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Development of the UF LISA Benchtop Simulator for Time Delay Interferometry


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Thanksgotomycolleague,IraThorpe,forhisexcellentworkinthelab.Withouthishelp,portionsofthisresearchwouldnothavebeencompleted.SpecialthanksgotoDanielShaddockforhostingmyvisitstoJPLandforhisguidanceandsupport.Ithankmyadvisor,GuidoMueller,forhishelp,support,andoverseeingthisproject.ThanksgotoMichaelHartmanwhohelpedwithbuildingvariouselectronicsandaidedinimplementationofthebenchtopsimulator.ThanksalsogotoWanWufortheuseofhisphase-lockloopboardduringthetwo-armTDIexperiment. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii KEYTOABBREVIATIONS ........................... xii KEYTOSYMBOLS ................................ xv ABSTRACT .................................... xvi CHAPTER 1BACKGROUND:GRAVITATIONALWAVESANDTHEIRDETECTION 1 1.1GravitationalRadiation ........................ 1 1.2WavesinGeneralRelativity ...................... 3 1.3Detection ................................ 4 1.3.1LaserInterferometerGravitational-WaveObservatory(LIGO) 7 1.3.2LaserInterferometerSpaceAntenna(LISA) ......... 8 2LISAINTERFEROMETRY ......................... 11 2.1OpticalBenchSignals ......................... 12 2.2LaserFrequencyCorrection ...................... 14 2.2.1LaserFrequencyStabilization ................. 15 2.2.2Arm-locking ........................... 15 2.2.3TimeDelayInterferometry ................... 15 3UFBENCHTOPSIMULATOR ....................... 20 3.1FrequencyStabilization ......................... 20 3.1.1OpticalCavities ......................... 20 3.1.2ZerodurCavities ......................... 22 3.1.2.1Opticalcontacting .................. 24 3.1.2.2Cavityparameters .................. 29 3.1.3Pound-Drever-Hall(PDH)LockingScheme .......... 30 v

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....................... 35 3.1.4.1Expectedstability .................. 36 3.1.4.2Controlelectronics .................. 38 3.1.5Results .............................. 42 3.2DesignofBenchtopSimulator ..................... 49 3.3ElectronicPhaseDelay(EPD) ..................... 51 3.3.1Hardware ............................. 53 3.3.2TimingError .......................... 53 3.4Phasemeter ............................... 55 3.5TDIExperiments ............................ 56 3.5.1FirstElectronicExperiment .................. 56 3.5.2InitialOpticalExperiment ................... 59 3.5.2.1Experimentalsetup .................. 59 3.5.2.2Analysisofsignals .................. 60 3.5.2.3Suppressionlimit ................... 62 3.5.3Optically-SplitExperiment ................... 65 3.5.3.1Timingerrorestimate ................ 66 3.5.3.2Non-integerlinearshiftofdata ........... 69 3.5.3.3Interpolation ..................... 71 3.5.4One-ArmExperiment ...................... 73 3.5.5Two-ArmExperiment ...................... 79 3.5.6Improvements .......................... 87 3.6FutureExperiments ........................... 88 3.6.1Second-generationTDI ..................... 88 3.6.2LowLightInterferometry .................... 89 3.6.3InsertionofGWsignals ..................... 89 4CONCLUSION ................................ 90 APPENDIX AVACUUMSYSTEM ............................. 92 BLINEARSPECTRALDENSITIESUSINGTHEDISCRETEFOURIERTRANSFORM ........... 94 REFERENCES ................................... 96 BIOGRAPHICALSKETCH ............................ 99 vi

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Table page 3{1Propertiesofstandardquality,expansionclass0Zerodur. ......... 23 3{2ParametersofthecavitiesusedtostabilizeRLandL1. .......... 29 3{3PZTtuningcoecientsofthelasersusedforpre-stabilization. ...... 38 vii

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Figure page 1{1Agreementbetweenthegeneralrelativisticallypredicteddecayinorbitwiththeobservationaldataforabinarypulsar ............... 2 1{2Michelsoninterferometerwiththebeamsplitterattheoriginandtheendmirrorslocatedalongthex-andy-axes. .................. 5 1{3Timeseriesshowingtheeectofagravitationalwaveonaringoftestmasses ..................................... 7 1{4ThetwoLIGOsitesandtheotherlarge-scaleinterferometricdetectors 8 1{5RepresentationoftheLISAspacecraftinorbit. .............. 9 1{6DepictionofLISA'sheliocentricorbit .................... 10 2{1ALISAproofmassontheopticalbench .................. 11 2{2ViewoftheY-tubeinsidetheLISAspacecraft ............... 12 2{3SimpliedPhaseAinterferometrydesignofthebenchandtelescope. .. 13 2{4SimpleconsiderationoftheLISAspacecraftasaMichelsoninterferometer. 16 3{1Fieldsofanopticalcavity .......................... 21 3{2ReferencecavitiesmadefrommirrorsopticallycontactedtoZerodurspacers ..................................... 23 3{3Cleaningopticsinaclass100cleanroom. .................. 26 3{4Samplesofopticcontaminantsasviewedthroughamicroscope. ..... 28 3{5ThePDHsetup ................................ 31 3{6AbeaminfrequencyspaceafterpassingthroughanEOM ........ 32 3{7PlotofPDHerrorsignal ........................... 34 3{8Experimentalsetupforfrequencystabilitymeasurement ......... 36 3{9Zerodurcavitiesinsideofvelayersofthermalshielding ......... 37 3{10CircuitschematicoftheRLPDHboard. .................. 39 viii

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.................. 39 3{12MeasuredtransferfunctionoftheRLPDHboard ............. 40 3{13MeasuredtransferfunctionoftheL1PDHboard ............. 40 3{14ClosedlooptransferfunctionofRL. ..................... 41 3{15ClosedlooptransferfunctionofL1. ..................... 41 3{16Frequencystabilizationresults ........................ 43 3{17Frequencystabilizationresultsplottedwiththefree-runningnoiseandtheLISArequirement ............................ 44 3{18Frequencystabilityresultsfromboththecounterandthephasemeter .. 45 3{19LengthstabilityofZerodurcavity ...................... 45 3{20PlotoftheerrorpointnoiseofthePDHsignal. .............. 46 3{21Temperaturestabilityofthecavities .................... 47 3{22OptoCadlayoutofbenchtopexperiment. .................. 50 3{23Schematicofelectronicphasedelay(EPD)technique. ........... 51 3{24SchematicofhowtheEPDunitisusedtomakeasignalequivalenttoaLISAsignal. .................................. 52 3{25Schematicoftimingintheexperiment. ................... 54 3{26Operationalschematicofphasemeterdesign ................ 55 3{27SetupfortherstelectronicTDIexperiment ................ 56 3{28BaselineofelectronicTDIexperiment. ................... 58 3{29ResultsoftheelectronicTDIexperiment. ................. 58 3{30Schematicofinitialopticalexperiment. ................... 59 3{31Phasesignalsfromtheopticalexperiment. ................. 61 3{32Thetime-delayedcombinationforvariousvaluesofshift. ......... 62 3{33Phasenoiseresultsfromtheopticalexperiment. .............. 63 3{34Timingsuppressionlimitsforvariousvaluesof. ............ 64 3{35Experimentalsetupforoptically-split,two-signalTDItest. ........ 65 ix

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........... 67 3{37Linearspectralnoisedensityofthetime-delayedcombinationfromtheoptically-splitexperimentandthesuppressionlimitsduetotimingresolution ................................... 68 3{38Estimateofthetimingerror,,fortheoptically-splitexperiment. ... 69 3{39Plottoshowlinearinterpolationofdataanadditional2106secfromthemeasureddata. .................... 70 3{40Linearspectralnoisedensitiesofthetime-delayedcombinationsusingtheoriginalsamplingandforinterpolatingtogiveanadditional2106secshift. ...................................... 70 3{41Sectionof10-pointinterpolateddata. .................... 71 3{42LSDsofthetime-delayedcombinationforvariousvaluesofshiftonthe10-ptinterpolateddata. ........................... 72 3{43ComparisonoftheLSDsofthetime-delayedcombinationsoftheoriginalandtheinterpolateddata .......................... 73 3{44RepresentationoftheLISAcongurationsimulatedwiththeone-armmeasurement ................................. 74 3{45Experimentalsetupfortheone-armmeasurement. ............. 75 3{46Transferfunctionofthecontrolelectronicsofthephaselockloopusedintheoneandtwo-armexperiments. .................... 76 3{47LSDsofsignalsfromtheone-armmeasurement. .............. 77 3{48ComparisonofLSDsofthetime-delayedcombinationfortheone-armmeasurementwiththeEPDunitremovedfromdierentpartsoftheexperiment. .................................. 78 3{49RepresentationoftheLISAcongurationsimulatedwiththetwo-armmeasurement ................................. 79 3{50Experimentalsetupforthetwo-armmeasurement. ............. 80 3{51Photographofthebenchtopsimulatorduringthetwo-armexperiment. 82 3{52LSDsofsignalsfromthetwo-armmeasurement .............. 83 3{53LSDsofsignalsinthetwo-armexperimentelectronically-splitandoptically-split ................................. 84 x

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................................. 86 3{55Comparisonofthetime-delayedcombinationsfromthetwo-armexperimentwithvariousdelaysimplemented. ...................... 86 4{1Suppressionofnoisefromtheinputsignalstothetime-delayedcombination. ................................. 90 4{2Frequencystabilityresultsfromboththecounterandthephasemeter .. 91 xi

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ADC:analog-to-digitalconverterAIGO:AustralianInternationalGravitationalObservatoryAOM:acousto-opticmodulatorBS:beamsplitterDAC:digital-to-analogconverterDAQ:dataacquisitionandcontrolDI:de-ionizedE&M:electromagnetismEMRI:extreme-massratioinspiralEOM:electro-opticmodulatorEPD:electronicphasedelayESA:EuropeanSpaceAgencyFPGA:eldprogrammablegatearrayFSR:freespectralrangeFWHM:fullwidthhalfmaximumGEO:British-GermanCooperationforGravitationalWaveExperimentGRS:gravitationalreferencesensorGSFC:GoddardSpaceFlightCenterGW:gravitationalwaveJPL:JetPropulsionLaboratoryL1:laser1L2:laser2L3:laser3 xii

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xiv

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31 ].Figure 1{1 showstheexperimentaldataforthispulsaroveratimespanofmorethan25yearsalongwiththepredictionfromgeneralrelativity.Forthismonumentaldiscoveryverifyingrelativity,HulseandTaylorreceivedthe1993NobelPrizeinPhysics.Obviously,wewouldliketobeabletodetectthesegravitationalwavesdirectly,throughexperimentalmeansandstudythemtolearnmoreabouttheirsources.Gravitationalwavesopenupawholenewwindowtoviewastronomicalobjectsandhence,shouldgreatlyaidinunderstandinghowtheseobjectsexistandinteract. 1

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Figureshowingtheagreementbetweenthegeneralrelativisticallypredicteddecayinorbitwiththeobservationaldataforabinarypulsar.FigurecopiedwiththekindpermissionoftheAstronomicalSocietyofthePacicConferenceSeries:JoelM.WeisbergandJosephH.Taylor,TheRelativisticBinaryPulsarB1913+16,ASP-CS302(2003).

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(r21

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gravitationalradiation.Thereducedquadrupolemomentisdenedas 3r2)(r)dV(1{6)whereistheKroneckerdeltafunction,ristheradialcoordinate,andisthedensityfunction.Thegravitationalradiationisthendescribedby Rc4I(1{7)whereGisthegravitationalconstantof6:67261011Nm2=kg2andRisthedistancebetweenthesourceandtheobserver.Usingthemomentofinertiaforatypicalbinarysystem,thestraincanbeapproximated 1{2

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Michelsoninterferometerwiththebeamsplitterattheoriginandtheendmirrorslocatedalongthex-andy-axes. Sincelighttravelsatc,eventsofthelaserbeamaredescribedby (1{10) where

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wherewewilllookatwavespropagatinginthez-direction.Thetermsofthestrainaredened t=h0sin(f0) (t)=t2c =h02c sin(f0) 1{3

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Timeseriesshowingtheeectofagravitationalwavepropagatinginthedirectionperpendiculartothepageonaringoftestmasses. Severaleortsareunderwayworldwidetodirectlydetectgravitationalwaves.Ground-baseddetectorsliketheLaserInterferometerGravitational-WaveObservatory(LIGO)useinterferometrytodetectsuchminutelengthchanges.TheLaserInterferometerSpaceAntenna(LISA)willsimilarlyuseinterferometrybutoveralargerbaselineinspace.Inadditiontointerferometricdetectors,therearealsoeortsunderwaytodetectgravitationalradiationwithresonantmass,orbar,detectorsandalsothroughdirectpulsartiming.

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AdvancedLIGOincludeanincreaseinlaserpowerfrom10Wto200W,largertestmassesfrom10kgto40kg,andfurtherimprovementstotheseismicisolation.ThedesignsensitivityofAdvancedLIGOwillbemorethanafactoroftenbetterthaninitialLIGObutrequiresthattheassociatedeectsoftheimprovements,suchasincreasedthermallensingduetoincreasedlaserpower,behandledappropriately.TheLIGOdetectorscomprisethelarge-scaleinterferometricdetectorsinNorthAmericabutthereareotherground-baseddetectorsinEuropeandAsiaincludingGEOinGermany,VIRGOinItaly,andTAMAinJapan.TherearealsoplansforanadditionaldetectorAIGO,tobebuiltinthesouthernhemisphereinAustralia. SchematicshowingthetwoLIGOsitesaswellastheotherlarge-scaleinterferometricdetectorsinGermany(GEO),Italy(VIRGO),andJapan(TAMA).LISA,shownintheupper-left-handcorner,willcomplementthesedetectorsbymakingmeasurementsinspace.

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RepresentationoftheLISAspacecraftinorbit. atriangularformationactingassemi-independentMichelsoninterferometers.ThebaselinedistancebetweenspacecraftwillbevemillionkilometersandLISAmustbeabletomeasurechangesinthisdistancewithanaccuracyoftensofpm=p

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LISA'sheliocentricorbitisatanangleof60totheecliptic. galaxiesandprovideacomparisonforthegreatadvancesinnumericalrelativityonthemodelingofsuchsystems.Perhapsofgreatestinterest,LISAwillalsobesensitivetostellar-massblackholesinspiralingtosuper-massiveblackholes,calledextreme-massratioinspirals(EMRIs),whichprovideatest-particlecaseforstronggravity.ThemeasurementsofLISAwillalsoaidintheeorttoputupperlimitsonthegravitationalwavebackgroundremainingfromtheformationoftheUniverse.Itisexcitingtohavetheopportunitytoworkonopeningupthisnewwindowintothephysicsofthesevarioussystems.

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RepresentationofaLISAproofmass,madeofagold-platinumalloy,ontheopticalbench. 11

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Viewofthe\Y-tube"insidetheLISAspacecraftthathousesthetwotelescopesandthetwoopticalbencheswiththeproofmasses.Eachopticalbenchandtelescopeassemblyispointedatoneofthetwofarspacecraft. thespacecraftisdesignedtomonitorthedistancebetweenproofmassesattheendsofeachoftheLISAarmsanddetectchangesinthisdistancefrompassinggravitationalwaves. 2{2 .Theinitialbaselinedesign(PhaseA)hadthelightfromthefarlaserreectingotheproofmassandthencombiningwiththeleakageeldfromthelocallaser,asshowninFigure 2{3 .Inthecurrentdesign,referredtoasstrap-down,thelightfromthefarlaserdoesnotreectooftheproofmassbutratheroamirrorattachedtotheopticalbenchandaseparatemeasurementismadeoftheproofmasswithrespecttothebench.

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SimpliedPhaseAinterferometrydesignofthebenchandtelescope.Note:nobacklinkinterferometryisshownhere.PDstandsforphotodiode,BSisapolarizingbeamsplitter,and=4isaquarter-waveplate. Ineithercase,themainmeasurementateachbenchisthephotodiodemeasurementbetweenthelocallaserandthefarlasersuchthat cfL(2{2)

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wherefisthefrequencyofthelaserandListhearmlength.Thephasestabilityisthen c(fL+Lf)=4 cfL(L L+f f)(2{3)Ifneitherthearm-lengthstabilityorfrequencystabilitytermisdominant, L=f f)f=fL L(2{4)ThelaserusedisaNd:YAGat1064nm.Thismeansthatthefrequencystabilityofthedetectorinordertoseeastrainof1021needstobelessthan6107Hz=p 16 ].However,pre-stabilizationstillleaveseightordersofmagnitudeoffrequencynoisetoovercome.Arm-lockingisatechniquethatusestheLISAarmasastablereference.ThearmlengthchangeisonayearlyorbitcyclesointheLISAband,thisdistanceprovidesanexcellentreference.However,arm-lockingposeschallengesforthecontrolleranditisnotcurrentlyconsideredinthebaselinedesignasthemethodtoreducealltheordersofmagnitudeneededinfrequencynoise.TDI,whichwillbediscussedbelowinSection 2.2.3 ,willberesponsibleforreducingthelaserfrequencynoisetothelevelofneededsensitivity.

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3{17 ofSection 3.1.5 ).Thisrequirementwasoriginallysetbyexpertsintheeldsurmisingthelimitthesetechniquescouldachieve.Sincethattime,severalexperimentsworld-widehaveshownthatwithsomecare,thislevelofstabilityisreadilyachievable.OurfrequencystabilizationsystematUFisdescribedinSection 3.1 andtheresultsareshowninSection 3.1.5 25 ].ThefeedbackcontroltolockalasertoaLISAarmwillbeinsensitiveatfrequenciesofthereciprocaloftheround-triptimeforthatarm,whichisroughlyevery30mHz.Onewaytogainsomeresponseatthesefrequenciesistousecommon-modearm-lockingwhereyoualsouseanarm-lockingsignalfromthesecondarmwhichwillhaveinsensitivitiesatslightlydierentfrequencies.IfsuchaschemeweretobeimplementedinLISA,itisadvantageoustohavetheorbitsofthespacecraftsuchthatthedierencebetweenarmlengthsismaximizedratherthanminimized.Duetotheseissues,arm-lockingpossessomechallengesforimplementationbutholdspromisingresults.Currently,arm-lockingisincludedintheLISAbaselinebutnotresponsibleforreducinganypartofthefrequencynoiseasfurtheranalysisandexperimentationisperformed.

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SimpleconsiderationoftheLISAspacecraftasaMichelsoninterferometer.Thesolidlinesrepresenttheround-tripsignalsineacharm,thedashedlinesarethesesignalsdelayedbythetimeoftheotherarm. spacecraftasasimpleMichelsoninterferometer,asshowninFigure 2{4 .Themainspacecraft,1,actsasthebeamsplitterwitharmsgoingouttothetwofarspacecraft,2and3.Foradditionalsimplicity,assumethelasersonthefarspacecraftarephase-lockedtotheincomingbeams;theythenactastranspondersandcanbeconsideredequivalenttomirrors.Bytakingthesignalfromspacecraft2anddelayingitthetimetospacecraft3andlikewisedelayingthesignalfromspacecraft3bythetimetospacecraft2(thedashedlinesinFigure 2{4 ),thencombiningallofthesesignals,thesignalsineacharm\travel"thesamedistanceandwehaveregainedthenoiserejectionofanequal-armconguration.Letusconsiderthisargumentinmoredetail.Theeldfromthemainlaseronspacecraft1canbeexpressed [E1(t)]SC2=Aei!1(t3)ei1(t3)eih3(t)(2{6)

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whereiisthelighttraveltimealongthearmoppositespacecrafti(i=Li=c)andhiisthechangeinphaseduetoagravitationalwavealongarmi.AnalogoustoEquation 2{5 ,theeldofthelaseronSC2is [E2(t)]SC1=Aei!1(t3)ei2(t3)eih3(t)(2{9)Atthemainspacecraft,thisincomingeldiscombinedwiththelocallaser.Fortheinterferometry,thesignalofinterestisthephaseofthiscombinedeld,S2. (2{10) Similarly,thesignalfromSC3willbe

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combinationhastheform[ 2 ] (2{13) whichisindependentofthelaserphasenoise.Thiscombinationissometimesreferredtoasan\eight-pulse"TDIsignalfortheeightdistincttimesagravitationalwavewouldappearinthesignal.ThereisadditionalcomplexityfromthissimplerealizationtotheactualinterferometrybetweentheLISAspacecraft.InthePhaseAdesign,anadditionalinterferometrymeasurementismadeatthebackoftheproofmasswiththelocallaserandthelaserfromtheotherbenchonthesamespacecraft[ 18 ].Thisback-sidemeasurementisincludedintheTDIcombinationandremovesthenoiseduetotheopticalbench[ 28 ].ThebaselinedesignhasbeenalteredfromthePhaseAdesigntoseparatethemeasurementofthelow-powerincomingbeamwiththelocalbeamfromthereectionotheproofmass.Referredtoas\strap-down,"thebeamfromthefarspacecraftreectsoofamirrorattachedtotheopticalbench,isthenmeasuredwiththelocalbeamandaseparatemeasurementwiththetwohigherpowerbeamsonthespacecraftismadeoftheproofmasswithrespecttotheopticalbench.Anadditionaldesignchange,called\frequencyswapping"or\cross-over,"usesthelaserfromtheotherbenchtointerferewiththeweakincomingbeam.Sincethelaserontheotherbenchisatadierentfrequencythanthelocallaser,thestraylightonthebenchwillnolongeraddnoisetothemeasurement.Also,theabovediscussionofrst-generationTDIassumesastaticLISAconstellation.However,inorbitthespacecraftwillbemovingwith

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respecttoeachotherandthetimetotraverseeacharmwillbeafunctionoftime,(t).Second-generationTDIhasadditionaltermstocompensatefortheseeects[ 23 29 ].

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3{1 ,thelighteldsofanopticalcavitycanbedescribedby 20

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Diagramofeldsforanopticalcavitymadeoftwomirrorswithtransmissionandreectioncoecientstandr,respectively.Einistheinputeld,Eristhereectedeld,Etisthetransmittedeld,andEaandEb,whicharemeasuredattheinsideedgeoftherstmirror,makeupthecirculatingeldinsidethecavity. where!=2=fisthelaserfrequencyandriandtiarethereectionandtransmissioncoecientsofmirrori.Foralosslesscavity,r2+t2=1.ThereectionandtransmissionofasurfaceisRr2andTt2,respectively.Writingthelighteldsintermsoftheinputeld,Ein,thetransmittedandreectedeldsbecome Whenconsideringthetransmittedandreectedelds,theaboveequationsgivethetransferfunctionstobeappliedtotheincomingeld. Assumingalosslesscavity,thetransferfunctionofthereectedeldcanbesimpliedto 2!L c=2n;n2N(3{5)

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whichgives T(3{10)whereTisthetransmissivityofthemirrors. 3{1 .Threespacerswerecutfromonesquarerod,38:10mmonaside,resultinginspacersoflengths208:28mm,231:14mm,and259:08mm.Eachspacerhasaholeofdiameter8mmboredthroughthecenteranda2mmairholefromthecenterboretothesidetoallowairowafterthemirrorsarebondedon

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ReferencecavitiesmadefrommirrorsopticallycontactedtoZerodurspacers. Table3{1. Propertiesofstandardquality,expansionclass0Zerodur.Ourspacersweremadefrommelt#F0159,reference#240224.isthecoecientoflinearthermalexpansionfrom0Cto50C,isthedensity,isthethermalconductivityat20C,andcthermisthethermalcapacity,orspecicheatcapacity. Coef.ofExpansion 800J=(kgK)

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theends.Theresonantfrequencyofthecavitydependsonthestabilityoftheopticalpathlength(OPL)withinthecavity,whichdependsonthegeometricalstabilityofthespacerandofthebondsholdingthemirrorstothespacer.Ourinitialattemptataligningthecavitiesprovedproblematic.Oneofthecavitiesshoweda\doublebeating"ofeachmode.Byadjustingthe=4-waveplatebeforethecavityandwatchingtherelativeamplitudesofthetwosameresonantmodes,wewereabletodeterminethattherewassomekindofpolarizationeectcausingslightlydierentresonantfrequenciesforthetwoorthogonalpolarizations.Inordertodetermineiftheseproblemswerecausedbythemirrors,weremovedthemfromtheZerodurspacerandplacedthemonasteelcavitytomeasurethenesse.TheFSRofthemetalcavityis681MHz;measuringthelinewidthgaveanesseofroughly2500.Itriedvariousmethodsofcleaningmirrorsinthelabandtooknumerouslinewidthmeasurementsgivingvaluesforthenesserangingfromafewhundredtoafewthousand-alllowerthanexpected.Inmanyinstances,contaminantscouldbeseenonthemirrors,evenaftercleaningsoitwasdecidedtomovetoacleanroomtocleanandbondthematerials.Aftercleaninginthecleanroomwiththeproceduredescribedinthenextsection,themeasurednesseimprovedtovaluesbetween5000and7000. 9 ]

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14 15 ].IfthispressureisduetovanderWallsattraction,theequationabovethengivescorrespondingseparationsof1:91:2nmforfusedquartz(H'0:61019J).Thisgivesanestimateofthetolerableseparationbetweenourmirrorsandspacersduetocontaminantsorsurfaceroughness.

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Cleaningopticsinaclass100cleanroom.LocatedatthefarleftisaowhoodwithaDIwatertapanddrain.Thetableholdsanultrasonicbathandhotplateforsoakingopticsaswellasahighintensitycoherentlightsourceforexaminingoptics. fastdryingtime,reducingthechanceofdusttocollectontheevaporatingliquid.Thelenstissuesusedwerecleanroomratedforclass100;standardlenstissuesproducebersanddust.Mycleaningprocedureisasfollows: 1. PlaceopticinaLiquinoxsolution.Besuretoplacelenstissuebetweenthepetridishorbeakerandtheopticinordertoprotectitssurface.Heatthesolutionto60Candletsit,coveredifovernight.Soakingtimemayvaryandthisstepmayneedtoberepeatedseveraltimes.Note:thereareanecdotalaccountsofLiquinoxdegradingcoatingssoitisadvisedtostartwithaverylowconcentrationofsolutionandincreasebysmallincrementsonlyifneeded. 2. RemoveopticfromsolutionandrinsewithDIwater.Standtheopticonitssidewhilepreparingfornextstep.

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3. DipalenstissueintheLiquinoxsolution.Wipeeachfacesurfaceoftheopticwithone,soft,evenstrokemovinginonedirection.RisewithDIwaterandagainstandonitsside. 4. WipeeachfacesurfaceoftheopticasdescribedinthepreviousstepwithaDI-wettedcleanroomlenstissue. 5. RinseeachfacesideoftheopticunderrunningDIwaterforatleast10seconds.Standtheopticonitssideontopofalenstissue. 6. Wipeeachfacesurfaceoftheopticintheabovemannerwithanacetone-wettedlenstissuetocompletethedryingoftheoptic.Alsowipethenon-facesidestoremoveanyremainingwaterdrops.Settheopticbackonitssideonadryspotofthetissue.Note:trytominimizethetimebetweensteps2-6suchthatwaterandsolutiondoesnothavetimetodryontheoptic,possiblyleavingsolventsandcaptureddustonthesurface. 7. Foldacleanroomlenstissueandplaceinforceps.Takecaretohaveanevenamountofthetissueexposedalongtheedgeoftheforceps.Useadropperbottletowettheedgeofthetissuewithacetone.Wipethisedgealongeachsideoftheopticinoneevenstroke,startingwiththefacetobecontactedrst. 8. Examineopticunderamicroscope.Ifnotadequatelyclean,repeatpreviousstepandreexamineorstartfromthebeginningagain.Evenafterthoroughcleaning,therewilllikelybesomeartifactsontheoptic'ssurface.SamplesofvariousobjectsareshowninFigure 3{4 .Oncethemirrorsareclean,preparetheendsoftheZerodurspacerinthesamemannerandcarefullyplacethemirrorovertheendholeofthespacer.Sincethespacersaretoolargetobeusedwiththemicroscope,theendfaceshavetobeinspectedbyeye.Useofabright,coherentlightanddarkbackgroundsurfacecanbehelpfulinseeingartifactsonthesurfaceaswellasfringesbetweenpartially

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Samplesofopticcontaminantsasviewedthroughamicroscope.(a)Microscopesetat20toexamineoptics.(b)Small\E"of\EPLURIBUSUNUM"onadimetoshowscale.Thedothasadiameterofroughly225m.(c)Specofdirtordust.(d)Fibersfromnon-cleanroomlenstissue.(e)Contaminatedevaporationmarksfromsolvent.(f)Smallscratch.

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contactedsurfaces.Sometimesopticalcontactingwilloccurononlyaportionofthemirrorsurface,whichcanbeseenbyalackoffringesbetweenthesurfacesinonesection.Pressingrmlybutcarefully,moveoutfromthecontactedarea,pushingallofthefringestotheedgesofthemirror.Whenthebondoccurs,thefringeswillnotreappearandpullingonthemirrorwillnotbeabletoseparateitfromthespacer.Ifthebonddoesnotoccurproperly,cleantheopticsrepeatedlyandtrycontactingagainuntilthebondholds. 3{2 .Withinthevacuumtank,thecavityfortheRLislocatedverticallyabovethecavityforL1.ThecavityforRLislongerthanthecavityforL1.Bothcavitieshaveaatfrontmirrorandcurvedbackmirrorsothewaistoftheresonantmodeofthecavity,!c,islocatedatthefrontmirror.Thecavitygparameters,g1andg2,aregivenby[ 21 ] Ri(3{13)whereListhelengthofthecavityandRiistheradiusofcurvatureofmirrori.Acavitywillbeastableresonatorwhen 0g1g21(3{14) Table3{2. Parametersofthecavitiesusedtostabilizethereferencelaser(RL)andlaser1(L1).Listhelengthofthespacer,R1istheradiusofcurvatureofthefrontmirror(bothhaveats),R2istheradiusofcurvatureofthebackmirror,g1g2isastabilityfactor,!cisthewaistoftheresonantmodeofthecavity,FSRisthefreespectralrange,fcavityisthemeasuredlinewidth,andFisthenesse. Laser R1 260mm 290:9m L1 225mm 463m

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Forbothofourcavities,g1=1sog1g2issimplygivenbyg2.Thecavitywaistoftheresonatorisgivenby[ 21 ] s (g1+g22g1g2)2(3{15)forthecaseofouratfrontmirrors,thisexpressionsimpliesto r 3{7 and 3{8 aboveinSection 3.1.1 onopticalcavities.Thelinewidthofthecavitieswasfoundbyanalyzingthetransmittedresonant00modeofthecavity.Whilesweepingthefrequencyofthelaseroverthe00mode,thetimeoftheFWHMoftheintensityisrecorded.Thesweeprateandthetuningcoecientofthelasercontrollerarethenusedtocalculatethechangeinfrequencythatcorrespondstothemeasuredtime.UsingEquation 3{10 andthenessevaluestosolveforthetransmissivity, 7 ].Figure 3{5 showstheexperimentalsetup.ThelaserpassesthroughaFaradayisolator,actingasanopticaldiode,andthenaportionofthebeamispickedoforusewithothermeasurements.Inorder

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SchematicofthePDHsetupusedtolockthefrequencyofthelasertotheresonantfrequencyofthereferencecavity.ThelaserbeamgoesthroughaFaradayisolator,powerbeamsplitter(BS),electro-opticmodulator(EOM),polarizingBS,and=4-waveplatebeforereachingthecavity.ThesignalfromthephotodiodethatreceivesthereectedcavitylightisdemodulatedwiththedrivingfrequencyoftheEOMtoformthePDHerrorsignal.Theerrorsignalisshapedwithcontrolelectronicsandinputtothelasercontroller. tocreatethePDHerrorsignal,thelaserisphasemodulatedusinganelectro-opticmodulator(EOM).Alasereld,Elaser=E0ei!t,thathaspassedthroughanEOMwhichisdrivenwithafrequencycanbeexpressedwiththemathematicalform (3{19) Inthisform,itiseasytoseethattheEOMaddstwo\sidebands"infrequencyspace(locatedat!)tothemain,orcarrier,beam(at!),asshowninFigure 3{6

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RepresentationofabeaminfrequencyspaceafterpassingthroughanEOMwithmodulationfrequency. Thelaserthentransmitsthroughthepolarizingbeamsplitter,a=4-plate,andthroughavacuumwindowtothecavity.Thereectedcavityeldwillagaingothroughthe=4-plateandnowreectatthepolarizingbeamsplittertoreachaphotodiode.Whenthelaseristhesamefrequencyasthecavityresonance,almostallofthecarrierwilltransmitthroughthecavityandaslongasthemodulationfrequency,,islargerthanthecavitylinewidth,thesidebandswillbereectedandincidentonthereectedlightphotodiode.Allowingforthecarrierandthesidebandstoberesonant,theeldincidentonthephotodiodecanbedescribedby[ 5 ] 3{4 .Theintensityonthephotodiodeisthenthemagnitudeofthiseld

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Weareinterestedinthetime-varyingcomponentsoneglectingtheDCtermandsecond-ordertermsthephotodiodesignalcanbewritten ThephotodiodesignalismixedwiththedrivingsignalfromtheEOM,pullingoutthetermproportionaltosin(t).Hence,theleadingACtermoftheerrorsignalis c(3{25)torstorder.Thisresultsintheerrorsignalbeingoftheform (1r1r2)2!L c(3{26)Forsymmetrically-coatedcavitieswherer1=r2r, T!L c(3{27)

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RewritingEquation 3{9 suchthat T(3{28)theerrorsignalcanbewritten PlotofPDHerrorsignalasgiveninEquation 3{24 foracavitywithaFSRof600MHzandE0=1,m=0:1,r1=r2=0:99. TheerrorsignalforacavitywithaFSRof600MHzisshowninFigure 3{7 .Asshownintheplot,whenthelaserfrequencymatchesthefundamentalmodeofthecavity,theerrorsignaliszero.Ifthefrequencymovesslightlyothispoint,theerrorsignalmovesquicklytoanon-zerovaluesincetheslopeisverysteepand

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suppliesacorrectingsignaltomovethelaserbacktotheresonantfrequency.Aslongasthegainandbandwidthofthecontrolelectronicsthatusetheerrorsignaltosupplyavoltagetothelasercontrollerareadequate,thelaserwillbelockedtothecavityindenitely. 19 ].TheGSFCandUFexperimentsareverysimilarexceptthattheGSFCexperimentusesCorning'sULEglassasthecavityspacerandhastwo,smaller,separatevacuumtanks,eachcompletewiththesamevelayersofgold-coatedstainless-steelthermalshieldingtohousethetwocavities.ThetanksalsohaveanadditionallayerofexteriorthermalshieldingbutthetemperatureofthelabatGSFChasmorevariationthanthelabatUFduetotheheatingandcoolingsystemsofthebuildingandthelocationoftheexperimentnearanexteriorwallofwindows.ULEandZerodurarecomparablematerialswithsimilarthermalexpansioncoecientssowedecidedtouseZeroduratUFtotestiftherewereanymeasurabledierencesbetweenthematerials.ThefrequencystabilityoftheZerodurcavitiesismeasuredbyrecordingthebeatfrequencybetweenthetwolaserswhichareindependentlylockedtothecavities,asshowninFigure 3{8 .Ifeachofthecavitieshasaverystableresonantfrequency,thenthedierencefrequencybetweenthelaserswillremainconstant.Bylookingatthechangesofthedierencebeatnote,wegetameasureoftherelativestability.Aglassplateisusedtopick-opartofthebeamofeachlaserwhichiscombinedatapowerbeamsplitterandincidentonaphotodiode.ThebeatfromthephotodiodeismeasuredwithaHP53181Afrequencycounterthatcanmeasuresignalsupto225MHz.

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Experimentalsetupforfrequencystabilitymeasurement.Thebeatnotebetweenthelasersoftwo,independentPDHsystemsismeasuredwithafrequencycounter. L=T0:02106 Lf21014

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Zerodurcavitiesinsideofvelayersofgold-coatedstainlesssteelthermalshielding(coversofshieldsareremovedforthepicture)locatedinsideofalargevacuumtank.Oncethetankispumpeddown,thetemperaturestabilityinsidetheshieldsisexpectedtobeontheorderofK=p

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17 ].FromtheinputoftheerrorsignalofthePDH,theanalogelectronicsgiveoutputstothePZTandtothetemperaturelaseractuators.Figures 3{12 and 3{13 showthemeasuredtransferfunctionsandFigures 3{14 and 3{15 areclosedlooptransferfunctionmeasurements.UnitygainoccursforbothsystemsinthelowkHz. Table3{3. PZTtuningcoecientsofthelasersusedforpre-stabilization. Laser Serial# PZTTuningCoef. RL 2258 4:653366MHz=V 2142 2:109725MHz=V

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CircuitschematicoftheRLPDHboard. CircuitschematicoftheL1PDHboard.

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MeasuredtransferfunctionoftheRLPDHboard,PZToutputwithintegratoro. MeasuredtransferfunctionoftheL1PDHboard,PZToutputwithintegratoro.

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ClosedlooptransferfunctionofRL.A20kHzlow-passlterhadtobeinsertedbetweentheRLPDHPZTcontrolsignalandtheRLPZTactuatortosuppressoscillationsappearingatmultiplefrequencies.Unitygainisnear2kHz. ClosedlooptransferfunctionofL1.Unitygainforthesystemisnear3kHz.

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3{16 .Thecorrespondinglinearspectralnoisedensityisshowninthebottomplot.Athighfrequenciesofourmeasurementband,ourstabilizationresultsareroughly10Hz=p 3{31 andonthesameorderorslightlybetterthansimilarstabilizationexperimentsperformedaroundtheworld.ThereappearstobenoappreciabledierencebetweenourZerodurmeasurementsandtheGSFCULEmeasurements.Figure 3{17 alsoshowsthisresultplottedwiththefrequencynoiseofafree-runninglasertoillustratethelevelofnoisesuppressionpre-stabilizationprovides.ThisgurealsoplotstheLISAprestabilizationrequirementwhichoursystemmeetsatalmostallfrequencies.Thefastestrateatwhichdatacanbereadfromthefrequencycounteris1Hz.UsingdatatakenfortheopticalTDIexperimentinSection 3.5.3 thatisreadwiththephasemeter(PM),wehaveameasurementofthefrequencynoiseathigherfrequencies.ThefrequencystabilityofthesystemmeasuredwithboththecounterandreadthroughthephasemeterisplottedinFigure 3{18 .Thedatathatoverlapsbetweenthetwomeasurementmethodsnear0:1Hzgivesaconsistentresult.Inthephasemeterdata,thereisalargenoisestructurearound10HzthatisnotsuppressedbythePDHelectronics.NoisespikessuchastheseareseenthroughouttheTDIdatainSections 3.5.2 3.5.5 .Computingtheassociatedlengthchangeofthecavityfromthefrequencyresultsgivesvariationsontheorderoffm=p 3{19

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Frequencystabilizationresultsoveraweekendrunofthreedays.Thetopplotshowsthetimesseriesofthebeatnotebetweenthetwostabilizedlasersrecordedfromthecounter.Thebottomplotisalinearspectraldensityoftheresidualtimeseriesafterapplyingaquadratict.

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FrequencystabilizationresultsfromFigure 3{16 (orange)plottedwiththefree-runningnoise(blue)andtheLISArequirementforpre-stabilization(red).

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Frequencystabilityresultsfromboththecounterandthephasemeter(PM). LengthstabilityofZerodurcavitycomputedfrommeasuredfrequencystability.

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Figure 3{20 showsaplotofthePDHerrorpointnoise.TheLSDswereconvertedtofrequencyusingtheampliederrorsignalslopesof28:0V=HzforRLand43:6V=HzforL1.Sincethespectrumfallsrightatwhatweseeforthefrequencystabilityofthecavitiesabove2mHz,oursystemiscurrentlygainlimitedatthosefrequencies.WeareintheprocessoftransitioningourcontrolelectronicsfromananalogtodigitalsystemwhichwillgreatlyfacilitatereshapingthePDHcontrolelectronicstohavehighergainwithoutexcitinghighfrequencyresonancesandtosuppressthenoisestructuresseenathigherfrequenciesinthedataread PlotoftheerrorpointnoiseofthePDHcontrolsignal.Thetopplotshowsthetimeseriesoftheerrorsignalsandthebottomplotisthelinearspectraldensitiesoftheirresidualsfromaquadratict.Sincetheerrorpointnoiseisatthelevelofthefrequencystabilizationofthecavitiesabove2mHz,oursystemiscurrentlylimitedbyinadequateelectronicgainatthesefrequencies.

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fromthephasemeter.Weshouldthenseeimprovementinthefrequencystabilityabove2mHzuntilanothernoisesourceisreached.Wealsoexpectthatusingdigitalelectronicswillhelpalleviatetheproblemswithglitchesthatsometimesoccurintheerrorsignal,severalofwhicharevisibleinthetop,time-seriesplotofFigure 3{20 .Ifourstabilityresultswerelimitedtotemperaturevariationsofthecavity,Figure 3{21 showsthespectrumofthetemperaturestabilityofthecavityitself.Sinceweknowourresultsareerrorpointnoiselimitedabove2mHz,weknowthatthetemperaturestabilityofthecavitiesisatleastasgoodaswhatisplottedforthesefrequencies.Thethermalmassofthespacerprovidesalow-passlterforthetemperaturevariationsofthevacuumenvironment.Yet,fromtheresultsofthe Temperaturestabilityofthecavitiesasinferredfromthelengthstabilityresults.Sincethespectrumisprimarilyerrorpointnoiselimited,weknowthatthetemperaturestabilityisintheKrange,asexpected.

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cavitystability,wecanconjecturethatthetemperaturestabilityofourvacuumenvironmentisintheK=p Acrosswherecthermisthethermalcapacity,orspecicheatcapacity,listhelengthoftheobject,isitsthermalconductivity,andAcrossisthecross-sectionalarea[ 13 ].Approximatingthatthemajorityoftheconductionoccursfromheatingattheends[ 19 ], 3{1 forthevaluesofthespacermaterialproperties,ourcavitieshaveatimeconstantofroughly

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whichisacornerfrequencyof50Hz.Similarly,theradiativetimeconstantisgivenby[ 13 ] 3{22 showsthegraphicaloutputofthemodel.

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OptoCadlayoutofbenchtopexperiment.

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ThesepowerlevelsarereachedbyreectingRLandL1onceoofglassplatesbeforethecavityopticsandeachbeamhastworeectionsoofglassplatesbeforereachingaphotodiode.Inorderforgoodmode-matchingatthephotodiodes,theRLandeachotherlasertravelthesamepathlengthtoreachagivendetector.ThisLISAsimulatorcanbeusedtotestmanyaspectsofLISAinterferometry;Ihavebeendevelopingittotestrst-generationTDIalgorithmsusingLISA-likesignals. 3{23 ,thesignalisdigitized,storedinamemorybuerforaspeciedamountoftime,andthenregenerated. Schematicofelectronicphasedelay(EPD)technique.ADCisanalog-to-digitalconverterandDACisdigital-to-analogconverter.ImagecourtesyofJ.IraThorpe. TheinputsignalfortheEPDisalwaysaphotodiodesignalofabeatbetweentwolasers.Suchasignalwillbeoftheform 27 ].

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where!iisthefrequencyandi(t)isthephasenoiseoflaseri.Dening!20=!2!0and20(t)=2(t)0(t), 3{24 ,mimicsanactualLISAsignalasdescribedinSection 2.1 ontheopticalbenchsignals.Fortheinterferometry,weareinterestedinthephaseofthesignalsolookingattheargument,thephasesignalfromtheEPDis (3{39) Thepromptsignalbetweenlasers1and0issimply SchematicofhowtheEPDunitisusedtomakeasignalequivalenttoaLISAsignal.

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Dening!21=!20!10anddroppingtheconstantterm!20, 2{1 withoutthephasechangeduetoagravitationalwave,h(t).ThesignalgeneratedwiththeEPDcontainsadditionaldierencetermstogetthissameformbutsincethelasersallhavethesamenoisecharacteristicsastheLISAlasers,thesignalusingtheirdierenceswillaswell. 3.5.1 wasperformedusingtheMicrostarcard.InordertoimprovetheEPD,weacquireddedicatedhardwarefromPentekthathasa100MHzsamplingrate.Onthisboard,wecandelaysignalswithamaximumfrequencyof5MHz,onfourchannels,upto6seconds.ThePentekboardconsistsofacarrierboardthathas1GBofSDRAMwithtwodaughterboards,oneeachhousingthe14-bitADCandthe16-bitDAC.ThereareveFPGAsonthePentekboard:twoonthedaughterboardwiththeADC,twoonthecarrierboard,andoneonthedaughterboardwiththeDAC.Inthisconguration,thelimitationoccursattheVIMinterfacesbetweenthedaughterboardsandthecarrierboardwheredatatransfercanoccuratamaximumrateof33MHz.AlloftheopticalexperimentswereperformedusingthePentekEPDunit.

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SchematicrepresentationoftimingintheTDIexperiments.Thex'srepresentwhenthesignalsaresampledbytheADCoftheDAQwheretsamp=1=fsamp. acquisition(DAQ)processthateventuallyrecordsthesignal.Hence,mostlikelythisdelayintheexperimentwillnotfallatexactlyoneoftherecordedsamplingpoints,ntsamp;n2N,butrathersomewhereinbetweenasshowninFigure 3{25 .ThemaximumtimethatcouldbebetweenEPDandtheclosestsamplingpointoftheDAQis1 2tsamp.WhenanalyzingthedatafromtheTDIexperiments,wetrytoshiftthedelayedsignalbackbyexactlyEPDinordertoseethesamenoisestructureatthesamepointasinthepromptsignal.Withthepost-processdatashift,shift,limitedtothetimingoftherecordeddatapoints,therewillbeanerrorbetweenwhatwecanachievepost-processandwhattheactualdelaywasintheexperiment.Wedenethistimingerroras =jEPDshiftj(3{43)Thesuppressionlimitduetothistimingerror,discussedinSection 3.5.2.3 ,isanimportantfactorintheopticalTDIresults.ThistimingerrorisnotjustanartifactoftheEPDprocess,butoccursinLISAaswell.ThetimethattheLISAlasereldstaketotravelthearmswillalmostcertainlyfallinbetweensamplingpointsoftheLISAsignalsandwillbeafactorintheimplementationofTDIandarm-lockingforthemission.

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Operationalschematicofphasemeterdesign.ImagecourtesyofJ.IraThorpe. 3{26 ,theinputsignalatfrequencyfandphasei(t)ismultipliedbythesineandcosinegeneratedbyanNCOwithphasem(t).Themultipliedsignalsarelow-passlteredtoremovethesecondharmonicandgivetheamplitudeandtheresidualphaser(t),ofi(t)fromm(t).ThisresidualphaseisfedbacktotheNCOtoadjustitsoutputfrequencyandaslongasthefeedbackgainsH(f)areappropriate,themodelphasewillstayclosetotheinputphaseandr(t)<1cycle.Byaddingtheresidualphasetothemodelphase,wegettheoutputphaseo(t).The 27 ].

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ltersinthephasemeterbegintoeectthesignalsaround100Hzandsomeofthephaseinformationislost,whichshouldberememberedwhenlookingattheanalysisoftheopticalTDIexperiments.Overthemeasurementbandusedintheexperimentsdiscussedbelow,thephasemetershowsperformanceatLISAlevelsof105cycles=p Schematicofsetupfortherstelectronicexperiment.VCOisvoltage-controlledoscillator,SRSisStanfordResearchSystems,andRPD-1andSRA-6refertomixermodelnumbers.

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AdetailedexperimentalschematicisshowninFigure 3{27 .Afunctiongeneratorat10kHzsuppliesthesignalforS1(t)whichisthenputthroughthedelayunitandltered.Avoltage-controlledoscillator(VCO)isusedtogenerateS2(t).TheVCOusedisaMinicircuitsZOS-50whichoperatesintherangeof25-50MHzsoanadditionaloscillatorwasusedtomixdownthesignalto10kHz.The10kHzVCOsignalislteredandampliedbyafactoroftwowithaStanfordResearchSystemspreamplier(SRSpreamp.)togiveS2(t).Thissamesignalismixedwiththedelayedfunctiongeneratorsignal,S1(tEPD),lteredwithaSRSpreamp.,andusedtoformthecontrolsignalfortheVCO.TheVCOrequiresanon-zerocontrolsignalsothemixedsignalisaddedtoaconstant2:0VsignalwithanotherSRSpreamp.The10kpot.onthemixedsignalallowsforchangingthegaininthefeedbacktolocktheVCO.Thecontrolsignalphase-lockstheVCOtoS1(tEPD)andhenceshowsupinS2(t).ThesignalsS1(t)andS2(t)arerecordedwithadataacquisitionboard.Inputtingadelayof2.0secondsintotheMicrostarcardresultedinadelaybetweensignalsS1(t)andS2(t)of2:0016sec.ThistimewasdeterminedbychangingtheamplitudeofS1andseeingwhenthechangeappearedinS2.TheplotinFigure 3{28 showsthebaselinenoiseoftheexperiment.ThesignalS2(t)isnoisierthanS1(t)duetothenoiseoftheVCO.Sincethisnoiseisnotcommonamongthesignals,thecombinedsignalS1(t)S2(t)islimitedbythisnoise.Totestthenoiserejectionofperformingthecombination,a20degreephasemodulationat30HzwasintroducedtotheoscillatordrivingS1(t)whichisthenimposedonS2(t)throughtheVCOcontrolsignalEPDsecondslater.Figure 3{29 showsthatthecommon30Hznoiseissubtractedinthecombinedsignal.

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Baselineofelectronicexperiment.Thetopplotisatimeseriesofthesignalsandthebottomplotistheirlinearspectraldensities. Resultsoftheelectronicexperiment.Theplotshowsonlythefrequenciesaround30Hztoshowthenoisecancellationthatoccurred.

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3.5.2.1ExperimentalsetupFortherstopticalexperiment,thebeatnotebetweenthetwocavity-stabilizedlaserswaselectronicallysplitandusedtogiveapromptanddelayedsignal,asshownintheexperimentalschematicofFigure 3{30 .Thephotodiodeof32MHzismixeddowntoroughly800kHzwithafunctiongenerator(oroscillator).ThedemodulatedsignalislteredwithaStanfordResearchSystems(SRS)preamplierwithonepoleat1MHzandelectronicallysplitbyusingtheamplierstwooutputs.ThedelayedsignalisformedbypassingoneofthesesignalsthroughthePentekEPDunitwithadelayof2seconds.Bothsignalsareagainmixeddowntoroughly9kHzwithanotherfunctiongeneratortoberecordedonaNationalInstrumentsPCI-6036EDAQatan80kHzsamplingrate.ThetwofunctiongeneratorsandtheEPDunithavetheirtimebaseslinkedtoacommonrubidiumstandard.Afterrecordingthesevoltagesignals,theyareruno-linethroughtheSimulinksoftwarephasemetertogivethephasesofthepromptanddelayedsignals,p(t)andd(t),respectively.Figure 3{31 (a)showsthephasesignalsafterbeingrunthroughthephasemeter.Theslopeofthesignalsisduetothedierencebetweenthesignalfrequencyandtheinitialmodelfrequencysuppliedtothephasemeter. Schematicofinitialopticalexperiment.The32MHzphotodiodesignalisdemodulatedto800kHz,ltered,andsplit.ThepromptsignalandsignaldelayedwiththeEPDaredemodulatedto9kHz,recorded,andrunthroughthephasemeter.LPFstandsforlowpasslterandNIDAQistheNationalInstrumentsdataacquisitioncard.

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3{31 (a)showstheprompttruncated,time-shifteddelayed,andtime-delayedcombinationphasesignals.Figure 3{31 (b)showsthesethreesignalsagainwiththepromptanddelayedsignalsdetrendedandun-truncatedorshifted.Theinputphasesignalsaregoingovermanycyclesyetthesamestructureisseeninboththepromptanddelayedphasesatatimedierenceof2sec.AllphasesignalsfromtheopticalTDIexperimentsexhibitthesamebehavior.Thevalueofshiftthatgivesthegreatestnoisesuppressionisnotexactlythesameasthe2secdelaytimethatisspeciedtotheEPD.DuetotheinternaldigitizationoftheEPDunitanddelaysthroughtheEPDADC,DACandpossiblyothercomponentsintheexperiment,theoptimumshiftisslightlymorethansimply2:0sec.Inordertondthebestsuppressionofthetime-delayedcombination,theaboveprocedureofsubtractingthetruncatedsignalsisdoneforvariousvaluesofshiftandthelinearspectraldensities(LSDs)arecomputedandcompared.AplotofthesevariousLSDsisshowninFigure 3{32 andclearlyshowsthatthemaximumphasenoisecancellationwasreachedwithshift=2:0005625sec.Thetime-delayedcombinationsignalthatisshowninbothplotsofFigure 3{31 wascomputedusingthisoptimumvalue.Thetime-delayedcombinationinFigure 3{31 showsvirtuallynonoiseatthisscale.Aswithndingtheoptimumvalueofshift,computingtheLSDsofthe

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Phasesignalsfromtheopticalexperiment.Thepromptphase,p(t),isinblue;thedelayedphase,d(t),isingreen;andthetime-delayedcombination,S(t),isinred.(a)Thephasesignalsafterrunningthroughthesoftwarephasemeterandapplyingthetimeshift.(b)Thephasesignalsbutwiththepromptanddelayedphasesignalslinearlydetrendedtomoreclearlyshowtheirstructure.Noticethatthetwosignalsarenoisyyetfolloweachotherwitha2seclag.Thetime-delayedcombinationiscomputedfromthenon-detrendedsignalsandshowsfarlessnoise.

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LSDsofthetime-delayedcombinationforvariousvaluesofshift.Thebestnoiserejectionisreachedforshift=2:0005625secwhichiswhatisplottedinthepreviousgureandusedforallcalculations. signalsishighlyillustrative.Figure 3{33 showsthephasenoiseLSDsofbothinputsignalsandthetime-delayedcombination.Thepromptanddelayedsignalshavespectrathatarenearlyidentical,bothfollowingnoisethatisroughly1 3{44 andassumingd(t)=p(t+EPD),thetime-delayed

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Linearspectraldensitiesofthephasenoiseoftheprompt(inblue),delayed(ingreen),andtime-shiftedcombination(inred)phasesignals.

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combinationcanbeexpressedintermsofthetimingerrorinEquation 3{43 as 3.3.2 ,themaximumtimingerroris1 2tsamp.AlloftheopticalTDIexperimentsare TimingsuppressionlimitsofEquation 3{47 forvariousvaluesofplottedalongwiththespectrumofthetime-delayedcombination(Shifted&Subtractedshowninred).

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digitizedat80kHzsotsamp=12:5sec.ThetimingsuppressionlimitofEquation 3{47 for=1 2tsamp=6:25106secisplottedinblackalongwiththelinearspectraldensityofthetime-delayedcombinationsignalinredinFigure 3{34 .Thetime-delayedcombinationresultsliebelowtheworst-casetimingerrorsuppressionlimitsowecansurmisethatEPDisclosertoadatapointthaninthemiddleoftwopoints.ToestimatehowfarofromEPDthetime-delayedcombinationresultsare,suppressionlimitsfortwomorevaluesofarealsoplottedinFigure 3{34 .Thehighernoisestructuresofthetime-delayedcombinationtcloserto=1 8tsamp=1:5625sec,showninpurple.However,thebaselineofthetime-delayedcombinationsitsatahigherlevel,closertothesuppressionlimitof=3 8tsamp=4:6875105sec,suggestingthatwearestartingtoseeothernoisesourceintheexperiment.Toimprovethenoiserejectionofthetime-delayedcombinationwhereitistimingsuppressionlimited,wewouldhavetointerpolatebetweendatapointstoreducethevalueof. 3{35 isverysimilartotheelectronicallysplitexperiment.The Experimentalsetupforopticallysplittwo-signalTDItest.Twophotodiodesmeasurethe35MHzbeatsignalandbothsignalsaredemodulatedto800kHz.OnesignalisdelayedwiththeEPDunitandthenbothsignalsaredemodulatedto10kHz,recorded,andrunthroughthephasemeter.VCOstandsforvoltage-controlledoscillator,LPFislow-passlterandNIDAQisaNationalInstrumentsdataacquisitionboard.

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opticalbeatsignalisat35MHzsoeachphotodiodesignalismixeddownwithavoltagecontrolledoscillator(VCO)toroughly800kHzandeachsignalislteredwithaSRSpreampwiththecornerfrequencysetat1MHz.OneofthesignalsisdelayedwiththeEPDunit2seconds,bothsignalsaremixeddownwithafunctiongeneratortoroughly10kHz,ltered,andthenrecordedusingthesameNationalInstrumentsDAQwithasamplingrateof80kHz.TheEPDunit,oscillator,andalsotheDAQhavetheirtimebaseslinkedtoacommonrubidiumstandard.Thesignalswererecordedfor145secondswhichisnearthecomputationallimitofthephasemeterduetothelargelesize.OncethesignalswererecordedwiththeDAQ,thedatawastakeno-lineandrunthroughthesamephasemetertogivethepromptanddelayedphase,p(t)andd(t),respectively.Theprocedureforcomputingthetime-delayedcombinationS(t)isidenticaltoaboveinSection 3.5.2.2 .Iagaincomputedthetime-delayedcombinationforvariousvaluesofshiftandfoundtheoptimalsuppressionforthesamevalueof2:0005625sec.ThelinearspectralnoisedensitiesareplottedinFigure 3{36 .Thereisagaingoodnoisesuppressioninthetime-delayedcombination,reachingjustoversixordersofmagnitudenear20mHz.However,theredoesappeartobeadditionalnoiseintheexperimentatthelowerfrequenciesthatarenowvisiblebytakingdataanadditional100seconds.Thismaybeduetothesignalsbeingtakenfromdierentphotodiodesalthoughitisdiculttosaywithoutothermeasurementsattheselowerfrequencies. 2tsampduetothetimingresolutionexpressedinEquation 3{47 iscomputedfortheoptically-splitexperimentandplottedalongwiththespectrumofthetime-delayedcombinationinFigure 3{37 .Thespectrumofthetime-delayedcombinationliesmostlybelowthetimingsuppressionlimitof=1 2tsamp,asitdidfortheelectronically-splitexperiment.

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Linearspectraldensitiesoftheprompt,delayed,andtime-delayedcombinationsignalsfortheoptically-splitexperiment.

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Inordertoestimatethetimingerror,Equation 3{47 canbesolvedfortogivetheexpression: =1 3{38 .Fromthespectrumofthetime-delayedcombination,wecanseethatthereisanadditionalnoisesourcebetweenabout40and400mHz,whichisalsoshowninanincreaseinthevalueofforthesefrequencies.Lookingbetween400mHzand100Hzwheretheltersinthephasemeterdonotaectthesignals,aroughestimateofis2106sec.Thetimingsuppressionlimitfor=2106secisplottedalongwiththetime-delayedcombinationandworst-casetimingerrorsuppressionlimitinFigure 3{37 astheoptimizedsuppressionlimit.Asyoucan Linearspectralnoisedensityofthetime-delayedcombinationandsuppressionlimitsduetotimingresolutionfromEquation 3{47

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Estimateofthetimingerror,,fortheopticallysplitexperiment.Between400mHzand100Hz,isroughly2106sec. see,thereisgoodagreementbetweenthisoptimaltimingsuppressionlimitandthespectrumofthetime-delayedcombination. 3{39 showstheresultinginterpolateddataset,stillwith1:25105secbetweenpointsbuteachpointislocated2106secfromthemeasureddata.Usingthisshifteddatasetofthedelayedphaseandtheoriginalpromptphase,Iagainrecomputedthetime-delayedcombination,shiftingthedatasetsthesame2:0005625secwhichresultsinatotalshiftof2:0005645sec.Thelinearspectraldensitiesofthetime-delayedcombinationsfortheoriginaldatasetsandthedatawithonesetadditionallyshiftedisshowninFigure 3{40 .Lookingagainatthefrequenciesbetween400mHzand100Hz,theLSDofthetime-delayedcombinationusingtheinterpolateddata

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Plottoshowlinearinterpolationofdataanadditional2106secfromthemeasureddata. Linearspectralnoisedensitiesofthetime-delayedcombinationsusingtheoriginalsamplingandforinterpolatingtogiveanadditional2106secshift.

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showsmorenoiserejection.Thus,thetimingerrorhasbeenreducedandweareclosertothetruevalueofEPDwithshift=2:0005645sec. Sectionof10-pointinterpolateddata. Inordertoreducethetimingerrorfurtherandimprovethetime-delayedcombinationnoisesuppression,Iinterpolatedadditionaldatapointsbetweenthesampledpoints.Inordertogettoaresolutiongreaterthanwhatwascomputedintheabovesectionwithanon-integerlinearshiftof2sec,thereneedstobeatleastsevenadditionalpoints(1:25105sec=2106sec=6:25).Usinglinearinterpolation,Icreatedanadditionalninepointsbetweeneachmeasuredpoint,makingtenpointsforeach12:5sec.Sincetheoriginalphasedataleswereverylargeandcumbersometoworkwith,Iinterpolatedonlyasmallsectionoftheoriginalphasedata,workingwiththerstroughly5%.Figure 3{41 showsasectionoftheinterpolateddata.Boththepromptanddelayedphasedatawereinterpolated,ineectcreatingadatasetwithanewsamplingrateof800kHz.Thetime-delayedcombinationwascomputedforvaluesofshiftaround

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2:0002645sec,asshowninFigure 3{42 .Thelowestcurve,andhencebestnoisesuppressionisachievedwithshift=2:00056375sec.NoticethattheLSDforshift=2:00056500secofonemorepointisclosertothelowestLSDthanthatforshift=2:00056250secofonelesspoint.Therefore,theresultsfromtheinterpolationsuggestthatthetruevalueofEPDliesbetween2.00056375and2:00056500sec,ttingnicelywithourestimateof2:0005645seccomputedintheprevioussection.Figure 3{43 plotstheoriginaltime-delayedcombinationspectrumandthelowesttime-delayedresultfortheten-pointinterpolateddata.TheLSDfortheten-pointinterpolationresults,liesbelowtheoriginalresultsinseveralareas.Intheremainingopticalexperiments,theone-armandtwo-armsetup,ananalysisofinterpolationwillnotbeperformed.Greaternoisecancellationmaybeachievedfortheseexperimentsaswell;however,itwouldimprovebylessthananorderofmagnitudeandfollowalongthesamelinesasdiscussedhere.Inallmeasurements,thetimingerrorwillbelessthan6:25sec. LSDsofthetime-delayedcombinationforvariousvaluesofshiftonthe10-ptinterpolateddata.

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ComparisonoftheLSDsofthetime-delayedcombinationsoftheoriginal80kHzdata(red)andtheinterpolated800kHzdata(cyan). Themethoddescribedhereforinterpolationisaverysimpleapproach;however,amoresophisticatedformofinterpolationwillbenecessaryfortheLISAdatastreams.TheLISAdatastreamswillbedown-sampledtoreducethebandwidthneededtosendthesignalsbacktoearth.Thisdown-samplingrateis3HzwhichisfarbelowthetimingresolutionneededinorderforTDItocancelthenoisetotherequiredlevel.Amethodofinterpolation,suchasthosebasedwithfractional-delayltering,willbeappliedtotheLISAdatastreams[ 24 ], 3{44

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RepresentationoftheLISAcongurationsimulatedwiththeone-armmeasurement.ThetimebetweenspacecraftiscreatedwiththeEPDunit. ThedetailsoftheexperimentalsetupareshowninFigure 3{45 .The12MHzbeatsignalbetweenthetwocavitystabilizedlasersismeasuredattwophotodiodesandbothsignalsaredemodulatedwithafunctiongeneratorto800kHzandlteredwithSRSpreamps.Oneofthesignalsisdelayedfor2secondswiththePentekEPDunitandthenusedtophase-lockL2.Thephase-lockerrorsignalisformedbymixingtheL2beatsignalwiththedelayedsignalfromL1.Thephase-lockloopboardusestheerrorsignaltogiveaninputtotheL2laserPZTcontrollertochangeitsfrequency.Theelectronicsaredesignedtokeepthiserrorsignalatzerosooncetheservoislocked,L2willbethesamefrequencyatthedelayedL1signal,inthiscase800kHz.Thetransferfunctionofthephase-lockloopboardisshowninFigure 3{46

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Experimentalsetupfortheone-armmeasurement.Thebeatsignalbetweenthetwocavitystabilizedlasersisdelayedandusedtophase-lockanadditionallaser,L2.ThesignalfromL2isalsodelayedandthenrecordedalongwiththepromptL1-RLbeat.LPFstandsforlow-passlter,PLLisphase-lockloop,andNIDAQisNationalInstrumentsdataacquisition.

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Transferfunctionofthecontrolelectronicsofthephaselockloopusedintheoneandtwo-armexperiments. recordedonthesameNationalInstrumentsDAQaswithallotheropticalTDIexperiments.Bothfunctiongenerators,EPDclock,andDAQtriggeringarelinkedtoarubidiumstandard.ThetworecordedsignalsarerunthroughthesameSimulinksoftwarephasemeterandthesameprocessofndingtheoptimumtime-delayedcombinationofEquation 3{44 isperformedasinthepreviousopticalexperiments.Theoptimumsuppressionwasreachedwithshift=4:0011375secwhichisonedatapointmorethan2shiftofthepreviousexperiments.Thephasenoisespectraoftherecordedsignalsandtime-delayedcombinationareplottedinFigure 3{47 .Again,weseeexcellentnoisesuppressionfromtheinputsignalsalthoughthereisincreasednoiseatlowfrequencies.Theincreasednoiseatlowfrequenciesfromtheoptically-splitexperimentisduetotheinabilityofthephase-locklooptoimposephasechangesdowntothelevelseenbefore.Irantheone-armexperimentadditionaltimeswithonlytherstdelayandalsowithalldelaysremoved.Figure 3{48 showsthephasenoise

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LSDsoftheprompt,delayed,andtime-delayedcombinationsignalsfromtheone-armmeasurement.

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ComparisonofLSDsofthetime-delayedcombinationfortheone-armmeasurementwiththeEPDunitremovedfromdierentpartsoftheexperiment. spectrumofthetime-delayedcombinationforthesedierentexperiments.Thereisnoappreciabledierencebetweentheresultswithonedelayandwithtwodelays.Thedierenceintheshapeofthesedelayedcurvesfromthenon-delayeddataislikelyduetoachangeingainsettingsofthephase-lockloopboardbetweendataruns.Also,theEPDunitproducesasignalwithslightlylessthanhalfoftheinputamplitude.Inthesedierentone-armmeasurements,thepromptsignalfromL1hadthesameamplitudebutitwasnotpossibleforthesignalfromL2todosoaswell.Hence,thenoiseinL2withnodelayswouldhaveappearedslightlylargerthantherunswithdelays.Thisgureexempliesthatparticularlyasthecomplexityofthebenchtopsimulatorincreases,therearenon-stationarysystems

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present.However,westillseenearlyveordersofmagnitudecancellationoflaserphasenoiseasareproducibleresult. 3{49 andisanalogoustothecongurationdiscussedinSection 2.2.3 RepresentationoftheLISAcongurationsimulatedwiththetwo-armmeasurement.ThetimebetweenspacecraftiscreatedinthelabwiththeEPDunit. Theexperimentalsetupisanalogoustotheone-armsetupwiththesymmetryofanadditionalarm.ReferringtoFigure 3{50 ,the59MHzbeatsignalbetweenthetwocavity-stabilizedlasersismeasuredattwophotodiodesandbothsignalsaredemodulatedwithafunctiongeneratorto800kHzandltered.TheltersusedareStanfordResearchSystemspreampliers(SRSpreamps)andeachonehastwo

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Experimentalsetupforthetwo-armmeasurement.Theincorporationofthefourthlaser,L3,isidenticaltothatforL2exceptforadierentdelaytimeofthesignals.LPFstandsforlow-passlter,PLLisphase-lockloop,andNIDAQisNationalInstrumentsdataacquisition.

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outputs,onewith50impedanceandtheotherwith600.TheEPDinputsare50sothe50outputfromeachoftheStanfordampsisusedtodelaythesignalsforphase-lockingL2andL3.ThepromptsignalfromL1istakenwithoneofthe600outputs.Thephase-locksignalforL2isdelayedthroughtheEPDunitwith2seconds;thephase-locksignalforL3isdelayed3seconds.Thesamephase-lockelectronicsareusedtolockL2asintheone-armmeasurementsandananalogboardwiththesamedesignandsimilarperformanceisusedtophase-lockL3.L2andL3beatwiththecavity-stabilizedreferencelaserontwophotodiodeseach;oneisusedtoformthephase-locklooperrorsignalandtheothersignalisdelayed.ThesignalfromL2isdelayedthesame2secondsandL3isdelayed3seconds.WeinitiallyexperiencedsomeproblemsusingallfourchannelsoftheEPDunitwithimplementingthesecondsetofchannelsatadierentdelaytime.Thenewlyutilizedsecondsetofchannelsgaveanincreasednoiseoortothesignalsandproducedglitchesthatwerevisiblewhenmonitoringthesignalsonanoscilloscope.AlteringofthedelaycodeontheEPDunitxedthisproblemandweareabletodelayfourchannelswitheachsetoftwochannelsatadierentdelaytimewithnoadditionalnoise.ThedelayedsignalsfromL2andL3andthepromptsignalfromL1arealldemodulatedwithanotherfunctiongeneratordownto10kHz,ltered,andrecordedwiththeNationalInstrumentsDAQ.Whiletheschematicshowsallthreesignals,onlytwoofthemcanberecordedatatimesincetheDAQhasamaximumsamplingrateof200kHzforallchannelsandthephasemeterrequiressignalssampledat80kHz.Again,bothfunctiongenerators,EPDclock,andDAQtriggeringarelinkedtoacommonrubidiumstandard.ThetwosignalsthatarerecordedatatimearerunthroughtheSimulinksoftwarephasemeterandanalyzedinthesamefashion.

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Photographofthebenchtopsimulatorduringthetwo-armexperiment.L1isintheleft-foregroundwithRLandL2successivelybehindit.L3isthelaserontherightnearthetankholdingthereferencecavities.

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LSDsofthetwodelayedsignalsandtheirtime-delayedcombinationfromthetwo-armmeasurement. Inthisexperiment,bothofthesignalsrecordedfromeacharmaredelayed.ThesignalrecordedfromL2,S2(t),isdelayed22'4secfromthepromptL1signal,S1(t).Similarly,S3(t)isdelayed23'6secfromS1(t).Therefore,S3(t)isdelayed2322'2secfromS2(t).Thetime-delayedcombinationbetweenthephasesofS2(t)andS3(t),2(t)and3(t),isformedbypost-processshifting3(t)backbythedierenceinthetotaldelaytimes:

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LSDsofsignalsinthetwo-armexperimentwiththesignalselectronically-splitfromonephotodiodeandwithtwophotodiodesusedtooptically-splitthesignals. L3andL1itisshift31=6:003625sec.Forthetwodelayedsignals,shift23=1:9994125secwhichisonedatapointlessthanshift31shift21.Figure 3{52 showsthelinearspectraldensitiesofthetwodelayedsignalsandthistime-delayedcombinationusingthisshift.Eventhoughourdatanowincludesanotherlaserandadditionalelectronics,westillseeexcellentphasenoisesuppressionthatagainreachesnearly5ordersofmagnitudeat0:1Hz.IntheinitialopticalTDIexperimentswithoutanyphase-locking,weconjecturedthatsplittingthesignalopticallymayhaveaddedsomeadditionalnoiseatlowfrequencies.Suchnoisewouldlikelybeduetopointingonthesmallareaphotodiodes.Althoughthenoiseseenintheoptically-splitexperimentwasatafewtimes104cycles=p

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arecomparedwiththeresultsoftheexperimentwithtwophotodiodesinFigure 3{53 .Thetworesultsareroughlythesame,withtheelectronically-splitrunbeingslightlyhigherthantheoptically-splitrun.Thisislikelyduetotwothings.First,byusingonlyonephotodiodeateachbeatandelectronicallypower-splittingthesignals,theamplitudelevelswillbeslightlydierentthanfortheoptically-splitsignals.Generally,Iwouldexpecttheelectronically-splitsignalstobebetterbalancedwitheachotherthantheoptically-splitsignalsandhencehavelowernoisebuttheremayhavebeendierentcablesordierentlevelsintheStanfordampsorsomethingelseslightlydierenttocauseaslightlyelevatednoiseduringthatrun.Second,whentheelectronically-splitexperimentwasrun,wewerestillexperiencingproblemswiththeEPDunitsoallofthesignalsfromL2andL3weredelayedthesametime.ByhavingadierentdelaytimeforthesignalsofL3intheoptically-splitexperiment,wemayjusthavegottenluckyandhadasamplingpointfallclosertotheactualdelayoftheexperiment,causingthetimingerrortobelessandhencethesuppressionslightlygreater.Ifeelthatthisisthelikelycase.Interpolationcouldbeusedtodigintothedatasetsfurtherandtrytoconrmthisbutthedierenceisminimalandtheexperimenthasshownthatoptically-splittingthesignalsisnotaddingnoisetotheexperiment.Tofurtheranalyzethetwo-armsystem,themeasurementswerealsorunwithonlytherstdelayandwithnodelaysimplemented.Figure 3{54 showstheresultsfortheexperimentswiththeEPDunitbypassed.Theresultsforthetwoarms,L1-L2andL1-L3,arenearlythesamewithanydierenceprobablyduetothedierenceinphase-lockloopsusedtoimposethephaseofL1ontoL2andL3.Thismeasurementessentiallygivethephase-lockloopperformance.Identicalmeasurementswerealsoperformedwiththerstdelayimplementedandresultsshowednoappreciabledierencefromtheresultswithnodelays.Thesetime-delayedcombinationresultsfromtheexperimentswithnodelaysandonlythe

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LSDsofthesignalsinthetwo-armexperimentwithnodelaysimplemented. Comparisonofthetime-delayedcombinationsfromthetwo-armexperimentwithvariousdelaysimplemented.

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rstdelayareplottedalongwiththetime-delayedcombinationresultsforthefull,two-armwithtwodelaysinFigure 3{55 .Allcurvesfollowasimilarshapeandarewithinanorderofmagnitudefromoneanother.Sincedierentexperimentaldelaytimeswereimplementedforthesedierentcurves,muchofthedierenceisduetotheresultingdierenttimingerrors,,inthedierentcombinations. 2{10 and 2{11 ofSection 2.2.3 .Inthetwo-armexperimentdescribedabove,directlycombiningthesignalsfromL1andoneof

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thefarspacecraftlasersgivesasignalatDCsinceahomodynephase-lockisused.Thesignalsarenoisysoasimpleanalogphasemeterwillwrapandthesoftwarephasemeterthatisusedrequires10kHzsignalssowearecurrentlyunabletoreadthephasefromsuchsignals.ItriedtoformtheLISAsignalsbyaddinganadditionaldemodulationafterthedelayofthesignalfromL1tolockthefarspacecraftlasersat500kHz.Thisthenallowedmetochangethenaldemodulationstageto310kHzagainproducingsignalsat10kHztoberecordedbutthefrequenciesmustnothavebeenfarenoughapartandthephasemeterwasnotabletoreadoutthecorrectphase.WiththenewEPDdesignundernaldevelopmentthatusesaphasemetertodown-samplethedata,itwillbemucheasiertocreatethesefullLISAsignalsbyincludingthefrequencyosetinthephasemeter.ThephasemeterontheEPDallowsfortheoutputsignaloftheEPDtoautomaticallybeatafrequencymuchdierentthantheinputsignal(ontheorderofMHz,aswilloccurinLISAwiththeDopplershifts)withouttheadditionaldemodulationstagethatcausedproblemsforme.Thesignalsproducedshouldhavenoadditionalnoise,unlessthephasemeteraddsit,andwillbeabletobeusedtodirectlytestthefullTDIalgorithm. 27 ],Whilefurtherstudiesofrst-generationTDIandarm-lockingwillbepursued,thereareadditionalmodicationsthatcanbemadetothesimulatortoallowforbroadeningthescopeofitsapplicability. 23 29 ],To

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bringthiseecttothebenchtopsimulator,acousto-opticmodulators(AOMs)canbeaddedtothebeamlinestoproducetheDopplershiftsbetweenspacecraftandtheEPDcanbeprogrammedtoproduceanon-constantdelay.

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4{1 ,therstresultsofapplyingTDI-likecombinationstoopticalsignalsshowssixordersofmagnitudeoflaserphasenoisesuppressiontonearlyreachLISAphasenoiserequirements.Theadditionalcomplexityofthefull,two-armsetupofthesimulatorstillachievesresultsthatareonlyanorderofmagnitudefromLISArequirements.TheUFLISAbenchtopsimulatorisavaluabletoolforinvestigatingmanyaspectsofLISAinterferometry.ThebenchtopsimulatormakesuseofanelectronicphasedelaytechniquetosimulatethelonglighttraveltimebetweenLISAspacecraftaswellasaLISA-likephasemetertomeasuresignals.The Suppressionofnoisefromtheinputsignalstothetime-delayedcombination. 90

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foundationofthesimulatoristwo,cavity-stabilizedlaserswhichproviderealistic,LISA-likefrequencynoisetoallmeasurements.Thefrequencystabilityresultsmeasuredwiththecounteratlowfrequenciescanbecomplementedbyconvertingthephasenoisereadwiththephasemeterathigherfrequencies.Figure 4{2 showsboththesesetsofdatawhichlineupwellattheiroverlappingfrequencies.Themeasuredfrequencystabilityofoursystemis10Hz=p Frequencystabilityresultsfromboththecounterandthephasemeter(PM).

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20 ] c=r M=464m=s(A{2)wherethegeneralgasconstantR=8:3145kgm2=(s2molK),themeanmolarmassforairM=0:0288kg=mol,androomtemperatureisT=293K.Forthe4inchdiameteropeningofaLF100ange, 92

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wherelisthelengthofthepipe.Thetotalconductanceoftheangeandpipeisgivenbytheadditionofthereciprocals: 1 1

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94

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whichgivesthewidthofeachfrequencybin.Therstnon-zeroabscissavalueofthetransformeddatagivesfres.Inordertoputthelinearspectrumintoadensity,dividebythesquarerootofthefrequencyresolution,givingthefamiliarunitof\perrootHertz."Inordertoreducetheeectofthediscontinuityofthedatanotrepeatingexactlywithinthesamplingofthetransform,fres,itishelpfultowindowthedatainthetimedomainbeforeperformingthetransform.XmgracehasmanydierentwindowswhichmaybeselectedintheFouriertransformdialogbox.Forourcalculations,weusetheHanningwindow[ 11 ] 2[1cos(2j N)];j=0:::N1(B{3)ItisimportanttonotethatXmgracedoesNOTnormalizethewindowfunctions.FortheHanningwindow,theresultantDFTneedstobemultipliedbyanormalizationfactorofp

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[1] A.AbramoviciandJ.Chapsky,FeedbackControlSystems:AFast-TrackGuideforScientistsandEngineers,KluwerAcademic,Boston,2000. [2] J.W.Armstrong,F.B.Estabrook,andM.Tinto,\Time-delayinterferometryforspace-basedgravitationalwavesearches,"Astrophys.J.527,pp.814-826,1999. [3] JohnWilliamBertholdIII,Ph.D.Dissertation:DimensionalStabilityofLowExpansivityMaterials{TimeDependentChangesinOpticalContactInterfacesandPhaseShiftsonReectionfromMultilayerDielectrics,UniversityofArizona,1976. [4] [5] E.Black,NotesonthePound-Drever-Halltechnique,LIGOtechnicalnoteLIGO-T980045-00-D,April16,1998. [6] ThomasDelker,Ph.D.Dissertation:DemonstrationofaPrototypeDual-RecycledCavity-EnhancedMichelsonInterferometerforGravitationalWaveDetection,UniversityofFlorida,DepartmentofPhysics,February14,2001. R.W.P.Drever,J.L.Hall,F.V.Kowalski,J.Hough,G.M.Ford,A.J.Munley,andH.Ward,\LaserPhaseandFrequencyStabilizationUsinganOpticalResonator",Appl.Phys.B31,pp.97-105,1983. [8] G.R.Fowles,IntroductiontoModernOptics,2ndEd.,GeneralPublishingCompany,Toronto,Canada,1975. [9] V.Greco,F.Marchesini,andG.Molesini,\OpticalcontactandvanderWaalsinteractions:theroleofthesurfacetopographyindeterminingthebondingstrengthofthickglassplates,"JournalofOpticsA:PureandAppliedOptics,3,pp.85-88,2001. 21 ]. 96

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[10] E.Hecht,Optics,ThirdEd.,Addison-Wesley,Reading,Massachusetts,1998. [11] G.Heinzel,A.Rudiger,andR.Schilling,\SpectrumandspectraldensityestimationbytheDiscreteFouriertransform(DFT),includingacomprehensivelistofwindowfunctionsandsomenewat-topwindows,"InternaldocumentoftheMax-Planck-InstituteforGravitationalPhysics,HannoverGermany,February15,2002. [12] J.Hough,D.Robertson,H.Ward,andP.McNamara,\LISA-theinterferometer,"Adv.SpaceRes.,Vol.32,No.7,pp.1247-1250,2003. [13] F.P.IncroperaandD.P.DeWitt,FundamentalsofHeatandMassTransfer,4thEd.,JohnWiley&Sons,NewYork,1996. [14] S.S.KachkinandY.V.Lisitsyn,\Optical-contactbondingstrengthofglasscomponents,"Sov.J.Opt.Technol.,47,pp.159-161,1980. [15] S.S.Kachkin,G.V.Listratova,andV.A.Ryzhakova,\Eectofscaleandtimefactorsonthemechanicalstrengthofanopticalcontact,"Sov.J.Opt.Technol.,56,pp.110-112,1989. [16] V.LeonhardtandJ.B.Camp,\Spaceinterferometryapplicationoflaserfrequencystabilizationwithmoleculariodine,"OpticalSocietyofAmerica,2005. [17] LightwaveElectronicsSeries125/126User'sManual,D-0864Rev.B,June22,2002. [18] LISAPre-PhaseAReport,2ndEd.,MPQ233,1998. [19] G.Mueller,P.McNamara,I.Thorpe,andJ.Camp,\LaserfrequencystabilizationforLISA,"NASATechnicalPublicationTP-2005-212790,2005. [20] PfeierVacuumtechnicaldocument,\WorkingwithTurbopumps:Introductiontohighandultrahighvacuumproduction,"PT0053PE,September,2003. [21] A.E.Siegman,Lasers,UniversityScienceBooks,Sausalito,CA,1986. [22] P.R.Saulson,FundamentalsofInterferometricGravitationalWaveDetectors,WorldScientic,Singapore,1994. [23] D.A.Shaddock,M.Tinto,F.B.Estabrook,andJ.W.Armstrong,\DatacombinationsaccountingforLISAspacecraftmotion,"Phys.Rev.D,68,061303,2003. [24] D.A.Shaddock,B.Ware,R.E.SperoandM.Vallisneri,\Post-processedtime-delayinterferometryforLISA,"Phys.Rev.D,70,081101,2004.

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[25] B.S.Sheard,M.B.Gray,D.E.McClellandandD.A.Shaddock,\LaserfrequencystabilizationbylockingtoaLISAarm,"Phys.Lett.A,320,9,2003. [26] S.T.ThorntonandA.Rex,ModernPhysicsforScientistsandEngi-neers,SaundersCollegePublishing,FortWorth,1993. [27] JamesIraThorpe,Ph.D.Dissertationinpreparation,UniversityofFlorida,August,2006. [28] M.TintoandJ.W.Armstrong,\Cancellationoflasernoiseinanunequal-arminterferometerdetectorofgravitationalradiation,"Phys.Rev.D,59,102003,1999. [29] M.Tinto,F.B.EstabrookandJ.W.Armstrong,\TimeDelayInterferometrywithMovingSpacecraftArrays,"Phys.Rev.D,69,082001,2004. [30] A.Vecchio,\LISAobservationsofrapidlyspinningmassiveblackholebinarysystems,"Phys.Rev.D,70,042001,2004. [31] J.M.WeisbergandJ.H.Taylor,\TheRelativisticBinaryPulsarB1913+16,"ASP-CS,302,2003. [32] W.Winkler,K.R.DanzmannandR.Schilling,\Heatingbyopticalabsorptionandperformanceofinterferometricgravitationalwavedetectors,"Phys.Rev.A,44,11,1991. [33] H.D.YoungandR.A.Freedman,UniversityPhysics,9thEd.,Addison-WesleyPub.Co.,Reading,Mass.,1996.

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IgrewupmostlyinMichigan,nishingmyhighschooleducationatEastKentwoodHighSchoolnearGrandRapids.IthenattendedNorthParkUniversityinthegreatcityofChicago,earningaB.S.inphysicsandaB.A.inmathematicsin2000.IreceivedthegreathonorofbeingchosenfortheNorthParkUniversityOutstandingSeniorAward.Duringthesummers,IparticipatedinresearchwithanengineerattheFloridaSolarEnergyCenter,aspartoftheREUprogramatKittPeakNationalObservatory,andaftergraduationatGoddardSpaceFlightCenterintheLabforExtraterrestrialPhysics.Inthefallof2000,IbegangraduateschoolintheDepartmentofPhysicsatUF.Iinitiallybegandoingresearchwiththeinstrumentationgroupintheastronomydepartmentbutswitchedtotheexperimentalastrophysicsresearchgroupinphysicstowardtheendofmysecondyear.IbeganlearningaboutgravitationalwavedetectionworkingwiththeLIGOgroupandhadtheopportunitytovisitbothdetectorsitesaswellasparticipateintherstsciencerun.Myadvisor,apost-docintheLIGOgroupatthattime,receivedaprofessorshipinthedepartmentandwebegantobuildtheLISAlab.MuchofmyresearchwasconductedasaNASAHarriettG.JenkinsPre-doctoralFellow;anotherofmygreathonors.Asafellow,IhadtheopportunitytocompeteforsummerresearchgrantstospendtimeataNASAfacility.IwasawardedgrantstoworkatGoddardSpaceFlightCenterandtwiceattheJetPropulsionLaboratorytolearnmoreabouttheLISAmissionandcurrentresearch.Theseexperienceswereverybenecialandhelpedtoshapemythesisresearch. 99


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Permanent Link: http://ufdc.ufl.edu/UFE0013832/00001

Material Information

Title: Development of the UF LISA Benchtop Simulator for Time Delay Interferometry
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013832:00001

Permanent Link: http://ufdc.ufl.edu/UFE0013832/00001

Material Information

Title: Development of the UF LISA Benchtop Simulator for Time Delay Interferometry
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013832:00001


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Full Text










DEVELOPMENT OF THE UF LISA BENCHTOP SIMULATOR
FOR TIME DELAY INTERFEROMETRY
















By
RACHEL J.CRUZ


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006


































Copyright 2006

by

Rachel J. Cruz



































I dedicate this work to my husband, Alberto, whom I met in graduate school, and

to my dog, Ducati, who found me while I was in graduate school.















ACKNOWLEDGMENTS

Thanks go to my colleague, Ira Thorpe, for his excellent work in the lab.

Without his help, portions of this research would not have been completed. Special

thanks go to Daniel Shaddock for hosting my visits to JPL and for his guidance

and support. I thank my advisor, Guido Mueller, for his help, support, and

overseeing this project. Thanks go to Michael Hartman who helped with building

various electronics and aided in implementation of the benchtop simulator. Thanks

also go to Wan Wu for the use of his phase-lock loop board during the two-arm

TDI experiment.


















TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS .......... .. iv

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

LIST OF FIGURES ......... . .. viii

K(EY TO ABBREVIATIONS ....... ... .. xii

K(EY TO SYMBOLS ........ .. .. xv

ABSTRACT ............ ........... xvi

CHAPTER

1 BACKGROUND: GRAVITATIONAL WAVES AND THEIR DETECTION 1

1.1 Gravitational Radiation . ...... .. 1
1.2 Waves in General Relativity ...... ... :3
1.3 Detection ... . .. .. .. 4
1.3.1 Laser Interferometer Gravitational-Wave Observatory (LIGO) 7
1.3.2 Laser Interferometer Space Antenna (LISA) .. .. .. .. 8

2 LISA INTERFEROMETRY ....... .. .. 11

2.1 Optical Bench Signals ........ .. .. 12
2.2 Laser Frequency Correction . ..... .. 14
2.2.1 Laser Frequency Stabilization .... .. 15
2.2.2 Arm-locking ......... ... 15
2.2.3 Time Delay Interferometry ... .. .. .. 15

:3 ITF BENCHTOP SIMULATOR . ...... .. 20

:3.1 Frequency Stabilization ..... .. .. 20
:3.1.1 Optical Cavities ....... .. .. 20
:3.1.2 Zerodur Cavities ....... .. .. 22
:3.1.2.1 Optical contacting .... .. .. 24
:3.1.2.2 Cavity parameters .... .. .. 29
:3.1.3 Pound-Drever-Hall (PDH) Locking Scheme .. .. .. .. :30











:3.1.4 Experimental Setup . .... .. :35
:3.1.4.1 Expected stability .... .. .. :36
:3.1.4.2 Control electronics .... .. .. :38
:3.1.5 Results ............. ........ 42
:3.2 Design of Benchtop Simulator ...... .. 49
:3.3 Electronic Phase Delay (EPD) ...... .... 51
:3.3.1 Hardware ......... ... 5:3
:3.3.2 Timing Error ......... .. 5:3
:3.4 Phasenieter ......... . 55
:3.5 TDI Experiments ......... .. 56
:3.5.1 First Electronic Experiment .... .. .. 56
:3.5.2 Initial Optical Experiment ..... ... .. 59
:3.5.2.1 Experimental setup .... .. .. .. 59
:3.5.2.2 All ll-k- of signals .... .. .. 60
:3.5.2.3 Suppression limit .... ... .. 62
:3.5.3 Optically-Split Experiment .... ... .. 65
:3.5.3.1 Timing error estimate .... .. .. 66
:3.5.3.2 Non-integer linear shift of data .. .. .. 69
:3.5.:3.3 Interpolation ...... .. 71
:3.5.4 One-Arnt Experiment . .... .. 7:3
:3.5.5 Two-Arnt Experiment . .... .. 79
:3.5.6 Iniprovenients ......... .. 87
:3.6 Future Experiments ......... .... 88
:3.6.1 Second-generation TDI ..... .. 88
:3.6.2 Low Light Interferometry ...... .. 89
:3.6.3 Insertion of GW signals ...... .... 89

4 CONCLUSION ......... ... 90

APPENDIX

A VACITITA SYSTEM ......... .. 92

B LINEAR SPECTR AL DENSITIES
RISING THE DISCRETE FOURIER TRANSFORM .. 94

REFERENCES ........ ........ 96

BIOGRAPHICAL SK(ETCH ......... .. 99
















LIST OF TABLES

Table pagfe

:31 Properties of standard quality, expansion class 0 Zerodur.. .. .. .. 2:3

:32 Parameters of the cavities used to stabilize RL and L1... .. .. 29

:33 PZT tuningf coefficients of the lasers used for pre-stabilization. .. .. :38

















LIST OF FIGURES

Figure page

1-1 Agreement between the general relativistically predicted decay in orbit
with the observational data for a binary pulsar ... .. .. 2

1-2 Michelson interferometer with the beamsplitter at the origin and the end
mirrors located along the x- and y-axes. .... .. 5

1-3 Time series showing the effect of a gravitational wave on a ring of test
masses..................... 7

14The two LIGO sites and the other large-scale interferometric detectors. 8

1-5 Representation of the LISA spacecraft in orbit. .. .. .. 9

1-6 Depiction of LISA's heliocentric orbit ..... .... 10

2-1 A LISA proof mass on the optical bench .... .. 11

2-2 View of the Y-tube inside the LISA spacecraft .. .. .. .. 12

2-3 Simplified Phase A interferometry design of the bench and telescope. 13

2-4 Simple consideration of the LISA spacecraft as a Michelson interferometer. 16

3-1 Fields of an optical cavity ......... .. 21

3-2 Reference cavities made from mirrors optically contacted to Zerodur
spacers ......... .. 23

3-3 Cleaning optics in a class 100 cleanroom. .... .. 26

3-4 Samples of optic contaminants as viewed through a microscope. .. .. 28

3-5 The PDH setup ........... ..... .... 31

3-6 A beam in frequency space after passing through an EOM .. .. .. 32

3-7 Plot of PDH error signal ......... .... 34

3-8 Experimental setup for frequency stability measurement .. .. .. .. 36

3-9 Zerodur cavities inside of five 1 e. ris of thermal shielding .. .. .. .. 37

3-1 Circuit schematic of the RL PDH board. .........3










3-11 Circuit schematic of the L1 PDH board. .. .. .. 39

3-12 Measured transfer function of the RL PDH board .. .. .. 40

3-13 Measured transfer function of the L1 PDH board ... .. .. 40

3-14 Closed loop transfer function of RL. .... .. 41

3-15 Closed loop transfer function of L1. ..... .. 41

3-16 Fr-equency stabilization results . ..... .. 43

3-17 Fr-equency stabilization results plotted with the free-running noise and
the LISA requirement ... ... 44

3-18 Fr-equency stability results from both the counter and the phasemeter 45

3-19 Length stability of Zerodur cavity ...... .... 45

3-20 Plot of the error point noise of the PDH signal. .. .. .. 46

3-21 Temperature stability of the cavities ..... .. 47

3-22 OptoCad layout of benchtop experiment. .. .. .. 50

3-23 Schematic of electronic phase delay (EPD) technique. .. .. .. 51

3-24 Schematic of how the EPD unit is used to make a signal equivalent to a
LISA signal. ... . .. 52

3-25 Schematic of timing in the experiment. .... .. .. 54

3-26 Operational schematic of phasemeter design ... ... .. 55

3-27 Setup for the first electronic TDI experiment .. . .. 56

3-28 Baseline of electronic TDI experiment. ..... .. 58

3-29 Results of the electronic TDI experiment. .... .. 58

3-30 Schematic of initial optical experiment. ..... .. 59

3-31 Phase signals from the optical experiment. .... .. 61

3-32 The time-d. I we 4 combination for various values of Tshirt. . . 62

3-33 Phase noise results from the optical experiment. .. .. .. 63

3-34 Timing suppression limits for various values of A-r. .. .. 64

3-35 Experimental setup for optically-split, two-signal TDI test. .. .. .. 65










3-36 Linear spectral densities of the prompt, d. 1 li-o I and time-d. 1 li-. I
combination signals for the optically-split experiment .. .. .. 67

3-37 Linear spectral noise density of the time-d. 1 li-. I combination from
the optically-split experiment and the suppression limits due to timing
resolution ......... .. . 68

3-38 Estimate of the timing error, A-r, for the optically-split experiment. .. 69

3-39 Plot to show linear interpolation of data an additional
2 x 10-6se SCfTOm the measured data. .. .. .. 70

3-40 Linear spectral noise densities of the time-d. 1 li-. I combinations using
the original sampling and for interpolating to give an additional 2 x10-686C
shift. ............ ............ 70

3-41 Section of 10-point interpolated data. .... ... 71

3-42 LSDs of the time-d. 1 li-. I combination for various values of Tswef on the
10-pt interpolated data. ........ .. 72

3-43 Comparison of the LSDs of the time-d. 1 li-. I combinations of the original
and the interpolated data ........ .. 73

3-44 Representation of the LISA configuration simulated with the one-arm
measurement ........ . .. 74

3-45 Experimental setup for the one-arm measurement. ... .. .. 75

3-46 Transfer function of the control electronics of the phase lock loop used
in the one and two-arm experiments. ..... .. 76

3-47 LSDs of signals from the one-arm measurement. .. .. .. 77

3-48 Comparison of LSDs of the time-d. 1 li-. I combination for the one-arm
measurement with the EPD unit removed from different parts of the
experiment. ......... . 78

3-49 Representation of the LISA configuration simulated with the two-arm
measurement ... . .. 79

3-50 Experimental setup for the two-arm measurement. .. .. .. 80

3-51 Photograph of the benchtop simulator during the two-arm experiment. .82

3-52 LSDs of signals from the two-arm measurement .. .. .. 83

3-53 LSDs of signals in the two-arm experiment electronically-split and
optically-split ......... . 84











3-54 LSDs of the signals in the two-arm experiment with no d. 1 li-
implemented. ......... . 86

3-55 Comparison of the time-d. 1 li-. I combinations from the two-arm experiment
with various d. 1 .va implemented. ...... .. 86

4-1 Suppression of noise from the input signals to the time-d. 1 li-. I
combination. ......... . 90

4-2 Fr-equency stability results from both the counter and the phasemeter 91















K(EY TO ABBREVIATIONS


ADC: analog-to-digital converter

AIGO: Australian International Gravitational Observatory

AOM: acousto-optic modulator

BS: heant splitter

DAC: digital-to-analog converter

DAQ: data acquisition and control

DI: de-ionized

E&M: electrontagnetism

EAIRI: extrenme-nlass ratio inspiral

EOM: electro-optic modulator

EPD: electronic phase delay

ESA: European Space Agency

FPGA: field prograninable gate array

FSR: free spectral range

FWHM: full width half nmaxiniun

GEO: British-Gernian Cooperation for Gravitational Wave Experiment

GR S: gravitational reference sensor

GSFC: Goddard Space Flight Center

GW: gravitational wave

JPL: Jet Propulsion Laboratory

L1: laser 1

L2: laser 2

L3: laser 3










LIGO: Laser Interferometer Gravitational-Wave Observatory

LISA: Laser Interferometer Space Antenna

LPF: low pass filter

LSD: linear spectral density

LTP: LISA Technology Package

NASA: National Aeronautics and Space Administration

NCO: numerically controlled oscillator

NI: National Instruments

NPRO: non-planar ring oscillator

OPL: optical path length

PBS: polarizing beam splitter

PCI: peripheral component interconnect

PD: photodiode

PDH: Pound-Drever-Hall

PLL: phase-lock loop

PM: phasenieter

PZT: piezoelectric

RF: radio frequency

R FAM: radio frequency amplitude modulation

R L: reference laser

SC: spacecraft

SDR AM: synchronous dynamic random access nienory

SRS: Stanford Research Systems

TAMA: Japanese Interferonietric Gravitational-Wave Project

TDI: time delay interferometry

TEC: thermoelectric cooler

ITF: University of Florida










ULE: ultra-low expansion

VCO: voltage-controlled oscillator

VIM: velocity interface module

VIRGO: Italian-French Laser Interferometer Collaboration















K(EY TO SYMBOLS


c: The speed of light 3 x 108 m/s

6,,,:The K~ronecker delta function

G: The gravitational constant 6.6726 x 10-" Nuz2 2
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT OF THE ITF LISA BENCHTOP SIMULATOR
FOR TIME DELAY INTERFEROMETRY

By

Rachel J. Cruz

?1.i- 2006

C'I I!1-: Guido Mueller
Co-CHl I!1-: David Reitze
Major Department: Physics

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

to detect gravitational radiation currently scheduled for launch in 2015. LISA

will be sensitive in the frequency hand of 3 x 10- to 1 Hx to complement the

higher frequency observations from currently running ground-based detectors.

LISA expects to detect gravitational waves from sources such as galactic binaries,

merging super-massive black holes, and extreme-mass ratio inspirals. LISA uses

interferometry to monitor the distances between drag-free proof masses located

within the three LISA spacecraft; however, many orders of laser frequency

noise must he overcome in order to have the required sensitivity to make the

observations.

At the University of Florida, our group has been developing a LISA benchtop

simulator to investigate various aspects of LISA interferometry. Our simulator

uses cavi' i---0 .Ill~l.:ed lasers to produce signals with realistic, LISA-like frequency

noise. Our frequency stabilization results presented reach nearly 10 Hxl/ A at

100 mHx, which is similar to what is expected in LISA. In order to mimic the










long light travel time between spacecraft, we use a newly developed electronic

phase delay technique. Along with the LISA-like phasemeter that we use to read

out the signals, these components provide a solid basis for the LISA simulator.

I use our benchtop simulator to implement and test time delay interferometry

(TDI), a method responsible for reducing the noise in the LISA signals. These

experiments are also valuable system-level tests of the simulator. The results

of my TDI experiments show at 1 Hz about four orders of magnitude of laser

phase noise suppression to reach a few times 10-s <1 tI. 1,
at LISA requirements. At lower frequencies of the experiment, near 10 mHz, our

implemented TDI shows over six orders of magnitude suppression but with a noise

level of a few times 10-4 (1 I~ lz* .


XV11















CHAPTER 1
BACKGROUND: GRAVITATIONAL WAVES AND THEIR DETECTION

1.1 Gravitational Radiation

Newtonian physics has in most cases been an adequate description of how

mass interacts on Earth; however, Newton's laws do not hold true on cosmological

scales. The publication of Einstein's paper on gravitational waves in 1916 ushered

in the understanding that space and time are entwined concepts in the physical

world, not independent quantities as understood before.

Gravitational waves are seen as one of the fundamental pieces of our

theoretical picture of the universe. Einstein's predictions have been assumed to

be true for several decades. In 1975, Russel Hulse and Joseph Taylor discovered

a binary pulsar that, unlike previous pulsars observed, showed a decreasing

orbit over time. The fact of note is that this decreasing orbital period measured

experimentally exactly matches the predictions front loss of energy through

the emission of gravitational radiation given by the theory of general relativity

[31]. Figure 1-1 shows the experimental data for this pulsar over a time span of

more than 25 years along with the prediction front general relativity. For this

nionuntental discovery verifying relativity, Hulse and Taylor received the 1993

Nobel Prize in Physics.

Obviously, we would like to be able to detect these gravitational waves directly,

through experimental means and study them to learn more about their sources.

Gravitational waves open up a whole new window to view astronomical objects and

hence, should greatly aid in understanding how these objects exist and interact.




















O



S -5 ---5









SGeneral Relativity prediction






--35




1975 1980 1985 1990 1995 2000
Year

Figure 1-1. Figure showing the agreement between the general relativistically
predicted decay in orbit with the observational data for a binary
pulsar.
Figure copied with the kind permission of the Astronomical Society of
the Pacific Conference Series: Joel 1\. Weisherg and Joseph H. Taylor,
The Relativi~stic B.:,:r a, Pulswer Bl918 16. ASP-CS 302 (2003).









1.2 Waves in General Relativity

Analogous to E&M, the curvature of space-time described by the gravitational

field does not change instantaneously at all points in space when the source moves

but rather the information is limited to propagate at the speed of light, c.

From the theory, the space-time interval ds between two points is given in

general form by



where g,, is the metric and contains all the information about the space-time

curvature. In looking for gravitational radiation, we are interested in the small

perturbation, h,,, caused by a wave. The metric can then be described as


g,, = rlp, + hyV (1-2)


where rly, is the flat-space Minkowski metric. Known as the strain, h,, satisfies

the weak field limit of Einstein's field equation with a certain gauge choice and

coordinates to yield
1 82
(V2 C 2 ~ (1-3)

which has the familiar solution


h(t) = hei(k-x)t (4


which is a plane wave propagating through space in the k direction where the

frequency of the wave is given by

clk|
f = (1-5)
2xr

Recalling from E&M, the dominant term in radiation results from the time

variation of the electric dipole moment. However, since mass has only one charge,

there cannot be a contribution to the gravitational radiation from that term,

leaving the quadrupole term to contribute the strongest allowed component of










gravitational radiation. The reduced quadrupole moment is defined as


io=(~X"-x,- ,T2)pp~r~dV (1-6)


where by, is the K~ronecker delta function, r is the radial coordinate, and p is the

density function. The gravitational radiation is then described by


hy, RC4 P

where G is the gravitational constant of 6.6726 x 10-11 NVm2/kg2 and R is the

distance between the source and the observer. Using the moment of inertia for a

typical binary system, the strain can be approximated


& ~T"T (1-8)
roR

where ro is the half distance between the two orbiting objects and r, is the

Schwarzschild radius of each mass. Plugging in reasonable numbers for such an

astronomical system yields a strain of


he 1 x10-21 _19)


Due to the fact that the leading term for gravitational radiation is quadrupole,

most sources of gravitational waves are astronomical objects rotating around each

other or rotating objects with .-i-mmetry. Typical objects expected to be observed

include binary neutron stars and black holes.

1.3 Detection

Let us consider a Michelson interferometer with its arms aligned along the x-

and y-axes and the beam-splitter located at the origin, as shown in Figure 1-2.























IU -



Figure 1-2. Michelson interferometer with the beamsplitter at the origin and the
end mirrors located along the x- and y-axes.


Since light travels at c, events of the laser beam are described by


ds2 = guedX~LdXV = 0

=(rl4, + hy,: .i, IIdXV = 0 (1-10)


where
-100 0

0 100
rlsL =
0 01 0

0 00 1

and making the gaugfe choice of a transverse traceless

00 0 0

h = 0 hll h12 0
0 h12 11 0

100 0 0


(1-11)


coordinate system


(1-12)









where we will look at waves propagating in the z-direction. The terms of the strain

are defined

hll = h (z ct) (1-13)



h12 h (xZ Ct) (114)

where h+ and hx are two orthogonal polarization states. Considering light in one of

the arms, wi along the x-axis. The space-time interval along this arm is


C2 t2 + 116 li2x ft 2 = 0 (1-15)


By integrating this equation, we can solve for the time to traverse the arm:

dt 1 nixx(1-16)


where L is the length of the arm and -rL is the time to travel L. Performing the

same integration from TL to Tround--trip and doing some math, it can be found

that the difference in the total round trip time in the interferometer due to the

gravitational wave is
sin(x f ro) irfr 17
at = b-ro ei"(-7
x f To
where -ro is the unperturbed round-trip time. Therefore, the resulting phase

difference is
2xrc 2xrc sin(k f To) irfr 18
af (t) = at = bro ext 1-8
A x rf To

Since the value of h is on the order of 1 x 10-21, the phase difference will also be

a very small quantity. Performing the integration along the y-axis will yield the

same result but with a negative sign. A gravitational wave with polarization h+

will stretch one arm as it compresses the other and then compress the first as it

stretches the other, as illustrated in Figure 1-3.















Time
phase: 0 7/2 7C 37C /4 2 x

Figure 1-3. Time series showing the effect of a gravitational wave propagating in
the direction perpendicular to the page on a ring of test masses.


Several efforts are underway world wide to directly detect gravitational

waves. Ground-based detectors like the Laser Interferometer Gravitational-Wave

Observatory (LIGO) use interferometry to detect such minute length changes. The

Laser Interferometer Space Antenna (LISA) will similarly use interferometry but

over a larger baseline in space. In addition to interferometric detectors, there are

also efforts underway to detect gravitational radiation with resonant mass, or bar,

detectors and also through direct pulsar timing.

1.3.1 Laser Interferometer Gravitational-Wave Observatory (LIGO)

The Laser Interferometer Gravitational-Wave Observatory (LIGO) is funded

by the National Science Foundation and includes two detector sites in the United

States in Livingfston, Louisiana and Hanford, Washington. The LIGO detectors

are 4 km long Michelson interferometers. The Hanford detector also has a 2 km

interferometer alongside the 4 km, allowing for the possibility of measurements in

coincidence of all three detectors. LIGO is looking for gravitational wave signals in

the frequency range of 10 Hz to 10 kHz from sources such as coalescing compact

binary systems of neutron stars and black holes, supernovae, rotating pulsars,

and the stochastic background. In late 2005, the detectors reached their design

sensitivity, a great achievement in technology and engineering.

After running at design sensitivity for several years, a substantial upgrade of

the detectors is planned for 2009, termed Advanced LIGO. MIi r~ improvements for










Advanced LIGO include an increase in laser power from 10 IT to 200 IT, larger test

masses from 10 kg to 40 ky, and further improvements to the seismic isolation. The

design sensitivity of Advanced LIGO will be more than a factor of ten better than

initial LIGO hut requires that the associated effects of the improvements, such as

increased thermal lensing due to increased laser power, he handled appropriately.

The LIGO detectors comprise the large-scale interferometric detectors in North

America but there are other ground-based detectors in Europe and Asia including

GEO in Germany, VIRGO in Italy, and TAMA in Japan. There are also plans for

an additional detector AIGO, to be built in the southern hemisphere in Australia.





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Figure 1-5. Representation of the LISA spacecraft in orbit.


a triangular formation acting as semi-independent 1\ichelson interferometers. The

baseline distance between spacecraft will be five million kilometers and LISA must

he able to measure changes in this distance with an accuracy of tens of pm/A~H.

LISA will be in a heliocentric orbit, either 1 I__;k-:: or preceding the earth hv 200,

with the plane of the LISA constellation at an angle of 600 to the ecliptic. Over the

course of a year, the LISA constellation will rotate as it orbits. This will allow for

directional information of sources through the resulting non-homogeneous antenna

pattern of the detector.

LISA is designed to perform low frequency observations, in the range of

3 x 10- to 1 Hx. One of the exciting aspects of LISA is that in the detector's

frequency hand, there are many compact binary systems in our own Galaxy

that are already cataloged. This is a blessing in that once LISA is operational,

it should definitely detect gravitational waves from these verification binaries;

however, it is a curse in that these sources will provide a background noise to the

other signals. LISA, unlike the ground detectors, should also be able to detect

gravitational radiation from merging super-massive black holes located in distant





























Figure 1-6. LISA's heliocentric orbit is at an angle of 600 to the ecliptic.


galaxies and provide a comparison for the great advances in numerical relativity

on the modeling of such systems. Perhaps of greatest interest, LISA will also be

sensitive to stellar-mass black holes inspiraling to super-massive black holes, called

extreme-mass ratio inspirals (EMRIs), which provide a test-particle case for strong

gravity. The measurements of LISA will also aid in the effort to put upper limits on

the gravitational wave background remaining from the formation of the Universe.

It is exciting to have the opportunity to work on opening up this new window into

the physics of these various systems.















CHAPTER 2
LISA INTERFERO1\ETRY

As with all interferonietric gravitational wave detectors, LISA will monitor

the distance between freely-falling proof (or test) masses. Once set into orbit,

the spacecraft will fly around the proof masses, shielding them from solar winds

and other possible non-gravitational disturbances. The spacecraft will do this

by monitoring its position with respect to the proof mass with a combination

of capacitive sensors and optical interferometry in a gravitational reference

sensor (GRS) in order to allow for drag-free control. This demanding portion

of the technology will be tested in the LISA Pathfinder mission carrying the

LISA Technology Package (LTP) to be launched in 2009. The LISA spacecraft

house a total of six proof masses, two in each spacecraft, with each proof mass

being oriented at one of the two far spacecraft. The interferometry between




















Figure 2-1. Representation of a LISA proof mass, made of a gold-platinunt alloy,
on the optical bench.




























Figure 2-2. View of the "Y-tube" inside the LISA spacecraft that houses the two
telescopes and the two optical benches with the proof masses. Each
optical bench and telescope assembly is pointed at one of the two far
spacecraft.


the spacecraft is designed to monitor the distance between proof masses at the

ends of each of the LISA arms and detect changes in this distance from passing

gravitational waves.

2.1 Optical Bench Signals

Each proof mass is located on an optical bench along with various optics as

well as a telescope that is used to focus the incoming beam from the laser on the

far spacecraft. The two optical benches and telescope structures on each spacecraft

are located in a "Y-tube," shown in Figure 2-2.

The initial baseline design (Phase A) had the light from the far laser reflecting

off the proof mass and then combining with the leakage field from the local laser,

as shown in Figure 2-3. In the current design, referred to as strap-down, the light

from the far laser does not reflect off of the proof mass but rather off a mirror

attached to the optical bench and a separate measurement is made of the proof

mass with respect to the bench.













PD


BS



A/4 A/4





Laser 1

Figure 2-3. Simplified Phase A interferometry design of the bench and telescope.
Note: no back link interferometry is shown here. PD stands for
photodiode, BS is a polarizing beam splitter, and A/4 is a quarter-wave
plate.


In either case, the main measurement at each bench is the photodiode

measurement between the local laser and the far laser such that


S(t) = wlt12~( 2 r 1l(t) + h(t) (2-1)


where w12 1S the difference frequency between the local laser (Laser 1) and the

far laser (Laser 2), 4i is the phase noise of laser i, -r is the time for light to travel

between the local and far spacecraft (-r = L/c), and & is the change in phase

due to a gravitational wave. This signal is then a measure of an unequal-arm

interferometer with length mismatch equal to the length of the arm between the

two spacecraft. Therefore, the photodiode will see the full noise of each laser and

have to measure the phase of this noisy signal to LISA requirements.

The phase of the signal from one of the arms is given by


= 4 fL (2-2)









where f is the frequency of the laser and L is the arm length. The phase stability

is then
4x 4x bL 6 f
50 (6L L~) L( + ) (2-3)
c c L f

If neither the arm-length stability or frequency stability term is dominant,

6L 6 f f6L
4 5 f =(2-4)
L f L

The laser used is a Nd:YAG at 1064 nm. This means that the frequency stability of

the detector in order to see a strain of 10-21 needs to be less than 6 x 10-7 Hz/~H

in order to detect the desired gravitational radiation.

2.2 Laser Frequency Correction

With such a strict requirement on laser frequency noise, laser phase noise is

all is r~ issue for LISA interferometry. Within the project there are three main

methods of phase noise reduction and correction: pre-stabilization, arm-locking,

and time delay interferometry (TDI). A method of pre-stabilization using

reference cavities is discussed in the next section. Another possible method of

pre-stabilization that may be used is to lock the lasers to a resonant transition

of molecular iodine. Iodine stabilization provides certain benefits over cavities,

mainly that it provides an absolute frequency reference, yet achieves laser frequency

stability on the same order as for cavities [16]. However, pre-stabilization still

leaves eight orders of magnitude of frequency noise to overcome. Arm-locking is

a technique that uses the LISA arm as a stable reference. The arm length change

is on a yearly orbit cycle so in the LISA band, this distance provides an excellent

reference. However, arm-locking poses challenges for the controller and it is not

currently considered in the baseline design as the method to reduce all the orders

of magnitude needed in frequency noise. TDI, which will be discussed below in

Section 2.2.3, will be responsible for reducing the laser frequency noise to the level

of needed sensitivity.










2.2.1 Laser Frequency Stabilization

Each laser will be stabilized to an external reference or phase-locked to another

laser which is prestabilized. For cavity stabilization, different ultra-low expansion

materials and bonding techniques may be used. Whichever method is finally

chlosen mrust meet thle requirements of 30 1 + (10-3 Hz/A7T (plotted in

Figure 3-17 of Section 3.1.5). This requirement was originally set by experts in the

field surmising the limit these techniques could achieve. Since that time, several

experiments world-wide have shown that with some care, this level of stability is

readily achievable. Our frequency stabilization system at UF is described in Section

3.1 and the results are shown in Section 3.1.5.

2.2.2 Arm-locking

An additional possible reference for laser stabilization is the length of the

LISA arm itself [25]. The feedback control to lock a laser to a LISA arm will

be insensitive at frequencies of the reciprocal of the round-trip time for that

arm, which is roughly every 30 mHz. One way to gain some response at these

frequencies is to use common-mode arm-locking where you also use an arm-locking

signal from the second arm which will have insensitivities at slightly different

frequencies. If such a scheme were to be implemented in LISA, it is advantageous

to have the orbits of the spacecraft such that the difference between arm lengths is

maximized rather than minimized.

Due to these issues, arm-locking posses some challenges for implementation but

holds promising results. Currently, arm-locking is included in the LISA baseline but

not responsible for reducing any part of the frequency noise as further analysis and

experimentation is performed.

2.2.3 Time Delay Interferometry

The concept motivating TDI is to synthetically create an equal-arm configuration

to regain common mode rejection of laser frequency noise. Consider the LISA















L,







Figure 2-4. Simple consideration of the LISA spacecraft as a Michelson
interferometer. The solid lines represent the round-trip signals in each
arm, the dashed lines are these signals d.l 1 i-c II by the time of the other
arm.


spacecraft as a simple Michelson interferometer, as shown in Figure 2-4. The

main spacecraft, 1, acts as the beam splitter with arms going out to the two

far spacecraft, 2 and 3. For additional simplicity, assume the lasers on the far

spacecraft are phase-locked to the incoming beams, they then act as transponders

and can be considered equivalent to mirrors. By taking the signal from spacecraft

2 and delaying it the time to spacecraft 3 and likewise dl 1 i-ing the signal from

spacecraft 3 by the time to spacecraft 2 (the dashed lines in Figure 2-4), then

combining all of these signals, the signals in each arm It l.-. the same distance

and we have regained the noise rejection of an equal-arm configuration.

Let us consider this argument in more detail. The field from the main laser on

spacecraft 1 can be expressed


Ei(t) = Aeihteis,(t) (2-5)


where to is the angular frequency of the laser, ~(t) is the laser phase noise, and A is

a general amplitude. This field at spacecraft 2 is then


[El(t)]SC2 = Aeier(t-73)el 1 -73) i3t)


(2-6)









where -ri is the light travel time along the arm opposite spacecraft i (-ri = Li/c) and

hi is the change in phase due to a gravitational wave along arm i. Analogous to

Equation 2-5, the field of the laser on SC2 is


E2(t) = Aei"2teis,(t) (2-7)


The laser on spacecraft 2 is phase-locked to the incoming beam from SC1, hence,

the phase of the laser on SC2 is


2~t -o173 1 ~ 73) h3 () (2-8)


The field of the far laser back at the main spacecraft is then


[E2 (lSC1 = Aei@ (t-73) i2 (-73) i3 t) (2-9)


At the main spacecraft, this incoming field is combined with the local laser. For the

interferometry, the signal of interest is the phase of this combined field, S2-


S2 (t) = Arg {[E2 t~SC2 -[El(t)]sci}

= 11t + o173 2 7~t 3) h3( h3( 73

S2 (t) = 2Lo173 1 1lt l(t 273) h3( h3( -3) (2-10)


Similarly, the signal from SC3 will be


S3(t) = -2Lo172 1lt 1(t 272) + 2(t h2( -2) (2-11)


As you can see, if the spacecraft formed an equal arm interferometer where

-r2 = 3, then combining S2 and S3 WOuld cancel all terms except for those due to a

gravitational wave. While a straight combination of the signals from each arm with

unequal arm lengths will not cancel the laser phase noise, it is possible to get this

result with a different linear combination. The first-generation Michelson-X TDI










combination has the form [2]


S2 (t) S3 (t) S2 (t 272) + S3 (t 273) (2-12)


Applying this algorithm to the signals yields


ht ,()3h(t--, 2 3( 72 2 73 3,(t- 27r3) + 2(t 272)

-h3 3 72) (- 73 -72) (2-13)


which is independent of the laser phase noise. This combination is sometimes

referred to as an I!,!l -pulse" TDI signal for the eight distinct times a gravitational

wave would appear in the signal.

There is additional complexity from this simple realization to the actual

interferometry between the LISA spacecraft. In the Phase A design, an additional

interferometry measurement is made at the back of the proof mass with the local

laser and the laser from the other bench on the same spacecraft [18]. This back-side

measurement is included in the TDI combination and removes the noise due to

the optical bench [28]. The baseline design has been altered from the Phase A

design to separate the measurement of the low-power incoming beam with the

local beam from the reflection off the proof mass. Referred to as -lI Ilp-down,"

the beam from the far spacecraft reflects off of a mirror attached to the optical

bench, is then measured with the local beam and a separate measurement with

the two higher power beams on the spacecraft is made of the proof mass with

respect to the optical bench. An additional design change, called "frequency

swappingt or "cross-over," uses the laser from the other bench to interfere with

the weak incoming beam. Since the laser on the other bench is at a different

frequency than the local laser, the stray light on the bench will no longer add noise

to the measurement. Also, the above discussion of first-generation TDI assumes

a static LISA constellation. However, in orbit the spacecraft will be moving with






19


respect to each other and the time to traverse each arm will be a function of time,

-r(t). Second-generation TDI has additional terms to compensate for these effects

[23, 29].















CHAPTER 3
UF BENCHTOP SIMULATOR

At the University of Florida, we are developing a benchtop experiment to

simulate many aspects of LISA interferometry. The foundation of the simulator is

LISA-like lasers that are frequency stabilized to a pair of reference cavities. The

simulator makes use of an electronic phase delay (EPD) technique in order to

create long light travel times in the signals and reads these signals with a LISA-like

phasemeter. I have developed the frequency stabilization system as the foundation

of the first generation of the benchtop simulator and run a system-level test of the

electronic and optical components by performing TDI-like experiments. My focus

of the benchtop is for use of implementation of first-generation TDI, however, the

simulator can be used to investigate many issues pertaining to LISA interferometry,

especially as the complexity of the simulator is developed to include more aspects

of the spacecraft.

3.1 Frequency Stabilization

For our stabilization system, we use a stable cavity as a frequency reference.

3.1.1 Optical Cavities

Referring to Figure 3-1, the light fields of an optical cavity can be described

by


E, = itlE4, + rEb

Eb = r2Eaei2LW/c

Et = it2EneiLs/c

E, = itlEb 71Ein (3-1)
















t2, r2


Figure 3-1.






where w/2x


Diagram of fields for an optical cavity made of two mirrors with
transmission and reflection coefficients t and r, respectively. Ei, is
the input field, E, is the reflected field, Et is the transmitted field, and
E, and Eb, Which are measured at the inside edge of the first mirror,
make up the circulating field inside the cavity.


=f is the laser frequency and ri and ti are the reflection and


transmission coefficients of mirror i. For a lossless cavity, T2 + 2 = 1. The reflection

and transmission of a surface is RE T 2 and TE -2, Opciey rtn h ih

fields in terms of the input field, Ei,, the transmitted and reflected fields become

-tlt26iwL/c
Et= E

Er Ei12 (3-2)


When considering the transmitted and reflected fields, the above equations give the

transfer functions to be applied to the incoming field.


-tlt26iwL/c
1 rlr26i2wL/c

1 rlr26i2wL/c


T (w)


(3-3)


Assuming a lossless cavity, the transfer function of the reflected field can be

simplified to

Tr (LO) =

From the above equations, resonant behavior will occur when

2wL
= 2xu, nN E


(3-4)


(3-5)









which gives

f = n-LL nLEN (3-6)

The quantity c/2L is then the spacing between resonant axial modes and is defined

as the free spectral range or FSR.


FSR =(3-7)
2L

Another important characteristic of a cavity is the finesse, FT. The finesse is the

ratio of a cavity's free spectral range to its bandwidth or linewidth which is the

FWHM of the intensity of Et.
FSR
F = (3-8)
a cavity
The finesse is also given by

F = (3-9)
1 get
where get is the magnitude of the round-trip gain. Assuming no losses in the cavity,

get = TIT2 and for symmetric coatings rl = r2. We can then approximate


Fe (3-10)


where T is the transmissivity of the mirrors.

3.1.2 Zerodur Cavities

Our cavity spacers are made from Schott's Zerodur glass (Schott North

America, Inc., Elmsford, NY). The batch of Zerodur used for our spacers is certified

to have a mean coefficient of linear thermal expansion of a~ = 0.02 x 10-6 OK)-1

from 000 to 5000 (expansion class 0). This and other properties of the glass are

listed in Table 3-1. Three spacers were cut from one square rod, 38.10 mm on a

side, resulting in spacers of lengths 208.28 mm, 231.14 mm, and 259.08 mm. Each

spacer has a hole of diameter 8 mm bored through the center and a 2 mm air hole

from the center bore to the side to allow airflow after the mirrors are bonded on












































Figure :32. Reference cavities made front mirrors optically contacted to Zerodur
spacers .


Table :31.


Properties of standard quality, expansion class 0 Zerodur. Our spacers
were made front nelt #F0159, reference #240224. c0 is the coefficient
of linear thermal expansion front 000 to 5000, p is the density, a is the
thermal conductivity at 2000, and Crizerm~ is the thermal capacity, or
specific heat capacity.


SCoef. of Exspansion a Density po Thermatl Cond. m | Thermal Cap. Ctrizrnr
S2.0 x 10-8 K-l 2.5:3 x 10 kg/mn" 1.46 Wl/(m K) 800 J/(kg K)










the ends. The resonant frequency of the cavity depends on the stability of the

optical path length (OPL) within the cavity, which depends on the geometrical

stability of the spacer and of the bonds holding the mirrors to the spacer.

Our initial attempt at aligning the cavities proved problematic. One of the

cavities showed a "double heatingt of each mode. By adjusting the A/4-wave plate

before the cavity and watching the relative amplitudes of the two same resonant

modes, we were able to determine that there was some kind of polarization effect

causing slightly different resonant frequencies for the two orthogonal polarizations.

In order to determine if these problems were caused hv the mirrors, we

removed them from the Zerodur spacer and placed them on a steel cavity to

measure the finesse. The FSR of the metal cavity is 681 MHx; measuring the

linewidth gave a finesse of roughly 2500. I tried various methods of cleaning mirrors

in the lah and took numerous linewidth measurements giving values for the finesse

ranging from a few hundred to a few thousand- all lower than expected. In many

instances, contaminants could be seen on the mirrors, even after cleaning so it was

decided to move to a clean room to clean and hand the materials. After cleaning

in the cleanroom with the procedure described in the next section, the measured

finesse improved to values between 5000 and 7000.

3.1.2.1 Optical contacting

Optical contacting (a.k.a. black magic) is a method of handing without using

any form of solution or bonding agent. Two very flat pieces of glass will be drawn

together if the pieces can get close enough for inter-molecular attraction to occur.

The potential per unit area due to van der Waals attraction is [9]


u(D) =(3-11)
127rD2

where D is the distance between the two surfaces and H is the Hamaker constant

given by the densities of the two materials, p and pt, and the interaction constant,










C, such that H = xr2pplC. For most glass substances, the Hamaker constant is on

the order of 10-19 J. The force per unit area is then

Bu(D) H
F (D) =(3-12)
8D 6;D3

As you can see, there is a strong dependence on the separation distance of the

materials to the bonding strength. Little experimental work on the measurement

of the bonding pressure between two glass plates has been done but there are a

couple publications that give typical values in the range of 5 20 x 10s Pa [14, 15].

If this pressure is due to van der Walls attraction, the equation above then gives

corresponding separations of 1.9 1.2 nm for fused quartz (H ~ 0.6 x 10-19 J). This

gives an estimate of the tolerable separation between our mirrors and spacers due

to contaminants or surface roughness.

Cleaning of optics When optical contacting, the distance between the two

surfaces needs to be very small in order to form the bond. If there is dirt, dust, or

many scratches on an optic, the distance between surfaces is increased and they

will not adhere.

Our first cavity mirrors were from an old order placed with a company called

REO and had been sitting in their plastic packaging for two to three years. This

packaging likely contaminates the optics over time and we experienced great

difficulty in optical contacting these mirrors to the Zerodur spacers. Looking at

the mirrors which had been cleaned in the lab under a microscope, many particles

of dirt and dust, fibers from tissues, and smears of solvent propagated the surface,

cleaning the mirrors in a standard lab setting was not sufficient.

I did not experience success with optical contacting until we moved the entire

cleaning procedure into a rated clean room. One of the key elements is to be in a

low-particulate environment as well as to use filtered, de-ionized (DI) water and

extremely low particulate solvents. Acetone proved to be the best solvent for its



































Figure 3-3. Cleaning optics in a class 100 cleanroom. Located at the far left is a
flow hood with a DI water tap and drain. The table holds an ultrasonic
bath and hot plate for soaking optics as well as a high intensity
coherent light source for examining optics.


fast drying time, reducing the chance of dust to collect on the evaporating liquid.

The lens tissues used were cleanroom rated for class 100, standard lens tissues

produce fibers and dust. My cleaning procedure is as follows:

1. Place optic in a Liquinox solution. Be sure to place lens tissue between the

petri dish or beaker and the optic in order to protect its surface. Heat the

solution to 6000 and let sit, covered if overnight. Soaking time may vary and

this step may need to be repeated several times. Note: there are anecdotal

accounts of Liquinox degrading coatings so it is advised to start with a very

low concentration of solution and increase by small increments only if needed.

2. Remove optic from solution and rinse with DI water. Stand the optic on its

side while preparing for next step.










3. Dip a lens tissue in the Liquinox solution. Wipe each face surface of the optic

with one, soft, even stroke moving in one direction. Rise with DI water and

again stand on its side.

4. Wipe each face surface of the optic as described in the previous step with a

DI-wetted cleanroom lens tissue.

5. Rinse each face side of the optic under running DI water for at least 10

seconds. Stand the optic on its side on top of a lens tissue.

6. Wipe each face surface of the optic in the above manner with an acetone-wetted

lens tissue to complete the drying of the optic. Also wipe the non-face sides

to remove any remaining water drops. Set the optic back on its side on a dry

spot of the tissue. Note: try to minimize the time between steps 2-6 such that

water and solution does not have time to dry on the optic, possibly leaving

solvents and captured dust on the surface.

7. Fold a cleanroom lens tissue and place in forceps. Take care to have an even

amount of the tissue exposed along the edge of the forceps. Use a dropper

bottle to wet the edge of the tissue with acetone. Wipe this edge along each

side of the optic in one even stroke, starting with the face to be contacted

first.

8. Examine optic under a microscope. If not adequately clean, repeat previous

step and reexamine or start from the beginning again.

Even after thorough (1 ...il_ there will likely be some artifacts on the optic's

surface. Samples of various objects are shown in Figure 3-4.

Once the mirrors are clean, prepare the ends of the Zerodur spacer in the

same manner and carefully place the mirror over the end hole of the spacer. Since

the spacers are too large to be used with the microscope, the end faces have to

be inspected by eye. Use of a bright, coherent light and dark background surface

can be helpful in seeing artifacts on the surface as well as fringes between partially
























(a) 1\ieroscope (b) Diine


(c) Spec of dirt (d) Tissue fibers


(e) Evaporation marks (f) Scratch


Figure 3-4. Samples of optic contaminants as viewed through a microscope.
(a) 1\icroscope set at x 20 to examine optics. (b) Small "E-" of
"E-PLITRIBITS-UNI1\I" on a dime to show scale. The dot has a
diameter of roughly 225 pm. (c) Spec of dirt or dust. (d) Fibers from
non-cleanroom lens tissue. (e) Contaminated evaporation marks from
solvent. (f) Small scratch.









contacted surfaces. Sometimes optical contacting will occur on only a portion of

the mirror surface, which can be seen by a lack of fringes between the surfaces

in one section. Pressing firmly but carefully, move out from the contacted area,

pushing all of the fringes to the edges of the mirror. When the bond occurs, the

fringfes will not reappear and pulling on the mirror will not be able to separate it

from the spacer. If the bond does not occur properly, clean the optics repeatedly

and try contacting again until the bond holds.

3.1.2.2 Cavity parameters

The parameters of the two reference cavities used to prestabilize the reference

laser (RL) and laser 1 (L1) are listed in Table 3-2. Within the vacuum tank, the

cavity for the RL is located vertically above the cavity for L1. The cavity for RL

is longer than the cavity for L1. Both cavities have a flat front mirror and curved

back mirror so the waist of the resonant mode of the cavity, w,, is located at the

front mirror. The cavity g parameters, gl and g2, arT glVen by [21]


ge = 1 (3-13)
Ri

where L is the length of the cavity and Ri is the radius of curvature of mirror i. A

cavity will be a stable resonator when


0< 2l~I 34

Table 3-2. Parameters of the cavities used to stabilize the reference laser (RL) and
laser 1 (L1). L is the length of the spacer, R1 is the radius of curvature
of the front mirror (both have flats), R2 is the radius of curvature of the
back mirror, gig2 is a Stability factor, wc is the waist of the resonant
mode of the cavity, FSR is the free spectral range, a feavity is the
measured linewidth, and FT is the finesse.

Laser L R1 R2 g192 co' FSR a fmntw 7
RL 260 mm 00 0.5 m 0.48 290.9 pm 576 M~Hz 50 kHz 11,500
L1 225 mm 00 2.0 m 0.8875 463 pm 647 M~Hz 140 kHz 4,600










For both of our cavities, gl = 1 so glg2 1S simply given by g2. The cavity waist of

the resonator is given by [21]

LA gig2( gl192
Lo'2 (315)
Tr (gl + 9 29192 2

for the case of our flat front mirrors, this expression simplifies to





The free spectral range (FSR) and finesse (FT) of a cavity are given in Equations

:37 and :38 above in Section :3.1.1 on optical cavities. The linewidth of the cavities

was found by analyzing the transmitted resonant 00 mode of the cavity. While

sweeping the frequency of the laser over the 00 mode, the time of the FWHM of

the intensity is recorded. The sweep rate and the tuning coefficient of the laser

controller are then used to calculate the change in frequency that corresponds to

the measured time. Using Equation :310 and the finesse values to solve for the

transmissivity,

TRL 274

TL1 684

The cavity for R L was constructed with new mirrors and the vendor AT Films

specifies the transmissivity to be near :360 ppm. The cavity for L1 was constructed

using the old R EO mirrors from Goddard which, as discussed previously, had

difficulties with contamination of the coatings.

3.1.3 Pound-Drever-Hall (PDH) Locking Scheme

The lasers are frequency stabilized by locking them to the resonant frequency

of an optical cavity. The method used to lock the lasers is commonly referred to

as the Pound-Drever-Hall (PDH) technique [7]. Figure :35 shows the experimental

setup. The laser passes through a Faraday isolator, acting as an optical diode, and

then a portion of the beam is picked off for use with other measurements. In order


















Faraday EUM C
Isolator
expermentPolarizinlg
BS


Schematic of the PDH setup used to lock the frequency of the laser to
the resonant frequency of the reference cavity. The laser beam goes
through a Faradayl~i isolator, power beam splitter (BS), electro-optic
modulator (EOM), polarizing BS, and A/4-wave plate before reaching
the cavity. The signal from the photodiode that receives the reflected
cavity light is demodulated with the driving frequency of the EOM
to form the PDH error signal. The error signal is shaped with control
electronics and input to the laser controller,


Figure 3-5.


to create the PDH error signal, the laser is phase modulated using an electro-optic

modulator (EOM).

A laser field, Elser = Eoeist, that has passed through an EOM which is driven

with a frequency a can be expressed with the mathematical form


Ecom = Elsereimsin(nt)


(3-18)


where m is the modulation index of the EOM. For m

Ecom ~ Elase [1+ im sin(Rt)] = Eoewt [1 (ei2t e-in2~

Ecom~~ ~Eeitmi(w+n)t _meiwnt]
2 2


(3-19)


In this form, it is easy to see that the EOM adds two "sidebands" in frequency

space (located at w + R) to the main, or carrier, beam (at w), as shown in Figure


3-6.





















J~~ coc+R frequency


Figure 3-6. Representation of a beam in frequency space after passing through an
EOM with modulation frequency R.


The laser then transmits through the polarizing beam splitter, a A/4-plate,

and through a vacuum window to the cavity. The reflected cavity field will again

go through the A/4-plate and now reflect at the polarizing beam splitter to reach a

photodiode. When the laser is the same frequency as the cavity resonance, almost

all of the carrier will transmit through the cavity and as long as the modulation

frequency, R, is larger than the cavity linewidth, the sidebands will be reflected

and incident on the reflected light photodiode. Allowing for the carrier and the

sidebands to be resonant, the field incident on the photodiode can be described by


[5]m

EPD = Eo [Tr ()eiwt mTr ( + )ei(w+")t mTr (Lc S2)ei(w-")t] (3-20)
2 2

where T, is the reflected field transfer function given in Equation 3-4. The

intensity on the photodiode is then the magnitude of this field


IPD = |EPD 2


(3-21)









We are interested in the time-varying component so neglecting the DC term and

second-order terms the photodiode signal can be written


IPD (t) ~V E; m{[T (Lo)Tr(Lo + S) Tr( )T\p*, (w )]in2t

+[T ~ ~ ~ lyW (w)T,( + )-T(wT( )]e-inq)
= E m{R [T~ u) (0 )\ T*(W)Tr(W + S2)] cos(St)


+I[,() (wr ,\)/ T o r(w)T,(w + R)]sin(Rt)} (3-22)


The photodiode signal is mixed with the driving signal from the EOM, pulling

out the term proportional to sin(SOt). Hence, the leading AC term of the error

signal is

E = E2 mIm[T (0) (0/ C) -T()r( ) (3-23)

The goal of PDH locking is to keep the laser on resonance with the cavity so

consider the error signal near resonance. When the carrier is resonant the transfer

function of the reflected field for the sidebands will be nearly one.


E~res = E mll m[T,(0) T (0)] = 2E mIm[T,(0)]W (3-24)


Linearizing T,(w) for small w yields

ri -T2 ? 2r 02 b Lo'
Tr (6w)= + 2i (3-25)
1 rir2 ( ri172) C

to first order. This results in the error signal being of the form


Eres = 4E ~m 2 / (3-26)
(1 rzlr2)2 C:

For symmetrically-coated cavities where rl = T2 r,

r 6w L
E~res =-4E ~m r (3-27)
T c











Rewritingf Equation 3-9 such that


F TT T
1 r2 T


(3-28)


the error signal can be written

4E ~m
Ers 6f (3-29)
a cavity

where A feavit, is the linewidth or FWHM of the cavity. Control electronics are used

to implement this error signal such that the frequency of the laser stays locked to

the resonant frequency of the cavity.


PDH Error Signal


5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7
Frequency (Hz] x 108


Figure 3-7. Plot of PDH error signal as given in Equation 3-24 for a cavity with a
FSR of 600 M~Hz and Eo = 1, m = 0.1, ri = r2 = 0.99.



The error signal for a cavity with a FSR of 600 M~Hz is shown in Figure 3-7.

As shown in the plot, when the laser frequency matches the fundamental mode of

the cavity, the error signal is zero. If the frequency moves slightly off this point,

the error signal moves quickly to a non-zero value since the slope is very steep and


S0.02

S0
IE
g -0.02










supplies a correcting signal to move the laser back to the resonant frequency. As

long as the gain and bandwidth of the control electronics that use the error signal

to supply a voltage to the laser controller are adequate, the laser will be locked to

the cavity indefinitely.

3.1.4 Experimental Setup

Before starting the LISA lab at UF, we were involved in a frequency stability

experiment at NASA Goddard Space Flight Center (GSFC) [19]. The GSFC and

UJF experiments are very similar except that the GSFC experiment uses Corning's

UJLE glass as the cavity spacer and has two, smaller, separate vacuum tanks, each

complete with the same five 1 u. ris of gold-coated stainless-steel thermal shielding

to house the two cavities. The tanks also have an additional le -;r of exterior

thermal shielding but the temperature of the lab at GSFC has more variation

than the lab at UF due to the heating and cooling systems of the building and the

location of the experiment near an exterior wall of windows. ULE and Zerodur are

comparable materials with similar thermal expansion coefficients so we decided to

use Zerodur at UF to test if there were any measurable differences between the

materials.

The frequency stability of the Zerodur cavities is measured by recording the

beat frequency between the two lasers which are independently locked to the

cavities, as shown in Figure 3-8. If each of the cavities has a very stable resonant

frequency, then the difference frequency between the lasers will remain constant.

By looking at the changes of the difference beat note, we get a measure of the

relative stability. A glass plate is used to pick-off part of the beam of each laser

which is combined at a power beamsplitter and incident on a photodiode. The beat

from the photodiode is measured with a HP 53181A frequency counter that can

measure signals up to 225 M~Hz.



















Laser Isolator L
Polarizinlg

Counter\
Ph~otodiod
Polarizing
Laser 1 _
Isolator
Glass
Plate W wave
Pplate








Figure 3-8. Experimental setup for frequency stability measurement. The beat
note between the lasers of two, independent PDH systems is measured
with a frequency counter.


3.1.4.1 Expected stability

In order to simulate a space-like environment, the cavities are measured in

a vacuum and are also surrounded by 1 e r~s of thermal shielding. We expect the

temperature stability of the environment inside the vacuum chamber and shields to

be at least pK/A~H. From the thermal stability of the spacer, the relative length

stability of the cavity is


=7 aT = 021- =C 2 x10-14! (i30)
L K /j

which then gives a resonance frequency stability of

6L 2 x 10-14 3 x 10s m/s
6 f = fm = 5.6 Hz/A ~ (3-31)
L H 1064 nm













































Figure 3-9. Zerodur cavities inside of five lIwr-is of gold-coated stainless steel
thermal shielding (covers of shields are removed for the picture) located
inside of a large vacuum tank. Once the tank is pumped down, the
temperature stability inside the shields is expected to be on the order
of p-K/A~H .









3.1.4.2 Control electronics

Our lasers use a gain medium of Nd:YAG in a non-planar ring oscillator

(NPRO ) design which emits at 1064 nm. The laser controllers have a fast

and a slow frequency actuator. The fast frequency tuning works by applying a

voltage to a piezoelectric element (PZT) mounted on the
PZT contracts or expands the < ti--r I1 producing a strain which then changes the

frequency. For modulations less than 10 kHz, the PZT actuator is tunable over

at least 30 M~Hz. For smaller modulation depths, the actuation will continue to

100 kHz. Above 100 kHz the PZT will still operate but the tuning coefficient

becomes non-linear. The PZT generates heat when it operates and can excite PZT

resonances so it is important to not drive it too hard, modulation signals should be

kept under +50 V at 100 kHz. The slow actuator works by applying a voltage to

a thermoelectric cooler (TEC) located under the laser crystal. The TEC changes

the temperature of the crystal to physically change the length of the cavity as well

as the index of refraction. As the temperature of the < !i--r I1 increases, it expands

and the index of refraction increases, both making the optical cavity longer and

decreasing the output frequency. The time constant for using the TEC is rather

large, one to ten seconds, but has a tuning range of 30 GHz [17].

From the input of the error signal of the PDH, the analog electronics give

outputs to the PZT and to the temperature laser actuators. Figures 3-12 and 3-13

show the measured transfer functions and Figures 3-14and 3-15 are closed loop

transfer function measurements. Unity gain occurs for both systems in the low

kHz.

Table 3-3. PZT tuning coefficients of the lasers used for pre-stabilization.

Laser Serial # PZT Tuning Coef.
RL 2258 4.653366 M~Hz/V
L1 2142 2.109725 M~Hz/V












47nF


10nF 20kGZ


220 V


IN 2

15kG

:10kGZ


5k200kGZ

3 9kG


3.3kG~2


4.7kGl


100nF


2.2MD


27ka~


182ki~
10kil J ~~


10kGZ
_wv


oF lk1
10 10 OF
5knG 330kG~2


3.9kG~2


Figure 3-10. Circuit schematic of the RL PDH board.


5nF

9.4kG 2
4.7kG~ 1kZ
IN m


n
82kG~


4.7nF10kI

880kG
500kGZ


2.4kG2~


1 kG 6.2K~2

1 00nF


3.3ka~


20kGZ


4.7kG~2


2.2MD 3


20kGZ
200k


3.3kil2


10kGZ
~M~TEMP OUT


1LF ~kG2

15kGZ 340kG2~1I.F


Figure 3-11. Circuit schematic of the L1 PDH board.


PZT OUT


TEMP OUT


PZT OUT





















1~ "`-
rla'41r


in
,,t 1 n, I~LHII
i ildnilrl Fan. L~9 Mjp 151
Blive t 1Hz 178.298 deg
r i~.,,,,;..,~ ,..-~ U~c. U~rl- ---- -~ ---------- --


:O L-~------
dsgplHt
1 T~nt~pr Rlne Ilnma Phace


-----------------~
rlnrlloslsa7:29 ~L QOL-1 FZf


36.783798 Hz 9.922 dB


-ITransfer Func. Log Mag 134s
BLive 1 Hz 178.626 deg
220
degi

deg/


20


degl 1 Hz 100kz
Transfer Func. Unwrp. Phase 134s

10/20/05 15:33:36


Figure 3-13. Measured transfer function of the L1 PDH board, PZT output with
integrator off.


ALive


Figure 3-12. Measured transfer function of the RL PDH board, PZT output with
integfrator off.


A Live


100 Ibz(


1Hz


~i


-~


_____

rsls '*"t













S1.7073527 kHz 15.731 dB _


B Live 55.908102 kHz -261.018 deg



deg/





degl 1 Hz 100kz
Transfer Func. Unwrp. Phase 256 s
11/07/05 18:45:30

Closed loop transfer function of RL. A 20 kHx low-pass filter had to
be inserted between the RL PDH PZT control signal and the RL PZT
actuator to suppress oscillations appearing at multiple frequencies.
Unity gain is near 2 kHx.


10~s


STransfer Func. Log Mag 173s
BLive 1 Hz -92.732 deg

deg I11 iii i -61



deg/dly ::::i :::. ::: ::


-500
degl 1 Hz 100kz
Transfer Func. Unwrp. Phase 173s
10/21/05 11:20:08


A Live


100 Miz


1Hz
Transfer Func. Log Mag


256 s


Figure :314.


3.0538555 kHz


22.499


A Live


100 Ilz


1Hz


Figure :315. Closed loop
:3 kHx.


transfer function of L1. Unity gain for the system is near










3.1.5 Results

The recorded frequency between the two cavi' ---1 .Ill~.:ed lasers over a long

weekend is shown in the top plot of :316. The corresponding linear spectral noise

density is shown in the bottom plot. At high frequencies of our measurement

hand, our stabilization results are roughly 10 Hxl/AH. This value is nearing the

stability predicted by thermal fluctuations in Equation :3-31 and on the same

order or slightly better than similar stabilization experiments performed around

the world. There appears to be no appreciable difference between our Zerodur

measurements and the GSFC ULE measurements. Figure :317 also shows this

result plotted with the frequency noise of a free-running laser to illustrate the level

of noise suppression pre-stabilization provides. This figure also plots the LISA

prestabilization requirement which our system meets at almost all frequencies.

The fastest rate at which data can he read from the frequency counter is 1 Hx.

Using data taken for the optical TDI experiment in Section :3.5.3 that is read with

the phasemeter (Pl\), we have a measurement of the frequency noise at higher

frequencies. The frequency stability of the system measured with both the counter

and read through the phasemeter is plotted in Figure :318. The data that overlaps

between the two measurement methods near 0.1 Hx gives a consistent result. In

the phasemeter data, there is a large noise structure around 10 Hx that is not

suppressed by the PDH electronics. Noise spikes such as these are seen throughout

the TDI data in Sections :3.5.2-3.5.5.

Computing the associated length change of the cavity from the frequency

results gives variations on the order of fm/M~H, as shown in Figure :319.























Zero~dur Beat Note
9-9-05
1.2812+a8II III

L.2810es+08 --

S1.2808e+08-.


L.2804c+0)8--


1.28001+08IIIII
0 50000 100000 150000 200000 250000
Time [see]




0 sanxrhed .,



zi

01)001o 0.001 0 01 0.1
Frequency [Hz]




Figure 3-16. Frequency stabilization results over a weekend run of three dl
The top plot shows the times series of the heat note between the
two stabilized lasers recorded from the counter. The bottom plot is
a linear spectral density of the residual time series after applying a
quadratic fit.






44
















Free-running vs. Cavity Stabilized Laser




-..i..' Free-running
-LISA requirement
le+06 -

le 0 - - -- - - -







100

... .. .. .. . ..... :... :. .. ...... ... : ::. :. 1







p-taiition(rd)












Free-running vs. Cavity Stabilized Laser






I: ii::~:::::i:::i: i:: i::i:i:l: i:iii~:::::i::3: i:i:i:iiil:::::i:::i i:i:ii::~:::::3:::i :i::i:i:i:i'
I:iiil::::::::: il:::::::::i i:::i:::l::::::::-Il .---.--.
::.:r: ...---. :::i::::::::::::~: ::::::: ::l:::::::::::i:: :::: :::::::r:::::: i iiii
i... i.i..ii".....'...' '.'."--T......i...: i i iiii

)-- :-:-- ::i --- :---- :-:-::::1--------- ----:-r:,
):I ::: )::: l:::[l::l::::::i:::l "
I ... :... :.:.::::J......:...: .i..i.::::
i i jjjj! i i i i iiii
-:-::::1 ii i iiii-

-:---i--: -:-:-:-,,I-- ---,----:-i :-iiil-----i---,-: ,i::l--------i-:-: i :- ,iil---
:I:: :: I: :1: t I:::
---i I -'----1::::::::::: ::::::::::::: ::::l:::::i:
11 111 11
...'...'.:.i jiiil: :::::::I: i:: I::::::I: :i iijjj[ iiiiiiii:i iiji:iiiiliiiiiiii:: i.:.:
ii:iiiiii: i:iii:iiiliiiiiiiiiiiijii :iiiili :'::': ---'---'-" '::':':': :::::::::: :::::
I- ---i---i-i-i i"'l-----'---'-'-i ii-i-i-!------i---i- i-i i-iii! ----'---:- :-'-:-iii!--- -'---' '-:-""1
.:...I..: .:.:.:.rri.. ...'.....i.' i riii.....i...r.:.1 11::1.....1...I. i .1......:...1. 1. 1.1[[1.....' ... '.i .I11I.....1...1 I.i.iiiii..
i. ...:........ .:::1. ...:...:.- ::.:.:.1. ....:...:. :.::.:::I ....:...:. :.:.:.:::1... .:...: :.:.::::1...
ii i i iii ::::
I- ;;;r- ---i----:-i i-i-i-i- ----:---- -:;-;;;I ------: i-iiil--- iiiir------i
'' """"' """"' """"' """"' """"'
.:...:..: :::::::~~I:: Illilll~lili ~iiiil ...:...:... :::::::I:::::::::::: ~:: ::~::t::::::::::: :::::::::r:::::::::~ ::::::::~::::::::::: :::.:
::: : : __-- i ::: i -------_- t----- -:::I:::::: i:: I::i::: -:: :::::1::: ---


00001


1


1000- -



100 ----



10 -


0.0001 0.001 0.01


0.1 1 10 100 1000 10000
Frequency [Hz]


Figure 3-18. Frequency stability results from both the counter and the phasemeter
(PM).


Zerodur Length Stability
Cavity of 225nun on 9-9-05


~ le-l?

T

%1,-11
8
~
u
.%
z


Frequency [Hz]


Figure 3-19. Length stability of Zerodur cavity computed from measured frequency
stability.










Figure 3-20 shows a plot of the PDH error point noise. The LSDs were

converted to frequency using the amplified error signal slopes of 28.0 pV/Hz for

RL and 43.6 pV/Hz for L1. Since the spectrum falls right at what we see for the

frequency stability of the cavities above 2 mHz, our system is currently gain limited

at those frequencies. We are in the process of transitioningf our control electronics

from an analog to digital system which will greatly facilitate reshaping the PDH

control electronics to have higher gain without exciting high frequency resonances

and to suppress the noise structures seen at higher frequencies in the data read


Zerodur PDH Error Signals
2-23-06

4. L1


0.0De400



-2.000c+0 -

0 1000 2000 3000 4000 5000 6000
Time [see]




z'0oii~ii~iiii~iiii~ii~i
0.0001 0 001 0.01 0.1 1i~iiiiiii
'::':!ii~i:::::::i:: F:r equency [Hz]::



Fiur 320 Potofth rro po :intnie ftePD otrlsgal h tppo
sh wsth t iimesei es of th ro in l n t eb to lti h
linear sectral dnsities f their esidualsfrom a qadratic it. Sinc




Figre32. Potofthe error point noise is a the lee o he fnrequnc stgablizahetion l


of the cavities above 2 mHz, our system is currently limited by
inadequate electronic gain at these frequencies.











from the phasemeter. We should then see improvement in the frequency stability

above 2 mHz until another noise source is reached. We also expect that using

digital electronics will help alleviate the problems with glitches that sometimes

occur in the error signal, several of which are visible in the top, time-series plot of

Figure 3-20.

If our stability results were limited to temperature variations of the cavity,

Figure 3-21 shows the spectrum of the temperature stability of the cavity itself.

Since we know our results are error point noise limited above 2 mHz, we know that

the temperature stability of the cavities is at least as good as what is plotted for

these frequencies. The thermal mass of the spacer provides a low-pass filter for the

temperature variations of the vacuum environment. Yet, from the results of the



Inferred Temperature Stability
Cavkcy of 225nun on 9-9-05

-smoothed
0.0001









le-06 .




10- 0001 0 001 0.01 0 1
Frequency [Hz]



Figure 3-21. Temperature stability of the cavities as inferred from the length
stability results. Since the spectrum is primarily error point noise
limited, we know that the temperature stability is in the pK range, as
expected.









cavity stability, we can conjecture that the temperature stability of our vacuum

environment is in the pK/4~H range, as expected.

There are several noise factors which may be what we see limiting the

stabilization spectrum below 2 mHz. Radio frequency amplitude modulation

(RFAM) can enter into the PDH error signal through misalignment of the EOM.

Laser intensity noise can affect the stability of the laser-cavity system in several

different v- .0~. Power absorption in the mirrors and scattering into the spacer

expands the materials, changing the resonance frequency of the cavity. This will

mostly provide a constant drift of frequency over time but if the light into the

cavity is chr lrging_ the expansion of the materials will be non-constant and result in

frequency noise. This noise is especially probable to be one that we see since one of

the cavities has the old REO mirrors on it that have been thoroughly cleaned but

are still very likely to have coatings with a higher absorption coefficient, increasing

the effect of intensity variations.

Effects from temperature changes of the cavity will not occur instantaneously

but more slowly over time, filtering out effects on short timescales. In general, a

time constant is given by -r = RC. The thermal capacitance is C = mass ceaerm

anld for con~duction, ReoaCtIonL~I -L whe(rel (LLcller is the thermn l capacity, or

specific heat capacity, I is the length of the object, a is its thermal conductivity,

and Aross is the cross-sectional area [13]. Approximating that the 1!! .0 i;r~y of the

conduction occurs from heating at the ends [19],

pcthermL2
Teondnction ~= I\2 (3 32)


Referring to Table 3-1 for the values of the spacer material properties, our cavities

have a time constant of roughly


TeondnctionL1 2.34 x 104 3 )
I eondnctionRL 1 .75 x 104










which is a corner frequency of a 50 pHz. Similarly, the radiative time constant is

given by [13]
pcenermAeross L
Tradiative (3-34)
4GsBT3Asurf



Tradiatives,2.3x 0
T radiativeRL 2.54 x 103
which is a corner frequency of a 0.4 mHz.

Importantly, our results for frequency stabilization are analogous to what is

expected for the lasers aboard the LISA spacecraft. Hence, our stabilized lasers

provide a good foundation for LISA interferometry experiments with the benchtop

simulator.

3.2 Design of Benchtop Simulator

In order to test aspects of LISA interferometry, such as TDI, we are developing

a benchtop simulator. The goal of the simulator is to provide a testing ground for

implementing TDI and arm-locking with realistic laser phase noise. The simulator

consists of a laser for each spacecraft and an additional laser that is combined

with each laser to beat the fields down into the RF band for detection on fast

photodiodes.

A model of the benchtop simulator was created using OptoCad, a Gaussian

mode propagation program. Figure 3-22 shows the graphical output of the model.]

The simulator is laid out on a standard 4 x 8 ft optical table, with nearly half of it

supporting the vacuum tank which houses the reference cavities.

Each laser is running at roughly 90 mW with glass plates used to attenuate the

power to roughly 3 mW into the cavities and less than 1 mW onto the photodiodes.


1 Portions of the OptoCad model were completed by Aaron Worley.















































































































4
a

rl
9
.tl
D





L,









p
o
4
o

,a


00O
O O
O V










These power levels are reached by reflecting RL and L1 once off of glass plates

before the cavity optics and each beam has two reflections off of glass plates before

reaching a photodiode. In order for good mode-matching at the photodiodes, the

RL and each other laser travel the same path length to reach a given detector.

This LISA simulator can be used to test many aspects of LISA interferometry,

I have been developing it to test first-generation TDI algorithms using LISA-like

signals.

3.3 Electronic Phase Delay (EPD)

In order to test certain aspects of LISA interferometry, it is important to

model the long light travel time between spacecraft. To accomplish this goal, we

use an electronic phase delay (EPD) technique.2 As depicted in Figure 3-23, the

signal is digitized, stored in a memory buffer for a specified amount of time, and

then regenerated.



Analog nput 1 emoryr Buffer > Analog Output
Anti-Aliasing ~jL'Reconstruction
Filter Filter

Figure 3-23. Schematic of electronic phase delay (EPD) technique. ADC is
analog-to-digital converter and DAC is digital-to-analog converter.
Image courtesy of J. Ira Thorpe.


The input signal for the EPD is alr-ws- a photodiode signal of a beat between

two lasers. Such a signal will be of the form


SPD t) = sinwR 2 2~~ Sinwo 0 0 (t)] (3-36)


2 Development and in-depth analysis performed by J. Ira Thorpe, see [27].










where wi is the frequency and i(t) is the phase noise of laser i. Defining

20, 2 0~-~ and #20~t 2 0at ~~





This signal at the output of the EPD will then be


SPD:EPD~t = 20~wo~ 20) + 7~ ~ 3


where -r is the delay time of the EPD. Mixing this db 1 4~I signal with a prompt

signal from another laser beat, as shown in Figure 3-24, mimics an actual

LISA signal as described in Section 2.1 on the optical bench signals. For the

interferometry, we are interested in the phase of the signal so looking at the

argument, the phase signal from the EPD is


S2 w~ 20 20) + (t -r) (3-39)


The prompt signal between lasers 1 and 0 is simply


Silt) = wl0t + tlo(t) (3-40)


Mixing these two signals yields


S(t) = S2(t ') Silt) = w,20 10) 20 wlo 10a~ r -~ot 3


Analog
Lasr 2Signal I~ I~nir Ba e Analog
SPD S Aa-4ul Rt140L Sigal~
Fibrer Filler
ILaerB 0.~ S2(t-T)
1 EPD Unit
PD
Laser 1 D" sot


Figure 3-24. Schematic of how the EPD unit is used to make a signal equivalent to
a LISA signal.










Definingf awo21 = o20 1o'0 and dropping the constant term W207,


S(t) = aw21t 20 7,( 10 -r (t) (3-42)


This signal is analogous to the LISA signal on the optical bench given in Equation

2-1 without the phase change due to a gravitational wave, h(t). The signal

generated with the EPD contains additional difference terms to get this same form

but since the lasers all have the same noise characteristics as the LISA lasers, the

signal using their differences will as well.

3.3.1 Hardware

Our first EPD unit was on a Microstar 200 kHz PCI card. The Microstar

card could delay a maximum signal frequency of 30 kHz, on two channels, up to

80 seconds. The first electronic experiment in Section 3.5.1 was performed using

the Microstar card. In order to improve the EPD, we acquired dedicated hardware

from Pentek that has a 100 M~Hz sampling rate. On this board, we can delay

signals with a maximum frequency of 5 M~Hz, on four channels, up to 6 seconds.

The Pentek board consists of a carrier board that has 1 GB of SDRAM with two

daughter boards, one each housing the 14-bit ADC and the 16-bit DAC. There

are five FPGAs on the Pentek board: two on the daughter board with the ADC,

two on the carrier board, and one on the daughter board with the DAC. In this

configuration, the limitation occurs at the VIM interfaces between the daughter

boards and the carrier board where data transfer can occur at a maximum rate

of 33 M~Hz. All of the optical experiments were performed using the Pentek EPD

unit .

3.3.2 Timing Error

As discussed, the EPD unit will produce a signal with del li TEPD Which is

a function of the storage time in the memory buffer and the time for the signal

to traverse the ADC and the DAC, all independent of the timing of the data










EPD

X XXX time


samp

Figure 3-25. Schematic representation of timing in the TDI experiments. The
x's represent when the signals are sampled by the ADC of the DAQ
where tsamp samp*,~


acquisition (DAQ) process that eventually records the signal. Hence, most likely

this delay in the experiment will not fall at exactly one of the recorded sampling

points, utsamp, ne N but rather somewhere in between as shown in Figure 3-25.

The maximum time that could be between TEPD and the closest sampling point of
the DAQ is Itsa

When analyzing the data from the TDI experiments, we try to shift the

d.l I- we signal back by exactly TEPD in order to see the same noise structure at the

same point as in the prompt signal. With the post-process data shift, unhift, limited

to the timing of the recorded data points, there will be an error between what we

can achieve post-process and what the actual delay was in the experiment. We

define this timing error as

aTr = |7EPD Tshiftl (38)

The suppression limit due to this timing error, discussed in Section 3.5.2.3, is an

important factor in the optical TDI results.

This timing error is not just an artifact of the EPD process, but occurs in

LISA as well. The time that the LISA laser fields take to travel the arms will

almost certainly fall in between sampling points of the LISA signals and will be a

factor in the implementation of TDI and arm-locking for the mission.










Input
A,(t)sin 2;rft+4(t)]-




LPF



NCO LFScaling




Feedback





Figure 3-26. Operational schematic of phasemeter design.
Image courtesy of J. Ira Thorpe.


3.4 Phasemeter

The phasemeter is an I,Q demodulation phasemeter with a tracking NCO

to avoid problems with wrapping. This design is VeTy Similar to the LISA

phasemeter that is currently being designed at JPL. Referring to Figure 3-26, the

input signal at frequency f and phase 44(t) is multiplied by the sine and cosine

generated by an NCO with phase '. (t). The multiplied signals are low-pass

filtered to remove the second harmonic and give the amplitude and the residual

phase 4,(t), of 44(t) from '. (t). This residual phase is fed back to the NCO to

adjust its output frequency and as long as the feedback gains H( f) are appropriate,

the model phase will stay close to the input phase and 4,(t) < 1 (I;. 1. By

adding the residual phase to the model phase, we get the output phase #o(t). The


3 Development and in-depth analysis performed by J. Ira Thorpe, see [27].











filters in the phasemeter begin to effect the signals around 100 Hz and some of

the phase information is lost, which should be remembered when looking at the

analysis of the optical TDI experiments. Over the measurement band used in the

experiments discussed below, the phasemeter shows performance at LISA levels of

10-s i (1, I
3.5 TDI Experiments

The benchtop simulator with EPD and phasemeter can be used to test many

aspects of LISA interferometry. My purpose with the design was to setup a test

of implementing first-generation TDI on LISA-like optical signals. At the heart of

TDI is combining time-shifted versions of the d. 1 li-. I and prompt signals. Hence,

the first generation of experiments looks at just recombining a d. 1 li-. I signal

shifted back in time with its prompt counterpart to see the level of noise rejection.

3.5.1 First Electronic Experiment

To become familiar with the process of forming TDI-like combinations, I

ran a first experiment with signals generated electronically. Using the Microstar

card to implement the delay and a simple I/Q demodulation phasemeter (without

feedback), I performed an initial two-signal test.






SRS Preamp.A
1 kHz LPF 10OSRS Preamp.
A a xl Por DC Filter x1
Term A-(-B)
SR rarnp
30 kHz LPF Power
6.4V e x2Supply
Supply
Voltagee
Colrhol



Figure 3-27. Schematic of setup for the first electronic experiment. VCO is
voltage-controlled oscillator, SRS is Stanford Research Systems, and
RPD-1 and SRA-6 refer to mixer model numbers.









A detailed experimental schematic is shown in Figure 3-27. A function

generator at 10 kHz supplies the signal for Silt) which is then put through the

d. 1 li- unit and filtered. A voltage-controlled oscillator (VCO) is used to generate

S2(t). The VCO used is a Minicircuits ZOS-50 which operates in the range of 25-50

MHz so an additional oscillator was used to mix down the signal to 10 kHz. The

10 kHz VCO signal is filtered and amplified by a factor of two with a Stanford

Research Systems preamplifier (SRS preamp.) to give S2(t). This same signal is

mixed with the d. 1 li-. I function generator signal, St (t TEPD), filtered with a SRS

preamp., and used to form the control signal for the VCO. The VCO requires a

non-zero control signal so the mixed signal is added to a constant 2.0 V signal with

another SRS preamp. The 10 kR pot. on the mixed signal allows for changing the

gain in the feedback to lock the VCO. The control signal phase-locks the VCO to

Silt TEPD) and hence shows up in S2(t). The signals Silt) and S2 1) arT TOCOrded

with a data acquisition board. Inputting a delay of 2.0 seconds into the Microstar

card resulted in a delay between signals Silt) and S2(t) of 2.0016sec. This time was

determined by changing the amplitude of S1 and seeing when the change appeared

in S2-

The plot in Figure 3-28 shows the baseline noise of the experiment. The signal

S2 1) 1S IlOiSier than Silt) due to the noise of the VCO. Since this noise is not

common among the signals, the combined signal Silt) S2 1 Tr) is limited by this

noise. To test the noise rejection of performing the combination, a 20 degree phase

modulation at 30 Hz was introduced to the oscillator driving Silt) which is then

imposed on S2(t) through the VCO control signal TEPD Seconds later. Figure 3-29

shows that the common 30 Hz noise is subtracted in the combined signal.













Electronic TDI
tau = 2.0016 sec., no modulation, slope = -9.3 mHz








-- S1(t)
- S2(t)
-- S1(t)-S2(t-tau)-0.5


S-0).4

S-0.6

-0.8


10 20 30 40
Time [seconds]

Noise Spectrum


50 60 70


0.0001


1 10 100 1000
Frequency [Hz]


Figure 3-28. Baseline of electronic experiment. The top plot is a time series of the
signals and the bottom plot is their linear spectral densities.



Electronic TDI
tau = 2 0016 sec 20 deg = 0 056 cycle modulation at 30 Hz

S1(t

SS2(t)
SS1(t)-S2(t-tau)
0 01 -




0 001 -




0 0001 -




29 9 30 30 1 30 2
Frequency [Hz]


Figure 3-29. Results of the electronic experiment. The plot shows only the
frequencies around 30 Hz to show the noise cancellation that occurred.










3.5.2 Initial Optical Experiment

3.5.2.1 Experimental setup

For the first optical experiment, the beat note between the two cavi' i---r I1.ill.:ed

lasers was electronically split and used to give a prompt and d.l 1i signal, as

shown in the experimental schematic of Figure 3-30. The photodiode of 32 M~Hz

is mixed down to roughly 800 kHz with a function generator (or oscillator). The

demodulated signal is filtered with a Stanford Research Systems (SRS) preamplifier

with one pole at 1M~Hz and electronically split by using the amplifiers two outputs.

The d.l 1i signal is formed by passing one of these signals through the Pentek

EPD unit with a delay of 2 seconds. Both signals are again mixed down to roughly

9 kHz with another function generator to be recorded on a National Instruments

PCI-6036E DAQ at an 80 kHz sampling rate. The two function generators and

the EPD unit have their time bases linked to a common rubidium standard. After

recording these voltage signals, they are run off-line through the Simulink software

phasemeter to give the phases of the prompt and d.l 1i signals, 4,(t) and fa(t),

respectively. Figure 3-31(a) shows the phase signals after being run through the

phasemeter. The slope of the signals is due to the difference between the signal

frequency and the initial model frequency supplied to the phasemeter.




Phtdid 1Mz80 k;Hz D ,,,a Prompt
32 MHz 80 ~ LPF PF 9kZ80 kl;zPhs
sampling
gPP) too k;Hz rate ll ll~~~sea elae
Oscilaor 7tell i2 egc LPF I -- ***t~s Ph ase



Figure 3-30. Schematic of initial optical experiment. The 32 M~Hz photodiode
signal is demodulated to 800 kHz, filtered, and split. The prompt
signal and signal d. 1 li- with the EPD are demodulated to 9 kHz,
recorded, and run through the phasemeter. LPF stands for low pass
filter and NI DAQ is the National Instruments data acquisition card.









3.5.2.2 Analysis of signals

To compute the TDI-like combination, the first 'rsave ~~ 2 sec of the d. 1 li-. I

phase data are removed, yielding fa(t Tshift) where 'rsaffe is the post-process shift

imposed on the data set. Likewise, the last 'rsave ~~ 2 sec of the prompt phase data

are removed so that the data sets are the same length. Then the shifted d. 1 li-- II

phase is subtracted from the prompt phase to form the time-d. 1 li-. I combination:


S(t) = 4,(t) fa(t Tshift) (3-44)


Figure 3-31(a) shows the prompt truncated, time-shifted d.l 1i .Id and time-d. 1 li-. I

combination phase signals. Figure 3-31(b) shows these three signals again with

the prompt and d. 1 li-. I signals detrended and un-truncated or shifted. The input

phase signals are going over many cycles yet the same structure is seen in both the

prompt and d.l 1 i-o I phases at a time difference of 2 sec. All phase signals from the

optical TDI experiments exhibit the same behavior.

The value of Tesits that gives the greatest noise suppression is not exactly the

same as the 2 sec delay time that is specified to the EPD. Due to the internal

digfitization of the EPD unit and d. 1 .x~ through the EPD ADC, DAC and

possibly other components in the experiment, the optimum reshif is slightly

more than simply 2.0 sec. In order to find the best suppression of the time-d. 1 li-. I

combination, the above procedure of subtracting the truncated signals is done for

various values of I-saiti and the linear spectral densities (LSDs) are computed and

compared. A plot of these various LSDs is shown in Figure 3-32 and clearly shows

that the maximum phase noise cancellation was reached with 'rhift = 2.0005625 sec.

The time-d. 1 li-. I combination signal that is shown in both plots of Figure 3-31 was

computed using this optimum value.

The time-d. 1 li-. I combination in Figure 3-31 shows virtually no noise at this

scale. As with finding the optimum value of Tshift, computing the LSDs of the











Time-shifted Phase Data


15
Time [sec]


Figure 3-31.


Phase signals from the optical experiment. The prompt phase, 4,(t),
is in blue; the d.l 1 i-o I phase, fa(t), is in green; and the time-d. 1 li-. I
combination, S(t), is in red.
(a) The phase signals after running through the software phasemeter
and applying the time shift.
(b) The phase signals but with the prompt and d.l 1 i-o I phase signals
linearly detrended to more clearly show their structure. Notice that
the two signals are noisy yet follow each other with a 2 sec lag. The
time-d. 1 li-. I combination is computed from the non-detrended signals
and shows far less noise.


Phase Data


Time [sec]










LSDs of Shifted&Subtracted Signals for Varying Shifts
2.000525 sec
-2.0005375
-2.00055
10 2.0005625
I ~2.000575







100 10 1
Freqency[Hz
Fiue -2 L~ o h tie- B( cminaionfrvrosvle f!s h








signals an2 Ld the time-deli 1 d ~ I combination. h promp vando delun d ~signal have





spectra that are nearly identical, both following noise that is roughly f. The sp~ikes

in the input signals are noise in the frequency stabilization system that is not

suppressed by the PDH electronics. This type of structure is seen in a similar form

in all measured optical signals. The noise in the time-d.l 1 li. combination cr li--

level at a few times 10-s 5 (1, I/ lzH with the exception of the structure from

the noise spikes. At low frequencies of the measurement, there is over five orders of

magnitude noise suppression with the time-d. 1 li-. I combination.

3.5.2.3 Suppression limit

There is a limit to how much noise suppression can be gained from the

recombination of the time-shifted signals. If we shift the data set by an integer

number of points, the limit comes from the time resolution of the data. Referring

back to Equation 3-44 and assuming 4d(t) = p(t + TEPD), the time-d. 1 li-. I

























































Figure 3-33. Linear spectral densities of the phase noise of the prompt (in blue),
de i-. II (in green), and time-shifted combination (in red) phase
signals .


* * * * *


Optical Beat 11-28

-prompt
-delayed
'_____~:j~i~l~l~j7_~litimi e delayed comb. ?


.''''''''''










.* jjjjjjjjijjjj~


Iu
N
2

V 103
G),


1.0 8
101


1


T


102
Frequency [Hz]









combination can be expressed in terms of the timing error in Equation 3-43 as


S(t) = ,(t) ,(t a-r) (3


We are interested in the noise of this signal in the frequency domain:


dtS ~(t,e-i xft 1 e-i2"fA] (3


and we measure real signals so computing the magnitude of the above equation

yields

|S(f )| = 2sin(7r f ar)|p(f )| (3

giving a relation of the phase noise of the time-d. 1 li-. I combination from the


-45)


_46)


-47)


phase noise of its constituent signals and the timing error. As discussed in Section

3.3.2, the maximum timing error is ~tsam. All of the optical TDI experiments are

Cosmparison o~f Experimental Result with Theo~retical Suppressio~n Limit
Due to Timing Resolution (Sampling rate of 80 kHz)
SShifted&Subtracted
SSup. limit for res. of 1/2 pt
Sup. limit for res. of 3/4 of 1/2 pt i
N Sup. lirnit for res. of 1/4 of 1/2 pt



Iu 10 -



I In I I I II I I I r I r




100 10'
Frequency [Hz]

Figure 3-34. Timing suppression limits of Equation 3-47 for various values of A-r
plotted along with the spectrum of the time-d. 1 li-. I combination
(Shifted & Subtracted shown in red).










digitized at 80 kHz so isamp = 12.5 psec. The timing suppression limit of Equation

3-47 for aTi = ~tsamp = 6.25 x 10-6 SCC is plotted in black along with? the linea~r

spectral density of the time-d. 1 li-. I combination signal in red in Figure 3-34. The

time-d. 1 li-. I combination results lie below the worst-case timing error suppression

limit so we can surmise that TEPD is ClOSer to a data point than in the middle

of two points. To estimate how far off from TEPD the time-d. 1 li-. I combination

results are, suppression limits for two more values of a-r are also plotted in Figfure

3-34. The higher noise structures of the time-d. 1 li-. I combination fit closer to

aTr = ~tsam = 1.5625 psec, shown in purple. However, the baseline of the

time-d. 1 li-. I combination sits at a higher level, closer to the suppression limit of

aTr = ~tsam = 4.6875 x 10-s sec, -II---- -r;.-:: that we are starting to see other

noise source in the experiment. To improve the noise rejection of the time-d. 1 li-. I

combination where it is timing suppression limited, we would have to interpolate

between data points to reduce the value of A-r.

3.5.3 Optically-Split Experiment

I repeated the above optical experiment but added an additional photodiode

to split the signals optically rather than electronically. The experimental setup

shown in Figure 3-35 is very similar to the electronically split experiment. The

NI
1 MF 800 kHz 28 kF 10 krHz Pas
VCO Os-samplingp
1 M~z EPD 30 k~z ateDead
PhoodidesLPF tau= 2 sec LPF '"" hs
35 MlHz


Figure 3-35. Experimental setup for optically split two-signal TDI test. Two
photodiodes measure the 35 MHz beat signal and both signals are
demodulated to 800 kHz. One signal is d. 1 li-. I with the EPD unit
and then both signals are demodulated to 10 kHz, recorded, and run
through the phasemeter. VCO stands for voltage-controlled oscillator,
LPF is low-pass filter and NI DAQ is a National Instruments data
acquisition board.










optical beat signal is at 35 MHz so each photodiode signal is mixed down with a

voltage controlled oscillator (VCO) to roughly 800 kHz and each signal is filtered

with a SRS preamp with the corner frequency set at 1 M~Hz. One of the signals is

d.l 1 i-o I with the EPD unit 2 seconds, both signals are mixed down with a function

generator to roughly 10 kHz, filtered, and then recorded using the same National

Instruments DAQ with a sampling rate of 80 kHz. The EPD unit, oscillator, and

also the DAQ have their time bases linked to a common rubidium standard. The

signals were recorded for 145 seconds which is near the computational limit of

the phasemeter due to the large file size. Once the signals were recorded with the

DAQ, the data was taken off-line and run through the same phasemeter to give the

prompt and d.l 1 i-o I phase, 4,(t) and fa(t), respectively.

The procedure for computing the time-d. 1 li-. I combination S(t) is identical

to above in Section 3.5.2.2. I again computed the time-d. 1 li-. I combination for

various values of Tssest and found the optimal suppression for the same value of

2.0005625 sec. The linear spectral noise densities are plotted in Figure 3-36. There

is again good noise suppression in the time-d. 1 li-. I combination, reaching just over

six orders of magnitude near 20 mHz. However, there does appear to be additional

noise in the experiment at the lower frequencies that are now visible by taking

data an additional 100 seconds. This may be due to the signals being taken from

different photodiodes although it is difficult to ;?i without other measurements at

these lower frequencies.

3.5.3.1 Timing error estimate

The worst-case suppression limit of AT = tsamp due to the timing resolution

expressed in Equation 3-47 is computed for the optically-split experiment and

plotted along with the spectrum of the time-d. 1 li-. I combination in Figure 3-37.

The spectrum of the time-d. 1 li-. I combination lies mostly below the timing

suppression limit of ATr = tsamp, aS it did for the electronically-split experiment.

















































Frequncy [Hz]


Figure 3-36. Linear spectral densities of the prompt, d. 1 lid~I and time-d.l l- .Id
combination signals for the optically-split experiment.


LSbs of Signals

-prompt
delayed
time-del.comb. -


S100



S10.

~:102
0.


105


104


; ,


-2


1.0-9 t
1O










In order to estimate the timing error, Equation 3-47 can be solved for A-r to

give the expression:
1 S(f)
aTr = sin-] I (3-48)


Using this algorithm, A-r is computed for the optically-split experiment assuming

that the time-d. 1 li-. I combination results are timing resolution suppression limited

so thait ilthey are beuedfr ~sl~ i. T he resultling Ar is plottedl in Figurre 3-38.

From the spectrum of the time-d. 1 li-. I combination, we can see that there is an

additional noise source between about 40 and 400 mHz, which is also shown in

an increase in the value of A-r for these frequencies. Looking between 400 mHz

and 100 Hz where the filters in the phasemeter do not affect the signals, a rough

estimate of A-r is 2 x 10-6 sec. The timing suppression limit for ATr = 2 x 10-6 ScC

is plotted along with the time-d. 1 li-. I combination and worst-case timing error

suppression limit in Figure 3-37 as the optimized suppression limit. As you can

Time-Delayed Combination and Suppression Limits
sup. limit for 1/2 data pt.
-time-delayed comb.
Optimized sup. limit ;




O -41
S 10 ..i i


Z 10 .....





10- 100 101 102
Frequency [Hz]

Figure 3-37. Linear spectral noise density of the time-d. 1 li-. I combination and
suppression limits due to timing resolution from Equation 3-47.










10-4 Delta Tau







10-o









1 102 100 101 102
Frequency [Hz]

Figure 3-38. Estimate of the timing error, A-r, for the optically split experiment.
Between 400 mHz and 100 Hz, A-r is roughly 2 x 10-6 ScC.


see, there is good agreement between this optimal timing suppression limit and the

spectrum of the time-d. 1 li-. I combination.

3.5.3.2 Non-integer linear shift of data

In order to get a data set at a non-integer sampling, I linearly interpolated the

d.l 1 i-o I phase data to produce a set where each point (except for the last one) is

shifted in time 2 x 10-686C fTOm its measured point. Figure 3-39 shows the resulting

interpolated data set, still with 1.25 x 10-s sec between points but each point is

located 2 x 10-6 ScC fTOm the measured data. Using this shifted data set of the

d.1 li-. I phase and the original prompt phase, I again recomputed the time-d. 1 li-. I

combination, shifting the data sets the same 2.0005625 sec which results in a

total 'rsaire of 2.0005645 sec. The linear spectral densities of the time-d. 1 li-. I

combinations for the original data sets and the data with one set additionally

shifted is shown in Figure 3-40. Looking again at the frequencies between 400 mHz

and 100 Hz, the LSD of the time-d. 1 li-. I combination using the interpolated data













Phase Interpolation

4 -*-original sampling

\ interpoated alt 2e-6se


-2.4635

-2.464


a,-2.4645

LO -2.465

-2.4655


-2.4665


9.4 9.6 9.8 10
Time [sec]


10.2 10 4
x 10`


Figure 3-39. Plot to show linear interpolation of data an additional
2 x 10-6 ScC fTOm the measured data.


Time-Delayed Combinations


10





10

Z

1 1


10- 100 101
Frequency [Hz]


Figure 3-40. Linear spectral noise densities of the time-d. I
the original sampling and for interpolating to
2 x 10-6 ScC Shift.


li- II combinations usingf
give an additional










shows more noise rejection. Thus, the timing error has been reduced and we are

closer to the true value of TEPD With ~shift = 2.0005645 sec.

3.5.3.3 Interpolation


10 pt interpolation
-2.158
-@-original sampling
-2.1585 j interpolated data
-2.159-

-2.1595-

a, -2.16-

S-2.1605-


-2.1615-

-2.162-

-2.`1625-

6.5 7 7.5 B 8.5 9 9.5 10
Time [sec] x 10.


Figure 3-41. Section of 10-point interpolated data.


In order to reduce the timing error further and improve the time-d. 1 li-. I

combination noise suppression, I interpolated additional data points between the

sampled points. In order to get to a resolution greater than what was computed

in the above section with a non-integer linear shift of 2 psec, there needs to

be at least seven additional points (1.25 x 10-s sec/2 x 10-6 sec = 6.25).

Using linear interpolation, I created an additional nine points between each

measured point, making ten points for each 12.5 psec. Since the original phase

data files were very large and cumbersome to work with, I interpolated only a

small section of the original phase data, working with the first roughly 5' Figure

3-41 shows a section of the interpolated data. Both the prompt and d.l 1 i-o I phase

data were interpolated, in effect creating a data set with a new sampling rate of

800 kHz. The time-d. 1 li-. I combination was computed for values of Tshiti around










2.0002645 sec, as shown in Figure 3-42. The lowest curve, and hence best noise

suppression is achieved with ~shift = 2.00056375 sec. Notice that the LSD for

Tshift = 2.00056500 sec of one more point is closer to the lowest LSD than that

for ~shift = 2.00056250 sec of one less point. Therefore, the results from the

interpolation so-----~ -1 that the true value of TEPD lieS between 2.00056375 and

2.00056500 sec, fitting nicely with our estimate of 2.0005645 sec computed in the

previous section. Figure 3-43 plots the original time-d. 1 li-. I combination spectrum

and the lowest time-d.l 1 i-o I result for the ten-point interpolated data. The LSD for

the ten-point interpolation results, lies below the original results in several areas.

In the remaining optical experiments, the one-arm and two-arm setup, an

analysis of interpolation will not be performed. Greater noise cancellation may

be achieved for these experiments as well, however, it would improve by less than

an order of magnitude and follow along the same lines as discussed here. In all

measurements, the timing error will be less than 6.25 psec.

Time-Delayed Comb. with 10pt interpolation
for various values of tau

-2.00056125 sec
103 2.0005625
-- 2.00056375
N I 2.000565










10 10 0



Frequency [Hz]

Figure 3-42. LSDs of the time-d. 1 lid~I combination for various values of reshts on
the 10-pt interpolated data.










Time-Delayed Combinations

- no interpol., 2.0005625 secl
10pt interpol., 2.00056375


10-4 :: :::







0- 10-




100 101 102
Frequency [Hz]

Figure 3-43. Comparison of the LSDs of the time-d.ll- I & combinations of the
original 80 kHz data (red) and the interpolated 800 kHz data cyann).


The method described here for interpolation is a very simple approach,

however, a more sophisticated form of interpolation will be necessary for the

LISA data streams. The LISA data streams will be down-sampled to reduce the

bandwidth needed to send the signals back to earth. This down-sampling rate is

3 Hz which is far below the timing resolution needed in order for TDI to cancel

the noise to the required level. A method of interpolation, such as those based with

fractional-delay filtering, will be applied to the LISA data streams [24],

3.5.4 One-Arm Experiment

The one-arm experiment takes the d. 1 li-. I optical signal used in the first

optical experiments to phase-lock an additional laser. The signal from this laser,

L2, is also d.l 1 i-o I and then recorded. In this way, the experiment recreated one

arm of the LISA constellation where the far spacecraft is transponder locked to the

main spacecraft laser, L1, as shown in Figure 3-44.














EPD




1l)


Figure 3-44. Representation of the LISA configuration simulated with the one-arm
measurement. The time between spacecraft is created with the EPD
unit.


The details of the experimental setup are shown in Figure 3-45. The 12 M~Hz

beat signal between the two cavity stabilized lasers is measured at two photodiodes

and both signals are demodulated with a function generator to 800 kHz and

filtered with SRS preamps. One of the signals is d.l 1 i-o I for 2 seconds with the

Pentek EPD unit and then used to phase-lock L2. The phase-lock error signal

is formed by mixing the L2 beat signal with the d. 1 li-. I signal from L1. The

phase-lock loop board uses the error signal to give an input to the L2 laser PZT

controller to change its frequency. The electronics are designed to keep this error

signal at zero so once the servo is locked, L2 will be the same frequency at the

d. 1 li-. I L1 signal, in this case 800 kHz. The transfer function of the phase-lock

loop board is shown in Figure 3-46.4 L2 beats with the cavi' i---r .1.1I11. :d reference

laser on two photodiodes, one is used to form the phase-lock loop error signal

just discussed and the other signal is d.l 1 i-o 1 2 seconds with a second channel

of the EPD unit. The d. 1 li-. I signal from L2 and prompt signal from L1 are

demodulated down to 10 kHz with another function generator, filtered, and



4 Wan Wu had developed a phase-lock design for experiments pertaining to his
research. Our board is based on his design.













r-----~


Delayed Prompt
Phase Phase


Experimental setup for the one-arm measurement. The heat signal
between the two cavity stabilized lasers is d. 1 li-. I and used to
phase-lock an additional laser, L2. The signal from L2 is also d. 1 li-. I
and then recorded along with the prompt L1-RL heat. LPF stands
for low-pass filter, PLL is phase-lock loop, and NI DAQ is National
Instruments data acquisition.


Figure 3-45.











A Live 21.405457 Hz 20.401 dB




dB/dl5



dBl 100 mHz 102.4kz
Transfer Func. Log Mag 461 s
BLive 100 mHz 179.737 deg
200
deg




100


degl 100 mHz 102.4kz
Transfer Func. Unwrp. Phase 461 s
2/23/06 14:52:02

Figure 3-46. Transfer function of the control electronics of the phase lock loop used
in the one and two-arm experiments.


recorded on the same National Instruments DAQ as with all other optical TDI

experiments. Both function generators, EPD clock, and DAQ tri r--;ingl~ are

linked to a rubidium standard. The two recorded signals are run through the

same Simulink software phasemeter and the same process of finding the optimum

time-d. 1 li-. I combination of Equation 3-44 is performed as in the previous optical

experiments. The optimum suppression was reached with 'rhift = 4.0011375 sec

which is one data point more than 2'rsits of the previous experiments. The phase

noise spectra of the recorded signals and time-d. 1 li-. I combination are plotted

in Figure 3-47. Again, we see excellent noise suppression from the input signals

although there is increased noise at low frequencies.

The increased noise at low frequencies from the optically-split experiment is

due to the inability of the phase-lock loop to impose phase changes down to the

level seen before. I ran the one-arm experiment additional times with only the

first delay and also with all d. 1 .va removed. Figure 3-48 shows the phase noise






















lo


100








S.c-10:
0.


10o
Frequency [Hz]


Figure 3-47. LSDs of the prompt, delai d\~ and time-dell I\dI combination signals
from the one-arm measurement.


One Arm with Two Delays










One Arm Measurements
10-2




II




S10-4




0_ 10 :
-no delays
Only first delay
-both delays
1 0 * * * * * *
101 100 101 102 103 104
Frequency [Hz]

Figure 3-48. Comparison of LSDs of the time-d. 1 li-. II combination for the one-arm
measurement with the EPD unit removed from different parts of the
exp eriment .


spectrum of the time-d. 1 li-. Il combination for these different experiments. There is

no appreciable difference between the results with one d.l 1 i- and with two d. 1 .1- .

The difference in the shape of these d. 1 li-. I1 curves from the non-d. 1 li-o I1 data

is likely due to a change in gain settings of the phase-lock loop hoard between

data runs. Also, the EPD unit produces a signal with slightly less than half of

the input amplitude. In these different one-arm measurements, the prompt signal

from L1 had the same amplitude but it was not possible for the signal from L2 to

do so as well. Hence, the noise in L2 with no d. 1 .1-< would have appeared slightly

larger than the runs with d. 1 .1-<. This figure exemplifies that particularly as the

complexity of the benchtop simulator increases, there are non-stationary systems










present. However, we still see nearly five orders of magnitude cancellation of laser

phase noise as a reproducible result.

3.5.5 Two-Arm Experiment

For the two-arm experiment, the one-arm setup is repeated with an additional,

fourth laser to generate signals from a second arm. The remaining two channels

of the EPD unit were utilized with a different delay so that the signals from

the two arms appear to have traveled different distances. With these optics and

electronics, we can generate signals equivalent to those from all three spacecraft

of LISA, utilizing the entire constellation. Using the realization of the laser on

spacecraft 1 being the master laser with the lasers on the two far spacecraft, 2 and

3, transponder locked to L1, the two-arm experiment models the LISA constellation

shown in Figure 3-49 and is analogous to the configuration discussed in Section

2.2.3.


(;2




T2= Z 2 sec e
EPD

Z3 = 3 sec

Figure 3-49. Representation of the LISA configuration simulated with the two-arm
measurement. The time between spacecraft is created in the lab with
the EPD unit.


The experimental setup is analogous to the one-arm setup with the symmetry

of an additional arm. Referring to Figure 3-50, the 59 M~Hz beat signal between

the two cavi' ---r1 .Ill. e:d lasers is measured at two photodiodes and both signals are

demodulated with a function generator to 800 kHz and filtered. The filters used

are Stanford Research Systems preamplifiers (SRS preamps) and each one has two






80













800 k-Hz | Phto-
]59 MlHz I



Func. Gn



1 MHz 1 MIHz
LPF LPF
LL800 krHz

EPD 2,7 krHz EPD EPD I EPD 27 kEHz
tau LPF tau tan tau LPF

2sec 2 sec 3 sec 3 sec

Func. Gen.



30 krHz 28 rHz 28 kiHz
LPF LPF LPF
10 kHz

NIDAQ: 80 krHz sampling rate









L2 Delayed L1 Prompt L3 Delayed
Phase Phase Phase


Figure 3-50. Experimental setup for the two-arm measurement. The incorporation
of the fourth laser, L3, is identical to that for L2 except for a
different delay time of the signals. LPF stands for low-pass filter,
PLL is phase-lock loop, and NI DAQ is National Instruments data
acquisition.










outputs, one with 50 R impedance and the other with 600 R. The EPD inputs are

50 R so the 50 R output from each of the Stanford amps is used to delay the signals

for phase-locking L2 and L:3. The prompt signal from L1 is taken with one of the

600 R outputs. The phase-lock signal for L2 is d. 1 li-, I1 through the EPD unit with

2 seconds; the phase-lock signal for L:3 is dl 1 i-o 11 :3 seconds. The same phase-lock

electronics are used to lock L2 as in the one-arm measurements and an analog

board with the same design and similar performance is used to phase-lock L:3. L2

and L:3 heat with the cavi' i---r .Ill~l.:ed reference laser on two photodiodes each;

one is used to form the phase-lock loop error signal and the other signal is d.l 1 i-o I

The signal from L2 is d. 1 li-. I1 the same 2 seconds and L:3 is dl 1 i-o 11 :3 seconds.

We initially experienced some problems using all four channels of the EPD unit

with implementing the second set of channels at a different delay time. The newly

utilized second set of channels gave an increased noise floor to the signals and

produced glitches that were visible when monitoring the signals on an oscilloscope.

Altering of the delay code on the EPD unit fixed this problem and we are able to

d. 1 li- four channels with each set of two channels at a different delay time with no

additional noise. The d.l 1 i-o I signals from L2 and L:3 and the prompt signal from

L1 are all demodulated with another function generator down to 10 kHx, filtered,

and recorded with the National Instruments DAQ. While the schematic shows all

three signals, only two of them can he recorded at a time since the DAQ has a

maximum sampling rate of 200 kHx for all channels and the phasemeter requires

signals sampled at 80 kHx. Again, both function generators, EPD clock, and DAQ

tri c'l~--;ingl~ are linked to a common rubidium standard. The two signals that are

recorded at a time are run through the Simulink software phasemeter and analyzed

in the same fashion.







82













ma
So

co

















c:O


r
bD

"o

'f c


E i








Am~












102


10


9: 100





Zj 10-3





10-6


**:*t :


-


n


-


-


-~ :


10 1


Figure 3-52. LSDs of the two dI li-. I signals and their time-d. 1 lid~I combination
from the two-arm measurement.



In this experiment, both of the signals recorded from each arm are de1 li-. I

The signal recorded from L2, S2 1), iS d.l 1 ~1. 22-r 4 ScC fTOm the prompt L1

signal, St (t). Similarly, S3 1) is d.l 1i .1. 273 ~ 6 sec from St (t). Therefore, S3 1) is

d.l 1 i-o 1 273 272 ~ 2 sec from S2(t). The time-d. 1 li-. I combination between the

phases of S2(t) and S3() ~2(t) and ~3 1), iS formed by post-process shifting ~3(

back by the difference in the total d.l 1 i- times:


S(t) = 2,t 3( Tshift23)


(3-49)


where ~shite is attempting to approximate 2(-r3 -r2). The experiment was run

recording L2 and L3 each with L1 to find the optimal post-process delay time in

each arm. Between L2 and L1, the optimum Tshiftzl = 4.004200 sec and between


Two-Arm Measurement


101 102
Frequency [Hz]