figure E.5. From these we see that:

- VCMON for f < ab fRC is a good indicator of the mass motion and has a sensitivity ~ 0.11 V/ftm.

- VCMO for f > ab fRC is a good measure of mass motion with a gain = 0.8 V//,m.

- VCMO for fRC

- 142 - 9. -

8.8- 4 ? -

7.0 1.F U-- U,- 2.:L-

5.-- a- a 5 B 3.. w -

g.e-- 0.0 -. 0 .0 -2.8 -1.8 0.8 1.8 2.8 3. .0

FREQUENCY CLOGCMz)) FREQUENCYCLG z)

8.6 I I I I

7.5)

7.8

'.5

a. 5.5 6 .

5.8 2.8 -- 3 88 18 28 3 - .5 F -2.8 -1.8 8.8 1.0 2.9 3.8 0 FREQUENCY CLOG(Mz)) FREQUEP4CY CLOS(H:))

-Vc X x

Figure E.5 Transfer Functions from x to VCMo and VCMON

- 143 - E.3 Equations of Motion For the geometry and definition of the variables see Figure E.2. For the center-of-mass or "pendulum" motion we have:

x 1 + x 2 F 1 +F 2 + 2(E.27) 2 M 2 where Wp = natural frequency of pendulum mode

X1 , X2 = sensor-plate-mass displacement from equilibrium

F 1, F 2 = Forces applied by the driving plates

Likewise, for rotations about the center-of-mass, or "rocking" motion, we have:

-2 o + -(F - F ) = _WR 21 1 2 (E.28) where WR = natural frequency of rocking mode

X1 -X2 d d = driving plate seperation

I = moment of inertia = 1(w 2 + h2 )

To damp these modes we apply forces:

-F = aGpGHV[CcsSXI 1 + Ccssx 2] + aGRGHV[CSsx1 - Ccssx 2 ]

-F 2 = aGPGHV[CcssX1 + Ccssx 2 ] + aGRGHVCcssX2 - ccssXi(

where G H V = high voltage amplifier gain a = drive-plate force constant, section E.1.2 Cs = capacitive sensor constant, section E.2.3 Gp, GR = pendulum and rocking damping gains S = iW

- 144 - Substituting these expressions for the forces into equations E.27 and E.28 we get the characteristic equations of the damped harmonic oscillators:

Pendulum 2 a 2 mode s + 4-M CCs-GHV-GP s + wp =0 (.o Rocking 2 d2 2 ode s + a--C GHv-GR mode I 5 - s + WR =0 In general, for a harmonic oscillator:

2 +s+O = 0 (E.31) Q Thus, the Qs of the pendulum and rocking oscillators are given by:

WP M = wp M QP 4aCcsGHvGp (E.31) WR I QR = 2W d 2 aCcsGHvGR

Typically d ~ h/2 and with I= i(w 2 + h2 ) we can write:

QR = QP 2 +] (E.33) GR wp 3 h

The last factor above gives the additional gain needed to damp the rocking mode to the same Q as the pendulum mode. Typically WR/WP is 5 to 10; w/h varies depending on the mass (EM or CM) and particular plates (X, Y, Z). Its values are 0.35, 1.65, and 1.71 for EM-X, EM-Y, and EM-Z plates; 1.0 for all CM plates. The equations above can be used to get a good idea of the kinds of gains required for a given Q.

- 145 - E.3.1 Exact Damping Equations and the Root-Locus

The previous assumption that V 1 = Cc sxi represents an ideal velocity sensor. For the actual CDT systems, Ce, has been approximated as Qdx Tb and a rough estimate of the gains needed can be made. The exact damping behavior can be calculated using the correct CDT transfer function, equation E.26 for VCMO. In this case the complete damping loop can be drawn as:

-- GH a e (E.34)-HVF6t

a dV (s+1/RC) M dx (s + (ab + 1)/RC)(s + iwp)(s - iwp)

(For the rocking system Gp = GR 3[(W)2 + 1]

WP WR

The exact poles and zeroes of the closed loop system can then be calculated through the usual root-locus techniques; an example for the pendulum and rocking modes is given in Figure E.6 with the parameter values given below.

Parameters used in the Root-Locus Plots

a = 1.8 cgs force/lab volt M = 8000 g dV = 9.7 V/cm a d x = 800 ab =40 RC =1 sec GHV = 26 w/h = 0.35 (EM - X axis) WR = 29 rad/sec (4.6 Hz) wp = 3.0 rad/sec (0.48 Hz)

- 146 - I I I I I I I I

4.0 - ' 3.5 - Cr 3.0 - Irk 2.5 -

2.0 -

0.s -

PQ YJ

5. 90 -B. 0 -2.0 -1.0 0.0 . REAL 5 C

S IMUM 4. '4oz e UaL.5 AsOAD-

3.0 - tn 20 z 2. I

1. 0 -

PRockto i naode ( e 0.0

-1.01 I I I I I I -6.0 -s.0 -4.0 -3.0 -2.0 -1.0 0.13 1.0 REAL 5 C10)1

Figure E.6 Root-Loci for the Pendulum and Rocking Modes

- 147 - E.4 Damping System Noise There are three noise sources specifically related to the damping system described here. They are: 1) Conversion of capacitor plate motion to mass motion through the electrostatic force gradient.

2) Conversion of capacitor plate motion to mass motion through the resulting voltage changes on the capacitor plates due to the operation of the damping circuit. 3) Voltage noise on the capacitor plates which drives the mass. The first has been calculated in section E.1.2, equations E.10-12 and has the transfer function:

Xmass 2 (from E.12) Xplate S2 for typical parameter values. The second noise mechanism can be calculated straight forwardly:

Xmass _ dV a 180 (E.35) = a-2-GP-2-GHV~~(.) Xplate dx Ms 2 S2 where it has been assumed that s > (ab + 1)/RC and the motions are common mode. This second term is larger than the first; however, it can be filtered with a low-pass filter (see below), unlike the first. Finally, there are 3 main sources of voltage noise in the damping system, as indicated in Figure E.7. For s > (ab + 1)/RC the noise on the HV plates is:

2 v2 vHVamp 2 2 2 v 2 + aCvFETa2C2v2 + b a c2 n OPampam (E.36) where c. = c/V/2 with c = 2GpGHV; the factors of V/2 occur because the noise from two CDTs or HV amplifiers add in quadrature. The mass motion due to this noise voltage is simply:

2a xm Ms2 vp (E.37)

The noise terms have typical values:

vFET ; 6 nV/Vfii

vopamp ~ 40 nV/v/Iz vHvamp ~ 5 pV/vi/i

- 148 - giving vp ~ 175 ttV/N/i, dominated by the VFET term. Converting this noise to mass motion gives an xm of 10-16 cm//ilz at 5.6 kHz; 10-15 cm// Hz at 1.7 kHz. These are bothersomely high. The voltage noise on the plates can, however, be rolled off at high frequencies. For example, the addition of two poles of low-pass filter at 70 Hz only slightly effects the damping system performance (see Figure E.6) and reduces the 10-16 cm/v/Hz crossing frequency to 560 Hz, see Figure E.8. These poles also reduce the effects of the second noise source described above to less than those of the first for f > 700 Hz.

- 149 - 100 1000 10000 p- 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 I 1 1 1 1 - I I I IJ.A.J-LLAII CD ...... w ......

......

CU ...... rU

...... CD ...... L ...... Z ...... -GD ko ...... , ......

...... L ...... t...t ..

N CU ...... r ru -7- ;Oor-- ...... : ...... t... 0 OD ...... -- 4 ...... -CD ...... *f f T ......

...... i ...i-q ...... : ..... C 0 CU ...... : ..... -ro

0 Ln 1.-" ...... 1...... ; ..... CD 0 CD ...... -- 4 ...... OD ...... da; ......

CU ...... 7...... fu

...... CS) CD- ...... I ...... : ...... I ..... f .... 1: ...... f...... t ...... I ...... -QD ;.Z; ......

......

CU ...... t ...... T ...... ru

2 3 4I 5I 6' ' 7' ' 8' T9 I 2 3 4 5 6 7 8 9 100 1000 10000 Frequency (Hz)

Figure E.8 End Mass Motions Produced by some Electronics Noises

-150- E.5 Parameter Trade-offs The selection of the four parameters a, b, c, and WRC is made using the following approximate relationships:

WRC - mir imum frequency that can be damped

ab WRC - ma imum frequency that can be damped

b - gi es the relative amounts of VFET and VOPamp noise in the output

a -. c M- con verts VFET to noise on plate

ccx d Ma bwRC,1 - the mass damping is proportional to this

Now choose a, b, c, and WR by:

picking b so that VFET dominates (b=u.05)

pick WRC < 3 rad/sec (0.5 Hz) (WRC = 1)

pick ab WRC < 30 rad/sec (5 Hz) (ab==40, a=800)

pick c/bwoc to damp pendulum mode (c=50)

- 151 - Appendix F The RF Modulation/Demodulation Scheme and its Noise Sources

F.1 Overview The scheme described here is used to interrogate the interferometer fringe position through the use of a modulation technique. A block diagram is shown in Figure F.1. Sinusoidal RF drive is applied to the internal Pockels cells and thus modulates the output of the interferogram. As can be seen in Figure F.2, when at a minimum or maximum the result of the modulation is a signal containing even harmonics of the modulating frequency. As the fringe moves from the minimum or maximum, the amplitude of the even harmonics decreases and odd harmonics appear. In fact the amplitude of the component at the fundamental is proportional to the sine of the fringe phase shift from minimum. The interfering light is incident on the photodetector whose output is then filtered and demodulated to look at the fundamental and 2 nd harmonic terms. This scheme has the advantage that laser noise at the modulating frequency is the relevant noise source. The modulation frequency is chosen sufficiently high so that the laser noise has contributions only from shot noise. A variety of noise sources are present within this scheme. In addition to the laser shot noise are thermal noise from the photodetector load resistor, RF amplifier noise, mixer noise and any amplitude noise that might be on the laser beam at the modulating frequency. In general, the exact sensitivity of the demodulated signal to optical phase shift, as well as the importance of the various noise terms, vary with the interferometer contrast K and the amplitude of the phase modulation 6. By plotting the total noise vs. 6 it is possible to pick the 6 that provides the optimum signal-to-noise ratio.

F.2 The Modulation/Demodulation Scheme The power on the photodetector as a function of the relative phase of the two interfering optical beams is given by:

P(0) = Po - KPo cos! (F.1)

where Po is the phase-averaged power on the detector, and K is the interferometer contrast defined by:

K = Pmax - Pmin (F.2) Pmax + Pmin

Phase modulation of amplitude 6 and frequency wm is applied by the internal Pockels cells. The total phase between beams is then given by:

- 152 - PRotoccrej-nt PockeIs CQ-1IS RF Armpli{Ler1 F7I-QG GAp RL

Dov 6 ter(

DC ofJ R F

SvQ~ y- 0-) RGm&jk~~

DEC-' PoczJe Fe4cc

DLorn 0I RE 1OAU\&Lioy /De-vmoduvoator Sc~ei~ c6 e-m e- P

D (4 _L L-)

P,(I-K) ------> I tThr~e{~.c

Figure F.2 Operation of the Fringe Interrogation Scheme

0(t) = Osignai(t) + Omod (t) (F.3) = Osignal (t) + 6 sinlmt

and thus,

P(t) = Po - KPocos(signai(t) + 6sinwmt) (F.4a) - P0 - KPoJo (6) cosdsignai

+ KPo2J1 (6) sin qsignai (t) sin wmt (F.4b) - KPo2J2 (6) cos ksignai(t) sin 2wmt 3 4 + terms at wm, Wm, . - - where an expansion in harmonics of wm has been used. The Jn are the Bessel functions of the first kind. The amplitude of the sinWmt term is proportional to sin Osignal and, thus, serves as the measurement output. The power at the photodetector is converted to a current through the detector conversion constant:

CA/W = hv e Amps/Watt (F.5)

with r7 the quantum efficiency of the photodetector. For a quantum efficiency of one, every incident photon produces a single photo-electron. The photocurrent is converted to a voltage by the load resistor. The voltage is amplified, filtered, and demodulated both at wm and at 2Wm. The resulting

- 154 - mixer output signals are referred to as "the w (mixer) output" and "the 2w (mixer) output". The voltage at the w output is given by:

VW = 2KPoJ1 (6) sin qsignal CA/W - RL - GA - GM (F.6)

The sensitivity of the w output to a change in qsignal is maximum for Osignal = 0, in which case:

dV~ VPR =aO = 2KPoJ(6) CAw -RL- GA -GM (F.7) ddsignal

Note that VPR, "volts-per-radian", is simply the amplitude of the output sinusoid and can be easily measured.

F.3 Noise Terms The equivalent displacement noise for several noise terms related to the modulation scheme is calculated below. The noise voltage at the W output is calculated and then converted to a displacement through the VPR, the wavelength A, and the number of bounces in the delay lines Nb. For an output noise voltage vn, the equivalent mass displacement noise is: 1 A v, xm = (F.8) Nb 27r VPR Several noise sources are now treated in turn.

F.3.1 Shot Noise The shot noise due to the photocurrent is given by:

n = 2eI (F.9) where i2 is the noise power spectral density in amps/A/uz, and I is the time averaged photocurrent. This current is given by the DC terms of equation F.4b when Osignal is small:

I = (PO - KPoJo(6)) - CA/W (F.10)

The resulting voltage noise at the mixer output is given by: o- 2 - 2e(Po - KPoJo(t))CA/W - RL - GA - GM (F.11)v sb =

where the additional factor of two in the radical is due to the doubling of the effective bandwidth, as noise power at frequencies both above and below wm are combined in the output. The displacement noise due to shot noise is thus:

- 155 - ...... 0

E ..

(-4, 0

c-n

K~o.98

0.0 0 2040.6 0.8 1-0 1-2 1.4 1.61.8 Delta (radians) Figure F.3 Shot Noise Limit vs. 6 for several values of K

1 A V1-KJ o(6) hi Xm, shot - b 27 _ F.2 Nb 2ir KJ 1 (6) .Po

This is plotted in Figure F.3 as a function of 6 for several values of K. The desire- ability of high contrast is clear. For the ideal case in which K -- 1, the optimum 6 approaches zero. In the limit as their argument goes to zero, the Bessel functions are approximated by:

Jo(6)~ 1 - (6/2)2 (F.13) Ji (6)~ 6/2 and, thus, the ideal high contrast shot noise limited displacement noise is given by:

1 A 2h Xm, ideal shot - (F.14) Nb 27r 7lPtotaF where Ptotai = 2PO is the equivalent input laser power for a lossless interferometer.

F.3.2 Thermal and Mixer Noise Thermal noise of the input resistor, as well as noise sources in the following amplifier and mixer stages, generate a fixed, i.e., independent of K, 6, and PO, noise voltage at the w output. In the case of thermal noise and mixer noise (or input

- 156 - amplifier noise suitably referred to the mixer output), the w mixer output noise is given by:

Vtherm,mixer = V 4kTRL + Vamp GA - GM (F.15) and the equivalent displacement noise:

8kTRL + 2v2mp 6 Xm, therm,mixer 2KP0 J 1 ( )CA/WRL (F.16)

For these noise terms there is an advantage in using a large load resistor RL. How large an RL is practical depends on the limitations imposed by the photodetector RC time constant set by RL, and dynamic range limitations in the RF amplifier. In addition, once the thermal and mixer noises are less than typical shot noise, further increase of RL is unproductive.

F3.3 RF Amplitude Noise If the input laser noise at wm has a component that is proportional to the light intensity, i.e., laser excess noise or amplitude noise due to external phase modulation, this term will become important as 6 is increased. This noise is parametrized by vperw, the voltage noise at the output per watt of average detector power. The noise at the w output is thus:

vRFampl = vperw ' PO(1 - KJo(6)) (F.17)

and the displacement noise is:

vperw Po(1 - KJo(6)) xm, RFampl = 2KPoJ1(6)CA/WRLGAGm (F.18)

F.3.4 The Total Noise vs. Delta Plots The various noise contributions described above can be plotted as a function of 6. This clearly shows the important noise sources and the value of 6 that gives the lowest total noise. Two such plots are shown. The one for data taken on June 9, 1984, Figure F.4, has an important contribution from amplitude noise due to external wideband phase modulation being applied. For large 6 the amplitude noise term dominates but, for low 6, the thermal and mixer noises provide a floor. The second, from June 8, 1985, Figure F.5, has the amplitude noise term essentially reduced to zero because the use of pseudo-random phase modulation does not introduce a significant amplitude noise.

- 157 - 0 .0 0.2 0.4 0 .6 0.8 1.0 1.2 1.4 1.6 1.8 IIIIIIII III III IIIII-I 111 1.11 111111111 ...... - ID * ...... * ...... * ...... j ...... Lot In - ---.:, -- -.- - - ...... Sho:t No ...... sip ...... M i x'e r:* No Ise: N ...... RmpA I-tude: %...... Noise: ; ...... CD Q ...... v-4 ...... E ...... in - ...... 4-) ...... E

ftj - ...... * ...... * ...... ra

...... 0 0 ...... I ...... 0 ...... :- ...... &.- .- ...... ; ...... - ,j ...... :...... U1 ...... *0*4, 14044.0. 4. : .-. -......

...... % ...... cu - ...... 7 ...... -ru

0.0 0.2 0.4 0 .6 0.8 1.0 1.2 1.4 1.6 1.8 Delta (radians)

Figure F.4 Parameters are: Po 3.0 mW, Contrast 0.900 Amp. noise 3.00 microV/rHz per mW, Mixer noise = 1.30 mIcroV/rHz RRF = 240.0, GRF = 11.5, GMIX = 40.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ...... - ta ...... -OD ...... -Ij ...... -0 ...... %...... %...... Sho:t No:1 ...... sb ,...... x* e r: o i se: In N , ...... A mp:l itude N 1 *e: *...... L q-4 Go- ...... CA r, ...... E %0- ...... % ...... -Ij In - ...... 4--) t ...... -H M ...... E -r-4 oi- ...... %......

...... I-- ...... %...... 0 ...... z...... -Ij ...... -0% In - ...... %...... % ...... -CA ...... * ......

......

...... cu - ......

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 .6 1.8 Delta (radians)

Figure F.5 Parameters ore: Po = 11.5 mW, Contrast = 0.730 Amp. noise 0.00 microV/rHz per mW, Mixer noise = 0.75 microV/rHz RRF = 173.0, GRF = 5.6, OMIX = 35.0 Appendix G Driving the Internal Pockels Cells

The internal Pockels cells serve two purposes. First, they apply the sinusoidal RF phase modulation required by the fringe interrogation scheme. This is at 5.38 MHz, a frequency where the laser amplitude noise is shot noise limited. Second, a "DC" (DC to 50 kHz) feedback voltage can be applied to the Pockels cells as part of the loop feedback system that holds the interferometer to a null fringe. The RF voltage applied to the cells is typically 80 Vpp (6 = 0.9 radian) and the DC feedback is 75 V. In principle a single amplifier could provide these signals; however in practice seperate RF and DC amplifiers are used. The circuit described here combines the RF and DC signals and provides resonant step-up for the RF signal.

The circuit used is shown in Figure G.1. The resonant circuits, L 1C1 , are used to short the DC HV amplifiers at the RF signal frequency. Likewise, L 2C 2 , ensures a low impedence at low frequencies but has little effect on the RF signal. The circuit can thus be analyzed at low frequencies and at the RF modulation frequency. Equivalent circuits for these cases are shown in Figure G.2. At low frequencies the circuit presents a simple capacitive load to the HV amplifiers. The voltage on each Pockels cell can be independently controlled. The use of two HV amplifiers can reduce the effective Pockels cell noise voltage by 2, and halve the capacitive load seen by each amplifier, potentially increasing the useful bandwidth. At the RF modulation frequency the circuit can be considerably simplified because of the action of L1 C1 . Two stages of voltage step-up are employed. First, the drive voltage is increased by an RF transformer; this also provides isolation of the RF drive from the rest of the circuit. The voltage is again increased through a resonant capacitive step-up circuit. This circuit has two frequencies of interest:

fL= 1

22r 2/LR(CP1 + CP2 ) Cp C + Cp1 C 11 fH - 1 CR 1 1 27r/LRCR CP 1 + CI1 CP 1 + C 1 For the first of these, the step-up ratio is large and the input impedence goes to zero, both depending on the Q and sensitive to slight detuning. The second, at which the circuit operates, has a finite step-up ratio determined by the capacitor values and a large input impedence proportional to the circuit Q. The resultant step-up ratio is:

VP Cp + C1 1 2 (G.2) Vin Cr i + CP2

- 160 - Essentially all the RF input power is dissipated by the finite Q of the resonant inductor, LR. This leads to an input impedence on resonance of:

R Zin =NT L (G.3) 2 CR (1 + C11 +C2) CP1+CP2

For a typical Q value of 60 this is well matched to the 50 ohm load expected by most RF amplifiers.

OPI D C. I c Lp C1 11 Cz, L., T LI

N I Tc sA\-ir'z KF IN

Typical values for wm/27r = 5.38 MHz.

Cpj = 64 pf = 50 pf CP 2 CK -- 91 fi Ci' = 256 pf kce so0on<:.e

C1 2 = 200 pf LR = 9.6 pH LK R ~ 19 K-o C1 = 27 pf L, = 32.4 pH

C 2 = 330 pf L2 = 2.65 pAH qir

N = 13 Figure G.1 Pockels Cell RF and DC Drive Circuit

- 161 - c~L cP1 DCI. CT: C7T D C .- 4 C, ITT

Dc- ~~ C-'L(- C- U L-L

P c LR R*-S ;e t9 Ks-. PsI z ZLINE -- z'"60SL -N -9a-

RF L-(VOM ey t- C..'Lrcv Lt

Figure G.2 RF and DC Equivalent Circuits

- 162 - Appendix H The Loop System

H. Overview The interferometer is operated so that the output is held very near an interference minimum. This gives advantages of signal to noise, A/D dynamic range and rejection of low frequency input laser amplitude noise. A servo system has been designed to achieve this locking of the differential motion of the two arms of the interferometer to a fraction of a fringe. The fringe shift from minimum is sensed through the modulation/demodulation scheme described in Appendix F. Corrections to keep the phase at a minimum can be applied through the internal Pockels cells and through the electrostatic pusher plates to the masses directly. All three of these loop components introduce complications. Because the output of the w mixer is proportional to the sine of the fringe phase, highly nonlinear behavior results when the fringe phase deviates from null by more than -r/2. The Pockels cells have a finite dynamic range of about r/2; when their operation in the loop requires they exceed this range the system becomes nonlinear. Finally, the mass feedback force is limited and asymmetrically clipped and the displacement dynamic range falls as the square of the frequency. Because of these nonlinearities and limitations the loop system operates in two regimes: 1) The system is "locked" and linear analysis applies when the phase changes at the photodetector have an amplitude much less than 7r/2, the Pockels cells feedback is less than 7r/2, and the mass feedback voltages are unclipped. 2) The system is "unlocked" when, without or inspite of the loop's action, the fringe phase goes through many radians, or any of the signals become nonlinear, i.e., the mass feedback voltage clips. In either case the overall structure of the complete loop system is the same: it is made up of an "inner" Pockels cell loop and an "outer" mass loop. A block diagram of the system is shown in Figure H.1. The "Pockels" loop uses the w mixer output as an error signal and applies a "DC" (< 100 kHz) feedback voltage to the Pockels cells. This holds the interferometer output to a null. Because of the operation of the loop, the voltage applied to the Pockels cells is proportional to the phase shift between the two interfering beams, i.e., the relative displacement of the masses. This voltage can then be used to apply feedback through the electrostatic plate system to the masses and reduce the relative motions. This feedback constitutes the "mass" loop. Even though the Pockels loop serves as the sensor for the mass loop, there is little coupling between the loops; each can be considered separately.

- 163 - P ckvds Cclls H V 0 _Amp

M rxeFr;.* PF+ D- RFDr;vwe

DS 5

7 ?APTP Por-e-ts - Po e- s Loo,p HV Amp. -Z.G OuT Is>k5 Eoreiin Sysitxcn

" - Mc5 5 Loo o" p Fri o j x 10 COYY\ LY\SOPtr C ;rc- .?,00 Hiz Hs t-pss MC oYres5 L0A oe Fco r IW R s

systern Ei ur e. 1. i LF0FS7 H0L DC SPoc0 ' Y-) ,------~ -- -) -)mPExtb-rnoj M'fcos fyJ o

/o c- - - K

(sIs Ii~ C c Gmv,4

IA. ~. Tcc~f\S~~ Ky-c-LVJ-\s OdLA- V Vc-t P Loo~G pyi~'WS Table H.1 Parameters of the Loop System

General :

A - 514.5 nm - Wave length of the laser light Nb -56 - Number of passes in the delay line

Pockels Loop :

VPR - 55.0 V/rad - w mixer sensitivity Gp - 1.0 - Gain setting on Pockels compensation box 1 sp - 27r x 194 sec- - Pockels compensation pole GPHV - 10.3 - Gain of Pockels HV amplifier )3 - 8.9 x 10-10 m/V - Coefficient for a single Pockels cell

Mass Loop

sfrn - 27r 1.65 kHz - Fringe circuit pole GFRN - 0.033 - Gain of the fringe circuit GRKN - 17.8 (25dB) - "Red knob gain", sets the mass loop gain GATT - 0.82 - Due to finite output Z of compensation box GMHV - 23.5 - Gain of the mass high voltage amplifiers a - 2.2 x 10- 5 kgm - Driver plate coefficient, force/volt M - 8 kg - Mass of an end mass WO - 27r x 0.44 sec-1 - Pendulation frequency of an end mass Q - 2.0 - Q of the damped pendulum mode

- 166 - H.2 Linear Analysis of the Loop System The feedback systems are presented in the Bode analysis formalism; each component of the loop is characterized by a transfer function as a function of complex frequency, s =iw. The gain and phase of the open loop transfer functions are of particular interest. Figure H.2 gives the system block diagram with the transfer functions of the components indicated. Table H.1 presents typical values of the loop parameters for this model.

H.2.1 Pockels Loop The Pockels loop consists of the RF w mixer, the Pockels feedback gain and compensation box, the Pockels "DC" high voltage amplifier, and the two internal Pockels cells. The w mixer output is characterized by its sensitivity to optical phase shift, given by VPR, in units of volts per radian (equation F.7). ihe Pockels cells are characterized by a coefficient 3, which gives the change in optical length per applied volt per cell. The high voltage amplifer, which applies feedback to the Pockels cells, has a gain GPHV. The mixer and the high voltage amplifier have flat frequency response to 20 kHz and, thus, the loop transfer function is determined by the compensation. The Pockels cell loop is compensated by a single pole at 194 Hz. The open loop gain of the loop is given by:

Pockels o.l.g. = VPR - TPLC(s) - GPHV ' 2 7 (H-1) A

with TPLC(s) 10GP 1+ s/sr

and s, = 27r x 194 sec- 1

This is plotted in Figure H.3 for typical operating values of VPR and Gp. As shown, a fairly high unity gain frequency is achieved, about 20 kHz. This is limited primarily by the Pockel HV amplifier; the loop oscillates for unity gain frequencies above 50 kHz. When in lock, i.e. with the pockels feedback phase within r/2, this loop holds the interferometer phase to within (7r/2)/(Pockels o.l.g.) radians of null. Of interest to the mass loop is the transfer function from interferometer phase shift to the voltage at the Pockels high voltage amplifier input. This is given by:

VPamp.in _ VPR - TPLC (s) Obefore pockels 1 + Pockels o.l.g. For frequencies below 1 kHz, of importance to the mass loop, this is effectively independent of the Pockel loop gain and compensation, as long as the loop gain is high. In this case:

- 167 - 0.0 ...... Ln LUJ LUJ -3. 0 ...... 0 -4 ...L.10 ......

LUJ ...... U-1 a: .0 1 ...... N......

-2 .0 -1.0 0.0 1.0 2.0 3.0 L.O 5.

FREQUENCY (LOG(Hz))

q-4

-4

...... LU

=1 -4 -J

-1.0

-2.0L -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

FREQUENCY (LOG(Hz)) Figure H.3 Pockels Cell Loop-Open Loop Gain

- 168 - Pe VPamp.in 1(H.3) VPReff - Obefore/pockels GPHV (H.3)

This allows the gain and compensation of the two loops to be treated as essentially uncoupled.

H.2.2 Mass Loop Because of the limited dynamic range of the Pockels loop alone, r/2 of optical phase, and the large motions of the masses at low fequencies due to ground vibration coupling, additional feedback must be applied to hold the interferometer to a fringe. This is accomplished by appling a force directly to one of the end it-asses. Mass feedback is applied directly to the mass through a set of two electrostatic drive plates. These drive plates are driven by high voltage amplifiels whose output is a DC bias level plus the input multiplied by a gain. Changes in the voltage about the bias point can effectively push or pull the mass. These >ame amplifiers and plates are used by the damping system to damp the pendulum and "rocking" modes of the mass; the matrix box serves to sum the damping and loop signals and apply them to the high voltage amplifiers. The matrix box input "Vxptp" is the input for the mass loop feedback and applies a common mode signal to the two HV amplifers with a gain of one. The basic transfer function of the mass loop is given by the transfer function from a voltage at Vxptp to the resulting motion of the mass, in units of meters per volt. The resulting transfer function is given by:

TM(s) = 2 GMHV M S2 + Wl

This transfer function is shown in Figure H.4 for the parameter values as tabulated. Though modeled as a simple pendulum response, there is suspicion that resonances associated with the suspension system affect the exact TM(s), introducing resonances and non-1/f behavior. In order to have a well behaved servo loop, compensation is introduce into the loop. This compensation is the sum of three terms. Because the basic transfer function approaches 180 degrees above the pendulum resonance, phase lead compensation (differentiation) is introduced into the loop at these frequencies. Below resonance phase lag (integration) can be used to increase the very low frequency gain. Finally, a constant term is used to approximately match the Q of the compensation to that of the pendulum at resonance. A schematic of this compensation is given in Figure H.5. The resulting transfer function is plotted in Figure H.6 and is:

- 169 - .I ...... U.'

......

U.' 'i

-0.

-1.0 0.0 1.0 2.0 3.0 '. 5.0

FREQUENCY (LOG (Hz))

4La ......

-1.0 ...... 7 , . .*......

......

...... -2.51 .1

......

~ ~L.

0 -1.0 0.0 1.0 2.0 3.0 L.O 5.8

FREQUENCY (LOG(Hz)) Figure H.4 Mass Loop-Basic Transfer Function, TM(s)

- 170 - 91K

394~C I K.B

0.0 clayo

3 1 n koo 0 zo qe 3lo

(Au 0.C

0.Lf In LUI LU 0.2 1......

0.0 LUI ...... -0.2 - - - . -. U-I

-2.0 -1.0 0.0 1.0 2.0 3.0 t.0 5.0

FREQUENCY (LOG(Hz))

B. 0 1

2.0

.5 -......

0.0 ......

.0 -1.0 0.0 1.0 .0 .0 .0 FREQUENCY (LOGCHz')

0 Figure H.6 Transfer Function of Mass Compensation, TMLC(S), RKN -1.0

- 172 - TMLC (S) +s/(27r x 0.092) 1 + s/(27r x 74) 1+ s/(27r x 2.9) 1+ s/(27r x 498) + 0.36 (H. 5) 1 + s/(27r x 2.65)_ 1+ s/(27r x 0.053)

The only other frequency dependent component in the mass loop is a single pole at 1.65 kHz in the fringe circuit. Details and a schematic of the fringe circuit are presented in Section H.3 describing the nonlinear operation of the loops. The effect of this pole combined with the compensation transfer function is shown in Figure H.7. The transfer function for the mass loop open loop gain is given by:

2ir Mass o.l.g. = VPRff -TFRN(s) ' TMLC(s) - GRKN - GATT T M(s) - Nb - (H1.6) and is plotted in Figure H.8 for the typical parameter values. The DC gain of the mass loop is - 10 and, as operated, the unity gain frequency is a 200 Hz. The open loop phase crosses -180 degrees at ; 680 Hz. In operation the loop oscillates at a 500 Hz when the loop gain is increased. The model would suggest this is due to the fringe circuit pole pushing the loop phase close to -180 degrees, however resonances in the suspension may also be responsible.

- 173 - 0.G

In 0.2 .--- - . ..------. - -

0.0

-0.2 LUJ

U-I

-1.0 .0 -1.0 0.0 1.0 2.0 3.0 L.0 5.0

FREQUENCY (LOG(Hz))

-0.5 -.-- -. -...... -1.0

------1.5 - - -

......

*......

-. 0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

FREQUENCY (LOG(Hz))

Figure H.7 Effect of the Fringe Circuit Pole. TMLC - TFRN . GATT

- 174 - 2.0 K 1.5 -

U, 1.0 1 1 0.5

0.0 LU.

-0.5 U, C -1.0 ...... -1.5

-2.0 .0 -1.0 0.0 1.0 2.0 3.0 .0 5.

FREQUENCY (LOGrHz))

I -~

LUJ 0c.01......

- -

- 1.5 -2 . -1.0 0.0 1.0 2.0 3.0 .0 5.0

FREQUENCY (LOG(.Hz)) Figure H.9 Mass Loop-Open Loop Gain

- 175 - H.2.3 Transfer Functions of Interest- Signal and Noise in the Loops Given that the loops are operational, an output signal must be chosen. In addition, the transfer function from mass motion to this output as well as the effects of noise sources on this output must be evaluated. Two signals are particularly suited for this purpose, the error signals for each of the loops: the w mixer output Ve, and the Pockels amplifier input voltage VPamp.in. The transfer function from mass motion to each of these is given by:

VW N 2r VPR A (H.7) xm (s) = 1 + Mass o.l.g. 1 + Pockels o.l.g.

VPamp.in -VW - (s) - TPLC (s) (H.8) Xm Xm These are plotted in Figures H.9 and H.10. The main difference in shape between these is in the region of frequencies above 300 Hz, i.e., when the mass loop gain is less than one, and the Pockels loop gain is no longer constant. Because noise spectra typically rise towards lower frequencies, the w mixer output is favored for A/D dynamic range reasons. When, ideally, the noise spectrum is flat, the VPamp.in output is preferable. These transfer functions are used to correct the observed voltage spectra and obtain the correct equivalent mass displacement spectra. For the data taken with the instrument, the 2 x Gp output of the Pockels compensation box is used. This output has the same mass motion to voltage transfer function as the V, output with an additional factor of 2 x Gp. Because GP is chosen to keep VPR x GP constant for different light levels, modulation amplitudes, contrasts, etc. the transfer function to the 2 x Gp output (and the VPamp.in output as well) is constant and the A/D system gain does not have to be adjusted, once set. Some of the possible noise sources in the loop system are indicated on the transfer function block diagram. Mixer noise, from the RF fringe interrogation scheme, includes laser shot noise as well as photodetector load resistor thermal noise, RF preamp noise, and actual mixer board noise. For typical parameters (2PO = 23 mW, K = 73%, 6 = 0.9 rad) vmixer = 1.2 pV/v Hz, 0.95 from shot noise and 0.75 from all other sources. The remaining three sources shown, vpamp, vcompl, and vcomp2, are amplifier noises associated with the indicated components. The effect of the noise terms can be evaluated by calculating the transfer function which gives the equivalent mass motion per volt of noise. These transfer functions depend on the particular noise source as well as the chosen signal output. Table H.2 gives the results for the above mentioned sources and outputs. Measured values of these noise sources under typical data taking conditions are used in Table H.3 to calculate their effect on the measurements. The ideal shot noise limited noise level of 2.4 x 10-17 m/v/Hz is increased to 3.8 x 10-17 m/ Hz, primarily by the non-shot mixer terms and the Vcomp2 noise term.

- 176 - LU .

0LU.. .

U1.

f . . N ...

0 . 1 ......

0.0 . -. 0 -1.0 0.0 1.0 2.0 B.0 4.0 5.0

FR.EQUENCY (LOG(H:-))

......

1 ......

I...... 1.8

zRQEC-i. 0.0 .0 (LGH7).2.0 . . 4... Fiur H.1rnfrFncinfo.as oint ,

- 177 - 2.0

1.5 LU LU LUj 1 . S

---- ...... ---; ...... L 0.5 U.1 =m 0.0,

-2.3 -1.0 0.0 1.0 2.0 B.0 4.0 5.0 FRElUENCY (LOGiH))

*1 C'

'1 .* . . . .

1 L4...... 1 . . LU .

*1 *~i CL- C.

1.0~

0.8 -2.0 -1.0 0.0 1.0 2.0. 0 0 C.O 5.0 FREQUENCY (LOG(Hz)) Figure H.10 Transfer Function from Mass Motion to Vpamp.in

- 178 - Note that the effect of a noise term can depend dramatically on its location in the loop and the chosen output. The VPamp.in output has the least sensitivity to amplifier noises. In fact, even the Pockels amplifier noise can be overcome by taking the output at the Pockels cell drive voltage itself with a suitable low noise amplifier. These avenues have not been pursued because there is little to be gained (3.9 to 3.2) at present light levels and there is much noise above shot noise at all but the highest frequencies.

- 179 - P-Yi fA-t

Mvw Va.Gp L\IOL5-

Sour ML

r'Le-r VPR NL ,

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P rw\ r,

-Tx I -\ .JL lcO\Yr\s S-f- F u (jLr.\ -1 'osSCo 0 v V) C" Q NI lS L s #0 o EQlv ,

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C o w 7L 0 Y\-5 0 f'o- ry\ C J Q- ),ru v 0, b r I kW~ ( LtVAAuSS

(~~so I-]~ N) 'I L+ 4/A4 1 ,Z7-- I Ty b ,

T 15i - -o\s t Ss 4\Vb1 'PD-Z- O-S (, 0 YV\T UAS VA+ 6 Y) U

T ie H.3 A- 5 ir To,~ -D w4j L L \4 v R---5 V !5 H.3 Nonlinear Operation of the Loop System

H.3.1 The Fringe Circuit Before getting locked in the linear regime, it is neccessary for the loop circuits to be able to achieve lock inspite of large initial mass motions. The problem in this case is the cyclical nature of the w mixer output with mass motion which leads to a similarly non-monotonic signal at VPamp.in. The feedback applied to the mass is thus useless. To get around this the "fringe" circuit of Mark Herald (Hereld 1984) is used to produce a monotonic voltage vs. position signal from VP-amp.in when the motions exceed a fringe. Though the output is not ideal, it none-the-less is able to damp the mass to the point where the motions are within the linear range. The circuit as implemented has the added advantage of requiring no external decision making or changes of wiring as it operates. The circuit used is show schematically in Figure H.11. This is the Hereld circuit with slight modifications as indicated. The operation of the original circuit is explained with reference to Figure H.12. The "w" input is available with a gain of +1 and -1 and the sign of the 2w input switches which gain is applied to the output summing amplifier. The series capacitors and grounding switches ensure that at each 2w transition the voltage stays where it is. The effect is to remove the discontinuities from the w - sign2w wave form. The resulting signal is monotonic in fringe phase. This signal can be used to damp the masses. However, with the mass loop gain alone, the size of the ground noises present are such that excursions from null are large enough to couple low frequency laser amplitude noise into the output. The Pockels loop is introduced to increase the effective gain of the loop holding the w mixer output to a null. In the frequency range of most disturbances (< 200 Hz) the Pockels loop gain is constant and about 120. The operation of the Pockels loop severely distorts the w and 2w mixer outputs as shown in Figure H.13. However if the Vpamp.in signal is used as the "w" input to the fringe circuit, the output shown is obtained. Two slight modifications are made to the circuit, both due to the fast edge in the Vpamp.in signal. First, during one phase of the 2w signal the output is held fixed, this prevents the edge from appearing at the output. Secondly, the 2w signal is delayed slightly to ensure that the edge occurs during the ignored phase. When locked, the fringe circuit has a fixed gain of 1/30 and, because of the 910 ohm resistor in series with the 0.1 /f capacitor, a single pole at 1.65 kHz. This pole frequency could be moved to higher frequencies by decreasing the resistor value. The resistor is present due to paranoia.

H.3.2 Motions that cause Loss of Lock Given the operation as described above it is possible to quantify the kinds of motions needed to "unlock" the loop system. Unlocking occurs when any one of the signals saturates. Note that unlocking does not, typically, destroy the loop

- 182 - IOKK

s A

3.0

Fr H

Figure H.11 The Modified Hereld Fringe Circuit

- 183 - W, ~7 U-L /000, N

Z000-- Ll L/

c fCv Lt Octvt

Figure H.12 Fringe Circuit Operation a la Hereld

- 184 - ~~rr\,~$ \T :4V~~ ~ cL-i.

r NO

%/ %4

Pck. Lfm

C- V it

CFf-A~~

(A.)r-\

cu'v

Figure H.13 Fringe Circuit Operation with the Pockels Loop

- 185 - operation; rather it indicates periods of non-linear behavior. The limits most likely to be encountered are imposed by the feedback amplifiers: the mass HV amplifier and the Pockels HV amplifier. The Pockels cell amplifier has a limited dynamic range of about 75 V. This corresponds to approximately ir/2 radians of optical phase shift. The size of the signal on the Pockels cells when the loop is locked is given by the unlocked mass motion divided by the mass open loop gain. Thus the mass open loop gain determines, with the Pockels cell dynamic range, the size of motions that will saturate the Pockels amplifier. In Figure H.14 is plotted the maximum amplitude of mass motion allowed as a function of frequency, given by:

7r A 1 xmax = Nb . (1 + Mass o.l.g.) (H.9) 2 27r Nb The mass high voltage amplifier has a linear dynamic range at low frequencies on the order of 200 V. In the low frequency range where the mass loop gain is high this can be related to the maximum allowed mass motion by:

xmax = 2- 2 + QsM +wO2200 V (H.10)

This is plotted in Figure H.15. In addition to the hard clipping of the mass voltage, it should be pointed out that the force applied by the plate is proportional to the square of the plate voltage. Thus for large voltage excursions the feedback signal is not symmetric for a symmetric mass motion. For example, if the plates are biased at 400 V, a symmetric mass motion might produce a feedback waveform going from 200 V to 530 V, i.e., +130 V and -200 V. The resulting distortion will produce false signals at the harmonics of the actual motion.

- 186 - -0. G 1 -3 VO

-o

0 ...... -GI 1.2 - to

1-4 I 4t

-1.8 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

FREQUENCY (LOG(Hz))

Figure H.14 Mass Motion Limit due to Pockels HV Amplifier Range

-0.5 Co ...... X.~ - . -...... - ......

-1.5 0

9-4 -2.8 I X No-(% W\

-2.5 -- -- .-.. -- - -.. . . -- -. . -...... -- ......

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

FREQUENCY (LOG(Hz)) Figure H.15 Mass Motion Limit due to Mass HV Amplifier Range

- 187 - Appendix I Attenuation of Scattered Light Noise with Phase Modulation

1.1 The Problem Scattered light noise has been observed in a variety of precision laser interferometric instruments, in particular those designed to detect gravitational radiation. A method to diagnose and partially eliminate the effects of scattered light was described by Schilling et al. 1981 (MPQ81). Their technique of applying sinusoidal external phase modulation to the input beam of the interferometer succeeds in eliminating the effects of scatterers located at certain well-defined distances. However, it does not eliminate the effects of all scatterers spaced at multiples of a single distance nor, in general, more than two unrelated scatterers. In order to overcome these limitations the two techniques described here were developed; each attenuates scattering effects from a continuum of scatterer path lengths. Scattered light noise refers to the signal produced at the output of an interferometer by stray, coherent beams which interfere with the main beams. Often the phase of these undesired beams is time-varying due to relative motions of the scatterer and the main optics and thus gives rise to a time varying "noise" at the output. The schemes to combat this noise take advantage of the fact that the scattered light has taken a path different from that of the main beams of the interferometer and that this difference can be large: 1/AT = c/Az < 100 MHz. Thus it is possible to modulate the phase of the scattered beam relative to the main beams D by impressing phase modulation on the input laser beam 0m:

$(t) = qm(t - Ar) - om(t) (I.1) Pictorially, as described in MPQ81 and shown in Figure I.1, the scattered light vector of constant amplitude is rotated in phase by the external modulation. The goal is to choose the external modulation such that the vector's time average is zero over times short compared to the measurement time. In this way the scatterer's power is translated to frequencies away from those of interest. (Note that the philosophy here is opposite to standard modulation/demodulation schemes in which the signal is modulated and the measurement made at the modulation frequency; here the noise is modulated and the measurement made near DC.) The time average of the unit scatterer vector is given by the two integrals

T +00 S(Ar) = 1fsinD(t)dt = J sino P(4) d4 = sinD 0 -00 (1.2) T +00 C(AT) = 1Jcos (t)dt = J cos D P(f) dD = cos 4 0 -00

- 188 - E~

LKtt )V4,&5 _C. _ RrI

Figure 1.1 Effect of a Scattered Light Field on the Main Beam Phase where P(4) is the probability density of D. The magnitude of the remaining scatterer component at "DC" is given by -/S 2 + C2 ; ideally 0,m(t) is chosen such that S = C = 0. The action of a candidate phase modulation can be displayed by plotting the probability density of 4 for a given amplitude of modulation and scatterer delay and also by plotting the attenuation in dB, 20logio /C 2 + S2 , as a function of scatterer delay, Ar.

1.2 Single Tone Modulation For comparison, the original modulation scheme of MPQ81 is described briefly. The modulation is taken to be sinusoidal :

om = ti sin(Wmt) (1.3)

and thus

~(t) = m(t - Ar) - Om(t) = 2/-t sin(wmAr/2) cos(wmt) (1.4) = 1 '(Ar) coA(wmt)

where st'(Ar) = 2ptsin(wmAr/2). The integrals then become:

'2r/Wm S (Ar) = sin(IL'(Ar) cos wmt)dt = 0 27r I 0 27r/wm (1.5) C(Ar) = J cos (pt'(Ar) coswmt))dt = Jo(iz'(Ar)) 0

Thus scatterers with Ar such that ji'(Ar) is a zero of Jo are completely eliminated; their energy now appearing around Wm. Scatterers with other values of Ar are

- 189 - 1.5 i .1

[A s 1-3 8 T '/3 ...... 1.0

;4: v7 ...... 0.5 ......

......

s O.S 1.0 1.5

scn t tERF-R PATH OELAT METERS)

00 0 .0 ......

ID ......

-2 .0 10 ......

13 -3 .0 .jc ...... zi Z' ko -4 .0 ......

-9.0 -1.5 4.0 1.0 I.S 0.9 0.2 9.4 0.9 6.8 1.0 1.2 1.4 1.6 1.8 r.0 Sln(phl) ,O-tsSCATTEREP PAT" DELAY METERS)

F'L 2- S , , ,As t, - A a J r, a u c)n, : Eff, I on onJ Sco, A+ -Y) kf o,t Lo Y\ attenuated to varying degrees. The phase probability density as well as the attenuation versus AT are plotted in Figure 1.2 for this modulation. Note that the whole attenuation pattern is periodic in Ar with the period of the single tone modulation.

- 191 - 1.3 Gaussian Noise Modulation The desire to attenuate scatterers independent of their Ar suggests using a on whose properties are independent of t. Noise sources are an obvious candidate and of interest is their probability density, P(rm). For Ar greater than Tr, the correlation time of the noise, the probability density of D is given simply as:

P(4) = P(Om) convolved with P(m) (I.6)

One way to ensure a zero value of the averages S and C is to generate a P() which is uniform on the circle. It is quite remarkable that if P(D) is Gaussian, the resultant distribution, as it "wraps" around, approaches a uniform distribution quickly and monotonically as the Gaussian width /ct is increased. Explicitly:

P (4 m) = 1 e_ 0/2J12 (I) P21 ep where , is the rms phase modulation. Then for Ar > r, we have:

P(@) = 1 e 2 /42 (1.8) 27ry The S and C averages are then :

+00 S(Ar) = sin P(4) dd = 0 00 (1-9) +00 C(Ar) = cos D P(D) d =e-A2

-00

Thus Gaussian white noise can very effectively attenuate scatterers at any location provided only that Ar > Tc. Figure 1.3 shows the distribution of (D for two values of I; the approach to uniformity is quite striking. This type of modulation was first considered from the perspective of its effect on the laser line width. Analysis (Weiss 1982) shows that the laser carrier is suppressed by the same e-" 2 factor and the remaining power spread out over a broad bandwidth. This method is equivalent, then, to using a very broadband light source.

- 192 - 1.5

. .- 1

-5.5-

;td-er -

-1 S -i.B -u.S g.g 0.5 1.0 1.5

Sin(hi) c)

15.31 ROd. per V 3.02 S.: Ove = -e.0!, 0.064 V-jno =O.O

1.5.

* .. y*--

-1 . - :0 -0 . 0 0 .5 I. .

SinCph i) c

Dl - = 15.31 nod. pe, V ga , ave = -003 e~g -o - ~g

Figure 1.3 The Effect of Gaussian Noise Modulation Distribution of scatterer vector for a) A = 0.85 and b) pt =1.70.

- 193 - 1.4 Pseudo-random Digital Modulation In the MPQ81 paper the seed for the digital scheme presented here is given: "If one, for instance, could achieve that the scattered light vector for half a cycle points into its original direction and for the other half of the cycle into the opposite direction" then the averages S and C will vanish. Such strict alternation carl be achieved for one specific scatterer by applying square wave modulation of peak-to- peak amplitude 7r/2 with a period 2Ar. However, if Om is chosen randomly every Atd to be either 0 or 7r then, for Ar > Atd, P has the following distribution:

P( = 0) = 1/2 P(4 = ir) = 1/2 and thus on the average the desired condition is met, and S = C = 0. Such a scheme still modulates the crystals with broadband noise and the next step is to choose a periodic pseudo-random Om(t).

For the case where qm(t) takes on only the values of 0 and 7r the S and C integrals can be simplified:

S =0

-T C T cos(qOm(t - Ar')) cos(Om(t)) dt 0 If we now consider the sequence a(t) which takes on -1 and 1 when 0m(t) takes on 0 and ir we have :

T C = a(t -Ar) a(t) dt = A(AT) (I.12) 0 and the autocorrelation function of a(t), A(AT), determines the amount of scatterer supression. There is a set of pseudo-random sequences, a(t) with period p = 2N - 1, which have very well-defined randomness properties (Golomb 1967). In particular their autocorrelation function is given by:

1 - (1 + 1/p)[ ____A - np] np < ArdAT < np + A(Ar) = -1/p np +1< A

with n=0,1,2,... (I.13)

- 194 - Timie C10 ld g -urits)

-......

e...... 2 . . .. .2 .5 .. . .

SCATTER!R PATH DELAY CMETZRS)

Figure 1.4 Digital Pseudo-Random Noise a) shows two cycles of pseudo-random noise with p = 31 b) shows its autocorrelation function c) plots the scatterer attenuation versus Ar - 195 - Two cycles of a sequence with p=31 are shown in Figure 1.4 along with its autocorrelation function and the corresponding attenuation versus AT. Because the autocorrelation function is independent of AT in a wide range, this modulation will attenuate most scatterers by a factor of 1/p. In the special case of a set of scatterers spaced at multiples of a single Ar, on average a small fraction, a 2/p, of the scatterers will not be attenuated by the full 1/p factor.

1.5 Experimental Details and Results The apparatus used to produce a phase modulated beam is shown in Figure 5. The laser light is passed through two LiTaO 3 Pockels effect crystals, each with dimensions 1 mm x 1 mm x 25 mm, by a pair of lenses. This arrangement gives a total phase shift of ir for 43V applied to the crystals. An RF amplifier (ENI model 525LA, 1-500 MHz, 50 dB gain, 25 Wrms) drives the crystals and a termination load through a 50 ohm coaxial cable. The crystal holder was designed to have low stray capacitance and inductance and presents a simple capacitance to the circuit at RF frequencies. System roll-off occurs at f ; 250 MHz primarily due to the capacitance of the crystals. The noise sources used to generate random and pseudo-random noise are shown in Figure 1.6. Band-limited white noise is generated by a reverse biased noise diode and amplified by standard RF components. The noise source is flat. from 0.1 MHz to ; 500 MHz. The pseudo-random sequence is generated by a slLi1 register with feedback (Damashek 1976) implemented in Schottky TTL and runs to clocking frequencies of 50 MHz (Atd = 20 ns). The phase modulator was used to modulate the input light of the MIT prototype gravitational wave detector. This detector is a Michelson interferometer with multi-pass delay lines in each arm, very similar in design to that described by Billing et al. 1979. The delay line is 1.46m long and currently set for 56 passes giving a two pass time of 9.7 ns and a complete round trip time of 270 ns. At present the scattered light noise is the dominant noise in the instrument at frequencies above 2.5 kHz. Single tone modulation proved ineffective, presumably due to the variety of scatterers in the delay line system. Shown in Figure 1.7 is the effect of Gaussian white noise on the output spectrum of the interferometer in the frequency band dominated by scattere.id light noise; the e- 2 dependence is demonstrated. (For this demonstration the scattering has been accentuated through deliberate misalignment of the optics.) The Gaussian modulation works very well but has suffered froi two drawbacks. First, the application of broadband noise to the phase modulatinig crystals can excite a variety of non-linear responses producing amplitude noise as well as phase modulation of the beam. This amplitude noise term exceeds the laser shot noise floor and is thus the dominant noise term in the system. Second, the optical system which follows the modulator must be free of reflections, stray beams, and birefringent materials as these can all convert the impressed phase noise into amplitude noise.

- 196 - oL'sd vo L t TsD3

R F Am p.

Figure 1.5 Phase Modulator

Ap,mP at 50MH > GovsLf

Figure 1.6 Sources of Phase Modulation

- 197 - 0 = . ------. -- - -- .....- -..--...... %1...... -7 - -7- .---8......

. ... . *19.9 ......

t.4 2.C 2.8 3.1 3.2 3.9 3.6 3.8 1.8 41.2 4.9 .S iKHZ Frjue-rc-y, LD.o He iOKH0

Figure 1.7 The Effect of Gaussian Modulation on the Antenna Spectrum

Curves from top to bottom: No Modulation e- 2 = 1/2 =- -6 dB e-A2 = 1/3 ;- -10 dB e-A2 = 1/12 =- -21 dB The straight lines are separated by the predicted amounts of attenuation. The noise floor is due to amplitude noise generated in the phase modulator.

- 198 - -9.9 -

-g .5 ---- -:------:----- .---- -:------:------.--- - ..-- . ------:----.. c.J ...... 0

0 -J

C. 0 ......

-11.5

-13.0

-11.0

- 2.6 2.8 3.8 3.2 3.11 3.6 3.8 4.8 4.2 4.11 4.6

Ais, Fr-eyncq, Lo5io H-F

Figure 1.8 Effect of Digital Modulation on Antenna Spectrum

Upper curve: No modulation Lower curve: period 15 modulation Straight lines are separated by the factor 1/15.

- 199 - These are not fundamental limitations of the scheme but have prompted a search for a periodic and yet broadband scheme. The results of applying the digital psuedo-random modulation are shown in Figure 1.8 for a digital period of p = 15; the observed attenuation agrees with that expected. Though the pseudo-random technique has performed as expected it puts a high demand on the RF amplifier. In order to generate a clean square wive the amplifier must have response to 3-5 times 1/Atd, and good phase response at L(w frequencies. Thus, to use a sequence of length p, an amplifier must be rated froi I/(3pAtd) to 3/Atd. Our amplifier and HF roll-off limit us to p=31. The techniques described here do allow a continuum of scatterers to be attenuated. The various schemes each have their own advantages and disadvantages. The properties of the schemes are summarized in Table I.1.

Table 1.1 Comparison of Scattered Light Schemes

Single Tone Gaussian Pseudo-random

Range of Ar Discrete All Ar> c np+l < Ag

Amount of Complete exp(-pt 2 ) 1/p suppression

Scatterer at Wm from DC to at Harmonics noise appears BWeffective of 1/(p - Atd)

Amplitude of p > 1.20 y = 2.15 y= 1.57 modulation for -40 dB for all p

RF power > 5.4 W 17.3 W 9.2 W

- 200 - Appendix J Derivations Related to the Matched Filter

J.1 The Matched Filter gives the Optimum SNR A justification for the fact that the matched filter gives the optimum SNR can be made by considering the signal and template below (this "derivation" is that given by Baldinger and Franzen 1956):

Signal 0 0 00 0 0 --- O

Template 0 - - - b 0 0 0 0 00a --- O

What value of b maximizes the SNR for detection? When the signal and template are aligned the output of the correlator is:

S = 1 x 1 + axb (J.1)

Now, assuming there is no signal but only noise which is white (i.e., uncorrelated from sample to sample) and has an rms value a per sample, the rms noise out of the correlator is:

Noise = v/(i x a)2 + (b x a) 2 (J.2) where the fact that the variance of the sum of independent random variables is the sum of the variances has been used. The signal to noise ratio is thus:

SNR 1+ab (J.3) aV1+b 2 Requiring that the derivative with respect to b be zero, i.e., finding the value of b that maximizes the SNR, yields b=a . Thus, the optimum template is the matched template.

- 201 - J.2 The SNR Formula for Digitized Signals Before deriving the SNR formula it is useful to introduce the concept of a normalized template (VanTrees 1968). For the case of a digital template the normalization requirement is:

Nt 2i = 1 (J.4) N The normalized template has two useful properties: first, the rms value of the output noise is equal to that of the input noise, provided the input noise is uncorrelated between samples; second, if a matched signal is A times the template and is detected by the template, the output is A. These results can be easily shown:

U-out = Z(tirin)2 = Ocin V>tL = Uin (J.5) Correlator output = (ti - Ati) = A tI = A (J.6)

Now, assume that white noise with a single sided power spectral density of VN is sampled at a rate 1/tsample and is band-limited at the Nyquist frequency, 1/2tsample. The rms noise of each sample point is:

1 U = VN (J.7) 2 tsample

This is also the noise out of a normalized template, ti. If a matched signal, Ati, is detected the output is A and the SNR is:

2 = A V tsample SNRSNR=A (J.8)

The "energy" of the signal is:

2 Es - (Atj) tsample = A 2 tsample (J.9)

Solving this for A in terms of E. and tsample and substituting it into equation J.8 gives the desired result:

SNR = 2E2 VN

- 202 - J.3 Calculating SNR/SNRopt for Mismatched Templates The reduction in SNR due to a mismatched template can be calculated from the normalized form of the template and signal (note that here the template and signal waveforms are interchangeable). For continuous signals the normalization condition is: I U 2 (t) dt = 1 (J.10) and the reduction in SNR is:

SNR U (t+ T) dt (J.11) SNRoptimum 2 where the value of T is chosen to maximize the ratio; for many types of mismatch the value of T is evident from inspection.

J.3.1 Mismatch of the Component Pulse Shape The effect on the SNR of an error in the shape of a component pulse of a signal can be demonstrated by comparing several types of shapes. Shown below are three possible pulse shapes, their normalized equations, and the resulting values of the overlap integral, equation J.11.

{ J2sinx O

- 203 - J.3.2 Mismatch in the Number of Component Pulses If the template and signal differ in the number of component pulses a reduction in SNR occurs. Consider the waveforms shown below; NHC speciries the number of identical component pulses in each waveform.

NHTC- U' larger

. U 2 N11C~1 1 ~ NJ -j 'sm1ller

Define integrals over the time of one component pulse:

I = U (t)dt; 12 = f U2(t)dt; Ii,2 = Ul(t)U 2 (t)dt (J.12) a pulse a pulse a pulse

Then the reduction in SNR is given by:

SNR f U1U 2dt NHCsmallerI1,2 SNRoptimum f Uldt f Ujdt /NHClargerIli NHCsmallerI2 (J.13) NHCsmaller 11,2 NHClarger VII2

The first term above expresses the reduction in SNR due to the mismatch in the number of component pulses; the second term is the reduction due to the shape error, as previously discussed.

- 204 - Appendix K The Data Taking System

The data taking system is designed to record the interferometer output, housekeeping information, and timing information on tape for later analysis. A block diagram of the system is shown in Figure K.1. The components of the system are described below.

K.1 A/D Systems High speed continous data from the interferometer and, optionally, related signals are digitized to 12 bit accuracy by a Data Translation model DT3362 A/D operating with the DT3369 dual port memory. The dual port structure allows continous data taking: as data is stored in one buffer of the dual port memory via the A/D data port, previous data samples in the other buffer are transferred to tape via the 11/23 port. The requirement that a buffer be transferred to tape faster than it is filled limits the system to a total rate of 20 kwords/sec. Thus if a single channel is recorded, a Nyquist frequency of 10 kHz results for the data. If more channels are recorded the Nyquist frequency is reduced proportionally. To prevent the aliassing of input frequencies above the Nyquist frequency into the data, a four pole low-pass filter is used at the input of the A/D. The corner frequency and gain of this filter can be switched by commands sent over a serial line from the computer. For a 10 KHz Nyquist frequency this filter was chosen to have a -6 dB point at ; 6 kHz. Housekeeping information is read in from a second A/D system which is made up of four 12 bit A/Ds, each fitted with a 16 input multiplexer; this allows 64 inputs. Signals on these inputs include the mass positions from the capacitive displacement transducers, the RF drive level on the central mass Pockels cells, the DC light intensity at the interferometer output ports, and the temperature. When data is taken with a 10 kHz Nyquist frequency, the housekeeping signals are sampled every 0.82 seconds.

K.2 Clock and Timing Board Timing and A/D triggering are accomplished with a rubidium clock and a programmable counter board, the M-Timer (Model 140-00 by Codar Technology, Inc., Longmont, Colorado). The M-Timer configuration is shown in Figure K.2. It is set-up to have a time-of-day clock, a 48 bit counter, and several divide-by- N rate generators. Each of these uses the 5 MHz from the rubidium clock as its fundamental timing input; they are discussed, in turn, below. The time-of-day clock keeps 24 hour time. It is sychronized to WWV time broadcasts by triggering an osciliscope with the seconds clicks of the clock alarm output and observing the advance/delay of the received WWV seconds tone burst. A program allows the time of the clock to be shifted to bring them into synchrony,

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9 FL j c e- K -.?- La C)fK~~~~cvUJ Ko2nRYLa~?--1 0S to r&1 C o u r\ tp-- (- -, with WWV time lagging our clock time by about 10 mS (light travel time is 5 mS per 1000 miles; the distance to WWV in Fort Collins, Colorado, is 1700 miles). This time is good to 10 mS, limited by the resolution of the clock as configured and fluctuations in the ionospheric path length from Fort Collins. (It was exciting to then receive the Canadian time signals from CHU and observe an 8 mS offset.) In principle, this clock would be adjusted from time to time to maintain WWV time, compensating for any frequency error in the rubidium clock. However, over the course of the data runs no resetting was needed. The 48 bit counter directly counts the rubidium 5 MHz standard. The counter's range is 2.8 x 1014 counts or 1.8 years at a 5 MHz counting rate. The counter, once started, is never reset or modified and thus represents a master time reference for the data, allowing data even a week or more apart to be accurately phased. The rate generators, like the 48 bit counter, are started and not modified. Each generator outputs a pulse every N counts of the 5 MHz. When data is taken, a selected generator's output is enabled and triggers the A/D system. Thus the sampling phase for data taken at a given frequency is fixed for all data sets independent of gaps in the data. The expectation is that this will allow simplifications in the analysis schemes for periodic signals. Table K.1 gives the rates available during the data runs. The two strange rates were chosen to keep the alias of a 620 kHz, origin unknown, signal near DC or the Nyquist frequency, and thus out of the frequency range of interest.

K.3 The Data Format The 64 housekeeping values, the 15 M-Timer values and the "number of blocks written to tape so far" are read at each change of the dual port memory buffer and are written to the tape along with the high speed A/D data. The size of the tape buffer is 16384 words, and thus a complete cycle of two A/D buffers and two sets of housekeeping/timing consists of 32768 words of which 32608 are data; this unit has been termed a "chunk". The format of the data tapes is given schematically in Figure K.3.

- 208 - Table K.1 Data Taking Parameters

N fsample fNyquist fAA filter Time per Housekeeping Recording -6 dB Time Sample Rate Chunk

85 58.8 kw/s 29.4 kHz 20.0 kHz 2.9 min/disk 3.6 Hz 0.55 sec

250 20.0 kw/s 10.0 kHz 6.5 kHz 16 min/tape 1.2 Hz 1.63 sec

855 5.8 kw/s 2.9 kHz 2.2 kHz 19 min/tape 1.1 Hz 1.87 sec (3 channels) 57 min/tape 0.36 Hz 5.62 sec (1 channel)

- 209 - RT-11 1 block (512 bytes)

File 1 block Name

* EOF *

Header 1 block - Data Taking Parameters Block

32608 A/D Samples

1 "Chunk" of Data = 128 blocks

2:x 80 Words of Housekeeping and Timer Information

Figure K.3 Format of the Data Tapes

- 210 - Appendix L Assigning ho Values to the Events

The path of the (potential) gravity wave signal through the instrument is shown schematically in Figure L.1. Given the output value from a template, the equivalent ho of the incident event can be evaluated by undoing the these steps. The conversion from the output value of a template filter to the equivalent ho of the incident pulse is a function of the template, as this determines the frequency and shape (i.e., NHC) of the waveform. Table L.1 presents a list of the templates and frequencies assigned to them as well as values of the parameters used in the conversion. These correct for the effects of the template correlation, the anti-aliassing filter and the loop system of the interferometer. Corrections are not made for the two high-pass filters. The equation in Table L.1 shows in detail the steps in the conversion. The SNR of the event is multiplied by the rms output value of its template-this gives the correlation output in ADU. The TmpAmpCor term converts this output of the correlation to the equivalent amplitude of the input waveform, assuming it is sinusoidal and matched in frequency to the template. (Intuitively this correction should be of order 1/(NHC x N1 ). The V/ADU term contains the A/D gains, etc., and converts the units to volts. The AACor term corrects for the four pole anti-aliasing filter and has little effect except for templates at 10 and 5 kHz. The final term, VtohCoef, is the result of evaluating equation H.7 in Appendix H at the frequency of the pulse, and gives the conversion from voltage at the interferometer output to the equivalent strain (i.e., displacement divided by arm length) of an incident gravity wave. To check this procedure, the rms values at the outputs of the set of NHC=6 templates were used to estimate the input spectrum using the relation:

hN = ho, SNR=1 NRC (L.1) which follows from equations 8.1 and 6.1. The resulting noise levels are tabulated in Table L.2 and plotted in Figure L.2; they show reasonable agreement with the spectrum.

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Figure L.1 Signal Chain from Gravity Wave to Detected Event

- 212 - Table L.1 Values of the correction terms ho, event = SNRevent TmprMS- TmpAmpCor -V/ADU - AACor -VtohCoef

ITMP Tmp ID Fequiv. TmpAmpCor AA Cor VtohCoef strain/volt

1 101 10.000 1.0000 6.602 2.35 x 10-12 2 301 2.500 0.4142 1.022 8.72 x 10-12 3 601 1.250 0.2112 1.001 1.6,1 x 10-11 4 102 10.000 0.5000 6.602 2.35 x 10-12 5 202 5.000 0.3536 1.350 4.46 x 10-12 6 312 2.500 0.2071 1.022 8.72 x 10-12 7 522 1.429 0.1235 1.002 1.46 x 10-11 8 832 0.909 0.0782 1.000 2.09 x 10-11 9 104 10.000 0.2500 6.602 2.35 x 10-12 10 204 5.000 0.1768 1.350 4.Akc x 10-12 11 214 3.333 0.1443 1.069 6.60 x 10-12 12 314 2.500 0.1036 1.022 8.72 x 10-12 13 424 1.667 0.0747 1.004 1.2' x 10~11 14 624 1.250 0.0528 1.001 1.64 x 10-11 15 834 0.909 0.0391 1.000 2.0w x 10-11 16 206 5.000 0.1179 1.350 4.46 x 10-12 17 216 3.333 0.0962 1.069 6.60 x 10-12 18 316 2.500 0.0690 1.022 8.72 x 10-12 19 416 2.000 0.0542 1.009 1.08 x 10-11 20 426 1.667 0.0498 1.004 1.28 x 10-11 21 626 1.250 0.0352 1.001 1.64 x 10-11 22 736 1.000 0.0293 1.001 1.95 x 10-11

Table L.2 Inferred hN values from measured ho,rms values

Template fkHz ho,rms hN X10-1 5 x10-1 6 / Hz 206 5 4.3 1.05 216 3.3 7.7 2.3 316 2.5 11.3 3.9 416 2.0 19.2 7.4 426 1.7 27.3 11.5 626 1.25 14.6 7.1 736 1.0 8.3 4.5

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Figure L.2 hNValues Corresponding to the RMS Values from the NHC=6 Templates.

-214-