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Design and Performance of High Laser Power Interferometers for Gravitational-Wave Detection

Permanent Link: http://ufdc.ufl.edu/UFE0043578/00001

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Title: Design and Performance of High Laser Power Interferometers for Gravitational-Wave Detection
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Dooley, Katherine
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: gravitational-waves -- interferometer -- optics
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A prediction of Einstein's general theory of relativity, gravitational waves (GWs) are perturbations of the flat space-time Minkowski metric that travel at the speed of light. Indirectly measured by Hulse and Taylor in the 1970s through the energy they carried away from a binary pulsar system, gravitational waves have yet to be detected directly. The Laser Interferometer Gravitational-wave Observatory (LIGO) is part of a global network of gravitational-wave detectors that seeks to detect directly gravitational waves and to study their sources. LIGO operates on the principle of measuring the gravitational wave's physical signature of a strain, or relative displacement of inertial masses. An extremely small effect whose biggest of expected transient signals on Earth is on the order of one part in $10^{23}$, gravitational-wave strain can only be measured by detectors so sensitive to displacement as to encounter the effects of quantum physics. To improve their sensitivities and to demonstrate advanced technologies, the LIGO observatories in Hanford, WA and Livingston, LA underwent an upgrade between fall 2007 and summer 2009 called Enhanced LIGO. This study focuses on the experimental challenges of one of the goals of the upgrade: operating at an increased laser power. I present the design and characterization of two of the interferometer subsystems that are critical for the path towards higher laser power: the Input Optics (IO) and the Angular Sensing and Control (ASC) subsystems. The IO required a new design so its optical components would not be susceptible to high power effects such as thermal lensing or thermal beam drift. The ASC required a new design in order to address static instabilities of the arm cavities caused by increased radiation pressure. In all, I demonstrate the capability of an interferometric GW detector to operate at several times the highest of laser powers previously used.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Katherine Dooley.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Reitze, David H.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043578:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043578/00001

Material Information

Title: Design and Performance of High Laser Power Interferometers for Gravitational-Wave Detection
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Dooley, Katherine
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: gravitational-waves -- interferometer -- optics
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A prediction of Einstein's general theory of relativity, gravitational waves (GWs) are perturbations of the flat space-time Minkowski metric that travel at the speed of light. Indirectly measured by Hulse and Taylor in the 1970s through the energy they carried away from a binary pulsar system, gravitational waves have yet to be detected directly. The Laser Interferometer Gravitational-wave Observatory (LIGO) is part of a global network of gravitational-wave detectors that seeks to detect directly gravitational waves and to study their sources. LIGO operates on the principle of measuring the gravitational wave's physical signature of a strain, or relative displacement of inertial masses. An extremely small effect whose biggest of expected transient signals on Earth is on the order of one part in $10^{23}$, gravitational-wave strain can only be measured by detectors so sensitive to displacement as to encounter the effects of quantum physics. To improve their sensitivities and to demonstrate advanced technologies, the LIGO observatories in Hanford, WA and Livingston, LA underwent an upgrade between fall 2007 and summer 2009 called Enhanced LIGO. This study focuses on the experimental challenges of one of the goals of the upgrade: operating at an increased laser power. I present the design and characterization of two of the interferometer subsystems that are critical for the path towards higher laser power: the Input Optics (IO) and the Angular Sensing and Control (ASC) subsystems. The IO required a new design so its optical components would not be susceptible to high power effects such as thermal lensing or thermal beam drift. The ASC required a new design in order to address static instabilities of the arm cavities caused by increased radiation pressure. In all, I demonstrate the capability of an interferometric GW detector to operate at several times the highest of laser powers previously used.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Katherine Dooley.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Reitze, David H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043578:00001


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DESIGNANDPERFORMANCEOFHIGHLASERPOWERINTERFEROMETERSFOR GRAVITATIONAL-WAVEDETECTION By KATHERINELAIRDDOOLEY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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2011KatherineLairdDooley 2

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Formygrandmother 3

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ACKNOWLEDGMENTS Iwasoneofthefewveryluckygraduatestudentswhohadtheprivilegetospendmy graduateschoolyearsattheLIGOLivingstonObservatory.Igottolearnhowtheinterferometer worksrst-handandIhadtheprivilegeofgettingtoknowandworkwithsomanyoftheworld's topinterferometerexperts.Manythankstomyadvisor,DaveReitze,forsupportingmydesireto resideatthesiteandforspendingayeartherehimselfhelpingmegetmythesisworkunderway andwell-denedforthenextseveralyears.Thankyou,too,forallofthephonecallsoverthe years. Iwouldn'thavemadeitallthiswayifitweren'tformyteachers,friends,andfamilywho haveservedasrolemodelsthroughoutmylife,makingmefeelcompletelycomfortabletopursue myinterests,includingthoseinmathandscience.SpecialthankstomyUncleTom,Jamie Lombardi,BenHummon,andMrs.Matts.Mostofall,Ithankmyparents,JanineandAlan,for theirsupportandloveallalongtheway. MyLIGOcareerwouldnotbeifitweren'tfortheserendipitoustimingofchattingwithmy lateneighbor,NoahGoldsh,wholearnedofmySURFworkandcontinuedinterestinLIGO. NoahintroducedmetoAndriGretarssonwhointurnrecommendedmetotheUFgroup.Thank you,Andri.IamthankfulforGuidoMueller,DaveReitze,andDavidTanner'sgoodjudgement torecruitmeonly2monthsbeforetheschoolyearstartedanddespitethefactIhadalready acceptedgradschooladmissionelsewhere.Thatwasthebestthingthatcouldhavehappenedfor me. Thankyoutoallofmyfellowgraduatestudentswhohelpedgurethingsoutwithme andwhomadeworkingatthesitesallthemorefun:TobinFricke,NicolasSmith,DanHoak, AnamariaEfer,RyanDeRosa,JeffKissel,andRupalAmin.Tobindeservesextrarecognition forallthathetaughtmeandforhisendlesspatienceandencouragement. RanaAdhikarialwaysmadesureIhadonetoomanyinterestingprojectstoworkonand wouldbugmeonmyprogress.Itwasnicetobelookedafter.Thankyou,Rana,andthanksfor alsoteachingmetowritegoodelogentries.ThankstoSamWaldman,LisaBarsotti,andMatt 4

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Evansfordemonstratinghowtobegoodscientistsandforallthetimeyouspentatthesites. ThankstoRaiWeissforhisenthusiasm,formakingmemakethatrstoutlineofmythesis,and foralwaysbeinganadvocateforusstudents. ThankstotheLLOstaffandtheUFLIGOgroup,especiallyTomEvansforlettingme borrowhisjugglingclubsandGuidoMuellerforbeingsoreasonable.AndDavidFeldbaumsays ImustincludeathankyouforhimbecauseImadehimsearchtheelogforaspiriconBCSpicture onetoomanytimesforme.Thankyou,David! Finally,thisthesiswouldnotbeifitweren'tforValeraFrolov,whoIthanksincerely forwelcomingmeasafellowcommissionerandforbeingarolemodelonhowtoapproach scienticwork.Hetaughtmetothinkcriticallyandcouldalwaysbecountonwhetheritbeday, night,orweekend,toofferadviceandthought-provokingdiscussions. ThisworkwassupportedbytheNationalScienceFoundationthroughgrantsPHY-0855313 andPHY-0555453.LIGOwasconstructedbytheCaliforniaInstituteofTechnologyand MassachusettsInstituteofTechnologywithfundingfromtheNationalScienceFoundationand operatesundercooperativeagreementPHY-0757058. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 10 LISTOFFIGURES ....................................... 11 LISTOFABBREVIATIONS .................................. 14 ABSTRACT ........................................... 16 CHAPTER 1THESEARCHFORGRAVITATIONALWAVES .................... 18 1.1TheTheoryofGravitationalRadiation ....................... 18 1.2Sources ....................................... 20 1.3MethodsofDetection ................................ 21 1.4StateofGround-basedInterferometry ....................... 21 1.5MotivationforthisWork .............................. 23 1.5.1TheInputOpticsHighPowerStory ..................... 24 1.5.2TheAngularSensingandControlHighPowerStory ............ 25 2LASERINTERFEROMETERSFORGRAVITATIONAL-WAVEDETECTION .... 26 2.1MeasuringGravitational-waveStrainwithLight .................. 26 2.2Power-recycledFabry-PerotMichelsonInterferometers .............. 27 2.2.1DCReadout ................................. 28 2.2.2DARM .................................... 29 2.2.3DARMandStrainOpticalGain ....................... 30 2.3SignalVersusNoise ................................. 30 2.3.1Noise .................................... 30 2.3.2NoiseFloor ................................. 31 2.3.2.1Displacementnoiseoor .................... 31 2.3.2.2Sensingnoiseoor ........................ 32 2.4ControllingtheInterferometer ........................... 32 2.4.1RFSidebands ................................ 32 2.4.2DigitalControlinLIGO ........................... 33 2.4.3MirrorSuspensionandActuation ...................... 33 2.5Summary ...................................... 34 3INPUTOPTICSDESIGNANDCHARACTERIZATION ................ 36 3.1FunctionoftheInputOptics ............................ 36 3.1.1Electro-opticModulator ........................... 36 3.1.2ModeCleaner ................................ 37 6

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3.1.3FaradayIsolator ............................... 38 3.1.4Mode-matchingTelescope ......................... 38 3.2ThermalProblemsinInitialLIGO ......................... 38 3.3EnhancedLIGOInputOpticsDesign ........................ 40 3.3.1Electro-opticModulatorDesign ....................... 40 3.3.2ModeCleanerDesign ............................ 43 3.3.3FaradayIsolatorDesign ........................... 45 3.3.3.1Thermalbirefringence ...................... 45 3.3.3.2Thermallensing ......................... 47 3.3.3.3Polarizers ............................ 47 3.3.3.4Heatconduction ......................... 48 3.3.4Mode-matchingTelescopeDesign ..................... 49 3.4PerformanceoftheEnhancedLIGOInputOptics ................. 49 3.4.1OpticalEfciency .............................. 49 3.4.1.1Modecleanerlosses ....................... 50 3.4.1.2Faradayisolatorlosses ...................... 52 3.4.2FaradayIsolationRatio ........................... 52 3.4.3ThermalSteering .............................. 54 3.4.4ThermalLensing .............................. 54 3.4.5Mode-matching ............................... 57 3.5ImplicationsforAdvancedLIGO .......................... 58 3.6Summary ...................................... 59 4ANGULARMOTIONOFTHEINTERFEROMETERMIRRORS ........... 61 4.1ToleranceforAngularMotion ........................... 61 4.2SourcesofAngularMirrorMotion ......................... 62 4.2.1GroundMotion ............................... 63 4.2.2CoilActuators ................................ 64 4.2.3NoisefromAngularControl ........................ 66 4.2.4RadiationPressure .............................. 67 4.3TheMirrorasaTorsionPendulum ......................... 67 4.4OverviewofInterferometerAlignment ....................... 69 4.5TheAngularSensingandControlServo ...................... 74 4.5.1TheWavefrontSensingScheme ....................... 74 4.5.2TheDigitalPath ............................... 76 4.5.3OpticalLeverCompensation ........................ 76 4.6AngularControlLimitations ............................ 78 5THEEFFECTOFHIGHLASERPOWERONINTERFEROMETERALIGNMENT 80 5.1TheRadiationPressureAngularSpring ...................... 80 5.1.1DiagonalizingtheModiedEquationsofMotion ............. 81 5.1.2SoftandHardModes ............................ 84 5.1.3PoleAnalysis ................................ 89 5.2Implications ..................................... 89 7

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6ANGULARSENSINGANDCONTROLCHARACTERIZATIONANDPERFORMANCEINTHERADIATIONPRESSUREEIGENBASIS .............. 92 6.1TheASCChangeofBasis ............................. 92 6.1.1WFSInputMatrix .............................. 93 6.1.2WFSOutputMatrix ............................. 94 6.1.3DiagonalizingtheWFSDriveMatrix .................... 95 6.2SensingMatrixStability .............................. 96 6.3InputBeamMotion ................................. 98 6.4TheMarginally-stablePowerRecyclingCavity .................. 101 6.4.1PowerScaling ................................ 103 6.4.2SidebandImbalance ............................. 105 6.5WFSServoOpenLoopTransferFunctions ..................... 105 6.6ResidualAngularMotion .............................. 107 6.7ASCtoDARMNoiseBudget ............................ 109 6.8SeismicFeed-forwardtotheASC ......................... 118 6.9ExperimentalMeasurementoftheRadiationPressureAngularSpring ...... 120 6.10Summary ...................................... 121 7CONCLUSIONS ..................................... 124 7.1HigherpowerinEnhancedLIGO .......................... 124 7.2Summary ...................................... 126 APPENDIX AINPUTOPTICSSUPPORTINGMATERIAL ...................... 128 A.1PhaseModulation .................................. 128 A.2ModeCleanerPole ................................. 128 A.3GaussianBeamonaSplitPhotodetector ...................... 129 A.4BeamPropagationFormalism ............................ 130 A.5BeamDriftCalibration ............................... 132 A.6CarrierMode-matchingintotheInterferometer .................. 133 A.6.1InterferometerVisibility ........................... 133 A.6.2ImpedanceMatching ............................ 134 A.6.3Mode-matching ............................... 135 A.7OverlapIntegrals .................................. 136 BANGULARSENSINGANDCONTROLCALIBRATIONS .............. 138 B.1BeamSpotMotion ................................. 138 B.1.1MovingtheBeam .............................. 138 B.1.2MeasuringHowMuchtheBeamHasMoved ................ 139 B.2AngularMirrorMotion ............................... 140 B.2.1ETMandITMOpticalLevers ........................ 141 B.2.2RM,BS,andMMT3OpticalLevers .................... 142 8

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B.3WFSErrorSignals ................................. 143 B.4AngularOpticalGain ................................ 143 CANGULARSENSINGANDCONTROLSUPPORTINGMATERIAL ......... 145 C.1OpticalLeverOpenLoopTransferFunction .................... 145 C.2MisalignedCavityAxis ............................... 145 C.3PowerinaMisalignedCavity ............................ 146 C.3.1DisplacedCavity .............................. 147 C.3.2TiltedCavity ................................. 148 C.3.3DisplacedandTiltedCavity ......................... 149 C.4InitialDCAlignmentoftheInterferometer ..................... 149 C.5Photodiodes ..................................... 150 C.6WFSControlFilters ................................. 152 C.7SeismicSpectra ................................... 153 REFERENCES ......................................... 156 BIOGRAPHICALSKETCH .................................. 160 9

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LISTOFTABLES Table page 3-1ComparisonofselectedpropertiesoftheInitialandEnhancedLIGOEOMcrystals .. 43 3-2EnhancedLIGOInputOpticspowerbudget. ....................... 50 3-3AbsorptionvaluesfortheLivingstonandHanfordmodecleanermirrors ........ 52 5-1GeometricparametersoftheLIGOarmcavityeigenmodes ............... 83 5-2Torsionalspringconstantsforthesoftandhardcavitymodes .............. 87 5-3Opto-mechanicalparametersfortheLIGOLivingstonandLIGOHanfordcavities ... 88 5-4Conditionsontotaltorsionalconstantfordeterminingsystemstability ......... 89 6-1WFSopticalgainmatrix ................................. 94 6-2WFSoutputmatrix .................................... 95 6-3Mirrorgainsfordiagonalizationofdrivematrix ..................... 96 6-4Actualeigenbasismotionduringsensingmatrixexcitations ............... 96 A-1Mirrorradiiofcurvatures. ................................. 132 B-1Beamspotmotioncalibrations .............................. 140 B-2Opticallevercalibrations ................................. 143 B-3DemodulationchaincalibrationforeachquadrantofeachWFS ............. 143 10

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LISTOFFIGURES Figure page 1-1Depictionofstrain .................................... 20 1-2StrainsensitivitiesofLIGO-VIRGOcollaborationinterferometers ........... 23 2-1Power-recycledFabry-PerotMichelsonlaserinterferometer ............... 28 2-2TheDCreadoutdarkfringe ................................ 29 2-3SimplesketchofaLIGOsuspension ........................... 34 3-1BlockdiagramoftheInputOpticssubsystem. ...................... 37 3-2BeamprolethroughtheInputOptics .......................... 39 3-3EnhancedLIGOInputOpticsopticalandsensingconguration ............. 41 3-4PhotographsoftheEnhancedLIGOHAM1InputOptics insitu ............. 42 3-5Electro-opticmodulatordesign .............................. 44 3-6Faradayisolatorphotographandschematic. ....................... 46 3-7Photographofanindium-wrappedTGGcrystal ..................... 48 3-8Datafromthemodecleanerabsorptionmeasurement .................. 51 3-9Faradayisolatorisolationratioasmeasuredinairandinvacuum ............ 53 3-10ModecleanerandFaradayisolatorthermaldriftdata. .................. 55 3-11Proleathighandlowpowersofmodecleanertransmittedbeam ............ 56 3-12Faradayisolatorthermallensingdata ........................... 57 4-1Typicalangularmotionofthecoresuspendedmirrorsintheabsenceofinterferometriccontrol ......................................... 63 4-2Contributionofseismicnoisetoopticallevererrorsignal ................ 65 4-3TorquetopitchtransferfunctionofaLIGOcoreoptic .................. 70 4-4LayoutofASCsensorsandthemirrorstheymustcontrol ................ 71 4-5Schematicofthealignmentsensingandcontrolsystem,viewedastwodifferentunits. 71 4-6Beamcenteringservoimageofbeamsplitter ....................... 73 4-7ASCcontrolservo ..................................... 75 4-8Opticallevercompensationscheme ............................ 77 11

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4-9WFSerrorsignalanddarknoise ............................. 78 5-1Illustrationoftheorthogonalmodesofcavitytilt .................... 84 5-2Controlsviewofadditionofradiationpressuretothependulumtransferfunction ... 85 5-3Torsionalspringconstantsofanopticallycoupledcavity ................ 87 5-4Singlecavityopto-mechanicaltransferfunction ..................... 88 5-5Polesofthetorquetopitchopto-mechanicaltransferfunction .............. 90 6-1Impressionofinputbeammotiononthecoremirrors .................. 99 6-2ComparisonofWFSerrorsignals(theresidualmotion)duringatimeofnormaloperationandatimewhenthecommonWFSgainswere2 5 higherthannominal .... 100 6-3Theoreticaldependenceofpowerrecyclingcavitypoweron g -factorandmirrorangle 102 6-4MeasureddependenceoftheWFSerrorsignalsonthepowerrecyclingcavitygeometry 104 6-5Openloopgains(pitch)ofthe5WFSloopsasmeasuredwith6Winputpower. .... 106 6-6Openloopgains(pitch)ofthedifferentialsoft(WFS1)loopasmeasuredatfourdifferentpowers. ....................................... 107 6-7BeamspotmotionontheITMsandETMsduringa16Wlock ............. 108 6-8AngularmotionsuppressionduetotheASC ....................... 110 6-9IndividualmirrormotionwithandwithoutASC ..................... 111 6-10WFStoDARMtransferfunctions ............................ 113 6-11OptictoDARMnoisebudget ............................... 114 6-12WFStoDARMnoisebudget ............................... 115 6-13TotalWFSandopticallevernoisecontributiontoDARMduringa16Wlockatnight 116 6-14EffectoftheWFS1lowpassltercutofffrequencyonstrainsensitivity. ......... 117 6-15DemonstrationofpotentialreductionofWFSerrorsignalsusingseismicfeed-forward 119 6-16Demonstrationofradiationpressureeigenbasistorquetoangletransferfunctionmeasurement .......................................... 120 6-17Hardopto-mechanicalmodemeasurementandtforseveralpowers. .......... 122 6-18Softopto-mechanicalmodemeasurementandtforseveralpowers. .......... 123 7-1HistogramofinputpowersusedduringS6 ........................ 125 12

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7-2Timeseriesofinterferometersignalsshowingatypicallocklossandre-lockfollowed byanincreaseofpowerto14W ............................. 125 7-3Zoomoftheshot-noise-limitednoiseoorsoftheInitialLIGOandEnhancedLIGO detectors. ......................................... 126 A-1Livingstonmodecleanerintensitynoisetransferfunction ................ 129 A-2Endofan8.7WlockatLivingstononFeb.23,2010 .................. 134 A-3Interferometerreectivityduetoimpedancemismatch ................. 136 B-1DiagramofmirrorandOSEMgeometry ......................... 139 B-2Opticallevercalibrationdata. ............................... 142 C-1Opticalleveropenlooptransferfunction ......................... 146 C-2Schematicofabasicphotoconductivephotodiode. .................... 151 C-3WFSdigitalcontrollters. ................................ 152 13

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LISTOFABBREVIATIONS A2LAngletolength ADCAnalog-to-digitalconverter ASAnti-symmetric ASCAngularSensingandControl BCSBeamcenteringservo BSBeamsplitter CWPCalcitewedgepolarizer DACDigital-to-analogconverter DARMDifferentialarm DCDirectconversion;Directcurrent DKDPDeuteratedpotassiumdihydrogenphosphate DOFDegreeoffreedom EOMElectro-opticmodulator ETMEndtestmass FIFaradayisolator FPMFabry-PerotMichelson GWGravitationalwave HEPIHydraulicexternalpre-isolator HWPHalf-waveplate IOInputOptics ISCInterferometersensingandcontrol ITMInputtestmass L2ALengthtoangle LIGOLaserInterferometerGravitational-waveObservatory LHOLIGOHanfordObservatory LLOLIGOLivingstonObservatory 14

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LSCLengthSensingandControl LVEALargevacuumequipmentarea MCModecleaner MICHMichelson MMTMode-matchingtelescope NSPOBNormalizedbeamsplittersidebandpick-off OSEMOpticalsensorandelectro-magneticactuator POBBeamsplitterpick-off PSLPre-stabilizedlaser PRCPowerrecyclingcavity PRMPower-recycledMichelson QPDQuadrantphotodiode QRQuartzrotator RMSRootmeansquare REFLReectedbeam RFRadiofrequency RMRecyclingmirror ROCRadiusofcurvature RPRadiationpressure SNRSignaltonoiseratio SPOBBeamsplittersidebandpick-off TFPThinlmpolarizer TGGTerbiumgalliumgarnate TMTestmass UGFUnitygainfrequency VIRGOVariabilityofSolarIrradianceandGravityOscillations WFSWave-frontsensor 15

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DESIGNANDPERFORMANCEOFHIGHLASERPOWERINTERFEROMETERSFOR GRAVITATIONAL-WAVEDETECTION By KatherineLairdDooley December2011 Chair:DavidReitze Major:Physics ApredictionofEinstein'sgeneraltheoryofrelativity,gravitationalwaves(GWs)are perturbationsoftheatspace-timeMinkowskimetricthattravelatthespeedoflight.Indirectly measuredbyHulseandTaylorinthe1970sthroughtheenergytheycarriedawayfromabinary pulsarsystem,gravitationalwaveshaveyettobedetecteddirectly.TheLaserInterferometer Gravitational-waveObservatory(LIGO)ispartofaglobalnetworkofgravitational-wave detectorsthatseekstodetectdirectlygravitationalwavesandtostudytheirsources. LIGOoperatesontheprincipleofmeasuringthegravitationalwave'sphysicalsignature ofastrain,orrelativedisplacementofinertialmasses.Anextremelysmalleffectwhosebiggest ofexpectedtransientsignalsonEarthisontheorderofonepartin10 23 ,gravitational-wave straincanonlybemeasuredbydetectorssosensitivetodisplacementastoencountertheeffects ofquantumphysics.Toimprovetheirsensitivitiesandtodemonstrateadvancedtechnologies, theLIGOobservatoriesinHanford,WAandLivingston,LAunderwentanupgradebetween fall2007andsummer2009calledEnhancedLIGO.Thisstudyfocusesontheexperimental challengesofoneofthegoalsoftheupgrade:operatingatanincreasedlaserpower. Ipresentthedesignandcharacterizationoftwooftheinterferometersubsystemsthatare criticalforthepathtowardshigherlaserpower:theInputOptics(IO)andtheAngularSensing andControl(ASC)subsystems.TheIOrequiredanewdesignsoitsopticalcomponentswould notbesusceptibletohighpowereffectssuchasthermallensingorthermalbeamdrift.The ASCrequiredanewdesigninordertoaddressstaticinstabilitiesofthearmcavitiescaused 16

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byincreasedradiationpressure.Inall,IdemonstratethecapabilityofaninterferometricGW detectortooperateatseveraltimesthehighestoflaserpowerspreviouslyused. 17

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CHAPTER1 THESEARCHFORGRAVITATIONALWAVES Einstein'spredictionsofgeneralrelativityopenedtothescienticcommunityawholenew windowofhowtolookattheuniverse.Justasscientistshadbeenbuildingdetectorstoobserve directlyopticalandmicrowaveradiation,theynowhadthetheoryinhandtothinkaboutbuilding detectorsforgravitationalradiation.Gravitationalwaves(GW)aredynamicstrainsinspace-time thattravelatthespeedoflightandaregeneratedbynon-axisymmetricaccelerationofmass. JosephWeberoftheUniversityofMarylandandJohnWheelerofPrincetonUniversity introducedtheeldofgravitationalwaveastronomyinthe1960s.Weberbuiltaresonantbar,the rstinstrumentdesignedtodirectlyobservegravitationalwaves[ 1 ].Althoughhisbarnevermade apositivedetection,theinterestindirectlydetectinggravitationalwavespersisted,andnewand moresensitivedetectordesignswereconceived. Themostpromisingofnewdetectordesignsformeasuringagravitationalwave'sdistortion ofspace-timeprovedtobealaserinterferometer.RobertForwardofHughesSpacecraftbuiltthe rstbenchtopprototypeinthe1970s[ 2 ].RaiWeissofM.I.T.andRonDreverofCaltechwith theaidofothersdevelopedthisconceptintowhatisbecomingaworldwidearrayoflargescale interferometers. Theeldofground-basedgravitational-wavephysicsisrapidlyapproachingastatewitha highlikelihoodofdetectingGWsforthersttime.Suchadetectionwillnotonlyvalidatepart ofEinstein'sgeneraltheoryofrelativity,butinitiateaneraofastrophysicalobservationofthe universethroughGWs.Arstdetectionisexpectedtowitnessaneventsuchasabinaryblack hole/neutronstarmerger.Thischapterprovidesthetheoreticalframeworkofgravitationalwave generationandpresentsvariouswaystodetectGWs,includingthecurrentstatusofaneffortto doso.Iexplainthepurposeofthisdissertationinthecontextofthesecurrenteffort. 1.1TheTheoryofGravitationalRadiation Gravitationalradiationisaperturbation | h | 1totheatspace-timeMinkowskimetric = diag ( # 1 1 1 1 ) [ 3 ].Themetricdescribingspace-timeinthepresenceofgravitational 18

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radiationistherefore g = + h (11) Asinelectrodynamicswhereonehasfreedominchoosingthevectorpotential A forcalculating themagneticeld B = ! A ,onealsohasfreedomingeneralrelativityinchoosingtheformof h foreaseofcalculation.Aconvenientandpopularchoiceiscalledthetransverse-traceless (TT)gaugeinwhich h = " # 0000 0 h + ( t ) h ( t ) 0 0 h ( t ) # h + ( t ) 0 0000 $ % % % % % % % & (12) wherethe + and representtwolinearlyindependentpolarizations.Withoutlossofgenerality, weconsiderthe h + polarizationintheexamplethatfollows. Foragravitationalwavetravelingalongthe z -axis,thespace-timemetricisgivenby: ds 2 = # c 2 dt 2 +[ 1 + h + ( t )] dx 2 +[ 1 # h + ( t )] dy 2 (13) ThissaystheTTcoordinatesystemisstretchedalongthe x axisandandcompressedalongthe y axisbyafactorof 1 h + ( t ) $ 1 1 2 h + ( t ) (14) Therefore,fortwofreemasseslocatedaproperdistance L fromoneanotheralongeitherthe x -axisorthe y -axis,theirseparationismagniedbythefactorinEq. 14 inthepresenceofa gravitationalwave.Theircoordinateseparations,however,remainconstant.Thegravitational waveperturbationisadimensionlessstrain: h + ( t )= 2 L ( t ) L (15) where L ( t ) isthechangeinseparationbetweenthefreemasses,asillustratedinFig. 1-1 19

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Figure1-1.Depictionofstrain. 1.2Sources Anyobjectwithanacceleratingmassquadrupolemomentgeneratesgravitationalwaves. Thetypicalstrainamplitudes,however,areextremelytiny:abinarysystemofcoalescing1 4M % neutronstarsintheVirgoCluster(adistanceof15Mpc)wouldproduceamaximumGWstrain onEarthofonly10 # 21 at800Hz[ 4 ].Thestrainisproportionaltosourcemass, M ,andvelocity, v ,andinverselyproportionaltothedistancefromtheobserver, R : h $ GMv 2 Rc 4 (16) Thequantity G / c 4 iswhatsetsthescaleforstrainamplitudesbecauseofhowsmallitis: 8 26 10 # 45 m # 1 kg # 1 s 2 .Consequently,themostpromisingsourcesofdetectablegravitational wavesarenearby,fast-moving,massiveastrophysicalobjectsthatinclude supernovae[ 5 ] binarystars(orbitingorcoalescing)[ 6 ] spinningneutronstars[ 7 ] cosmological/astrophysicalbackground[ 8 ] andcanbecategorizedasproducingperiodic,burst,orstochasticGWs. Stablyorbitingbinarystarsystemscomprisedofblackholesorneutronstarsaswellas rapidlyspinningnon-axisymmetricpulsarsareconsideredperiodicsourcessincetheywill produceGWsofrelativelyconstantfrequency.ThesereliablesourcesofGWsrequirealong integrationtimetopickouttheirsignalabovenoise.Supernovaeareburstsourcessincethe gravitationalcollapsewillproduceashort-lived,unmodeledemissionofGWs.Binariesintheir naltensofmillisecondsofinspiralalsofallintothiscategory.Finally,theanisotropiesinthe 20

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inationoftheuniversetogetherwiththehumofalldistantastrophysicalsourceswillcreate astochasticbackgroundofradiation.Coherentcross-correlationbetweenmultipledetectorsis necessaryformeasuringtheconstantamplitude,broad-spectrumGWbackground[ 9 ]. Directlydetectinggravitationalradiationfromanysuchsourcewillrevealinformationthat electromagneticradiationcannotconvey.ThefrequencyoftheGWtellsaboutthedynamical timescaleofthesource.OnlythroughGWradiation,forexample,canmassandspinproperties ofablackholeberevealed.Arstdetectionisexpectedtowitnessaneventsuchasabinary blackhole/neutronstarcoalescence[ 10 ]. 1.3MethodsofDetection Inordertodetectdirectlyagravitationalwave,theinstrumentmustbesensitivetostrain. Weber'sbarandlaserinterferometersbothaccomplishthisrequirement.Thereisathirdmethod ofdetection,however,thathasalreadyprovedsuccessful,althoughthedetectionisnotdirect. HulseandTaylorobservedtherateofchangeoftheorbitalperiodofabinarystarsystem, demonstratingbeautifullyapreciseagreementwiththepredictionsofGRshouldtherateof changebeduetogravitationalradiation[ 11 12 ].AwardedtheNobelPrizefortheirwork,Hulse andTaylor'sindirectevidenceofGWshasfueledtheeldtoproduceadirectdetection.Newer methodsunderactiveresearchincludepulsartiming[ 13 ]andB-modemeasurementsofthe cosmicmicrowavebackgroundpolarization.Foranapproachableoverviewofthehistoryofthe eld,includingdetectordesignchoicesandestimatedGWstrainamplitudesofvarioussources, refertoRef.[ 14 ]. 1.4StateofGround-basedInterferometry Anetworkofrstgenerationkilometer-scalelaserinterferometergravitational-wave detectorscompleteditsintegrated2-yeardatacollectionrunin2007,calledS5.Theinstruments were:theAmericanLaserInterferometerGravitational-waveObservatories(LIGO)[ 15 ],one inLivingston,LAwith4kmlongarmsandtwoinHanford,WAwith4kmand2kmlong 21

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arms;the3kmFrench-ItaliandetectorVIRGO[ 16 ]inCascina,Italy;andthe600mGermanBritishdetectorGEO[ 17 ]inRuthe,Germany.Multipleseparateddetectorsincreasedetection condencethroughsignalcoincidenceandimprovesourcelocalizationthroughtriangulation. TherstgenerationofLIGO,knownasInitialLIGO,achieveditsdesigngoalofsensitivity toGWsinthe40Hz-7000Hzbandwhichincludedarecordstrainsensitivityof2 10 # 23 / & Hz at155Hz.However,onlytheclosestofsourcesproduceenoughGWstraintoappearinLIGO's band,andnogravitationalwavehasyetbeenfoundintheS5data.AsecondgenerationofLIGO detectors,AdvancedLIGO,hasbeendesignedtobeatleastanorderofmagnitudemoresensitive atseveralhundredHzandaboveandincludeanimpressiveincreaseinbandwidthdownto 10Hz,dramaticallyincreasingthechancesofdetection.ThebaselineAdvancedLIGOdesign [ 18 ]improvesuponInitialLIGObyfeaturingbetterseismicisolation,theadditionofasignal recyclingmirrorattheoutputport,homodynereadout,andanincreaseinlaserpowerfrom10W to165W. TotestsomeofAdvancedLIGO'snewtechnologiessounforeseendifcultiescouldbe addressedandsothatamoresensitivedatatakingruncouldtakeplace,increasingthechances ofdetection,anincrementalupgradetotheinterferometerswascarriedoutafterS5[ 19 ].This project,EnhancedLIGO,culminatedwiththeS6sciencerunfromJuly2009toOctober2010. Anoutputmodecleanerwasdesigned,builtandinstalled,andDCreadoutoftheGWsignalwas implemented[ 20 ].AnAdvancedLIGOactiveseismicisolationtablewasalsobuilt,installed,and tested[ 21 ,Ch.5].Inaddition,the10WInitialLIGOlaserwasreplacedwitha35Wlaser[ 22 ]. Accompanyingtheincreaseinlaserpower,thetestmassThermalCompensationSystem[ 23 ],the AlignmentSensingandControl,andtheInputOpticsweremodied. Asofthewritingofthisdissertation(September2011),constructionandinstallationof AdvancedLIGOisunderway.Thevacuumsystemsarebeingretro-ttedtoaccompanythenew layout,andatLLOthe165Wlaserhasbeeninstalled.Atbothsites,thenewseismicisolation platformsandmulti-levelsuspensioncagesarebeingmass-produced.By2012,therstof thesuspendedmirrorswillbeinstalledandtestingbegun.Simultaneously,VIRGOandGEO 22

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10 1 10 2 10 3 10 24 10 22 10 20 10 18 strain [Hz 1/2 ] frequency [Hz] GEO600 (2006) GEO HF VIRGO (2009) VIRGO+ Initial LIGO (2007) Enhanced LIGO (2010) Advanced LIGO Figure1-2.StrainsensitivitiesofLIGO-VIRGOcollaborationinterferometers.Solidlinesshow achieveddetectornoiseoorsanddashedlinesshowdesignnoiseoorsfor near-futuregenerationinterferometers. arebothundergoingtheirownupgradesaswell[ 16 24 ].Figure 1-2 showstheachievedand theoreticalfuturenoisecurvesofthisnetworkofground-basedGWdetectors. 1.5MotivationforthisWork Thepurposeofthisworkistodemonstratethecapabilityofaninterferometricgravitational wavedetectortooperateatseveraltimesthehighestoflaserpowerspreviouslyused.Froma na vestandpoint,morepowerisdesirablesincestrainsensitivityimprovesby & P inthehigh frequency( > 200Hz)shot-noise-limitedregion.However,asdetectorsbecomemoresensitive atlowfrequencies( < 40Hz)intheyearstocome,radiationpressurenoisewillbecomethe dominantnoisesourcethere,makinghighlaserpoweroperationadesigntrade-off.Currently, seismicnoiselimitslowfrequencysensitivity,soexploringthetechnicalworldofincreasingthe laserpowerisafruitfuladventure. 23

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Morepowerintroducesradiationpressureandthermallyinducedsideeffectsthatmustall beaddressedforeffectiveinterferometeroperation.Concernsaboutthepracticaldifcultiesof handlinghighpowereffectsaboundedduringInitialLIGOwhenoperatingatthedesignpower of10Wprovedmoredifcultandlessstraight-forwardthanexpected.ToachievetheAdvanced LIGOdesignsensitivity,anambitious160Wofinputpowerisneeded.Withoutanunderstanding ofthethermalandradiationpressureproblemsat10W,AdvancedLIGObecomesadaunting goal. TheworkpresentedinthisdissertationwascarriedoutduringEnhancedLIGOtoverifyand investigatethepredictedandunforeseeneffectsofasmuchas25Woflaserpower.Italsoserved thepurposeofenablingtheoperationofLIGOathigherpowersandrecordstrainsensitivities.I presentthedesignandthemeasurementsImadeoftheperformanceoftwooftheinterferometer subsystemsthatareaffectedbyanincreaseinlaserpower:theInputOpticsandtheAngular SensingandControl.Ishowthatthethermalandradiationpressureeffectsonthesesubsystems arewellunderstood.ThisworkontheEnhancedLIGOdetectorsinformsdesignchoicesfor AdvancedLIGO. 1.5.1TheInputOpticsHighPowerStory TheperformanceoftheInitialLIGOInputOpticsdegradedwithinputpowerastheresult ofabsorbingtoomuchheat.Particularissuesthatneededtobeaddressedforanyfurtherincrease inpowerincludedthermalsteeringofthebeamrejectedbytheinterferometer,adecreaseinthe opticalisolation,andthermallensingthatdrovethespatialmodeofthebeamdirectedatthe interferometerawayfromoptimal.WereplacedtwoofthekeyInputOpticscomponentsand modiedtheothers.IdescribethedesignoftheimprovedInputOpticsforEnhancedLIGO whichincludeslessabsorptiveopticalcomponentsinordertoconquerthermalissuesatthe sourceandchangestothedesignarchitecturethatcompensateforanyresidualeffects.Ialso presentthesetofmeasurementsImadetocharacterizetheInputOpticsperformancewithupto 30Winputpower.IshowthatwecanexpectthedesignoftheEnhancedLIGOInputOpticsto alsoperformwellforAdvancedLIGO. 24

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1.5.2TheAngularSensingandControlHighPowerStory Radiationpressurecreatestorques,along-knownconcept,andtheopticaltorque'sability tode-stabilizeopticalcavitieswasrstrecognizedin1991bySolimenoetal.[ 25 ].However, thetheoryofradiationpressure'seffectonangularmechanicaltransferfunctionswasnotfully appreciateduntil2006,inthepaperbySidlesandSigg[ 26 ].Theconcernarosethatradiation pressuremightbethefactorlimitingInitialLIGO'sabilitytoincreasetheinputpower.Eiichi Hiroseshowedthattheopticaltorquewaspresentandmeasurable,butthatitwas not limitingat InitialLIGO'spower[ 27 ]. 1 Theconcernoftheopticaltorque'sroleincavitydynamicsshifted toEnhancedandAdvancedLIGO,whichweredesignedtooperateatfourtimesand20times thelaserpowerofInitialLIGO,respectively.LisaBarsottidevelopedanumericalmodelof theangularsensingandcontrolsforEnhancedLIGO,specicallyincludingradiationpressure torque.Sheshowedthat,inprinciple,theradiationpressuretorquecanbecontrolledwithout detrimentalconsequencestothesensitivityofthedetector[ 28 ].WeimplementedBarsotti's theoreticalcontrolschemeandImeasureditsperformancewithupto20Wofinputpower, demonstratingathoroughunderstandingoftheprinciplesatworkandprovidingcondenceinthe abilitytocontrolradiationpressuretorquesinAdvancedLIGO.IalsoimproveduponHirose's measurementoftheopticalangular(anti-)spring.Inaddition,throughpost-analysisofangular data,IdemonstratethepotentialofatechniquethatmaybeusedinAdvancedLIGOforreducing theangularcontrolsignals. 1 Infact,aftertheEnhancedLIGOlaserwasinstalled,andbeforeanychangesweremadeto theASC,wesuccessfullyoperatedtheinterferometerwith14Winputpower. 25

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CHAPTER2 LASERINTERFEROMETERSFORGRAVITATIONAL-WAVEDETECTION Weshowinthischapterhowalaserinterferometercandetectgravitationalwavestrain,and wepresentthebasicdesignprinciplesoftheLIGOdetectors.Wemotivatethedesireforhigher laserpower,andintroducesomeofthedetailsoftheinterferometerthatarerelevantforlater chapters.Toreduceclutter,Idonotspecifythepolarizationofthegravitationalwavestrainand usesimply h foritssymbol. 2.1MeasuringGravitational-waveStrainwithLight ConsideringasimpleMichelsoninterferometerconsistingofalaser,abeamsplitter,and twoendmirrorseachadistance L fromthebeamsplitter,onecanunderstandintuitivelywhyan interferometercandetectgravitationalwaves.Ifanappropriatelypolarizedgravitationalwaveis present,itwillstretchonearmandcompresstheother.Fortwowavepacketsleavingthebeam splitteratthesametime,eachheadingdownadifferentarm,theroundtriptraveltimeforthelight travelingdownthestretchedarmislongerthanthatforthelighttravelingdownthecompressed arm.Forthestretchedarmtheroundtriptraveltimeis: t stretched = 2 L c 1 + h 2 # (21) andforthecompressedarmtheroundtriptraveltimeis: t compressed = 2 L c 1 # h 2 # (22) Astationaryclockatthebeamsplittercould,inprinciple,measurethenon-zerodifferencein arrivaltimes, t = 2 Lh / c ,ofthetwodifferentwavepackets. 1 1 Itshouldbenotedthat h istreatedasaconstantinEqs. 21 and 22 .Weusetheapproximationthatthegravitationalwavewavelength # gw ismuchlargerthantheMichelsonarmlength L Thismeansthatthetemporalvariationof h ( t ) isnegligibleduringthetimeittakesthephotonto makeitsroundtrip. 26

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Inpracticewesendacontinuouselectromagneticwaveintotheinterferometer.Thedifferenceintraveltimesturnsintoadifferenceinphaseofthebeamsreturningtothebeamsplitter: $ BS MICH = % t = 2 L c % h (23) where % istheangularfrequencyofthelaserlight.Ref.[ 29 ]providesanicediscussionabout whyGWdetectorsshouldworkfromtheviewpointofgaugeinvariance.Wenowintroducethe modiedMichelsoninterferometerusedinLIGO,andinthiscontextcontinuethediscussionof strainmeasurementandsensitivity. 2.2Power-recycledFabry-PerotMichelsonInterferometers TheLIGOdetectorcongurationisapower-recycledFabry-PerotMichelson(FPM)laser interferometerasdepictedinFigure 2-1 .Abeamsplitter(BS)directs1064nmlightfroma diode-pumped,poweramplied,andintensityandfrequencystabilizedNd:YAGlasertothe Fabry-Perotarms,whicharemadeofaninputtestmassmirror(ITM)andanendtestmassmirror (ETM).Botharmsareoflength L $ 4kmandaresettomaintainnearlyperfectdestructive interferenceoftherecombinedlightattheanti-symmetric(AS)port,whereaphotodetectoris placedtomeasureanychangeinpower.Apowerrecyclingmirror(RM)atthesymmetricport directstheconstructively-interferedlightbackintotheinterferometer. TheFabry-PerotarmsareamodicationtotheMichelsonthatincreasesthechangeinphase measuredattheASportcomparedtothatforasimpleMichelson.Ratherthanmakeasingle roundtripdowneacharm,thelightistrappedbytheFabry-Perotcavity,experiencingmany roundtripsbeforereturningtothebeamsplitterandinterferingwiththelightfromtheotherarm. TheeffectisthatEq. 23 fortheFabry-PerotMichelsonincludesafrequency-dependentphase gainfactor, g $ ( f ) : $ BS FPM = 2 L c % g $ ( f ) h ( f ) (24) ForEnhancedLIGO, g $ = 137atDCandfallsoffas1 / f after85Hzduetothestoragetime ofthelightinthearmcavities.ThepowerrecyclingmirrorisamodicationtotheMichelson 27

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Figure2-1.Power-recycledFabry-PerotMichelsonlaserinterferometer. interferometerthatincreasesthecirculatingpowerbyafactorof g 2 cr $ 40.Detailsarefoundin Appendix A.6.2 2.2.1DCReadout ThegravitationalwavereadoutinEnhancedLIGOwasnotoperatedpreciselyatthedark fringeattheASport.Instead,itusedasmalloffsetfromthequadraticminimumsothatsmall changesinphaselinearlyproducepowerchangesasisdepictedinFigure 2-2 .Theoffsetused was $ 0 $ 6 10 # 5 rad.ThetechniqueisaformofhomodynedetectioncalledDCreadout.For detailsoftheimplementationofDCreadoutinEnhancedLIGO,refertoRef.[ 30 ]. WithaDCoffset,theelectriceldattheASportis E AS = E BS sin ( $ 0 + $ ) .Squaringthe electriceldandexpandingabout $ 0 ,wedeterminethepowerincidentonthephotodetector, P AS : P AS = P BS sin 2 ( $ 0 + $ ) (25) $ P BS sin 2 ( $ 0 )+ 2 P BS $ 0 $ (26) 28

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Figure2-2.TheDCreadoutdarkfringe.TheASportisnotkeptatthedarkfringe,butisslightly offsetby $ 0 .Changesinphaseatthebeamsplitterarealinearfunctionofpower. Thersttermontherighthandsideoftheexpanded P AS istheDCpowerduetothestaticoffset fromthefringe.Thesecondtermontherighthandsidedescribeshowachangeinphaseatthe beamsplitterisconvertedtoachangeinpower: dP AS d $ BS = 2 P BS $ 0 (27) Thisrelationshipislinearandproportionaltothepoweratthebeamsplitter.Throughoutthis dissertation,whenwerefertosignalsfallingina linearregime ,wemeanthattheyaresmall enoughtobewellmodeledbyatangenttotheactualresponse,justasforthecasedescribedhere regardingsmallphasesignals. 2.2.2DARM Thedifferentialarmlength(commonlyknownasDARM)isofcentralinterest.Thisisthe lengthdegreeoffreedomaffectedbygravitationalwaves.Itisdenedas DARM: = L # : = L x # L y (28) where L x and L y arethelengthsofthe x -armand y -arm,respectively.Whenthereisnogravitationalwave, L # = 0,butinthepresenceofagravitationalwave,theDARMsignalis: L # = Lh (29) Weseethat L istheconversionfactorbetweenGWstrainandDARM. 29

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2.2.3DARMandStrainOpticalGain TheDARMopticalgaintellshowdifferentialdisplacementisconvertedtopowerattheAS portandhasunitsofWattspermeter.CombiningEqs. 24 27 ,and 29 ,theDARMopticalgain atDCoftheLIGOinterferometerwithDCreadoutis: dP AS dL # = 4 c P BS $ 0 % g $ (210) Likewise,thestrainopticalgainis: dP AS dh = 4 c P BS $ 0 L % g $ (211) 2.3SignalVersusNoise FromEq. 211 wenotefourfundamentalwaystoincreasethestrainopticalgainfora constantDARMoffsetandthereforeproducemorepowerattheASportforagivenGWstrain: Increasethearmlength. Increasethepoweratthebeamsplitter. IncreasethephasegainoftheFabry-Perotarms. Increasethelaserfrequency. Althoughaddressingeachofthesepointswillimprovethestrainsignal,ourabilitytodetect gravitationalwavestrainisalsodependentonthedetectornoiseswhichwillmaskaweakGW signal.Nomatterhowlargeasignalonemighthave,itwillnotbefoundcondently,oratall, ifthereistoomuchnoise.Forathoroughoverviewoftheinterferometernoiseanalysisand examplesofwaystoimprovestrainsensitivity,refertoRef.[ 31 ]. 2.3.1Noise Thesourcesofnoisewhichcontaminatethedetector'soutputcanbegroupedintotwo categories: displacementnoise sensingnoise Displacementnoisesarethosethatcreaterealmotionofthemirrors,whilesensingnoisesare thosethatariseintheprocessofmeasuringtheelectriceldatthedetector'soutput. 30

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Theprimarydisplacementnoisethatplaguesterrestriallaserinterferometersismotionof theground,i.e.seismicnoise.Thermalmotionofthemirrorsandtheirsuspensionsareanother sourceofdisplacementnoise[ 32 ].Theprimarysensingnoisesareelectronics(dark')noise duetothermalnoiseinresistorsandelectronicampliers,andshotnoisewhicharisesfromthe Poissonstatisticsofphotonarrivalatthephotodetector.Shotnoiseappearsasauctuatingpower withamplitudespectraldensity P SN = 2 Ph p (212) where P isthemeanpoweronthephotodiode, h p isPlanck'sconstant,and isthefrequencyof theincidentlight.Shotnoiseisspectrallywhite.Thedetectorelectronicsaretypicallydesigned sothatelectronicsnoiseisneverlimiting. 2.3.2NoiseFloor Thedetector'snoiseoorislimitedbyseismicnoisebelow40Hzandbyshotnoiseabove 200Hz.Ingeneralweendeavortopushthenoiseoordownasfaraspossiblesothatany underlyingGWsignalswillberevealed.Whetherlimitedbydisplacementnoiseorbysensing noise,thenoiseoorcanbeloweredbyincreasingthelengthofthearms,whichactsasthe conversionfromstraintodifferentialdisplacement(Eq. 29 ) 2 .Furtherimprovementsrequire consideringthenoisesourcesindividually. 2.3.2.1Displacementnoiseoor Atfrequencieswherethenoiseoorislimitedbydisplacementnoise,simplyincreasing theDARMopticalgain(Eq. 210 )willnothelp.Themirrordifferentialdisplacements,whether duetogravitationalwavesorduetogroundmotion,areconvertedintopowerattheASportin theexactsameway.Reductionofdisplacementnoisesreliesprimarilyonthedevelopmentof moresophisticatedseismicisolationsystemsandmirrorsuspensionarrangements.Notethat 2 InthecontextofLIGO,increasingthearmlengthsisnotactuallyanoption. 31

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improvementsinthedisplacementnoiseoorareonlylinearwithrespecttoarmlength,whereas eachadditionalstageofseismicisolationcanpotentiallyreducethedisplacementnoiseby1 / f 2 2.3.2.2Sensingnoiseoor ThenoiseoorduetosensingnoiseisimprovedbyincreasingtheDARMopticalgain.In particular,thecontributionduetoshotnoisemaybefoundbydividingtheshotnoiseamplitude spectraldensitybythestrainopticalgain: h shot = $ h p 2 P BS c 4 &$ 0 Lg $ (213) Hereweseethattheshotnoiselimit(calibratedineffectivestrain)dropswithincreasesin theopticalgain.Increasingthepowerintheinterferometerimprovestheshotnoiselimitbecause theopticalgainincreasesmorequickly( # P BS ,seeEq. 210 )thantheshotnoiseamplitude (whichgoeslike & P BS ,assumingtheDARMoffsetisheldconstant). 2.4ControllingtheInterferometer Theabilityoftheinterferometertooperateasdescribedaboverequiresthatthemany interferometercavitiesbeheldonresonance.Themotionofthemirrorsintheabsenceof controlismuchtoolargeontheorderof1m,afullwavelength!tomaintainresonance.The motionoftheinterferometermirrorsmustthereforebecontrolled.Afeedbackcontrolsystemis implementedtoholdthesystemsufcientlyneartheintendedoperatingpoint(forDARM,within 10 # 13 m)sothattheresponsetoresidualdeviationsremainslinear.Calibrationofthedetector musttakeintoaccounttheactionofthecontrolsystem[ 33 ]. 2.4.1RFSidebands Thevariouslengthandangulardegreesoffreedomaresensedthroughtheuseofradiofrequency(RF)sidebandsonthecarrierlight.Thestandardtechniqueforlockingopticalcavities isthePoundDreverHallmethodoflaserfrequencystabilizationasexplainedinRef.[ 34 ]. LIGOusesthreesetsofRFsidebandsat24.4MHz,33.3MHz,and61.1MHzforlockingthe Fabry-Perotarmcavities,inputmodecleaner,andpowerrecyclingcavity,respectively.An electro-opticmodulater(EOM)phasemodulatesthe1064nmcarrierlighttoproducesthese 32

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referenceeldsthatarecriticalforinterferometercontrol.Phasemodulationisexplainedin Appendix A.1 2.4.2DigitalControlinLIGO Althoughtheinterferometerisananaloginstrument,itisinterfacedthroughadigitalcontrol system. 3 Theanalogsensorsignalsaresentthroughanalog-to-digitalconverters(ADCs),are digitallyltered,andthensentthroughdigital-to-analogconverters(DACs)beforereturningto theinterferometer'sactuatorsascontrolsignals.Thedigitalcontrolsystemallowscomplexlters tobeimplementedandtunedfromacomfortablecontrol-roomenvironment. ThevariousLIGOsubsystemsoperateatdifferentsamplerates.Thelengthsensingand control(LSC)subsystem,whichmeasuresandcontrolsDARMinadditiontootherlength degreesoffreedom,operatesat16384samples/second,whiletheangularsensingandcontrol (ASC)system,whichmaintainsmirroralignment,operatesat2048samples/second.Inaddition totheall-importantDARMchannel,manyotherauxiliarydatastreamsarepermanentlyrecorded. 2.4.3MirrorSuspensionandActuation Theprimaryinterferometeropticsaresuspendedinvacuumsothattheyactlikefreemasses inthehorizontalplaneatthefrequenciesintheGWdetectionband,andsothattheyareisolated fromgroundmotion.Eachmirrorishungfromasinglewirethatloopsaroundthebottomofthe barrelofthemirrorasshowninFigure 2-3 .Stand-offsgluedjustabovethemirror'scenterof massonbothsidesofthebarrelmarkthenalpointofcontactofthewirewiththemirror,and bothendsofthewireareclampedtothetopofasuspensioncage. Eachmirrorisequippedwithfouropticalsensorandelectro-magnetic(OSEM)actuators forroughsensingandnecontrolofthemirrorpositionandorientation.Magnetsarrangedto formthefourcornersofasquarearegluedonthemirror'sbacksurfacewhichareenvelopedby 3 Thereareaselectfewcontrolsystemsthatremaincompletelyanalog,likethelaserintensity stabilizationservo(ISS).WhenthefrequenciesofinterestextendbeyondseveraltensofthousandsofHz,theuseofcomputersbecomesimpractical. 33

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Figure2-3.SimplesketchofaLIGOsuspension. theOSEMsolenoidcoil.Thecurrentsthrougheachcoilmaybedrivenindependently.Length controlofthecavities,forinstance,sendscurrentofthesamemagnitudethrougheachcoilon agivenmirrortoprovideapistonforceforchangingthemirror'sposition.OSEMsensingis accomplishedthroughsimpleshadowsensors. Toavoidthermalnoise,themirrorsuspensionsaredesignedtominimizedissipation. Dampingforthelargeopticsisthereforeachievedthroughelectronicservos.Motionoftheoptics correspondingtoachangeincavitylengthisdampedatalltimesbysimplevelocitydamping servosthatusetheOSEMsensorsandactuators.Angularmotionissensedandcontrolledatall timesviaopticalleverswhichprovidevelocitydampingbetween0.2Hzand2Hz. 4 TheOSEM signalsalsoprovideangularfeedback,butonlywhentheinterferometerisnotlocked. 2.5Summary ThemodiedMichelsoninterferometerprovidesarobustfoundationonwhichtobuilda gravitationalwavedetectorinwhichuctuatinggravitationalwavestrainsaretransducedinto measurableopticalpoweructuations.Fortheinterferometertooperateproperly,themirrors positionsandorientationsmustbecontrolled.Thenoiseooroftheinterferometermaybe 4 TheopenlooptransferfunctionoftheopticalleverservoisdescribedinAppendix C.1 34

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improvedintheshotnoisedominatedregimebyincreasingthelaserpowercirculatinginthe interferometer.Increasesinlaserpower,however,createseveralchallenges.Twosuchchallenges areaddressedinthisthesis:copingwiththehigherpowerintheinterferometer'sInputOptics (Chapter 3 )anddealingwithradiationpressureinducedangularinstabilitiescausedbyhigh powerintheinterferometer'sarmcavities(Chapters 4 5 ,and 6 ). 35

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CHAPTER3 INPUTOPTICSDESIGNANDCHARACTERIZATION 3.1FunctionoftheInputOptics TheInputOptics(IO) 1 isoneoftheprimarysubsystemsoftheLaserInterferometer Gravitational-waveObservatory(LIGO)interferometers.Itspurposeistodeliveranaligned, spatiallypure,mode-matchedbeamwithphase-modulationsidebandstothepower-recycled Fabry-PerotMichelsoninterferometer.TheIOalsopreventsreectedorbackscatteredlightfrom reachingthelaseranddistributesthecontrolsidebandsreectedfromtheinterferometer(designatedthe reectedport )tophotodiodesforsensingandcontrollingthelengthandalignmentof theinterferometer.Inaddition,theIOprovidesanintermediateleveloffrequencystabilization andmusthavehighoverallopticalefciency.Itmustperformthesefunctionswithoutlimitingthe strainsensitivityoftheLIGOinterferometer.Finally,itmustoperaterobustlyandcontinuously overyearsofoperation.TheconceptualdesignisfoundinRef.[ 35 ]. AsshowninFigure 3-1 ,theInputOpticssubsystemconsistsoffourprinciplecomponents locatedbetweenthepre-stabilizedlaserandthepowerrecyclingmirror: electro-opticmodulator(EOM) modecleanercavity(MC) Faradayisolator(FI) mode-matchingtelescope(MMT) EachelementisacommonbuildingblockofmanyopticalexperimentsandnotuniquetoLIGO. However,theirrolesspecictothesuccessfuloperationofinterferometryforgravitational-wave detectionareofinterestanddemandfurtherattention.Here,webrieyreviewthepurposeof eachoftheIOcomponents;furtherdetailsaboutthedesignrequirementsareinRef.[ 36 ]. 3.1.1Electro-opticModulator TheLengthSensingandControl(LSC)andAngularSensingandControl(ASC)subsystems requirephasemodulationofthelaserlightatRFfrequencies.Thismodulationisproduced 1 TheInputOpticswasoriginallycalledtheInput-OutputOptics(IOO). 36

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Figure3-1.BlockdiagramoftheInputOpticssubsystem.TheIOislocatedbetweenthe pre-stabilizedlaserandtherecyclingmirrorandconsistsoffourcomponents: electro-opticmodulator,modecleaner,Faradayisolator,andmode-matching telescope.Theelectro-opticmodulatoristheonlyIOcomponentoutsideofthe vacuumsystem.Diagramisnottoscale. byanEOM,generatingsidebandsofthelaserlightwhichactasreferencesagainstwhich interferometerlengthandanglechangesaremeasured[ 37 ].Thesidebandlightmustbeeither resonantonlyintherecyclingcavityornotresonantintheinterferometeratall.Thesidebands mustbeoffsetfromthecarrierbyintegermultiplesoftheMCfreespectralrangesothatneither MClengthuctuationsnorphasemodulationofthesidebands(duetophasenoiseoftheRF oscillator)areconvertedtoamplitudemodulation. 3.1.2ModeCleaner Stablyalignedcavities,limitednon-mode-matched(junk)light,andafrequencyand amplitudestabilizedlaserarekeyfeaturesofanyultrasensitivelaserinterferometer.Themode cleaner,attheheartoftheIO,playsamajorroletothiseffect. Athree-mirrortriangularringcavity,themodecleanersuppresseslaseroutputnotin thefundamentalTEM 00 mode,servingtwomajorpurposes.Itenablestherobustnessofthe ASCsincehigherordermodeswouldotherwisecontaminatetheangularsensingsignalsofthe interferometer.Also,allnon-TEM 00 lightonthelengthsensingphotodiodes,includingthose usedfortheGWreadout,contributesshotnoisebutnotsignalandthereforediminishesthesignal tonoiseratio.Themodecleaneristhuslargelyresponsibleforachievinganaligned,minimally shot-noise-limitedinterferometer. 37

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TheMCalsoplaysanactiveroleinlaserfrequencystabilization[ 37 ],whichisnecessaryfor ensuringthatthesignalattheanti-symmetricportisduetoarmlengthuctuationsratherthan laserfrequencyuctuations.Inaddition,theMCpassivelysuppressesbeamjitteratfrequencies above10Hz. 3.1.3FaradayIsolator Faradayisolatorsarefour-portopticaldeviceswhichutilizetheFaradayeffecttoallowfor non-reciprocalpolarizationswitchingoflaserbeams.Anybackscatterorreectedlightfrom theinterferometer(duetoimpedancemismatch,modemismatch,non-resonantsidebands,or signal)needstobedivertedtoprotectthelaserfrombackpropagatinglight,whichcanintroduce amplitudeandphasenoise.Thisdiversionofthereectedlightisalsonecessaryforextracting lengthandangularinformationabouttheinterferometer'scavities.TheFaradayisolatorfulls bothneeds. 3.1.4Mode-matchingTelescope Thelowest-ordermodecleanerandarmcavityspatialeigenmodesneedtobematchedfor maximalpowerbuildupintheinterferometer.Themode-matchingtelescopeisasetofthree suspendedconcavemirrorsbetweenthemodecleanerandinterferometerthatexpandthebeam fromaradiusof1.6mmatthemodecleanerwaisttoaradiusof37mmattherecyclingmirror asshowninFigure 3-2 .TheMMTshouldplayapassiverolebydeliveringproperlyshapedlight totheinterferometerwithoutintroducingbeamjitteroranysignicantaberrationthatcanreduce modecoupling. 3.2ThermalProblemsinInitialLIGO TheInitialLIGOinterferometerswereequippedwitha10Wlaser,yetoperatedwithonly 7Winputpowerduetopower-relatedproblemswithothersubsystems.TheEOMwaslocated inthe10Wbeamandtheothercomponentsexperiencedanywhereupto7Wpower.The7W operationallimitwasnotduetothefailureoftheInputOptics;however,manyaspectsoftheIO performancediddegradewithpower. 38

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0 5 10 15 20 25 30 35 40 45 50 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 distance from MC waist [m] beam radius [m] MMT1 MMT2 MMT3 RM Figure3-2.BeamprolethroughtheInputOptics.Thestartingpointisthemodecleanerwaist andthechangesintrajectoryareduetothemode-matchingtelescopemirrors. OneoftheprimaryproblemsoftheInitialLIGOInputOptics[ 38 ]wasthermaldeection ofthebackpropagatingbeamduetothermally-inducedrefractiveindexgradientsintheFaraday isolator.Asignicantbeamdriftbetweentheinterferometer'slockedandunlockedstatesled toclippingofthereectedbeamonthephotodiodesusedforlengthandalignmentcontrol. Ourmeasurementsdeterminedadeectionofapproximately100rad/WintheFI.Thiswas mitigatedatthetimebythedesignandimplementationofanactivebeamsteeringservoonthe beamcomingfromtheisolator. TherewerealsoknownlimitstothepowertheIOcouldsustain.Thermallensinginthe Faradayisolatoropticsbegantoaltersignicantlythebeammodeatpowersgreaterthan10W, leadingtoaseveralpercentreductioninmodematchingtotheinterferometer[ 39 ].Additionally, theabsorptiveFIelementswouldcreatethermalbirefringence,degradingtheopticalefciency andisolationratiowithpower[ 40 ].TheInitialLIGONewFocuselectro-opticmodulatorshad anoperationalpowerlimitofaround10W.Therewasahighriskofdamagetothecrystalsunder 39

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thestressofthe0.4mmradiusbeam.Also,anisotropicthermallensingwithfocallengthsas severeas3.3mat10WmadetheEOMsunsuitableformuchhigherpower.Finally,themode cleanermirrorsexhibitedhighabsorption(asmuchas24ppmpermirror),enoughthatthermal lensingoftheMCopticsatEnhancedLIGOpowerswouldinducehigherordermodalfrequency degeneracyandresultinapower-dependentmodemismatchintotheinterferometer[ 41 42 ].In fact,asinputpowerincreasedfrom1Wto7Wthemodematchingdecreasedfrom90%to83%. InadditiontothethermallimitationsoftheInitialLIGOIO,opticalefciencyindelivering lightfromthelaserintotheinterferometerwasnotoptimal.OfthelightenteringtheInputOptics chain,only60%remainedbythetimeitreachedthepowerrecyclingmirror.Moreover,because only90%atbestofthelightattherecyclingmirrorwascoupledintothearmcavitymode,room wasleftforimprovementintheimplementationoftheMMT. 3.3EnhancedLIGOInputOpticsDesign TheEnhancedLIGOIOdesignaddressedthethermaleffectsthatcompromisedthe performanceoftheInitialLIGOIO,andaccommodateduptofourtimesthepowerofInitial LIGO.Also,thedesignwasaprototypeforhandlingthe165WlaserplannedforAdvanced LIGO.BecausetheadversethermalpropertiesoftheInitialLIGOIO(beamdrift,birefringence, andlensing)areallattributableprimarilytoabsorptionoflaserlightbytheopticalelements, theprimarydesignconsiderationwasndingopticswithlowerabsorption[ 39 ].BoththeEOM andtheFIwerereplacedforEnhancedLIGO.OnlyminorchangesweremadetotheMCand MMT.AdetailedlayoutoftheEnhancedLIGOIOisshowninFigure 3-3 andphotographsarein Figure 3-4 3.3.1Electro-opticModulatorDesign Wereplacedthecommercially-madeNewFocus4003resonantphasemodulatorof InitialLIGOwithanin-houseEOMdesignandconstruction.Bothanewcrystalchoiceand architecturaldesignchangeallowforsuperiorperformance. TheEnhancedLIGOEOMdesignusesacrystalofrubidiumtitanylphosphate(RTP),which hasatmost1/10theabsorptioncoefcientat1064nmofthelithiumniobate(LiNbO 3 )crystal 40

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Figure3-3.EnhancedLIGOInputOpticsopticalandsensingconguration.TheHAM1(horizontalaccessmodule)vacuumchamber isfeaturedinthecenter,withlocationsofallmajoropticssuperimposed.HAM2isshownontheright,withits components.Thesetablesareseparatedby12m.Theprimarybeampath,beginningatthepre-stabilizedlaserandgoingto thepowerrecyclingmirror,isshowninredasasolidline,andauxiliarybeamsaredifferentcolorsanddotted.TheMMTs, MCs,andsteeringmirror(SM)aresuspended;allotheropticsarexedtotheseismicallyisolatedtable.Thelaserand sensinganddiagnosticphotodiodesareonin-airtables. 41

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Figure3-4.PhotographsoftheEnhancedLIGOHAM1InputOptics insitu withadrawingofthe beampathsuperimposed.PhotographscourtesyofKatherineDooley. 42

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Table3-1.ComparisonofselectedpropertiesoftheInitialandEnhancedLIGOEOMcrystals, LiNbO 3 andRTP,respectively.RTPwaspreferredforEnhancedLIGObecauseofits lowerabsorption,superiorthermalproperties,andsimilarelectro-opticproperties[ 39 ]. unitsLiNbO 3 RTP damagethresholdMW/cm 2 280 > 600 absorptioncoeff.at1064nmppm/cm < 5000 < 500 electro-opticcoeff.( n 3 z r 33 )pm/V306239 dn y / dT 10 # 6 /K5.42.79 dn z / dT 10 # 6 /K37.99.24 fromInitialLIGO.At200WtheRTPshouldproduceathermallensof200mandhigherorder modecontentoflessthan1%,comparedtothe3.3mlenstheLiNbO 3 producesat10W.The RTPhasaminimalriskofdamage,sinceithasbothtwicethedamagethresholdofLiNbO 3 andissubjectedtoabeamtwicethesizeofthatinInitialLIGO.RTPandLiNbO 3 havesimilar electro-opticcoefcients.Also,RTP's dn / dT anisotropyis50%smaller.Table 3-1 comparesthe propertiesofmostinterestofthetwocrystals. WeprocuredtheRTPcrystalsfromRaicolandpackagedthemintospecially-designed, custom-builtmodulators.Thecrystaldimensionsare4 4 40mmandtheirfacesarewedged by2 85 ( andanti-reection(AR)coated.Thewedgeservestoseparatethepolarizationsand preventsanetaloneffect,resultinginasuppressionofamplitudemodulation.Onlyonecrystal isusedintheEOMinordertoreducethenumberofsurfacereections.Threeseparatepairs ofelectrodes,eachwithitsownresonantLCcircuit,areplacedacrossthecrystalinseries, producingthethreerequiredsetsofRFsidebands:24.5MHz,33.3MHZand61.2MHz.A diagramisshowninFigure 3-5 .Reference[ 43 ]containsfurtherdetailsaboutthemodulator architecture. 3.3.2ModeCleanerDesign Themodecleanerisasuspended12.2mlongtriangularringcavitywithnesse F =1280 (refertoAppendix A.2 forameasurementofthenesse)andfreespectralrangeof12.243MHz. Thethreemirrorarchitecturewasselectedoverthestandardtwomirrorlinearltercavitybecauseitactsasapolarizationlterandbecauseiteliminatesdirectpathbackpropagationtothe laser[ 44 ].Apick-offofthereectedbeamisnaturallyfacilitatedforuseingeneratingcontrol 43

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A B Figure3-5.Electro-opticmodulatordesign.A)ThesingleRTPcrystalissandwichedbetween threesetsofelectrodesthatapplythreedifferentmodulationfrequencies.The wedgedendsofthecrystalseparatethepolarizationsofthelight.Thep-polarized lightisusedintheinterferometer.B)Aschematicforeachofthethreeimpedance matchingcircuitsoftheEOM.Forthethreesetsofelectrodes,eachofwhichcreates itsown C crystal ,acapacitorisplacedparalleltotheLCcircuitformedbythecrystal andahand-woundinductor.Thecircuitsprovide50 $ inputimpedanceonresonance andarehousedinaseparateboxfromthecrystal. signals.Apotentialdownsidetothethreemirrordesignistheintroductionofastigmatism,but thiseffectisnegligibleduetothesmallopeningangleoftheMC. TheMChasaround-triplengthof24.5m.Thebeamwaisthasaradiusof1.63mmand islocatedbetweenthetwo45 ( atmirrors,MC1andMC3.SeeFigure 3-3 ).Aconcavethird mirror,MC2,18.15minradiusofcurvature,formsthefarpointofthemodecleaner'sisosceles triangleshape.ThepowerstoredintheMCis408timestheamountcoupledin,equivalentto about2.7kWinInitialLIGOandatmost11kWforEnhancedLIGO.Thepeakirradiancesare 32kW/cm 2 and132kW/cm 2 forInitialLIGOandEnhancedLIGO,respectively. 44

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Themodecleanermirrorsare75mmindiameterand25mmthick.Thesubstratematerial isfusedsilicaandthemirrorcoatingismadeofalternatinglayersofsilicaandtantala.Inorder toreducetheabsorptionofheatinthesematerialsandthereforeimprovethetransmissionand modalqualityofthebeaminthemodecleaner,weremovedparticulatebydragwipingthe surfaceoftheMCmirrorswithmethanolandopticaltissues.Themodecleanerwasotherwise identicaltothatinInitialLIGO. 3.3.3FaradayIsolatorDesign TheEnhancedLIGOFaradayisolatordesignrequirednotonlytheuseoflowabsorption optics,butadditionaldesignchoicestomitigateanyresidualthermallensingandbirefringence. Inaddition,trade-offsbetweenopticalefciencyintheforwarddirection,opticalisolationinthe backwardsdirection,andfeasibilityofphysicalaccessofthereturnbeamforsignalusewere considered.TheresultisthattheEnhancedLIGOFaradayisolatorneededacompletelynew architectureandnewopticscomparedtoboththeInitialLIGOFIandcommerciallyavailable isolators. Figure 3-6 showsaschematicoftheEnhancedLIGOFaradayIsolator.Itbeginsandends withlowabsorptioncalcitewedgepolarizers(CWP).BetweentheCWPsisathinlmpolarizer (TFP),adeuteratedpotassiumdihydrogenphosphate(DKDP)element,ahalf-waveplate(HWP), andaFaradayrotator.Therotatorismadeoftwolowabsorptionterbiumgalliumgarnet(TGG) crystalssandwichingaquartzrotator(QR)insidea7-discmagnetwithamaximumeldstrength of1.16T.TheforwardpropagatingbeamuponpassingthroughtheTGG,QR,TGG,andHWP elementsisrotatedby + 22 5 ( # 67 5 ( + 22 5 ( + 22 5 ( = 0 ( .Inthereversedirection,therotation throughHWP,TGG,QR,TGGis # 22 5 ( + 22 5 ( + 67 5 ( + 22 5 ( = 90 ( .TheTGGcrystalsare non-reciprocaldeviceswhiletheQRandHWParereciprocal. 3.3.3.1Thermalbirefringence ThermalbirefringenceisaddressedintheFaradayrotatorbytheuseofthetwoTGGcrystals andonequartzrotatorratherthanthetypicalsingleTGG[ 45 ].Inthisconguration,anythermal polarizationdistortionsthatthebeamexperienceswhilepassingthroughtherstTGGrotator 45

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Figure3-6.Faradayisolatorphotographandschematic.TheFaradayisolatorpreservesthe polarizationofthelightintheforward-goingdirectionandrotatesitby90degreesin thereversedirection.LightfromtheMCentersfromtheleftandexitsattheright towardstheinterferometer.Itisideallyp-polarized,butanys-polarization contaminationispromptlydiverted 10mradbytheCWPandthenreectedbythe TFPanddumped.Thep-polarizedreectedbeamfromtheinterferometerentersfrom therightandisrotatedtos-polarizedlightwhichispicked-offbytheTFPandsentto theInterferometerSensingandControl(ISC)table.AnyimperfectionsintheFaraday rotationoftheinterferometerreturnbeamresultsinp-polarizedlighttraveling backwardsalongtheoriginalinputpath.PhotographcourtesyofKatherineDooley. 46

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willbemostlyundoneuponpassingthroughthesecond.Themultipleelementsinthemagnet requiredalargermagneticeldthaninInitialLIGO.The7-discmagnetis130mmindiameter and132mmlongandplacedinhousing155mmindiameterand161mmlong.TheTGG diameteris20mm. 3.3.3.2Thermallensing ThermallensingintheFaradayisolatorisaddressedbyincludingDKDP,anegative dn / dT material,inthebeampath.AbsorptionoflightintheDKDPresultsinade-focusingofthe beam,whichpartiallycompensatesforthethermalfocusinginducedbyabsorptionintheTGGs [ 46 47 ].Theopticalpathlength(thickness)oftheDKDPischosentoslightlyover-compensate thepositivethermallensinducedintheTGGcrystals,anticipatingotherpositivethermallenses inthesystem. 3.3.3.3Polarizers Thepolarizersused(twoCWPsandoneTFP)eachofferadvantagesanddisadvantages relatedtoopticalefciencyintheforward-propagatingdirection,opticalisolationinthereected direction,andthermalbeamdrift.TheCWPshaveveryhighextinctionratios( > 10 5 )and hightransmission( > 99%)contributingtogoodopticalefciencyandisolationperformance. However,theangleseparatingtheexitingorthogonalpolarizationsoflightisverysmall,onthe orderof10mrad.Thissmallanglerequiresthelighttotravelrelativelylargedistancesbefore wecanpickoffthebeamsneededforinterferometersensingandcontrol.Inaddition,thermally inducedindexofrefractiongradientsduetothe4.95 ( wedgeangleoftheCWPsresultinthermal drift.However,theCWPsfortheEnhancedLIGOFaradayhaveameasuredlowabsorptionof 0.0013cm # 1 withanexpectedthermallensof60mat30Wanddriftoflessthan1.3 rad/W [ 39 ]. Theadvantagesofthethinlmpolarizeroverthecalcitewedgepolarizerarethatitexhibits negligiblethermaldriftwhencomparedwithCWPsanditoperatesattheBrewsterangleof 55 ( ,thusdivertingthereturnbeaminaneasilyaccessibleway.However,theTFPhasalower transmissionthantheCWP,about96%,andanextinctionratioofonly10 3 47

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Figure3-7.PhotographofTGGcrystalwithindiumfoilwrapping.Photographcourtesyof KatherineDooley. Thus,thecombinationofCWPsandaTFPcombinesthebestofeachtoprovideahigh extinctionratio(fromtheCWPs)andeaseofreectedbeamextraction(fromtheTFP).The downsidesthatremainwhenusingbothpolarizersarethatthereisstillsomethermaldriftfrom theCWPs.AlsothetransmissionisreducedduetotheTFPandtothefactthatthereare16 surfacesfromwhichlightcanscatter. 3.3.3.4Heatconduction Faradayisolatorsoperatinginavacuumenvironmentsufferfromincreasedheatingwith respecttothoseoperatinginair.Convectivecoolingatthefacesoftheopticalcomponentsisno longeraneffectiveheatremovalchannel,soproperheatsinkingisessentialtominimizethermal lensinganddepolarization.IthasbeenshownthatFaradayisolatorscarefullyalignedinaircan experienceadramaticreductioninisolationratio( > 10-15dB)whenplacedinvacuum[ 48 ]. Thedominantcauseisthecouplingofthephotoelasticeffecttothetemperaturegradientinduced bylaserbeamabsorption.AlsoofimportanceisthetemperaturedependenceoftheVerdet constantdifferentspatialpartsofthebeamexperiencedifferentlinearpolarizationrotationsin thepresenceofatemperaturegradient[ 49 ]. ToimproveheatconductionawayfromtheFaradayrotatoropticalcomponents,wedesigned housingfortheTGGandquartzcrystalsthatprovidedimprovedheatsinkingtotheFaraday rotator.WealsowrappedtheTGGswithindiumfoilaspicturedinFigure 3-7 toimprovecontact 48

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withthehousing,andwecushionedtheDKDPandtheHWPwithindiumwireintheiraluminum holders.Thishastheadditionaleffectofavoidingthedevelopmentofthermalstressesinthe crystals,anespeciallyimportantconsiderationfortheveryfragileDKDP. 3.3.4Mode-matchingTelescopeDesign Themodematchingintotheinterferometer(atLivingston)wasmeasuredtobeatbest90% inInitialLIGO.BecauseofthestringentrequirementsplacedontheLIGOvacuumsystemto reducephasenoisethroughscatteringbyresidualgas,standardopto-mechanicaltranslatorsare notpermittedinthevacuum;itisthereforenotpossibletophysicallymovethemodematching telescopemirrorswhileoperatingtheinterferometer.Throughacombinationofneedingtomove theMMTsinordertotthenewFaradayisolatoronthein-vacuumopticstableandadditional measurementsandmodelstodeterminehowtoimprovethecoupling,anewsetofMMT positionswaschosenforEnhancedLIGO.Fundamentaldesignconsiderationsarediscussedin Ref.[ 50 ]. 3.4PerformanceoftheEnhancedLIGOInputOptics ThemostconvincinggureofmeritfortheInputOpticsperformanceisthattheEnhanced LIGOinterferometersachievedlow-noiseoperationwith20Winputpowerwithoutthermal issuesfromtheIO.Additionally,theInputOpticswereoperatedsuccessfullyuptotheavailable 30Wofpower.(InstabilitieswithotherinterferometersubsystemslimitedtheEnhancedLIGO sciencerunoperationto20W.)WepresentinthissectiondetailedmeasurementsoftheInput OpticsperformanceduringEnhancedLIGO.Specicmeasurementsandresultspresentedin guresandthetextcomefromLivingston;performanceatHanfordwassimilarandisincludedin tablessummarizingtheresults. 3.4.1OpticalEfciency TheopticalefciencyoftheEnhancedLIGOInputOpticsfromEOMtorecyclingmirror was75%,amarkedimprovementovertheapproximate60%thatwasmeasuredforInitialLIGO. Asubstantialpartoftheimprovementcamefromthediscoveryandsubsequentcorrectionofa 6.5%lossatthesecondofthein-vacuumsteeringmirrorsdirectinglightintotheMC(referto 49

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Table3-2.EnhancedLIGOInputOpticspowerbudget.Errorsare 1%,exceptfortheTFPloss whoseerroris 0 1%.Thecompositemodecleanertransmissionisthepercentageof poweraftertheMCtobeforetheMCandistheproductoftheMCvisibilityand transmission.InitialLIGOvalues,whereknown,areincludedinparenthesesandhave errorsofseveralpercent. LivingstonHanford Modecleanervisibility92%97% Modecleanertransmission88%90% CompositeMCtransmission81%(72%)87% Faradaytransmission93%(86%)94%(86%) -Thinlmpolarizerloss4.0%2.7% IOefciency(PSLtoRM)75%(60%)82% Figure 3-3 ).A45 ( reectingmirrorhadbeenusedforabeamwithan8 ( angleofincidence. LossesattributabletothemodecleanerandFaradayisolatoraredescribedinthefollowing sections.AsummaryoftheIOpowerbudgetisfoundinTable 3-2 3.4.1.1Modecleanerlosses ThemodecleanerwasthegreatestsinglesourceofpowerlossinbothInitialandEnhanced LIGO.Themodecleanervisibility,denedhereas visibility = P in # P reected P in (31) theratiooftheamountoflightcoupledintotheMCtotheamountimpingingtheMCinput mirror,was92%.Visibilityreductionistheresultofhigherordermodecontentof P in andmode mismatchintotheMC.Thevisibilitywasconstantwithin0.04%upto30Winputpowerat bothsites,providingapositiveindicationthatthermalaberrationsintheMCandupstreamwere negligible. OfthelightcoupledintotheMC,88%wastransmitted,correspondingtoanaverage lossof98ppmpermirror.Thescatterlossisexpectedtobe22ppm/mirrorbasedonthe mirrors'measuredrootmeansquaresurfacemicroroughnessof rms < 0 4nm[ 51 ].Partof thediscrepancybetweenexpectationandmeasurementwasdeterminedtocomefrompooror damagedARcoatings.Wemeasureda1.3%reectionfromtheARcoatingsonMCmirrorsat 50

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0 5 10 15 20 25 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 Mode Freq Shift (Hz) Time (h) Raw Data 0 5 10 15 20 25 0.5 0 0.5 BSC1&3 dT (F) 0 5 10 15 20 25 0 2 4 6 P MC (W) Time (h) f=28164 Hz (MC1) f=28209 Hz (MC2) f=28237 Hz (MC3) Figure3-8.Datafromthemodecleanerabsorptionmeasurement.PowerintotheMCwascycled between0.9Wand5.1Wat3hourintervals(bottomframe)andthechangein frequencyofthedrumheadmodeofeachmirrorwasrecorded(topframe).The ambienttemperature(middleframe)wasalsorecordedinordertocorrectforits effects. bothLivingstonandHanford,atransmittedpowerlossequivalentto10ppmofintracavityloss permirror. AnothersourceofMClossesisviaabsorptionofheatbyparticulatesresidingonthe mirror'ssurface.Wemeasuredtheabsorptionwithatechniquethatmakesuseofthefrequency shiftofthethermallydrivendrumheadeigenfrequenciesofthemirrorsubstrate[ 52 ].The frequencyshiftdirectlycorrelateswiththeMCabsorptionviathesubstrate'schangeinYoung's moduluswithtemperature, dY / dT .Aniteelementmodel(COMSOL)wasusedtocomputethe expectedfrequencyshiftfromatemperaturechangeofthesubstrateresultingfromthemirror 51

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Table3-3.AbsorptionvaluesfortheLivingstonandHanfordmodecleanermirrorsbefore(in parentheses)andafterdragwiping.Theprecisionis 10%. mirrorLivingstonHanford MC12.1ppm(18.7ppm)5.8(6.1ppm) MC22.0ppm(5.5ppm)7.6(23.9ppm) MC33.4ppm(12.8ppm)15.6(12.5ppm) coatingabsorption.Themeasuredeigenfrequenciesforeachmirroratroomtemperatureare 28164Hz,28209Hz,and28237Hz,respectively. Wecycledthepowerintothemodecleanerbetween0.9Wand5.1Wat3hourintervals, allowingenoughtimeforathermalcharacteristictimeconstanttobereached.Atthesametime, werecordedthefrequenciesofthehighQdrumheadmodepeaksasfoundinthemodecleaner frequencyerrorsignal,heterodyneddownby28kHz.Figure 3-8 showsthemeasurementdata. Correctingforambienttemperatureuctuations,wendafrequencyshiftof0.043,0.043,and 0.072Hz/W.Asaresultofdrag-wipingthemirrors,theabsorptiondecreasedforallbutone mirror,asshownforbothHanfordandLivingstoninTable 3-3 3.4.1.2Faradayisolatorlosses TheFaradayisolatorwasthesecondgreatestsourceofpowerlosswithitstransmissionof 93%.Thiswasanimprovementoverthe86%transmissionoftheInitialLIGOFI.Themostlossy elementintheFaradayisolatorwasthethinlmpolarizer,accountingfor4%oftotallosses. TheintegratedlossesfromARcoatingsandabsorptionintheTGGs,CWPs,HWP,andDKDP accountfortheremaining3%ofmissingpower. 3.4.2FaradayIsolationRatio TheisolationratioisdenedastheratioofpowerincidentontheFaradayinthereverse direction(thelightreectedfromtheinterferometer)tothepowertransmittedinthereverse directionandisoftenquotedindecibels:isolationratio=10log 10 ( P in # reverse / P out # reverse ) .We measuredtheisolationratiooftheFaradayisolatorasafunctionofinputpowerbothinairprior toinstallationand insitu duringEnhancedLIGOoperation. 52

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0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 power in Faraday [W] Faraday isolation ratio [dB] in air data in vacuum data Figure3-9.Faradayisolatorisolationratioasmeasuredinairpriortoinstallationand insitu in vacuum.Theisolationworsensbyafactorof6uponplacementoftheFaradayin vacuum.Thelineartstothedatashowaconstantin-airisolationratioandan in-vacuumisolationratiodegradationof0.02dB/W. Tomeasurethein-vacuumisolationratio,wemisalignedtheinterferometerarmssothat theinputbeamwouldbepromptlyreectedoffofthe97%reectiverecyclingmirror.Thisalso hastheconsequencethattheFaradayisolatorissubjectedtotwicetheinputpower.Ourisolation monitorwasapick-offofthebackwardstransmittedbeamtakenimmediatelyaftertransmission throughtheFaradaythatwesentoutofavacuumchamberviewport.Refertothe"isolation checkbeam"inFigure 3-3 .Theinairmeasurementwasdonesimilarly,exceptinanopticslab withareectingmirrorplaceddirectlyaftertheFaraday. Figure 3-9 showsourisolationratiodata.Mostnotably,weobserveanisolationdecreaseof afactorofsixuponplacingtheFaradayisolatorinvacuum,aresultconsistentwiththatreported byRef.[ 48 ].Inairtheisolationratioisaconstant34.46 0.04dBfromlowpowerupto47W, andinvacuumtheisolationratiois26.5dBatlowpower.Theunderlyingcauseistheabsence ofcoolingbyairconvection.IfweattributethelosstotheTGGs,thenbasedonthechangein TGGpolarizationrotationanglenecessarytoproducethemeasuredisolationdropof8dBand thetemperaturedependenceoftheTGG'sVerdetconstant,wecanputanupperlimitof11K 53

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onthecrystaltemperaturerisefromairtovacuum.Furthermore,adegradationof0.02dB/Wis measuredinvacuum. 3.4.3ThermalSteering Wemeasuredthe insitu thermalangulardriftofboththebeamtransmittedthroughthemode cleanerandofthereectedbeamfromtheFaradayisolatorwithupto25Winputpower.Justas fortheisolationratiomeasurement,wemisalignedtheinterferometerarmssothattheinputbeam wouldbepromptlyreectedoffoftherecyclingmirror.TheFaradayrotatorwasthussubjected toupto50WtotalandtheMCto25W. Pitchandyawmotionofthemodecleanertransmittedandinterferometerreectedbeams wererecordedusingthequadrantphotodiode(QPD)ontheInputOpticstableandtheRF alignmentdetectorsontheInterferometerSensingandControltable,asseeninFigure 3-3 TherearenolensesbetweentheMCwaistanditsmeasurementQPD,soonlythepathlength betweenthetwowereneededtocalibrateinradiansthepitchandyawsignalsontheQPD.The interferometerreectedbeam,however,passesthroughseverallenses.Thus,raytransfermatrices andthetwoalignmentdetectorswerenecessarytoextracttheFaradaydriftcalibration.Detailsof thecalibrationmethodarepresentedinAppendix A.4 Figure 3-10 showsthecalibratedbeamsteeringdata.Theangleofthebeamoutofthe modecleanerdoesnotchangemeasurablyasafunctionofinputpowerinyaw(4.7nrad/W)and changesbyonly440nrad/Winpitch.FortheFaradayisolator,werecordabeamdriftoriginating atthecenteroftheFaradayrotatorof1.8rad/Winyawand3.2rad/Winpitch.Therefore, whenrampingtheinputpowerupto30Wduringafullinterferometerlock,theupperlimiton thedriftexperiencedbythereectedbeamisabout100rad.Thisisathirty-foldreductionwith respecttotheInitialLIGOFaradayisolatorandrepresentsafthofthebeam'sdivergenceangle, ( div =490rad. 3.4.4ThermalLensing Wemeasuredtheprolesofboththebeamtransmittedthroughthemodecleanerandthe reectedbeampickedoffbytheFaradayisolatoratlow( 1W)andhigh( 25W)input 54

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5000 5500 6000 6500 7000 7500 8000 300 200 100 0 100 200 300 time [sec] angle [urad] 5000 5500 6000 6500 7000 7500 8000 0 5 10 15 20 25 30 input power [W] FI pitch FI yaw MC pitch (x10) MC yaw (x10) input power A 0 10 20 30 40 50 60 50 0 50 100 150 200 power in MC and FI [W] angle [urad] FI pitch FI yaw MC pitch MC yaw B Figure3-10.ModecleanerandFaradayisolatorthermaldriftdata.A)Angularmotionofthe beamattheMCwaistandFIrotatorastheinputpowerisstepped.Thebeamis double-passedthroughtheFaradayisolator,soitexperiencestwicetheinputpower. B)AveragebeamangleperpowerlevelintheMCandFI.Lineartstothedataare alsoshown.TheslopesforMCyaw,MCpitch,FIyaw,andFIpitch,respectively, are0.0047,0.44,1.8,and3.2rad/W. 55

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5.6 5.65 5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 0 200 400 600 800 1000 beam radius (um) distance from MC waist (m) 0.2 W data w 0 = 105 um at 5.811 m, M 2 = 1.02 26.3 W data w 0 = 103 um at 5.811 m, M 2 = 0.96 Figure3-11.Proleathighandlowpowersofapick-offofthebeamtransmittedthroughthe modecleaner.Theprecisionofthebeamproleris 5%.Withintheerrorofthe measurement,therearenoobviousdegradations. powerstoassessthedegreeofthermallensinginducedintheMCandFI.Again,wemisaligned theinterferometerarmssothattheinputbeamwouldbepromptlyreectedofftherecycling mirror.WepickedoffafractionofthereectedbeamontheInterferometerSensingandControl tableandofthemodecleanertransmittedbeamontheInputOpticstable(refertoFigure 3-3 ), placedlensesineachoftheirpaths,andmeasuredthebeamdiametersatseverallocationson eithersideofthewaistscreatedbythelenses.Achangeinthebeamwaistsizeorpositionasa functionoflaserpowerindicatesthepresenceofathermallens. AsseeninFigure 3-11 andFigure 3-12 ,thewaistsofthetwosetsofdataarecollocated: nothermallensismeasured.FortheFaradayisolator,thedivergenceofthelowandhighpower beamsdiffers,indicatingthatthebeamqualitydegradeswithpower.The M 2 factorat1Wis 1.04indicatingthebeamisnearlyperfectlyaTEM 00 mode.At25W, M 2 increasesto1.19, correspondingtoincreasedhigher-order-modecontent.Thepercentageofpowerinhigher-order 56

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94.9 94.95 95 95.05 95.1 95.15 95.2 95.25 95.3 0 100 200 300 400 500 600 700 800 beam radius (um) distance from MC waist (m) 1W data w 0 = 139 um at 95.029 m, M 2 = 1.04 25W data w 0 = 137 um at 95.027 m, M 2 = 1.19 Figure3-12.Faradayisolatorthermallensingdata.With25WintotheFaradayisolator (correspondingto50Windoublepass),thebeamhasasteeperdivergencethana pureTEM 00 beam,indicatingthepresenceofhigherordermodes.Errorsare 5 0% foreachdatapoint. modesdependsstronglyonthemodeorderandrelativephasesofthemodes,andthuscannotbe determinedfromthismeasurement[ 53 ]. Theresultsforthemodecleanerdataareconsistentwithnothermallensing.Thehighand lowpowerbeamprolesarewithineachother'serrorbarsandwellbelowourrequirements. 3.4.5Mode-matching Wemeasuredtheeffectivenessofthemode-matchingtelescopebytakingtheratioofpower atthereectedportwhenalloftheinterferometercavitiesareonresonancetothepowerinthe reectedbeamwhenthecavitiesareunlocked.Sincetheimpedancematchingisnearperfect, alllightatthereectedportduringinterferometerlockisattributabletoamodemismatch. Initially,anywherebetween10%and17%ofthelightwasrejectedbythecavityduetopoor, power-dependentmodematching.Aftertranslatingthemode-matchingtelescopemirrorsduring 57

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avacuumchamberincursionandupgradingtheotherIOcomponents,theratiowemeasured was8%independentofinputpower.TheMMTsucceedsatcoupling92%ofthelightintothe interferometeratalltimes,markingbothanimprovementinMMTmirrorplacementandsuccess ineliminatingmeasurablethermalissues.Appendix A.6 presentsdetailsofthemode-matching measurement. 3.5ImplicationsforAdvancedLIGO AswithotherAdvancedLIGOinterferometercomponents,EnhancedLIGOservedasa technologydemonstratorfortheAdvancedLIGOInputOptics,albeitatlowerlaserpowersthan willbeusedthere.TheperformanceoftheEnhancedLIGOInputOpticscomponents,at20W ofinputpowerallowsustoinfertheirperformanceinAdvancedLIGO.Therequirementsforthe AdvancedLIGOInputOpticsdemandareforsimilarperformancetoEnhancedLIGO,butwith almost8timesthelaserpower. TheEnhancedLIGOelectro-opticmodulatorshowednothermallensing,degradedtransmission,nordamageinover1millionhoursofsustainedoperationat30Woflaserpower. MeasurementsofthethermallensinginRTPatpowersupto160Wshowarelativepowerlossof < 0 4%,indicatingthatthermallensingshouldbenegligibleinAdvancedLIGO.Peakirradiances intheEOMwillbeapproximatelyfourtimesthatofEnhancedLIGO(a45%largerbeamdiameterwillsomewhatoffsettheincreasedpower).TestingofRTPat10timestheexpectedAdvanced LIGOirradianceover100hoursshownosignsofdamageordegradedtransmission. Themodecleanershowednomeasurablechangeinoperationalstateasafunctionofinput power.ThisbodeswellfortheAdvancedLIGOmodecleaner.ComparedwiththeEnhanced LIGOmodecleaner,theAdvancedLIGOmodecleanerisdesignedwithalowernesse(520) thanInitialLIGO(1280).For150Winputpower,theAdvancedLIGOmodecleanerwill operatewith3timesgreaterstoredpowerthanInitialLIGO.Thecorrespondingpeakirradiance is400kW/m 2 ,wellbelowthecontinuouswavecoatingdamagethreshold.Absorptioninthe AdvancedLIGOmodecleanermirroropticalcoatingshasbeenmeasuredat0.5ppm,roughly fourtimeslessthanthebestmirrorcoatingabsorptioninEnhancedLIGO,sotheexpected 58

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thermalloadingduetocoatingabsorptionshouldbereducedinAdvancedLIGO.Thelarger AdvancedLIGOmodecleanermirrorsubstratesandhigherinputpowersresultinasignicantly highercontributiontobulkabsorption,roughly20timesEnhancedLIGO,howevertheexpected thermallensingleadstosmallchange( < 0 5%)intheoutputmode[ 42 ]. TheEnhancedLIGOdataobtainedfromtheFIallowsustomakeseveralpredictionsabout howitwillperforminAdvancedLIGO.Themeasuredisolationratiodecreaseof0.02dB/Wwill resultinalossof3dBfora150WpowerlevelexpectedforAdvancedLIGOrelativetoitscold state.However,theAdvancedLIGOFIwillemployan insitu adjustablehalfwaveplatewhich willallowforapartialrestorationoftheisolationratio.Inaddition,anewFIschemetobetter compensateforthermaldepolarizationandthusyieldhigherisolationratiosmaybeimplemented [ 54 ].Themaximumthermallyinducedangularsteeringexpectedis480rad(usingadriftrate of3.2rad/W),approximatelyequaltothebeamdivergenceangle.Thishassomeimplications fortheAdvancedLIGOlengthandalignmentsensingandcontrolsystem,asthereectedFI beamisusedasasensingbeam.OperationofAdvancedLIGOathighpowerswilllikelyrequire theuseofabeamstabilizationservotolockthepositionofthereectedbeamonthesensing photodiodes.Althoughnomeasurablethermallensingwasobserved(nochangeinthebeam waistsizeorposition),themeasuredpresenceofhigherordermodesintheFIathighpowersis suggestiveofimperfectthermallenscompensationbytheDKDP.Thisfaultpotentiallycanbe reducedbyacarefulselectionofthethicknessoftheDKDPtobettermatchtheabsorbedpower intheTGGcrystals. 3.6Summary Insummary,wehavepresentedacomprehensiveinvestigationoftheEnhancedLIGOInput Optics,includingthefunction,design,andperformanceoftheIO.Severalimprovementsto thedesignandimplementationoftheEnhancedLIGOIOovertheInitialLIGOIOhaveleadto improvedopticalefciencyandcouplingtothemaininterferometerthroughasubstantialreductioninthermo-opticaleffectsinthemajorIOopticalcomponents,includingtheelectro-optic modulators,modecleaner,andFaradayisolator.TheIOperformanceinEnhancedLIGOenables 59

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ustoinferitsperformanceinAdvancedLIGO,andindicatesthathighpowerinterferometrywill bepossiblewithoutseverethermaleffects. 60

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CHAPTER4 ANGULARMOTIONOFTHEINTERFEROMETERMIRRORS Forlighttoresonateintheinterferometer,themirrorsneedtopointatoneanotherand remainstationarywithrespecttothispointing.Itisnecessarytoactivelyalignthemirrorsfor severalreasons: tondtheoptimalDCalignmentoftheinterferometer tosuppressanymotionthatresultsfromexternaldisturbances tocounteractahighpowerstaticinstability Thereare16angulardegreesoffreedom(DOFs)thatneedtobesensedandcontrolled,making theASConeofthemostcomplexinterferometersubsystems.ThischapterpresentstheASC design,anddiscussesthecausesofmirrorangulardisplacementandtheeffectsofresidualmirror motionontheinterferometer. 4.1ToleranceforAngularMotion Inamostgeneralsense,itisthecontrolledangularmotionofthemirrorsthatmatters,which isincontrastwithDARMforwhichboththecontrolledanduncontrolledlengthdisplacements matter.Theresidualangularmotionsmustbesmallenoughtobothkeeptheinteferometerlocked andtonotimpactstrainsensitivity.Therequirementsforhowmuchmotionistolerablestem fromtwoeffectsofmisalignmentthatdirectlycoupletostrainsensitivity:failuretoachieve maximumpowergain,andangletolengthcoupling.Themisalignmenttolerancesaredictated bywhatisnecessarytopreventthestrainsensitivityoftheperfectlyalignedinterferometerfrom degradingbymorethan0.5%inthedetectionbandof40to7000Hz[ 55 ]. Becausethestrainsensitivityisproportionaltothepoweratthebeamsplitter(referto Eq. 213 ),adecreaseincirculatingpowerdirectlyresultsinadecreaseofshot-noise-limited DARM.Furthermore,differingpoweructuationsinthetwoarmcavitiesresultsinachanging contrastdefect,adifferenceintheamountoflightreturningfromonearmcomparedtotheother. AchangingcontrastdefectcreatespoweructuationsattheASport,makingitindistinguishable fromgravitationalwaves.Also,theDCpowerofthecontrastdefectcontributestoincreasing theshotnoisenoise-oor.PoweructuationscouplequadraticallytoDARM,andinthepower 61

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recyclingcavityinparticular,poweructuationsmakeforinconsistentsignaltonoiseratiosfor thesignalsthatdependonsidebandpower.Tomaintainapowerbuildupwithin1%ofmaximum, thecoreopticsmusthaveanangulardisplacementoflessthan10 # 8 radrmswithrespecttothe cavityaxis[ 56 ].Thederivationofpowerbuildupasafunctionofmirrorangledisplacementis foundinAppendix C.3 Anotherdeleteriouseffectofpooralignmentisangletolengthcoupling.Whentheaxisof rotationofamirrorcoincideswiththecenterofthebeam,anytiltofthemirroraboutthisaxis doesnotaffectthepathlengthofthereectedbeam.However,ifthereisamismatchbetween rotationaxisandbeamlocation,thenthelightwillpickupalongitudinalphaseshiftwhenthe mirroristilted.Duringafullinterferometerlock,thisisrecordedbyDARM.Explicitly,whenthe beamislocatedadistance x awayfromthecenterofthemirror,anangulardisplacementofthe mirror ( aboutitscenterresultsinapathlengthchangeofthebeam L = x ( (41) Therefore,thealignmentspecicationsmustincludenotonlytolerablelevelsofangularmotion, butrequirementsforthephysicalcenteringofthebeamspotsonthemirrors.AsdetailedinRef. [ 56 ],thebeamsmustbecenteredonthecoreopticswithin1mm.AtDC,forx=1mmand ( = 10 # 7 rad, L = 10 # 10 mwhichisfourordersofmagnitudebelowtheDARMrmsof10 # 6 m. 4.2SourcesofAngularMirrorMotion Thereisacontinuousstreamofchangingtorqueinputstothemirrorsthatcausethem totwistandturninpitchandinyaw.Sometorqueinputsexistregardlessofthestateofthe interferometer,whileothersareadirectconsequenceofthecontrolsystems.Theprimarytorque inputsareintroducedhere,andfurtherdiscussionofsomeofthemisfoundlaterinthechapter. Thelistincludes: groundmotion coilactuators(lengthtoangle) noiseimpressionfromtheangularcontrolsystem radiationpressure 62

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10 2 10 1 10 0 10 1 10 2 10 11 10 10 10 9 10 8 10 7 10 6 pitch [rad/rtHz] frequency [Hz] ITMX ITMY ETMX ETMY RM BS MMT3 Figure4-1.Typicalangularmotionofthecoresuspendedmirrorsintheabsenceof interferometriccontrol.VelocitydampingprovidedbytheOSEMsandtheoptical leversispresent.Oncetheinterferometerislocked,theOSEMdampingisturnedoff. GroundmotionatthetimeofthismeasurementisshowninFigure C-4 4.2.1GroundMotion Themostegregiousofthesetorqueinputsisgroundmotionthatmakesitswaythrough themultiplestagesofseismicisolationtothemirrorsuspensionsandthencetothemirrors. Groundmotionistheonlysourceofangularmotionthatispresentregardlessofthestateof interferometeroperation.Anexampleoftheshapeandamountofangularmotionexperienced bythecoreopticsduetoseismicnoiseduringarelativelyquietseismictimeisshowninFigure 4-1 1 .Thermsmirrormotionisoftheorder10 # 7 rad.Thisisthemotionthatneedstobe controlledinterferometrically. 1 Thesespectraincludetheeffectoflocalvelocitydampingbecausetheopticalleversare alwayson.SeeSection 2.4.3 63

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Iobtainedthesespectrausingtheopticalleversaswitnessesofthemirrormotion,whichin turnisexpectedtocomefromtheground'smotions.Theopticalleversignal,may,however,be contaminatedbyopticalleversensing(electronics)noiseoracousticcouplings.Itistherefore informativetoembarkonastudyofhowmuchoftheopticalleverspectraareinfactdueto groundmotion. IusedamethodofWienerltering[ 57 ]toshowthatthemotionofFigure 4-1 ,indeed originatesprimarilyfromtheground.TheWienerlteristhebestestimateofthecontribution ofonesignaltoanother.Thecontributionsofthethreeseismometerstoeachlargeopticoptical leverareshowninFigure 4-2 .Weseethattheopticalleversignalsarealmostentirelyexplained bygroundmotion,withthenotableexceptionsbeingapeakbetween3and4HzforITMY,and 0.3and0.7HzfortheRM,andbroadbandextramotionfortheBS. Furthermore,therelativemagnitudesofthecontributionsfromthethreeseismometersmake sense.Greaterthan1Hz,coherenceisgreatestwiththenearestseismometer,andlessthan1Hz, seismometerscontributemoreequally(exceptfortheITMs,whichareeverywhereverycoherent withthecornerstationseismometer). 4.2.2CoilActuators Imperfectionsintheforcesappliedbytheactuatorsontherearofthetestmassescan convertpiston(puretranslation)drivesintotorque.Thelengthsofthecavitiesarecarefully controlled(that'swhatwestrivetobemostsensitiveto!)andanyimbalancesbetweenthefour electromagnetsonasinglemirrorwillcreateacouplingfromlengthdrivetoangle(L2A).This effectismeasurable,andtheniscarefullytunedoutthroughselectingappropriatedigitalgains foreachofthecoils.Relativegainvariationis10%ofaverage.Theabilitytotunethegains perfectlyislimitedandtheresiduallengthtoanglecouplingisabout1%.Forthetypicalrms lengthdriveof1monacoreopticandOSEMsseparatedbyadistanceof & 2 R where R isthe radiusoftheoptic,the1%L2Acouplingresultsina10 # 8 radiandisplacement: ( = 0 01 ) 10 # 6 m & 2 ) 0 125m $ 10 # 8 rad (42) 64

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10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] ITMX [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic 10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] ITMY [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic 10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] ETMX [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic 10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] ETMY [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic Figure4-2.Contributionofseismicnoisetoopticallevererrorsignal(calledresidual).The interferometerwasunlockedandopticalleverandOSEMACdampingpresent. 65

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10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] RM [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic 10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] BS [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic 10 1 10 0 10 1 10 11 10 9 10 7 frequency [Hz] MMT3 [rad/rtHz] residual corner seismic x end seismic y end seismic all seismic Figure 4-2 continued. 4.2.3NoisefromAngularControl Theangularcontrolsystem,whichstrivestocounteracttheabovetorqueinputstoreduce angularmotion,introducesangularmirrormotionitself.Theprimarywayitcontributesnoiseis throughimperfectsensingoftheangulardisplacements.Thealignmentisalsounder-controlled, whichendsupallowingthecontrolsystemtoimpressinputbeammotiononthemirrors.These issuesareexplainedinmoredetailinSection 4.6 66

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4.2.4RadiationPressure Radiationpressurecreatesatorquewhenthebeamimpingesthemirroroff-center.Theforce onthemirrorduetoradiationpressureisderivedfromthechangeinmomentumofaphoton uponreectionoffthemirrorandresultsin: F rp = 2 P c (43) where P isthepowerofthelightreectedbythemirrorand c isthespeedoflight.Becausethe beamofphotonsstrikesthemirrorperpendiculartoitssurface,thetorqueexertedonamirrordue toradiationpressureis ) rp = 2 Px c (44) where x isthedistanceofthebeamfromthemirror'scenterofmass.Fora40kWbeam1mm off-center,thetorqueisontheorderof10 # 7 Nm,yieldinganangulardisplacementoftheorder 10 # 7 radasdeterminedbythependulumtorquetoangletransferfunction.RefertoSection 4.3 Amongstthevarioustorqueinputs,radiationpressureplaysauniqueroleinmirrormotion becausethetorqueitexertsdependsontheanglesofthemirrors.Aresultofthegeometric couplingbetweenbeamdisplacementsandmirrorangles,radiationpressurethereforeactsasan angularspring.Itisbesttreatednotasanexternaltorque,butasamodicationtothependulum torquetoangletransferfunction.Chapter 5 dedicatesadiscussiontothephysicsofradiation pressuretorques.Inall,radiationpressureshapestheangulardynamicsofthemirrorsinLIGO andplaysanimportantroleinthedesignofanangularcontrolsystem. 4.3TheMirrorasaTorsionPendulum Inordertodesignacontrolsystemthatreducestheangularmotionoftheinterferometer mirrorstothelevelsnecessaryforstableinterferometeroperationandminimalimpactonstrain sensitivity,theangularresponseofthemirrorstoexternaltorquemustbefullyunderstood.A modelofthemirror'storquetoangletransferfunctionisthusrequired.Tostart,themirrorsin LIGOmayberegardedastorsionpendula.Themirrormaytwistanangle ( aboutahorizontal 67

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axispassingthroughitscenterofmasstocreatemotionin pitch andaboutaverticalaxispassing throughitscenterofmasstocreateamotionin yaw Theangularequationofmotionofthemirrorisgovernedbythesumofalltorquesonthe mirror.First,let'sconsiderthemostsimplisticscenariowherethereisonlyapendulumrestoring torque ) p = # p ( ,where p isthependulum'storsionalconstant.Theequationofmotionis I ( + p ( = 0 (45) whichhasasolutionof ( ( t )= sin ( % 0 t ) ,where % 0 = p / I istheresonantangularfrequency and I isthemirror'smomentofinertia.Thependulumtorsionalconstantservestomakethe mirroroscillateindenitelyaboutitsequilibriumpositionupontheslightestofdisplacements. Weareparticularlyinterestedinthependulum'sangularresponsetoanexternaltorque,such asseismicnoise.Inordertocalculatethetorquetoangletransferfunction,wemustincludean externaltorqueterm, ) ext ,intheequationofmotion: I ( + + ( + p ( = ) ext (46) Avelocitydampingterm, + ,isalsoincluded.TakingtheLaplacetransformtoconvertfromthe timedomaintothefrequencydomain,wehave: Is 2 % + + s % + p % = ) ext (47) where s isacomplexparameter.Thetransferfunctionisthendenedas H ( s ) : = % ( s ) ) ext ( s ) = 1 Is 2 + + s + p (48) Weareonlyinterestedinexaminingthetransferfunctionforapuresinewaveexcitation, e i % t ,so wesubstitute s = i % toget H ( % )= 1 / I % 2 0 # % 2 + i +% / I (49) 68

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Theresonantfrequencyofthisdampedsystemcanbecomputedbyndingthe % atwhichthe amplitudeofthetransferfunction, [ I 2 [ % 2 # % 2 0 ] 2 + + 2 % 2 ] # 1 / 2 ,ismaximized: % res = % % 2 0 # + 2 2 I 2 (410) Wenotethatdampingreducestheresonantfrequency,althoughtheeffectisusuallyinsignicant. Aquantitythatismorefamiliarthan + fordescribingthelossesofasystemwithareal resonanceisthequalityfactor, Q : = % res / FWHM,whereFWHMisthefull-width-half-maxof thetransferfunction'samplitude-squaredresonance.Whenthelossesaresmall, % res $ % 0 and FWHM $ + / I [ 58 ,23-4].Thequalityfactoristhenwellapproximatedby Q = p I / + .The transferfunctionwrittenintermsof Q is H ( % )= 1 / I % 2 0 # % 2 + i %% 0 / Q (411) Figure 4-3 showsthependulumtorquetoangletransferfunction(forpitch)usingthe parametersofaLIGOcoreoptic.Forexternaltorquesappliedtothemirrorwellaboveits resonantfrequency,themirroractslikeafreemass,onethatisnotheldinplacebysuspension wiresnorsubjecttodamping.Fortorquesappliedtothemirrorbelowitsresonantfrequency,the mirror'sangleisdeterminedbytheinverseofthetorsionalconstant. 4.4OverviewofInterferometerAlignment Thereare8mirrorswhosepitchandyawanglesmustbesensedandcontrolled.Thesensing isaccomplishedby8sensors,whichfallintothreegroups: cameraimage(BS):sensesthepositionofthebeamontheBS quadrantphotodiodes(QPDX,QPDY):sensethepositionofthelighttransmittedthrough theETMs wavefrontsensors(WFS1,WFS2 2 ,WFS3,WFS4):sensetheangularmisalignmentofthe cavitieswithrespecttotheirinputbeams 2 TwosignalsarederivedfromWFS2. 69

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10 1 10 0 10 1 10 2 10 0 10 2 frequency [Hz] magnitude [rad/Nm] TF, =0.72075, =0.006 TF, =0 (free mass) 1/ Figure4-3.TorquetopitchtransferfunctionofaLIGOcoreoptic(blue).Theopticactslikea freemassathighfrequencies(red)andtheDCmagnitudeofthetransferfunctionis determinedbytheinversetorsionalconstant(green).Adampingconstant + = 0 006 ( Q = 32)wasselectedforpictorialrepresentationonly.Theresonantfrequencyof LIGOcoreopticsinyawis0.5Hz. Figure 4-4 showsthelocationsofthesesensorsandthe8mirrorstheymustcontrol.The alignmentschemecanbesimpliedbyconsideringitintwobasicunits:theinputbeamand thepower-recycledFabry-PerotMichelson(FPM).Thealignmentofthelatterisself-contained (withtheexceptionoftheBS)andusestheWFStokeepthemirrorsalignedtooneanother fromDCuptoseveralHz.ThealignmentoftheinputbeamtothepowerrecycledFPMunit isaccomplishedviatheBScameraimageandQPDXatfrequencieswellbelowthependulum resonances.Finally,aspartoftheinputbeamalignment,QPDYkeepsthebeamthatisreected fromtheBSpointedatthey-arm.ThealignmentschemeisdepictedinFigure 4-5 andexplained inmoredetailintheremainderofthissection. 70

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Figure4-4.LayoutofASCsensors(WFS,QPDs,andcamera)andthe8mirrorstheymust control(ITMs,ETMs,MMTs,BS,andRM). A B Figure4-5.Schematicofthealignmentsensingandcontrolsystem,viewedastwodifferentunits. A)TheQPDservoandbeamcenteringservo(BCS)togetherdirecttheinputbeamon minutetimescales.B)Thewavefrontsensingservomaintainsthealignmentofthe power-recycledFPMmirrorswithrespecttoeachotheruptoseveralHz. 71

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Theself-containedalignmentofthepower-recycledFabry-PerotMichelsonisrealized throughthesetofwavefrontsensors(WFS) 3 whichprovidethemostsophisticatedformof measuringangularmotionofthemirrorsandwhoseprincipleofoperationisdescribedin Section 4.5.1 .Thepitchandyawmotionsofthevemirrorsinthisunit,ETMX,ETMY,ITMX, ITMY,andthePRM,aresensedbythepitchandyawofveWFSsignals,WFS1Q,WFS2I, WFS2Q,WFS3I,andWFS4I,whereIandQdenotein-phaseandquadraturedemodulation, respectively.TheseWFSlookatlightattheASport,atthereectedport,andinthepower recyclingcavity.Theirerrorsignalsareusedtocontroltherelativemotionsofthesevemirrors uptoacoupleHz.Theoriginaldesignofwavefrontsensingforapower-recycledMichelsonwith Fabry-PerotcavitiesinthearmswascreatedandtestedbyMavalvala[ 59 60 ]. TheMMT-directedinputbeamandthepower-recycledFPMunitneedtobealignedtoone anothersuchthattheinputbeamisperfectlyreecteduponitself.Overminutetimescales,the inputbeamfollowstheinterferometerandoverfastertimescalestheinterferometerfollowsthe inputbeam.Thisisachievedviaablendofallsensors.Thelowfrequencymatchingoftheinput beamtotheinterferometerisrealizedthroughthepitchandyawsignalsofQPDX,aQPDwhich monitorsthepositionofthelighttransmittedthroughthex-arms,andthepitchandyawsignalsof acamerathatmonitorsthelocationofthebeamspotonthebeamsplitter.Thesetwoalignment sensorsadjustthepointingofMMT1andMMT3atabout30mHz.Thecamera'sbeamcentering servo(BCS)worksbytakingtheimageofthespeckleoflightreectedoffofthebeamsplitteras showninFigure 4-6 andfeedingitintoalabviewprogramwhichintegratestheintensityofthe imagetoidentifythecoordinatesofthecenterofthebeamspot.Thesecoordinatesarecompared withahardcodeddesiredcenterlocation.Amirrorupstream,MMT1,ismovedtoredirectthe inputbeam,minimizingthedifferencebetweenthedesiredandactualbeamspotlocationonthe BS.TheQPDservosworkinasimilarfashion.Thehigherfrequencymatchingoftheinputbeam 3 Pronounced"woofs". 72

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Figure4-6.Imageofbeamonbeamsplitterasusedinthebeamcenteringservo.Thebeam appearsstretchedbecausethecamera'sviewingangleisat45 ( withrespecttothe mirrorsurface.Thecolorscaleisarbitrary,butorangeisstrong,violetisweak. andtheinterferometertooneanotherisachievedbythereectedportwavefrontsensors(WFS3 andWFS4),uptoacoupleHz. Theoneadditionalstepneededforfullinterferometeralignmentistomaintaintherelative alignmentofthey-armtothex-armasthex-armandinputbeamtogethermovearound.This isaccomplishedthroughthepitchandyawsignalsofQPDY,theQPDthatmonitorsthelight transmittedthroughthey-arm,whichsensehowthebeamsplittershouldbepointed.QPDXalso sendsasignaltotheBStocompensateforthesignalitsendstoMMT3. Allmirroranglesareofcourseinter-dependentandtheymusttrackeachother.However, aroughhierarchyofwhofollowswhocanbeestablishedbecauseultimatelytheinterferometer isboltedtothegroundandnecessarilymaintainssomeDCorientation.Thisorientationcomes fromQPDXandQPDY,whicharephysicallyattachedtopiersstandingonthegroundandforce thebeamstransmittedthroughtheETMstostayputatacertainlocationontheirsensors.In all,theinputbeammustmakeittothosetwoexactplacesandtheothermirrorsarelefttoline themselvesupaccordingly. 73

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ThisalignmentprocessinvolvingtheWFS,QPDs,andBScamerareliesontheentire interferometeralreadybeinglocked.Itmanagesthecontinuousne-tuningofmirroranglesso thatmaximalpowerbuildupintheinterferometerismaintained,andsothattheinterferometer doesnotwanderfromitslinearoperatingpoint(describedinSection 2.2.1 ).Howtoachieve theinitialalignmentofallofthemirrorsisaninterestingprocessinitselfandisdocumentedin Appendix C.4 4.5TheAngularSensingandControlServo Whentheinterferometerislocked,theopticalleversandthewavefrontsensorsprovide simultaneousfeedbacktothemirrorangles.Theopticalleversprovidelocaldamping,andthe WFSmaintainboththealignmentofthepower-recycledFPMwiththeinputbeamandthe internalalignmentofthepower-recycledFPM.Thestudyandcharacterizationoftheelementsof thisangularcontrolsystemwhenthelaserpowerisincreasedisthetopicoftheremainderofthis chapterandthenext.First,itishelpfultopresentanoverviewoftheservowithouttheeffectsof radiationpressure. Figure 4-7 isablockdiagramofthemajorcomponentstotheangularcontrolservo.The torsion-likependulaoftheinterferometeraresubjecttoexternaltorquewhichisconvertedto mirrorangle.Theinterferometer,inturn,turnsthemirroranglesintoTEM 10 / TEM 01 modes atitsvariousports,andthesemodesareconvertedintoerrorsignalsbythewavefrontsensors. Thevoltagesproducedbythewavefrontsensorsaredigitizedandmanipulatedbythefront-end computerstocreatecontrolsignalsthatarethenconvertedbacktoanalogsignalsforactuationon themirrors.TheresultofthisprocessisasuppressionoftheerrorsignalseenbytheWFSand thereforeasuppressionofthephysicalrelativemirrormotions. 4.5.1TheWavefrontSensingScheme Thewavefrontsensorsprovidethemostsophisticatedformofmeasuringangularmotionof themirrors.TheyarequadrantphotodiodesequippedwithRFelectronicsthatrelyonthePound DreverHalllockingschemetoproduceerrorsignals[ 61 ].WFSaretheangularequivalentof thesingleelementPDsusedforlengthsensing.Theirframeofreferenceisthefundamental 74

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Figure4-7.ASCcontrolservo,showingboththeWFSandopticalleverloops.Theexternal torque,torsionpendulum,interferometer,andWFSareanalog;allotherblocksare digital.Mirroranglesareconvertedbytheinterferometerintosignalsthewavefront sensorscandetect.TheseWFSerrorsignalsaredigitized,ltered,andconvertedinto analogcontrolsignalsforindividualmirrors.Thedetailsofeachofthedigitalblocks areexplainedinChapter 6 Gaussianmodeoftheinterferometercavities;angularmisalignmentsofthecavitiesgenerate TEM 01 andTEM 10 spatialmodesofthecarrierlightasexplainedinRef.[ 62 ]andderived inAppendix C.3 .TheoverlapofthehigherordermodecarriereldwithaTEM 00 reference sidebandeldproducesanexcessofpowerononehalfofthedetectorcomparedtotheother whendemodulatedatthesidebandfrequency.Thisistheerrorsignal.Thesidebands Reference[ 63 ]containsathoroughdescriptionofthewavefrontsensingscheme.In summary,theideaisthattheWFSarelocatedwhereopticalgain,theamountoflaserpower producedatsomeinterferometerportforagivenphysicalchangeofsomeaspectoftheinterferometer,ishigh.TheWFSopticalgainin,particular,isdeterminedbyhowmuchTEM 01 /TEM 10 modeshowsupatthelocationsoftheWFSwhenamirror(orspeciccombinationofmirrors) movesinangle.OneWFSisplacedattheanti-symmetricportwheredifferential-modesignals aretransmitted,oneisplacedatapick-offofthebeamfromtherecyclingcavitywhichcontains 75

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commonanddifferentialinformation,andtwoareplacedatthereectedport,wherecommonmodesignalsaretransmitted.ThepreciselocationsaredeterminedfromtheGouyphasesofthe lightanddetailsarefoundinRef.[ 28 ]. 4.5.2TheDigitalPath Thedigitalportionoftheservo,fromthepowerscalingthroughtheopticallevercompensation,experiencestwochangesofbasiswhichareeffectedbytheinputmatrixandtheoutput matrix.TheinputmatrixtakestheappropriatelinearcombinationofWFSerrorsignalstocreate errorsignalsforwhichcontrolltersmaybedesignedeasily.Theoutputmatrixcreatesalinear combinationofthedigitalcontrolsignalsinordertodistributethemappropriatelytospecic mirrors.Boththepowerscalingandthemirrorgainsarediagonalmatricesthatprovideminor modicationstotheamplitudeoftheerrorandcontrolsignals,respectively.Thepowerscaling isupdatedinrealtimetocompensateforanychangesinlaserpowerontheWFS.Themirror gainsarehard-codedscalingfactorstoreectdeviationsfromthetheoreticalcavitygeometry. SeeSections 6.4.1 and 6.1.3 ,respectively,fordetails. 4.5.3OpticalLeverCompensation Eachofthelargeopticshasitsownopticalleverservothatprovidesvelocitydamping. BecauseboththeWFSandtheopticalleverscontrolthemirrormotionwhentheinterferometer islocked,wemustconsidertheinteractionofthetwoservos.Whetheroneviewstheinteraction astheWFScontrollingtheoptical-lever-controlledmirrorsorastheopticalleverscontrolling theWFS-controlled-mirrorsisarbitrary.Forthepurposeofexplanationhere,Iusetheformer viewpoint. Figure 4-8 zoomsinontheportionoftheFigure 4-7 controllooppicturethatshowsthe opticallevercontrollingthependulum.Thetorquetoangletransferfunctionofthependulum isourplant, P ,theopticallevercontrolis O ,andtheopticallevercompensationisshownas 1 + OP .Aresultofbasiccontroltheoryisthattheopticallevercontrolledpendulumloopcanbe 76

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A B C Figure4-8.Opticallevercompensationscheme.SubguresA,B,andCareequivalent.A)Zoom ofthebottompartofFigure 4-7 P isthependulumand O istheopticalleverdigital controllters.B)Replacementoftheopticallevercontrolledplantwiththeclosed loopgainrepresentation.C)Opticallevercompensationremovestheopticallevers fromthepictureofthependulum.Externaltorqueisstillsuppressedbytheoptical leverclosedloopgain. representedbyitsclosedlooptransferfunction: TF closed = P 1 + OP (412) asshowninFigure 4-8 B. Onewillnoticethatitisnotacoincidencethattheopticallevercompensationhasthe sameformasthedenominatoroftheclosedlooptransferfunction.Together,theycanceleach 77

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10 2 10 0 10 2 10 8 10 6 10 4 10 2 WFS1Q [Volts/rtHz] frequency [Hz] dark residual 10 2 10 0 10 2 10 8 10 6 10 4 10 2 WFS2I [Volts/rtHz] frequency [Hz] dark residual 10 2 10 0 10 2 10 8 10 6 10 4 10 2 WFS2Q [Volts/rtHz] frequency [Hz] dark residual 10 2 10 0 10 2 10 8 10 6 10 4 10 2 WFS3I [Volts/rtHz] frequency [Hz] dark residual 10 2 10 0 10 2 10 8 10 6 10 4 10 2 WFS4I [Volts/rtHz] frequency [Hz] dark residual Figure4-9.WFSdarknoisecomparedtotypicalerrorsignal.Theexcesssignalabovethedark noiseinWFS3IandWFS4Iabove20Hzislikelyacousticnoise,althoughthis hypothesishasnotbeenveried.WFS1andWFS2areonaoatingtableinasound proofchamber,whileWFS3andWFS4areonanon-seismicallyisolatedtable withoutasoundproofenclosure.Seismicnoiseattimeofresidualspectraisin Figure C-5 otherout,andallthatremainsastheplantisthesimplependulum.Includingtheopticallever compensationintheASCloopisnotnecessary,butitisausefultechniqueforsimplifyingthe designprocessoftheWFScontrollters.Oneneedonlyconsiderhowtocontrolthesimple pendulumtorquetoangletransferfunction,ratherthantheoptical-levercontrolledpendulum. 4.6AngularControlLimitations Thelimitsforhowwellwecancontroltheangularmotionoftheinterferometerare governedbyhowwellweareabletosensetheangularmotion.Severalofthewavefrontsensors' signalsaredark-noise-limitedabove20to25Hz,asseeninFigure 4-9 .Moreover,dependingon 78

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thepowerlevel,WFS1Qmayinsteadbelimitedbyshotnoise(refertoEq. 212 ).Anycontrol signalderivedfromfrequenciesinthesensingnoiselimitedregimewillimpressthesensornoise onthemirrors.Theimpressionofsensornoisecannotbeavoidedentirelyinthepresenceof feedback,butcanbemitigatedbyincludingamongstthecontrolltersasteepcut-offbeginning atfrequencieswheresensornoisedominates. Besidesthesensingnoise,therearealsosometimesrealsignalsthatresultinmoreharm thangoodwhenusedasfeedback.TheHAMseismicisolationtablesusedbytheInputOptics (thecoreopticsaresuspendedfromBSCtables)havestackmodesat0.8to3Hzthatringup theMMTs.Atlowfrequencies,around1Hz,someoftheWFSsignalsaredominatedbythese angulaructuationsoftheinputbeam.Theresultingattemptofthemirrorstofollowtheinput beamjitterleadstoamagnicationofthemotionbecauseofthedrasticallydifferentlength scales.Largepoweructuationsinbotharmsandthepowerrecyclingcavityensue,leadingto departurefromthelinearerrorsignalregimeandoftenlockloss. Otherlimitationstothereductionofmirrormotionresultfromthenatureofcontrolloops. Thecut-offlter,forexample,reducesthephasemarginoftheopenloopgain,necessarily pushingdowntheunitygainfrequency(UGF)andthereforethemagnitudeofsuppressionat allfrequenciesbelowtheUGF.Alessaggressivecut-offlter,whileimprovingtheservo's stabilityandallowingforhigheroverallgains,leadstomoreimpressionofsensingnoiseonthe optics.Also,ifthephasemarginoftheloopislow,mirrormotionwillbeampliedthroughgain peaking. Analpointthatshouldbenotedisthatthereisanangulardrivetolengthcouplingdueto OSEMimbalancesthatresultsfromthesameprincipleofthelengthtoanglecouplingpresented inSection 4.2.2 .Inall,wewillshowinChapter 6 thatitisultimatelytheWFSnoiseoor thatdeterminesthebestpossibleachievablesuppressionbecausethroughangularcontrolthe impressionofWFSsensingnoiseplaysamajorroleinthereductionofDARMsensitivityupto 55Hz. 79

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CHAPTER5 THEEFFECTOFHIGHLASERPOWERONINTERFEROMETERALIGNMENT Thetorqueinducedbyradiationpressure,asintroducedinSection 4.2.4 ,couplestheangular motionofthesuspendedmirrors,complicatingtheplantforwhichcontrolsmustbedesigned. Thederivationoftheangularresponseofthemirrorstoexternaltorqueinthepresenceof radiationpressureispresentedinseveralpublications,butIprovidemyownderivationherefor completeness.Radiationpressuretorqueis,afterall,thefoundationforthiswork'sinvestigation ofhighpowereffectsintheangularsensingandcontrol.Wederiveasetofeigenfunctionsthat diagonalizethelinearcavity'sresponsetoradiationpressureanddescribethemirrors'equations ofmotioninthisneweigenbasis.Weshowthatthetorquetoangletransferfunctionsofthenew eigenmodesaremodiedsuchthatonemodeisstaticallyunstableatEnhancedLIGOpowers. 5.1TheRadiationPressureAngularSpring Thegeometricaxisofacavityformedbytwosphericalmirrorsisdictatedbythelinejoining thetwocentersofcurvature.Onlyifthemirrorsarepointeddirectlyatoneanotherwillthecavity geometricaxispassthroughthecentersofthemirrors.Shouldalaserbeamresonateinthecavity, itwilldosoalongthisgeometricaxis.Thus,ifthemirrorsaretiltedawayfromoneanother,the beamspotoneachmirrorwillnotbecentered.Therelationshipbetweenthepositionsofthe beamsonthemirrorsrelativetocenter, x i ,andtheanglesofthemirrors, ( i ,isgivenby: # x 1 x 2 $ % & = L 1 # g 1 g 2 # g 2 1 1 g 1 $ % & # ( 1 ( 2 $ % & (51) The g -factorisdenedas g i = 1 # R i / L where R i isthemirrorradiusofcurvature,and L isthe lengthofthecavity.The g -factoris0 73(0 71)fortheLLO(LHO)ITMand0 54(0 45)forthe LLO(LHO)ETM. Wesawinthepreviouschapterthattheradiationpressuretorqueonamirrordependson thepositionofthebeamonthemirror, ) rp = 2 Px / c (Eq. 44 ).BasedonEq. 51 theradiation pressuretorqueonamirrorthatispartofaFabry-Perotcavityisthereforedependentontheangle 80

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ofboththemirrorofinterestandthesecondmirrorformingthecavity: # ) rp 1 ) rp 2 $ % & = 2 PL c ( 1 # g 1 g 2 ) # g 2 1 1 g 1 $ % & # ( 1 ( 2 $ % & (52) Thisismoresuccinctlyexpressedas ) rp = # K rp ( (53) where K rp isthe torsionalstiffnessmatrix .Equation 53 istheexpressionthatdescribesthe radiationpressureangularspring. 5.1.1DiagonalizingtheModiedEquationsofMotion Theradiationpressurespringmodiesthependulumangularequationofmotionand thereforethetorquetoangletransferfunctionthroughtheadditionofanangle-dependenttorque term.Re-writingEq. 46 inmatrixformandwiththeradiationpressurespringterm,thetwo equationsthatdescribethemotionoftwomirrorsformingaFabry-Perotcavityis: I ( + ( + p ( # 2 PL c ( 1 # g 1 g 2 ) # g 2 1 1 g 1 $ % & ( = ) ext (54) I + ,and p are2 2diagonalmatricesand ( and ) ext are2 1vectorsasintheprevioussection. Duetothenon-diagonalmatrixinEq. 54 ,themotionsofeachofthemirrorsformingthecavity aretiedtooneanother.Thenaturalwaytoworkwithsuchasystemistorotatethecoupled equationsintoanewbasis.Theresultingde-coupledequationsofmotionwilldescribedspecic combinationsofmirrortiltsinsteadofthetiltofanindividualmirror.Vectorsintherotatedbasis arewrittenwithprimes. InordertodecouplethetwoequationsofEq. 54 ,weneedtodiagonalize K rp .The subscripts a and b areusedtodenotetheelementsofthediagonalizedbasis,tocontrastthe1and 81

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2whichdenotethemirrorbasis.Ignoringtheconstantsofmatrix K rp ,itseigenvaluesare # a = g 1 + g 2 + ( g 1 # g 2 ) 2 + 4 2 (55) # b = g 1 + g 2 # ( g 1 # g 2 ) 2 + 4 2 (56) anditseigenvectorsare v a = # 1 g 1 # g 2 + & ( g 1 # g 2 ) 2 + 4 2 $ % & (57) v b = # # g 1 + g 2 # & ( g 1 # g 2 ) 2 + 4 2 1 $ % & (58) Therefore,thematrix S = v a v b ( = # 1 # g 1 + g 2 # & ( g 1 # g 2 ) 2 + 4 2 g 1 # g 2 + & ( g 1 # g 2 ) 2 + 4 2 1 $ % & (59) diagonalizes K rp suchthat S # 1 K rp S = D = # # a 0 0 # b $ % & = # g 1 + g 2 + & ( g 1 # g 2 ) 2 + 4 2 0 0 g 1 + g 2 # & ( g 1 # g 2 ) 2 + 4 2 $ % & (510) Thematrixofeigenvectors, S ,isthebasistransformationmatrix.Itservestodenethetorque andanglevectorsinthenewbasis.Forexample, ( = # ( a ( b $ % & = S # 1 # ( 1 ( 2 $ % & = S # 1 ( (511) RearrangingEq. 510 totheform K rp = SDS # 1 andsubstitutingitintoEq. 54 ,wehave: I ( + ( + p ( # 2 PL c ( 1 # g 1 g 2 ) SDS # 1 ( = ) ext (512) 82

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Table5-1.GeometricparametersoftheLIGOarmcavityeigenmodes. x i arethebeamlocations onthemirrorsrelativetocenter, a isthecavityaxisdisplacementatthewaist,and is thecavityaxisanglewithrespecttoalinejoiningthecentersofthemirrors. DifferencesbetweenLLOandLHOarisefromthemirrorsateachsitehavingdifferent radiiofcurvature.Quantitiesareexpressedasafunctionoftheamountoftiltina particularmode. LLOLLOLHOLHO cavityparameterunit v a mode v b mode v a mode v b mode | x 1 | mm/urad9.882.448.202.51 | x 2 | mm/urad10.842.229.352.20 | a | mm/urad10.171.018.481.34 | | urad/urad0.241.170.291.18 Multiplyingontheleftby S # 1 ,takingadvantageofthediagonal I + ,and p matrices,andusing S # 1 tochangethebasisofeachofthevectors,thede-coupledequationsofmotionare: I ( + ( + p ( # 2 PL c ( 1 # g 1 g 2 ) # # a 0 0 # b $ % & ( = ) ext (513) Theradiationpressuretorsionconstant, rp ,is rp = # 2 PL c ( 1 # g 1 g 2 ) # (514) where # = # a or # b ,dependingonthemodeinquestion. TheangularmotionoftheFabry-Perotcavityisnolongerdescribedbythemotionsofitsindividualmirrors.Duetoradiationpressure,thecavityistreatedasaunitandthetwoorthogonal modesofangularmotionarecombinationsofthetwomirrors'angles.Theeigenvectors v a and v b describethesetwosetsoforthogonalmirrortilts,andtheeigenvalues # a and # b (alongwiththeir commonconstants)quantifythemagnitudeoftheradiationpressuretorsionalspringconstantfor eachofthemodes.Whiletheequationsofmotionhadbeenidenticalforeachoftheindividual mirrors,thedecoupledequationsinthepresenceofradiationpressurebreaksthatsymmetry. Table 5-1 outlinesthecharacteristicsofthesetwoeigenmodesforthespecicgeometryof theLIGOarmcavities.Theamountofbeamdisplacementoneachmirrorisgivenasafunction oftheamountoftiltinoneeigenmodeortheother.Furthermore,theamountofcavityaxis 83

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Figure5-1.Illustrationoftheorthogonalmodesofcavitytilt.Theupperdiagramshowstilts givenbyeigenvector v b andthelowerdiagramshows v a displacement a andcavityaxistilt isalsocalculatedforeacheigenmodeusingthegeometric relationshipbetweenasetofmirrortiltsandtheircavityaxisasderivedinAppendix C.2 .Figure 5-1 illustratesacavityineachofthetwoeigenmodeswhenusingtheparametersfromTable 5-1 5.1.2SoftandHardModes Thetorquetoangletransferfunctionofeachoftheseeigenmodeshasthesameformasthat ofasinglependulum(Eq. 48 ),butthetorsionconstantismodied.Moreimportantly,thespring constantismodieddifferentlyforeachmode,yieldingdistinctbehaviorsofthetwoeigenmodes. Inthissection,weanalyzethesebehaviorsandaccordinglyintroducethenames soft and hard to useinplaceof a and b fordescribingthetwomodes. 84

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Figure5-2.Demonstrationofhowradiationpressuremodiesthetorquetoangletransfer functionofaFabry-Perotcavity'seigenmodes. JustasinSection 4.3 ,wecantaketheLaplacetransformofeachoftheequationsinEq. 513 togetthegeneralformofthemodaltorquetoangletransferfunction: H ( s )= % ( s ) ) ext ( s ) = 1 Is 2 + + s + p + rp (515) Figure 5-2 showsthecontroltheoryviewoftheadditionoftheradiationpressurespringconstant tothetransferfunction. Themagnitudeandsignofthetotaltorsionalspringconstant, tot = p + rp ,conveys criticalinformationaboutthestabilityofthecavityandthenatureofitsresponsetoexternal torque.Recallingtheequationofanangularspring, ) = # tot ( ,arestoringtorqueisprovided onlyif tot > 0,whichisequivalenttotheconditionforstability.If tot < 0,thespringisan anti-spring,resultinginanunstable,run-awaysituation.Furthermore,while tot ispositive,its magnitudedirectlyrelatestothestiffnessofthespring. 85

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Thestabilitycriteriaforthecoupledcavityeigenmodesdependontherelationshipbetween p and rp : stable: k tot > 0 = + 2 PL c ( 1 # g 1 g 2 ) # < p (516) unstable: k tot < 0 = + 2 PL c ( 1 # g 1 g 2 ) # > p (517) Thependulumspringconstant, p ,isalwayspositive,sowecanconcludewithcertaintythatthe cavityeigenmodeisstableaslongasthequantityontheleft-handsideofEq. 517 isnegative. However,ifthisquantityispositive,thenitsmagnitudecomparedto p determinesstability. Because P L ,and c areallpositivenumbersandthe g -factorisrestrictedto0 < g 1 g 2 < 1 1 ,the signoftheleft-handsideisdeterminedsolelybythatof # .Fromthe g -parameterrestriction, itcanbeshownthat # a isalwayspositiveandthat # b isalwaysnegative.Therefore,themode whosemirroranglesaredescribedby v a iseitherstableorunstable,andthemodedescribedby v b willalwaysbestable. Theprecisesituationforthepotentiallyunstablemodedependsontheonenon-constant variable,thecirculatingpower P .Thereisacriticalpoweratwhich rp =* p ,andatany greaterpower,instabilityensues.Ingeneral,aspowerincreases,thetotalspringconstantforthe potentiallyunstablemodedecreases,creatingasofterspring,andthetotalspringconstantfor theunconditionallystablemodeincreases,creatingastifferspring.Thusarisetheterms soft and hard todescribethetwoeigenmodesthathavebeenreferredtoby v a and v b ,respectively. Figure 5-3 showsthedependenceof tot oncirculatingpowerforthesoftandhardmodes ofaLIGOarmcavity.Withoutpowerinthecavity,themodesareidenticalandtheirspring constantsaresimplythatoftheindividualpendula.Thesymmetry-breakingeffectofradiation pressurecomesintoplayassoonaslightresonatesinthecavity:thehardmode'sspringconstant increasesandthesoftmode'sspringconstantdecreases.Thecriticalpoweratwhichthesoft 1 Thisisthenecessaryconditionforatwomirrorresonatortoformastableperiodicfocusing system.[ 64 ,p.747] 86

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0 5 10 15 20 25 30 35 40 45 3 2 1 0 1 2 k tot [Nm/rad] Circulating power [kW] hard mode soft mode Figure5-3.Torsionalspringconstants(pitch)ofanopticallycoupledcavityforLLOparameters. Thesoftmodeisunstablewhenthespringconstantisnegative. Table5-2.Torsionalspringconstants(pitch)forthesoftandhardmodesofatypicalInitialLIGO powerandthehighestofEnhancedLIGOpowers.ThesoftmodeinEnhancedLIGO isunstable.The p valuesassumearesonantfrequencyof0.6Hz. P circ p tot ,softmode tot ,hardmode InitialLIGO9kW0.721Nm/rad0.0734Nm/rad0.867Nm/rad EnhancedLIGO40kW0.721Nm/rad-2.18Nm/rad1.38Nm/rad modebecomesunstableis10kW,whichcorrespondstoapproximately6Winputpower(for EnhancedLIGOefciencies)totheinterferometer.Abovethecriticalpower,radiationpressure createsanopticalanti-spring. Table 5-2 highlightsthevaluesofthespringconstantsforthetypicalpowerthatwasused inInitialLIGO(9kW)andforthehighestofpowersachievedinEnhancedLIGO(40kW).The correspondingtransferfunctionsforthesespringconstantsisfoundinFigure 5-4 .Theresonant frequency, % 0 = tot / I ,increaseswithpowerforthehardmodesanddecreasesforthesoft modes.Once tot becomesnegative,asisthecasefortheEnhancedLIGOsoftmode,thereis norealresonantfrequency.Asummaryoftheopto-mechanicalparametersforLLOandLHOis foundinTable 5-3 87

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10 2 10 1 10 0 10 1 10 2 10 0 10 2 magnitude [rad/Nm] 10 2 10 1 10 0 10 1 200 150 100 50 0 frequency [Hz] phase [degrees] 0 W 9 kW (soft mode) 9 kW (hard mode) 40 kW (soft mode) 40 kW (hard mode) Figure5-4.Singlecavityopto-mechanicaltransferfunctionforpitch.Theresonantfrequency increaseswithpowerforthehardmode,butdecreasesforthesoftmode,eventually becomingimaginary. P circ = 9kW(5.25Winput)wasatypicaloperatingpowerfor InitialLIGOand P circ = 40kW(23.5Winput)isthehighestofpowersreachedfor EnhancedLIGO. Table5-3.Opto-mechanicalparametersfortheLIGOLivingstonandLIGOHanfordcavities. Differencesresultbecausethemirrorsateachsitehavedifferentradiiofcurvature. parameterlabelLLOLHO ITM g -factor g 1 0.730.71 ETM g -factor g 2 0.540.45 armlength L 3995m3995m testmassmomentofinertia I 0.0507kgm 2 0.0507kgm 2 pendulumtorsionconstant p 0.72Nm/rad(pitch)0.72Nm/rad(pitch) 0.50Nm/rad(yaw)0.50Nm/rad(yaw) softmodeeigenvector v a (1,1.10)(1,1.14) hardmodeeigenvector v b (-1.10,1)(-1.14,1) powerwhensoftmodeisunstable P crit 10.0kW(pitch)11.6kW(pitch) 7.0kW(yaw)8.0kW(yaw) 88

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Table5-4.Conditionsontotaltorsionalconstant tot fordeterminingsystemstability. tot conditionpole s + impulseresponse tot < 0realpositivestaticallyunstable tot = 0zero 0 < tot < + 2 / 4 I realnegativestabledecay tot > + 2 / 4 I realnegative,andimaginarystable,oscillatory 5.1.3PoleAnalysis Onenalcommentabouttheanalysisofthemodiedtransferfunction(Eq. 515 )isthatthe twopoles, s = s = # + + 2 # 4 I tot 2 I (518) provideanalternativewaytoviewthestabilityofthesystem.Aslongasthepolesarenegative, theimpulseresponsewilldecayorbesinusoidal.However,ifapoleispositive,thesystem's motionwillexperienceexponentialgrowth.Theconstraintsfor s tobeinaparticularhalfof thes-planeareeasilyderivedfromEq. 518 .Notethat s # willalwaysbeinthelefthalfofthe planeandthat s + isthepolethathasthepotentialoffallingintherighthalfoftheplane.Table 5-4 showhowthes-planelocationsfor s + dependon tot .Thesignof tot determinesstability,as expected,andweseethatthenatureofthestableresponsedependsonthedampingcoefcient. Figure 5-5 plotsthepolelocationsforarangeof tot experiencedwhilepoweringupEnhanced LIGO. 5.2Implications TheEnhancedLIGOgoalofincreasingtheinputpowerto30WfromtheInitialLIGO7W makesradiationpressuretorquescrossintotherealmofsignicance.Inparticular,thesoftoptomechanicalmodewhichjustapproachedinstabilityforInitialLIGOpowersactuallybecomes unstableforEnhancedLIGOpowers.Thetransferfunctionsforwhichcontrolsmustbedesigned arenolongerthoseofapendulumwithresonanceat0.6Hz(pitch)or0.5Hz(yaw),butthoseof thesoftandhardeigenmodes,whoseresonanceschangewithpower.ForEnhancedLIGO,the angularcontrolservoplantofFigure 4-7 istreatedasthecavity'sradiationpressure-modied torquetoangletransferfunction,notasasimplestand-alonependulumtransferfunction.An 89

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! 8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8 real imaginary k=1.5 k=1 k=0.5 k=0 k= 0.5 k= 1 k= 1.5 k= 2 Figure5-5.Polesofthetorquetopitchtransferfunctionasafunctionoftorsionalconstant, tot Crossesshow s + andasterisksshow s # .Polesintherighthalfofthes-planeindicate thesystemisunstable. elegantimplicationofthepurelygeometricdescriptionofthecavityaxisisthattheradiation pressureeigenmodesareorthogonalindependentlyofpower.Therefore,themodesremain independentandthecontrolsystemneednotbeupdatedwithpowerchanges. Onecomplicationofhavingaradiationpressuremodiedandthereforepowerdependent plant, P rp ,isthattheopticallevercompensation(describedinSection 4.5.3 )isnolongervalid. Thecompensationof1 + OP ishard-codedinthedigitalcontrolsystem,wherethemodelfor P isthatofasimplependulum.Theonlywaytoachieveperfectcompensationwouldbeto loadanewmodelof P intothecompensationlterbankforeachnewpowerandforeachoptic. DoingthisinEnhancedLIGOwasnotpractical,soonlyatlowpowersdoestheopticallever 90

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compensationactuallycancelouttheeffectoftheoptical-lever-controlledradiationpressuremodiedplant, P rp / 1 + OP rp ,andleavesimply P rp fortheWFStocontrol.Itthisturnsoutthatas powerincreases,theimperfectcompensationactuallymaketheloopsmorestable[ 65 ]. ThesensorsinInitialLIGOwerenottunedtospecicallylookforthecombinedmirror motionsthatcreatethesoftandhardmodes.Theonlywaytoprovideadequatecontrolforboth modeswouldbetoincreasethegainofalloftheangularcontrolloops.Becausesomeofthe sensorsarenotasgoodasothers,thiswouldresultinexcessiveimpressionofsensornoise onDARM.Tominimizeimpactonstrainsensitivitywhilereducingtheangularmotionofthe interferometer'smirrorstothelevelsnecessaryforstableoperation,weneedtopickoutthe combinationofsensorsthattogethersensespecicallythehardmodeorthesoftmode,and thendesigncontrolsthatspecicallyaddressthecharacteristicsofjustonemode.Thisisthe foundationoftheASCworkforEnhancedLIGO:switchingtheWFScontroltotheradiation pressureeigenmodebasis,andincreasingthegainsofonlythoseloopsthatrequireit. 91

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CHAPTER6 ANGULARSENSINGANDCONTROLCHARACTERIZATIONANDPERFORMANCEIN THERADIATIONPRESSUREEIGENBASIS TheprimaryaspectoftheEnhancedLIGOASCupgradewastoswitchthecontrolservo fromthesensorbasistothenaturalradiationpressureeigenmodebasis,andtokeepthecontaminationtoDARMataminimum.Ipresentinthischapterthedesignofthenewbasisand measurementsImadetocharacterizeitanditseffectonDARM. Acriticalaspectofthecharacterizationofanysystemistocalibratethedatainphysical unitstofacilitatecomparisontomodelsandtomakemeaningfulstatements.BecausetheLIGO dataiscollecteddigitally,theunitsarenaturallyindigitalcounts.Partofmyworkwastherefore tocalibrateeachoftherelevantASCchannelstophysicalunits.Iincludethedetailsofthe calibrationsinAppendix B Also,asshowninSection 4.2.1 almostallofthemirrormotionisinfactduetotheground. Therefore,themeasurementsImadeoftheASCareverysensitivetotheparticularstateof seismicnoise.IincludeinAppendix C.7 seismicspectrafromthetimeofeachmeasurementI present. 6.1TheASCChangeofBasis ThebasisforangularcontrolinInitialLIGOwasthatofthephysicalsensors.Thewavefront sensorsarelocatedsuchthattheysensecommonanddifferentialETMandcommonand differentialITMangularmotions.Theinputmatrixwasdiagonalandthecontrollterswere designedtofeedbacktothosesetsofmotionsviatheoutputmatrix.Thisdoesnot,however,lend itselftoeasilyhandlingradiationpressuretorque.Becausethesensorbasisisnottheradiation pressureeigenbasisofSection 5.1.1 ,eachcontrolservohandledcombinationsofthesoftand hardmodes.Duetoradiationpressure,eachmirrorhastworesonances,amorecomplicatedplant thanthatofferedbytheeigenbasiswhichhasasingleresonanceforeachmode.Thechangein controlbasisfromInitialLIGOtoEnhancedLIGOwasaratherstraightforwardoperationof changingonlytheASCinputmatrixandtheASCoutputmatrixasdescribedinthefollowing subsections.RefertoFigure 4-7 forthelocationsofthesematricesintheservo. 92

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AlthoughwechangedthemeaningoftheWFScontrolsignalsforEnhancedLIGO,we didnotupdatetheconventionfortheirnamingasseeninthechannelnames:WFS1,WFS2A, WFS2B,WFS3,andWFS4.Thiscanbemisleadingbecauseofthesimilaritytothesensornames (WFS1Q,WFS2I,WFS2Q,WFS3I,andWFS4I),potentiallyleadingonetoassumeaone-to-one correspondence.Althoughaone-to-onecorrespondencewasaccurateforInitialLIGO,itisnot trueforEnhancedLIGO.Therefore,forclarityinthisdissertation,IrefertotheWFScontrol signalsbytheradiationpressureeigenbasisdegreesoffreedom(DOFs)theyrepresent: WFS1 differentialsoft(dSoft) WFS2A commonsoft(cSoft) WFS2B differentialhard(dHard) WFS3 commonhard(cHard) WFS4 recyclingmirror(RM) Differentialandcommonrefertothecomparisonofthesignofthemodeineacharm. 6.1.1WFSInputMatrix Theopticalgainforeachoftheoptics'motionsisnotconcentratedatanyoneparticular port,althoughitcanappearmorestronglyinonelocationcomparedtoanother.Inordertomake useofasmuchoftheinformationaspossible,wemustuseallsensorsthatwitnessaparticular motion.However,whentherearemultiplesignalsataparticularplace,theWFSerrorsignalstell usthesumofallopticalgainsatitslocation.Theamountofopticalgainateachdetectorforeach motionatoneparticularfrequencyformsthe sensingmatrix .Theinverseofthesensingmatrix isknownasthe inputmatrix ,whichtellshowtotaketheappropriateweightedsumofsignalsin ordertoreconstructaparticularmotion.Theprocedureformeasuringthesensingmatrixisas follows: exciteoneofthemirrors(orspeciccombinationofmirrors)atfrequency f demodulateeachoftheWFSsignalsat f normalizetothephaseoftheexcitationreadback repeatforeachmirror(orsetsofmirrors) Akeyaspectofthemeasurementisthatanotchlteratfrequency f isengagedsothecontrol servodoesnotsuppressourexcitation. 93

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Table6-1.Angularopticalgainat9.7HzinunitsofVoltsperdegreeoffreedommicroradian (pitch).Numbersin gray arethemeasurementresultsthathavecoherenceslessthan 0.9.Boxeshighlighttheelementsactuallyusedinthecontrolsystem.Allother elementsaresettozero. WFS1QWFS2QWFS2IWFS3IWFS4I 2.0 0.03 0.06 -0.008 0.01 dSoft 0.31 -0.03 -0.04 0.002 -0.01 dHard 0.02 -0.01 0.18 -0.02 -0.10 cSoft 0.17 -0.01 -0.21 0.007 -0.12 cHard 0.09 -0.01 -0.21 0.04 -0.21 RM Anexamplecalibratedsensingmatrixintheradiationpressureeigenbasisastakenduring a10WlockisshowninTable 6-1 1 Rowsrepresentexcitationandcolumnsarethewavefront sensors.Beforeinvertingtocreatetheinputmatrix,thesmallestoftheelements(whichare moreorlessequivalenttotheelementsforwhichopticalgainisalsoexpectedtobeweak), arearticiallysettozero.Thisavoidsthecontaminationofstrongsignalsbythosewithweak signal-to-noiseratios.Theelementsthatremainafterthisprocessarehighlightedbyboxes.Note thatthesensingmatrixisinfactcomposedoftwosub-matrices:oneforthedifferentialdegrees offreedom,andoneforthecommondegreesoffreedom.Also,WFS1Qhasparticularlystrong signalcomparedtotheotherwavefrontsensors.WeseeinSection 6.5 thatthisallowsusto providemuchmorecontroltothedifferentialsoftmodecomparedtotheothermodes. 6.1.2WFSOutputMatrix TheWFSoutputmatrixdetermineshowtoconverttheradiationpressureeigenbasiscontrol signalsintoindividualmirrorcontrolsignals.Itisthebasistransformationmatrix, S ,asdened inEq. 59 .Thematrixisarbitrarilynormalizedsothelargestelementis1,anditisrepeatedwith appropriatesignchangestoformdifferentialandcommonsoftandhardmodesofthetwoarms. TheoutputmatrixisshowninTable 6-2 ,wherethe r is0.91forLivingstonand0.87forHanford. 1 RefertoAppendix B.3 and B.4 foradescriptionofhowthecalibrationisdone. 94

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Table6-2.WFSoutputmatrix(pitch).FortheLivingstoncavitygeometry r = 0 91andfor Hanford r = 0 87. dSoftdHardcSoftcHardRM 1r1r0ETMX -1-r1r0ETMY r-1r-10ITMX -r1r-10ITMY 00001RM 6.1.3DiagonalizingtheWFSDriveMatrix Themirrorgainmatrix(introducedinFig. 4-7 )isadiagonalmatrixthatmodiesthe amplitudeofthecontrolsignaltoeachopticbysomefactorclosetounity.Itcorrectsfor experimentallymeasureddeviationsfromthetheoreticaloutputmatrix.Thepurposeofthemirror gainmatrix, M ,istoensurethatwhenaparticularWFSisdriven,onlyitsDOFisexcited.We callthematrixofobservedDOFmotionsperWFSexcitationthedrivematrix, D Themirrorgainmatrixiswhatwetunetomakethedrivematrixdiagonal.Thedrivematrix canberepresentedas: D = UMC (61) AWFScontrolsignalismultipliedrstbytheoutput(control)matrix, C ,thenbythemirror gains,andthenputintohardwarewithanunknowntransferfunction, U ,tocreateaphysical torqueonthemirrors.Tomake D diagonal,westartwith M = 1 ,experimentallyconstructthe drivematrix,andcalculatethenewmirrorgainmatrix, M new ,whichmusthavetheproperty: M new = U # 1 C # 1 = MCD # 1 C # 1 (62) Notethatthesetofunknowntransferfunctions U iseliminated. Imeasured D byrecordingthedemodulatedopticallevererrorsignalsduringthesensing matrixmeasurement(refertoSec. 6.1.1 ).Becausetheopticalleversprovidearecordofthe motionofthetestmasses,combiningtheircalibratedresponsesviatheoutputmatrix(Table 6-2 ) 95

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Table6-3.Mirrorgainsfordiagonalizationofdrivematrix. ETMXETMYITMXITMYRM 1.331.380.960.871.0 Table6-4.Actualeigenbasismotionduringsensingmatrixexcitationsaswitnessedbytheoptical levers.Columnsareexcitationsandrowsarethepitchmotionsat9.7Hzinunitsof rad.Themirrorgainsareselectedtomakethismatrixasdiagonalaspossible. dSoftdHardcSoftcHardRM 5 1e-06 -5.2e-086.1e-07-3.8e-08-1.8e-07dSoft -3.4e-07 5 0e-06 7.3e-07-1.0e-062.4e-07dHard -4.1e-07-3.3e-08 5 9e-06 6.8e-072.5e-07cSoft -6.4e-07-5.9e-071.1e-06 5 7e-06 4.7e-07cHard -1.6e-07-1.8e-06-5.5e-072.6e-06 5 6e-06 RM allowsonetoconverttheindividualmirrormotionsduringeachDOFexcitationtomotionsinthe radiationpressureDOFbasis. 2 ThemirrorgainmatrixisshowninTable 6-3 .The30%differencewiththemodelisaresult ofuncertaintyinthecavitygeometry.Theresultingdrivematrix,correspondingtothesame measurementtimeasthesensingmatrixofTable 6-1 ,ispresentedinTable 6-4 .Columnsare excitationsandrowsarethepitchmotionsat9.7Hzinunitsofrad.Theabsoluteamplitude ofDOFmotionsforeachexcitationisnotofsignicance;onlytheamplitudesineachcolumn shouldbecompared.Ideally,thisdrivematrixshouldbediagonalanditisuptoatmostafactor oftwo. 6.2SensingMatrixStability Astheinterferometerconditionschange,sodoesthesensingmatrix.Theinverseofthe sensingmatrix,theinputmatrix,ishardcodedinthedigitalcontrolservoandnotactively updated.Therefore,itcanbeexpectedthattheASCperformancemaynotbestable. Bydesign,theinputmatrixisnotexactlytheinverseofthesensingmatrix,meaningthe systemisnotcompletelydiagonal.Forexample,theinputmatrixtimesthesensingmatrixis,by 2 TheopticallevercalibrationprocedureandmeasurementresultsarefoundinAppendix B.2 96

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design: " " # 0 7900 3000 01 0000 001 000 0001 00 00001 0 $ % % % % % % % % % % & (63) Overtime,thesensingmatrixchangesenoughthatthesystemisevenlessdiagonal.Only10 minutesafterhavingmeasuredandcreatedtheinputmatrix,theproductoftheinputmatrixwith anewlymeasurednewsensingmatrixis: " " # 0 7900 3200 01 0000 # 0 0201 000 0 # 0 0201 00 02 00 0200 051 03 $ % % % % % % % % % % & (64) Afteroneweek,itis: " " # 0 6700 2700 00 9100 03 # 0 03 0 0900 8400 00 1500 830 27 00 0600 071 04 $ % % % % % % % % % % & (65) Andafterthreeweeks,itis: " " # 1 400 5500 01 80 # 0 090 15 # 0 0802 000 0 # 0 2201 40 64 00 300 # 0 722 9 $ % % % % % % % % % % & (66) 97

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Despitethesesignicantchanges,theinterferometerremainedstableandthesensitivity remainedconstant.ThisshowsthattheASCisaveryrobustsensingandcontrolsystem. 6.3InputBeamMotion ThebeamcenteringandQPDservosoperateuptoonlyabout50mHz,meaningthebeamcenteringdegreeoffreedomisuncontrolledathigherfrequencies.Becausebeamspotmotionon thetestmassescouplestoDARM,anythingthatcausesthebeam'spositiononthetestmassto changeontimescalesfasterthanhalfaminutebecomesitselfadirectnoisesourceforDARM. TheHAMseismicisolationtablesfromwhichtheinputopticsaresuspendedhaveresonant "stack"modesfromabout0.8Hzto3Hz.Theexcesstablemotionatthesefrequenciesis transmittedtotheMMTs.Jitteronthepointingoftheinputbeamisthusaprimarycontenderfor beamspotmotiononthetestmasses. Thewavefrontsensorservosarethemechanismbywhichinputbeammotionisimpressed onthetestmasses;theyareresponsible,amongotherthings,formakingtheinterferometerfollow theinputbeamuptoseveralHz.TheWFSdetectdifferencesbetweentheangleofthecavity(as determinedbytheanglesofthemirrors)andtheangleofthebeamimpingingthecavity.Ifeither theinputbeamorthecavityanglechanges,theWFSwillmovethemirrorstocorrectforthe anglemismatch.Thus,evenifthemirrorsareperfectlyquiet,anon-stationaryinputbeamwill resultinmirrormotionandmirrormotioninturncreatesbeamspotmotionasseenfromEq. 51 Imeasuredtheimpressionoftheinputbeammotiononthemirrorsbyincreasingthe gainofthecommon-degree-of-freedomWFSservos(cHard,cSoft,RM)forabout10minutes. Comparingtheamountofangularmotionofthemirrorsaswitnessedbytheopticalleversfrom thistimeofhighcommonWFSgaintoatimewithnominalWFSgainandsimilarseismic motion,wecanseetheeffectdirectly.Figure 6-1 showscomparisonspectra,demonstratinghow thereishighertestmassmotionaround1HzwhenthecommonWFSgainsarehigher.Therms mirrormotionalsoincreasesbyabout20%. ItispossiblefortheextramirrormotiontoresultfromgainpeakingoftheWFSservos. However,aplotoftheWFSerrorsignalsduringthetimeofnominalgainandhighgainshows 98

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0.5 1 2 3 4 5 10 9 10 8 10 7 ETMX [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 9 10 8 10 7 ITMX [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 9 10 8 10 7 ETMY [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 9 10 8 10 7 ITMY [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 9 10 8 10 7 RM [rad/rtHz] frequency [Hz] high gain nominal Figure6-1.Inputbeammotionimpressiononthecoremirrors(pitch).Residualmirrormotionas witnessedbytheopticalleverswhenthecommonWFSgains(cSoft,cHard,RM)are increasedto2 5 nominaliscomparedtoresidualmirrormotionwhentheWFS gainsarenominal.Dashedlinesaretheroot-mean-squareoftheamplitudespectral densityintegratedfromtheright.Bothspectracomefromatimeofsimilarseismic activity(typicalweekdayafternoonnoise),showninFigure C-6 99

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0.5 1 2 3 4 5 10 10 10 9 10 8 10 7 10 6 Differential soft [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 10 10 9 10 8 10 7 10 6 Common soft [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 10 10 9 10 8 10 7 10 6 Differential hard [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 10 10 9 10 8 10 7 10 6 Common hard [rad/rtHz] frequency [Hz] high gain nominal 0.5 1 2 3 4 5 10 10 10 9 10 8 10 7 10 6 RM [rad/rtHz] frequency [Hz] high gain nominal Figure6-2.ComparisonofWFSerrorsignals(theresidualmotion)duringatimeofnormal operationandatimewhenthecommonWFSgainswere2 5 higherthannominal. Thisexcludesgainpeakingasacauseoftheextramirrormotionwitnessedduringthe timeofhighgain(Figure 6-1 ).Dashedlinesaretheroot-mean-squareofthe amplitudespectraldensityintegratedfromtheright.Figure C-6 showstheground motionspectraatthetimeofthismeasurement. 100

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thatthereisnot,infact,anyevidenceofgainpeaking.ThecommonWFSspectrainFigure 6-2 donotshowanyextranoisewhentheirgainsarehigher.Itisworthnotingthatthehigher gainisevidentinthecommonWFSspectrabytheextrasuppressionseenbelow1Hz.Also, thedifferentialWFSspectraareunchanged,asexpected.Itcanthereforebeconcludedwith reasonablecertaintythattheincreaseintestmassmotionbetween1and2Hzduringthistestis indeedduetotheWFSimpressinginputbeammotiononthemirrors. 6.4TheMarginally-stablePowerRecyclingCavity Thepowerrecyclingcavity(PRC)isthelinearcavityformedbytheRMandITMs.Because theradiusofcurvatureofboththeRMandtheITMspointsinthesamedirectionandthewaistis welloutsidetheRayleighrangeofthemirrors,thecavityisgeometricallyunstable.Forexample, initscoldstateatLLOthe g -factorofthecavityis1.00005andatLHOit's1.00003.Thebeam inthePRCisnotspatiallycontainedandthecavityisdegeneratewithrespecttohigherorder modes.TheheatingoftheITMsfromthekilowattsofpowerinthearmcavitiestogetherwiththe ITMthermalcompensationsystem(TCS)servetheroleofmakingthePRCgeometricallystable forinterferometeroperation.TheheatingandcoolingoftheITMsisaverycomplicatedprocess andthereforenotveryprecise,sothevalueofthehotPRC's g -factorisusuallynotconstant. Thechanging g -factorhaspotentiallysevereconsequencesfortheASC.Becauseofits geometry,thepowerbuild-upinthePRCisverysensitivetoboththemirroranglesandthe g factor.Poweructuationisdetrimentalbecausethesignaltonoiseratiosofthesensorsthatprobe thePRClightdegradeduetothepresenceofincreasedjunklightthatcontributesshotnoisebut notsignal.WFS1Q,WFS2I,andWFS2QarethemostsensitivetothePRCbecausetheirsignals arederivedfromthe25MHzsidebands.Theirsensitivitytomirrormotionisthereforesubjectto change.Becauseachievingaatpowerbuild-upinthePRCisadifculttask(toomuchmotion inthePRCisquiteoftenacauseoflocklosswhenmakingmeasurements),wemustupdatethe real-timecontrolsystemtoreecttheirchangingsensitivities.Otherwise,themirrorangleswill notbecontrolledaccurately. 101

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10 9 10 8 10 7 10 6 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RM [rad] PRC power g = 1 0.1 g = 1 0.01 g = 1 0.001 g = 1 0.0001 Figure6-3.Dependenceofpowerbuild-upinthepowerrecyclingcavityonthePRC's g -factor andtheRMtilt.TCSisnecessaryforstabilizingthePRC'sgeometryandtherefore itssensitivitytomirrormotion.Forsimplicity,theITMisassumedstationaryinthese plots. Anestimateoftheexpectedpoweructuationsbasedonthe g -factorandRMmotionisa straightforwardexercisewhenusingEq. C4 andEq. C31 asderivedinAppendix C .Ifwe estimatethe g -factorsoftheRMandITMas g RM = 1 + and g ITM = 1 # (where = 6 10 # 4 forLLOthecoldstate)andapproximatethedistanceofeachmirrortothecavitywaistas z becausethetwomirrorsareveryclosetoeachothercomparedtothewaistlocation,thenEq. C4 reducesto: # a PRC PRC $ % & = # z ( 2 + ) / z ( 2 # ) / # 1 / # 1 / $ % & # ( RM ( ITM $ % & (67) Figure 6-3 plotsthepowerinthePRCasafunctionof ( RM forseveralvaluesof ,demonstratingthesensitivityofthePRCtotheITMheating.Forexample,thetypicalRMangular 102

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displacementof10 # 7 radresultsina66%powerlosswhenthePRC g -factorisverynearinstabilitywithavalueof1 # 0 0001.Onlyasthe g -factormovesfurtherfrom1doestheangular motionoftheRMhavelessandlessofaneffectonthepowerbuild-up. 6.4.1PowerScaling Thesignalatthewavefrontsensorsisproportionaltotheamplitudeofthesidebands,or thesquarerootofthesidebandpower.Thus,asthePRC g -factorandthereforethepowerinthe recyclingcavitychanges,sodotheWFS1andWFS2opticalgains.Inordertocompensatefor this g -factordependence,wemultiplytheWFS { 1Q,2I,2Q } errorsignalsinreal-timeby 1 P in & NSPOB 350 # 1 / 2 (68) andWFS3IandWFS4Iby1 / P in .NSPOBisthenormalizedsidebandpowerinthePRCas measuredbythe2 f -demodulatedPOBsignal,andthe350isthereferenceNSPOB,treatedas nominal.Thus,duringinterferometeroperation,allWFSsignalsarenormalizedtoinputpower andarenotdependentonthePRCpower.ThiscorrectiontotheWFSsignalsiscalledpower scaling. ToverifythattheWFSopticalgainsdoindeedscalewiththesidebandpowerasexpected, ItrackedtheWFSopticalgainas g changes.Iexcitedthreeofthetestmasses(ETMX,ITMX, RM)atthreedifferentfrequencies(9.7Hz,10.7Hz,and11.7Hz,respectively)duringafull interferometerlockandchangedtheTCSsettingssothatoverthecourseof15minutesthe g -factorsteadilychanged.DemodulatingeachoftheWFSsignalsateachofthethreeexcitation frequenciesasafunctionoftimeshowshowthestrengthofthesignalattheWFSduetothe motionofthesethreemirrorschanges.Tocompensateforthedifferenceinpendulumresponses totheexcitations,Imultipliedthedemodulatedsignalsforaparticularexcitation f by ( f / 9 7 ) 2 Ialsonormalizedtheresponsebythephaseofthemirror'smotionaswitnessedbytheoptical levers. TheresultsareshowninFigure 6-4 ,andincludeaplotofhowNSPOBchangedovertime. Asexpected,WFS1Q,WFS2I,andWFS2QshowdependenceonthePRCpower,andtherefore 103

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0 5 10 15 200 250 300 350 time [min] PRC power [a.u.] 220 240 260 280 300 320 340 0.2 0 0.2 0.4 0.6 0.8 1 WFS1Q [V/urad] PRC power [a.u.] ETMX ITMX RM 220 240 260 280 300 320 340 0.2 0 0.2 0.4 0.6 0.8 1 WFS2I [V/urad] PRC power [a.u.] ETMX ITMX RM 220 240 260 280 300 320 340 0.2 0 0.2 0.4 0.6 0.8 1 WFS2Q [V/urad] PRC power [a.u.] ETMX ITMX RM 220 240 260 280 300 320 340 0.2 0 0.2 0.4 0.6 0.8 1 WFS3I [V/urad] PRC power [a.u.] ETMX ITMX RM 220 240 260 280 300 320 340 0.2 0 0.2 0.4 0.6 0.8 1 WFS4I [V/urad] PRC power [a.u.] ETMX ITMX RM Figure6-4.WFSopticalresponsetotestmassmotionasafunctionofpowerrecyclingcavity geometry.WFS1Q,WFS2I,WFS2Qaremoresensitivetotestmassmotionasthe powerintherecyclingcavityincreases.Therefore,toachieveadependablefeedback system,wescaletheerrorsignalsinreal-time,forcingtheirresponsestobeatwith power.ThisrangeofPRCpowerislowfornormaloperations. 104

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the g -factor.TheWFS3andWFS4sensingelementsareat.Fittinglinestoeachofthetracked elements,wendagoodtwiththeexpectedpowerof1 / 2dependence. 6.4.2SidebandImbalance AnotherimportanteffectofthePRContheASCsignalsisthebalancingoftheupperand lower24.4MHzsidebandamplitudes.SPOBistheproductoftheamplitudeoftheupperand lowersidebands,butthetotalsidebandpoweristhesumofthepowerinthelowersidebandand thepowerintheuppersideband.Therefore,iftheupperandlowersidebandsarenotthesame, SPOBdoesnotaccuratelyrepresentthepowerinthecavity.ThiscreatesinaccuraciesintheWFS powerscaling. Wesetupatemporaryopticalspectrumanalyzeratapickoffoftheanti-symmetricport beaminordertomeasureamplitudesofthesidebands.WithoutanyTCS,wesawthatwith lessthan6Winputpower,thelowersidebandissmallerthantheuppersideband,at6Wthe amplitudesareequal,andabove6W,theuppersidebandissmallerthanthelower.Thus,ifTCS isnottunedperfectlyatalltimes,wecanexpectunequalsidebands. 6.5WFSServoOpenLoopTransferFunctions Theopenlooptransferfunction,oropenloopgain(OLG),isaninformativemeasureofthe characteristicsofacontrolservo.Itistheproductofeachoftheelementsoftheloop,andisoften summarizedasbeingtheproductoftheplant, H ,withthecontrollters, G : OLG = HG (69) Inourcase,theplantistheradiation-pressure-modiedpendulumtransferfunction(Eq. 515 ) andthecontrollters(foundinAppendix C.6 )arethedigitalltersbetweentheinputandoutput matrices.TheamplitudeoftheOLGtellsthesuppressionthecontrolloopprovidesandthephase tellsaboutthestabilityofthesystem. Wemeasuretheopenlooptransferfunctionwhiletheloopisclosed.Thisisdoneby injectingaswept-sineexcitationjustbeforethedigitalltersandtakingtheratioofthesignals justbeforeandjustaftertheexcitation.Figure 6-5 showstheopenlooptransferfunctionsof 105

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10 1 10 0 10 1 50 0 50 100 magnitude [dB] Differential soft Common soft Differential stiff Common stiff RM 10 1 10 0 10 1 250 200 150 100 50 0 phase [deg] frequency [Hz] Figure6-5.Openloopgains(pitch)ofthe5WFSloopsasmeasuredwith6Winputpower. eachofthewavefrontsensorloopsasmeasuredduringa6Wlock.Asanticipatedfromthe largedifferentialsoftsignalseenbyWFS1inthesensingmatrixmeasurement(Table 6-1 ), thatisthemodeforwhichwecananddoprovidethestrongestsuppression.However,itisalso conditionallystable(duetoapoleatzerothatisengagedaftertheservoisturnedon),asseenby thephasedroppingbelow # 180 ( attwodifferentfrequencies.Theunitygainfrequency(UGF) mustbeheldbetweenroughly2Hzand10Hz.Asshownhere,thedSoftUGFisat5Hz.Forthe otherdegreesoffreedom,theUGFsareallbetween0.8and1Hz.Eachloophasaphasemargin of30 ( to40 ( Figure 6-6 showsmeasurementsofhowthedSoft(WFS1)OLGchangeswithpower.The 6WmeasurementdataisthesameasthatpresentedinFigure 6-5 .From1Wto14Winput powerthesuppressionprovidedbythecontrolloopremainsconstant,butthestabilityoftheloop 106

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10 1 10 0 10 1 20 0 20 40 60 80 magnitude [dB] 1W 6 W 10W 14W 10 1 10 0 10 1 250 200 150 100 phase [deg] frequency [Hz] Figure6-6.Openloopgains(pitch)ofthedifferentialsoft(WFS1)loopasmeasuredatfour differentpowers. changes.ThephasemarginattheUGFdecreasesbyabout10 ( andathird # 180 ( phasecrossing appearsatabout500mHz.Thedigitalltersarexedwithpower,sotheobservedchangesare duetoachangingplant.Thisispreciselytheresultoftheradiationpressureangularspring.I presentthedirectmeasurementoftheplant'smodiedtransferfunctioninSection 6.9 6.6ResidualAngularMotion TheresidualbeamspotmotiononthetestmassesisshowninFigure 6-7 .Weseethe rmsbeamspotmotionontheETMsis1mmandontheITMsitis0.8mmwhichmeetsthe requirementsofSection 4.1 .Thebeamspotmotioncalibrationmethodandresultsarepresented inAppendix B.1 107

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10 2 10 1 10 0 10 4 10 3 pitch [m/rtHz] frequency [Hz] ITM ETM 10 2 10 1 10 0 10 4 10 3 frequency [Hz] yaw [m/rtHz] ITM ETM Figure6-7.BeamspotmotionontheITMsandETMsduringa16Wlockatnight.Ground motionatthetimeofthismeasurementisshowninFigure C-7 108

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WequantifytheeffectoftheASContheradiationpressureeigenbasisdegreesoffreedom bycomparingspectraoftheresidualeigenbasismotionduringlocktotheequivalenteigenbasis motionaswitnessedbytheopticalleversoutoflock.Thecomparisonscannotbeperfectbecause thespectraarenecessarilytakenatdifferenttimesandthereforewithdifferentseismicnoise conditions.However,theeffectoftheASCisstarkatlowfrequencieswheregainishigh,asis seeninFigure 6-8 .At0.01Hzangularmotionofalldegreesoffreedomissuppressedbyatleast oneorderofmagnitude.Thetypicalresidualrmsangularmotionis10 # 7 rad/ & Hz.Theeffectof thehighgainforthedifferentialsoftdegreeoffreedom,inparticular,isseenhere,whereorderof magnitudesuppressionisseenalreadyat1Hz. Themagnitudesofthebeamspotmotionandtheresidualmirrormotionareconsistentand reasonable.Forexample,for10 # 7 radofsoftorhardmodemotioninonearm,themaximum cavitytiltanddisplacementare0.12radand1.02mm,respectively,asfoundinTable 5-1 Figure 6-9 showsthesamedataasFigure 6-8 exceptinthemirrorbasisinsteadofthe radiationpressureeigenmodebasis.TheASCon/offcomparisonofmirrormotionisinteresting becauseitshowsthatthemirrorsactuallymovemorewithrespecttothegroundwhentheyare controlledbytheASCthanwhentheyarenotcontrolledbytheASC.Thisistobeexpected becausethegroundmotionisdifferentateachmirrorandthejoboftheASCistocontrolthe motionsofthemirrorswithrespecttoeachother,notwithrespecttotheground. 6.7ASCtoDARMNoiseBudget OneofthemostimportantguresofmeritfortheASCishowmuchnoiseitcontributes toDARM.AsintroducedinSection 4.1 ,thecombinationofbeamspotmotiononthemirror togetherwiththeangularmotionofthemirrorcreatesanangletolength(A2L)coupling.Aslong asthelengthdisplacementduetothiscouplingiswellbelowthedesireddisplacementsensitivity, theASChasdoneagoodenoughjob.However,beingofthemostcomplexofinterferometer systems,theASCdoesinfactlimitthestrainsensitivity. Tomeasurethecoupling,abroadbandnoiseinjectionratherthanatypicalswept-sine injectionisnecessarybecauseofthenon-linearityoftheA2Lprocess(inparticular,itisthe 109

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10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 Differential soft [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 Common soft [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 Differential stiff [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 Common stiff [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 RM [rad/rtHz] frequency [Hz] no ASC with ASC Figure6-8.Demonstrationofangularmotionsuppressiondownto4mHzduetotheASC.The backgroundmotion("noASC")istheRPeigenbasisreconstructionofopticallever signalswhentheinterferometerisnotlocked.Dataaretaken45minutesapart,and thegroundmotionsareshowninFigures C-4 and C-5 .Thedifferencesinground motionexplainsthediscrepanciesbetween1Hzand10Hz. 110

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10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 ITMX [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 ITMY [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 ETMX [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 ETMY [rad/rtHz] frequency [Hz] no ASC with ASC 10 2 10 1 10 0 10 1 10 2 10 10 10 8 10 6 RMmir [rad/rtHz] frequency [Hz] no ASC with ASC Figure6-9.IndividualmirrormotionwithandwithouttheASC.Themirrorsmovemorewith respecttotheirlocalgroundswhentheinterferometeriscontrolledthanwhenthey're ontheirown.Dataaretaken45minutesapart,andthegroundmotionsareshownin Figures C-4 and C-5 111

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convolutionofthebeamspotandmirroranglespectra).Inaddition,forthemeasurementitself tobelinear,theamplitudeofthenoiseinjectionmustbeonlylargeenoughforaneffecttobe observed;itmustnotoverwhelminglydominate.Therefore,forbothDARMandtheappropriate ASCchannel,Isubtractedquadraticallyaquiet,no-noisespectrum, q ,takenimmediatelyprior tothemeasurementfromthespectrumwithnoise, n ,soastonotassumethattheentiretyofthe observedeffectisduetotheexcitation: N = n 2 # q 2 (610) ThetransferfunctionfromtheASCtoDARMisthen: ASC DARM = N DARM N ASC (611) whichcanbemultipliedbyanASCsignalatanytimetodetermineanoisebudget. TherearetwoplacesintheASCloopfromwhereImadetransferfunctionmeasurementsby puttingina40to110Hzbroadbandexcitation:thesuspensionangularcontrolpoint, 3 andthe WFSerrorpoint. 4 Eachprovidesultimatelythesameinformation,butallowstheASCnoisesto bebrokendownindifferentbases.TheviewusingtheWFS(radiationpressureeigenmode)basis isusefulforcommissioningpurposes,whereastheopticbasisisbettersuitedforincludingthe ASCnoiseinfullinterferometernoisebudgetplots.Theprimaryreasonsarethatthesuspension controlchannelisrecordedtodiskanditalsoservesasthetransferfunctionfortheopticallever noisetoDARM.Figure 6-10 showsthesetofWFStoDARMmeasuredtransferfunctions,asan example.TheunitsareDARMmetersperWFSerrorpointdigitalcounts,sothatthedigitalWFS errorsignalmaybemultipliedbythetransferfunctionatanytime. Thenoisebudgetforbothpitchandyawatatimewhentheinterferometerwaslockedwith 14WinputpowerisshowninFigure 6-11 fortheopticsbasisandinFigure 6-12 fortheWFS 3 i.e. L1:SUS-ETMX ASCPIT EXC 4 i.e. L1:ASC-WFS1 PIT EXC 112

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40 50 60 70 80 90 100 110 10 19 10 18 10 17 10 16 frequency [Hz] amplitude [m/ct] dSoft data 9.2564e 09 f ^ ( 5.128) cSoft data 1.5724e 09 f ^ ( 4.8595) dHard data 2.4572e 09 f ^ ( 4.8871) cHard data 5.3343e 06 f ^ ( 6.8643) Figure6-10.ASCtoDARMtransferfunctionforfourofthevewavefrontsensorloops.The RMtoDARMtransferfunctioncouldnotbemeasuredbecausethecontributionis toosmall.ThettedcurvescanbemultipliedbytheWFSerrorsignalsatanytime tocalculatetheASCnoisecontributiontoDARM. basis.Eachoptic'scontributiontoDARMisthesamewithinaboutafactoroftwo,exceptfor theRM,whichisnotincludedintheseplots.Wewerenotabletomeasurethetransferfunction forRMmotiontoDARMbecausesolargeofanexcitationwasrequiredtoseeaneffectthatthe interferometerwouldloselock.IprovedthisindicationthattheRM'scontributiontoDARMis indeedinsignicantbyobservingnochangeinDARMuponturningofftheWFS4cut-offlters. IntheWFSbasis,thesoftmodescontributemoretoDARMthanthehardmodes. Becausetwodifferentangularcontrolsignals,theWFSandtheopticallevers,independently contributetothesuspensionangularcontrol,wecanseparatetheircontributionsinthenoise budget.Furthermore,wecomputethequadraturesumofthepitchandyawcontributions, assumingthesetwodegreesoffreedomarede-coupled,thuscreatinganupperlimitfortheASC 113

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40 50 60 70 80 90 100 110 10 22 10 21 10 20 10 19 10 18 10 17 10 16 frequency [Hz] amplitude [m/rtHz] ETMX ETMY ITMX ITMY total noise DARM A 40 50 60 70 80 90 100 110 10 22 10 21 10 20 10 19 10 18 10 17 10 16 frequency [Hz] amplitude [m/rtHz] ETMX ETMY ITMX ITMY total noise DARM B Figure6-11.OptictoDARMnoisebudgetduringa14Wlock.A)Pitch.B)Yaw. 114

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40 50 60 70 80 90 100 110 10 22 10 21 10 20 10 19 10 18 10 17 10 16 frequency [Hz] amplitude [m/rtHz] dSoft cSoft dHard cHard total noise DARM A 40 50 60 70 80 90 100 110 10 22 10 21 10 20 10 19 10 18 10 17 10 16 frequency [Hz] amplitude [m/rtHz] dSoft cSoft dHard cHard total noise DARM B Figure6-12.WavefrontsensortoDARMnoisebudgetduringa14Wlock.A)Pitch.B)Yaw. 115

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10 2 10 3 10 22 10 20 10 18 10 16 10 14 frequency [Hz] amplitude [m/rtHz] DARM WFS noise optical lever noise Figure6-13.TotalWFSandopticallevernoisecontributiontoDARMduringa16Wlockat night.Pitchandyawcontributionsareaddedinquadratureundertheassumption theyarede-coupled.Seismicspectraatthetimeofthismeasurementarefoundin Figure C-8 contributiontotheDARMnoisebudget.Figure 6-13 showsthenalsummaryofASCnoisein DARMfora16Wlockatnight. Theimportantmessageisthattheangularsensingandcontrolis,infact,alimitingnoise sourceforfrequenciesbetween20and55Hz.TheASCbecomeslessandlessofaprimarynoise sourceasfrequencyincreases,andby100HztheASCnoiseoorisafactorof10belowDARM. Thespecicstructureofthenoisecontributions,includingtheapparentnotches,isadirectresult oftheshapeofthecontrollters.ImperfectionsintheestimateofDARMbelow50Hzarise becausethetransferfunctionisnotperfectlystableintime.Theseismicnoisecontributionto DARM(notshown)doesinfactsitjustbelowtheASCoor,sotheinterferometersensitivityto GWsisnotdramaticallyhinderedbytheASC.Anexample,however,ofhowtoreducetheASC noiseoor,isfoundthroughevaluatingtheWFScontrollters. 116

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20 30 40 50 60 70 80 90 100 10 19 10 18 10 17 10 16 10 15 10 14 frequency [Hz] displacement sensitivity [m/rtHz] S5 best WFS1 cutoff f = 35 Hz WFS1 cutoff f = 30 Hz Figure6-14.EffectoftheWFS1lowpassltercutofffrequencyonstrainsensitivity. Thecut-offfrequencyofthelowpassltersfortheWFScontrolareofparticularimportance intheDARMnoisebudget.Thelowpasslterisnecessaryforsuppressingtheimpressionof sensingnoiseonsuspensioncontrolsignals.Steepeningthecut-offfrequencyresultsinless sensingnoiseimpression,buteachpoleusedtoachievethesteeperdrop-offintroducesanextra 90 ( ofphaseloss.Likewise,loweringthecut-offfrequencyreducesnoiseimpression,butit pushesthephaselosstolowerfrequencies.Theeffectisadecreaseinphasemarginofthe WFSloopsforaparticularUGF,whichleadstogainpeakingandagreaterlikelihoodofloop instability.Anebalancemustthereforebefoundbetweenloopstabilityandnoiseimpression. BecauseWFS1(differentialsoft)isthedominantcontibutortothenoisebudget,weputforth efforttotuneitslowpasslter'sfrequency.Figure 6-14 demonstratestheeffectonDARMof decreasingtheWFS1cut-offlterfrequencyfrom35Hzto30Hz. 117

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6.8SeismicFeed-forwardtotheASC WeshowinSection 6.7 thatangularcontrollimitsthestrainsensitivityatlowfrequencies andweseefromFigure 1-2 thatAdvancedLIGOaimstoimprovethelowfrequencysensitivity byuptotwoordersofmagnitude.Althoughmuchoftheimprovementcomesfrombetter seismicisolation,itisprudenttodevelopmethodstoreducetheangularcontrolnoiseforfuture interferometers.ApromisingtechniquethatIinvestigatedusingEnhancedLIGOdataisseismic feed-forwardtotheASC. TheconceptistouseWienerltering[ 57 ]topredicttheASCsignalsfromseismometer signalsandfeed-forwardthelteredsesimometerdatainrealtime.TheWienerltercoefcients arechosentominimizethemeansquaredifferencebetweentheWFSerrorsignalsandthe lteredseismometersignals.Forthistowork,theremustbeacorrelationbetweentheseismic andASCsignals.IdesignedFIRWienerltersfromsimultaneousseismometerandASCerror signaltimeseriesandfoundthatwithasufcientlylongWienerlter(ontheorderof1024 taps)and30minutesworthofdata,wecanaccuratelypredicttheASCerrorsignalfrom0.1to 20Hz.Figure 6-15 showsthecomparisonofthenominaldSoft(WFS1)errorsignalwiththe reducederrorsignalthatonecanhopetoachievethroughfeed-forward.Calledtheresidual,this reducederrorsignalistheresultofsubtractingtheseismometer-predictedASCerrorsignalfrom theoriginalerrorsignal.Weseethatifimplemented,feed-forwardmayreducethermsangular mirrormotionbyafactoroftwo.Furtherstudiesmayrevealevenagreaterreductioncanbe achieved.Oneaspectoffeed-forwardthatstillrequiresinvestigationisthestationarityofthe Wienerlterovertime. Thereareseveralwaysinwhichfeed-forwardmightbeaccomplished.Thetwoprimary optionsaretosendtheseismometer-predictedASCdrivetoeitherthemirrorcoilsortothe hydraulicexternalpre-isolator(HEPI)seismicisolationtables.Thebenetofbothofthese methodsisthattheywouldphysicallyreducethemirrormotionandthereforereducetheneed forASCfeedback.Thiswouldallowformoreexibleloopshapedesignsuchaslowercut-off ltersandresultinlessimpressionofASCnoiseinDARM.Thesimplestofthesetwooptionsis 118

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10 2 10 1 10 0 10 1 10 11 10 10 10 9 10 8 frequency [Hz] dSoft [rad/rtHz] original residual Figure6-15.DemonstrationofpotentialreductionofWFSerrorsignalsusingseismic feed-forward. sendingthesignaltothecoilsbecausethetransferfunctionfromthecoilstothemirrormotion iswellknown.Adisadvantageisthatthetotalcoilcurrentswouldnotbereduced,meaningthe noisefrommagneticdomainipping,Barkhausennoise,wouldnotbereduced.Feeding-forward toHEPIwouldreducetheBarkhausennoise,butwouldrequireacarefullymeasuredtransfer functionfromHEPItoangularmirrormotion.Seismicfeed-forwardtoHEPItoreduceLSC signalswasdemonstratedduringEnhancedLIGO[ 66 ]. AnalternativemethodtoreducingtheASCnoiseinDARMdoesnotinvolvetheseismometersorfeed-forwardtotheASC,butfeed-forwardtoDARMitself.Fromacarefullymeasured ASCtoDARMtransferfunction,thepredictedcontributionofASCnoiseinDARMcanbe subtractedfromDARMinrealtime.Thistypeoffeed-forwardwasimplementedinInitialand EnhancedLIGOfortheMichelson(MICH)andpowerrecyclingcavity(PRC)loops[ 67 ,Ch.2]. Anyofthesemethodsmaybeimplementedinpost-processingoftheDARMdata,butthere arebenetstodoingitinreal-time.Theprimaryadvantageistheactualreductionofmirror 119

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Figure6-16.Demonstrationofradiationpressureeigenbasistorquetoangletransferfunction measurement.ThroughaproperchoiceofmeasurementlocationswithintheASC servo,theplant'stransferfunctioncanbesingledout. motionwhichwouldmakeforanall-aroundmorestableinterferometer.Second,eliminating knownnoisesfromthestrainspectrumisveryusefulforcommissioning.Itallowsustothensee othernoisesinreal-timeandtobemoreefcientattheendlessgameofnoisehunting. 6.9ExperimentalMeasurementoftheRadiationPressureAngularSpring Idirectlymeasuredtheexpectedradiation-pressure-modiedtorquetoangletransferfunctionoftheLIGOarmcavitiesthatIrstintroducedinChapter 5 .Throughaclevermeasurement method,Iwitnessboththestableandunstablemodeswithoutanypost-data-takingmanipulation. Thedigitalcontrolsysteminwhichtheangularcontrolfeedbacksystemisimplemented providesaconvenientmilieuinwhichtomeasuretheresponseoftheoptomechanicalsystem. Byinjectingadisturbancesomewhereintheloopandmeasuringtheresponseatselectedpoints intheloop,wecanproduceameasurementoftheoptomechanicalsystemthatisnotsensitive tothedetailsofthecontrolsystem.Hereweusethissystemtoproducemeasurementsofthe opto-mechanicalplantatseveraldifferentoperatingpowers,demonstratingthemodicationsdue 120

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toradiationpressure,i.e.thesoftandhardmodes,sometimesalsoreferredtoastheSidles-Sigg effect. ThevariouselementsoftheplantandthecontrolsystemaredepictedinFigure 6-16 .For thismeasurement,Itooktransferfunctionsfromthetorqueinputtotheresultingangulardisplacement(asmeasuredbytheWFS),bothintheradiationpressureeigenbasis.Simultaneously, Iinjectedanexcitationintothecontrollegoftheservoloop,asisdoneforthesensingmatrix mesaurementdescribedinSection 6.1.1 .Theresultingmeasurementreproducesthetransfer functionoftheopto-mechanicalplant,independentofthecontrolsystem. ResultsareshowninFigures 6-17 (hardmode)and 6-18 (softmode);least-squarests ofsecond-ordertransferfunctionsaremadetothedata.Inthehardmodeplot,wecanclearly seetheincreaseoftheresonantfrequencywithpower,from 0 65Hzat1Winputpowerto 0.95Hzat10Winputpower.Simultaneously,inthesoftmodeplot,weseetheresonance decreaseinfrequencyasthepowerisincreasedfrom1Wto6W.Whentheinputpoweris increasedto10Wandbeyond,theresonancedisappears;theplanthasbecomestatically unstable. ThesemeasurementsshowaclearconrmationoftheSidles-Siggtheoryanddemonstratea successfulpower-independentdiagonalizationofthesensingandcontroloftheopto-mechanical plant. 6.10Summary TheASCperformedasitneededtoforEnhancedLIGOtooperateathigherpowerswithout introducinganexcessofcontrolnoise.TheASCdoeslimitstrainsensitivityatthelowestendof thedetectionband,butweofferpossiblesolutionstodecreasethatnoiseinfuturedetectors.We directlymeasuredtheradiationpressureangularspring,conrmingourtheoretcialunderstanding ofthebasicphysicsthatdrivestheASCdesign.AlthoughtheASCinneitherbasewaslimiting us,theexperiencegainedforAdvancedLIGOispriceless.AdvancedLIGOwillhaveheavier mirrorswithadifferentgeometrysuchthatradiationpressuretorquewillnotplaysolargearole [ 68 ]anditspowerrecyclingcavitywillbestable[ 18 ]. 121

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10 1 10 0 10 1 10 2 10 1 10 0 10 1 10 2 magnitude [rad/Nm] frequency [Hz] 1 W 6 W 10 W 10 1 10 0 10 1 225 180 135 90 45 0 45 frequency [Hz] phase [degrees] Figure6-17.Hardopto-mechanicalmodemeasurementandtforseveralpowers. 122

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10 1 10 0 10 1 10 2 10 1 10 0 10 1 10 2 magnitude [rad/Nm] frequency [Hz] 1 W 6 W 10 W 14 W 10 1 10 0 10 1 225 180 135 90 45 0 45 frequency [Hz] phase [degrees] Figure6-18.Softopto-mechanicalmodemeasurementandtforseveralpowers. 123

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CHAPTER7 CONCLUSIONS 7.1HigherpowerinEnhancedLIGO ThebroadresultoftheInputOpticsandAngularSensingandControlupgradesalongwith theworkofmanyotherpeopleonmanyothersubsystemswassuccessfuloperationofboththe HanfordandLivingstoninterferometersatthehighestoflaserpowerseverused.Thetypical inputpowersusedduringInitialLIGOwerebetween5and7WandduringEnhancedLIGO typicalinputpowerswerebetween8and20W,asshownbythehistogramsinFigure 7-1 AsnapshotofhowtheinterferometerreactstoincreasesinlaserpowerisfoundinFigure 7-2 ,whichshowsthetimeseriesofseveralchannelsduringatypicallocklossandre-lockofthe interferometerfollowedbyanincreaseininputpowerto14W.Theincreaseintheuctuations ofNSPOB,thesidebandpowerinthePRC,priortothelocklossat t = 250secindicatesPRC instability.TheMCregainslockalmostimmediatelyandthepowerisincreasedto1W.The denseregionofNSPOBsignalshowstheashesofresonanceastheinterferometertriesto acquirelock,whichitultimatelydoesataround t = 500sec.Thelaserpowerissubsequently increasedto8W,andpulsesofTCSpowerturnedontohastentheITMcoolingprocessin preparationforthelaserpowerincreaseto14W.Allalong,thepowerinthearmcavities increasesaswellasthatreectedfromtheinterferometer.NSPOBisabitmore"hairy"at14W thanitisat8W,butthislockisotherwisestableandprovidesaneutronstarbinaryinspiralrange of15Mpcandlastsfor5hoursuntilanearthquakekillsit. Theimprovementinstrainsensitivityintheshot-noise-limitedregionduetooperatingat higherlaserpowerisseeninFigure 7-3 .ThereisafactoroftwoimprovementfromInitialLIGO toEnhancedLIGOfromabout300Hzon.Thecorrespondingbestneutronstarbinaryrange increasedfrom15to20Mpc.Althoughmorepowerimprovestheshot-noise-limitedregionof thestrainspectrum,itdoesintroducemanycomplicationsforinterferometeroperationthatmust becarefullyaddressed. 124

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0 4 8 12 16 20 1000 2000 3000 4000 LHO science mode hours input power [W] 0 4 8 12 16 20 1000 2000 3000 4000 LLO science mode hours input power [W] Figure7-1.HistogramofinputpowersusedduringS6ateachsite. 500 1000 1500 2000 2500 2 0 2 4 6 8 10 12 14 16 time [sec] [counts] PWRSET NPTRX/100 NSPOB/100 TCSX pwrset TCSY pwrset REFL DC Figure7-2.Timeseriesofinterferometersignalsshowingatypicallockloss(atabout250sec) andre-lock(atabout500sec)followedbyanincreaseofpowerto14W.They-axis isinunitsofWattsfortheinputpower,TCSX,andTCSYtraces.Theothertracesare displayedindigitalcounts.DataisfromJuly23,2009. 125

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10 2 10 3 10 23 10 22 10 21 strain [Hz 1/2 ] frequency [Hz] Initial LIGO Enhanced LIGO Advanced LIGO design Figure7-3.Zoomoftheshot-noise-limitednoiseoorsoftheInitialLIGOandEnhancedLIGO detectors.TheimprovedInputOpticsandAngularSensingandControlenabledthe increaseinpowerforEnhancedLIGO.Theshot-noise-limitedstrainsensitivity improvedbyafactoroftwo. 7.2Summary WedescribedthedesignoftheEnhancedLIGOInputOpticsandAngularSensingandControlsubsystemsandpresentedmeasurementscharacterizingthesystemsandtheirperformances whenoperatingwithrecordlaserpowers.Upgradestothetwosystemswerenecessaryforallowinghigherlaserpowers,forimprovingtheefciencyofsendinglightintotheinterferometer, andforkeepinglightintheinterferometeronceitisthere.Higherpowerintheinterferometer improvestheshot-noise-limitednoiseoor,andwesucceededatoperatingtheinterferometers withmorethantwicethehighestofpowersachievableduringInitialLIGO.TheEnhanced LIGOshot-noise-limitedsensitivitydidindeedreachrecordlevels,improvingthechancesof gravitational-wavedetection. 126

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Inaddition,wedirectlymeasuredthestableandunstableopto-mechanicalmodesof theFabry-Perotarmcavities.Wewitnesstheexpectedeffectofradiationpressuretorque, demonstratingaclearunderstandingofthephysicsthatwillaffectfuturegenerationsoflaser interferometersforgravitational-wavedetection.Furthermore,wesuccessfullycontrolledthe stableandunstableopto-mechanicalmodeswithoutcontaminatingthegravitational-wavereadout inthefrequencyrangeofinterest. 127

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APPENDIXA INPUTOPTICSSUPPORTINGMATERIAL A.1PhaseModulation Phasemodulationmultipliescarrierlightwitheld E 0 e i % t by e i & sin ( $ t ) ,where & isthe modulationindexand $ isthefrequencyofthephasemodulation.UsingtheJacobi-Anger expression, e iz sin ( = ( n = # J n ( z ) e in ( (A1) where J n aretheBesselfunctions,wecanwritetherstfewterms(n=0,1,-1)ofthephasemodulatedeld: E modulated = E 0 J 0 ( & ) e i % t + E 0 J 1 ( & ) e i ( % + $ ) t + E 0 J # 1 ( & ) e i ( % # $ ) t + ... (A2) Weseethatbothanupperandlowerprimarysidebandarecreated,withfrequencies % + $ and % # $ .Phasemodulationdoesproduceaninnitenumberofsidebands,yettheamplitudeofthe Besselfunctiondecaysrapidlywithhigher | n | ,soonlythisrstsetofsidebandsaresignicant. A.2ModeCleanerPole Opticalcavitiesactaslowpassltersforintensityvariationsofthelightsentintothem.The modelforanintensitynoisetransferfunctionofacavityisthatofasinglepole: E after E before = 1 1 + s / s 0 = s 0 s 0 # s (A3) where s isacomplexparameter.However,weareinterestedinonlypurelysinusoidalvariations inintensitysowelet s bepurelyimaginary, s = i % ,where % isanangularfrequency. WemeasuredtheintensitynoisetransferfunctionoftheLivingstonmodecleanerupon completionoftheEnhancedLIGOInputOpticsupgrade.Wemodulatedtheintensityofthelaser lightgoingintotheMCbyinjectingaswept-sineexcitationin L1:PSL-ISS EXC andmeasured thepowervariationofthelightintwoplaces:beforeandafterthemodecleaner.Weuseda singlephotodetector(PDA55)inordertoeliminatethePDresponse,andthereforemadethe measurementtwice.Weensuredtherewas1VDConthePDinbothlocations. 128

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10 2 10 3 10 4 10 5 10 1 10 0 magnitude frequency [Hz] 10 2 10 3 10 4 10 5 100 80 60 40 20 0 phase [deg] frequency [Hz] data fit FigureA-1.Livingstonmodecleanerintensitynoisetransferfunction.Redopencirclesaredata; solidbluelineisasinglepolet.Relatingtparameterstothemodel,thepole frequencyis f p = 4762Hz. Figure A-1 showsthetransferfunctiondataandthet(tobothmagnitudeandphase simultaneously).Thethasapolefrequencyof f p = 4762Hz.The1 / e ringdowntimeofthe modecleaneristherefore ) = 1 / 4 & f MC = 16 7sandthenesseis F = FSR / 2 f p = 1282. A.3GaussianBeamonaSplitPhotodetector ThepowerperareaofaGaussianbeamtravelingalongthe z -axisis p ( x y )= 2 P 0 & w 2 exp & # 2 x 2 w 2 exp & # 2 y 2 w 2 (A4) 129

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where w isthebeamradiusat z .Thishasbeennormalizedsuchthat ( # ( # p ( x y ) dxdy = P 0 Then,forabeamdisplacedby x 0 fromthecenterofasplitphotodetector,thepowerontheleft sideis P left = 2 P 0 & w 2 ) x 0 # exp & # 2 x 2 w 2 dx ) # exp & # 2 y 2 w 2 dy (A5) = % 2 & P 0 w & ) 0 # exp & # 2 x 2 w 2 dx + ) x 0 0 exp & # 2 x 2 w 2 dy (A6) = % 2 & P 0 w w 2 % & 2 + w 2 % & 2 ) & 2 x 0 / w 0 exp + # t 2 dt (A7) = P 0 2 1 + erf & 2 x 0 w -. (A8) whereerf ( t 0 ) 2 & & ( t 0 0 exp + # t 2 dt .Thepowerontherightsideis: P right = P 0 2 1 # erf & 2 x 0 w -. (A9) Wecreateanormalizedyawas YAW = P left # P right P 0 = erf & 2 x 0 w (A10) UsingtheTaylorseriesexpansionoftheerrorfunction,wehavearstorderestimateforthe relationshipbetweennormalizedyawandbeamdisplacement x 0 forabeamofradius w : x 0 YAW $ w 2 % & 2 (A11) Thesameequationholdstrueforpitch. A.4BeamPropagationFormalism FortheinputbeammodelandfortheInputOpticsbeamdriftcalibrations,theABCDmatrix formalismisausefultooltopropagateaGaussianbeam.IchoosetoignorethefactthattheMC beampassesthroughthesubstrateofMC3onitswaytotheFaraday.Ialsotreatthebeamsplitter asaatmirrorandignorethepresenceofitssubstrate.Iusethethicknessofthelargeoptic substrates, t = 0 01m,andaccountforindexofrefractioneffectswhenpassingthroughoptics. 130

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Signsofradiiofcurvaturearedenedperthefrontfaceoftheoptic;forexample,allmainLIGO opticshaveapositive R Forabeamthatstrikesaatinterfaceandexitsatacurvedinterface(ie.forward-going transmissionthroughRM): # AB CD $ % & = # 10 ( n 2 # n 1 ) / Rn 1 n 2 / n 1 $ % & # 1 t 01 $ % & # 10 0 n 1 / n 2 $ % & (A12) Forabeamthatstrikesacurvedinterfaceandexitsataatinterface(ie.transmission throughETM): # AB CD $ % & = # 10 0 n 2 / n 1 $ % & # 1 t 01 $ % & # 10 ( n 1 # n 2 ) / # Rn 2 n 1 / n 2 $ % & (A13) Forabeamthatstrikesaatinterface,travelsthroughthesubstrate,reectsoffthebackofa curvedinterface,travelsthroughsubstrateandexitsattheoriginalatinterface(ie.singlebounce offRM): # AB CD $ % & = # 10 0 n 2 / n 1 $ % & # 1 t 01 $ % & # 10 2 / R 1 $ % & # 1 t 01 $ % & # 10 0 n 1 / n 2 $ % & (A14) Finally,forpromptreectionoffacurvedinterface(ie.reectionoffMMTs): # AB CD $ % & = # 10 # 2 R 1 $ % & (A15) andforpropagationadistance d throughvacuum: # AB CD $ % & = # 1 d 01 $ % & (A16) Foreachofthese n 1 = 1istheindexofrefractionofvacuumand n 2 = 1 44963istheindex ofrefractionofthefusedsilicausedfortheoptics.Table A-1 showstheradiiofcurvatureofeach oftheopticsforbothsites. 131

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TableA-1.Mirrorradiiofcurvatures. opticL1ROC[km]H1ROC[km] MC10.01725 MC280 MC30.01725 MMT16.766.77 MMT23.163.17 MMT325.1625.04 RM15.7814.40 BS-189-336 ITMX14.76013.910 ITMY14.52013.600 ETMX8.7307.260 ETMY8.7207.320 A.5BeamDriftCalibration WeusetheABCDmatrixformulationtoconvertpitchandyawdataofWFS3andWFS4 intoapositionandangleattheFaradayisolator.Thebasicrelationshipbetweenbeamdisplacementandangleatonelocationtodisplacementandangleatanotherlocationisgivenby: # x 1 x 1 $ % & = # AB CD $ % & # x 0 x 0 $ % & (A17) Forthisapplication,wewanttorelatethebeampositionsontheWFS, x 3 and x 4 ,tothe beampositionandangle, x FI and x FI ,attheFaradayisolator.Usingonlythetopequationof Eq. A17 sincetheWFSaresensitivetobeampositiononlyandnotangle,wecanwriteanew relation # x 3 x 4 $ % & = # A 3 B 3 A 4 B 4 $ % & # x FI x FI $ % & (A18) where A 3 B 3 and A 4 B 4 aretheAandBABCDmatrixelementsforthebeampathsfromthe FaradayisolatortoWFS3andWFS4,respectively.Takingtheinverseandwriting x 3 and x 4 asa functionofthepitchandyawrecordedbytheWFS(seeAppendix A.3 ),theusefulequationis # x FI x FI $ % & = # A 3 B 3 A 4 B 4 $ % & # 1 # w 3 0 0 w 4 $ % & 1 2 % & 2 # DOF 3 DOF 4 $ % & (A19) 132

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where w 3 and w 4 aretheradiiofthebeamateachWFSandDOFcanmeanPITorYAW. A.6CarrierMode-matchingintotheInterferometer Whenacavityislockedtoaninputbeam,allofthelightimpingingthecavityiscoupled intoitifthecavityisimpedancematchedandiftheinputbeamandthecavityaremode-matched. Whentheserequirementsarenotmet,thenlightisreectedfromthecavity.Thecavityvisibility isaquantitythatsummarizesthecompoundeffectofthesesourcesofreectedlight. 1 By measuringtheinterferometervisibilityandbythemeasuringtheimpedancemismatch,wecan deducethecarriermode-matchingintotheinterferometer. A.6.1InterferometerVisibility Thevisibilityisameasureofhowmuchcarrierlightisreectedfromthelockedinterferometercomparedtohowmuchcarrierlightissentin.Tomeasurethevisibility,weneedtoknow onlyafewnumbers.Wemusthaveameasureofhowmuchlightissenttotheinterferometer fornormalizationpurposesandwemusthaveameasureoftheDCreectedpowerwhenthe interferometerisbothlockedandnotlocked(alllightisreectedoffoftheRM).Thevisibilityis thengivenby: visibility = 1 # P REFL locked P REFL unlocked P IN unlocked P IN locked (A20) Wehavetwomeasuresofhowmuchlightisbeingsentintotheinterferometer(apick-off ofthelightbeforeitentersthevacuumandapick-offoftheMCtransmittedlight)andseveral ofhowmuchlightisreected.AnexampleshowingsomeofthesesignalsforLLOisshown inFigure A-2 .Thelockstretchendsat t = 5min.Notethattheamountofreectedlightis increasinguptotheendofthelockastheinterferometerislosingstability.Whenlockislost,the commonmodeservokicksthemodecleaneroutoflocktoo,andtheMCtranspowerdropsto0. About15secondslatertheMCrelocksandthenthepowerintoitincreases.About15seconds 1 Whentherearesidebands,asisthecaseforLIGO,thesidebandsarealsoasourceofreectedlight.Wemeasuredthat6%to8%ofthepowerinthereectedbeamduringlockissidebandcontent,andthatthe25MHzsidebandvisibilityis87%. 133

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3 3.5 4 4.5 5 5.5 6 6.5 7 50 0 50 100 150 200 250 300 350 400 time [minutes] arbitrary units interferometer reflected power input power (x10) [Watts] MC transmitted power FigureA-2.Endofan8.7WlockatLivingstononFeb.23,2010.Themodecleanerre-locksat 0.5Wabout15secondsafterlocklossandthenthepowerisincreasedto2W. Comparisonoftheminimumintereferometerreectedpowerduringlockandthe maximumreectedpoweroutoflockprovidesameasureofinterferometermode matching. afterthat,at5.5minutes,theMCWFSturnon,improvingthealignmentoftheMCtotheinput beamandweseeanotherstepinthepowergettingthroughtheMC.Theinterferometerisstill notlocked,soalllight(exceptfor2 7%)isreectedoffoftherecyclingmirror.Thedownward spikesinthereectedlighttracearetheresultofinterferometerashes,instancesofallmirrors liningupcorrectlytoletsomelightin. Forthisparticularexample,thevisibilityis92.1%.Whenevaluatedforasamplingoflock lossesthroughouttheEnhancedLIGOrun,theaveragevisibilityis91.84% 0.07%. A.6.2ImpedanceMatching Acavityisimpedancematchedwhentheinputandoutputcouplershavethesamereectivity.Ifthereisadifferencebetweenthereectivitiesofthetwomirrors,thecavityisover-or under-coupledandlightwillbereected.Treatingtheinterferometerarmsasasinglemirror 134

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thatformsacavitywiththeRM,wewanttheRMtransmissiontomatchthetransmissionofthe arms.Thismustincludealllossesintheinterferometersuchasabsorption,scattering,andETM transmission.DesignestimatesresultedinanRMpowertransmissionof2.7%.Iflossesdonot equal2.7%,thenthereisanimpedancemismatchandwewillseelightatthereectedport. Theamplitudereectivityoftheinterferometeris: r ifo = r arms # r rm 1 # r rm r arms (A21) Thecompositearmcavityamplitudereectivityis r arms andtheRMamplitudereectivityis r rm = & 0 973.Itisnotsosimpletoknowwhat r arms isinpractice.Aprecisemeasureofall lossesinthearmswouldbeneeded.Therefore,weturntowriting r arms intermsofaquantitythat wecanmeasure,thepowerrecycledMichelsoncarriergain G cr : G cr = g 2 cr = & t rm 1 # r rm r arms 2 (A22) Experimentally,therecyclinggainismeasuredas G cr = T rm NPTRX + NPTRY 2 (A23) whereNPTRXandNPTRYarechannelsrecordingtheamountoflighttransmittedthrough theETMs,normalizedsuchthatNPTRX=NPTRY=1duringasinglearmlock.ForLivingston, G cr = 39. Figure A-3 shows R ifo = r 2 ifo asafunctionof G cr .CurvesforacoupledifferentRM reectivitiesareshowntogiveanideaofhowtheinterferometerreectivitywouldchangefor minormis-approximationsoftheRMreectivity.Wendthattheimpedancemismatchfor Livingstonisonly0.07%. A.6.3Mode-matching Anydifferencebetweentheinterferometervisibilityandwhatisexpectedfromimpedance mismatchcanbecontributedtoimperfectmodematching.ForLivingston,theinterferometeris 135

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20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 G cr R ifo T rm = 2.6% T rm = 2.7% T rm = 2.8% FigureA-3.Interferometerreectivityduetoimpedancemismatch.Thepercentageofpower incidentontheRMthatisreectedbytheinterferometerisafunctionofcarrier recyclinggainandRMreectivity.Thecarrierrecyclinggainis39forLivingston. Therecyclingmirrorpowertransmissionisnominally2.7%. nearlyperfectlyimpedancematched,soalllightatthereectedportisduetoimperfectmode matching.Therefore,theLLOmodemismatchduringS6was8%. A.7OverlapIntegrals Ameasureofthemodematchingcanbegivenbytheamountofpowercoupledfromone modeintoanother.Thisiscalculatedasthesquareoftheoverlapintegraloftwoelds, 1 and 2 ,foraparticularz-axis(propagationdirection)cross-section: P = . | / 2 = & ) # ( x ) ( x ) dx 2 (A24) WeareinterestedinthelowestorderHermite-Gaussianmode: ( x z )= u 0 ( x z )= & 2 & w 2 0 1 / 4 & q 0 q ( z ) 1 / 2 exp & # ikx 2 2 q ( z ) (A25) whichcanberewrittenasafunctionof x and q using q 0 = i Im ( q ) ,and w 2 0 = # 2 q 0 i / k : u 0 ( x q )= & # k Im ( q ) & q 2 1 / 4 exp & # ikx 2 2 q (A26) 136

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Eq.( A26 )isnormalizedsuchthat u 0 | u 0 / = 1. Wewanttoknowthesquareoftheoverlapintegralfortwoeldsgivenbydifferent q parametersatonelocation z .First,theoverlapintegral: q 1 | q 2 / = ) # u 0 ( x q 1 ) u 0 ( x q 2 ) dx (A27) = & k 2 Im ( q 1 ) Im ( q 2 ) & 2 q 2 1 q 2 2 1 / 4 ) # exp & # & # ik 2 q 1 + ik 2 q 2 x 2 dx (A28) = & # k 2 Im ( q 1 ) Im ( q 2 ) & 2 q 2 1 q 2 2 1 / 4 / 0 0 1 & 2 # ik 2 q 1 + ik 2 q 2 3 (A29) =[ Im ( q 1 ) Im ( q 2 )] 1 / 4 $ 2 q 1 q 2 4 1 / q 2 # 1 / q 1 5 (A30) Then,thepowerisgivenby: | q 1 | q 2 / | 2 = 2 Im ( q 1 ) Im ( q 2 ) % q 1 q 1 q 2 q 2 2 1 q 2 # 1 q 1 32 1 q 2 # 1 q 1 3 (A31) = 2 Im ( q 1 ) Im ( q 2 ) | q 2 # q 1 | (A32) NotethatEq.( A32 )simpliesto1when q 1 = q 2 asexpectedandthatthiswholeformulation assumesthatthebeamsarepropagatingalongthesame z -axis. 137

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APPENDIXB ANGULARSENSINGANDCONTROLCALIBRATIONS Thetypicalmethodofcalibratingadigitalchannelistoinjectasignalofknownamplitude intothesystemandtaketheratiowiththeamplitudeofthedigitalmeasurementofthesignal.I describeinthisappendixthecalibrationsImadeofsomeoftheangularsensingchannels. B.1BeamSpotMotion AquantityofinterestishowmuchthebeammovesontheITMsandETMs.Itisthis beamspotmotionwhich,togetherwiththemirrorangularmotion,createsalengthsignalthat contributesnoisetoDARM.Anelegantwayoffollowingthemotionofthebeamonthetest massesistotrackpickoffsofthelighttransmittedorreectedfromthemirrors.Wehavesuch signalsnaturallyavailablefortheETMsandITMsfromtheQPDswhichareotherwiseused forASCsensing.Forexample,QPDXandQPDYseethelighttransmittedthrougheachofthe ETMsandWFS2seesthepickoffoflightfromthewedgeofITMX. TocalibratethecountsoftheQPDandWFS2pitchandyawerrorsignals, 1 Imovedthe beamaknowndistanceonthetestmass, x ,andrecordedthecorresponding y oftheQPDand WFS2readback.Theratio x / y isthecalibrationfromcountstometers.Thedetailsofthe procedurearedescribedbelow. B.1.1MovingtheBeam Movingthebeamonthemirrorsinacontrolledfashionisstraightforwardbecauseofthe ASCsystem.AllthatweneedtodoisintroduceanoffsettothesetpointoftheofthebeamcenteringaspectoftheASCservo.FortheETMsweputaDCoffsetinthe L1:ASC-QPD { X,Y } { PIT, YAW } { OFFSET } channelandfortheITMswechangedthe X and Y targetsofthebeamsplitter beamcenteringservo. 1 L1:ASC-QPDY { PIT,YAW } IN1 and L1:ASC-WFS2 DC { Pitch,Yaw } Mon 138

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FigureB-1.GeometryofOSEMsandmirrorasusedforcalculatingthelocationoftheaxisof rotationwhenthetorquesareunequal. B.1.2MeasuringHowMuchtheBeamHasMoved Themoredifculttaskismeasuringjusthowmuchthebeamhasmoved.Forthis,wemake useoftheleverarmmechanismofangletolengthcouplingasexplainedinSection 4.1 .The conceptofthemeasurementistomovetheaxisofrotationofthemirrorsothatitpassesthrough thecenterofthebeam.WeusetheOSEMstochangethelocationoftheaxisofrotation,andwe useDARMtodeterminewhentheaxisisalignedwiththebeamcenter.Forexample,ifwedrive thetoptwoOSEMsmorethanthebottomtwoOSEMs,we'vecreatedanaxisofrotationthatsits belowthecenterofmass.Theresultofsuchtuningisaneffectiverebalanceofthecenterofmass ofthemirrorsothatitisalignedwiththecenterofthebeam.Theprocedureis: Shakethemirroratsomefrequency f (weuse39.5Hz)duringafulllock DemodulateDARMat f forseveraldifferentsetsofOSEMgains FitaquadratictothedemodulateddatatopinpointtheOSEMgainsthatminimizethe couplingtoDARM RelatingtheOSEMgainstoabsolutebeampositiononthemirrorrequiresonlythegeometryofthemirrorandOSEMsetupassketchedinFigure B-1 .WeestimatetheOSEMlocations asbeingontheedgeofthemirrorsuchthatthelength d ofonesideofthesquarethattheyform isgivenby d = & 2 R ,where R = 12 5cmistheradiusofthemirror.Then,collapsingthefour OSEMsintoarepresentativetwoatthecentersoftwooppositesidesofthesquareandassigning 139

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TableB-1.CalibrationstobeusedwiththeQPDX,QPDY,andWFS2DCpitchandyawerror signalsforameasureofbeamspotmotion. ETMXETMYITMs pitch1 03 10 # 5 m/ct1 21 10 # 5 m/ct5 52 10 # 2 m/ct yaw0 88 10 # 5 m/ct0 80 10 # 5 m/ct4 79 10 # 2 m/ct themgainsof1 + and # ( 1 # ) foraforce F ,wecanevaluatewherethepivotpoint x is locatedbysettingthesumofthetorquesequaltozero: F [ 1 + ] x = F [ 1 # ][ d # x ] (B1) Therefore,thebeamlocationrelativetocenter, x ,is x : = d 2 # x = d 2 (B2) andforachangeinapitchoryawcoilgain,thechangeinbeamposition, x ,is: | x | = | gain | R & 2 (B3) ThenalcalibrationsofthesechannelsareshowninTable B-1 2 B.2AngularMirrorMotion Theopticalleversprovideastraightfowardmeasureofindividualmirrormotion.The channelsIcalibratedwereoftheform L1:SUS-ETMX OPLEV { P,Y } ERROR ,theopticallevererror signalsforeachofthelargeoptics.Imadeuseofthedependenceofpowerinamisalignedcavity (refertoAppendix C.3 )tocalibratetheETMandITMopticallevers,andusedalessprecise, rudimentarymethodtocalibratetheRM,BS,andMMT3opticallevers. 2 AminortechnicalityisthatsincetherearenoltersbetweentheQPDerrorsignalsandthe offsetchannel,theirunitsareexactlythesame.Thus,calculatingmetersofbeamspotmotionas afunctionofoffsetservestocalibratetheerrorpoint.Forconvenience,thisiswhatIdid. 140

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B.2.1ETMandITMOpticalLevers Icalibratedthearmcavityopticalleversbytrackingthepowerlossinthelockedarmasone ofitsmirrorsistilted.Theclosedformexpressionforcavitypowerasafunctionofmirrortiltis derivedinAppendix C.3 .Allthatisneededisaquadraticttothedatacollected.Fromthet parameters,Icandeterminethefactor, ( / y ,whichconvertsthedigitalcountsoftheoptical leverchannel, y ,tounitsofradians. Tomakethemeasurement,Ilockedasinglearmandmaximizedthepowerbuildup.Then Islowlysteppedthepitchoryawpointingofoneofthemirrorsawaytoonesideofresonance, andthenbackandtotheotherside,repeatingthisseveraltimes.Allthewhile,Irecordedthe opticallevererrorsignalofthemirrorwhoseangleIwaschanging,andthepowerinthearmas determinedfromtheamountoflighttransmittedthroughtheETMs. 3 FromEq. C31 ,weseethatthepowerinthearm, P ,isafunctionoftheform P = P max exp [ # b ( y # y 0 ) 2 ] (B4) where y 0 istheDCoffsetoftheopticalleverchanneland b isrelatedtophysicalcavityaxis displacement a andtilt by by 2 =( a / w 0 ) 2 +( / ( 0 ) 2 .Inordertorelatetheopticalleversignal, y ,tophysicalcavityparameters,wedivideby ( 2 andrearrangetoget: ( y = & b & a / ( w 0 2 + & / ( ( 0 2 # 1 2 (B5) Thetermsinthenumeratorsontherighthandsidearexedconstantsbasedonthecavity geometryandcanbecalculatedusingEq. C4 .Themeasurementdataandtsareshownforboth pitchandyawinFigure B-2 .TheETMopticalleversmakeuseofabroaderrangeofopticallever signalthandotheITMs.(Alsonotethatthemaximumpowerinthey-armisabout10%lessthan 3 L1:LSC-NPTR { X,Y } OUT16 141

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! 0.4 0.2 0 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 pitch optical lever readback [cts] arm power [a.u.] 0.4 0.2 0 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 yaw optical lever readback [cts] arm power [a.u.] ETMX ITMX ETMY ITMY FigureB-2.OpticallevercalibrationdataandtstoEq. B4 thatinthex-arm.ThisistrueatbothHanfordandLivingston,andisduetotheprioritygivento thex-arminthealignmentscheme,asexplainedinAppendix C.4 .) B.2.2RM,BS,andMMT3OpticalLevers TocalibratetheRM,BSandMMT3opticallevers,Iusedmyowneyesandthecamera imagesandknowndimensionsoftheETMbeamcages.Withtheinterferometerunlocked,I movedtheopticsinpitchandyaw,trackingthebeam'smovementontheETMcages.Inorder tocalibratetheRM,IusedthereectionoffITMYtotheRMandontotheETMYcage.The BSandMMT3requiredonlystraightshotstotheETMs.Foryaw,Imovedthemirrorsuntilthe beamwascenteredoneachverticalsuspensionpost,andforpitchImovedthebeamfromthe centerofthemirrortothetopofthecage.Thebeammovesby x = 2 ( onacrossthecagewhen themirrormovesby ( ,sowithsmallangleapproximations,themirrorangleissimply x / 2 L where L isthedistancefrommirrortoETMcage. Thenal ( / x calibrationsofallopticalleversareinTable B-2 142

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TableB-2.Calibrationstobeusedwiththeopticallevererrorsignalsforameasureofangular mirrormotion.Unitsarerad/ct. ETMXETMYITMXITMYRMBSMMT3 pitch49.443.014.915.661.947.357.4 yaw50.743.320.120.242.563.555.5 TableB-3.DemodulationchaincalibrationforeachquadrantofeachWFS.UnitsareV/count. q1q2q3q4average WFS10.350.320.340.350.34 WFS28.88.68.78.58.7 WFS36.45.85.85.75.9 WFS46.35.45.37.46.1 B.3WFSErrorSignals TheWFSerrorsignals 4 arephysicallyWattsofpoweratthedetectors,whichtheWFS electronicsconvertintoavoltage.ToturnWFScountsintovoltageofsignalattheoutputofthe detector,wemustbacktrackthroughtheelectronicsandcalibratetheWFSdemodulationchain. TheanalogtodigitalRFchainfortheWFSincludesademodulationboard,awhitening board,ananti-aliasboard,andtheADC.Icalibratedthischainbyinjectingasinewaveofthe samefrequencyasatypicalWFSsignal,yetofknownvoltageintotheWFSdemodulationboard. Comparingthepeaktopeakvoltageofthisinputsinewavetothepeaktopeakamplitudeof theresultingdigitalcountssignalprovidestheVoltspercountconversion.Thecalibrationsare presentedinTable B-3 Itshouldbenotedthatthedemodulationchaincalibrationnumbersforallquadrantsofa particularWFSdiffernomorethan20%fromtheaverage.Thedemodulationchaindoesnot signicantlydistorttheerrorsignals. B.4AngularOpticalGain ThecalibratedWFSsensingmatrix(usingthecalibrationpresentedinSec. B.3 )canbe usedinconjuctionwiththesimultaneouslymeasureddrivematrix(asdescribedinSec. 6.1.3 ) 4 i.e. L1:ASC-WFS1 QP 143

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tocalculatetheangularopticalgain.Eachrowofthesensingmatrixmustbedividedbythe measuredamplitudeofthatrow'sDOFexcitation,whichisfoundasthediagonalelementsof Table 6-4 .Theresultofdoingsogivestheangularopticalgainoftheinterferometerintermsof WFSVoltsperradian. 144

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APPENDIXC ANGULARSENSINGANDCONTROLSUPPORTINGMATERIAL C.1OpticalLeverOpenLoopTransferFunction TheopticalleveropenlooptransferfunctionforETMXisshowninFigure C-1 .The measurementwasmadewhentheinterferometerwasunlockedandonlytheopticalleverand OSEMdampingloopson.Becauseofthetwounitygaincrossings,oneat2.2Hzandoneat 0.2Hz,theopticalleverloopprovidesonlyAC(velocity)damping.ThereisnoDCcontrol. Themodelusesthependulumtorquetoangletransferfunctionastheplantandthelters usedduringEnhancedLIGOasthecontrol.Iusethemeausurementtotunethepedulum parametersinthemodeltobestdeterminetheactualdampingcoefcientandresonantfrequency. ForETMX,wendthepitchresonantfrequencytobe0.65Hz,8%differentfromthetheoretical 0.6Hz.Also,wemeasuretheETMXdampingcoefcienttobe0.02. C.2MisalignedCavityAxis HereIprovidethegeometricargumentthatshowshowtocalculatethetilt a anddisplacement ofacavityasafunctionofmirrormisalignment.Cavitytiltisdenedbytheangle formedbetweenthelinethatconnectsthetwobeamspots(asgivenbyEq. 51 )andtheline joiningthecentersofthemirrors.Cavitydisplacementusesthesametwolines,yetisdened bythedistancebetweenthematthelocationofthewaistoftheresonantspatialmode.Basedon puregeometry,thecavitydisplacementandtiltare: # a $ % & = 1 L # z 2 z 1 # 11 $ % & # x 1 x 2 $ % & (C1) where z i isthedistancetothewaistfrommirror i calculatedas: z 1 = g 2 ( 1 # g 1 ) L g 1 + g 2 # 2 g 1 g 2 (C2) z 2 = L # z 1 (C3) 145

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10 1 10 0 10 1 60 40 20 0 20 40 magnitude [dB] 10 1 10 0 10 1 600 400 200 0 200 phase [deg] frequency [Hz] model measurement FigureC-1.ETMXpitchopticalleveropenlooptransferfunction.Themodeloftheplantis tunedtomatchthedata,resultinginapitchresonanceof0.65Hzandadamping factorof + =0.02.TheUGFisat2.2Hzandthephasemarginis38 ( Clearly,wecancombineEqs.( 51 )and( C1 )toarriveatanequationdirectlyrelating mirrortilttocavitydisplacementandtilt: # a $ % & = 1 1 # g 1 g 2 # g 2 z 2 + z 1 z 2 + g 1 z 1 # g 2 + 1 # 1 + g 1 $ % & # ( 1 ( 2 $ % & (C4) C.3PowerinaMisalignedCavity I'llshowhowtocalculatethepowerinacavityasafunctionofcavityaxisdisplacement andtilt.CombinedwiththeresultsofEq. C4 wedeterminehowthepowerbuild-upinacavity dependsonasinglemirror'sangulardisplacement. 146

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Theeldofalowest-orderGaussianlaserbeamalongoneaxisatthebeamwaistis: ( x )= U 0 ( x )= & 2 & w 2 0 1 / 4 exp # & x w 0 2 (C5) where w 0 isthebeamwaistradiusand U 0 isthelowest-orderHermitepolynomial.TheHermitepolynomialsareorthonormal,ie. U i | U j / = ij .Forexample,thenexttolowestorder polynomialis: U 1 ( x )= 2 & w 2 0 # 1 / 4 2 x w 0 exp [ # ( x / w 0 ) 2 ]= 2 x w 0 U 0 ( x ) (C6) C.3.1DisplacedCavity Theeldofacavitywitha displaced z-axisatthecavitywaistis: ( x )= ( x # a ) (C7) = U 0 ( x # a ) (C8) = c 0 U 0 ( x )+ c 1 U 1 ( x )+ c 2 U 2 ( x )+ ... (C9) where a isthedisplacementoftheaxisand c i areconstants. Wewanttoknow c 0 ,theprojectionofthedisplacedcavityeldontothebeameld: c 0 = | */ (C10) = ) # ( x ) ( x ) dx (C11) = exp [ # a 2 / 2 w 2 0 ] (C12) Thepowerinthismodeisthesquareoftheoverlapofthetwoelds: P 0 = | | */ | 2 (C13) = exp [ # [ a / w 0 ] 2 ] (C14) Forthepurposeofwavefrontsensing,weneedtoknowtheamplitude, c 1 ,oftherstorder U 1 eld.ThiscanbeapproximatedasdemonstratedinAnderson[ 62 ]usingtheTaylorseries expansionoftheexponentialin ( x )= U 0 ( x # a ) ,assumingadisplacement a that'ssmall 147

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comparedtowaistsize w 0 ( x )= & 2 & w 2 0 1 / 4 exp # & x # a w 0 2 (C15) = & 2 & w 2 0 1 / 4 1 # & x # a w 0 2 + O ( a 4 ) (C16) = & 2 & w 2 0 1 / 4 & 2 xa w 2 0 & 1 # x 2 w 2 0 + ... + & 1 # x 2 w 2 0 + 1 2 x 4 w 4 0 # ... + O ( a 2 ) (C17) = & 2 & w 2 0 1 / 4 & 1 + 2 xa w 2 0 + O ( a 2 ) exp # & x w 0 2 (C18) = U 0 ( x )+ a w 0 U 1 ( x )+ ... (C19) Noticethatherewend c 0 = 1,whichisconsistentwiththeexactresultofEq. C12 whenwe applyour a 2 $ 0approximation.WendthattheamplitudeoftherstorderHermite-Gausseld foradisplacedcavityis c 1 = a / w 0 (C20) C.3.2TiltedCavity Theeldofacavitywitha tilted z-axisatthecavitywaistisatadmorecomplextoderive. Weassumethetilt, ,issmallsuchthatsin $ andcos $ 1.Also,weassumethebeam divergenceangle, ( 0 = # / & w 0 ,issmallsuchthatthewavefrontsnearthewaistcanbeconsidered paralleltooneanother. Here,theimportantquantitytoconsideristhephaseofthecavityeldatthecross-sectionof thebeamwaist.Thephaseiseitheradvancedorretardedcomparedtothatofthebeam: ( x )= ( x ) exp [ # ikz ] (C21) $ ( x cos ) exp [ # ikx sin ] (C22) $ ( x ) exp [ # ikx ] (C23) = U 0 ( x ) exp [ # ikx ] (C24) where k = 2 & / # and # isthewavelengthofthelaserlight. 148

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Theoverlapoftheeldsofthebeamandtiltedcavityisexp [ # 2 / 2 ( 2 0 ] .Thereforethe poweris: P 0 = exp [ # ( / ( 0 ) 2 ] (C25) AnexpansionoftheexponentialinEq. C24 forasmalltilt gives: ( x )= U 0 ( x )[ 1 + ikx + O ( 2 )] (C26) = U 0 ( x )+ ik w 0 2 U 1 ( x )+ O ( 2 ) (C27) Therefore,theamplitudeoftherstorderHermite-Gausseldforatiltedcavityis c 1 = ik w 0 / 2 (C28) C.3.3DisplacedandTiltedCavity Themostgeneralcase,ofcourse,istohaveacavityaxisthatisbothdisplaced and tiltedat thebeamwaist: ( x )= ( x # a ) exp [ # ik ( x # a ) ] (C29) Wend: | */ = exp & # a 2 2 w 2 0 exp & # 2 2 ( 2 0 exp & # ia x 0 ( 0 (C30) and P 0 = exp & # a 2 w 2 0 exp & # 2 ( 2 0 (C31) C.4InitialDCAlignmentoftheInterferometer Afteranykindofin-vacuumwork,theDCalignmentofthemirrorsisusuallytoopoor fortheinterferometertolock.Abootstrappingprocessoftweakingthealignmentbyhandis necessary,assumingthemirrorsstartoutpointingingenerallytherightdirection,asisusually thecase.AspointedoutinSec. 4.4 ,theQPDsattheendstationsarethexedreferencepoints fortheoverallalignment,sothisprocessbeginswithmakingsurethelightreachesthem.We thenadjusttherestofthemirrorstomaximizepowerbuild-upinthearmsandtomaximzespatial overlapofthelightreectedfromeacharm. 149

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Anoutlineoftheprocessispresentedhere."Misalign"meanstointentionallypointamirror sofarawayfromanyknowngoodpositionsastoeliminateitfromtheconguration."Align" and"restore"meantobringamirrororcongurationtothebestknownposition(s).Centering thebeamonamirrorisaccomplishedbyusingthesuspensioncagesurroundingthemirrorasa reference.CameraimagesandQPDreadbackprovidethesignalsusedforbeamcentering. X-arm restorethex-arm(misalignRM,ITMY,andETMY,alignITMXandETMX) useITMXtocenterthebeamonQPDX useETMXtocenterthebeamonITMX withx-armlocked,useMMT3tomaximizethex-armpowerbuild-up(NPTRX,canexpect about95%) savetheMMT3,ITMX,andETMXalignmentsettings Y-arm restorethey-arm(misalignITMXandETMX,alignITMYandETMY) useITMYtocenterthebeamonQPDY useETMYtocenterthebeamonITMY withy-armlocked,useBStomaximizethey-armpowerbuild-up(NPTRY,canexpect about90%) savetheBS,ITMY,andETMYalignmentsettings Relativex-armandy-arm noteASbeampositiononcamerawhiletogglingbetweenx-armandy-armlocks useETMstoalignthetwoASbeams restoretheMichelson(misalignETMs,alignITMs) useBStomakeASportasdarkaspossible re-doy-armalignmentifambitious Recyclingmirror restorethePRM(misalignETMs,alignITMsandRM) useRMtocenterbeamonETMYcage Restorefullinterferometeroffyougo! C.5Photodiodes ThebasicphotoconductivephotodiodeisdepictedinFigure C-2 .Anegativevoltageapplied totheanode,calledthebiasvoltage V bias ,doesnotyieldanycurrentacrossthephotodiodeuntil electronsarereleasedbytheenergyofphotonsstrikingthediode.Electronsowinthedirection 150

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FigureC-2.Schematicofabasicphotoconductivephotodiode. ofthecathodewhichisconnectedtoground,andproduceacurrent i + = q e h P / (C32) where q e istheelectroncharge, h isPlanck'sconstant, isthefrequencyoftheincidentlight, P isthepowerofthelight,and / isthequantumefciencyofthediode.Thequantity q e / h is knownasthe responsivity .ForLIGOwhere # = 1064nm,theresponsivityis0 86A/W. Tohaveanaccuratemeasureofthepoweronthephotodiode,wemustmeasuretheproduced current i + .Thisisaccomplishedbyinsertingaresistorinseriesbetweenthecathodeandground andmeasuringthevoltagedropacrosstheresistor,whoseresistanceiscalledthe transimpedance R t .Atypicalvoltmeter,despitehavinghighresistance,willinevitablydrawsomesmallamount ofcurrent,therebyunderestimatingthecurrentproducedbythelight.Thecleversolutionisto makeuseofanop-ampasavoltmeter.Op-ampshavetwoprimaryproperties: theinputsdrawnocurrent thevoltagesattheinputstotheop-amparethesame 151

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10 2 10 1 10 0 10 1 10 2 10 3 40 20 0 20 40 60 80 magnitude [dB] 10 2 10 1 10 0 10 1 10 2 10 3 200 100 0 100 200 phase [deg] frequency [Hz] dSoft cSoft dHard cHard RM FigureC-3.WFSdigitalcontrollters. Thevoltageattheinputtotheop-amp, V 1 ,isdeterminedbymeasuringthevoltageattheoutput oftheop-amp, V 2 ,andaccountingfortheop-amp'sgainsuchthat: V 1 = R a R a + R b V 2 (C33) Thephotocurrentistherefore i + = V 1 / R t C.6WFSControlFilters Figure C-3 isarecordofthedigitialltersusedfortheWFScontrolservoduringEnhanced LIGO. 152

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10 2 10 1 10 0 10 1 10 8 10 7 10 6 10 5 frequency [Hz] velocity [m/s] LVEAx LVEAy LVEAz ETMXx ETMXy ETMXz ETMYx ETMYy ETMYz FigureC-4.Groundmotionattimeofopticalleverspectrawhentheinterferometerwasunlocked (Figure 4-1 )andattimeofASCsuppressiondemonstration(Figure 6-8 ).GPStimeis 956751915(May1,201007:25CDT). C.7SeismicSpectra TherearethreeseismometersatLLOforthepurposeofmonitoringthegroundmotionat thecornerstation(LVEA)andtwoendstations.Thecalibrationforthedigitallycollecteddata is2 4 10 # 9 m/s/count.AsstatedintheintroductionofCh. 6 ,Iincludeheresnapshotsofthe groundmotionatthetimeofgroundmotion-sensitivemeasurements.Eachspectrarepresents 30minutesofdatacenteredaroundthetimeofthemeasurement.Therearethreedegreesof freedomforeachseismometer, x y ,and z .Theyarealignedwiththeinterferometer's x and y arm coordinatesystem. Althoughtheinterferometerisverysensitivetogroundmotion,therangeofseismicactivity forwhichitcanmaintainfulllockisreasonablylarge.Themicroseism(0.1-0.35Hz)variesby factorsofseveralseasonally(it'sworseinthewinter),andthe1-3Hzmotionvariesbyfactors ofseveralfromdaytonight.Seismicmotiontheinterferometertypicallycannothandleincludes eventslikeearthquakes(0.03-0.1Hz)andheavyactivityonsite(3-10Hz).Motionatthese frequenciesareotherwiseconstantandatalevelsoastonotaffectinterferometeroperation. 153

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10 2 10 1 10 0 10 1 10 8 10 7 10 6 10 5 frequency [Hz] velocity [m/s] LVEAx LVEAy LVEAz ETMXx ETMXy ETMXz ETMYx ETMYy ETMYz FigureC-5.GroundmotionattimeofASCsuppressiondemonstration(Figure 6-8 ).GPStimeis 956751915(May1,201008:15CDT). 10 2 10 1 10 0 10 1 10 8 10 7 10 6 10 5 frequency [Hz] velocity [m/s] LVEAx LVEAy LVEAz ETMXx ETMXy ETMXz ETMYx ETMYy ETMYz FigureC-6.Groundmotionattimeofinputbeammotionimpressionmeasurement(Figure 6-1 ). GPStimeis971128215(Oct.14,201016:50CDT). 154

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10 2 10 1 10 0 10 1 10 8 10 7 10 6 10 5 frequency [Hz] velocity [m/s] LVEAx LVEAy LVEAz ETMXx ETMXy ETMXz ETMYx ETMYy ETMYz FigureC-7.Groundmotionattimeofbeamspotmotionmeasurement(Figure 6-7 ).GPStimeis 956728935(May1,201001:02CDT). 10 2 10 1 10 0 10 1 10 8 10 7 10 6 10 5 frequency [Hz] velocity [m/s] LVEAx LVEAy LVEAz ETMXx ETMXy ETMXz ETMYx ETMYy ETMYz FigureC-8.GroundmotionattimeofASCtoDARMnoisebudgetplot(Figure 6-13 ).GPStime is958456964(May21,201001:02CDT). 155

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BIOGRAPHICALSKETCH KatherineLairdDooleywasborninRhinebeck,NYtoJanineLProtzmanandAlanP Dooley.ShegrewupinPoughkeepsie,NYwiththreeyoungerbrothers,Brian,Greg,andTim, andgraduatedfromSpackenkillHSin2002.ShewenttoVassarCollegeandgraduatedin2006 withamajorinphysicsandminorsinFrenchandmathematics.KatebeganherphysicsPhD programattheU.ofFloridainthefallof2006,andmovedtoBatonRouge,LAinthefallof 2007tocarryouthergraduateresearchatLIGOLivingston. 160