thesis by Martin Regehr

Signal Extraction and Control for anInterferometric Gravitational Wave Detector

Thesis by

Martin W. Regehr

In Partial Fulfillment of the Requirementsfor the Degreeof

Doctor of Philosophy

California Institute of TechnologyPasadena,California

1995

(SubmittedAugust 1, 1994)

To my Father

ii

Acknowledgements

I am indebted to my three advisors: to Ron Drever, who accepted me

into the LIGO team, to Fred Raab, without whose tremendous patience and

valuable suggestions my experiment might never have been completed, and whose

painstaking reading of this thesis eliminated countless errors therefrom, and to

Amnon Yariv, whose interest, enthusiasm, and kindness have been a constant

source of inspiration for me.

I am also grateful to the entire LIGO team, whose support, technical and

emotional, has been invaluable during my stay at Caltech, and especially to

Alex Abramovici, Jake Chapsky, Torrey Lyons, David Shoemaker, Lisa Sievers

and Stan Whitcomb, each of whom contributed very substantially to the work

described herein.

iii

Abstract

Large interferometersare currently under constructionfor the detectionof

gravitationalradiation. Thesewill containa numberof optical surfacesat each

of which the relativephaseof incidentbeamsmustbe kept strictly controlledin

order to achievehigh sensitivity.

Thetypeof interferometerconsideredhereconsistsof two Fabry-Perotcavities

illuminatedby a laserbeamwhich is split in half by a beamsplitter,togetherwith

a recycling mirror betweenthe laserand the beamsplitter, which reflectslight

returningfrom the beamsplitter towardthe laserbackinto the interferometer.A

schemefor sensingdeviationsfrom properinterferencehasbeenanalyzedandthe

adequacyof this methodfor incorporationin a controlsystemhasbeenevaluated.

The sensingschemeinvolves phasemodulatingthe laser light incident on the

interferometer,introducinganasymmetryin thedistancesbetweentheFabry-Perot

cavitiesandthe beamsplitter, anddemodulatingthe signalsfrom photodetectors

monitoring three optical outputs of the interferometer. Theseoptical outputs

are light returning to the laser, light extractedby a pick-off from betweenthe

recyclingmirror andthe beamsplitter, andlight leavingthe interferometerat the

beamsplitter.

The analysishas shown that the matrix of transfer functions from mirror

displacementto demodulatedsignal is ill-conditioned, that as many as threeof

iv

the transferfunctionsmay containright half planezeros,and that one of these

transferfunctionscanbe affectedby the modulationdepth. The performanceof

theclosed-loopsystem,however,neednot besignificantlyaffected,providedthat

certainconstraintsareobservedin the optical andelectronicdesign.

A table-topinterferometerhasbeenconstructed,to demonstratethefeasibility

of constructinga control systemusing this sensingschemeand to comparethe

responseof theinterferometerwith thatpredictedby calculations.Goodagreement

betweenthe experimentand the calculationhasbeenobtained.

v

Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Table of Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . xiv

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 DC Analysis . . . . . . . . . . . . . . . . . . . . . . . . 15

Photodetector, Mixer . . . . . . . . . . . . . . . . . 17

Fields in Optical Cavities . . . . . . . . . . . . . . 19

Fields in the Interferometer . . . . . . . . . . . . . 22

Derivatives of Mixer Outputs . . . . . . . . . . . . 31

Chapter 3 Interferometer Frequency Response . . . . . . . . . 40

Fabry-Perot Cavity . . . . . . . . . . . . . . . . . . 40

Response of the Complete Interferometer . . . . 53

vi

Chapter 4 Numerical Models . . . . . . . . . . . . . . . . . . . . 64

Chapter 5 Experiments with the Table-top Prototype . . . . . 72

Setup and Hardware . . . . . . . . . . . . . . . . . 73

Component Response Measurements . . . . . . 78

Response of a Fabry-Perot Cavity . . . . . . . . 81

Response of the Coupled-Cavity . . . . . . . . . 82

Response of the Complete Interferometer . . . . 86

Chapter 6 Design and Analysis of a Control System . . . . . 98

Feedback Configuration and Gain Constraint . 101

System Performance . . . . . . . . . . . . . . . 108

Numerical Example of a Control System

Design . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 7 Optical Design Considerations . . . . . . . . . . . 122

Common-mode Feedback Configuration . . . . 122

Asymmetry . . . . . . . . . . . . . . . . . . . . . . 123

Recycling Mirror Reflectivity . . . . . . . . . . . 125

Modulation Index . . . . . . . . . . . . . . . . . . 126

Arm Cavity and Recycling Cavity Lengths . . . 129

vii

Chapter 8 Summary and Conclusion . . . . . . . . . . . . . . 130

Robustness . . . . . . . . . . . . . . . . . . . . . . 131

Lock Acquisition . . . . . . . . . . . . . . . . . . . 134

Conclusion . . . . . . . . . . . . . . . . . . . . . . 135

Appendix A Shot Noise at the Mixer Output . . . . . . . . . . . 140

Appendix B Specification of Allowable RMS Deviations from

Perfect Resonance . . . . . . . . . . . . . . . . . . . 149

Introduction . . . . . . . . . . . . . . . . . . . . . . 149

Power in the Arm Cavities . . . . . . . . . . . . 150

Frequency Noise . . . . . . . . . . . . . . . . . . 151

Intensity Noise . . . . . . . . . . . . . . . . . . . . 152

Dark at the Antisymmetric Output . . . . . . . . 152

Summary . . . . . . . . . . . . . . . . . . . . . . . 153

Appendix C Alternative Feedback Configurations . . . . . . . 156

Appendix D Effect of Mixer Phase Error . . . . . . . . . . . . . 163

viii

List of Tables

Table 1.1 Possible optical parameters for a LIGO

interferometer. . . . . . . . . . . . . . . . . . . . . . . 13

Table 2.1 Vanishing terms in equation (2.6) for derivatives of

�� and �� with respect to the differential degrees of

freedom. . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 2.2 Vanishing terms in equation (2.8) for derivatives of

�� and �� with respect to the differential degrees of

freedom. . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 5.1 Results of Coupled-Cavity Experiments . . . . . . . 86

Table 5.2 Fixed Mass Interferometer Optical Properties . . . 87

Table 5.3 Sampled values of experimental (bold) and

calculated response. . . . . . . . . . . . . . . . . . . 94

Table 6.1 Loop shapes and performance predictions for

numerical servo design. . . . . . . . . . . . . . . . . 121

ix

List of Figures

Figure 1.1 Effect that a horizontally propagating gravitational

wave might have on a human being. . . . . . . . . . 2

Figure 1.2 Michelson interferometer. . . . . . . . . . . . . . . . . 2

Figure 1.3 Interferometer with Fabry-Perot arms. . . . . . . . . . 4

Figure 1.4 Interferometer with power recycling. . . . . . . . . . . 5

Figure 1.5 Interferometer, showing mirror suspensions. . . . . 6

Figure 1.6 Signal extraction scheme. . . . . . . . . . . . . . . . . 7

Figure 1.7 Feedback configuration. . . . . . . . . . . . . . . . . . 9

Figure 2.1 Example of a section of laser beam. . . . . . . . . 15

Figure 2.2 Partially transmitting mirror. . . . . . . . . . . . . . . 19

Figure 2.3 Fabry-Perot cavity. . . . . . . . . . . . . . . . . . . . . 20

Figure 2.4 Fields at in-line arm cavity. . . . . . . . . . . . . . . 22

Figure 2.5 Fields at beam splitter and pick-off. . . . . . . . . . 24

Figure 2.6 Compound mirror. . . . . . . . . . . . . . . . . . . . . 25

Figure 2.7 Equivalent optical configuration for the analysis of

sideband fields. . . . . . . . . . . . . . . . . . . . . . 28

Figure 2.8 Compound mirror for the RF sidebands. . . . . . . 30

Figure 3.1 Fabry-Perot cavity. . . . . . . . . . . . . . . . . . . . 41

x

Figure 3.2 Equivalent system for the purpose of deriving

common-mode response. . . . . . . . . . . . . . . . 57

Figure 4.1 Table of numerical (upper) and approximate analytic

(lower) derivatives of interferometer outputs, for the

optical parameters given in Table 1.1. . . . . . . . 69

Figure 4.2 Frequency response curves calculated using the

approximate analytical method and (dashed) the

strictly numerical model. . . . . . . . . . . . . . . . . 70

Figure 5.1 RF Distribution. . . . . . . . . . . . . . . . . . . . . . . 75

Figure 5.2 Schematic representation of a high voltage

amplifier and the dynamic signal analyzer. . . . . . 77

Figure 5.3 Displacement transducer characterization. . . . . . 79

Figure 5.4 Measuring the modulation index. . . . . . . . . . . . 80

Figure 5.5 Measurement of the response of a Fabry-Perot

cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 5.6 Coupled-cavity experiment. . . . . . . . . . . . . . . 83

Figure 5.7 Block diagram of the control system for the

coupled-cavity experiment. . . . . . . . . . . . . . . 84

Figure 5.8 Layout of complete interferometer. . . . . . . . . . . 88

Figure 5.9 Servo configuration of Table-top prototype.

Feedback amplifiers not shown. . . . . . . . . . . . 89

xi

Figure 5.10 Time record of lock interruption. . . . . . . . . . . . 91

Figure 5.11 Block diagram representing setup for closed loop

measurements. . . . . . . . . . . . . . . . . . . . . . . 92

Figure 5.12 Setup for measuring the factors��

and �� . . . . . 93

Figure 5.13 Plots of experimental and calculated (dashed)

response. . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 6.1 Equivalent block diagrams for the plant. . . . . . . 99

Figure 6.2 Closed-loop system. . . . . . . . . . . . . . . . . . . . 99

Figure 6.3 Closed-loop system with a diagonal controller. . 100

Figure 6.4 Closed-loop system with a “crossed” controller. . 100

Figure 6.5 Block diagram for analysis of system containing

frequency-independent plant. . . . . . . . . . . . . . 102

Figure 6.6 Block diagram used to analyze performance,

showing inputs driven by seismic disturbance. . . 108

Figure 6.7 Block diagram showing plant and one feedback

loop as a single block. . . . . . . . . . . . . . . . . . 110

Figure 6.8 Seismic spectrum. . . . . . . . . . . . . . . . . . . . 117

Figure 6.9 Block diagram for the propagation of seismic

disturbance. . . . . . . . . . . . . . . . . . . . . . . . . 118

xii

Figure 6.10 Loop gains (a)) and residual motion (b)) in

common-mode degrees of freedom. Curves

corresponding to the��

loop are dashed. . . . . 119

Figure 6.11 Contributions from loop 1 and loop 2 (dashed) to

residual deviations from resonance in loop 1 (a)

and in loop 2 (b). . . . . . . . . . . . . . . . . . . . . 120

Figure C.1 First alternative configuration: feed-back to beam

splitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure C.2 Second alternative feedback configuration. . . . 160

Figure D.1 Block diagram used to analyze the effect of mixer

phase error. . . . . . . . . . . . . . . . . . . . . . . . . 164

xiii

Table of Mathematical Symbols

Symbol Page number��,��

, � � , � � , � , �� , �� 6�� , �� , �� , �� 8��10��11� �10� �11�16�16� �

,� !#"�$

16%'&17� � , ��( 18) 19*19+

,+-,

21) $/.0�1��2 23) $43 , ) ,$53 236

, 7 25� 8,� 9

,� :<; � ,

� 8>=@? �,� 8>=0? �

,�BAC"D?E!

24)GFIH ,* &

23)GJ 25� 8K= 925

) $ � , ) ,$ � 27L 29+M3

,+ � 35

xiv

�� , �� 36� 41� �� 43�� 42 44� �� 44�

45 �� 53 �� 56�66�

,��

,� �

99 98!,! � , !#" 99$ � , $ " , �%� , & � , & " , � � , � " 101� � , � " 105' � , '(" , ) � , ) " 108�

109*, + � , + " 111

xv

1

Chapter 1 Introduction

In the year 1915, Albert Einstein publisheda theory which he called the

Generaltheory of Relativity. This theory interpretsthe force of gravity as a

distortion of spaceand time producedby objectswith mass. The theory also

predictsthat wave-likedistortionsof spaceandtime be able to propagateacross

the universe,in a fashionsimilar to the propagationof electromagneticwaves.

Thesewaveswould be producedby acceleratingmass,muchaselectromagnetic

wavesareproducedby acceleratingcharge. The direct detectionof thesewaves

would providea strongconfirmationof Einstein’stheory. Their observationwith

good signal-to-noiseratio could provide a wealth of new information about the

universe,sincemostanticipatedsourcesof gravitationalradiationaredifficult to

observeoptically.

Although the Generaltheory of Relativity is conceptuallyquite subtle, the

effect of gravitationalwaveson detectorsis simple: they producea fluctuating

shearstrain transverseto the directionof propagation.This is shownin Fig. 1.1

for a gravitationalwave propagatinghorizontally througha humanbeing. Also

visible from thesizeof theeffect* indicatedon this figure is thefact thata human

being is not nearlysensitiveenoughto detectgravitationalwavesdirectly.

* The sizeshownis the rms amplitudeof gravitationalwavesexpectedwith a meanfrequencyof threetimesperyear

in burstsfrom coalescingcompactbinaries.

2

Figure1.1 Effect that a horizontallypropagatinggravitationalwavemight haveon a humanbeing.

ωt=0 ωt=π/2 ωt=π ωt=3π/2

10 m-21

Figure 1.2 Michelson interferometer.

LaserIn-line arm

Perpendicular arm

PB

IBBS

Photodetector

Antisymmetric port(dark)

Thefact thatthewaveproducesa transverseshearstrainmakestheMichelson

interferometer(Fig. 1.2) an obvious candidatefor a detectorand in fact early

detectorsusedthis configuration1,2. Interferometerscurrentlybeingdevelopedfor

LIGO (Laser InterferometerGravitationalObservatory3) will be variants(to be

describedbelow) of the Michelson interferometer.

If thedetectoris operatedwith themirrorspositionedsothattheantisymmetric

outputis dark thena displacementof��

(about125 nanometersfor greenlight)

3

of eitherof the backmirrors PB and IB will causethe intensityat the output to

changeto full brightness.Oneeasyway to increasethe sensitivity is to increase

the length of the arms. The plannedLIGO detectorswill havearmsthat are 4

km long. Sincethegravitationalwavestrainsspace,stretchingit in onedirection

andcompressingit in the orthogonaldirection,the changein lengthof an arm is

equalto the size of the strain times the initial length of the arm.

Anotherway to increasethe sensitivityof the detectoris to havethe light in

eacharm reflect severaltimesfrom eachbackmirror. This canbe accomplished

by using an optical delay line4, wherethe light strikesthe mirror in a different

spot on eachtraversalof the arm, or by usinga “Fabry-Perot”resonantcavity5,

where light from many traversalsinterferesin a single spot on eachmirror. In

initial LIGO detectorsthe Fabry-Perotresonators(or “arm cavities”) are formed

by insertingpartially transmitting“front” mirrors PF and IF betweenthe beam

splitter and the back mirrors, as shownin Figure 1.3. Sucha cavity is said to

be resonantwhen light enteringthrough the partially transmitting front mirror

interferesconstructivelywith light traveling backand forth inside the resonator.

Nearresonancethephaseof thelight returningfrom theresonatoris very sensitive

to the lengthof the resonator,and the effect of the small displacementproduced

by a gravitationalwaveon the intensity at the output is magnified.

Larger signals can also be obtained by increasingthe amount of power

4

Figure 1.3 Interferometerwith Fabry-Perotarms.

LaserInline arm

Perpendicular arm

PB

IBBS

Photodetector


PF IF

incident on the beam splitter*. This is accomplished,in a techniquereferred

to as“power recycling,”6 by insertinga (partially transmitting)“recycling mirror”

R betweenthelaserandthebeamsplitterasshownin Figure1.4. If light returning

from the armsto the beamsplitter interferesdestructivelyin the directionof the

photodetector,thenby energy conservationthe light mustinterfereconstructively

in the directionof the laser. The recyclingmirror allows this light to be reused.

To do so,therecyclingmirror mustbeplacedsothat the light it reflectsbackinto

the interferometerinterferesconstructivelywith freshlight transmittedthroughit

from the laser.

This work addressestwo problemsthat exist with the simple interferometric

detectoroutlined thus far. The first problem is that the intensity at the output

is minimum in the absenceof a gravitationalwave and increaseswhen a grav-

itational wave distorts the arms in either direction. By symmetry,the signal is

* A morepreciseway to statethe advantageobtainedthus is to note that the signal to shotnoiseratio is improved.

We will discussthis in moredetail in chapter7.

5

Figure 1.4 Interferometerwith power recycling.

LaserInline arm

Perpendicular arm

PB

IBBS

Photodetector


PF IFR

XPB

XIB

at best a quadraticfunction of the gravitational-waveinduceddistortion; thus�� , where �� is the signal from the photodetector,and �� and

�! � arethe displacementsof the two backmirrors, asshownin Figure1.4.

The secondproblemis that it is virtually impossibleto build a mechanical

structuresufficiently stiff andthermallystableto hold themirrors in the locations

wherethe interferenceconditionsaresatisfied.(Thereare four suchinterference

conditions,one for eachoptical surfacewhereinterferencetakesplace: the two

front mirrors, thebeamsplitter,andtherecyclingmirror.) In practice,themirrors

areindividually suspendedfrom wires (Figure1.5)† in orderto attenuateseismic

noise which would otherwiseshakethe mirrors, imitating gravitationalwaves.

While thesepassivesuspensionsreduceseismicnoiseat higherfrequencies,they

can amplify low-frequencyseismicnoise.

† Drawing courtesyof Aaron Gillespie.

6

Figure 1.5 Interferometer, showing mirror suspensions.

mirrors

mirrorsLight Source

beamsplitter

photodetector

recycling mirror

One way to keep the interferometer resonating is to implement some sort

of sensing scheme which produces signals corresponding to the deviations from

perfect interference, and to use these signals to feed back to the positions of the

mirrors and to the laser frequency, in order to cancel any such deviations.

In 1991 S. Whitcomb suggested that the following scheme7,8,9, illustrated in

Figure 1.6, would accomplish this task. The phase of the laser light is modulated

at a radio frequency (“RF”), typically on the order of 10 MHz (we will use 12.5

MHz in examples throughout this text). The effect of this modulation is to impose

two sidebands on the light, one at 12.5 MHz above the optical “carrier” frequency

and one at 12.5 MHz below the carrier. The arm cavity lengths are chosen so that

the sidebands do not resonate in the arms, the “in-line” cavity is placed farther

from the beam splitter than the “perpendicular” cavity by a distance��

,

7

and the “average recycling cavity length”�� is arranged to be* � �� where

�� is the modulation wavelength �� (and � �� is the modulation

frequency). The effect of these choices of lengths is to allow both the carrier and

the sidebands to resonate in the recycling cavity, and to allow sideband light to be

transmitted to the antisymmetric port while that port remains dark for the carrier.

When the interferometer is distorted by a gravitational wave, some carrier light

exits the antisymmetric port, where it beats against the sidebands. This light is

detected and demodulated with a mixer (circled “x” in Figure 1.6).

Figure 1.6 Signal extraction scheme.

Laser

I Q

IQ

Q

90RFOscillator

Phase Modulation

I

o

L I

l I

l P

L P

v1

v2

v 3v 4

Pickoff

Isolator

* An average length of �� ! " ! for any integral � will work.

8

The insertionof a pick-off* into the recyclingcavity andan isolatorbetween

thephasemodulatorandtherecyclingmirror† producestwo moreopticaloutputs,

eachof which canbedemodulatedusingeitherthe inphasemodulatingsignal“I”

or the quadraturephasesignal “Q”, asshownin Figure 1.6. Four of the signals

obtainedthis way‡ (labelled �� through �� in Figure 1.6) can then be fed back

as shownin Figure 1.7. The inphasesignal from the isolator is fed back to the

laser frequency. The quadraturephasesignal at the isolator is fed back to all

four of the arm cavity mirrors in proportionsuchthat��

and��

arechangedby

equalandoppositeamounts.The inphasesignal from the pick-off is fed backto

the recyclingmirror, andthequadraturephasesignalfrom theantisymmetricport

is fed back in equaland oppositeamountsto the two arm cavity back mirrors.

Other feedbackconfigurationsare also possible;someof theseare discussedin

Appendix C.

* This pick-off neednot be a separateoptical element.From example,the anti-reflection-coatedsurfaceof the beam

splitter might be usableif it facesthe recyclingmirror. Evenif this surfacefacesawayfrom the recyclingmirror, useful

signalsarelikely to be availablein the light reflectedat this surface.This latter situationhasnot beenanalyzedin detail.

† In addition,LIGO will havea passiveopticalfilter, sometimescalleda “modecleaner,”installedbetweentheisolator

andthe recyclingmirror. This filter will be transparentto the carrierandthe RF sidebands,andits effect is irrelevantfor

mostof the issuesaddressedhere.

‡ The quadraturephasesignal at the pick-off containsthe sameinformation as the quadraturephasesignal at the

isolator,but generallywith lower signal-to-noiseratio. Thereis in addition an optical output availableat the otherside

of the pick-off (sincelight propagatesin both directionsin the recycling cavity); this output containsvirtually identical

signalsto the outputshown. Some(generallysmall) improvementsin someof the signalsare possibleby mixing down

the signalsfrom all of the optical outputsin both phasesandtaking appropriatelinear combinations.

9

Figure 1.7 Feedbackconfiguration.

I Q I QI Q

Photodiode with I & Q Demodulator

Laserfrequencyfeedback

Summing Node Inverting

Amplifiers

Let us restatein greaterdetail the problemto be solved.We will concentrate

on the problemof keepingthe above-mentionedinterferenceconditionssatisfied,

and show along the way that the configurationdescribedhere also meetsthe

requirementof producing a gravitational wave signal. For each interference

condition,we mustexaminetheconsequenceof allowing small errors,so thatwe

can determinethe deviationsfrom eachcondition that can be toleratedwithout

significantly degradingthe sensitivity of the detector.This issueis examinedin

Appendix B; the resultsare summarizedbelow.

A constrainton the cavity lengthsis due to the fact that if the arm cavities

deviatefrom resonancein the samedirection, the phaseof the light returningto

therecyclingmirror will change.Thiswoulddegradetheconstructiveinterference

betweenlight reflectedthereand light enteringfrom the laser. This setsa limit

10

of� ��

to the common-mode-displacementof the cavity backmirrors,or

equivalently,a limit on the common-modearm cavity phase*

�� (1.1)

� is definedas the sum of the arm cavity round-trip phases†:

�� !"�$#&%(')!"�+*,%(1.2)

The condition on the phaseof light returning to the recycling mirror also

constrainsthe common-moderecycling cavity phase,suchthat- ! - #.% ')! - *,%

� �0/1�2� (1.3)

(where- #.%,354"6 %87 # 4 -:9�;8<>=

and- *,%?354"6 %�7 * 4 -:9@;�<>=

).

If the arm cavities deviatefrom resonancein oppositedirections,then the

destructiveinterferenceat the antisymmetricport could be disturbed,allowing

excesslight to hit the photodetector‡. This setsa condition on the differential

* We will seethat using differential and common-modecombinationsof phasesis more convenientthan using the

individual phasesthemselves.

† Theround-tripphaseis thephasechangethe light accumulatesin travelling twice the lengthof thecavity, including

the Guoy phase,which is a phasedeficit due to the fact that the laser beam has a finite diameter. For exampleACBED�FHG�I DKJ B:LMG�N�OEPRQ>S

, whereI D FTGVUXWKY D

is the wave numberof the light. It will generallybe more useful

for us to work with theseround-tripphasesthanwith the distancesbetweenadjacentmirrors. We usethe subscript Z to

identify quantitiesrelatedto the carrier; later we will usethe subscripts[ andL [ for the upperandlower RF sidebands.

‡ Excesslight on the photodetectorrepresentsan unnecessarylossof light from the interferometerwhich reducesthe

optical efficiency andgeneratesadditionalshotnoise.

11

changein arm cavity phases:��

�� (1.4)

The requirementabove, to keep excesslight from the antisymmetricport

photodetector,alsoconstrainsthe differential recyclingcavity phase,suchthat� � � � ��

�� (1.5)

In thefollowing chapterswewill investigatethebehaviorof theinterferometer

output signals in responseto motion of the mirrors and changesin the laser

frequency.This detailedcharacterizationis necessaryfor the designandanalysis

of a control system. Thereare two parts to this analysis: an analytic approach

in which approximationsare madeand relatively simple expressionsdescribing

the interferometerresponsearederived,anda numericalapproach,within which

the approximationsare lesssignificantand which thereforecan be usedto test

the validity of the approximationsmadein the analyticderivations.The valueof

the analyticderivationis that it moreclearly expresseshow the choiceof certain

opticalparametersin theinterferometer(suchasmirror reflectivitiesandpositions)

affects critical aspectsof the behaviorof the instrument. For example,we will

seethat if the asymmetry�

is chosento be too large, the responseof � � to��

containsa right-half-plane-zero,which compromisescontrolsystemperformance.

This sort of conclusioncan be derived from numericalmodelsonly through a

tediousexplorationof parameterspaceandthe useof inductiveassumptions.

12

On the other hand, someaccuracyhas beensacrificed in the derivation of

the analytic model. Our goal will be to derive expressionswhich are accurate

to 10% over the rangeof frequenciesin which the servosfeeding back to the

mirror positionsare active. It is for honing thesepredictionsthat the numerical

modelsare useful.

The LIGO interferometersare not yet sufficiently well specifiedto merit a

detailednumericalanalysisandcontrol systemdesign*. Nonetheless,theexercise

of analyzinga possibleinterferometerand designinga control systemfor it is

useful, and will give insight toward a final design. A set of possibleoptical

parametersfor a LIGO interferometeris listed in Table 1.1. We will apply

our analytic resultsand our numericalmodel to this interferometerto explore

its behavior,and then we will designa control systemaroundit, to meet the

specificationslisted above.

It is virtually certainthat neitherthe optical parametersnor the specifications

will remain unchangedbetweenthe time of this writing and the time when a

LIGO control systemis installed, and for this reasonwe will emphasizemore

the methodsof analysisand designthan the detailsof our numericalexample.

The numericalmodel will provide examplesof the size of neglectedquantities

* For example,specificationsfor mirror surfaceerror have not beenset and possibletrade-offs betweencost and

quality arestill beingevaluated.† Thearmcavity lengthsaredisplacedslightly from thelengthwheretheRF sidebandswould beexactlyanti-resonant.Seealso the footnoteon p. 27, andSection7.5.

13

Table 1.1 Possibleoptical parametersfor a LIGO interferometer.

Quantity Symbol Value Units

Recyclingmirror

��5%

Arm cavityfront mirrors

��3%

Arm cavityBack mirrors

��Mirror(power)transmissions

BeamSplitter

� ��50%- /2

Pick-off reflectivity �� 0.1%

Loss in eachoptical element 100 ppm

In-line cavity �� 4002.004† metersArm cavitylength Perpen-

dicular cavity� 4001.996 meters

recyclingmirror toin-line cavity

� � 6.29 meters

Recyclingcavity lengths recycling

mirror toperpen-dicular cavity

� 5.71 meters

Laserpower�� photons/

second

Modulation index 0.1 radians

ModulationFrequency ! 12.5 MHz

comparedto relevant ones, and will give us some idea of the magnitudeof

the errors introducedby the approximationsmadewithin the analytic approach.

It is hopedthat, as the parametersand specificationsof LIGO interferometers

14

becomebetterdefined,thesesamemethodswill be valuabletools for designing

and analyzing control systems,and may even influence the choice of optical

parametersand the definition of performancespecifications.

15

Chapter 2 DC Analysis

We begin our analysisof the interferometerby obtainingthe derivativesof

the voltages �� with respectto the mirror positions.‡

Considera small region,as shownfor examplein Figure 2.1, inside one of

the laserbeampathswithin our interferometer.

Figure 2.1 Exampleof a sectionof laser beam.

Laser

PB

IBBS

PF IFR

Region containing a section of laser beam

It containselectromagneticradiation travelling in two directions,which we

will call the +z and —z directions,and the electric field due to that radiation

can be written:

�� !#"%$�& �'�� !)(+*-,/.�0�12$�&43 5!607$�& �'��

� !#(�*�,/.'"812$�& (2.1)

We will assumeuniform linear polarizationin the�� direction. For the remainder

of this work, we will concernourselvesexclusivelywith the complexfunctions

‡ Expressionssimilar to the onesin this chapterwerefirst derivedby A. Abramovici usinga differentapproach.

16

of time��

and��

, which we will refer to asthe field travelingin the

+z direction and the field traveling in the –z direction, respectively.The factor�� is usedto make the units of the fields �� ! #"%$ (the field magnitude

squaredequalsthe numberof photonsper second),which is convenientwhen

calculatingshot noise.

The light from the laser is phasemodulatedat someangular frequency &beforeenteringthe interferometer.This meansthat10 if

�('is the field from the

laserand�*),+ � is the field incident on the interferometer,

�*),+ �.- � '�/ )10�243�57698:<;=?>@�� 'BADC :BE�F>@�� ' / )�698 AGC :BEH4>@�� ' / ),6I8

�()�+ � ; A �*),+ � E / )�698 A �(),+ � E / ),698(2.2)

we call�(),+ � ; the carrier field, and

�(),+ � E and�*),+ � E the upper RF sideband

field and the lower RF sidebandfield, respectively. We will continue to use

numeral subscriptsto index different frequencycomponentsof the light, and

literal subscriptsto denotelocation.

In theremainderof this chapter,wewill first showhowthefieldsincidentona

photodetectoraffect thecorrespondingmixeroutputandhow thederivativesof the

mixer outputarerelatedto thederivativesof thosefields. Thenwe will analyzea

simpleFabry-Perotcavity to find thefieldsandtheir derivativeswithin thecavity.

Finally we will solvefor thefields in the interferometerandtheir derivatives,and

17

concludewith expressionsfor thederivativesof themixer outputswith respectto

thephases��

, and��

. Thesearetheneasilyinterpretedasthe response

in the mixer outputsto changesin mirror positionor laserfrequency.

2.1 Photodetector, Mixer

If we assumethat the photodetectorshaveunit quantumefficiency then the

photocurrentin a detectoris

�� !�� "�#� �

� � � � � � � � � � � � � �%$'&�� )(� � � � �"�#��$*&+� � (� � � � �� (� � � ��"�#�

(2.3)

We assumethat the mixersaredouble-balancedmixers,followed by low-passor

notchfilters to eliminateharmonicsof themodulationfrequency.To a reasonable

approximation§, the mixer multiplies its input by ,.-0/21�3 (or /5476�1�3 if the mixer is

fed by the quadraturephaselocal oscillatorsignal). By the Fouriershift theorem

(or “heterodyne”theorem11), the only frequencycomponentsof the photocurrent

which will passthroughthe low-passfilter unattenuatedare at frequenciesnear

§ The true effect of mostmixers is to multiply by somethingmoreclosely resemblinga squarewave. This haslittle

effect if themodulationindex is smallor if thephotodiodeis tuned,with a resonantfilter, to besensitiveonly to amplitude

modulationat frequenciesnearthe modulationfrequency.

18

�. The inphaselow-passfilter output then is

��

�� !�� #"%$ �'& (*)�+ � �-,/.0�-,

�1� �� 2�3�� (2.4)

wherefor conveniencewe havemodelledthelow passfilter asa devicewhich

averagesover an integral numberof modulationperiods 465$ .

Similarly,

�879�:�;�< =�> �� ?�2�3�� (2.5)

Note that for pure phasemodulation, for exampleif� � "A@CB , both outputs

vanish: �D�EF�87GIH .Now we canfind the rateof changeof thesemixer outputs.Suppose

��,� � ,

and� �J� are all functionsof someparameterK ; then

L �D�L K

�1� L ��L K

� � � �!� �J� � � �!��LL K� � � �2� �� (2.6)

In the specialcasewhere� � � �J� and both are pure imaginaryand their

derivativesarerealandequal,and��

is realandits derivativeimaginary,we have

L ��L K

�CMN�� MN�� L �9�L K

�� MN� � �L � �L K (2.7)

Recognizing�"PORQRS O QSDT as the rate of changeof the phaseof

��U, we seethat

in this specialsituation SDV;WSXT is proportionalto the differencebetweenthe rateof

changeof carrier phaseand the rateof changeof sidebandphase.

19

Similarly

��

��

�� (2.8)

and if ��

and ��

areequalandpure imaginary,and if !�

is real, we have

�"��#� � $�

��%� � ��

(2.9)

We seethat a quadraturephasesignal is producedif onesidebandgrowsandthe

other shrinks.

2.2 Fields in Optical Cavities

Now let us showhow the fields are relatedto the mirror positions.

Figure 2.2 Partially transmittingmirror.

AE v

E w

E x

+ -

Laser beam

Considerfirst a partially transmittingmirror A, illuminatedfrom theleft, asin

Figure2.2. Wedefineits reflectivity&(' (sometimescalled“amplitudereflectivity”)

astheratioof thereflectedfield !)

to theincidentfield +*

. Wewill alwaysmodel

our mirrors as having positive real reflectivity on one side, and we denotethat

side with a “+” as shown. We will also model the transmission,�' as being

positiveandreal; thenenergy conservationrequires12 thatwe assignnegativereal

reflectivity to the other side of the mirror, denotedby a “-”.

20

Figure 2.3 Fabry-Perotcavity.

A DE v

E w

E x

E y+ - + -

d

Next, considerthe cavity AD, shownin Figure 2.3.

It is easyto find the other fields in termsof��

:

�� (2.10)

��(2.11)

�� !��(2.12)

where " is the travel phasecorrespondingto the optical path from A to D and

back: " �$#�%'& # ")(+*-,/. . Solving:

�� !0 �1� � � � �� (2.13)

�� -2�3�4� �5��0 �6� 2��4� �� (2.14)

�� 87 � � � ��0 �6� � � � �5��

� � � � 7 �9� 7 � � � �5��0 �1� :�� 5�;� ��

(2.15)

21

To find derivatives,we define

�� (2.16)

then ��

��

�� "! (2.17)��

��#��

�$�%� � � �&� ! (2.18)��

��'�� !�

�� &� ! (2.19)

As we sawin (2.7), the rateof changeof the phaseof a field canbe a useful

quantity; we define

( ��#

�� )+* �

��#��

� ��,� � � � � � ��

(2.20)

and (.- �� '

�� )+* �

��'� �

� � !� � � � ��/ � � � � !� � � !� � � � 0��

(2.21)

and call thesethe bounce number* and augmented bounce number respectively.

For a resonantcavity, thebouncenumberis theratio of therateat which thephase

of the field returningto the front mirror changeswhenthe rearmirror moves,to

* The expression“bouncenumber” is somethingof a misnomerin this context,sinceit doesnot correspondto the

numberof times any physical quantity “bounces.” It deriveshistorically from an analogousquantity for “delay line”

gravitationalwavedetectors.

22

the rate at which that phasewould changeif the front mirror were absent.The

augmentedbouncenumberis the sameratio exceptthat it is defined for the field

��reflectedfrom the cavity.

2.3 Fields in the Interferometer

We can use theseexpressionsto solve rather easily for the fields in our

interferometerand also for their derivatives. We will first find the carrier fields

everywhere,then the sidebandfields. Thenwe will find the derivativesof these

fields. Finally we will assembletheseinto expressionsfor the derivativesof the

mixer outputs.

Figure 2.4 Fields at in-line arm cavity.

R

IF IB

PF

PB

BSE a

E b IΦ

Inline Cavity

+ - + -

First, considerthecarrierfields in anarmcavity, the in-line onefor example.

Using (2.15), and assumingthat the two arm cavitiesare identical, e.g., ��

23

�� , etc., we have

��

� � �� ! #"%$'&

(2.22)

where " $ is the round-tripphaseof the in-line arm cavity, andwe havedefined

�� as the reflectivity of an arm cavity. We choosethe length of the arm cavity

so that the carrier resonatesin it. This means

" $)( +* -,/.1032145.76 * & (2.23)

Then from (2.19),

88 "�$�(

� � (�9� (

: � � �� & � (2.24)

The arm cavity reflectivity andits derivativewill occurso frequentlythat we

assignthem specialsymbols†:

�� ( ��; * &

� ��

(2.25)

�=<� (: ��

� �� & � (2.26)

Next, we includethe beamsplitter andpick-off, illustratedin Figure2.5:

For conveniencewe will assumethat thebeamsplitter,pick-off andrecycling

mirror (identified by the subscripts“BS,” “p” and “R” respectively)are all

† The sign of >@?�A is chosenso that this quantity is positivefor an over-coupled,resonantcavity.

24

Figure 2.5 Fields at beam splitter and pick-off.

IF IB

PF

PB

E r

E symE anti

E f

E ret P

E ret I+

-+

-

BS

very close together*. The generalization to arbitrary component positions is

straight forward, but clutters the formulae. We make the approximation that

�� to get, for the carrier:

��

� �� "! �#��$�%'&)(+*�,.-(2.27)

� ��/�10��

� �� "! 02� $�% &)(3*�45-(2.28)

�7698 � ( �� :� � � �� "! �#� $;% &)(+*�,<- �=�9�>! 02� $�% &)(+*=4'-

(2.29)��?"@BA �� :� � � �

� ��9�>! 02��$;%C&)(3*=4C-ED �� >! �F�=$�%C&)(+*�,<-(2.30)

�HG �� JI �� 9�>! 02��$;% &)(3*=4C- D �=�9�>! �F��$;% &)(3*�,E-

(2.31)

* Here “very close together” means separated by distances small compared to KBLNMPO .

25

where, in the final equation,we have written��

for ��, using the convention

that uppercaseR’s and T’s equal the squaresof the correspondinglower case

quantities. Taking

�� (2.32)

we have

��

�� "!$#&%('*)+�,�"!-#/.0'(2.33)

In this lastequationwehavetheratiobetweenthelight to theright of therecycling

mirror travelling away from the recycling mirror, and the light returning to the

recycling mirror. Another way to think of this (shownin Figure 2.6) is that all

of the mirrors of Figure2.5 form a compoundbackmirror for the cavity having

the recycling mirror as its front mirror.

Figure 2.6 Compoundmirror.

IF IB

PF

PB

E r

E f

Compound mirror

E inc

E ref

26

Using (2.13) we have

��

�� !#" �� $%! (2.34)

and to achievemaximum power buildup in the recycling cavity,† we need to

arrangefor the secondterm in the denominatorto be asnearlyaspossibleequal

�. Hencewe choose

&(' � � &*) � �,+ (2.35)

so that

��-��.��(� �

�� /�0� � �� (2.36)

Using (2.14) and (2.15)

�21 ��.��(�3�

4� �0� � �� 0� � �� (2.37)

� ��5 1 ��(� � � � �� 0� � ��

� � � �0� � �� 0� � �� 6� � ��

(2.38)

wherethe last expressionwasderivedusingthe approximationthat lossesin the

recycling mirror can be neglected‡:� �� " � � �

.

† The ratio 778

is calledthe recycling gain or the recycling factor andis a measureof the benefitdeliveredby

the recyclingmirror.

‡ Lossesof a few tensof partsper million areanticipatedin the recyclingmirror, whereasthe total round-triplossin

the recyclingcavity is expectedto be a few percent.

27

Since��

,

��

! �"��#$�%�"� � (2.39)

and from (2.29)

��&��"'(�)�*�,+(2.40)

Now we turn to the sidebandfields in an arm cavity. We begin with the

uppersideband;the derivation for the lower sidebandis almost identical. The

armcavity lengthis chosensothat thesidebandsarealmostexactlyantiresonant§,

i.e. -/.10 +. Referringback to Fig. 2.4,

�"� 0�32 0��& 0

�4�"576 ��85 �"9! 6:��5;� 9

(2.41)

and �)<� 0=

= -/.10� 2 0��& 0

� > � 85 � 9? ! 6:� 5 � 9A@ 8

(2.42)

Becausethe sidebandsare nearly perfectly excludedfrom the arm cavities,

we canapproximatethe expressionsaboveas beingunity andzero, respectively

(in our numericalexample,their valuesare 0.99995and 0.008).

§ If they wereexactly antiresonantthen the RF sidebandsat BCED would all be resonant,causingthe interferometer

responseto be very sensitiveto arm cavity lengthandmodulationdepth.SeealsoSection7.5.

28

Figure 2.7 Equivalentoptical configurationfor the analysisof sidebandfields.

E r

E symE anti

E f

E ret P

E ret I+

-

Accordingly, Figure 2.7 showsthe interferometerwith the arm cavities re-

placedby mirrors with unit reflectivity. Then

��

��

��(2.43)

��

��

� ��! �(2.44)

�#" � ��$�

% ��

� ��'&� �

� � ! �(2.45)

��(*) � � � ��

+��

� � ! ��,�

��-�(2.46)

We havestatedalreadythat the averagedistancebetweenthe beamsplitter

andthearmcavity front mirrors is|| .0/21+34 ; becauseof theasymmetryit is .5/21+34 &768

|| Seefootnote,p. 7, andSection7.5.

29

for the in-line arm and�� for the perpendiculararm. Then

��

��! " �$#�

�% �'& � �(��

� ��) " �$#�

�*�+�-,(2.47)

where, . �/ .

In similar fashiononecanshowthat102 ��3� ,

. It thenfollows that46587 � 4:9 � ;=<�>@?BA�,

(2.48)

4DCFE�GIH 4D9 � J=;�<KAMLOND,

(2.49)

Similarly 4 587 �QP 4:9 P � ;�<R>F?BA�,

(2.50)4DCSE�GTH P 4D9 P �3J=;�<KAULOND,

(2.51)

Finally, as we did for the carrier, we find the sidebandfields everywhereby

consideringthepick-off andeverythingto its right asa compoundmirror (shown

in Figure2.8). The difference�V,

between �K

and 0W

affectsthe transmission

(to theantisymmetricport)of thismirror. Theaverageof��K

and10W

determines

whetherthe cavity formed by the compoundmirror and the recycling mirror is

resonant.

From (2.13) through (2.15):4:9 4DHOE /

� ;MXY Z X\[]<�>F?BA�, (2.52)

30

Figure 2.8 Compoundmirror for the RF sidebands.

E r

E f

Compound mirror

E inc

E ref

��

�� !�� (2.53)

��"$#&% ��'�( �)�

��* +��,��-�� .�,�� (2.54)

��/�0 ��'�( �1� � �

32� �� !�-�� 4�� !��

� � ��,��-�� .�,��-�

(2.55)

(the approximateequality aboveholds when we neglectlossesin the recycling

mirror) and using (2.49) and (2.52)

��5 �768� ��'�( �9�

: ;�� <�.��=?>�� @��.�,��-� (2.56)

Let us summarizethis in words. For the carrier and for the sidebands,the

field is amplified in the recycling cavity becausethe circulating field interferes

constructivelywith the incoming field. For the sidebands,the amplification is

mainly limited by the transmissionto the antisymmetricport, whereasfor the

carrier it is limited principally by loss in the arm cavities.

31

2.4 Derivatives of Mixer Outputs

Now we candifferentiatethefields andfind thederivativesof mixer outputs.

We will considerfirst the differentialsignals: the derivativesof the gravitational

waveoutput �� andtheisolatorquadratureoutput �� with respectto phasechanges

��and � � . Then we will calculatethe common-modesignals: the derivatives

of thepick-off andisolatorinphaseoutputs�� and �� with respectto� �

and � � .First we note that since

� �� (2.57)

we have ��

��

��

�� (2.58)

and that for ! constantand ! "� � � � �$# � �&% "' � ! ' � �)(+*&,.-� � �� # � ! '

(2.59)

so that �� /� � (2.60)

whetherthe independentvariable is " or# �

.

32

Differential Signals

We beginwith the gravitationalwavesignal. Dif ferentiating(2.29)

��

��

��

��

��

�� "!$#&%' �

(2.61)

and, using (2.8), we have(at the antisymmetricport)

�)(�*�� ,+�-

�� . ��0/ ��1��/�243

� �65798 � .;: 2 8 / .<: 2< =!��>� ' �

��0/��?� ' /

# %' �(2.62)

which can be further expandedusing (2.49) and (2.51). Equation (2.62) can

be understoodas follows. When �

deviatesfrom @ we ceaseto haveperfect

destructiveinterferenceat the antisymmetricport for the carrier. The sizeof this

deviationfrom perfectdestructiveinterferenceis proportionalto the rate# %' � at

which thecarrierfield returningto thebeamsplitter from thearmcavitieschanges

with ��

. This carrier light then beatsagainstthe sidebandswhich are present

becauseof the asymmetry,producingmodulationof the photocurrentat A , and

a signal at the mixer output.

A similar mechanismis responsiblefor the sensitivity to B � at the antisym-

metric port. When B � deviatesfrom 0 we alsoceaseto haveperfectdestructive

interferenceat the antisymmetricport for the carrier. Quantitatively,to find C�DFECHG�I ,

again differentiate(2.29)

33

��

�"!$#&% ��"'(#�)*! ��"!+#&% � ,.-�/ % � (2.63)

We now move on to 0�132054&6 . From (2.45) the reflectivity of the compound

mirror for the upper sidebandis 7 -98�:<; �>=@? ��A�B � ; using (2.19) with C 7 -D8�:�; �.=@? ��A<B � we have

�� FEHG �

�"!$#&% � , I 7 -J;LK$M =B �FN / I 7 -O8�:�; =O� (2.64)

Similarly, �� PEQG � �

�"!$#&% � � , I 7 -R;FK$M =B �FN / I 7 -98�:�; =O� (2.65)

Sincethe conditionsleadingto (2.9) are satisfied,we have

��S�� T�� PE3G ��"!$#&% � , I 7 - ;LK$M =�FN / I 7 - 8�:�; =O�

(2.66)

which can be further expandedusing (2.38).

In words, this sensitivity in��S

to��

is due to the fact that when��

changes,the reflectivity of the compoundmirror increasesfor one RF sideband

and decreasesfor the other. This causesa correspondingchangein the relative

size of the sidebandsreflectedfrom the recycling mirror and a signal according

to (2.9).

Thereis alsoa small dependencein��S

on U � . Becausethe derivative(2.42)

of thearmcavity reflectivity for thesidebandis not exactly V , it is non-negligible

34

whenit is responsiblefor theonly contributionto thesignal. We re-insert��and �� into (2.45) and differentiateto get (using (2.9))

�� ! �" ��#$ ��#% �'&)(+* "�-,/. � "

0 �1$24365)798-:�<; � 1 243�=?>@5A: � (2.67)

Common-mode Signals

Now let us look at thecommon-modesignals.We beginwith� � , the inphase

pick-off signal. Using (2.31) and (2.18) with

B ��C* "� & " (2.68)

to find DDFEAGHJILKH�MON)P K we have:

�� RQ

�-S+T�U "�-,V. � " �

0 1 0 3 � �� "W �X; � 1 2Y3 �F� " � (2.69)

Then, using (2.7), we obtain

�� RQZ� �[��A \" ��#% ��#%^] 3 � S+T�U "�-,V. � "

� S�T?U �',V. � _ "

� �� "�� " (2.70)

which canbe further expandedusing(2.25), (2.26),(2.39)and(2.54),andwhere

_ " �;

� S+T�U "� � S+T�U "�a` Q

�;Hcbedgf KH�MhNXP K

��a` Q

� S�T?U "�',V. � "

�;

; �F1 2Y3 �F� "

(2.71)

is thebouncenumberfor the carrierin the recyclingcavity. In words: When � Qchanges,the phaseof the carrier light changesin proportionto the factors i &kjP K i& P K

35

(which is just theaugmentedbouncenumberfor thecarrierin anarmcavity) and

��, generatinga signal accordingto (2.7).

When �� changes,the phasesof the carrierandof the sidebandsall change,

so that by (2.7) the signal is proportionalto the differencein the ratesof change

of thesephases.To find �� we have,using (2.31) and (2.45),

�� (2.72)

for the carrier and

�� !�� "��$#&%('*) (2.73)

for the sidebands;then using (2.18) and (2.7), we have

+-,�+ �-� �/. �021 �(35476 1�8 39476;: � .=<9>@? �

. �BA � �.C<;>D? 8. �BA � 8FE �G� � 8IH (2.74)

where

� 8 � JJ � K7��L#�%('*) (2.75)

�G�and

� 8 arethebouncenumberfor thecarrieranduppersideband(respectively)

in the recycling cavity. The bouncenumberfor the lower sidebandequalsthe

bouncenumberfor the uppersideband.

We now move on to, 8 , the inphasesignal at the isolator. This signal is

producedby thesamemechanismas ��I�NM2� exceptthat therateof changeof carrier

phaseis proportionalto�PO�

insteadof��

. � �@Q�NM2� is found using (2.31), (2.21),

36

(2.19), and (2.7):

��

��

�� "!�# � � �$�%� � �'& � (2.76)

�)(�*�+� � � � �,.- � !0/ & - * !1/ &

��2��3��

�� *��3�� * 4 ��

� � � (2.77)

To find 576985;:7< we use(2.31), (2.19), (2.45), and (2.7):

� (�*�>=?�@� � �, - � !1/ & - * !1/ &

�A��2��A�3��

�A�� *�A�3�� * 4 �� 4 �*

(2.78)

Herethemechanismis thesameastheonefor 576CB5;:7< , exceptthatagain,augmented

bouncenumbersare used. With the approximationthat the recycling mirror is

lossless,we can write

4 �� #

D �� 2��>=?�

��$�E� � �

! � � �$�%� � �;& !F# � � ��%� � �7&(2.79)

and

4 �* �#

D ��C *� ��C *�>=?�

�� $��GIH%JLK

! � � �$��GIH%JLK & !�# � � �� GMH�JLK &(2.80)

Both (2.77) and (2.78) can be expandedusing (2.38) and (2.55).

This completesour analysisof the signalsof interest.Onecanshowthat the

remainingeightderivativesall vanish.Table2.1showsthefactorswhichvanishto

make(2.6)vanish,andTable2.3showsthefactorswhich cause(2.8) to vanish.

37

Table 2.1 Vanishing terms in equation (2.6) for derivatives of�� and �� with respect to the differential degrees of freedom.

Output�� Deriv-

ative

with re-

spect

to:

�� ! #" �$�

��%� �� &�'� �(� � � �! #" �$�

) � *� � �+� �� +� �� ,� � � �! " ��

*� � �+� �� +� �-� �(� � � �! " ��

Table 2.2 Vanishing terms in equation (2.8) for derivatives of��. and �0/ with respect to the differential degrees of freedom.

Output�1 �32Deriv-

ative

with re-

spect

to:

54�76��$� �� 8 ��

� � � � � �� *��8 �'� � � � �! ��

) 4�76��$� *� � �+*8 ��

� � � � � �� *+*8 �'� � � � � ��

Finally we consider the effect of a change in the laser frequency. Since, for

example, :9 6 �$;=<?> 9 � ; )�@BA*C#D�FEG,HI > 9 � ; ) @BA*C#D (2.81)

38

changes in this phase can be effected either by a change in the length of the in-line

arm cavity or by a change in the laser frequency. Now

��

�� (2.82)

and ��

�� (2.83)

so that, for example

��

��

��

��

��

��

��

��

��

(2.84)

where we can make the approximation in the last line because�

and because�� ! ��"��#$! . Similarly, we will use

��&%��

��

��'%�� (2.85)

which is a very good approximation for the same reasons.

In chapter 4 we will compare, for our numerical example, the values of the

derivatives found using this analysis and the values found by a numerical model

which makes fewer approximations.

It is easy to generalize the analysis above, of the response to particular

combinations of mirror displacements, to the case of arbitrary combinations. We

39

simply note that the following combinations of mirror displacements and changes

in laser frequency produce no signal at all. They are in the kernel of the Jacobian

of interferometer outputs with respect to mirror position and optical frequency.

1. Displacement of the entire interferometer towards (or away from) the laser.

2. Displacement of the arm cavities towards the beam splitter and displacement

of the recycling mirror away from the beam splitter, all by equal amounts.

3. A change��

in the laser frequency, together with displacements of the

arm cavity input mirrors PF and IF away from the recycling mirror, through

distances �� and � �� , and displacements of the arm cavity back mirrors

PB and IB through distances of � � �� and � � �� respectively.

These three combinations have the property that they leave the relative phases of

any interfering beams unchanged.

From this observation we can conclude, for example, that a displacement of

the recycling mirror towards the beam splitter will generate the same signals as

an equal-sized displacement of all of the arm cavity mirrors towards the beam

splitter, since the two differ by an element (#2 above) of the kernel. Similarly,

displacement of the beam splitter and of the recycling mirror, such that � and � �change by equal and opposite amounts, differs only by an element of the kernel

from a displacement of the two arm cavities in opposite directions in the same

amounts, and the two combinations must produce the same signal.

40

Chapter 3 InterferometerFrequency Response

Implicit in the analysisof the last chapter,sincewe did not allow any of the

fields��

,��

, or��

to dependon time, is the assumptionthat changesin the

phases� and occur slowly. Supposeinsteadthat we want to know what the

responsein the output signalsis when the laser frequencychangesabruptly, or

whena mirror vibratesat a high frequency.This is the problemwe will analyze

here.

3.1 Fabry-Perot Cavity

As in Chapter2 we will begin by analyzinga single cavity in somedetail,

andthenextendingthe resultsto the completeinterferometer.The responseof a

Fabry-Perotcavity to backmirror motionhasbeenunderstoodfor sometime13,14,

but we will re-deriveit becauseour methodwill includetheRF phasemodulation

in a naturalway andwill be easilygeneralizedto morecomplexoptical systems.

Our cavity, shownin Figure3.1, consistsof two mirrors A andD, separatedby a

distance suchthat the carrier resonatesin the cavity but the two RF sidebands

do not.

41

Figure 3.1 Fabry-Perotcavity.

A DE T

E M

E R

LaserPhaseModulation Isolator

Photodetector

+ - + -

d

If both mirrors are stationary,then as in (2.2), the field at the back mirror

essentiallyconsistsof three frequencies:

�� (3.1)

Now supposethe back mirror is shakingwith small amplitudeat a frequency�( � � ), i.e.,

� �"!$#�% ��&'� ()� �+*,� ( -�.(0/2143 � # 5 1467()8 (3.2)

This motion modulatesthe optical pathwithin the cavity so that

��9:�.; � ��<�=�?>A@�BDCFEAGH*,�; � �� IJLKNMO(P�H�Q*,��LKRMS(P� � ��*,� (3.3)

Clearly if� �

containsthree frequencies

� � �� (3.4)

then��9

containsnine:

��9T�.; �"U KRMS()��V �� XW � Y�*[Z+� \�� LKRMS(P�� ]� � � W � � *SZ?�

42

�� !��"��#�� %$ �'&)(*�+��,-� �.� %$ �)(/� �� %$ �0&)(�12��,��43

(3.5)

We will call the additional six frequencies“audio sidebands”imposedon the

original three“RF frequencies”(the carrier and its two RF sidebands),and we

will identify themwith an additionalindex. Using this notation,we write:�576��5 � $ � $ �+�0&�(�12��,-� � ��5 $*$ �'&�(81��,-�

6$

9 : � $$

;<: � $� 5 9�; �0& 9 (81 ; ��,=� (3.6)

In thesummationabove,> indexesRF frequencyand ? indexesaudiosidebands.

To avoid ambiguitiesbetweenthe useof the “-” to indicatea negativeindex and

a subtraction,we will encloseany operationin parentheses;for exampleif > 6�@we would write

� & 9 � $ , for� � $ .

Now supposethe field�

incident on the moving mirror containsnine

frequenciesinsteadof only three(which in generalit will, sincemirror A reflects

�5backto mirror D). Thenby (3.3)

�5is a productof a factor (

� ) containing

nine termsanda factor A �B�� )�C� �B�� +��C� of threeterms,so� 5

has27

terms(at 15 distinct frequencies).However,12 of thesetermsareof order D ��E�Fandwe neglectthem. The remaining15 terms(at nine frequencies)are:

� 5 6HG�I$

9J: � $� 9 K � 9 (L� � ;M: � $ONP$ D

� 9�; �� 9JK EQ �0& 9 (�1 ; �R,=�(3.7)

43

Evidently the shakingof the mirror causes,for example,an amount

�� (3.8)

to be added to the incident� � ��

field upon reflection (and corresponding

quantitiesto be addedto the other audio sidebandfields). The moving mirror

actsasa sourceof this frequencycomponentof the light. As in (2.10)— (2.15)

we cansolve for the field at this frequencyeverywhere.At the two mirrors, we

have:

�� !�#" ��(3.9)

� " �� %$ ��(3.10)

Solving:

��" �� && $ � � � � � �'�(� �

��(3.11)

�) �� * �� '�(��+ -,/.& $ ��0��1� �'��

�2��(3.12)

if the carrier is resonantin the cavity,

32�� 54 687:9<;+=<>(9@?�4�A(3.13)

44

�� !��

(3.14)

"$#�%'&)(+* � "�#�%',+- (3.15)

"$#.%/& (+*1032 � 4 " #�%',+- 032 (3.16)

Then

576 �� 498;: " #�%',+-<0�2= > : >? " #.%',@-�A

��(3.17)

We call the ratio of the audio sidebandfield at the photodetectorto the source

A��

of that field the transmission function B �C� from the shakingmirror to the

optical output and write

5 6 �C� � B �C� A�C�

(3.18)

We find for the lower audio sidebandon the carrier

576 � # � �4D8 : " %',@-E0�2

= > : > ? " %',+-FA� # � (3.19)

Similar expressionscan be found for the other audio sidebands.

Givenfull informationaboutthespectralcontentof5 6

we still needto know

the mixer output when this light is presentat the photodetector.As before,the

photodetectoroutput is

45

��

��

��

� � �� !��"$#�%�

(3.20)

Again, as in (2.3), this containstermsat a numberof frequencies,of which only

thosenear & will bemixeddownto nearDC andpassthroughthe low-passfilter.

Moreover,of thosefrequencies,we areonly interestedin the onesat frequencies

of &('() and & ) becausethesewill bemixeddownto ) , andit is theresponse

in the mixer output at ) that we want*. Thesetermsare

� �+* %$�-,." 0/21 �3

��

3

�� 4�65� �+� � � � �� # � �� # �7��

�-�!"8#�%

' �95� � � �:� � #� �

� �� #;� �7��<� "8#�% �

(3.21)

Then

=�> ��?�+ @A%

%B� �� C* %$� ,!" �ED F�GIH & �JDLK;�JD

01 �3

��

3

�:�� B� 5� �� # � �� # '

� � � � �� #� 5�

� �� #;�� "M%

(3.22)

which is in the form

=�> ��?�+01 � N > � � "M% (3.23)

* The only othercomponentpassedby the lowpassfilter is at DC, correspondingto (2.4) and (2.5). We areusually

interestedin the responseto small motionsaroundthe point of perfectresonance,wherethis term vanishes.

46

with

��

��

�

�� !�#" (3.24)

so that� �

is the size of the responseand $&%(' � � � " is its phase.

Hencewe definethe transferfunction from mirror position to inphasemixer

output

) �(*�+-,/.� �0 (3.25)

which is a function of frequency.By a similar argument,we canshowthat

�21 .�

��

�

�� 3� �� 4� �� 5� (3.26)

In principle, this completesour analysisof the frequencyresponse.For a

givenamplitudeandfrequencyof mirror motionwe know how to find thespectral

contentof the light on the photodetectorand we know how to find the mixer

output given that spectralcontent. This algorithm is well suited to numerical

implementationandis in fact theoneusedin theprogramdescribedin Chapter4.

Let us examinethe expressionsfor� �

and� 1

morecarefully. Eachterm in

theseexpressionsis theproductof anRF field which is independentof theshaking

frequency6 and either an upperaudio sidebandfield or the complexconjugate

of a lower audiosidebandfield. Eachaudiosidebandfield is proportionalto the

47

transmission function from the shaking mirror to the optical output or to a sum

of transmission functions if several mirrors are shaking or if the shaking mirror

is illuminated from both sides. In any event, the transfer functions from�

to ��and �� are linear combinations of transmission functions and complex conjugates

of transmission functions.

It is often the case that a few reasonable approximations can reduce these

linear combinations to a very simple form. In our analysis of the interferometer

we will see that the transmission functions for the audio sidebands on the RF

sidebands are all frequency independent, and that the complex conjugate of each

transmission function for the lower audio sideband on the carrier differs at most

by a constant from the corresponding transmission function for the upper audio

sideband on the carrier. From these two facts we can state immediately that

� �� (3.27)

for some constants � and � (and a similar expression for �� ). The constants �and � can usually be found from the DC model of Chapter 2; the transmission

function �� will need to be derived separately.

For the simple cavity being analyzed here we assume that the RF sidebands are

essentially absent from inside the cavity so that the sources of audio sidebands

on the RF sidebands are negligible:

48

�� (3.28)

This means that the corresponding four terms in �� vanish, and

�� ! �� (3.29)

We also restrict our analysis to frequencies much smaller than the cavity free

spectral range:

"�# $ (3.30)

So that (3.17) becomes (by neglecting the exponential in the numerator† and

Taylor expanding the one in the denominator):

� � �� %'&)(+* �-,/.103254

$ 6 ( 6�7 * ��,/.80 �9��%:& ($ 6 ( 6�7

$$ � ,;..=<?>'@ �-�A�

(3.31)

where "CB'DFE ��5��GIH=GKJGIH=GKJ�0 .

For the lower audio sideband on the carrier,

� �� %:&)(

$ 6 ( 6!7$

$ � ,;..=<?>L@ � ��+�� (3.32)

since �NM ��is pure imaginary,

� ��+�� % 6!7PO8Q � �M �� % 6�7NO8Q �PM �� 9��(3.33)

† The exponential in the numerator only produces a small phase change in the frequency range of interest.

49

so�� and, using (3.29)

��

�"! #%$$'&)(+* (3.34)

For compatibility with notation used in references on control systems, we may

let , .-0/ and write

� 21436587:9<; ,�= � (3.35)

where

1>3?5+7:9<; ,@= A1 3?587:9 �CB�

��! D$�&)(E* (3.36)

and1 3?587:9 �CB GF

�FIH �

��J�C�K� �L� �M� � ��N�� O� �@�(3.37)

What if we were to shake the front mirror instead of the back mirror? We

could use the same method to answer this question, adding an additional source

for each of the audio sidebands (since they would now be produced at both

surfaces of the front mirror). This is in fact the approach taken in the program.

In our analytic derivation, however, this becomes quite tedious since we can no

longer neglect the audio sidebands on the RF sidebands, and since, for each audio

sideband on the carrier, there are different transmission functions from each of

the two sources to the photodetector.

50

Insteadwe use a simple thought-experimentto argue that if the carrier is

resonant,theresponseto motionof thetwo mirrorsmustbethesamein amplitude

(andoppositein phaseexceptfor a factorwhich is negligiblefor frequenciessmall

comparedto��

). Supposewe shakeboth mirrors with the sameamplitudeand

in thesamedirectionandwith a relativephasesuchthatmotionof thefront mirror

lagsmotion of the backmirror by a time delayof��

. Now if the backmirror,

mirror D, is moving with a certainvelocity whena particularbit of light reflects

from it, thenall of thefrequencycomponentsin thatpieceof light will beDoppler

shifted to new frequencies,only to be shifted back to their original frequencies

when the bit of light reflects from mirror A, after travelling the length of the

cavity, which takesa time��

. The light travelling towardsthe backmirror then

hasno audio phasemodulationbecauseit is a combinationof fresh light from

outsidethecavity andlight which hasbeenmodulatedby reflectionfrom theback

mirror and un-modulatedby reflection from the front mirror. The light inside

the cavity travelling towardsthe front mirror then hasa total phasemodulation

correspondingto onereflectionfrom a moving mirror, asdoesthe light reflected

from the outsideof the front mirror. All of the RF frequencycomponentsof the

light returningto the photodetectorthenhavethe sameaudiophasemodulation,

that amountcorrespondingto onereflectionfrom a moving mirror:

� � �� "!�� ! ��#� � �%$'&)(*��+,��-$'&)(*� ! ��+,�(3.38)

51

which producesno signal at the mixer output. Sincethe mixer output is linear

in (small) mirror motions,andsincethis combinationof mirror motionsproduces

no output,eithermotion by itself mustproducethe oppositesignalasthat which

the other motion by itself would produce.

Finally we want to show that the responseto laserfrequencymodulationis

the sameas the responseto motion of the back mirror; we will show that for

frequenciessmall comparedto��

,

�� (3.39)

The argumentis the following:

Modulating the laser such that�� "!#�%$'&#(*)��

, for exampleby

changingthelengthof thelasercavity, is equivalentto modulatingthelaserphase:

+-, �.�/� 0+-, ��21436587:9<;=?>(3.40)

with

@ �A�B) !#�(3.41)

andwe havealreadyseenthat the latterwill imposeaudiosidebandson the light.

Now thefield+DC

on thephotodetectorasbeforewill containninefrequencycom-

ponents.Theaudiosidebandson theRF sidebandswill befrequencyindependent,

52

whereastheaudiosidebandson thecarrierwill havesomefrequencydependence.

�� !�

" �#�

�� " �#� � � �� $�

�� &%'�(*)!+

" �,� (3.42)

where" �#� -

is againthe sourceof audiosidebands(in this casethey originate

in the laser).

Becauseof this, someof the termsin theexpressionfor .0/ will be frequency

independent,andsomewill havea�21 � � � %' (�)!+ frequencydependence:

. / � 34�65 �� %' (*)!+

-(3.43)

Now57� 3

, becausein the limit 8 9 the responsein . / to finite :<; is finite

(seefor example(2.82)); hencethe responseto finite-

vanishes.

. / �=3 � �� %' (*)!+

-

�?>�@ 38BA�CED

�� &%' (�)!+ :<;

(3.44)

We know (as in (2.81–2.85))what the low frequencylimit of this responseis:F�G /F ;

�IHJ@�KL

FMG /FON�IHJ@�K

L�

><PFMG /FMQ

�RK;

FMG /FMQ

(3.45)

Substituting,we get:

S DUT�VXW �RK;

S DYT[Z (3.46)

53

3.2 Response of the Complete Interferometer

At this point we arereadyto computethe frequencyresponsecorresponding

to the derivativeswe found in Chapter2. We will proceedin roughly the same

orderaswe did there,finding first theresponse,asa functionof frequency,in the

gravitationalwave output �� and in �� , the isolator quadraturephaseoutput, to

��and � � . Theseresponsesarerelativelysimplebecausefor eachof themeither

theaudiosidebandson thecarrieror theaudiosidebandson theRF sidebandscan

be ignored.Moreover,we will arguethat the transmissionfunction for the audio

sidebandson the RF sidebandsis frequencyindependent;this further simplifies

two of the transfer functions.

Next we will analyzethe responsein thecommon-modeoutputs�� and �� to

� �and � � . The responsesto � � arethe mostcomplexoneswe will encounter.

In theseresponses,all of theaudiosidebandscontributesignificantlyto thesignal.

Throughoutour analysisof frequencyresponse,we will restrictour attention

to frequenciesmuchsmallerthanthe free spectralrangeof an arm cavity:

�� (3.47)

(where �� ). This allows us to makea numberof simplifying

approximations,andmoreovergivesus all of the informationwe needto design

any of the servoloops which will feed back to a force on a mirror, sinceall of

theseloopsareconstrainedto haveunity-gainfrequenciesmuchsmallerthanthe

54

arm cavity free spectralrange,due to internal resonancesin the mirrors. The

loop which feedsback to the laserfrequencycanhavea bandwidthapproaching

or evenexceedingthearm cavity free spectralrange;hereour analysisis of little

help. For this one loop we rely on a numericalmodel (seeChapter4) to verify

that the interferometertransferfunction is sufficiently well behavedto permit us

to implementa stablecontroller.

Differential Signals

We beginour characterizationof the interferometerwith the signal in which

wewouldobservegravitationalwaves.SupposethemirrorsIB andPBareshaking

at frequency � with oppositephase. Eachwill be a sourceof audio sidebands

on the carrierandthesesidebandswill propagateout to the beamsplitter. At the

beamsplitter, they will interferealmostperfectlydestructivelyon the symmetric

side and almostperfectly constructivelyon the antisymmetricside. The reason

that the interferenceis not perfect is that one pair hashad to travel a distance

�farther than the other. For an asymmetryof 57 cm and an audio sideband

wavelengthof �� (correspondingto a shakingfrequencyof 10kHz),

a fraction �� "!$#&%(' )+*-, ) ��. / of the audio sidebandpower is diverted

from the antisymmetricport to the symmetricport. We will ignore this effect.

Againwe ignoretheeffectof thearmcavitybackmirrorson theRFsidebands.

Theneachtermin 0�1 is proportionalto thetransmissionfunctionfrom thesources

55

of audio sidebands (at the back mirrors) to the antisymmetric port:

�� (3.48)

where

�� ! � � � �"�#%$�& (3.49)

The response at this output to ')( is essentially frequency independent. To

see this we imagine shaking both mirrors of the in-line arm cavity in the same

phase, and at the same time shaking both mirrors of the perpendicular arm cavity

in the opposite phase. As we saw in the argument leading to (3.38), this produces

frequency-independent audio phase modulation of the light returning from each

cavity. The phase of the corresponding (frequency independent) audio sidebands

in the light returning from the in-line cavity are opposite in phase from the audio

sidebands in the light returning from the perpendicular arm cavity, and all of this

audio sideband light exits the interferometer at the antisymmetric port. Thus‡

� ��+*,� -� ��.*,�/�0�(3.50)

Now let us consider the response in 1,2 , the isolator quadrature phase output, to

3 ( and '4( . First we note that the audio sidebands on the carrier produced by

changes in3 ( and '4( interfere destructively on the symmetric side of the beam

splitter and therefore are absent at the isolator.

‡ The audio sidebands on the RF sidebands do not contribute to this signal since 57698�:<;>=,=@?BA

56

For the response to��

, we consider the compound mirror of Figure 2.8 as

the source of audio sidebands on the RF sidebands. For the cavity in which these

audio sidebands resonate (consisting of the recycling mirror and the compound

back mirror),

�� !�"� #%$'&)(*�,+.-0/ 21'354 17689;:�:=<?>A@ (3.51)

in our numerical example. This is far outside of the frequency range of interest

and we approximate

�� 4 BCEDGFIH � (3.52)

to get

>KJL!MON >PJLMON�Q�R(3.53)

For the response to S �, as in (2.66), the source is the small audio sidebands

put on the RF sidebands by motion of the arm cavity back mirrors. We repeat

the steps leading up to (3.17), using�UT�V : �XW �?Y�Z�[\�K],^_� for an RF sideband

in an arm cavity, to get

`ba T�T c�dfe�hg C;ikj DGlnm�o

� 4 d ,p e ��g C�i0j DGlrq TUT(3.54)

which is essentially constant in the frequency range of interest. The same thing

is true for the remaining audio sidebands on RF sidebands, and we conclude that

the response is frequency independent:

> JL!stN > JLstN�QfR(3.55)

57

Common-mode Signals

Next we turn to the responseto��

, which correspondsto commonmode

changesof arm cavity length. Whenthe arm cavity backmirrors shakein phase,

the audio sidebandson the carrier interfereconstructivelyat the symmetricside

of the beamsplitter. Becausethe carrier audio sidebandlight coming from the

two arm cavitieshasthe sameamplitudeandphaseandbecauseany of this light

reflectedback towardsthe cavitieswill be treatedin the sameway by the two

arm cavities,we can15, for the purposeof calculatingthe responseto this type

of mirror motion, replacethe beamsplitter and arm cavitieswith a single arm

cavity, as shown in Figure 3.2.

Figure 3.2 Equivalentsystemfor the purposeof deriving common-moderesponse.

R

BF

The mirrors R and F form a compoundmirror§16,17 with (using (2.15))

reflectivity from the right of

��

� � �� (3.56)

§ Implicit in this simplifying substitutionis the assumptionthat the reflectivity of the compoundmirror is frequency

independent.This is in fact a very good approximation.Letting �� "!$#$% (with �"!"#&%'�)(+*% in (2.16), (2.21) gives

roughly , -.�+!$#&% as the rateof changeof phaseof the reflectivity with � . This is exceedinglysmall comparedto �"/0!"1 ,

which is the rateof changeof round-tripphasedueto the lengthof the arm cavity.

58

so that

�� (3.57)

where

!#"�" � $&%('*)+$&,$&%('�)+$�,-&.�/10 (3.58)

Noting that

2 /436587:9 �;2�=< 0>7:91? ) ?A@ (3.59)

2 /4365�7#BC9 �;2�6< 0>7:BC91?4)?A@ (3.60)

we have immediately

�D�4E6��=�� 8��4E=��F�8� �� (3.61)

It is useful to write our expression for !:"�" , in terms of individual mirror parameters,

in a more compact form. We write

$&% � � GH% %� GI%J %J

(3.62)

where % � KL% G % is the loss in mirror F, and

G @ � � K�@ @ (3.63)

$ ) � GI)J )J (3.64)

59

to get

��

� � �� (3.65)

and*

�� "!$#&% � � � �� '� � �� () � ( (3.66)

Unlike our previousexpressionfor �*�� , theevaluationof this expressiondoesnot

requiresubtractingnearly equalnumbers.

Thetransferfunctionsfrom +-, , changesin which correspondto displacement

of all four arm cavity mirrors towards(or away from) the beamsplitter, are the

most complexof the oneswe will analyze. As we saw in the analysisleading

to (2.74), the low-frequencyresponsein the inphasepick-off signal . � is due to

the fact that whenthe recyclingcavity lengthchanges,the carrierphasechanges

in proportionto /10 andthe sidebandphaseschangein proportionto /32 , so that

a signal is producedaccordingto (2.7). To generalizethis result to the caseof

mirror motion at arbitrary frequencies,we first usethe audiosidebandpicture to

explainthesameresult. At very low frequencies(wherethetwo modelsoverlap),

the resultantphasorformedwhenthecarrier(or oneof theRF sidebands)andits

two audio sidebandsare summedis a phasorslowly rotating back and forth at

* The setof termsin parenthesesrepresentsthe setof “leaks” throughwhich carrier light is able to escapefrom the

system. If the laserwere abruptly shut off, it would take the field about1/10 of a secondto decayto 1/e of its initial

amplitude;in fact this stepresponserepresentsan alternateway of calculatingthe frequencyresponseto certaininputs.

60

the shakingfrequency. The frequency-dependentanalogof the statementabove

involving phaseschangingat differentratesis theremarkthat theaudiosidebands

on the RF sidebandsbeatagainstthe carrier,producinga signal which partially

cancelsthe signalproducedwhenthe audiosidebandson the carrierbeatagainst

the RF sidebands.

Now we seethat as the shakingfrequencyincreases,the contributionto the

signalfrom theaudiosidebandson theRFsidebandswill beconstant(asin (3.52))

whereasthecontributiondueto theaudiosidebandson thecarriermayhavesome

frequencydependenceproportionalto the transmissionfunctions��

and��

.

Anticipating that� ��

we write

�� "!$# � %�&('*),+.-%�&('�) � - # �# � # � (3.67)

Let us find��

. We considerthe arm cavities and beam splitter as the

compoundback mirror and sourceof audio sidebandsfor the cavity having the

recycling mirror and pickoff as its compoundfront mirror. The reflectivity of

this backmirror is frequencydependent.Using (2.15)with / ��0�� 2143 ��50687:9we get

;�< � 1=3 �"5>6?7:9$� � ;�@ A�B@ ;�CD ;E@F;�C � D �G��5>687:9H�� ;�< � D 7�I7�J &LK ++ JD 3 K ++ J

(3.68)

61

Now accordingto (3.11) and(3.14) with��

and* �� we have

�� "!#�$�&%(')�*�+-,�� .0/ �+�

� �� 1� �2!#�1� �

��4365587��93655#7:7/ ��

(3.69)

wherewe haveused† ��;� ��< �>=0?�@�� 7BA�>C �>=0?�@��>D �� . To find E �F<G� , we repeatthe above,

using � insteadof � . andconcludethat EIH�F<G� � E �� . Substitutinginto (3.67),

we get

JLKNM�O�P %�Q . � JLK�M>O$P"RTS �U�VQ WFXWYX < WGA�5#7:7 WGAWFX < WZA

�5#7�U� �

5#7[7(3.70)

Neglecting WGAW X < W A�5 7 next to WFXW X < W A

�5 7:7 we have

J\K�M>O$P %�Q . � JLK�M>O$P2R+S ��] X�^ ] A] X 5_7:7

�U� �5#7[7

(3.71)

To derive the responsein ` � to�aC

, we use exactly the sameargument,

substitutingaugmentedbouncenumbersbdc� and bdc� for the bouncenumbersb �

and b � , to get:

J K X O$P %�Q . � J K X O$P2R+S�e�gfh� WjiXW iX < W iA

�5_7:7 WeiAW iX < W iA

�5#7

��93B55#7[7J K X O$P_R+S

�e� �] iX ^ ] iA] iX 5#7[7

�� 5#7:7

(3.72)

* More relevanthereis the fact that k;l[m n;o1prq&s tvu;w[x"y[z|{~}�y6��U�_�~�;�~��:�† This can be shown to be equal to our previous expression for �8}[} by simplifying as we did in (3.65).

62

wherethe approximateequivalenceagainis derivedby neglecting�� next

to� �� .

Finally, since we intend to feed back to the recycling mirror, we needto

understandthe responseto recyclingmirror motion*. We define ��

to be the

set of transferfunctionsfrom recycling mirror displacement( �� "!$#&%(' ) to

the mixer outputs. We can seeimmediatelythat

)�*�� *�,+��(3.73)

by an argumentsimilar to the oneadvancedin Section3.1. Therewe found that

for a Fabry-Perotcavity, theeffect of motionof the front mirror differsonly by a

phasefrom theeffect of backmirror motion. If we shaketherecyclingmirror, the

armcavity front mirrors,andthearmcavitybackmirrorsall at thesamefrequency

andin thesamephase,thenagain,to a goodapproximation† we imposeanequal

audiophasemodulationon eachRF frequency,generatingno signal. This means

thatmotionof therecyclingmirror by itself producesa signalequalandopposite‡

to the signal producedby motion of the other four mirrors.

* Togetherwith the responseto beamsplitter motion of AppendixC, this will completethe analysisof the dynamic

responseof the interferometer.

† This result is exactif the mirror motionsarenot exactly in phase,but insteadthe arm cavity front mirrors leadthe

recycling mirror by the time it takeslight to travel acrosstheir respectiveseparations,and the arm cavity back mirrors

leadthe front mirrors by the time it takeslight to travel the lengthof an arm cavity.

‡ This sign differencedoesnot appearin equation(3.73) becausewe use the oppositedirection of motion for the

recyclingmirror to define - � .

63

In thenextchapterwewill compareplotsof thefrequencyresponsecalculated

accordingto our simplified analysisaboveto plots of the responsecalculatedby

a purely numericalmodel,which makesfewer approximations.

64

Chapter 4 Numerical Models

Two numericalmodelshavebeendevelopedfor the purposeof analyzingthe

optical system.

The first one,the “numericalDC model” calculatesthe samederivativesthat

we found in Chapter2§. Written in Mathematica,it finds symbolic expressions

for all of the relevantfields in the interferometer.It then differentiatesthem as

required to find the symbolic derivativesof the mixer outputswith respectto

the phases.BecauseMathematicadoesnot makethe approximationswe did, the

expressionsfor thesederivativesarequitelong, typically filling severalpages.Into

theseexpressionstheprogramsubstitutesnumericalvaluesfor mirror reflectivities

and phases,to generatea table of derivatives. The sameprogramis of course

alsoableto find valuesfor the fields at the outputs;thesenumbersareuseful for

shot noise calculations.

Thesecondmodel,called“Twiddle,” calculatesthefrequencyresponseof the

interferometer.To understandhow it works, we considerthe simpleFabry-Perot

cavity, fed by a sourceof light of unit magnitude.

The equationsfor this systemmay be written

��(4.1)

§ A modelwhich calculatedthemixer outputsignalsinsteadof their derivativeswaswritten in 1991by S. Whitcomb.

The modeldiscussedherewasadaptedfrom that earliermodel.

65

A DSource E a

E b

E c E e

E d E f

E g

E h

E i

φ 1 φ 2+ - + -

�� (4.2)

�� (4.3)

��! #"�$(4.4)

��%�� & '"�$(4.5)

��()�*��+,�.-(4.6)

� �� + ��-(4.7)

��-/��0��! #"�1(4.8)

� � �2��(3��! '"�1(4.9)

66

Thereis one equationfor the source,a pair of equationsfor the front mirror, a

pair of equationsfor the spacebetweenthe sourceand the front mirror, and so

on. Thesecan also be written in matrix form:

�

��...��

�

�...

(4.10)

and solved by matrix inversion. The Twiddle program constructsthe matrix

�from a set of commandsspecifying mirror propertiesand separations.For

example,a set of commandsfor constructingthe cavity aboveis:

s1 = source[gamma]

m1 = mirror[r1,t1]

connect[s1,1,m1,1,length1]

m2 = endmirror[r2,t2]

connect[m1,2,m2,1,length2]

Eachcommandabovecorrespondsto one group of equationsfrom (4.1) to

(4.9).

In addition to thesecommands,the program needsto be given a matrix

containingthe propagationphase,for the carrier and for eachRF sideband,for

eachconnectiondefined.The programthencalculatesthe propagationphasesfor

the audio sidebandsfrom the lengthsof the connections.When calculatingthe

audio sidebandfields, the equationsare slightly different becausethe sourceof

67

this light is in a different location. If the back mirror is shaking,then it is the

sourceof audio sidebands;equation(4.1) becomes

��(4.11)

and equation(4.6) becomes

�� (4.12)

where�

is proportionalto the amplitudeof the mirror motion and the RF field

��found in the previousstep. Thus to find the audio sideband,the program

inverts the samematrix (which howevercontainsdifferent valuesof �� and ��sincethe frequencyis different) and multiplies it with a different sourcevector

to solve for the fields:

��...��

��...��

(4.13)

In summary,Twiddle begins by constructingthe matrix�

from a set of

commandsspecifying mirror positions, reflectivities and transmissions,and a

matrix giving the propagationphasefor eachfrequency. It inverts this matrix

andmultiplies by the appropriatesourcevector to find the RF fields everywhere.

Next it formsthesourcevectorsfor theaudiosidebandsfrom theRFfieldsincident

on the mirror beingshaken.Finally, for eachfrequencyof interest,it insertsthe

appropriatepropagationphaseinto thematrix�

andsolvesfor theaudiosideband

68

fields everywhere.From the valuesof all of thesefields it calculatesthe mixer

output using|| (3.24) or (3.26).

As mentionedearlier,thenumericalmodelshavetheadvantageof beingmore

precisethan the analysesof Chapters2 and 3. Both numericalmodelsinclude

the effectsof loss in the recycling mirror andbeamsplitter andneitherneglects

the RF sidebandfields in the arm cavities. The numerical frequencyresponse

model is not limited to frequenciesmuchsmallerthan the free spectralrangeof

the arm cavitiesand is believedto be accurateup to a significantfraction of the

modulationfrequency.

It is instructive to seehow the calculationscompare. Table 4.1 showsthe

valuesof the derivativescalculatedin Chapter2, evaluatednumerically using

the parametersof Table 1.1, and the derivativescalculatedby the Mathematica

model describedabove. Evidently the amount of accuracysacrificed to the

approximationsmadein Chapter2 is not large.

Theplotsin Figures4.1and4.2showthecomparisonbetweentheapproximate

frequencyresponseanalysisof Chapter3 andthe Twiddle program.Eachplot is

labelledH_nm; the first index indicatesthe output, i.e.,��

through��

, and the

secondindex indicatesthe driven degreeof freedom,i.e., �� , �� , �� , and �� ,

in that order. We seefrom the plots that the approximateanalysisis goodup to

|| The program containsa generalizedform of theseequationswhich can include the effect of having more RF

frequencies,aswould be appropriatefor larger modulationdepth.

69

Figure 4.1 Tableof numerical(upper)and approximateanalytic (lower) derivativesof interferometeroutputs,for the optical parametersgiven in Table 1.1.

Derivativewith respectto��

� -52.6161-52.5585

-0.226582-0.221949

0. 0.

�� -4.06208-4.09635

0.006640.00676

0. 0.

�� 0. 0. -27.9861-28.1611

-0.213105-0.214438

Ouput

�� 0. 0. 0.0002000.000207

0.02631280.0271507

frequenciesof about1 kHz. The plot of H_11 extendsover a broaderfrequency

rangethanthe others;this is useful for the numericalservodesignof Chapter6.

The fact that the two curvesin this first plot agreeup to very high frequencies

is accidentalsincethe approximationsmadein deriving the approximateanalytic

expressionfor this responseare violated at high frequencies.

70

Figure 4.2 Frequency response curves calculated using the approximateanalytical method and (dashed) the strictly numerical model.

10-5

100

105

1010

0

100

200

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

100

102

104

106

108

10-10

10-5

100

105

Frequency (Hertz)

Log

Mag

nitu

de

H_11

10-2

10-1

100

101

102

103

104

-100

0

100

200

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-1

100

Frequency (Hertz)

Log

Mag

nitu

de

H_12

10-2

10-1

100

101

102

103

104

0

100

200

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-4

10-2

100

102

Frequency (Hertz)

Log

Mag

nitu

de

H_21

10-2

10-1

100

101

102

103

104

-100

0

100

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-3

10-2

10-1

Frequency (Hertz)

Log

Mag

nitu

de

H_22

71

10-2

10-1

100

101

102

103

104

0

100

200

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-1

100

101

102

Frequency (Hertz)

Log

Mag

nitu

de

H_33

10-2

10-1

100

101

102

103

104

100

150

200

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-1

100

Frequency (Hertz)

Log

Mag

nitu

de

H_34

10-2

10-1

100

101

102

103

104

-100

0

100

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-4

10-3

Frequency (Hertz)

Log

Mag

nitu

de

H_43

10-2

10-1

100

101

102

103

104

-100

0

100

Frequency (Hertz)

Pha

se (

degr

ees)

10-2

10-1

100

101

102

103

104

10-2

10-1

Frequency (Hertz)

Log

Mag

nitu

de

H_44

72

Chapter 5 Experiments withthe Table-top Prototype

Severaltable-topoptical configurationsof gradually increasingcomplexity

weresetup over the pastseveralyears.This setof experimentsculminatedwith

theconstructionof a completeinterferometerto bedescribedlater in this chapter.

The formal goal of theseexperimentswas to verify a reasonablenumberof the

predictionsof the low-frequencyoptical model#. A secondarymotivation was

simply to demonstratethe generalfeasibility of this extractionscheme.A large

numberof unmodelledeffects could in principle make the operationof sucha

configurationimpractical: extremesensitivity to misalignment,to asymmetryin

losses,to mirror imperfections,or difficulty in acquiring lock being important

examples. Although the successfulconstructionof a small scaleprototypeby

no meansproves that none of theseeffects will be important in a full-sized

interferometer,it nonethelessgives one somecomfort.

The first section of this chaptercontainsa descriptionof the parts from

which all of the experimentswere set up. The following sectionsdescribe

the experimentsin order of increasingcomplexity, from the method used to

characterizethe displacementtransducers,to the characterizationof a simple

# It is difficult to test the frequencyresponsemodel in a table-topexperimentbecausethe lengths(hencedelays)

are shorterand the losseslarger; both of theseeffects lead to optical characteristicfrequencieshard to attain with the

displacementtransducersused.

73

Fabry-Perotcavity, to the characterizationof the three-mirror“coupled cavity”

interferometer,andfinally, to the characterizationof the full interferometer.

5.1 Setup and Hardware

All of the experimentsdescribedin this chapterhave in common certain

aspectsof the setup and certain building blocks from which the optical and

control configurationswereassembled.A (CoherentInnova100) argon-ionlaser

operatingat 514 nm servedas the light source. This laser had beenmodified

by replacing its factory-installedmirrors with mirrors glued to piezo-electric

transducers,sothatthelengthof thelasercavity,andtherebythelaserwavelength

could be adjustedelectrically. Furthermoretwo Pockelscells (Gs̈angerPM 25)

outsideof the lasercavity were usedto effect high-frequencycorrectionsto the

laser wavelength. Thesefeedbackelementswere driven by a high-bandwidth,

low noiseamplifier in a configurationwhich madeit possiblesimply to connect

the demodulatedsignal from an appropriatephotodetectorto the amplifier input

in order to lock the laserfrequencyto a cavity. This setuphadpreviouslybeen

usedwith anotherprototypewithin the LIGO project, and was easily adapted

for the work describedhere. The laser light was phasemodulatedat 12.33

MHz by an additional Pockelscell (Gs̈angerPM 25), which was driven by a

power amplifier (LIGO 5 Watt Amp #9) and impedance-matchingtransformer

(LIGO Step-UpTransformer).Then the laserbeampassedthroughan acousto-

74

optic modulator(Intra-Action AFD 402) usedfor power stabilization,througha

single-modeopticalfiber to improvethebeamquality,andthroughapairof mode-

matchinglensesusedto producea beamwith a convenientwaistsizeandlocation.

Beforeenteringthefiber,aportionof thebeamwasdivertedto anopticalspectrum

analyzerwhich wasusedto checkwhetherthe laserwasoperatingsingle-mode,

andafter the fiber, a portion wasdivertedto a “referencecavity,” a Fabry-Perot

cavity to which the laserfrequencycould be lockedwhena stablefrequencywas

necessary(for exampleduring alignmentof the interferometer). Also after the

fiber, a strayreflectionfrom a mode-matchinglenscaughton a photodetectorwas

usedto stabilizethe power leavingthe fiber, by feedingthe photodetectorsignal

back to the acousto-opticmodulator. A roughly 4 by 10 foot areaof optical

tablewasreservedfor the constructionof variousinterferometers.A transparent

plastic cover with plastic curtainswas installedover this areato protectit from

air currentsand from dust. A small blower equippedwith a filter was installed

for the purposeof purging the enclosedvolumewith dust-freeair. In addition,a

cleanbenchin the room was operatedwheneverdoing so did not interferewith

theexperiments.Unfortunately,it wasnecessaryto shutoff both thecleanbench

and the blower wheneverit wasdesiredto havean interferometerresonatingon

the table.

An RF signal was usedfor phasemodulationand as a signal to drive the

mixers. Figure 5.1 shows the setup used to generateand distribute the RF

75

Figure 5.1 RF Distribution.

12.33 MHzOscillator

6-way Power SplitterLimiter/.Phase-ShiftersMixers

Step-up Transformer

5 Watt12 MHz Amp

Pockels Cell

modulationsignal.

A relatively small numberof building blockswasusedto assembleall of the

experimentslisted herein. Mirrors of variousreflectivitieswere used. Someof

the mirrors were mountedon piezo-electricstacks(“displacementtransducers”)

allowing them to be electrically displacedalong the beamaxis. Someof these

werealsosupportedin mirror mounts(Klinger SL25.4BD)providedwith piezo-

electric fine alignment controls (“orientation transducers”;Marshall Industries

#AE0505D08or AE0505D16, micrometeradaptermachinedin house). The

mirrors were 1 inch diametermirrors from various manufacturers(but mainly

from CVI), with thicknessrangingfrom 1/4 inch to 3/8 inch. Someof themirrors

weresuperpolishedbeforecoatingandsomewerenot, but thedifferencebetween

thesetwo typesof mirror wasfound to be of little importancein the presenceof

larger lossesdueto dust. The piezo-electricstacksweremadeby gluing together

(with Stycast#1266epoxy) two piezo-electricwashers(1 inch OD, 0.5 inch ID,

76

type 550, from PiezoKinetics) with brasswashersservingasthe electrodes.The

mechanicaldesignof the assemblyof mirror mounts,displacementtransducers,

and orientation transducerswas adopted,with only minor modifications, from

experiments18,19,20performedat the MassachusettsInstituteof Technology.

Two typesof photodiodeswere used: tunedRF photodiodesmadein house

(LIGO #5 and #6), and broadbandPIN photodiodes(ThorlabsDET1–Si); the

bandwidthof the latter type dependson the load impedance.Photodiodeswere

usedin two applications: low-bandwidthapplicationswherethe power at some

optical output was measured(here the Thorlabsdiodeswere usedwith a 5 k

ohmload),andRF applicationsin which fluctuationsin theoutputintensityat the

modulationfrequencyweremeasured(by demodulating).Eitherthelessexpensive

Thorlabs diodes terminatedin 50 ohms or the in-housetuned diodes (which

provided better signal-to-noiseperformance)were usedin the RF applications.

The high-bandwidthphotodiodeswere sometimesfollowed with an amplifier

(LIGO ComlinearE103 CompositeAmplifier, #MEZ 8/17/84, or 1/3 of LIGO

Triple W/B Amp) butalwayswith amixer. Doublebalancedmixers(Mini-Circuits

ZAY–1) were used, followed by low-pass filters to remove the modulation-

frequency componentsfrom the mixer output. The mixers were driven by

limiter/phaseshifters(LIGO #3,9,10and12), deviceswhich containanadjustable

delay line and a limiting and filtering circuit, so that the output is of fixed

amplitude(9 V peak to peak) but the phasedelay is adjustablewith a pair of

77

knobson thefront panel.Thephase-shiftersweredrivenby theoutputof a 6–way

powersplitter(Mini-Circuits ZFSC-6–1–BNC),in turn drivenby theRF oscillator

(LIGO 12.33MHz Oscillator#4). Generalpurposeadjustable-gainamplifierswere

usedin variousapplicationsand high-voltageamplifiers(LIGO Dual translation

PZT Driver, #1 and 2) were usedto drive the displacementtransducers.These

amplifiers(seeFigure 5.2 a)) havetwo inputs (a main input and a “test input,”

of which a linear combinationis formedinternally) andtwo outputs.Oneoutput

is the high voltageto be appliedto the displacementtransducer;the other is that

voltagedivided by 100 and is called the “output monitor” signal.

Figure5.2 Schematicrepresentationof a high voltageamplifier andthe dynamicsignalanalyzer.

a) High Voltage amplifier

input

test input

outputmonitor

HV output

Signal Analyzer

S1 d S2

b) Signal Analyzer

The standardbattery of test and measurementequipmentwas used: oscil-

loscopes,function generators,and a voltmeter. One pieceof equipmentwhich

deservesspecialmention is the HP3562Dynamic Signal Analyzer (Figure 5.2

b)). This apparatushasthe ability to generatea sweptfrequencysinewaveat its

78

output and simultaneouslytake the ratio of the componentsat that frequencyin

the two signalspresentat its inputs.

5.2 Component Response Measurements

Displacement Transducers

As mentionedearlier, severalof the mirrors in the interferometerwere sup-

portedon piezo-electrictransducers(displacementtransducers)so that theoptical

path lengths(and hencethe phases)could be modified by applying appropriate

voltagesto thesetransducers.The responseof eachdisplacementtransducerwas

characterizedbeforeinstallationinto the interferometer,for two reasons.First, in

orderto measurethe responseof the interferometerto changesin mirror position,

it is necessaryto know what thosechangesin positionare. Second,it wasfound

that the responseof mostof the displacementtransducerscontaineda large reso-

nanceat around25 kHz; the installationin the loop of a notchfilter, adjustedto

cancelthe resonanceaswell aspossible,allowedthe low frequencyloop gain to

be increasedsubstantiallywhile keepingthe loop gainat the resonancefrequency

below unity. The setupfor displacementtransducercharacterizationis shownin

Figure 5.3.

The signal from the photodetectorwas fed in to a differential amplifier (GP

in Figure 5.3), the other input of which was driven by an adjustablevoltage

source. Before closing the servo loop, the displacementtransducerwas driven

79

Figure 5.3 Displacementtransducercharacterization.

S1 d S2

ScopeSignal Analyzer

GP HV

with a large amplitudetrianglewaveandthe voltagesupplyadjustedso that the

differentialamplifieroutputwaszerowhenthesignalfrom thephotodetectorwas

halfway betweenits minimum andmaximumvalues �� and �� . To lock the

Michelson,the differential amplifier was fed back to the high voltageamplifier

driving the transducer.At this operatingpoint, the derivative of photodetector

signal with respectto mirror position is given by

��

�� (5.1)

��! #"%$�'&)(+*-,/.10 was then calculatedby dividing this into the responsemeasuredwith the

Michelsoninterferometerlocked,using the signal analyzer.**

** Moving the mirror througha distanceof severalwavelengths(the displacementtransducerstypically havea range

of 2354 ) providedanadditionalopportunityto measuretheresponseof thedisplacementtransducers.This wasdonesimply

by countingthenumberof bright fringesproducedat theoutputfor a givenvoltagechangeat thedisplacementtransducer.

Measurementsof displacementtransducerresponsedone this way generallyyielded valuesfor the responsethat were

abouta factor 1.7 larger thanthe responsemeasuredusingsmall driving voltages.

80

Modulation Index

The modulationindex was measuredby configuringa Fabry-Perotcavity as

an optical spectrumanalyzer* (Figure 5.4)

Figure 5.4 Measuringthe modulationindex.

Oscillo-scope

GP

HV

In

WaveformGen.

Trig

The oscilloscopein this setupwould showseparatepeaksfor eachfrequency

componentof the light as they individually resonatedin the cavity. The modu-

lation index was then calculatedfrom the relative heightsof the carrier and RF

sidebandpeaks.

Mixer Gain

The mixer gain wasmeasuredby driving the “RF” input (which is normally

connectedto the photodetector)with a sine wave of known amplitudeand at

a frequencyof 12.3 or 12.4 MHz (the mixer’s “Local Oscillator” input was

connectedto a phaseshifter as usual). Under theseconditionsthe (low-passed)

* The commercialspectrumanalyzersamplingthe beambeforethe fiber had insufficient resolutionto measurethe

modulationindex accurately.

81

mixer output was a sine wave at the difference frequency, a few tens of kilohertz.

The mixer response was calculated by taking the ratio of output amplitude to

input amplitude.

5.3 Response of a Fabry-Perot Cavity

An initial experiment consisted of measuring the response of a Fabry-Perot

cavity. The cavity was locked on resonance and the response measured by driving

the summing node, shown in Figure 5.5, and then taking the ratio of the two

outputs using the dynamic signal analyzer.Figure 5.5 Measurement of the response of a Fabry-Perot cavity.

Cavity

Piezomirror

Signal Analyser

AmplifiersPhotodiode

Mixer

low-passfilter

SummingNode

The expected result (calculated by multiplying together the gains of all of the

cascaded components) is easily found using (2.7) and (2.19); it is��

�� ! #"%$�&(')�+*-,�.�,�/�021�34�5�76��8934�+��(

:;-< =�<(5.2)

82

Where�� is thedisplacementtransducerresponse,�� is thevoltagefrom the

photodiodewhenthecavity is far from resonance,�� and �� areBesselfunctions

evaluatedat the modulation index, "! is the augmentedbouncenumber, and

#%$'& ��(�) and#%*+$-, � aremixer andamplifier gains. The measuredresult is

. � *+$/, �. � ,& (103254687�9 :

(5.3)

Theuncertaintyin themeasuredresult is duemainly to fluctuationsin alignment.

The fact that this is a moreaccurateway to measurethe productof the gainsof

all of the cascadedelements(than the individual measurementof eachfactor)

was important in obtaining more preciseresults in experimentswith the full

interferometer,describedbelow.

5.4 Response of the Coupled-Cavity

The “coupled-cavity” experimentconsistedof a three–mirrorinterferometer

as shown in Figure 5.6.

Theopticaloutputsweremixeddownandonewasfedbackto thethird mirror,

the other to the laser frequency. The responseof the interferometerwas again

measuredby injecting a swept-frequencysinewave into a summingnodein the

displacementtransducer-drivingamplifier and, using the signal analyzer,taking

theratio of thesignalfrom themixer to thesignalat thedisplacementtransducer.

The DC optical model describedin chapter4 was used to calculate the

expectedresponseof thecoupledcavity in theabsenceof anyservos.Now let us

83

Figure 5.6 Coupled-cavityexperiment.

FromRF oscillator

To laser frequencycontrol

piezo-electrically positioned mirror

Signal Analyzer

S 12

Recycling Cavity Arm

cavity

V1 V2HV

R F B

calculatetheresponsein thepresenceof theservos.We will showthat if thegain

in the loop feedingback to the laser is sufficiently large, then the measurement

describedabovedependsonly on the optical response,and not on the gain of

either loop. Let the optical responsebe the matrix� ��

�� so that

� �� (5.4)

Where

and � representchangesin thelengthof thearmcavityandtherecycling

cavity. We want to representthe interferometerasa two-input, two-outputdevice

whereoneof the inputs correspondsto changesin the laserfrequency,which is

equivalentto equal changesin both

and � . We do this by calling the new

system � :

��

� �

� ��

(5.5)

84

so that

��

�� (5.6)

where � is the cavity length and � the optical wavelength.

The two feedbackamplifierswe group into a “controller”

� � � � �� (5.7)

Figure 5.7 Block diagramof the control systemfor the coupled-cavityexperiment.

d fe

s1

s2

P

C

Then the closed-loopresponsecan be calculatedwith simple matrix manip-

ulations†: ��

�� ! #"� �� (5.8)

† Historically, in control theory, the points in the loop where signals are summedin have often been taken as

differencing,ratherthansummingnodes.This conventionbecomesawkwardif nodesexist at severalpoints in a loop or

if no nodesexist at all (the latter caseis of interestwhenanalyzingstability). We will usethe conventionthatno changes

of sign occur exceptwherethey are explicitly included. Becauseof this, someof the expressionswe derive may differ

by a sign from expressionsfound in sometextbooks.

85�� (5.9)

The experimentillustratedabovecorrespondsto this block diagramwith

�� (5.10)

the signal analyzerthen measures

� ��

� � �� "!� � �$# � �%�'& � � �)(�*�+�"!

� ��,& � � �(5.11)

wheretheapproximationis valid when� � is very large‡. As in section5.3 above,

the value of this expressionwas multiplied by the appropriatefactors such as

power on the photodiode, -/.0- � , the displacementtransducerresponse,and the

amplifier gains,and comparedto the measuredvalues.

Table5.1belowshowsthemeasuredandcalculatedresponsefor two different

valuesof the recycling mirror (labelled“R” in Figure 5.6) and severaldifferent

valuesof the modulationindex. The agreementwas good for small modulation

depthsand fair for larger modulation depths. It is suspectedthat the larger

discrepancyat larger modulationindex is due to the fact that this optical model

excludesthe effect of the higher harmonicsgeneratedin the optical spectrum

when the modulationindex is large.

‡ Specifically,for the approximationto be valid, we require 1 2�3416571 8:9;96<>=@?BA>CEDGFB1 and 1 2:361:5H1 IJ<4CK8@3;3MLN8:9;94FO1

86

Table 5.1 Resultsof Coupled-CavityExperiments

�� Response(dB)

Measured Calculated

8.9% 0.012 46 49

16.9% 0.012 31 32

16.9% 0.10 36 40

8.9% 0.17 56 58

16.9% 0.17 39 42

5.5 Response of the Complete Interferometer

Once the coupled-cavityexperimentwas deemedadequatelyunderstood,a

completeinterferometerwas set up. Its layout is shown in Figure 5.8. The

recycling cavity and both arm cavitieswere folded in half; the recycling cavity

becauseit neededto beslightly in excessof 6 m in lengthandthearmcavitiesto

allow theconvenientuseof cavity backmirrorswith thesamecurvatureasthatof

the recyclingmirror. Table5.2 showstheopticalpropertiesof the interferometer.

The loss in the arm cavitieswas mainly due to dust on the mirrors and in the

air, andthe loss in the recyclingcavity is presumedto be mainly dueto pick-off

reflectivity anddustandscatteringlosson the surfaceswithin that cavity*.

Alignment of eacharm cavity was doneby observingthe light transmitted

throughthebackmirror with anoscilloscopewhile driving thebackmirror with a

large amplitudetrianglewave. The oscilloscopewould showa numberof peaks,

* Light traveling aroundthe recycling cavity encountersan averageof 16 surfacesduring one round trip (counting

surfacesencounteredtwice astwo surfaces).

87

Table 5.2 Fixed Mass InterferometerOptical Properties

Fabry-PerotArmCavities

Front MirrorTransmission

9 %

Back MirrorTransmission

13 ppm

Losses 0.2 %

FringeVisibility 9 %

Cavity Gain 40

RecyclingCavity RecyclingMirrorTransmission

18 %

Cavity Losses(includingPickoff beams)

16 %

FringeVisibilty 88 %

RecyclingFactor 4

Beamsplitter Contrast 99 %

onefor eachresonanttransversemode,andonefor eachfrequencyof light incident

on the cavity. The heightof the peakcorrespondingto the carrier resonatingin

the(degenerate)01 and10 transversemodeswasthenminimizedby adjustingthe

orientationtransducervoltages.Therecyclingmirror wasalignedby driving both

it and the secondarm steeringmirror (SAS in Figure 5.8) with triangle waves

(at different frequencies)andagainminimizing the peakcorrespondingto the 01

and 10 transversemodes.

The feedbackconfigurationwhich was usedis shown in Figure 5.9†. This

† Althoughthefigureshowsloop 4 driving thebeamsplitter,this drive wasactuallyappliedto thesecondarmsteering

88

Figure 5.8 Layout of completeinterferometer.

Faraday Isolator

Mode-matchinglens

LEGEND

Photodetector

Phase modulation

Power stabilization

Single-mode fiber

Source of phase-modulated, power stabilized, spatially filtered light

Recyclingcavity

Arm cavities

Pickoff

δ

Asymmetryδ

Second arm steering mirror

SASSAS

configurationis quitedifferentfrom themorebalancedconfigurationwhich would

be usedin LIGO; howeverit holds the interferometeron resonanceand thus is

adequatefor thepurposeof verifying thecalculationsmadewith theopticalmodel.

Arranging for the interferometerto lock initially was a problemwhich was

only solvedby theapplicationof considerableamountsof patience.The“method”

usedto adjust the four gainsand mixer phasesto their operatingvalueswas to

observethe light transmittedthroughone of the cavity back mirrors with a pair

of photodiodesconnectedto a storageoscilloscope,while driving severalof the

mirrorswith low-frequencytrianglewaves.Thedrive andwhateverothersources

mirror (seeFigure5.8). The effect of driving the secondarm steeringmirror is nearly the sameas the effect of driving

the beamsplitter; onechangesonly the optical path length (��

) from the recyclingmirror to the first arm, and the other

changesonly the optical path length(��

) from the recyclingmirror to the secondarm.

89

of disturbancewere presentwould occasionallycausethe interferometerto pass

closeto resonance,and the storageoscilloscopewould thenrecordthe lengthof

time theservosheldit resonant,typically only a fractionof amillisecondwhenthe

variousconnectionswere first established.The operatorwould changea setting

on somegainor phasedial, andagainwatchtheoscilloscope,to seewhetherthis

changeimprovedor worsenedtheperformance.Theservoswereadjustedin order

of thedegreeto which theyarecritical to resonance:thelaserfrequencyservofirst

(becausethelaserfrequencycontainsthemostnoiseandbecausedeviationsin the

laser frequencypushthe interferometeraway from resonancemost effectively),

the servodriven by the gravitational-wavesignalsecond,and the remainingtwo

in either order.

Figure5.9 Servoconfigurationof Table-topprototype. Feedbackamplifiersnot shown.

I Q

I Q

I Q

Photodiodewith I & Q Demodulator

to laser frequency control

12

3

4

90

Oncethe amplifiersand mixers were properly set up, however,the interfer-

ometerwould acquirelock with remarkableease.The mechanismby which this

occursis not yet fully understood,but it was found that when the secondarm

steeringmirror was movedback and forth slowly with the other servosactive,

theotherservoswould acquirelock andhold it until thecontinuingmotionof the

secondarmsteeringmirror destroyedtheresonance(resonanceis relatively insen-

sitive to the positionsof the beamsplitter andsecondarm steeringmirror, aswe

seein AppendixB). This fact madeit easyto deviseanautomaticlock-acquisition

system: the power transmittedthrougheachof the two arm cavity backmirrors

is comparedto a threshold. When either power level is below threshold,the

secondarm steeringmirror displacementtransduceris driven by a trianglewave;

oncebothpowerlevelsexceedtheir respectivethresholds,thesecondarmsteering

mirror drive is disconnectedfrom the trianglewavesourceandconnectedto the

amplifier which suppliesthe appropriatefeedbacksignal. Figure5.10 is a setof

oscilloscopetracesshowingthe light levels in the interferometerandthe voltage

applied to the secondarm steeringmirror displacementtransducer. When the

interferometeris knockedout of lock (in this caseby rappingone’sknuckleson

theoptical tablecover),thevoltagebeginsto rampup anddown; onceresonance

is re-establishedthecircuitry disconnectstherampandreconnectstheservoloop.

Closed-loopresponsemeasurementswere made as shown in Figure 5.11.

Loop numberingis shownin Figure5.9. If a driving signald is appliedin loop�

91

Figure 5.10 Time record of lock interruption.

0 0.5 1 1.5 2 2.5 3 3.5 4-1

0

1

2

3

4

5

6

Beam splitter feedback

Arm 1 stored power

Arm 2 stored power

Recycling cavity stored power

Lock reacquired

Lockdisrupted

Time (seconds)

Arb

itrar

y U

nits

by the signal analyzer, and the analyzer inputs are connected to the displacement

drive signal of loop�

and to the mixer output in loop � respectively, then the

expected ratio* of the signals at the two outputs is

��

� � �� ! #" $ �&% �

� �� ' " �&�(5.12)

* Mathematica code was used to calculate the interferometer response, and Matlab code was used to calculate the

closed-loop response given the interferometer response. Matlab is a registered trademark of The MathWorks, Inc.

92

Figure 5.11 Block diagram representing setup for closed loop measurements.

A 2

A 3

G (laser feedback)1

G2

G3

G4

B 2

B 3

P(Interferometer)

A 4 B 4

A 1 B 1

Displacementtransducerresponse

mixer gain,modulation index,photodetector response

Signal Analyzer

dS1 S2

where P is a matrix representing the interferometer response, and�

, � , and � are

diagonal matrices representing the��

, � �, and � �

, respectively. The products

�� were found using

��

(5.13)

where ��! �� was measured, as shown in Figure 5.12, in each case using a single

Fabry-Perot cavity, whose response��"� �#��

is deemed understood.

The responses calculated (and measured) for the full interferometer, unlike

the coupled-cavity response, are not frequency independent. Although the plant is

still frequency independent, and the gain of the laser frequency servo large enough

93

Figure 5.12 Setupfor measuringthe factors��

and �� .

A i

Gi

B k

Pcavity

Signal Analyzer

dS1S2

to makechangesin it (with frequency)irrelevant,thegainsof theremainingloops

do changesignificantly with frequencyover the rangewherethe measurements

weremade.Figure5.13showsplotsof experimentalandcalculatedresponse(Hjk)

for measurementsmadeby driving loop�

andmeasuringthe responsein loop � ,

with both of thoseindices ranging from 2 to 4. Phaseplots are modulo 180�

and are shownfor referenceonly. The laserfrequencyloop was excludedfrom

thesemeasurementsbecauseit was difficult to achieveadequatesignal to noise

ratio without disrupting lock in any measurementwhere a signal was injected

into the laser frequencyloop*. In addition, samplesof the plotted curves(at 2

* Somereadersmay considerthis resultcounterintuitive(this authorcertainlydoes).How can it be that the noiseis

large enoughthat addinga signalcomparableto the noiseevenin a small frequencybanddisruptslock, yet small enough

that the entirenoisepowerover all frequenciesdoesnot disrupt lock? The answeris that the elementwhich behavesin a

non-linearmanner,disruptinglock, is in generallocatedat a differentpoint in the loop from the point wherewe measure

the output.

94

Table 5.3 Sampledvaluesof experimental(bold) and calculatedresponse.

MeasuredandCalculatedResponseat 2 kHz (dB)

Loop in which responsemeasured

2 3 4

2 38.3 37.7 9.86 16.4 15.4 17.8

3 26.9 24.6 19.0 22.1 23.6 23.4Loop

Driven4 24.6 25.8 27.3 31.8 36.3 36.9

kHz) are presentedin Table 5.3, in order to allow the data to be summarized

more compactly.

The experimentaluncertaintyestimatedfor theseresultsis about±3 dB, due

mainly to uncertaintyin the displacementtransducerresponse(about ±1 dB),

G

H

s

x

y n

The figure aboveshowsa block diagramof a systemwith two inputs,a signal � anda noisesource � . If the point in

the loop which limits the dynamicrange(the point wherenon-linearbehavioroccursmost easily) is at the input to�

,

thenthe conditionfor maintaininglock canbe written

��

� �� (5.14)

From which

� �� (5.15)

� �� (5.16)

andthe maximumcontributionto � �� that the inputscanproduceis given by ��!� and ��"��!� � for the signal and the noiseinput, respectively.The dynamicrangeconstrainedsignal-to-noiseratio can

takeany value,dependingon the distributionof gain within the loop.

95

uncertaintyin the lossesin the arm cavities* (about ±1.5 dB), and fluctuations

in alignment(about ±2 dB). All of the plots of experimentalversuscalculated

responseshow reasonableagreementwithin this uncertaintyexceptfor the case

of��

, wherethe discrepancyis perhapstwice the experimentaluncertainty. It

is not known whetherthis discrepancyis dueto a randomlargefluctuationin the

cumulativeerror or to an unmodeledsystematiceffect.

* The fractionaluncertaintyin the arm cavity roundtrip losswasperhaps+/-50%(dueto fluctuatingdustlevels),but

the correspondingfractionaluncertaintyin the recyclingcavity round trip lossonly about+/-15%, sincearm cavity loss

wasnot the dominantloss in the optical system.

96

Figure 5.13 Plots of experimental and calculated (dashed) response.

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

36

38

40

42

Frequency (Hertz)

Mag

nitu

de (

dB)

H22, run #4

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

103

104

0

10

20

30

Frequency (Hertz)

Mag

nitu

de (

dB)

H23, run #4

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

103

104

10

20

30

Frequency (Hertz)

Mag

nitu

de (

dB)

H24, run #4

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

10

20

30

40

Frequency (Hertz)

Mag

nitu

de (

dB)

H32, run #4

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

15

20

25

30

Frequency (Hertz)

Mag

nitu

de (

dB)

H33, run #4

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

10

20

30

40

Frequency (Hertz)

Mag

nitu

de (

dB)

H34, run #4

97

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

10

20

30

40

Frequency (Hertz)

Mag

nitu

de (

dB)

H42, run #4

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

0

20

40

60

Frequency (Hertz)

Mag

nitu

de (

dB)

H43, run #4

102

103

104

0

50

100

150

Frequency (Hertz)

Pha

se (

degr

ees)

102

103

104

20

30

40

50

Frequency (Hertz)

Mag

nitu

de (

dB)

H44, run #4

98

Chapter 6 Design and Analysisof a Control System

If we considerthe interferometeras a four-input, four-output device, then

the problem of designinga control systemis that of constructinganother4x4

devicewhich when connectedto thoseinputs and outputswill suppressseismic

disturbancesandkeepthe interferometeron resonance.At this level of generality

the problem is quite complex, and the only systematicapproachesknown are

purely numerical.Our approachwill be insteadto makea numberof simplifying

assumptionsaboutthe structureof the controller,to developsomeguidelinesfor

the designof a control system,and then to usenumericalmethodsto calculate

the expectedperformance.

First let us write the matrix of transferfunctionsof our interferometerin a

simpler form. We assumethat we can connectamplifierswith arbitrary (stable)

frequencyresponseand gain to the outputsof the interferometerand we will

choosethe gain and frequencyresponseof theseamplifiers to make the new

systemassimpleaspossible.This simplified systemwe will call the “plant.” Its

transferfunctionscan be written in matrix form:��

�

��

��

(6.1)

99

Figure 6.1 Equivalentblock diagramsfor the plant.

P

P+

P-

Figure 6.2 Closed-loopsystem.

C

P+

Sincethe plant is block-diagonal,it is equivalent,asshownin Figure6.1, to

two disconnectedsub-plants,thecommon-modesub-plant��

andthedifferential-

mode sub-plant��

.

There��

and��

correspondto theupperleft andlower right 2x2blocksof�

.

��is a simplesubsystemto controlbecauseit is nearlydiagonal;it representstwo

loopswhich arealmostindependent.��

on the otherhandis difficult to control

becauseit consistsof two rows which are barely linearly independent.We will

spendmost of the rest of this chapterconsideringthe problemof designingan

adequatecontrol systemfor��

.

This problem, then, consistsof finding a matrix of transferfunctions such

that the closedloop systemof Figure6.2 performsadequately.We will makean

effort to designa controller � which is diagonal,which meansthat it consistsof

two independentfeedbackamplifiers, �� and � asshownin Figure6.3.

100

Figure 6.3 Closed-loopsystemwith a diagonalcontroller.

P+

C1

C2

Figure 6.4 Closed-loopsystemwith a “crossed”controller.

P+

C1

C2

If we canachieveadequateperformancewith sucha controller,thenwe will

have avoided some complexity. Severalquestionsarise in the design of this

controller which do not arisein the designof a controller for the��

sub-plant.

For one thing, it is not obvious for the��

systemwhich output shouldbe fed

back to which input. For��

that choiceof connectionschemeis guidedby the

fairly intuitive rule that eachoutput is fed backto that input to which the output

is most sensitive.

In the��

systemit may be betterto connectthe feedbackin the “crossed”

configurationillustratedin Figure6.4. Thesystemobtainedthis way is equivalent

to the systemwe get by re-numberingthe rows of��

(i.e., interchanging�� and

� ) and connectingthe amplifiers in the “straight” configurationillustrated in

Figure 6.3.

101

Another importantconcernis that the transferfunctionsof��

may contain

right-half-planezeros,aswe canseefrom (3.71)and(3.72). It is well known21,22

that in a one-dimensionalservoloop, right-half-planezerosin the plant limit the

achievableunity-gain frequency.In our case,we may wish to designthe optical

systemto avoidsuchproblems.We will investigatetheeffectsof right-half-plane

zerosin the first sectionof this chapter.

In the secondsection,we will considerthe size of the performanceprice

we are paying by using a signal-extractionsystemwhich producesa poorly-

conditioned plant. Other signal extraction methods23, which at the expense

of increasedcomplexity producea very well-conditionedplant, are available

for sensingdeviationsfrom resonancein the interferometer,and in evaluating

the relative merits of the systempresentedherein againstsuch an alternative,

performancebenefitsare an importantcriterion.

At the end of this chapter,as a demonstrationof the designguidelineswe

will haveseen,we will designa control systemfor the exampleinterferometer

we havebeencarrying,andwe will evaluateits performancenumerically.

6.1 Feedback Configuration and Gain Constraint

The frequency-independent 2x2 plant

Considera systemas shownin Figure 6.5, with a plant

�� (6.2)

102

Figure6.5 Block diagramfor analysisof systemcontainingfrequency-independentplant.

P

C1

C2

where��

and��

areconstants.We will usevery simplefeedbackamplifierswith

onepole each(both at the samefrequency�� ) andwe will leaveourselves

the freedomto adjustthe polarity and gain of eachamplifier:

� �� (6.3)

and

� � � � � � �� (6.4)

where �� (for stableamplifiers), the �"! � �representpolarity switches,

andthe� ! representadjustablegain controls.Since

�and # areopen-loop

stable,the closed-loopsystemis stableif and only if† $&%('�) � � #+* containsno

right-half-planezeros24. Now

$,%-'�) � � #.* � � � �-��/� ��

� � � � � ��0� �� 1� �

�/� ��

� ��0� ��

� ) � � �(��

� �� "2�� 2��

�*

(6.5)

† We will continueto usethe feedbacksign conventionexplainedin the footnoteon page84.

103

We know that a polynomial with real coefficients has zeroeswith positive

real part if any pair of coefficients hasoppositesigns25 (for a quadratic,it can

be shownthat the latter is also a necessarycondition for the former). The last

coefficient (��) in thenumeratorpolynomial(in � ) is positive,sofor stability,

all of the remainingcoefficients must be positive.

Now supposewe choose� �� andwe want to startoff with � ��

andthen increasethe gain � � first. If we increaseit to the point where � ��

the systemwill be unstablesince the coefficient of �� becomesnegative. This

is an undesirablesituation becausewe would prefer to be able to increasethe

gains arbitrarily and enjoy the resulting improvementsin performance. In this

caseit is easyto understandwhat is wrong; asin a one-dimensionalservo,if the

sign of the feedbackis wrong, the systemwill be unstablefor gain exceedinga

certainthreshold.We needto choose� � � to avoid this problem. Similarly

we find that if � �� then we needto choose� �� to be allowedarbitrary

� � when � �� .

We might ask whether thesechoicesof polarity guaranteestability for all

valuesof thegains � � and � � . Theansweris “no”: As we canseefrom the last

coefficient of � � in (6.5), if � � � �� , the systemis unstableif

� � � ��

� � � � (6.6)

so again the gain is constrainedby stability considerations.It is easyto verify

104

thathadwe numberedour outputsdifferentlywe would havehada stablesystem

no matterwhat(non-negative)valueswe chosefor��

and��

. Equivalently,had

we hookedup the amplifiersin the crossedconfiguration,the correspondingterm

in the � � coefficient would havebeen � � � � and againthe systemwould have

beenstablefor any non-negative��

and��

.

With this motivation, we makethe following

Definition:

The closed-loopsystemconsistingof the plant and two controllers � � �� and � � � �� is gain constrained if � �"! �$# %&! %

suchthatthesystem

is stablefor all positive real valuesof��

and��

.

Now we can show the following:

Theorem 1:

The systemconsistingof the plant �% � �% � � andcontrollers � �'� �(��)��

and � �*� �+��)�� connectedin the straightconfiguration is gain constrainedif and

only if

� �-, � � � ��.'/10 (6.7)

Proof:

Sufficiency: � �&, � � � �2.3/40implies either

i. � �6570and � � � �8/70

or

ii. � ��/70and � � � �95:0

105

Casei) is the one we looked at in the exampleand we saw that it is gain-

constrained. In caseii), if we choose��

then the term��

is

negativeand the coefficient of �� becomesnegativefor large��

. If we choose

� � ��, the term

�� is negative.

Necessity:If��

we take� � � �

andthe sign of��

opposite

the sign of��

. Then all of the coefficients in the numeratorof (6.5) are non-

negativefor all valuesof� �

and��

.

Frequency-dependent plant

Now we will prove a weakerbut more relevantresult.

Theorem 2:

the systemconsistingof the plant‡

� �� !#"� �� !%$

(6.8)

and the controllers

& � � � � � �� !(' (6.9)

and

&��)� ��*�� !('

� (6.10)

‡ +*, canbe written in this form by factoring"" , out of (3.57),(3.61), (3.71),and(3.72).

106

connectedin the straightconfigurationis gain-constrainedif any of the following

is true:

i.��

ii.��

iii. � ��

Proof:

As before, we form

�� !#"%$'&(� ) �+*,��.- /��0

) ��*1�� 2- /� ��2- /� 0

�

- ) � * � ) � * � � � �.- /� � � � �2- /� ��.- /� 0

3

& ��.- /� 0

3 �4� ) �5*,� ) ��*1�6�� - ) �7*,� ) ��*1�8�� 8� $

-9�;: < ) �5*,� ) ��*1�� .- ��=� �

- ) �5*,� ) ��*1� ��=� ��

� �� $ >

� =- : ) �5*,� ) ��*1�6�� =

� � >��=

�- >

��=3

(6.11)

Now it is easyto seethat if anyof theconditionslisted in the theoremholds,

thenat leastoneof thecoefficientsof*,�

,*1�

, or*,�7*1�

will be negativeandthe

systemwill be unstablefor somevaluesof thesegains.

107

Implications for the Interferometer

Becauseof the way in which we numberedthe interferometeroutputs,the

feedbackconfigurationof Figure1.7 correspondsto the straightconfigurationfor

��. In Chapter7 wewill seethatthishasanimportantadvantageoverthecrossed

configuration(with the samenumbering). We let

�� (6.12)

Substitutingfrom (2.77), (2.78), (3.57) and (3.72), we get

��! " �! �� " # $ " (6.13)

and

�� "� � $�$ (6.14)

Similarly, letting

�&% �� %� �(')�� ' � � � �� (6.15)

and using (2.70), (2.74), (3.61), and (3.71), we have

�*%+�� " � �� " # $ " (6.16)

and

%+�� "� � $�$ (6.17)

Now the necessaryconditionsfor freedomfrom gain constraint(Theorem2)

become:

108

(from the third condition,and from (3.66) which implies �� )� �� (6.18)

which implies �� (since,from (2.17) and (2.75), �� );

then the first condition gives:

� � �

�� (6.19)

andthesecondconditionenforcesthesameconstraintasthefirst. We will explore

the consequencesof theseconditions for the optical design in more detail in

Chapter7.

6.2 System Performance

Let us now addressthe questionof how much performancewe sacrificeby

operatingwith a poorly conditionedplant. Figure6.6 showsa block diagramof

the closed-loopsystemwith and !� representingseismicdisturbancesdriving

the systemaway from perfect resonanceand " and "�� the residualdeviations

from resonancein the presenceof the servos.

Figure6.6 Block diagramusedto analyzeperformance,showinginputsdriven by seismicdisturbance.

P+

d2

d1

C1

C2

e1

e2

f 1f 2

109

We solve for�� :

�� (6.20)

�� (6.21)

�� (6.22)

If we let

� �� (6.23)

then

�� (6.24)

We will considerthis term first.

The smaller � � is, the bettera job our control systemis doing. Clearly the

larger we can make the elementsof�

, the better. In a real systemthere are

alwaysaspectsof the high frequencyperformanceof the systemwhich limit the

unity-gain frequencyand therebythe gain one can achieveat thosefrequencies

whereonewantsto suppressdisturbances.In our system,assumingthatit is not in

a gain-constrainedconfiguration,thesehigh frequencyfeatureswill bemechanical

resonancesat about 10 kHz for any loop driving a mirror, and propagation

delays in the optical path and wiring, which producesignificant phasedelays

110

Figure 6.7 Block diagram showing plant and one feedback loop as a single block.

P+

C1

R2

C2

R1d2 e2

(R C)*P1 +1

for frequencies exceeding about 1 MHz, for the loop feeding back to the laser.

In Figure 6.7 these are shown explicitly as additional factors��

and��

.*

When one of the outputs of the plant is connected back to one of its inputs

via controller � �the resulting one-input, one output system†

� � � � �� (6.25)

can be treated as a simple block from the point of view of analyzing loop 2. If we

assume that� � � � ��

is smooth and minimum-phase in the frequency range

where� �

is smooth, then its exact frequency response is irrelevant since we can

in principle cancel it by appropriate choice of frequency response of � �. Then

the minimum achievable value of � ��depends only on

� �, and, in particular, not

on whether or not�

is diagonal. The other contribution to � � is � ��, and we

* These factors have not been relevant until now because we model them as equalling unity at the frequencies where

we want to suppress seismic noise, typically frequencies of less than 10 Hz.

† For brevity, we will write �� and �� instead of �� "!$# and ��%�&! # throughout this section.

111

may askhow the sizeof this contributioncomparesto the other,sinceif we had

a diagonalplant, this contributionwould vanish. We canwrite�

explicitly:��

� �� (6.26)

where

�� ! (6.27)

and see that

�"� ��"�#� ��

� � � (6.28)

In otherwords,thefact thatourplantis notdiagonaldoesnotdegradeperformance

significantly if

�� (6.29)

A more useful expressioncan be derived by writing this in terms of the loop

gains, $ � and $ � :$ � � � �%��

�&� � �'� � � �� !��

(6.30)

wherewe haveassumedthat��

. Similarly

$ � ��%� � �� (��

(6.31)

112

and if��

we get the approximateresult:

��

�� (6.32)

In other words, we do not sacrifice a significant amount of performanceby

adopting this schemeas long as� �

is sufficiently large. The required ratio

of� �

to��

is��

; typically a few hundred*. This is easyto achievefor any

reasonableloop shapein loop 2 since the unity gain limitations on loop 1 are

so much less severe.

Next we look at loop 1 to seewhetherperformancethereis degraded.Here

� � ��(6.33)

and the relevant ratio is��

�� !��

(6.34)

Since�"�

is typically largerthan��

, this meansthataswe increase� �

we first start

to experiencea degradationin performancein loop 1† (over what it could have

beenwith a diagonalplant) and then we stop experiencingany degradationin

loop 2. Oneof the loopsalwaysexperiencessomedegradation.In the numerical

analysispresentedlater in this chapter,�#�

is very large, so that the performance

* In our numericalexample,$&%('*)�+ ),).-&/ and $.01'32 )�+ ).)465 .† The performancein loop 1 doesnot becomeworse as we increase7�% ; it just ceasesto improve. If we had a

diagonalplant, the performancein this loop would continueto improveaswe increasedthe gain.

113

in loop 1 is not as good as it would be with a diagonalplant. Nonetheless,

accordingto the predictionsof this numerical analysis,adequateperformance

can be achievedin both loops. If the specson loop 1 were much tighter or if

we found that we had trouble meetingthem, then we might want to consider

using a different signal extractionschemeor implementinga “decodingmatrix”

to improve the performancein loop 1.

A decodingmatrix is a set of amplifiersinterconnectedso as to form a 2x2

systemwhich, whenconnectedto the outputsof��

, producesa new plant, say

�� (6.35)

For an idealdecodingmatrix,��

would benearlydiagonal(i.e.,�� and

��

). Under thesecircumstancesthe noise injected into either loop by the other

would becomenegligible. For a morerealisticdecodingmatrix all of therelations

betweenthematrix��

andstability andperformancederivedaboveapplyequally

to��

, andperformancecanbe improvedif thenewplant is morenearlydiagonal

thanthe original plant. Whenanalyzingsucha system,it is especiallyimportant

to consider the imperfectionsin the componentsused to build the electronic

decodingmatrix, and possiblefluctuationsin the interferometerresponse,due

to misalignmentor deviationsfrom resonance.The designof an ideal decoding

matrix would be straight forward if one had ideal electroniccomponentsand a

114

non-varyinginterferometerresponse;a real devicewith imperfectionsmay yield

smaller performancebenefits.

6.3 Numerical Example of a Control System Design

Mainly for the sake of illustrating some of the ideas developedabove, a

controller was designedand evaluatednumerically*. The designwas done by

trial anderror. Polesandzeroswereaddedto thecontrollerandtheir frequencies

changeduntil adequateperformancewas achievedand all of the constraints

satisfied.The polesandzerosof the final design†26, aswell as the performance

predictedfor this design,are shownin Table 6.1.

Simplenumericalmodelswereusedto representthe plant andthe controller.

For theplant, the rationalfunctionapproximationderivedin Chapter3 wasused;

for thecontroller,a rationalfunction(pole/zero)representationwasused.Thusthe

numericalmodeldid not containthe effect of internal resonancesin the mirrors,

nor the effect of propagationdelays. That theseunmodeledeffects would not

causeinstability in a real systemwas checkedby inspectionduring the design

process.

* Almost all of the analysisdescribedin this sectionwasdoneusingMatlab code.

† L. Sieversfirst observedthat adequateperformancecould be achieved,and all of the designconstraintssatisfied,

with controllerscontainingonly real polesandzeros(i.e., no complexconjugatepairsof polesor zeros).

115

Constraints

Stability As mentionedabove,a mirror resonancecould causeinstability if the

loop gain at the resonantfrequencywere to exceedunity. Sincethe resonances

are expectedto havequality factorsof up to 107 and are expectedto occur at

frequenciesaslow as10 kHz27, this type of instability wasruled out by assuring

that the magnitudeof the loop gain in any of the loopsdriving a mirror wasless

than 10–7 at frequenciesexceeding10 kHz*28.

In the loop feeding back to the laser frequencythe criterion was simply

that the unity gain frequencybe less than 1 MHz. At that frequency80 ns of

delay(correspondingto 24 m of optical path),togetherwith 15 degreesof phase

changeseenin the high frequencyinterferometerresponse(H_11 of Figure4.2),

still leavesa phasemargin of up to 70 degrees.

The simplified rational function modelof the closedloop systemcould also

be unstableby itself. This was checkedby finding the eigenvaluesof the “A”

matrix in the state-spacerepresentation† of the closed-loopsystem,to makesure

that their real parts were positive.

Noise Oneadditionalconstraintwasobservedduring the design.As we sawin

Chapters2 and3, the gravitationalwaveoutput,in additionto beingsensitiveto

* A Mathematicaprogramwas written to checkthis particularconstraint. This was necessarybecausethe Matlab

code,usingthe state-spaceformalism,underestimatedthe attenuationat high frequencies.

† The (Matlab) softwareusedto do this part of the analysisanddesigndoesall of its calculationsin statespace.

116

��, is alsosensitiveto � � . If � � is controlledby a loopcontaininganoisysensor,

thenthe inducednoisein � � will contaminatethe gravitationalwavesignal*.

Quantitatively, we require, at frequencieswhere we want interferometer

performanceto be limited only by shotnoiseat the antisymmetricoutput (above

100 Hz),

�� (6.36)

where �� and �� are the shot-noiselimited sensitivity to � � at the

isolatorandto� �

at the antisymmetricoutput,and � � � is the open-loopgain in

the � � loop. In our numericalexample,this correspondsto

� � � � � � (6.37)

and we use � � �"!#�$�%� � as our constraint.

Performance

Figure6.8showstheexpectedseismicdisturbancedriving eachof theinterfer-

ometermirrors†29. To calculatethe rms deviationfrom resonance,this spectrum

is first multiplied by a number &(' correspondingto the numberof mirrors being

* In principle, this noise,being separatelymeasurablein the ) � loop, could be subtractedout of the gravitational

wavesignal. However,theprecisionwith which this canbedoneis limited by practicalconsiderations,andtheadditional

complexity is undesirable.

† This curvewascalculatedfrom a combinationof estimatedandmeasuredspectraof the seismicmotion, and from

the transferfunctionsof the mechanicalanti-seismicisolation.

117

perturbed,and multiplied at eachfrequencyby the closedloop gain of the sys-

tem. This set of operationsis representeddiagrammaticallyin Figure 6.9. This

producesthe spectrumof residualdeviations,which is then squared,integrated,

and raisedto the power of 1/2.

Figure 6.8 Seismicspectrum.

10-1

100

101

102

10-12

10-10

10-8

10-6

10-4

10-2

100

102

Frequency (Hz)

m/r

hz

Horizontal Seismic Noise Spectrum driving a LIGO test mass

Figure 6.10 shows the magnitudeof the open-loopgains, and the spectra

of residualmotion for the common-modedegreesof freedom. We seethat the

ratio of openloop gainseasilyexceeds��

at all frequencieswherethe seismic

disturbanceis significant,and,asexpected,theresidualdeviationsareduemainly

to thedisturbanceinput in loop 2. This is illustratedby Figure6.11,which shows

118

Figure 6.9 Block diagramfor the propagationof seismicdisturbance.

M1M2

M3M4

P

C

Input: Seismic Spectrum

Outputs: Spectra of residual deviations from resonance

the contributionsto the residualdeviationsin loop 1 and loop 2, due to seismic

disturbancesenteringloops 1 and 2.

Finally we note that although the performancespecifications set forth in

Chapter1 and Appendix B were only barely met in this design,considerable

improvementsarepossibleat the costof modestadditionalcomplexity. In loops

2 and3, which arelimited in bandwidthby mirror resonances,gain increasescan

beachievedby installingnotchfilters in theseloops,to partially canceltheeffects

of the resonances.The gain in loop 4 can be increasedwithout contaminating

thegravitational-waveoutputif subtractioncircuitry is installedto re-subtractthe

noiseintroducedby this loop from the gravitational-waveoutput.

119

Figure 6.10 Loop gains (a)) and residual motion (b)) in common-modedegrees of freedom. Curves corresponding to the

��loop are dashed.

10-1

100

101

100

105

1010

1015

Frequency (Hertz)

Log

Mag

nitu

de

a)

10-1

100

101

102

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Frequency (Hz)

Res

idua

l Mot

ion

(rad

ians

/roo

t(H

z))

b)

120

Figure 6.11 Contributions from loop 1 and loop 2 (dashed) toresidual deviations from resonance in loop 1 (a) and in loop 2 (b).

a)

b)

10-1

100

101

102

10-15

10-10

10-5

Frequency (Hz)

Con

trib

utio

n to

Res

idua

l Mot

ion

(rad

ians

/roo

t(H

z))

10-1

100

101

102

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Frequency (Hz)

Con

trib

utio

n to

Res

idua

l Mot

ion

(rad

ians

/roo

t(H

z))

121

Table 6.1 Loop shapes and performance predictions for numerical servo design.

polefrequen-cies (Hz)

zerofrequen-cies (Hz)

Unity-gainfrequency

Performance(RMS residualdeviation,radians)

Degree ofFreedom

� 0.0016,3 at 0.016

1.6, 1.6, 16,16,

300 kHz �

�

� 0.0016,2 at 0.0164 at 3000

1.6, 16 70 Hz �

�

�3 at 0.0164 at 3000

92,16, 16

60 Hz �

�

�2 at 0.0016,30, 30

3 9 Hz �

�

122

Chapter 7 Optical Design Considerations

In previous chapters,a number of somewhatarbitrary choicesof optical

parametersand layout have been made. In this chapterwe will investigate

the motivation behind someof thosechoices. In particular, we will consider

the choiceof common-modefeedbackconfiguration,of asymmetry,of recycling

mirror reflectivity, and of modulationdepth.

7.1 Common-mode Feedback Configuration

In Figure 1.7 we showedthe feedbackto the laser frequencycoming from

the isolatorinphaseoutputandthe feedbackto the recyclingmirror positionfrom

the pick-off inphaseoutput. In principle onecould reversetheseconnectionsfor

someoptical configurations. One compelling reasonto feed back to the laser

frequencywith �� is that the shot-noiselimited sensitivity to��

, is betterin ��

thanin �� (typically by a factorof about30). Sincewe insist on high gain in this

loop, any sensingnoise in this loop will be impressedon the laser frequency*.

This noisecan be measured,and it could in principle be subtractedout of the

gravitationalwave signal, but this is technically difficult, and would require a

separatelow-noise frequencyreference†.

* Although thegravitationalwaveoutputof an ideal interferometeris insensitiveto � , a smallasymmetryin thearm

cavitiescancompromisethis common-moderejection.

† The currentlyproposedmodecleaner(seefootnote,page8) is almostquiet enough,andcould be madesufficiently

quiet by narrowingits bandwidth.

123

7.2 Asymmetry

An immediateconsequenceof this choiceof feedbackconfigurationis that

(accordingto (6.18)) avoiding gain constraintwill require

�� (7.1)

which implies (using (2.71) and (2.75))

� �� (7.2)

A numberof other aspectsof interferometerbehaviorare affectedby the value

we choosefor theasymmetry� . Oneof theseis the shot-noiselimited sensitivity

in �� to �� , the gravitationalwave signal:

��! #"%$'&�)(+*%(+* ",$)&

- ( * �! "/.,0213$4& (7.3)

where

� ( * ( * ",$4&65 7�8%9 � 8,:,: ;=<?> 7A@CB�DFE � ; (7.4)

and

- (+* � 5G7 ;HJI � "LKM& I � "%KM& 7AN �7AEOB � �

7A@PB�DFE �7QEOB � � RTSU�WV� � (7.5)

Theaboveexpressionis complicatedandits optimizationis usuallydonenumeri-

cally. However, � affectsonly theterm > 7A@PB�DXE � ; in thenumeratorand Y[Z%\^]`_ �Y _a\cb �in the denominator.The numeratorincreaseswith 7A@PB�DFE � moreslowly thandoes

124

the denominator,so increasing�� improvesthe shot-noiselimited sensi-

tivity to��

. Now��

�! �"$#% &('*),+.-0/ �1# (7.6)

is maximum when

-2/ �1#43 &('*),+(7.7)

which correspondsto # 57698 in our numericalexample.However,the following

considerationsmotivateus to choosea smallervalue for # :

1. The vacuumenvelopemustbe able to accommodatethe test masspositions

correspondingto the desiredasymmetry.This may be easierwith a smaller

value of : .2. The shot-noiselimited sensitivity to

�<;in = � is degradedif the amountof

sidebandreflectedfrom the recycling mirror�?>A@ABC� �D�E�F is too small. From

(2.77) we can see that in fact the signal will vanish when�?>G@�B1� ��E�F 3H5 ,

which happenswhen-2/ �1#I3KJMLNPO , a value uncomfortablycloseto the value

in (7.7) above.

3. As mentionedabove,if we wish to be free from gain constraint,we must

choose-/ ��#RQ & �TS

; moreover,we will seein the next sectionthat we wish

to choose&(' & �TS

, so againthis motivatesthe useof a smallervalueof # .

4. The shot-noiselimited sensitivityto��

is quite a weakfunction‡ of # . This

‡ A numericalsurveyovera largerangeof valuesfor contrastdefect,recyclingmirror transmissionandlossesshowed

that the shotnoiselimited sensitivity to gravitationalwaveswasdegradedby at most5% when U wasreducedto 0.15.

125

is becausea lack of efficiency in transmissionto the antisymmetricport can

be partially compensatedby an increasein the modulationindex.

Fromtheaboveargumentsweconcludethatthechoiceof � is not critical, and

we chooseone which is roughly 75% of the value which would give optimum

shot noise performance.

7.3 Recycling Mirror Reflectivity

Therecyclingfactor (seefootnote,page26) is maximumfor �� , and

we would like to operateas closeas possibleto this value. Beforechoosing� �we will makesurethat condition (6.19) is not violated.

��

��

�� (7.8)

implies

� � � �� (7.9)

which is satisfiedif

� �!� � � �� (7.10)

But we have alreadychosento satisfy this in Section7.2, in order to achieve

"$# �� (no right-half-planezeroin %'&)(+*�,.-0/21 ). Wearefreeto chooseanyrecycling

mirror reflectivity we want. It is interestingto note that this would not be the

casewerewe to demandthat %'& 3�*,.-0/�1 alsobe free of right-half-planezeros.

126

7.4 Modulation Index

The shot noise limited gravitationalwave sensitivity of the instrumentis a

relatively sensitivefunction of the modulationindex�

, especiallyif �� is

large, and for large �� the optimum value of�

is also large. For large�

the approximation(2.2) is no longer valid, and one needsto add (at least) an

additional pair of terms:

�� ! #" �$��% ! &" ��'�� !'( &" �)��% *'+ &"(7.11)

Wecall thefrequencycomponentsseparatedfrom theopticalcarrierby ,�- second-

order RF sidebands; they arereflectedalmostperfectlyfrom the recyclingmirror

�/. ��01'�� !��&'

2(7.12)

since(by design)* they do not resonatein the arm cavitiesnor in the recycling

cavity. They arecapableof beatingagainstthe first-orderRF sidebandsto create

a signal at the mixer output. Generalizing(2.4):

3�46587 � �:9% ' � % � � �;9% � �/� � �:9� �<� � �;9� ��' (7.13)

If � % ' 5 ��' andthesefields aswell as �/� arereal, and if � % � 5 �<� then

3 4 5 , 7 � �<�� =' � (7.14)

Thefieldsat the isolatorsatisfytheseconditions,andtheaboveexplanationholds

for 3 � , the inphasesignal at the isolator. We seethat the signal generatedin 3 �* Seefootnotepage27.

127

whenthe carrierphasechangesis unaffectedby the presenceof the second-order

sidebands,so that��

is unaffected by the presenceof the second-order

RF sidebands.The contributionto��

due to the changein phaseof the

first-ordersidebandsis affected,however; the size of this contributionchanges

by a factor*

�� "! �

� � �#! � (7.15)

This contributionremainsfrequencyindependent,sincethesignalis dueto audio

sidebandson the first-order RF sidebands,and we accountfor it by defining$%'& � ��(*),+.-0/( )1+.-32 %54&

and replacing%64&

by$%7&

in all of our formulae. In

particular, (7.9) becomes

8�9 :<;=8�> �8�9 : ;@?BADCFE � � � � �#! �

� � �#! � G � (7.16)

Using (7.11), (7.12) and (2.38), we find

� � �#! �� #! �

H=�I�JK HD�D�JK � 8�9K: ; 8 > �

8�9 : ; 8 > � (7.17)

Now if (motivatedby the argumentsin Section7.2) we have 8�9 :<; ?LADCME GONthen regardlessof whether 8�9 : ; 8 > � is positive or negative†, (7.16) is harder

to satisfy than (7.10).

* The fields P / and P�Q / areunaffectedby the phasesin the interferometersincethey do not resonatein it.

† Thesetwo casescorrespondto havingthecarrierbe “undercoupled”or “overcoupled,”respectively,at therecycling

mirror.

128

A sufficient condition for this effect to be small is

��

� �� ! �"��#�$�� % (7.18)

or

� �#�� &(' )+*�&$, �% & ' )-*�&(, � (7.19)

Thiseffectof thesecond-orderRFsidebandsis inconvenientbecauseit implies

that the frequencyresponseof the interferometerdependson modulationindex.

Increasingthe modulationindex to optimize the sensitivitywill requirenot only

adjustingthe servogainsto compensatefor the changein overall response,but

will carry a risk of significantly degradingthe control systemperformance.

Onesolution,althoughitself inconvenient,is to amplitudemodulatethebeam

at .#/ before or after the phasemodulation occurs‡. If properly tuned, the

sidebandsappliedthis way will exactlycancelthosedueto thephasemodulation.

Another potential solution is to arrangethe arm cavity lengthsso that the

second-orderRF sidebandsare exactly resonantin the arm cavities. This possi-

bility hasnot beeninvestigatedin any detail.

‡ If an electro-opticmodulator is usedas the power-stabilizingfeedbackelement,then a convenientand power-

efficient way to apply this modulationwill be to insert an additional crystal driven at 0�1 betweenthe polarizers,or

to sum the power stabilizerand 021 signalselectronicallybeforeapplying them. A stand-aloneamplitudemodulatoris

power-inefficient becausethe beammustbe attenuatedto someextentfor the modulatorresponseto be quasi-linear.

129

7.5 Arm Cavity and Recycling Cavity Lengths

The arm cavity lengthwhich is easiestto analyzeis one for which the first-

order RF sidebandsare exactly antiresonant,i.e., a length of �� with

� a whole number. In this casethe sidebandsexperienceno phasechangeupon

reflectionfrom the arm cavitiesandthe averagelengthof the recyclingcavity is

chosento be � �� . Thenwhenthe carrier resonatesin all threecavities,

the sidebandsresonateonly in the recyclingcavity30. This configurationhasthe

disadvantagethat it allows the second-orderRF sidebandsto resonatein all three

cavitiesaswell, making the responseof the interferometerharderto understand,

andverysensitiveto smalldeviationsfrom thesedesignlengths.For this reason,in

our numericalexamplethearmcavity lengthswerechosento beslightly different

from thosewhich would makethe first-orderRF sidebandsexactlyantiresonant.

It is alsopossibleto usevirtually arbitraryarm cavity lengths,so long asthe

first-orderRF sidebandsdo not resonate§|| in the arm cavities. However, if the

sidebandsarefar from beingantiresonantin eitherarm,thentheywill experience

significantphasechangesuponreflectionfrom thatarm,andtheaveragerecycling

cavity length as well as the asymmetrymust be adjustedto compensatefor this

phase.

§ If the sidebandsresonatein the arm cavities,then the RF sidebandphasebecomesquite sensitiveto �� and the

plant becomesmore ill-conditioned.

|| To avoidanexaggeratedsensitivityto alignmentit is alsoimportantto avoidallowing thesidebandsto fall onto the

resonantfrequencyof other low-order transversemodes.

130

Chapter 8 Summary and Conclusion

Whenthesystemfor extractingsignalsdescribedhereinwasfirst conceived,a

numberof concernsexistedaboutwhethertheasymmetriclayoutwouldfit into the

proposedvacuumsystem,whetherthe asymmetrywould causeexcessivelosses

becauseof thedifferentmode-matchingdemandsimposedby thetwo arms#31, how

the systemwould respondto imperfectionsin the optical componentsor in their

alignment,andwhetherit would be possibleto acquirelock with sucha system.

In addition,it wasunclearwhetherthefact thattwo of theoutputsareproportional

to barely independentlinear combinationsof two of the degreesof freedom(that

the plant is ill-conditioned)would proveto be a formidabletechnicalproblem.

Since then, a numberof theseissueshave becomebetter understood. An

experimentalprototypehasdemonstratedthat,at leastfor aparticularsetof optical

parameters,the assumptionsmade in constructinginterferometermodels have

beenreasonable.The issueof whetherit is possibleto control the ill-conditioned

systemadequatelyhas beenresolved. The performancesacrificesinvolved are

understood,and the constructionof a working prototypeis further evidencethat

this problemis not critical. The dynamicsof the interferometerarealsobelieved

fairly well understood.Theapproximateresultsfrom a simpleanalysisagreewell

with an independentlyassemblednumericalmodel, and both predict behavior

# A. Abramovici hassinceshownthat this effect is completelynegligible.

131

which is simpleenoughnot to requireadditionalcomplexity from the controller.

Optical configurationscorrespondingto a plant which is hard to control have

beenidentified, and are easily avoided.

Other issuesremain less well settled. The following sectionsdescribea

numberof such issues.

8.1 Robustness

Although the prototype interferometerdescribedin chapter5 showed,for

one geometryand set of optical parameters,that the systemis not dangerously

sensitive to imperfections,very little was done to quantify the magnitudeof

the imperfectionsexisting there,and in any case,it would be difficult to make

predictionsfor the behaviorof a full-sized interferometerthis way. In its full

generality,the issueof interferometerbehaviorin the presenceof imperfections

is very complicated.However,with a combinationof analysisof imperfections

perceivedas likely to causetrouble, and careful prototyping, the risk that an

importantproblemmaysurfacevery latecanbereduced.Oneareawhich remains

largely unexploredconcernsrobustness.This is loosely definedas the ability

of a control systemto maintain good performancein the presenceof various

imperfections. Theseimperfectionscan include differencesbetweenactualand

ideal mirror reflectivities, misalignment,imperfectionsin electroniccomponents,

andeventhe effect on the interferometerresponseof deviationsfrom resonance.

132

Both experimentalprototypingandnumericalanalysisarepowerful methods

for testing robustness. The former automatically tests for the simultaneous

effect of a large numberof imperfections,whereasthe latter permitsa carefully

controlledcalculationof the effect of a relatively small numberof imperfections.

The systempresentedhereinmay merit someadditionalnumericalinvestigation

beforethe next stageof prototypingis attempted.Sometypesof imperfections,

which arelikely both to occurandto affect systemperformance,arelistedbelow.

Beam Splitter Reflectivity

Even with today’s bestcoating techniquesit is difficult to producea beam

splitter having a reflectivity differing by less than about5% from the specified

value of 50%. Consequentlythe RF sidebandssamplethe lengths��

and��

in different proportions; the effective averagerecycling cavity length and the

effective asymmetry � are both affected. The effect of an asymmetricbeam

splitter canbepartly cancelled32 by anappropriatemodificationof��

and��

; this

hasbeencheckedin a limited numberof numericaltests.

Mixer PhaseErr or

In our modeleachmixer is driven by a local oscillatorsignalwhich is either

exactlyin phasewith thephasemodulatingsignalor exactlyin quadraturephase.

In a real interferometer,thesephaseswill be affectedby propagationdelaysin

the optical beampath, in cables,and in circuitry. The output �� is particularly

133

vulnerableto this type of phaseerror becausethe inphasesignal at the same

photodetectoris more sensitiveto��

and to � � than the quadraturephaseis

to �� . In our numericalexamplethe performancein loop 4 is not significantly

degradedevenfor a mixer phaseerror as large as 20 degrees(the derivationof

this result is in Appendix D). Futuredesignchangescould howeverreducethe

toleranceto this type of error.

Changes in Interferometer Response due toDeviations from Resonance

For arbitrary positionsof the interferometermirrors, the output signalsare

non-linearfunctionsof thesepositions** andany substantialdeviationfrom reso-

nancecancausethe interferometerresponseto furthersmallmirror displacements

to be changed.This changein responsecan affect the performanceof the con-

trol system,allowing even larger deviationsto occur. This mutual dependence

of control systemperformanceandinterferometerresponsemakesthis a difficult

problemto analyze,and further protoypingmay be the mosteffective approach.

A preliminarynumericalexplorationwas doneto seehow the matrix of deriva-

tives (of chapters2 and4) is affectedby a few selectedtypesof deviationfrom

resonance. The magnitudeof the deviationsexplored was comparableto the

deviationspermittedwithin the specificationsof Appendix B. It was found that

for deviationsin � � alonetherewas no significantchangein the matrix, for si-

** For example,the outputsareperiodic in the round-tripphases.

134

multaneouschangesin��

and��

or in��

and � � , the matrix changed,but

the diagonalelementschangedby lessthan10%,andeachoff-diagonalelements

changedby lessthan 10% of the diagonalelementin the samerow. It remains

to be seenhow thesechangesin the plant matrix would affect performanceand

whetherothercombinationsof deviationsfrom resonancewill havemoreserious

effects on the plant matrix.

8.2 Lock Acquisition

Whenthe laserandcontrolelectronicsfor an interferometerarefirst switched

on, the mirrors are in essentiallyarbitrary positions,and, for an interferometer

with suspendedcomponents,moving with a wide rangeof velocities. In this

situation,the interferometeroutputsignalsarenon-linearfunctionsof the mirror

positions and of the laser frequency. When the interferometeris locked, the

mirrors and laserfrequencyarenot allowedto deviateexceptvery slightly from

resonance,and, to a goodapproximation,the signalsat the outputare relatedto

themirror positionsandlaserfrequencyby a (linear)matrix of transferfunctions.

Thetransitionfrom theformer“out-of-lock” stateto thelockedstateis difficult to

analyzebecauseof the non-linearresponseof the systemin the out-of-lock state.

Low-ordersystemshavebeenanalyzed33,34, but thegeneralizationto higherorder

or multi-dimensionalsystemsis not obvious.

135

It is in principlepossibleto modeltheout-of-lockstatenumericallyin thetime

domain,andthis hasbeendonewith limited successfor a servoloop controlling

a simpleFabry-Perotcavity*. The algorithmsusedarecomputationallyintensive

and a thoroughexplorationof initial-value spacefor a completeinterferometer

will requirea large amountof computertime. Model developmentis also time-

consumingbecausefew analyticor experimentalresultsexist to confirmor refute

any numerical result.

8.3 Conclusion

In addition to modelingefforts which will likely continuefor sometime, an

importantnextstepin thedevelopmentof this signalextractionandcontrolsystem

will be prototyping in the 40m interferometeron the Caltechcampus. In two

importantrespects,this instrumentis muchmorelike a LIGO interferometerthan

thetabletopprototypedescribedherein.Themirrorsaresuspended,andthelosses

andlengthsaresuchthat theopticaldynamicsoccurmuchmoreslowly. The fact

that themirrorsaresuspendedmeansthat thespectrumof theseismicnoiseto be

suppressedis quitesimilar to that in LIGO. Thefact that thedynamicsareslower

will makeit possibleto measurethe responseto mirror motion for comparison

with the numericaland analytic models.

* A programcalledSimulink wasused.Simulink is a trademarkof The MathWorks, Inc.

136

Cited References

1 R. Weiss, Quarterly Progress Report of the Research Laboratory of Electronics

of the Massachusetts Institute of Technology, 105, 54, (1972).

2 R. Forward, Phys. Rev. D, 17, 379, (1978).

3 A. Abramovici, W.E. Althouse, R.W.P. Drever, Y. Gursel, S. Kawamura, F.J.

Raab, D. Shoemaker, L. Sievers, R.E. Spero, K.S. Thorne, R.E. Vogt, R. Weiss,

S.E. Whitcomb, and M.E. Zucker,Science 256,325, (1992).

4 R. Weiss, Quarterly Progress Report of the Research Laboratory of Electronics

of the Massachusetts Institute of Technology, 105, 54, (1972).

5 R. W. P. Drever, G. M. Ford, J. Hough, I. M. Kerr, A. J. Munley, J. R. Pugh, N.

A. Robertson, and H. Ward, “A Gravity-Wave Detector Using Optical Cavity

Sensing,” 9th International Conference on General Relativity and Gravitation

at Jena, GDR, (1980).

6 R. W. P. Drever, in Gravitational Radiation, ed. N Deruelle and T. Piran, North

Holland, Amsterdam, (1983), p. 321.

7 L. Schnupp, unpublished, (1986), describes the use of a similar scheme for

the extraction of the gravitational-wave signal from a recycled Michelson

interferometer.

8 R. W. P. Drever (unpublished) suggested the adaptation of L. Schnupp’s scheme

to a recycled Michelson interferometer with Fabry-Perot arms (1991).

137

9 S. Whitcomb (private communication) realized that the scheme could be used

to sense all of the critical degrees of freedom (1991).

10 A. Yariv, Optical Electronics, 4th ed. Saunders College Publishing, Philadel-

phia, (1991), p. 331.

11 J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York

(1968), p. 9.

12 Z.Y. Ou and L. Mandel, Am. J. Phys. 57 ,66, (1989).

13 Y. Gursel, P. Lindsay, P. Saulson, R. Spero, R. Weiss and S. Whitcomb, The

response of a Free Mass Gravitational Wave Antenna, LIGO internal document,

(1983).

14 J. P. Monchalin and R. Heon, Appl. Phys. Lett., 55, 1612, (1989).

15 B. J. Meers, Phys. Lett. A, 142, 465, (1989).

16 B. J. Meers, Phys. Lett. A, 142, 465, (1989).

17 P. Fritschel, “Techniques for Laser Interferometer Gravitational Wave Detec-

tors,” MIT Ph.D. Thesis, (1992).

18 P. Fritschel, “Techniques for Laser Interferometer Gravitational Wave Detec-

tors,” MIT Ph.D. Thesis, (1992).

19 P. Fritschel, D. Shoemaker, R. Weiss, Appl. Optics, 31, 1412, (1992).

138

20 D. Shoemaker,P. Fritschel,J. Giaime,N. Christensen,R. Weiss,Appl. Optics,

30, 3133, (1991).

21 J.M. Maciejowski,Multivariable Feedback Design. Addison-Wesley,Reading

(1989), p 27.

22 M. Morari, andE. Zafiriou, Robust Process Control. Prentice-Hall,Englewood

Clif fs, (1989), pp. 339ff.

23 D. Shoemakerand R. Weiss, Analysis of an Externally Modulated Recycled

Interferometer, internal LIGO document(1992).

24 Morari, M. andZafiriou, E., Robust Process Control,Prentice-Hall,Englewood

Clif fs, (1989), p 220.

25 G. J. ThalerandR. G. Brown, Servomechanism Analysis. McGraw-Hill, New

York, (1953), pp 149–151.

26 L. Sievers,private communication,(1994).

27 A. Gillespie, private communication,(1993).

28 T. Lyons, private communication(1994).

29 L. Sievers,private communication(1994).

30 P. Fritschel,“Techniquesfor LaserInterferometerGravitationalWave Detec-

tors,” MIT Ph.D. Thesis,(1992).

31 A. Abramovici, private communication,(1991).

139

32 S. Whitcomb, private communication, (1991).

33 F. M. Gardner, Phaselock Techniques. Wiley, New York, (1979).

34 A. J. Viterbi, Principles of Coherent Communications. McGraw-Hill, New

York, (1966).

Appendix A Shot Noise at the Mixer Output

To quantify the noise performanceof an instrument, we must somehow

characterizethe randomprocess�� correspondingto the output in the absence

of any signal. For stationarynoise, this is most convenientlydone using the

one-sidedpowerspectrum�� of �� , definedastheFouriertransformof the

autocorrelation�� of �� :

�� (A.1)

�� ! "

# "�� %$'&%(*),+'-/.0� (A.2)

�� "

1�� 325476 98:�;�<.%� (A.3)

(As in the widely-usedtext by Papoulis1 we useboldfacedsymbolsfor random

variablesandthe notation � to meanexpectationvalueor ensembleaverage.)

The reasonthat this is a usefulway to characterizethe noiseis that if ��=� is the

input of a linear system > �� , and ? ��=� its output, then

�@A@ �� B� > �� & �� (A.4)

140

and � ��

��

�� !� ��(A.5)

Now if, for example,�"�$#%�

werean ideal “brickwall” bandpassfilter (onewhich

passesfrequencieswithin the bandpassunattenuatedbut attenuatesout of band

frequenciesinfinitely) with passband& ��')(�� +* in an ideal spectrumanalyzer,then

the expectedpowerout of that filter (expectedpower ‘‘in that frequencyband’’)

would be

�� , ��-�.�/

0�1

0+2� �!� ��3�4��65

(A.6)

The output 798 �:�+� of our model, in the absenceof signal, is not stationarysince

the photocurrentfluctuatesat twice the modulation frequencyand since it is

then demodulatedat the modulationfrequency. Similar problemsto this have

beenanalyzedpreviously2,3. Neither of thesesolutionswas derivedwithin the

formalismof Papoulis’stext, anda re-derivationin this morestandardlanguage

is useful.

We notethat 7$8 ��+� is cyclostationary;for any� � thestatisticsof 7$8 �:� � � arethe

sameasthoseof 7$8 �:� �<;>= � where= is theperiodof themodulation.Consequently,

the autocorrelation�@?BA�?CA!�:� ;"D (��+�@E� 7$8 �:� ;FD � 7�8 ��+� is also periodic in

�. For

141

an arbitrary cyclostationaryprocess�� with period � , if we define an average

autocorrelation

��

��

�� !� (A.7)

and its Fourier transform,the averagepower spectrum

" � �$#!�%�'&(

) (�� !�+*-,+.0/21430��5� (A.8)

thenthe latter alsohasthe importantpropertiesascribedto" � �$#6� above4. Now,

if 7 � �8� is again the output of a linear system 9 �$#!� to which �� is the input,

then�:;: �=< � hasthe significanceof being the time average(over the period � )

of the varianceof 7 � �� . In the exampleof the spectrumanalyzer,we expectthe

powerat theoutputof our filter to fluctuatewith period � , but theaverageof the

expectedpowerover an integralnumberof periodsis given by��:;: �>< � .

Our goal is to calculate"�?8@$?8@ ��#!� and

"�?+AB?8A �$#!� , the averagepowerspectrum

of the signal at the mixer output. In Chapter2 we modeledthe mixer as being

a devicewhich multiplies by CEDGFIH � or by F�JLK�H � and then averagesover some

time interval � , and for conveniencewe chose � to be an integral multiple of

the modulationperiod. We beginby defining M N MLO�CEDGF�H � and M>P MLOQF8J2KRH � to

be the signalafter multiplication by the appropriatesinusoidbut beforelow-pass

filtering, and finding" / @ / @ ��#�� and

" / A / A �$#!� .To calculate

� / @ / @ � �I� , we first find the expectedphotodetectorsignal MLO �� :142

�� ! #" � �%$� � � � �&�'� �� $�(� � � � �� #" � �%$�� )� � ��* �� * �,+.-#/102�3��45�,/76&8902�:� *

�+.-#/ 02�:�;4

�/�6�8 02�

(A.9)

Next we model the shotnoiseprocessasa processof Poissonimpulseswith

density < ��

< �� * � � * � +.-#/102�3�;4 � /76&8902�3� *�+�-=/ 02�>�;4

�/76&8 02�

(A.10)

Eachimpulsecorrespondsto the generationof onephotoelectronin our photode-

tector,andtheunitsof��

arephotoelectrons/secondbecauseof our definition(2.1)

of the fields. A Poissonprocess? �@�� is defined5 as a processwhich is constant

exceptfor unit incrementsat randompoints in time A � , wherethe densityof the

points A � is < �@�� , and the processof PoissonimpulsesB �� is its derivative6

B ��CED ? �F��D � (A.11)

i.e.,

B �@��G �H �� A � �;I

(A.12)

The autocorrelationof a non-uniformPoissonprocessis given by7:

"9J)J �� LK ��

�NM� < �F�� D � O2� �QP

� < �@�� D � ��9RS��

�@P� < �F�� D � O2� �FM

� < �@�� D � ��RS�� (A.13)

143

andthe autocorrelationof the derivativeof a randomprocessis given by8

�� (A.14)

Since � ��! #"��$�we haveonly to substituteinto the equationaboveto find

the autocorrelationwe seek.

�&%'%�� (�� )�� (A.15)

For��+*,��

,

�-��.0/

12 ��$�43'� 576

.98

12 ��$��3'�

(A.16)

� ��:��

�.0/

12 ��$�;3<� 2 ��

(A.17)

��

� 2 �� 2 ��(A.18)

Similarly, for� >= � �

,

�-��.�8

12 ��$�43<� 576

./

12 ��$��3'�

(A.19)

� ��

� 2 ��?�@6.0/

12 ��;3<� 2 ��?�

(A.20)

and we have

�� -��

� 2 �� 2 ��(A.21)

144

again. However,thereis a discontinuityin�� at �� , so

�� ! � �!" �$# � � �%" # � � �!"'& # � � �%"�( � � � � �)"+* (A.22)

We can use this result to find the averageautocorrelationof the inphase

signal ,�- :

��.0/�.1/!� � &324 � " �65 798 � � &32�";:%<>=@? � � &A2�" 78 � � "B:%<C=�? ��65 7D8 � � &32�" 7D8 � � " :%<C=@? � � &32�"E:�<>=@? ��GF9# � � &32�" # � � "H& # � � &32�"I( � 2@"�JK:�<>=@? � � &32�"E:�<>=@? �

(A.23)

and of the inphasemixer output L�-

�M.1/�.1/!� 2�" �NO

P

Q�M.1/ .1/)� � &A24 � "�R �

�NO

P

Q FS# � � &32�" # � � "T& # � � &A2'"�( � 2@"UJ�:�<>=@? � � &32�";:%<C=�? � R � *(A.24)

The first term in the abovewill turn out to be irrelevantto our analysis;it is

NO

P

Q FS# � � &A2'" # � � "�JK:�<>=@? � � &32�"E:�<>=@? � R �

�NV FSW�X � � & Y X �Q &ZY X Q X[� & X �� &]\ �� :�<>=@?^2& X � � &]\ �� :�<>= W ?^2_& X �� &+\ �� :�<>=@`>?^2aJ

(A.25)

The secondterm is

145

��

�� ! ��"�#��$�%�&'

( �) �� * � � *,+

)(A.26)

To find the averagepowerspectrum,we take the Fourier transform:

-�.0/�. / �21$� ( )435 3

6 .7/2.7/ � ��98 +;: .=<?> &@

( * � � *�+)

�BA�* + C �� )ED 1F�� A�* +� �GA�* � *H+I�B* ++ �KJ ++ �� )LD 1 �M�� * + C �KJ +C �� )LD 1 ) �M�� * ++ �KJ ++ �� )LD 1 N��M�

(A.27)

This powerspectrumhasfour sharpcomponents,oneat DC†, andthreemore

at harmonicsof the modulationfrequency,aswell asa broadbandcomponent.It

is only the broadbandcomponentwhich interestsus, sinceonly it falls into the

frequencyrangewherewe expectto detectgravitationalradiation. The lowpass

filter in ourmodelof thedetectionsystemwill leavethispartof thenoisespectrum

unaffected,and will attenuatethe very high frequencycomponents.Hencewe

† It is interestingto note that the DC componentof the abovespectrumwill vanishor be very small if our servos

areeffective since OQP and RQP areproportionalto the mixer outputs.

146

write ��

� �� ! �� (A.28)

For the averageshotnoisespectraldensityat the inphasemixer output.

We takethe sameapproachfor the quadraturemixer output. We write “ "$#&% ”

insteadof “ ')(*" ” in (A.23) and(A.24), andevaluatethe sameintegralsto get

� �$+,��+ �-��

� �� . �� (A.29)

We have,in the processof deriving this last resultagainthrown awaysharp

componentsin thespectrumat DC andat harmonicsof themodulationfrequency.

At either output, there could of coursebe light due to other sourcesthan

the idealizedfields we have modeledhere. Two examplesare light presentin

orthogonaltransversemodesbecauseof mirror imperfectons,and light from an

incandescentbulb which might beusedto performa shotnoisecalibration.When

there is excesslight on the photodetector,it frequently contributesonly to��

and we write

�� -��/� ��0-1�230-4-4 � � �5� � � �� 6�� . �� (A.30)

and

�� + � + �-��7� �80�1�230-4-4 � � �� . �� (A.31)

where�80�1�230-494 �

is the additionaloptical power(in units of photons/second).

147

148

Cited References

1 A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-

Hill, New York, (1965).

2 S. Whitcomb, unpublished, (1984).

3 T. M. Niebauer et al., Phys. Rev. A, 43, 5022, (1991).

4 Papoulis, pp. 449-451

5 Papoulis, p. 284.

6 Papoulis, p. 287.

7 Papoulis, p. 286 and equation (9–18).

8 Papoulis, pp. 316–317 and equation (9–79).

Appendix B Specification of Allowable RMSDeviations from Perfect Resonance

B.1 Introduction

In this appendixwe will seea set of specifications on allowed root-mean-

square(RMS) deviationsfrom resonance. Each specificationis motivated by

some mechanismthrough which a low-frequency(less than 10 Hz) deviation

from resonancedegradesthe in-band (100 Hz to 10 kHz) signal-to-noiseratio

eitherby reducingthesignalstrengthor by increasingthenoise.Sincethe signal

to noiseratio at the output dependson a numberof factors,only one of which

is the deviationfrom resonancebeingconsidered,settingspecificationson all of

thesefactorsinvolves an analysisof the costsandbenefitsof changingany one

of the specifications. A thoroughtreatmentof theseissuesis impossiblehere.

It is likely that as researchprogressesand new information becomesavailable,

the costsof changingcertainspecificationswill be reassessedand it is possible

that new mechanismswill be found which will imposeadditionalconstraintson

the specifications.

Our goal here will be only to derive a sample set of specifications for

illustrative purposes.It is hopedthat althoughthe numericalexamplesworked

out in this text will lose their applicability as thesenumbersevolve,the analysis

and designmethodsoutlined will remainrelevantand useful.

149

The remainderof this appendixdealswith four mechanismsby which devia-

tions from resonancecandegradeinterferometerperformance.Theseinvolve the

needto keep ample power circulating in the arm cavities, the needto prevent

laser frequencyor intensity fluctuationsfrom producinga signal in the gravita-

tional waveoutput,and the needto keepthe antisymmetricoutputdark.

B.2 Power in the Arm Cavities

As we sawin chapter3, thegenerationof audiosidebandson thecarrierin the

arm cavitiesproducesthe gravitationalwavesignal. A deviationfrom resonance

that reducesthepowerin thearmcavitiesby 10%would reducethegravitational

wave signal strengthby 5%. We will adoptthis numberas our specification on

power stability in the arm cavities.

From (2.13) we know that the power in a cavity AD (as in Figure2.3) is

��

�

��

� �� "!# � � � � !

$&%� (B.1)

and��'�)(+*-,/.&��'�10��2

for

$&% *-,/31� � �4�� (B.2)

We can derive a numberof specificationsby interpretingthis for the different

degreesof freedom.

150

If thearmcavity backmirrorsmoveequalamountsin thesamedirection,then

the relevantequivalentcavity is the one of Figure 3.2, which hasa compound

mirror consistingof the recyclingmirror anda front mirror; then

��

� � �! #"$� ��%!&(')%!*�+ (B.3)

For changesin , � we take ��-/.0�� and ��12.0��3)4 ; then

, � �5�� 6�7� ��384� � ��384

�� 6� �(B.4)

Thesetwo specificationsaresimilar to numbersquotedelsewhere1.

B.3 Frequency Noise

D. Shoemaker2,3 has shown that in order to avoid degradingthe expected

99% frequencynoiserejectionratio dueto the symmetryof the arm cavities,we

require*

9 �;:< 9 ��:= . � � � �/� � � #> (B.5)

If� �

satisfiesthe specificationabove,then we require

� �5�? ��A@(B.6)

* Shoemakerderivesthe constraintassumingthat all mirrors are stationaryexceptfor one arm cavity back mirror.

His result is easilygeneralizedto the oneshownhere.

151

B.4 Intensity Noise

Since the gravitational wave signal is proportional to the product of the

laserpower and��

, low frequencydeviationsin the latter can couple in-band

fluctuationsin the former into the gravitational wave output. If 100 mW of

laserpoweris pickedoff for the purposeof powerstabilization,thenthe relative

intensity noise(just after the pick-off, assumingthat it is limited by shot noise

in the referencephotodetector)will be

��

(B.7)

Now if thepick-off is locatedbetweenthemode-cleaner(seefootnoteon p.8)and

the recyclingmirror, thenthe low-passnatureof the recyclingcavity will reduce

this noiseat the beamsplitter by abouta factor of 50 ( "!� #!$! ):

�%��&�'(�� *)+�-, ��/.0. � ��

(B.8)

If our target interferometersensitivity is 1 �� (24365� ��

, correspondingto

� ��7(8 �9� �� .:2 � ��

, then we require

��<; 1 ��=2

(B.9)

B.5 Dark at the Antisymmetric Output

If the light returning from the two arm cavitiesdoesnot interfereperfectly

destructivelyat the antisymmetricoutput, then the additionalphotonshitting the

152

photodetectortherewill generateexcessshotnoise. It is believedthat about1%

of the power returningto the beamsplitter will arrive at the antisymmetricport

photodiodebecauseof mirror-imperfection-induceddifferencesin beamshapeor

intensity. We adoptthe requirementthat no morethanan additional0.1%of the

powerreturningto thebeamsplitterexit theantisymmetricport dueto deviations

in��

and � � .

For � ��, �� , so we require

�� ! �#"$�%�'&

(B.10)

and†

�� !)( &*� � �(B.11)

To derivethespecificationfor � � , wenotethatthephaseof thelight returning

to thebeamsplitter changes+-,��/.0 �213�4 0� 3�4 &*5%�

timesfasterfor � � thanfor��

,

so that the specificationmust be tighter by this factor:

� � !76 &�� 98(B.12)

B.6 Summary

For our final set of specifications,we choosethe tightestconstrainton each

of the degreesof freedom:

† Specifyingseparatelimits on : � and ; � , in order to ensuredarknessat the antisymmetricport, is a conservative

strategysince the power leaving the antisymmetricport is actually proportionalto : �=<?> 1@BADC ; � . A slightly more

sophisticatednumericalservomodelthanthe onein Chapter6 would showthat the effect of residualdeviationsin : � is

partially cancelledby opposingdeviationsproducedin ; � by the loop feedingbackto ; � .

153

�� (B.13)

� ��(B.14)

� �� (B.15)

� �� (B.16)

We demand that the RMS deviations in any of these degrees of freedom not

exceed the above-listed bounds.

154

155

Cited References

1 Technical Supplement to the LIGO Construction Proposal, internal LIGO doc-

ument, (1993), p. B-3.

2 D. Shoemaker, LIGO Working Paper #109, Internal LIGO document, (1991).

3 D. Shoemaker, Modulation and topology: Deviations from resonance, Internal

LIGO document, (1991).

Appendix C Alternative FeedbackConfigurations

In this appendixwe considertwo alternativefeedbackconfigurationsand

outline the consequencesof adoptingone of theseconfigurationsinsteadof the

one describedin Chapter1.

Thefirst modifiedconfigurationwe consideris shownin FigureC.1; it differs

from theconfigurationof Figure1.7 only in that thequadraturephasesignalfrom

the isolator, insteadof being fed back to the arm cavity mirrors, is fed back to

the beamsplitter and the recycling mirror in proportionsuchthat��

and��

are

changedin equaland oppositeamounts.

Figure C.1 First alternativeconfiguration: feed-backto beamsplitter.

I Q

I Q

I Q



Summing Node

Inverting Amplifier

Let us find the responseof the interferometerto this type of mirror motion.

Sincethebeamsplitteraffectsonly��

andsincethebeamsplitter is angledat 45� ,

156

this occurswhen the beamsplitter is displacedby�

as much as the recycling

mirror ( �� ) and we let

�� (C.1)

representthis combinationof displacements.To derivethe transferfunction from

� �to �� we first note that motion of the recycling mirror producesno signal

here, and we chooseto use that combinationof beam splitter and recycling

mirror motion which makesthe audio sidebandspropagatingtowards the two

input mirrors exactly equalin amplitudeandoppositein phase.This occursfor

�� . Then theseaudio sidebandswill interfere destructivelyon

the symmetricside of the beamsplitter and constructivelyon the antisymmetric

side, so that

"!$#&%(' )+* -,/.103254 �76 � (C.2)

"!$#8%(' � "%9';:5#=<?> � �=@��ACB�DFEE A�G. DFEE A(C.3)

This result hasno importanteffects on systemperformancesincewe still have

"! # % ' !$#�H�'.

Next we note that

!=I7%9' � !=I$J�' � !=I=J�'K<�>(C.4)

since,as before,the characteristicfrequenciesexperiencedby the sidebandsare

all very high.

157

Finally we look at the responsein �� and �� . The abovecombinationof dis-

placements,with�� , mustproducea frequency-independentsignal

in �� and �� becausetheaudiosidebandspresentat theseopticaloutputshavenot

interactedwith the arm cavities.Let us find the differencebetweenthis displace-

ment and �� . We want to considerthe beamsplitter and recycling mirror as a

compoundmirror which is a sourceof audiosidebands,andwe want to solvefor

theproportionof displacementfor which thesidebandsgeneratedin thedirection

of the perpendiculararm are exactly equaland oppositethosegeneratedin the

directionof thein-line arm. Motion of thebeamsplitteraloneproducessidebands

propagatingtowardsbotharms,in theamount � �� ! #"�$ � �&%('*) $ � � � � ��+ ,in thedirectionof theperpendiculararmandin theamount� � � �-�� "�$ � �.)�� + ,in thedirectionof the in-line arm. Therecyclingmirror whendisplacedproduces

sidebandsof the samesize � � �0/ ) $ � � �� (if we take � � )�� ) propagatingtowards

both arms. Setting the total sidebandspropagatingtowardsthe two armsequal

and opposite,and neglectingbeamsplitter loss, we get:

� � 21, 1 ' 1� � �4365 "�$

� �7� + ,(C.5)

for the ratio of displacementswhich producesa frequency-independentresponse

in �� and �� . Since

8:9&;6<-=�>@? 8:9-AB<-=�>C? ED(C.6)

158

we must have

��

�� !�� "� #%$�# $� # $�

&' ()(�+*-, �.�

(C.7)

and

� �0/1�!� � �� 0/1��"� # �# � # �

&' (2(�3*4, �.� (C.8)

Theseeffectscanbe summarizedby writing the new plant56

as follows

567698 : �<;=�>�!�

: � ;�/1�!�: :: :

56�? (C.9)

where56�?

differs from6�?

only in its upperright element.The analysisand

designmethodspresentedin Chapter6 still apply. The non-zeroelementsin the

upperright sub-blockrepresentadditionalnoisesummedinto the common-mode

subsystemfrom the differential subsystem.Thesewill degradethe performance

of the system. In a numericalanalysisusing the samecontrollersas in Table

6.6, the predictedperformanceis the sameas in that example,except that in

the @ 8 loop the RMS residualdeviationis increasedto � ACB � :?ED

. A word of

cautionis appropriatehere.The specsderivedin AppendixB implicitly assumed

very-low-frequencydeviationsfrom resonanceandtheirgeneralizationto dynamic

deviations is not obvious. Moreover, becauseof the method used to assess

performancenumerically,we implicitly changedthedynamicgeneralizationof the

performancespecificationswhenwe switchedplants,andthis is partly responsible

159

for the degradationin predictedperformance.Only a morecarefulderivationof

the specificationson allowabledeviationsfrom resonance,which specifies,as a

functionof frequency,theconstraintson mirror displacementsor changesin laser

optical frequency,can resolvethis issueproperly.

The secondalternativeconfigurationto considerinvolvesno feedbackto the

arm cavity back mirrors1. Insteadthe front mirrors are driven. This is shown

in Figure C.2.

Figure C.2 Secondalternativefeedbackconfiguration.

I Q

I Q

I Q



Summing Node

Inverting Amplifier

The correspondingplant matrix is

� ��

� ��

� ��

(C.10)

The main consequenceof this changeis to make the lower left elementof� ��

comparablein magnitudeto its lower right element. This meansthat residual

deviationsin loop 4 will producea largersignalin loop 3. This effect is unlikely

160

to be importantsincethe specificationsare tighter for deviationsin loop 4 than

in loop 3.

One additional considerationin analyzing thesealternativesis that in the

presenceof mixer phaseerror, the commonmode and differential subsystems

becomefully coupled.They canno longerbe analyzed(asin FigureD.1) astwo

subsystemswhich are independentexceptthat one injects noise into the other,

since signalsnow flow in both directions. Both performanceand stability can

be affected.

161

162

Cited References

1 L. Sievers, private communication, (1993).

Appendix D Effect of Mixer Phase Error

In this appendix,we will fill in the derivationof the effect of mixer phase

error mentionedin Chapter8. As mentionedthere,the output �� is particularly

vulnerableto this type of phaseerror becausethe inphasesignal at the same

photodetectoris moresensitiveto��

andto � � than the quadraturephaseis to

�� . This error transformstheinterferometerplantmatrix from theblock-diagonal

form of equation6.1 to the modified form‡

�� (D.1)

where

�� ! ��#"��$�%'&(*)�+

(D.2)

and+

is the phaseerror in the local oscillator at the � � mixer. If we neglect

the small coupling betweenloops 3 and 4, then the effect of this error can be

analyzedusing the block diagramof Figure D.1.

,�� - � . �0/21 � , � / � �� , ��3 . � . �- �. �0/ � , � / � �� , �46587 �� / 9587 :� � � &

(;)<+ />= 587 :� � � &(;)<+

(D.3)

‡ We neglectthe frequencydependenceof the plant sincewe seekonly a roughestimateof the sizeof the effect.

163

Figure D.1 Block diagram used to analyze the effect of mixer phase error.

P+

P44

d1

C4

d2

d4

e1

e2

e4

K

ε1K

C1C2

In the second line of this last equation, we have used the approximation that

�� , and in the third line we have substituted values from our numerical

example (where � � �� ). This is the result we use in Chapter 8.

164

thesis by Martin Regehr

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