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Laser Noise Mitigation through Time Delay Interferometry for Space-Based Gravitational Wave Interferometers Using the UF...

Permanent Link: http://ufdc.ufl.edu/UFE0044058/00001

Material Information

Title: Laser Noise Mitigation through Time Delay Interferometry for Space-Based Gravitational Wave Interferometers Using the UF Laser Interferometry Simulator
Physical Description: 1 online resource (186 p.)
Language: english
Creator: Mitryk, Shawn
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: The existence of gravitational waves was theorized in 1916 by Albert Einstein in accordance with the linearized theory of general relativity. Most experiments and observations to date have supported general relativity, but now, nearly 100 years later, the scientific community has yet devise a method to directly measure gravitational radiation. With the first attempts towards a gravitational wave measurement in the 1960s, many methods have been proposed and tested since then, all failing thus far to provide a positive detection. The most promising gravitational radiation detection method is through the use of a space-based laser interferometer and with the advancement of modern technologies, these space-based gravitational wave measurements will eventually provide important scientific data to physics, astro-physics, and astronomy communities. The Laser Interferometer Space Antenna (LISA) is one such space-based laser interferometer. LISA's proposed design objective is to measure gravitational radiation in the frequency range from 30 microHz to 1 Hz using a modified Michelson interferometer. The interferometer arms are 5 Gm in length measured between each of the 3 spacecraft in the interferometer constellation. The differential arm-length will be measured to an accuracy of 18 pm/sqrt(Hz) resulting in a baseline strain sensitivity of 4.8*10^(-21)/sqrt(Hz). Unfortunately, the dynamics of the spacecraft orbits complicate the differential arm-length measurements. The arms of the interferometer change in length resulting in time-dependent, unequal arm-lengths and laser Doppler shifts. Thus, to cancel the laser noise, laser beatnotes are formed between lasers on separate SC and, using these one-way laser phase measurements, one can reconstruct an equal-arm interferometer in post-processing. This is commonly referred to as time-delay interferometry (TDI) and can be exploited to cancel the laser phase noise and extract the gravitational wave (GW) induced arm-length strain. The author has assisted in the development and enhancement of The University of Florida Laser Interferometry Simulator (UFLIS) to perform more accurate LISA-like simulations. UFLIS is a hardware-in-the-loop simulator of the LISA interferometry system replicating as many of the characteristics of the LISA mission as possible. This includes the development of laser pre-stabilization systems, the modeling of the delayed inter-SC laser phase transmission, and the microcycle phase measurements of MHz laser beatnotes. The content of this dissertation discusses the general GW detection methods and possible GW sources as well as the specific characteristics of the LISA mission's design. A theoretical analysis of the phasemeter and TDI performance is presented along with experimental verification measurements. The development of UFLIS is described including a comparison of the UFLIS noise sources with the actual LISA mission. Finally, the enhanced UFLIS design is used to perform a second-order TDI simulation with artificial GW injection. The results are presented along with an analysis of relevant LISA characteristics and GW data-extraction methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shawn Mitryk.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Muller, Guido.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044058:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044058/00001

Material Information

Title: Laser Noise Mitigation through Time Delay Interferometry for Space-Based Gravitational Wave Interferometers Using the UF Laser Interferometry Simulator
Physical Description: 1 online resource (186 p.)
Language: english
Creator: Mitryk, Shawn
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: The existence of gravitational waves was theorized in 1916 by Albert Einstein in accordance with the linearized theory of general relativity. Most experiments and observations to date have supported general relativity, but now, nearly 100 years later, the scientific community has yet devise a method to directly measure gravitational radiation. With the first attempts towards a gravitational wave measurement in the 1960s, many methods have been proposed and tested since then, all failing thus far to provide a positive detection. The most promising gravitational radiation detection method is through the use of a space-based laser interferometer and with the advancement of modern technologies, these space-based gravitational wave measurements will eventually provide important scientific data to physics, astro-physics, and astronomy communities. The Laser Interferometer Space Antenna (LISA) is one such space-based laser interferometer. LISA's proposed design objective is to measure gravitational radiation in the frequency range from 30 microHz to 1 Hz using a modified Michelson interferometer. The interferometer arms are 5 Gm in length measured between each of the 3 spacecraft in the interferometer constellation. The differential arm-length will be measured to an accuracy of 18 pm/sqrt(Hz) resulting in a baseline strain sensitivity of 4.8*10^(-21)/sqrt(Hz). Unfortunately, the dynamics of the spacecraft orbits complicate the differential arm-length measurements. The arms of the interferometer change in length resulting in time-dependent, unequal arm-lengths and laser Doppler shifts. Thus, to cancel the laser noise, laser beatnotes are formed between lasers on separate SC and, using these one-way laser phase measurements, one can reconstruct an equal-arm interferometer in post-processing. This is commonly referred to as time-delay interferometry (TDI) and can be exploited to cancel the laser phase noise and extract the gravitational wave (GW) induced arm-length strain. The author has assisted in the development and enhancement of The University of Florida Laser Interferometry Simulator (UFLIS) to perform more accurate LISA-like simulations. UFLIS is a hardware-in-the-loop simulator of the LISA interferometry system replicating as many of the characteristics of the LISA mission as possible. This includes the development of laser pre-stabilization systems, the modeling of the delayed inter-SC laser phase transmission, and the microcycle phase measurements of MHz laser beatnotes. The content of this dissertation discusses the general GW detection methods and possible GW sources as well as the specific characteristics of the LISA mission's design. A theoretical analysis of the phasemeter and TDI performance is presented along with experimental verification measurements. The development of UFLIS is described including a comparison of the UFLIS noise sources with the actual LISA mission. Finally, the enhanced UFLIS design is used to perform a second-order TDI simulation with artificial GW injection. The results are presented along with an analysis of relevant LISA characteristics and GW data-extraction methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shawn Mitryk.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Muller, Guido.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044058:00001


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LASERNOISEMITIGATIONTHROUGHTIMEDELAYINTERFEROMETRYF OR SPACE-BASEDGRAVITATIONALWAVEINTERFEROMETERSUSINGTHE UFLASER INTERFEROMETRYSIMULATOR By SHAWNJ.MITRYK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012ShawnJ.Mitryk 2

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Idedicatethisworktomyparents,JeffandTerryMitryk,who mprovidedtheopportunity andneverquestionedmydesiretofulllmydreams,whatever theymaybe. 3

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ACKNOWLEDGMENTS I'dliketogiveaspecialthankstoeveryonewhohelpedmewit htheseprojects. GuidoMueller,whommanagedtobeanresearchadvisor,acare ermentor,andaclose colleague,allatthesametime,providedinsightanddirect iontowardsmyeducation andscienticpursuits.IraThorpewasessentialtomydevel opmentasaeffectiveand creativeproblem-solverandprovidedabaseformyfutureex perimentalinvestigations. ThankstoDylanSweeney,YinanYu,PepSanjuan,andSyedAzer forallowingmeto picktheirbrainsandforassistanceindeciperingcomplica tionswithmyexperiments. Also,I'dliketospecicallythankmyhigh-schoolphysicst eacher,Mr.StevenDesanto, whomnotedmytalentsinthesciencesandparticularly,inph ysics,andinspiredmeto cultivatethem.Andthankstoeveryoneelsewhoworkedinthe UF-LISAprojectaswell asthosethatcametovisitforkeepingmydaysinthelabinter esting. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................9 LISTOFFIGURES .....................................10 ABSTRACT .........................................13 CHAPTER 1INTRODUCTION ...................................15 1.1GravitationandGeneralRelativity ......................15 1.2TheGravitationalWaveSpectrum ......................16 1.3Space-BasedGravitationalWaveAstronomy ................17 1.4DetailsoftheDissertation'sContent .....................20 1.5NotetotheReader ...............................20 2GENERALRELATIVITYANDGRAVITATIONALWAVES .............21 2.1IntroductiontoGeneralRelativity .......................21 2.1.1NewtonianGravity ...........................21 2.1.2EinsteinFieldEquations ........................21 2.1.3ProperTimeInvariance ........................22 2.1.4Metrics,BlackHoles,andCurvature .................23 2.1.5Geodesics ................................27 2.1.6EinsteinFieldEquationsRevisited ..................28 2.2GravitationalWaveDerivation .........................29 2.2.1PolarizedPlaneWaveSolutions ....................29 2.2.2SpacetimeStrain ............................30 2.2.3Weak-eldGRMultipoleExpansion ..................31 2.3GravitationalWaveSourcesandDetectionMethods ............35 2.3.1StrainEstimation ............................35 2.3.2GravitationalWaveEvidence .....................35 2.3.3TheGravitationalWaveSpectrum ...................36 2.3.4LISAGravitationalWaveSources ...................37 2.3.4.1CompactBinaries ......................37 2.3.4.2BinaryBlackHoleMergers .................38 2.3.4.3ExtremeMassRatioInspirals ................39 2.3.4.4OtherSuggestedSources ..................39 3THELASERINTERFEROMETERSPACEANTENNA ..............40 3.1LISAOverview .................................40 3.2TheDisturbanceReductionSystem .....................42 5

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3.3TheInterferometryMeasurementSystem ..................42 3.4 CycleAccuracyPhaseMeasurement ....................47 3.4.1PhotodetectorNoise ..........................48 3.4.1.1Shot-Noise ..........................48 3.4.1.2DarkCurrentNoise .....................49 3.4.1.3Johnson-NyquistNoise ...................49 3.4.1.4FlickerNoise .........................50 3.4.1.5HeterodynePhaseMeasurements .............51 3.4.2 CyclePhasemeter ..........................51 3.4.2.1ClockNoise ..........................52 3.4.2.2ADCQuantizationNoise ..................53 3.4.2.3ADCAmplitudeNoise ....................53 3.4.2.4Clock-ADCTimingJitter ...................54 3.4.2.5DemodulationNoiseCoupling ...............55 3.4.3HeterodyneTime-DelayInterferometry ................58 3.4.3.1FiberNoise ..........................60 3.4.3.2Spacecraft/Proof-MassMotion ...............61 3.4.3.3Inter-SpacecraftMotion ...................62 3.4.3.4BasicTDI-CombinationsandConsiderations .......65 4TIMEDELAYINTERFEROMETERY ........................69 4.1LaserNoiseCancellation ...........................69 4.1.1LaserNoise ...............................70 4.1.2ClockNoiseTransfers .........................70 4.1.3RangingErrors .............................71 4.1.4FractionalDelayFilteringandInterpolation ..............72 4.2LaserPre-stabilization .............................73 4.2.1Pound-Drever-HallLocking ......................74 4.2.2ArmLocking ...............................75 4.3TDITheory ...................................77 4.3.1TDICombinations ...........................77 4.3.2SagnacCombinations .........................79 4.3.2.1TDISix-PulseCombinations ................79 4.3.2.2TDISymmetric-SagnacCombination ...........80 4.3.3MichelsonX-combinations .......................80 4.3.3.1TDIX 0.0 ............................81 4.3.3.2TDIX 1.0 ............................82 4.3.3.3TDIX 2.0 ............................84 4.3.4LISAOrbitalDynamicsandTDIDataAnalysis ............84 4.4Ranging .....................................85 4.4.1Pseudo-randomNoise(PRN)CodeRanging ............86 4.4.2Time-delayInterferometryRanging(TDIR) ..............88 4.4.2.1TDIRangingTone ......................89 4.4.2.2TDIRangingParameterSearchAlgorithm .........90 6

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5THELISAPHASEMETER ..............................94 5.1DigitialSignalProcessing(DSP)System ...................94 5.2 CyclePhaseMeasurements .........................94 5.2.1Design ..................................95 5.2.2PhasemeterReadouts .........................100 5.2.2.1PhaseQuantizationNoise .................101 5.2.2.2CICDownsamplingandAliasing ..............102 5.2.3PhasemeterTest-Measurements ...................103 5.2.4PhasemeterNoiseModel .......................106 5.2.5DifferentialandEntangledMeasurements ..............107 5.2.5.1DigitallySplitDifferentialNoise ...............107 5.2.5.2ElectronicallySplitDifferentialNoise ............109 5.2.5.3EntangledPhaseNoise ...................114 5.2.6ApplicationsinLISAandLIGO ....................115 5.3ADCNoiseEstimation .............................115 5.4TimingJitterExtraction .............................123 5.4.1PhaseDispersionMitigation ......................125 5.4.2AbsoluteTimingJitterExtraction ...................127 5.5PhasemeterPerformanceReview .......................129 6THEUNIVERSITYOFFLORIDALISAINTERFEROMETRYSIMULATOR ...132 6.1TheLISALaserTest-bench ..........................132 6.2TheElectronicPhaseDelay(EPD)Unit ...................135 6.2.1Design ..................................135 6.2.2Verication ...............................139 6.2.2.1Time-changingTimeDelay .................143 6.2.2.2GravitationalWaveInjection ................143 6.3UFLISSimulations ...............................145 6.3.1Arm-LockingStabilization .......................145 6.3.2TDISimulationOutline .........................149 7TIME-DELAYINTERFEROMETERYSIMULATONS ...............151 7.1TransponderTDISimulations .........................151 7.1.1Static-ArmTransponderSimulation ..................153 7.1.2Dynamic-ArmTransponderSimulation ................155 7.2LISA-like(Master-SlavePhaseLockedLaser)TDISimula tions ......158 7.2.1Static-ArmLISA-likeSimulation ....................159 7.2.2Dynamic-ArmLISA-likeSimulation ..................163 8CONCLUSION ....................................169 APPENDIX ATIMEVARYINGFRACTIONALDELAYINTERPOLATIONFUNCTION .....173 7

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BTDI2.0COMBINATIONFUNCTION ........................175 REFERENCES .......................................178 BIOGRAPHICALSKETCH ................................186 8

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LISTOFTABLES Table page 2-1PrimaryLISAvericationbinarysources ......................36 2-2Gravitationalwavefrequencyrangeofemission ..................37 3-1LISAcharacteristicsandrequirements .......................43 4-1OrbitaldynamicsapproximationsforTDIgenerations ...............79 6-1TDIexperimentalcharacteristics ..........................149 7-1TransponderTDI-rangingestimation ........................156 7-2LISA-likeTDI-rangingestimation ..........................165 9

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LISTOFFIGURES Figure page 1-1NASA-ESALISAsolicitation .............................18 2-1LISA/LIGOdifferentialarm-lengthchangesduetoGWpol arization .......30 2-2Atheoreticalmodelofabinarystarsystem ....................33 2-3AnoutlineofGWdetectionmethodsandassociatedfreque ncyranges .....37 3-1DiagramoftheLISAorbitaldynamics .......................41 3-2DiagramoftheLISAconstellation .........................45 3-3Diagramoftheinterferometrymeasurementsystem ...............59 4-1ModelofthePDHlockingscheme. .........................74 4-2DiagrammaticmodelsoftheTDI-XandSagnaccombination s ..........79 4-3Flowchartoftheranging-toneminimizationprocess ...............93 5-1AmodeloftheLISAphasemeter ..........................95 5-2CICltertransferfunctions .............................97 5-3TheoreticalPMreadoutdigitizationlimitations ...................102 5-4PhasemeternoisecausedbyCIClteraliasing ..................103 5-5SoftwarevericationofPMperformance ......................104 5-6HardwarevericationofPMperformance .....................105 5-7VericationofthePMfeedbacktransferfunction .................106 5-8Phasemeternoisemodel ..............................108 5-9ExperimentalmodelsofADCnoiseestimationmeasuremen ts ..........109 5-10PM/ADCQuantizationanddifferentialnoise ....................110 5-11ADCtimingjitternoiselimitations ..........................112 5-12ADCamplitudenoiselimitations ..........................113 5-13Entangledphasemeasurementresults .......................115 5-14N-stageCIClteraliasingandentangledphasemeasure ments .........116 5-15ModelforestimatingADCphaseandamplitudenoise ..............117 10

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5-16CommonsourceADCphaseandamplitudenoiseresults ............119 5-17Zero-crossingtimingjitterestimate .........................121 5-18ADCtimingjittercomparison ............................122 5-19Timingjittercalibrationmodel ............................125 5-20Temperaturedependentphasedispersion .....................126 5-21RFtransformerphaseloss .............................127 5-22ReplacementRFtransformerdispersionmitigation ................128 5-23Phasemeternoisemodel,estimation,andlimits ..................130 5-24ComparisonofthePMnoisecharacteristics ....................131 6-1UFLISlaserbenchtop ................................133 6-2Measurementsofcommonlyusedsources ....................134 6-3Modeloftheelectronicphasedelay(EPD)unit ..................136 6-4EPDunitdata-packingscheme ...........................137 6-5NoiselimitationsoftheEPDunit ..........................141 6-6EPDUnit'sphase-noisetransmissionaccuracy ..................142 6-7Timeseriesoftheinterpolateddelaydifference ..................144 6-8Interpolateddelaydifferentialmeasurementsandcorr ections ..........145 6-9Timeseriesofanarbitrarygravitationalwave ...................146 6-10Spectralcorrectionofanarbitrarygravitationalwav e ...............147 6-11Long-armhardware-basedsingle-arm-lockingexperim ent ............148 7-1ModeloftheTDI-Transponderexperimentalbenchtop ..............152 7-2Rawstatictransponderexperimentalresults ....................154 7-3Correctedstatictransponderexperimentalresults .................157 7-4Rangingtonecancellationspectralresults .....................158 7-5Dynamictransponderexperimentalresults .....................159 7-6TDIlaserphasenoisesuppression .........................160 7-7ModeloftheLISA-likeTDIexperimentalbenchtop ................161 11

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7-8StaticLISA-likeexperimentalresults ........................162 7-9Cross-correlatedTDIcombinations-StaticTransponde randLISA-like .....163 7-10DynamicLISA-likeexperimentalresults ......................166 7-11Cross-correlatedTDIcombinations-DynamicTranspon derandLISA-like ...167 7-12Confusionnoisetime-seriescomparison ......................168 8-1CompiledresultsandcomparisonwithTDIforLISA ...............170 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LASERNOISEMITIGATIONTHROUGHTIMEDELAYINTERFEROMETRYF OR SPACE-BASEDGRAVITATIONALWAVEINTERFEROMETERSUSINGTHE UFLASER INTERFEROMETRYSIMULATOR By ShawnJ.Mitryk May2012 Chair:GuidoM¨ullerMajor:Physics Theexistenceofgravitationalwaveswastheorizedin1916b yAlbertEinstein inaccordancewiththelinearizedtheoryofgeneralrelativ ity.Mostexperiments andobservationstodatehavesupportedgeneralrelativity ,butnow,nearly100 yearslater,thescienticcommunityhasyetdeviseamethod todirectlymeasure gravitationalradiation.Withtherstattemptstowardsag ravitationalwavemeasurement inthe1960s,manymethodshavebeenproposedandtestedsinc ethen,allfailing thusfartoprovideapositivedetection.Themostpromising gravitationalradiation detectionmethodisthroughtheuseofaspace-basedlaserin terferometerandwith theadvancementofmoderntechnologies,thesespace-based gravitationalwave measurementswilleventuallyprovideimportantscientic datatophysics,astro-physics, andastronomycommunities. TheLaserInterferometerSpaceAntenna(LISA)isonesuchsp ace-basedlaser interferometer.LISA'sproposeddesignobjectiveistomea suregravitationalradiation inthefrequencyrangefrom 30 Hz to 1 Hz usingamodiedMichelsoninterferometer. Theinterferometerarmsare 5 Gm inlengthmeasuredbetweeneachofthe3spacecraft intheinterferometerconstellation.Thedifferentialarm -lengthwillbemeasuredtoan accuracyof 18 pm = p Hz resultinginabaselinestrainsensitivityof 3.6 10 21 = p Hz Unfortunately,thedynamicsofthespacecraftorbitscompl icatethedifferential 13

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arm-lengthmeasurements.Thearmsoftheinterferometerch angeinlengthresulting intime-dependent,unequalarm-lengthsandlaserDopplers hifts.Thus,tocancelthe lasernoise,laserbeatnotesareformedbetweenlasersonse parateSCand,usingthese one-waylaserphasemeasurements,onecanreconstructaneq ual-arminterferometer inpost-processing.Thisiscommonlyreferredtoastime-de layinterferometry(TDI)and canbeexploitedtocancelthelaserphasenoiseandextractt hegravitationalwave(GW) inducedarm-lengthstrain. Theauthorhasassistedinthedevelopmentandenhancemento fTheUniversity ofFloridaLaserInterferometrySimulator(UFLIS)toperfo rmmoreaccurateLISA-like simulations.UFLISisahardware-in-the-loopsimulatorof theLISAinterferometry systemreplicatingasmanyofthecharacteristicsoftheLIS Amissionaspossible.This includesthedevelopmentoflaserpre-stabilizationsyste ms,themodelingofthedelayed inter-SClaserphasetransmission,andthe cycle phasemeasurementsof MHz laser beatnotes. ThecontentofthisdissertationdiscussesthegeneralGWde tectionmethodsand possibleGWsourcesaswellasthespeciccharacteristicso ftheLISAmission'sdesign. AtheoreticalanalysisofthephasemeterandTDIperformanc eispresentedalong withexperimentalvericationmeasurements.Thedevelopm entofUFLISisdescribed includingacomparisonoftheUFLISnoisesourceswiththeac tualLISAmission.Finally, theenhancedUFLISdesignisusedtoperformasecond-orderT DIsimulationwith articialGWinjection.Theresultsarepresentedalongwit hananalysisofrelevantLISA characteristicsandGWdata-extractionmethods. 14

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CHAPTER1 INTRODUCTION 1.1GravitationandGeneralRelativity Althougharst-ordertheoryofgravitywasestablishedin1 687withthepublication of'Principia[ 1 ]'byIssacNewtonandstartedtheeldofphysicsasweknowit today, physicistsarestilltryingtodenethedetailsofgravity' sinteractions.In1916,over200 yearslater,AlbertEinsteinpublishedhistheoryofgenera lrelativity(GR)whichdened theinteractionofmatterwiththecurvatureofspaceandtim e[ 2 3 ].Thisprovided physicistswithnewinsightsongravityandre-denedourco nceptsonthestructure oftheuniverse.EventhoughEinstein'stheorieshaveyetto bedis-proven,modern discoveriesareraisingnewquestionsandtestingthelimit sofEinstein'sequations. Forexample,thestandard-modelofparticle-physicsdoesn otincludeanexplanation forgravity,althoughsomephysicistshaveproposedtheexi stenceofa'graviton,'[ 4 ]the carrierofthegravitationalforce,despitethefactthatit wouldextremelydifculttodetect becauseofhowweaklygravityinteracts. 1 Proposalstoexplainanapparent'missing matter'ingalaxiesandtheacceleratedexpansionoftheuni versestatethatdark-matter anddark-energy[ 6 – 8 ]dominateover'light'matterandlargelydeterminethepas tand futureevolutionoftheuniverse.Ifdark-matterparticles 2 arediscovered,theexistence ofdark-matterraisesnewquestionsaboutthecompositiono fmatterintheuniverseand, ifnotdiscovered,mayindicateaneedtomodifyEinstein'st heories.Furthermore,the inabilitytoquantizegravityandcreateauniedtheoryoff orceswhichisconsistentwith relativityisarguablythegreatestdilemmaofmodern-dayp hysics. Thatsaid,Einstein'stheoriesongeneralrelativityhave, atthispoint,been supportedbyallexperimentaltestsandobservationsfromt hebendingoflightbya 1 Although,LISAcouldimprovetheupperboundonit'spossibl emass[ 5 ]. 2 Darkmatterparticlesinteractgravitationallybutnotele ctromagnetically. 15

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massivebody[ 9 ]tothepredictionandobservedevidenceforblackholes.No netheless, newquestionsarebeingraised.Howdophysicistsexplainbo thquantummechanics andrelativitysimultaneously?Istheresomewaytonda'gr andunication'theoryto explainbothphenomenon?Isthetheoryofgeneralrelativit yacompleteexplanationof gravityordoesithavetobemodiedinsomeway?Andnally,t hequestionwhichwill betheindirectfocusofthisdissertation,canweusespacebasedlaserinterferometry tomeasureagravitationalwaves(GWs)[ 10 ],thespace-timestraincausedbythe motionofmassivebodies,accuratelyenoughtolearnmoreab outthedetailsofgeneral relativity? 1.2TheGravitationalWaveSpectrum Observationally,thequestionofwhetherornotGWsexistha sbeenanswered. In1993,theNobelPrizeinPhysicswasawardedtoRussellHul seandJosephTaylor fortheindirectdetectionofGWsbydemonstratingthatther otationalenergylossof thebinarypulsarsystemPSRB1913+16equaledtheratepredi ctedthroughtheGW energylossgivenbyGR[ 11 ].Unfortunately,scientistshaveyettoachieveadirect detectionofGWs.Therstefforts,usingresonantWeberbar s[ 12 ],werenotnearly sensitiveenoughtomakeapositivedetectionandhaveaprim arydesigndisadvantage inthattheycanonlymeasureatasingleresonantfrequency. Moremodernattempts tomeasureGWsincludetheuseofgroundbasedinterferomete rssuchastheLaser InterferometerGround-BasedObservatory(LIGO)[ 13 14 ],pulsartiminganalysisusing radiotelescopessuchastheSquareKilometerArray(SKA)[ 15 16 ],andcryogenic resonantWeberbarssuchasALLEGRO[ 17 18 ]andminiGRAIL[ 19 ].Thusfar,all attemptshavefailedtomakeapositivedirectGWdetection. Ontheotherhand,LIGO andotherGWcollaboratorshavebeenabletosetupperlimits onthedistributionand amplitudeofmanyproposedGWsources[ 20 – 22 ]. ThejusticationforthemanyassortedeffortstomeasureGW sgoesbeyond braggingrightsforarstdetection.Asacomplimenttothee lectromagneticobservations 16

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oftheuniversefrommicrowavestox-rays,GWsprovideawhol enewspectrumthrough whichtoobserveastro-physicalevents.Themeasurementso fGWswillprovideamap ofblack-holespace-timesandverifyrelativisticblack-h olemodels[ 23 24 ],signicantly improvetheconstraintsontheHubbleconstant[ 25 ],andprovideearlywarningsystems forelectromagneticobservationsandcross-correlatedpa rameterestimation[ 26 27 ]. Mostimportantly,themeasurementsofaGWsignalcouldgive scientiststhenecessary informationtobeabletonarrowdownthemany,andgrowing,m oderntheoriesofgravity thatarebeingusedtoexplaintheaforementionedcomplicat ionstoEinstein'sGR. Fortunately,physicistsareworkingonamorepromisingdet ectionmethod: space-basedlaserinterferometerssuchastheLaserInterf erometerSpaceAntenna (LISA)[ 28 29 ],showninFigure 1-1 ,andit'sconceptualsuccessorsliketheNext GravitationalWaveObservatory(NGO)[ 30 ]. 1.3Space-BasedGravitationalWaveAstronomy Space-basedGWdetectorssuchasLISA,NGO,ortheDeci-hert zInterferometer GravitationalWaveObservatory(DECIGO)[ 31 ]havemanyaddedbenetsover ground-basedobservatories.Thegravitationally'quiet' environmentofspaceallows space-baseddetectorstogetawayfromtheseismicandgravi tygradientnoisethat limitthelow-frequencydetectioncapabilitiesofgroundbaseddetectors.Inaddition, space-basedsatellitesallowsforGigameter( Gm )baselinearm-lengths,incomparison tothefewkilometer( km )arm-lengthsofground-baseddetectors,decreasingthe requirementsonthedifferentialarm-lengthmeasurementr esolutiontoobtainan equivalentstrainprecision. TheGW-frequencymeasurementbandofspace-baseddetector s,fromabout 30 Hz to 1 Hz forLISAandfromabout 10 mHz to 100 Hz forDECIGO,hasanumber ofscienticallyinterestingGWsources.Thisincludes,bu tisnotlimitedto,compact galacticbinaries,extra-galacticbinaryblackholemerge rs,andextrememassratio inspirals(EMRIs)[ 33 ].MeasurementsofGWsfromcompactbinarysystemscould 17

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Figure1-1.NASA-ESALISAsolicitation:Thissolicitation ,releasedbyNASA,showsa modeloftheLISAinterferometrysetup,includingthe3spac e-craftandthe inter-space-craftlaserlinksusedtomeasuregravitation alradiation.Atthe bottomofthesolicitationisasimulationofaLISA-likeGWs ignalincluding theexpectedinstrumentalnoise;theobjectiveofthisthes isistore-createa hardware-in-the-loopsimulationmuchliketheoneshown.T heguresatin thetop,leftcornershowtheexpectedLISAsourcesincludin gblackhole binariesmergers,extreme-massratioinspirals,compacts tarbinaries,and nally,thegravitationalbackground,andpossibly,unkno wn, gravitational-quantumeffects.[ 32 ] 18

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beusedtoestimatethedensityofthesebinarysystemswithi nourgalaxy.TheGW measurementofabinaryblackholemergeralongwithanelect romagneticobservation, couldbeusedtoputamoreaccurateconstraintontheHubblec onstant[ 25 ].GWsfrom extrememassratioinspiralsprovideaperfectexperimenta ltest-benchtomapoutthe space-timecurvaturearoundablackholeandverifySchwarz schildandKerrblackhole metricsolutions[ 23 24 ]. Spacebasedgravitationalwavedetectorshavetheirowncom plicationsthough.The proof-masses,theobjectswithwhichthegravitationallyi nducedmotionismeasured, mustbekeptinagravitationalfree-fallandshieldedfroma nyothernon-gravitational forces,suchasthoseproducedbyelectromagneticradiatio nfromthesun.Also, becausetheproof-massesmustbeinfree-fall,theirindepe ndentgeodesicorbits causethearm-lengthsoftheinterferometertochangeovert ime.Asaresult,the common-modelaserphasenoiserejectioninherentinmostin terferometers,isnolonger maintained. Space-basedinterferometersmustmakeuseofasequenceofl aserphasenoise stabilizationtechniquesincombinationwithone-waylase rphasemeasurementsand thepost-processingremovaloflaserphasenoisetoaccurat elyextractGWsignals fromthephoto-detector(PD)signals[ 29 34 ].Laserpre-stabilizationtechniques includevariousmethodsoflockingthelaserfrequencytoas tablereference;this couldbeanultralowexpansion(ULE)glasscavityortheinte rferometerarm-length itself.Oncethelasershavebeenpre-stabilized,one-wayl aserphasemeasurements canbetakenalongtheindividualarmsoftheinterferometer .Bycombiningthese one-waylaserphasemeasurementsinparticulartime-shift edandtime-scaledlinear combinations,onecancancelthecommon-modelaserphaseno iseandextractthe phasemodulatedgravitationalwave.Thismethodoflaserno isecancellationinan unequal-arminterferometerisknownastimedelayinterfer ometry(TDI)[ 35 ]. 19

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1.4DetailsoftheDissertation'sContent Tobegin,wewilldiscussthebasicsofgeneralrelativity,d erivetheGWpropagation characteristics,outlineGWdetectionmethods,andprovid eanexampleofaninterferometricGWmeasurementofabinarysource.Wewillalsogivea briefintroduction oftheLISAdesignsensitivityandexaminetheexpectedLISA sourcesincludingtheir expectedstrainamplitude,frequency,anddetectionrates (Chapter 2 ).Wewillthen explainthedetailsoftheLaserInterferometerSpaceAnten naincludingtheDisturbance ReductionSystemandtheInterferometryMeasurementSyste m(Chapter 3 ).Chapter 4 providesamorein-depthexplanationoflaserphasemitigat ionmethodsofthe LISAInterferometryMeasurementSystem(IMS)andatheoret icalanalysisofTDI. Chapter 5 analyzesthedesign,performance,andlimitingnoisesourc esoftheUF-LISA phasemeter(PM)aswellaspresentsafewexperimentsandres ultsforADCnoise mitigationinLISA.TheUniversityofFloridaLaserInterfe rometrySimulator(UFLIS) alongwiththeelectronicphasedelay(EPD)unitandtheprestabilizedUF-LISAlaser bench-topisthenexploredindetail.Somebasicmeasuremen tsarethenpresented todemonstratethecapabilitiesofUFLIS(Chapter 6 ).Finally,thecompleteLISA constellationissimulatedtotestTDI1.0and2.0(Chapter 7 )linearcombinations andverifylasernoisecancellation,rangingestimation,a ndGWextraction. 1.5NotetotheReader Thereadershouldnotethat,althoughthisdissertationfoc usesonthedesign andscienceoftheLISAmissionaswellasreferencesLISAmis sioncharacteristics andpublications,thespecicationsforthedevelopmentof afuture,space-basedGW interferometermaydifferfromthosereferenced[ 36 ].Thismayincludechangesinthe arm-length,relaxedsensitivityrequirementsonthecompo nents,andareductionin thenumberoflaserlinksbetweenSC.Nonetheless,themeasu rementsandscience presentedinthisdissertationstillholdtrueandcanbeapp liedtoanyspace-based interferometrymission. 20

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CHAPTER2 GENERALRELATIVITYANDGRAVITATIONALWAVES ThefollowingintroductiontoGRasitpertainstogravitati onalwaveemissionand detectionwasderivedfromacombinationof[ 37 ],[ 38 ],and[ 39 ]alongwithadditional calculationsandconceptualrelationshipsprovidedbythe author. 2.1IntroductiontoGeneralRelativity 2.1.1NewtonianGravity SirIssacNewtonproposedthatthegravitationallyattract iveforcebetweentwo massiveobjects, m a and m b ,isgivenby: F ab = Gm a m b j r ab j 2 ^ r ab (2–1) where F ab istheforcevectorbetweentheobjects, !r ab isthedistancevectorconnecting thetwoobjects,and G isthegravitationalconstant.Onecanalsowritethegravit ational potential, V ( r ) ,surroundinganobjectofmass M as: V ( j r ij j )= GM j r ij j (2–2) where r ij isthedistancevectorconnectingthecenterofmassoftheob ject, x i ,to apointofinterest, x j .Theserelationshipsbecameuniversallyacceptedbasedon conrmedobservationsandpredictionsthroughoutthe1700 sand1800s,givingrise totheNewtonianinterpretationofgravity.However,somep henomena,suchasthe precessionoftheplanetMercury,werenotentirelyexplain edbyNewton'sLaws. 2.1.2EinsteinFieldEquations Itwasnotuntil200yearslaterwhentheTheoryofGeneralRel ativity,asproposed byAlbertEinsteinin1916[ 3 ],revealedthetruenatureofgravitybydeningthe interactionofspaceandtimewithenergyandmass.Morespec ically,GRassertedthat thecurvatureofatensorspacetimepotential, G ,denesthegravitationalforceonan objectwhile,atthesametime,themass/energydistributio noftheobject, T ,denes 21

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howthespacetimearoundtheobjectwilldistort.Thisissum medupbyEinstein'stensor eldequations: G = 8 G c 4 T (2–3) where G isthesymmetric,second-rank,covariant,Einsteintensor T isthe symmetric,second-rank,covariant,energy-momentum(ors tress-energy)tensor,and c isthespeedoflight.Generally, G ,canbeinterpretedasthecurvatureanddynamics ofspacetimewhiletheEinsteinstress-energytensor, T ,denesthedistribution andmomentumofmass/energybeingactedon,whilesimultane ously,changingthe spacetimearoundit.Thisself-interactivenatureofthe10 independentsecond-order, differentialEinstein'sequationsmakesthemextremelyha rdtosolvewithonlyafew speciccaseshavingacompleteanalyticsolution.2.1.3ProperTimeInvariance TobetterunderstandtheEinsteineldequationsandeventu allybeabletocompute thestrainandfrequencyofaGWsource,wemustrstintroduc e4-vectors,vector transformations,andexplainthespacetimemetric, g .Considera4-vectorfora parameterizedcurve, x ( )=[ ct ( ), x ( ), y ( ), z ( )] ,inaat,non-movingcoordinate basis,commonlyreferredtoasMinkowskispace: = 266666664 1000 0+10000+10000+1 377777775 (2–4) suchthatwecanwritethe'spacetimedistance'separatingt woinnitesimalpointsalong thecurve, dx = d ,as: ds 2 = dx dx = dx 2 + dy 2 + dz 2 c 2 dt 2 (2–5) 22

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usingEinsteinsummationnotation.This'spacetimedistan ce'canberelatedtothe propertimeandisinvariantundertransformationtoanewve ctorbasis x suchthat: ds 2 = c 2 d 2 = d s 2 = dx dx (2–6) Solvingfor d ,wecancalculatetheinstantaneouschangeinthepropertim einany referenceframeusingtheequation: d = 1 c r dx d dx d d (2–7) Usingthefactthatthisquantityshouldnotchangeasafunct ionoftheselected reference-frame,wecandeneatransformationfromonevec torspace, x ,toanother vectorspace, x ,usingtransformationmatrices, .Forexample,itcanbeshownthat thetransformationtoareferenceframemovinginthex-dire ctionatavelocityvwith respecttoastationaryreferenceframetakestheformofthe Lorentztransformation: = 266666664 cosh( ) sinh( )00 sinh( )cosh( )00 00100001 377777775 (2–8) where =tanh 1 ( v = c ) .ThistransformationisamemberofthePoincar etransformation groupwhichkeepsthepropertimeintervalinvariantunderr eferenceframetransformationsinaatspacetimeandisthebasisforSpecialRelati vity. 2.1.4Metrics,BlackHoles,andCurvature Transformationsina`curved'spacetimecanbedenedbyrep lacingtheMinkowski spacetimemetric, ,in( 2–6 ),withamoregeneralizedspacetimemetric, g ,suchthat theinvarianceequationbecomes: ds 2 = g dx dx (2–9) 23

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Thespacetimemetric, g ,isacovariant,second-rank,tensorwhichcontainsallthe informationaboutthestrain(currentdensity)anddynamic s(relativeparameterized density)ofspacetime. Someexamplesofcurvedanalyticmetricswhichhelprevealt henatureofthe metricdenitionofspacetimeincludethetheoreticalmode lsofstationary,non-rotating, blackhole,knownastheSchwarzschildmetric: g = 266666664 1 2 GM c 2 r 000 0 1 2 GM c 2 r 1 00 00+10000+1 377777775 (2–10) andthatofarotatingblackhole,knownastheKerrmetric: g = 266666664 h 1 2 GMr c 2 2 i 00 2 GM sin( ) c 2 2 0 2 00 00 2 r 2 0 2 GM sin( ) c 2 2 `00 1 2 r 2 [( r 2 + a 2 ) 2 a 2 sin 2 ( )] 377777775 (2–11) Theserelationshipsarewritteninsphericalcoordinatesw here M isthemassofthe blackhole, a istheratiooftheangularmomentumtothemass ( J = M ) pointinginthe ^ z -direction,and ( r )= r 2 2 GMr c 2 + a 2 (2–12) ( r )= r 2 + a 2 cos 2 ( ). (2–13) Computing ds 2 asgivenby( 2–9 )usingtheSchwarzschildmetric( 2–10 ),weobtain: ds 2 = 1 2 GM c 2 r dt 2 + 1 2 GM c 2 r 1 dr 2 + r 2 d 2 + r 2 sin( ) 2 d 2 (2–14) whichisamorecommonwaytowriteoutthemetricequation. 24

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Togainabetterintuitiononwhatthemetricrepresents,wec omparethese blackholemetricstotheatspacetimeMinkowskimetric( 2–4 ).Itwillbeshownin Chapter 2.1.5 thattherst-ordergravitationalacceleration,writteni ntermsofthe spacetimemetric,canbewrittenas: d 2 x i ( dx 0 ) 2 = @ i h 00 2 (2–15) where h isthedeviationfromtheMinkowskimetric, h = g (2–16) ReferencingtheSchwarzschildmetricforastationaryblac khole,asin( 2–10 ), weseethatthe dt 2 and dr 2 termsinthemetricconvergeto-1and+1,respectively, whilethegravitationalaccelerationapproacheszerointh elimitthat r !1 or M 0 ,resultinginMinkowski-likeatspacetime.Butas r isdecreased,thespacetime curvatureduetotheblackholeincreases,thusincreasingt hegravitationalacceleration. Ausefulwaytoanalyzetheseblackholesistoconsiderthera diusatwhichspacetime goesfrombeing'time-like', ds 2 > 0 ,to'space-like', ds 2 < 0 ,alsoknownastheevent horizon.Solvingfortheinversionpoint,or ds 2 =0 ,weobtaintheSchwarzschildradius: R S = 2 GM c 2 (2–17) Now,changingourfocustotheKerrmetric,givenby( 2–11 ),weseethatthe g and g termsarenolongerequalto1,indicatingthatthegravitati onaleldisnolonger sphericallysymmetricduetotheblackhole'sangularmomen tum.Additionally,ifwe computethegravitationalaccelerationneararotatingbla ckhole,asgivenby( 2–15 ), wewillseethatthecurvatureisgreatestnearthepolesofth erotatingblackholeand thatitdoesnotapproachinnityas r approacheszero.Thisisbecausetheblackhole's angularmomentumcausestheSchwarzchild-singularitytob estretchedintoacircular loop.Finally,oneofthemostinterestingaspectsoftheKer rblackholemetricisthe 25

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non-zero g t and g t termswhichrepresenttheangulardynamicsofspace-timedu eto therotationoftheblackhole,alsoreferredtoasframedrag ging.Thisframe-dragging concepthasrecentlybeenconrmedbyGravityProbeB[ 40 ]usingtheEarthasthe sourceofrotationalspace-timeframedragging. Anotherusefulwaytoanalyzethecurvatureofspacetimegiv enaparticularmetric, g ,isthroughtheuseoftheRiemannandRicciTensors.TheRiem annTensordenes howavectorchangesasitisparalleltransportedaroundacl osedcurvedenedby translationvectors, A and B ,andcanbegivenbythedenition: x = R x A B (2–18) Wewillrefrainfromgoingintothedetailsofthederivation andsimplyprovidea denitionfortheRiemanntensorintermsofpartialderivat ivesofaspacetimemetric, g ,whichmaybegivenby: R = @ @ + (2–19) where aretheChristoffelcoefcients,alsowrittenintermsofpa rtialderivativesof themetric, g ,as: = g 2 ( @ g + @ g @ g ), (2–20) andaredenedby: r x = @ x + x (2–21) Onecaneasilyseethatthiscurvaturetensorisdenedentir elyasafunctionof themetricandit'spartialderivatives.Althoughthisisth egeneralizeddenitionforthe curvatureonagivenmanifold,inGeneralRelativitywewill primarilybeconcernedwitha 26

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contractedversionoftheRiemanntensor,knownastheRicci Tensor: R = R (2–22) Bytrace-reversingtheRicciTensor,wenallyarriveatthe Einsteintensor: G = R 1 2 Rg (2–23) where,RistheRicciscalar,or: R = g R (2–24) TheEinsteintensorisasymmetric,second-ranktensorwhic hisequaltothenull tensorwhenspace-timeisatandhaszerodivergencesuchth at, r G =0 2.1.5Geodesics Nowthatcurvatureandthemetrichavebeensufcientlydisc ussed,thenext questionbecomeshowdoesthisspacetimecurvatureaffectt hemotionofaparticle? Anobjectinfree-fallwithinagivenmetricwillfollowapat hknownasageodesic.The geodesicequationgivesaparameterizedspace-timesoluti onforwhatisconsidereda 'straight'line,ortheshortestpathbetweentwopoints,wi thinagivenspace-timemetric, orcomparatively,Newton'sLawsforGR.Thegeodesicequati onofmotioniswrittenas: d 2 x d 2 + dx d dx d =0. (2–25) Byconsideringa'static'metricandnon-relativisticmoti on,theinterestingcomponentsoftheChristoffelcoefcientssimplifyto: 00 = g @ g 00 2 (2–26) Then,byevaluating( 2–20 )with g = h ,thegeodesicequationreducestothe aforementionedgravitationalaccelerationwithinagiven spacetimemetric,( 2–15 ). 27

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2.1.6EinsteinFieldEquationsRevisited Atthispointwehaveseenhowamassiveobjectdistortsthelo calspace-time metricaswellashowthismetricaffectsthemotionofanearb yparticle,butthebasis ofgravitationalwavegenerationexistsinthedynamicsofE instein'sequations,when massiveobjectsmovethroughspacetimecausingthemetrict ochange.Thebasisfor thederivationofEinstein'seldequationsliesintheatte mpttodynamicallyequatethe parameterizeddivergencefromaatMinkowskimetric, g ,orequivalently,the Einsteintensor, G ,tothematterenergydistributiontensor, T .Essentially,wewant theEinsteinianequivalentofPoisson'sequationforgravi tation: r 2 ( r )=4 G ( r ), (2–27) where isthe'gravitationalpotential'and isthemassdensity.Onecanmakea comparisonbetweenPoisson'sequationandEinstein'seld equations,againwrittenas: G = 8 G c 4 T (2–28) tobetterunderstandwhatthetermsrepresent.Astatic,rs t-ordercomparisonof theseequationswouldequatethemetric, g tothegravitationalpotential, ,the EinsteinTensor, G totheLaplacianofthegravitationalpotential, r 2 ,andthe stress-energytensor, T ,tothemassdensitydistribution, .Although,whenwe takeallthedynamicsoftheEinsteinequationsintoconside ration,wearerequired tosolve10second-orderinter-dependentdifferentialequ ations,whichhaveveryfew analyticsolutions.Fortunately,ifweconsidera'weak-e ld'expansion,thelinearizion ofEinstein'sequationswillprovideuswitharst-orderdy namicsolutionforthechange inthemetricdueachangeinthemassdistribution.Thesolut ionofthisweakeld expansionismanifestedaspolarizedplaneGWswhichpropag ateoutwardatthespeed oflightfromthegenerationsource. 28

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2.2GravitationalWaveDerivation 2.2.1PolarizedPlaneWaveSolutions Webegintheweak-eldsolutionforGRbydeningametricexp ansionasa Minkowskimetricwithasmallperturbationaddedtoit: g = + h + O [ h n ], (2–29) whileassuming h << 1 sothattheexpansionconverges.EvaluatingthisinEinstei n's equation,( 2–23 ),weobtainarelationshipforthelinearizedeldequation s: h h + h + h =16 T (2–30) where h isthetrace-reversed h ,givenby h = h h 2 (2–31) and h = @ h .Afterapplyingthegaugeconditionssuchthat: h =0, (2–32) weareleftwithaconciseformforthelinearizedeldequati ons: h =16 T (2–33) Althoughtheweak-gravitationaleldequationshaveanext radegreeoffreedom,using Maxwell'selectromagneticeldequationsasourguide,wee xpectagravitationalanalog tothepolarizedplane-wavesolutionsofelectrodynamics. Aswell,weshouldbeableto calculatetheluminosityandfrequencyofaparticularradi ationsourceusingamulti-pole expansion,asiscommonlydoneinelectrodynamics. TheGWsolutionsresultingfrom( 2–32 ),aftertransforming h tothetransversetracelessgauge, h TT ,wherethetemporalcomponentsarezero,canbebrokendown intotwobasispolarizationstates, h + and h .Notethat h TT isit'sowntrace-reverse, 29

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suchthat h TT = h TT ,eliminatingthetrace-reversedbasisneededtoformtheli nearized eldequations.WrittenexplicitlyinCartesiancoordinat es,aGWpropagatinginthe ^ z -directionwilltaketheform: h TT = 266666664 00000 h xx h xy 0 0 h yx h yy 0 0000 377777775 = 266666664 00000 < [ h + e i ( ct z ) ] < [ h e i ( ct z ) ]0 0 < [ h e i ( ct z ) ] < [ h + e i ( ct z ) ]0 0000 377777775 (2–34) where h + = istheGWpolarizationamplitudeand istheGWfrequency.Thecomplete analysisofthepolarizationstatesshowthataringofparti cleslaidinthex/yplanewillbe modulatedwiththepatternsshowninFigure 2-1 ,hencetheassignednames. Figure2-1.LISA/LIGOdifferentialarm-lengthchangesdue toGWpolarization:The affectsofthe h + = strainpolarizationsonaringofparticlesisdepicted.The LIGO(blue)andLISA(red)detectorsareoverlaidonthering toshowthe polarizationaffectsonthedifferentialarm-lengthchang esforeachofthe detectorstypes.TheLIGOdetectorsareonlysensitivetoas ingleGW polarization,inthiscase, h + ,whiletheLISAdetectorissensitivetoboth polarizationsbutwithareductionfactorof 2 = p 3 duetothe 60 o angle betweenthedifferentialarms.[ 29 ] 2.2.2SpacetimeStrain AnotherinterpretationofaGW'saffectonmatterisreprese ntedintermsofa space-timestrainwhichiscomputedfrom h .Inthisweakeldtransverse-traceless 30

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gauge,thegeodesicequationcanbereducedto @ 2 @ t 2 x = x 2 @ 2 @ t 2 h (2–35) Consideringtwoobjectsseparatedinthex-directionbyadi stance x o ,thegeodesic equationisthencalculatedtobe: @ 2 @ t 2 x ( t )= x ( t ) 2 @ 2 @ t 2 h x x + x ( t ) 2 @ 2 @ t 2 h x y (2–36) Tosimplifytheanalysis,wetake h =0 and h + > 0 .Thedistancebetweenthesetwo objectsina'at'spacetimeisgivenby: x ( t )= x o rrrr 1+ h + e i ( t ) 2 rrrr (2–37) Assuming h + << 1 ,wecandenetherelationship: x x o = h + (2–38) where x istheamplitudeoftheseparationdistancerangingfrom x o ( x = 2) to x o +( x = 2) .Thus,inthisway,themetriccanbeinterpretedasaspace-t imestrainwhich changesthedistancebetweentwoobjectsbyit'smagnitude. Themeasurementofthis changeindistanceisthebasisforinterferometricdetecti onofGWs. 2.2.3Weak-eldGRMultipoleExpansion ThemagnitudeofaGWsourcecanbecalculated,asiscommonly donein electrodynamics,intermsofGreen'sfunctionsgivenby: h ( x )= 16 G c 4 Z G ( x y ) T ( y ) d 4 y (2–39) Afterapplyingboundaryconditionsinthefar-eldapproxi mation,wecanwritethe Fourierdomainmetricperturbationas: e h ( x i )= 4 Ge i r c 4 r Z d 3 y e T ( y i ), (2–40) 31

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orthetime-domainperturbationas: h ( x i )= 4 G c 4 r Z d 3 yT ( ct r y i ). (2–41) Usingamulti-poleexpansionoftheradiatingsysteminthef ar-eldapproximation, wemaywritetheobservedstrainfromaquadrapolesourceas: h ij = 2 G c 4 r d 2 I ij dt 2 ( t r ), (2–42) where I ij isthequadrupolemomenttensor: I ij = Z d 3 xT 00 x i x j (2–43) and t r = ct r istheretardedtime.Themonopoleanddipoletermsdonotcon tributeto GWradiationwhilehigherordertermsfalloffas (1 = r ) ( n 1) andwillbeneglectedinthis far-eldapproximation. OneofthemostcommonastrophysicalGWsourcesarebinarysy stems,wheretwo massesareorbitingacommoncenterofmass.Givenabinarysy stemwithtwomasses, m 1 and m 2 ,andaseparationdistanceof 2 a ,wecanwritethequadrapolemomentof inertiaofthesystemas: I ij = a 2 266664 cos 2 ( t ) cos ( t ) sin ( t )0 cos ( t ) sin ( t ) sin 2 ( t )0 000 377775 (2–44) where =( m 1 m 2 ) = ( m 1 + m 2 ) isthereducedmassand = p ( GM ) = ( a 3 ) istheorbital frequencygivenbyKepler'sLaw, M = m 1 + m 2 = 2 a 3 = G [ 41 ].Evaluating I ij in( 2–42 ), 32

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theexpectedGWradiationstraincanbewrittenas: h ij ( t x i )= 4 G c 4 r 2 a 2 266664 cos (2 t r ) sin (2 t r )0 sin (2 t r ) cos (2 t r )0 000 377775 (2–45) = 4 G 2 m 1 m 2 c 4 ra 266664 cos (2 t r ) sin (2 t r )0 sin (2 t r ) cos (2 t r )0 000 377775 (2–46) Figure2-2.Atheoreticalmodelofabinarystarsystem:Twom asses, m 1 and m 2 rotate aboutacommoncenterofmasswithaseparationdistanceof 2 a .As rotationalenergyislostthroughGWradiationtheseparati ondistancewill decrease( 2–50 )whiletheangularfrequencyincreases( 2–52 ). TheinstantaneouspoweroutputthroughGWradiation,orrat herthegravitational luminosity,canbecalculatedbythetime-averagingoveras ingleorbitalperiodofthe radiatingsource,givenby: L GW = G 5 c 5 dI 3 ij dt 3 r 2 (2–47) Forabinarysystem,thisresultsinaluminositygivenby: L Binary = 32 5 2 M 3 a 5 G 4 c 5 f ( ), (2–48) 33

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where f ( ) isacorrectionfunctionbasedontheeccentricity, ,givenby: f ( )=[1+ 73 24 2 + 37 96 4 ][1 2 ] ( 7 = 2) (2–49) andmaycontributeanorderofmagnitudetotheresultinglum inosity[ 37 ]. AsaresultoftheGWenergyradiatedbythebinarysource,the orbitalkinetic energywilldecayresultinginareducedorbitaldistance,i ncreasedluminosity, andincreasedorbitalfrequency.Thesevaluescanbederive dintherstorder Post-Newtonianapproximationforacircularorbit ( =0) as: a = a o 1 t o 1 = 4 (2–50) L = L o 1 t o 5 = 4 (2–51) = o vuut 1 1 t o 3 = 4 (2–52) respectively.Inthisequation,thetime-scalingfactor, o ,isthe'mergertime'untilthe binarysysteminspiralsintoasinglecompactobject: o = 5 256 c 5 a 4 o G 3 M 2 (2–53) resultingfrominitialsystemvaluesof a o L o ,and o Post-NewtonianandNumericalRelativitySolutions HigherorderPost-NewtonianandNumericalRelativity(NR) solutionsprovidemore accuratedescriptionstothesebinaryin-spiralsystemsby takingintoaccountorbital circularizationandtherelativeangularmomentumofthein dividualin-spiralingobjects. Qualitatively,gravitationalradiationfromanin-spiral ingsystemisminimizedthrougha reductionintheeccentricityandcircularizationofthebi naryorbits.Therelativeangular momentumbetweenthestarsandtheangularmomentumofthebi narymaycausethe actualmergertimetoeitherleadorlagthenon-rotatingmer gertime, o .Thiscanalso 34

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causeanasymmetricgravitationalradiationwhichimparts alinearmomentumtothe binarysystem.Thesehigherordersolutionsarenecessaryi nordertoaccuratelyextract gravitationalwavesignalsfromthedetectorthroughwaveformmatching. 2.3GravitationalWaveSourcesandDetectionMethods 2.3.1StrainEstimation NowthatwehaveameasurefortheexpectedGWradiationfroma strophysical quadrapolesources,wecanestimatetheGWstrainfromparti cularsourcesbasedon theparametersdeterminedthroughelectromagneticobserv ations.Forexample,the primaryLISA'verication'binary,RXJ0806.3+1527,aknow nAMCVnbinarysystem withanorbitalperiodof 321 s [ 42 ],isexpectedtohavebinarystarswith m 1 =0.13 M J and m 2 =0.5 M J withanobservationdistanceof r =300 pc ,asseeninTable 2-1 UsingKepler'sLaw,wecancalculatetheseparationdistanc ebetweenthebinary starsas a RX J =37.97 10 6 m .IncludingtheotherknowncharacteristicsoftheRX J0806.3+1527system,wecancalculatethestrainmagnitude observedonEarthfrom ( 2–45 )as: j h j = 4 G 2 M c 4 ra =1.608 10 21 m m (2–54) Alargenumberofknowncompactbinarieshavestrainamplitu desontheorderof 10 22 10 21 m = m inthefrequencyrangefrom 0.1 10 mHz andareprimaryLISA sources.Threeothervericationbinariesandfourotherkn own,possiblesources areoutlinedinTable 2-1 [ 42 ].Furtherdetailsofthesesourcesarediscussedin Chapter 2.3.4 2.3.2GravitationalWaveEvidence ThecurrentevidenceforGWradiationfromastronomicalsou rcesisbasedonthe observationandorbitaldecaymeasurementsofbinarypulsa rsystems.Therstorbital decaymeasurementswereperformedbyHulseandTaylorusing thePSRB1913+16 binarypulsar.In1993,HulseandTaylorwereawardedtheNob elPrizeinPhysicsfor 35

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Table2-1.PrimaryLISAvericationbinarysources:Herewe listthetop8 most-probablevericationbinariesfortheLISA-GWdetect oraccordingto [ 42 ].ThesebinarieswillalsoassistinconstrainingtheHubbl econstant[ 5 ]. TheTDIexperimentsperformedinChapter 7 willusethedoublewhitedwarf RXJ0806.3+1527binaryasthesimulatedGWsource. Name f (mHz)SNR[mean:max]r(pc) m 1 m 2 RXJ0806.3+1527 6.22027[62:227]300 10000.130.2 0.5 V407Vul 3.51250[30:79]300 10000.0680.7 ESCet 3.22[19:62]350 10000.0620.7 AMCVn 1.94414[8:13]6060.140.85 HPLib 1.813[ < 3:5]1970.0320.57 4U1820-30 2.92[ < 3:5]8100 < 0.11.4 WZSge 0.04065[ < 3:5]43 < 0.11 > 0.7 KPD1930+2752 0.2434[ < 3:5]1000.50.97 showingtheorbitaldecayofthePSRB1913+16systemequaled thatpredictedby GR,thusprovidingindirectevidenceofenergylossthrough gravitationalradiation[ 11 ]. Todate,theorbitaldecayofthePSRB1913+16systemmatches thatpredictedby GRtowithin 0.2% [ 43 ].Otherastronomicalobservations,suchasthedoublebina ry pulsarPSRJ0737-3039,areprovidingeventighterconstrai ntsandfurthersupporting thevalidityofEinstein'sGR[ 44 45 ].DespitethecertaintyofGWexistence,adirect observationalongwithelectromagneticcounterpartswoul dprovidephysicistsvital informationinsupportoroppositionofmodernpost-Einste iniantheories. 2.3.3TheGravitationalWaveSpectrum AstronomicalGWsradiatefromawidevarietyofdynamicquad rapolesources includingbinaryinspiralsofcompactstarandblackholeme rgers,asymmetricspinning compactobjects,super-novastarcollapses,andblackhole capturesofcompactobjects. Itisalsoexpectedthatthereisagravitationalwavebackgr ound,muchlikethecosmic microwavebackground,resultingfromquantumuctuations shortlyafterthebig-bang andthesub-sequentialexpansionoftheuniverse.Eachofth esesystemsproduces GWswithinacharacteristicfrequencyrangefrom 10 18 10 9 Hz,eachwithan associatedpossiblemethodofdetection.Thefrequencyran gesandtheirproposed detectionmethodsareoutlinedinTable 2-2 anddepictedinFigure 2-3 36

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Table2-2.Gravitationalwavefrequencyrangeofemission: Thistableoutlinesthe gravitationalwaveemissionspectrum,theGWemissionsour ces,andthe associateddetectiontechniques.(BBHM:BinaryBlackHole Mergers;EMRI: ExtremeMassRatioInspirals) Freq.Range(Hz)NameSourcesDetectionMethod 10 18 10 7 VeryLowPrimordialGWs,PulsarTiming Freq.EarlyUniverseDynamics 10 4 10 0 LowFreq.BBHM,EMRIs,Space-based CompactBinaryStarsInterferometers 10 1 10 4 HighFreq.Supernova,Ground-based CompactBinaryInspiralsInterferometers 10 2 10 9 VeryHighBinaryInspiralHarmonics,ResonantDetectors Freq.TechnologicalApplications Figure2-3.OutlineofGWdetectionmethodsandassociatedf requencyranges:Very low-frequency(e.g.pulsartiming-SquareKilometerArray );Lowfrequency (e.g.space-basedlaserinterferometers-LaserInterfero meterSpace Antenna,NextGravitationalWaveObservatory);High-freq uency(e.g. ground-basedlaserinterferometer-LaserInterferometer GravitationalWave Observatory);Mid-frequency(e.g.space-basedresonantc avitiesDeci-HertzGravitationalWaveObservatory(DECIGO));Ver yhighfrequency (e.g.GWamplicationanddetectionthroughthemeasuredde formationsin aresonantgeometricalstructures-WeberBars/miniGRAIL: ). 2.3.4LISAGravitationalWaveSources Thelow-frequency/LISAmeasurementband,from 0.1 mHz to 1 Hz containsa wide-arrayofinterestinggravitationalwavesources.2.3.4.1CompactBinaries ThestrongestofthesesourcesaremonotonicGWsresultingf rominteracting andnon-interactingbinarystarsystemscomposedofbinari eswithoneorbothofthe starsbeingacompactwhitedwarf(WD)orneutronstar(NS)st ar.Althoughmany non-interactingWD/WD,WD/NS,NS/NSsystemshavebeenobse rved,noneareknown 37

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tobeintheLISAsensitivityrange.Atthesametime,therear emorethan10knownand electromagneticallyobservedinteractingAM-CVn(WDaccr etor)andcompactX-ray(NS accretor)binarysystemswhichareexpectedtobeexcellent vericationsourcesandare wellwithintheLISAstrainsensitivity.Thestrongestofth esevericationsystemswere introducedinTable 2-1 ThesciencemotivatingLISA'sobservationsofthesesource sincludeestimationsof thepopulationsofthesebinarysystemswithinourgalaxyan danimprovedconstrainton theHubbleconstant(alsoreferredtoasredshiftuncertain ty).Infact,LISAisexpected tobesensitivetosomanycompanybinarysystemsthattheirp opulationsareexpected toforma'confusionnoisebackground'duetotheinabilityt odifferentiatebetween individualsourcesforfrequenciesbelow 2 mHz [ 46 ]. 2.3.4.2BinaryBlackHoleMergers OneofLISA'smostinterestingastronomicalobservationsw illbethatofextragalacticnear-equal-mass, 10 3 M 10 7 M ,binaryblackholemergers.Thesesystems willappear,dependingonthetotalmassofthesystem,neart helow-frequencylimits oftheLISAmeasurementband,from 0.1 mHz to 1 mHz ,andwillincreaseinfrequency andstrainamplitudeasthesystemevolves.Thedynamicsoft hesebinaryblack holemergersisnotwelldened,requiringnumericalrelati vitysimulationstoproduce theoreticalmergerwave-forms.Observationsoftheseeven tswillprovideprecision testsoftheseextremeself-interactingspacetimesaswell asdetailsaboutblackhole formationandevolution.Althoughnosystemsareknown,eve ntrateestimatesrange from1to100speryeardependingonthepopulationandLISA's achievedstrain sensitivity.Asanexample,arecentmassiveblackholebina rysystemhasbeenfound, namedNGC-3393withbinarysystemcharacteristics, m 1 =3 10 7 M m 2 =1 10 6 M 2 a 5 10 18 m =162 pc r =50 Mpc [ 47 ].Ifthissystemwereclosertoit's mergertime,theexpectedLISAsensitivitycouldprovidea 10 5 to 10 6 signal-to-noise ratioontheGWemissionwaveformjustbeforetheblack-hole merger.Asaresult 38

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ofGW'snon-interactingproperties,cross-correlationsb etweentheGWsignaland electromagneticobservationsofthesebinaryblackholeme rgerswillallowLISAto estimateanabsolutedistancetothebinarysource,providi nganimprovedconstrainon theHubbleconstantwithanerrorof 1% 1 [ 25 29 ]. 2.3.4.3ExtremeMassRatioInspirals Binaryblackholemergerswithalargemassdifferenceratio ,otherwiseknownas extrememassratioinspirals(EMRIs),willbevisibleneart he 3 mHz cornerfrequencyof theLISAsensitivityband(Figure 8-1 )andaretheprimarydrivingmotivationtoimprove thepeakLISAsensitivity.EMRIsconsistofa 10 6 M massiveblackholebeingclosely orbitedbyand,eventuallymergingwith,asmaller 10 100 M blackhole.Thesmaller blackholewillactasatest-particletoprovideaspacetime mapinthevicinityofthe massiveKerrblackholebeyondthesingularity'sevent-hor izon[ 23 24 ]. 2.3.4.4OtherSuggestedSources Finally,LISAwillprovidesomeoftherstdirecttestsofne wphysicsincluding attemptstoprobethemicrowave-background-like,gravita tionalwavestochastic background.ThisGWbackgroundisconjecturedbasedonthei nationofrst-order phasetransitionsoftheearlyuniverse,shortlyaftertheb igbangandisexpectedto resultinawhite-noiseGWbackgroundthroughouttheuniver se.Formoreinformationon thisandtheotherLISA-liketestsofnewphysics,referto[ 48 ]. 1 CurrentmethodsofconstrainingtheHubbleconstantprovid eanerrorof 5% 39

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CHAPTER3 THELASERINTERFEROMETERSPACEANTENNA Amajorityofthecharacteristicsforthisdescriptioncanb efoundintheLISA NASA/ESAYellow-book[ 29 ]. 3.1LISAOverview TheLISAGWmeasurementschemeutilizesamodied-Michelso ndetection techniquebytakingone-waymeasurementsbetweenfreely-f allingproof-masses housedwithinthreespacecraft(SC).Thesespacecraftfoll owindependent,helio-centric orbitstrailing 20 o behindtheEarth 1 whilemaintaininganearlyequilateraltriangle congurationwhichisoff-setby 60 o totheorbitalplaneofthecenter-of-massasshown inFigure 3-1 .Housedwithineachofthesespace-craftaretwoproof-mass eswhich aremaintainedinagravitationalfree-fallbyadisturbanc ereductionsystem(DRS). Meanwhile,theinterferometricdistancebetweentheproof -massesonoppositeSC aremeasuredwithaninterferometrymeasurementsystem(IM S)asdiagrammedin Figure 3-2 .ThedataisthensenttoEarthtoformthepost-processedcom binations requiredtoextracttheGWsignals.Usingthisdesignbasis, thesuccessoftheLISA missiontowardsdetectingGWsdependsonaseriesofrequire ments(Table 3-1 )which aredenedtooptimallymeasurethepreviouslydiscussedGW sourceswhilestaying withintheboundsofcostandfeasibility.Generally,thepr imarysensitivitylimitingnoise sourcesintheLISAdesignaretheDRS'saccelerationnoisef or f < 3 mHz andthe IMS'ssensitivitynoisefor f > 3 mHz TheIMS'sinterferometrysensitivity, ~ x IMS ( f ) addstotheDRS'sacceleration noise, ~ a DRS ( f ) ,withtheascalingfactorof 2 =! 2 ,inroot-sum-squaretoproducethe 1 TheSCorbittheL5Lagrangepointtomaintainthestabilitya ndreducethe divergenceoftheorbits. 40

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Figure3-1.DiagramoftheLISAorbitaldynamics:Depictedi sadiagramoftheLISA constellation'sorbitaldynamicsshowingthetriangularc ongurationofthe SC,thegeodesicpathofanindividualSC,andtherelativean glebetween Earth'sorbitalplaneandtheLISAconstellation. effectivedifferentiallengthprecision: ~ x LISA ( f )= s ( ~ x IMS ( f )) 2 + 2 ~ a DRS ( f ) (2 f ) 2 2 (3–1) Whenwetakeintoconsiderationthesensitivitywithrespec ttotheGWsources, thislengthprecisionisscaledbyafactorof p 5 toaccountfora1-yearaverageoverthe 4 2 skyradiansandabyfactorof 2 = p 3 toaccountforthenon-orthogonal 60 o angle betweentheinterferometerarms.Inaddition,GW'swhichar esmallerinwave-length thantheLISA-armhaveareducedsensitivityduetoaGWalias ingtypeeffectresulting inmultipleGWoscillationsbetweenthetwoproof-masses.T hecombinedeffectsof thesescalingfactorsresultsinaLISAGW-to-lengthsensit ivityfunctiongivenby: T ( f ) p 5 2 p 3 s 1+ f 0.41 f o 2 (3–2) where f o = c = (2 L ) [ 29 ]. Usingthissensitivityfunction,theexpectedlengthpreci sionasaresultoftheIMS andDRSnoises,andtheLISAarm-length, L =5 Gm ,wecancalculatetheeffective strainsensitivity: ~ h LISA ( f )= T ( f ) ~ x LISA ( f ) L (3–3) 41

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Thissensitivityfunctionisplottedexplicitlyin cycles = p Hz inFigure 8-1 3.2TheDisturbanceReductionSystem InordertoaccuratelydetectGWs,wemustensurethatnoothe rnon-gravitational forcessuchaselectromagneticinteractionsorsolarradia tiondominatethedynamics oftheproof-masses'motion.Theproof-masses, 46 mm 3 cubes,willbecomposedof agold-platinumalloyandwillbeshieldedfromthesenon-gr avitationalforcesbythe DRS.TheSCandDRSthemselvesmustlargelyfollowthefree-f allingmotionofthese proof-masses,thus,micro-newtonthrustersareusedtomov etheSCandtrackthe proof-masses'geodesicpath.Capacitativesensorsensure thattheproof-massesdo nothitthewallsofthehousingand,attimes,intentionally actuatetheproof-massesto accountfortheindependentmotionofthetwoproof-masseso neachSC.Thegoalofall thesecomponentsworkingincollaborationistokeepthenon -gravitationalacceleration oftheproofmassesbelow: ~ a DRS ( f )= 3 fm = s 2 p Hz s 1+ f 8mHz 4 s 1+ 0.1mHz f (3–4) intheLISAmeasurementband.Thisdenesthelow-frequency sensitivitylimitof theLISAdetector.Testingtheabilitytoachievethisaccel erationnoisegoal,the LISAPathndermissionisbeinglaunched[ 49 ].CollapsingaLISA-armintoasingle spacecraft,theLISAPathndermissionwillattempttomeas urethedistancebetween twofree-fallingtestmassesusingheterodynedlaserelds ,providinganexcellent platformtotestmanyLISA-likecomplicationsandcharacte ristics. 3.3TheInterferometryMeasurementSystem TheLISAIMS'sprimaryobjectiveistomeasurethedifferent ialdistancebetween thefree-fallingproofmassestoanaccuracyof: ~ x IMS ( f )= 18 pm p Hz s 1+ 2.8 mHz f 4 (3–5) 42

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Table3-1.LISAcharacteristicsandrequirements:Therequ irementsoutlinedinthis table[ 29 ]denetheresultingLISAGW-sensitivitycurve(Figure 8-1 )basedon thepre-stabilizedlaserphasenoise,theinter-SCranging accuracy,the phasemetermeasurementprecision,theIMS'sTDInoiseextr actioncapability, andDRSisolatedtheaccelerationnoiseoftheproof-masses .Althougha futurespace-basedGWmissionmayusedifferentcharacteri stics[ 36 ],the requirementsspeciedinthisdescriptionfocusonaLISA-l ikescenario. ( f L =0.1 mHz f M =2.8 mHz f H =8 mHz ) CharacteristicSpecication LaserPre-stabilization 280 Hz p Hz q 1+ f M f 4 PhasemeterPrecision 1 cycle p Hz q 1+ f M f 4 IMSStrainSensitivity 18 pm p Hz q 1+ f M f 4 DRSAccelerationNoise 3 fm = s 2 p Hz r 1+ f f H 4 q 1+ f L f RangingAccuracy L =1 meter =3.3 ns Arm-Length L =5.0 0.1 Gm Light-TravelDelay =16.66 0.33 s RelativeVelocity v = 20 m = s = 66 ns = s EachcomponentintheIMSchainmustbetestedtoensurethatt hisrequirementis satisfactorilymet[ 29 ].Thisincludesalaserpre-stabilizationrequirementtos uppression theinherentfree-runninglaserphasenoise,aPD/phasemet erdifferentialphase measurementprecisionrequirementtoperformheterodynet ime-delayinterferometery, andarangingrequirementtoaccuratelyshiftandcancelthe residuallaserphasein theTDIcombinationsasshowninTable. 3-1 .Thesevaluesaredenedsuchthatthe residuallaserphasenoiseiscancelledbeyondtheshot-noi se( 3–8 )andacceleration noise( 3–4 )limits. First,the =1064 nm =282 THz ,lasersmustbepre-stabilizedtoanaccuracy of: ~ Pre Stab ( f )= 280 Hz p Hz s 1+ 2.8 mHz f 4 (3–6) bylockingtoafrequencyreference.Awell-knownmethodoff requency-referencinga laserisbyPound-Dever-Hall(PDH)lockingtoaULEcavity[ 50 – 52 ].Anothermethod, 43

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whichisalsobeingusedontheLISA-Pathndermission,isMa ch-Zehnderlocking. ThishasanadvantageoverPDHlockinginthatitusesthetwol asereldswhichare alreadypartoftheLISAdesign.Eitherofthesefrequencyre ferencemethodscouldbe assistedor,possibly,completelyreplacedbyusingtheLIS Aarmitselfasthefrequency referenceinatrackingtechniqueknownasarm-locking[ 53 54 ].Someoftheworkin thisdissertation(Chapter. 6.3.1 )willfocusontherstlong-armhardware-in-the-loop testsofthisarm-lockingtechnique. Thepre-stabilizedlasereldsareber-coupledthroughel ectro-opticalmodulators (EOM)toaddtheclock-noisetransfers(Chapter 4.3 )andSC-to-SClasercommunication signals.Eachofthesemodulatedeldsarethenber-couple dontoanULEoptical benchwhichdistributesthemtotheback-linkber,telesco pe,andthelocaloptical bench'sproofmass.Theback-linkbertransmitsthelaser eldtotheadjacentoptical benchonthesameSCwhilethetelescopetransmitstheeldto thetheadjacentSC asshowninFigure 3-2 .Thethreelasereldsoneachopticalbench(thelocallaser theadjacentoptical-bench'slaser,andtheadjacentSC'sl aser)areheterodynedto formthethreemainbeatnoteobservablesasdepictedwithmo redetailinFigure 3-3 Theseobservablescanbeinterpretedtorepresentthediffe rentiallaserphasebetween lasersonthesameSC, sr ,thelocal-SCtolocal-proof-massdistance, b sr 2 and thelocal-SCtothefar-SCdistance, s sr .Thesubscriptsallowustodifferentiate betweenthemeasurementsondifferentspacecraftwherethe 's'subscriptrefersto thesending SC s while'r'referstothereceiving SC r .Thetime-changingarm-lengths andresultinginter-SClight-traveltime-delays, q = q (0)+ q t ,areindexedsuch that SC q isopposite'Armq 'with q referringtoclock-wiselightpropagationand q 0 2 OtherTDIdescriptions[ 55 ]use sr or z sr forthesevariablesbutwewilluse b sr to avoidconfusionwiththelight-traveltimedelay, q ( t ) 44

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Figure3-2.DiagramoftheLISAconstellation:Amodelofthe completeLISA constellationdepictingthethreespace-craft,thesixind ividuallaser benchtops,theinter-SClaserlinks,andthenamesofeachof thearm-length lighttraveltimes, q = q 0 ,opposite SC q 45

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referringtocounter-clock-wisepropagation 3 asshowninFigure 3-3 .Fromeachof thesemeasurementsweareabletore-constructthedifferen tialdistancebetweenthe proof-massesonoppositeSCinpost-processing(Chapter 3.4.3 )and,thus,determine theGWstrain. Thecomplicationstothismeasurementschemearemany-fold .Noneofthe componentsinthelasertransferchaincanexceedaspecied requirementsothat addednoisetermsarenotintroducedintotheinterferometr ymeasurement.This includestheEOMsusedtoapplytheclocktransfersandSC-to -SClasercommunications, thebersusedtotransferthelasereldbetweenopticalben ches,andthetelescopes usedtotransferthelasereldsbetweentheSC.Theber-bac klinkmayintroduce 'non-reciprocal'(differentincounter-propagatingdire ctions)noise.Thetelescopelength maychangeovertime,causinganapparentarm-lengthchange .Inaddition,theangle betweentheopticalbenchesandtelescopepointingdirecti onmustbeactuatedto accountforthebreathingoftheconstellation. AnothercomplicationtotheIMSmeasurementisthereceived powerfromadjacent SC.Ofthe 2 W oflaserpoweremittedfromthelocalSC,thedivergenceofth elaser eldoverthe 5 Gm arm-lengthresultsinonly 100 pW beingreceivedontheadjacent SC'sphotodetector.Thisresultsinashot-noiselimitedhe terodynedeldwhichmustbe measuredtoanaccuracyof 1 cycle Generally,thebasisanddetailsoftheIMSsystemdescribed herefortheLISA missionareanalyzedandoutlinedin[ 29 ]and[ 34 ].Althoughnewproposalsforamore costeffectivedesignarebeingconsidered[ 30 56 57 ],theymuststillusethesameset ofprinciplestoperformgravitationalwavemeasurements. Thesechangesmayinclude 3 Notethatthefollowingdescriptionisdonewithoutalossof generalityby differentiatingbetweenindependent q ( t ) and q 0 ( t ) ,andaccountingfordifferent counter-propagatingtime-delaysalongthesamearmduetot heconstellation'sorbital rotation. 46

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shorterarm-lengths,thereductiontoatwo-armmeasuremen twithonlyfourinter-SC laserlinks,ageocentricorbit,ortheuseofasinglespheri calproof-mass. 3.4 CycleAccuracyPhaseMeasurement TheIMSdifferentialpath-lengthmeasurementprecision, ~ x IMS ( f ) ( 3–5 ),is primarilybasedonthecumulativeerrorfromthephoto-dete ctorsandphasemeters usedtomeasuretheheterodyned MHz beatnotes.Independentofshotnoises,the photodetectorerrormustbelessthan 3 pm whilethephasemetererrormustbeless than: ~ x PM ( f )= 1 pm p Hz s 1+ 2.8 mHz f 4 1 cycle p Hz s 1+ 2.8 mHz f 4 (3–7) Applyingtheexpectedshot-noiselimitation: ~ x Shot Noise ( f )= 7.7 pm p Hz (3–8) andthe 7 pm = p Hz path-lengthnoiserequirement,theoverallrootsumsquare d differentialphaseofeachpairoflasereldsismeasuredto anaccuracyof: ~ x Total ( f )= 11.7 pm p Hz s 1+ 2.8 mHz f 4 (3–9) Finally,theindividualmeasurementsareaddedinthelinea rtime-shiftedTDIcombinations toachievethe ~ x IMS ( f ) IMSrequirement( 3–5 ). Theselow-frequencyLISAbandrequirementsareusuallyhin deredbylong-term errorssuchassamplingbiases,temperaturedependentphas edispersion,and interferometriclengthchangeswhichcoupleintothephase measurements.For example,ifthevoltagebiaswhichisusedinsamplingandcon vertingthebeatsignal driftsoverthecourseofthemeasurement,thiswillresulti nanun-accountedphase coupling.Also,ifthetemperatureoftheltersorRFtransf ormersusedtoprepareand distributethebeatsignalchangesintime,thismayresulti natime-changingtransfer 47

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functionphaseresponse.Thedetailsofthesenoisesources areoutlinedinthefollowing sections.3.4.1PhotodetectorNoise Thephasenoiseofaphoto-detectorwithrespecttotheMHzhe terodynedbeatnote canbedeterminedbyalinearcombinationofthreeindepende ntnoisesources:(1) shotnoise,(2)Johnson-Nyquistnoise,and(3)ickernoise .Theshotnoiselimitation isdenedbythelimitednumberofphotonspersecondwhichar ereceivedonthe photodetector.Johnson-Nyquistnoiseresultsfromtemper aturedependenceof resistivedevicesinthephoto-detectorelectronicswhich maybeinuencedbyboth internalheatinganductuationsinthelaserpower.Flicke rnoisescaleswithapink, 1 = f ,ormoregenerally 1 = f n ,powerspectrumwhichresultfromacombinationof long-termprocesses.Thisincludesanynon-shot-noisebas edrelaxationprocessesor uctuationsinthesemi-conductorcharacteristicswhichr angefromwhitenoise( 1 = f 0 )to Brownian-quantumnoise( 1 = f 2 )[ 58 ]. 3.4.1.1Shot-Noise Thetheoreticalbasisforshotnoiseresultsfromtheinabil itytodistinguishbetween individualphotonswhenincidentonaphoto-currentproduc ingsemi-conductor.This well-understoodmeasurementlimitationpresentsitselfw ithaPoissoniandistribution andanuncertaintythatscaleswiththesquarerootofthenum berofphotonsin t measurementtime, N = p N t ,where N t = P Laser E photon t (3–10) foratotallaserpoweronthephoto-detector, P Laser ,withanaveragephotonenergy E photon = h .ExploitingHeisenberg'sUncertaintyPrinciple,wecanwr itetheRMSphase limitationas: RMS SN = 1 N = 1 p N t = r h P Laser t (3–11) 48

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Thephaseerror, ~ Shot Noise ( f ) ,presentsasawhitenoisespecturmdenedby RMS SN .EvaluatingthiswithLISA-likevalues, P =200 pW = c = =282 THz and t =1 s weobtainaphasenoiseof 5 cycles = p Hz .Thisisfarfromacomplete derivationwhichshouldalsoconsiderthephoton-electron conversionefciencies, ,but itgivesusanideaoftheexpectedphasenoise. Forcomparisontothedark-current,wecanextendthisresul tandwritethe shot-noisecurrenterroras: i RMS SN = p i SN = s Z f BW f BW ( e 2 ( f ) N ) df = p 2 e 2 f BW N (3–12) wherewehaveintegratedthelimitedefciencyelectro-cur rentovertheband-widthof thephoto-detector, f BW .[ 59 ] 3.4.1.2DarkCurrentNoise Thesemi-conductorsusedtoconvertthephotonsincidenton aphoto-diodetoan electro-currenthaveacharacteristicdark-currentwhich resultsfromrandomlycreated anddestroyedcurrentproducingelectronsinthesemi-cond uctor.Thisrandomprocess alsoresultsinashot-noiselikePoissonianerrorwhichcan causeRMSwhitenoise currenterrorsfrom1to500nA.[ 59 60 ]Theseprocessesresultinacurrenterrorgiven by: i RMS Dark = p i Dark = s Z f BW f BW ei dark ( f ) df = p 2 ei dark f BW (3–13) whichcanbelinearlycombinedwiththephoto-detector'ssh otnoisetondthetotal quantumnoise: i RMS Total = q i 2 SN + i 2 Dark = p 2 ef BW ( i dark +[ e N ] ) (3–14) 3.4.1.3Johnson-NyquistNoise Thermalheatingofthephoto-diode,whetherfrominternalo rexternalsources, causestheshuntresistance'sJohnson-Nyquistnoisetoadd errorstotheoutputcurrent. 49

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Thisnoisecouplesasafunctionofthetemperaturewithamag nitudegivenby: i RMS JN ( T )= q R f BW 0 4 Rk B T R = r 4 k B Tf BW R (3–15) whereRisthephoto-diode'sshuntresistance,Tisthetempe ratureinKelvin,and k B is Boltzmann'sconstant.[ 61 ] Explicitlystated,theseshotnoise,darkcurrentnoise,an dJohnson-Nyquistnoises areallwhitenoisesourcessuchthatthenoisecontribution increaseslinearlywiththe integrationtime.LISArefersallmeasurementstoa1second integrationtimeandplots thesewhitephasenoisespectraincycles/ p Hz 3.4.1.4FlickerNoise Flickernoisemayresultfromanumberofsourceswhichareal lcharacterizedby havinga 1 = f n powernoisespectrumwhere 0 < n < 2 .Thesearelargelyinuencedby long-termnon-quantumuctuations.Longtermtemperature variationsmaymodifythe photo-amplier'stransferfunctioncausingatime-changi ngphaseresponse,andthus,a longtermtemperaturecorrelatedphaseerror.Fluctuation sintheindividuallaserpowers willalsocoupleintothephase(Chapter 3.4.1.5 ). Long-termuctuationsinthelasereldintensitymightbea ttributedtolong termcharacteristicchangesinthecoherenteldproducing Nd:YAGlasercrystals includingatemperaturedependentcavitynesseandtheava ilabilityofexcitedstates toproducestimulatedemission.High-frequencycomponent softhelaserintensitywill alsocoupleintothemeasurementandaregenerallybasedont helaser'sresonant relaxationoscillation[ 62 63 ].Thisamplitude-phasecouplingcanalsobeintroduced byothersourcesincludingaphoto-detectorpolarizationd ependency,electronic noiseinthephoto-detectorsandADCsresultingfromLISA's digitizeddemodulation measurementscheme.Theill-denednatureoftheselong-te rmprocessesjusties furtherinvestigationwithlab-baseddifferentialPDmeas urements. 50

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3.4.1.5HeterodynePhaseMeasurements Considertwolasereldsgivenby: 1 ( t )= E 1 ( t ) e i 1 ( t ) t 2 ( t )= E 2 ( t ) e i 2 ( t ) t (3–16) Superimposingandcombiningtheselasereldsonaphoto-de tector,weobtaina currentgivenby: i beat ( t ) / ( 1 ( t )+ 2 ( t ))( 1 ( t )+ 2 ( t )) (3–17) /j E 1 ( t ) j 2 + j E 2 ( t ) j 2 + E 1 ( t ) E 2 ( t )cos([ 1 ( t ) 2 ( t )] t ) Inheterodyneinterferometry,theDCportionsof j E 1 ( t ) j 2 and j E 2 ( t ) j 2 aregenerally ACcoupledwithRFtransformers,althoughthesemaystillin troducehigh-frequency components.Combiningtheindependenteldterms, E Tot ( t )= E 1 ( t ) E 2 ( t )= E o (1+ E ( t )) ,andintroducingthemean MHz beatfrequency, = h 1 ( t ) (2)( t ) i ,we canpickoutthe, E 1 ( t ) E 2 ( t )cos([ 1 2 ] t ) termsbydemodulatingthisPDoutput with sin( t ) .IntegratingthedemodulatedDCoutput,wecangetameasure ofthe differentiallaserphase: ( t )= Z t 0 [ 1 ( ) 2 ( ) ] d (3–18) Variationsinthedifferentialeldintensityuctuation, E ( t ) ,atfrequenciesof could becoupledintothemeasurementsuchthatcompletedescript ionshouldbewrittenas: ( t )= Z t 0 [ 1 ( ) 2 ( ) ]+ E o E ( ) e i + e i 2 d (3–19) 3.4.2 CyclePhasemeter Thephoto-currentsproducedbythephoto-diodeswillbesam pledwith n -bit analog-to-digitalconverters(ADCs).Theclockedsamplin gprocesswhichdigitizesthe PDsignalstoperformthephasemetermeasurementsintroduc esit'sownindependent noisesources.The 2 20 MHz PD-beatnoteissampledwitha 50 MHz clockreference 51

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comparing ( t ) totheclockphase, CLK ( t ) ,whichisnotaperfecttimingreference, thus,introducingit'sownphasenoise.Thedigitizationpr ocessprovidesalimited resolutionwithwhichtomeasurethebeatnotephase,meanwh ile,voltage-biasnoise fromtheADCvoltagereferencecanalsocoupleintothephase measurement.Timing jitter,theerrorcausedbythetime-dependentdelaybetwee nvoltageconversiontime andtheclock'srising-edge,couplesintothemeasurementa sa 1 = p f phasenoiseand scaleswiththeinputbeatnotefrequency.Thesearealldisc ussedinmoredetailbelow. 3.4.2.1ClockNoise Digitizingthe 2 20 MHz PDsignalswitha 50 MHz clock,wesamplethePDcurrent drivenvoltagebiasacrossaloadresistoratthe'rising-ed ge'oftheclockoscillation.The PDvoltageiscomparedtoavoltagebiasedresistorbankwith intheADCsresultingin a n -bitread-outoftheloadresistancevoltage.Givenaninput oscillationwithanideal amplitudewrittenas: x in ( t )= A in sin(2 f in t + in ( t )) (3–20) wesampletheinputsignalwithaclocksourcegivenby: x CLK ( t )= A Clk sin(2 f Clk t + Clk ( t )). (3–21) where in ( t ) istheinputphasenoisewithrespecttotheinputfrequency, f in ,and Clk ( t ) istheclockphasenoisewithrespectto f Clk [ 64 ]. Usinganidealclockthephaseerrorinthedigitizationconv ersionwouldbelimited bythebit-resolution,discussedinthefollowingsection. Inpractice,thephasenoisein theclocksourcewithrespecttothetheoreticalclockfrequ encyisinterpretedasinput phasenoisescaledbytheratiooftheirfrequenciessuchtha t: Measured ( t )= Z t 0 in ( ) CLK ( ) d = in ( t ) f in f Clk Clk ( t ). (3–22) 52

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InLISA,theseclocknoisetermsmustbeaccountedforbyperf orminginter-SC clocknoisetransfersandmeasuringthedifferentialclock noisetermsagainstthelocal SC'sclock.Afterwards,thedifferentialphaseanddiffere ntialclockmeasurementsare interpolatedtocomparethemagainstoneglobalmastercloc k. 3.4.2.2ADCQuantizationNoise Whensamplingavoltagesignal, V in ( t ) ,withaclock, f Clk ,theADCcomparesthe inputvoltageagainsttheADCinternalvoltagebiasreferen ce.Thiscomparisonprocess producesaseriesofsamplesgivenby: x in [ n ]= V in ( n = f Clk ) V Bias (3–23) wherewehavetakenthebiasvoltagetobeaconstant,stabler eference.This-1to1 ratioisconvertedtoaseriesof n -bitvalueswitha 2 1 n resolution[ 65 ].Thus,giventhis bit-accuracyweobtainastandard-deviationinthemeasure damplitudegivenby: ~ ADCAmp Quant ( )= j V Bias j j V in j 2 1 n p 6 f s (3–24) Theextrafactorof 1 = p 6 comesfromthewhite-noisepropertiesofthequantization probabilitydensityfunction[ 65 66 ].Thefactorof V Bias = V in accountsforthefactthat thesignalamplitudemaynotspanthefull n -bitsoftheADC'sconversion-amplitude. Additionalquantizationerrorscanandwillbeappliedtoth ishard-warebasedquantization limitasthesedigitizedsignalsaremeasuredwithbit-limi tedxedpointprocessorsand rate-limitedread-outsintroducingtheirownquantizatio nerrors. 3.4.2.3ADCAmplitudeNoise Extendingthisdescription,weconsideructuationsinthe voltagebiasagainst whichthephoto-currentismeasured.Moregenerallyonecan writethebiasvoltageas: V Bias ( t )= V Bias o G ( t )+ V Gnd ( t ) (3–25) 53

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where G ( t ) isthegainfactoronthevoltagebiaswith h G ( t ) i' 1 and V Gnd isthevoltage ofthegroundreferenceincomparisontotheinput's'ideal' ground.Ignoringtheground termforthemoment,thisuctuationinthebiasvoltagescal estheamplitudeerroras: ~ x ADCAmp Quant ( )= ~ G ( ) j V Bias o j j V in j 2 1 n p 6 f s (3–26) whichshowshowthesegainfactorscancoupledirectlyintot hemeasuredamplitude. SinceLISAmeasurementsarefocusedonphasenoise,thecoup lingoftheseamplitude quantizationandbiasnoiseerrorsourcesintothephasemea surementsareanalyzedin Chapter 3.4.2.5 ItisoftendifculttodistinguishbetweentheADCamplitud enoise,thelaser intensitynoise,andthePDelectroniccurrentamplitudeno isesincetheyallcoupleinto thephasemeasurementwiththesamecharacteristics.Onthe otherhand,wecanget somemeasureoftheADCamplitudebiasnoisebyverifyingthe inversedependenceon theinputvoltage.Often-times,theselowfrequencyamplit udevariationsaredominated bytemperaturevariations;thus,thermalcorrelationcoef cientscanbemeasuredby observingthephasevariationwithachangeintemperature. Giventhesecorrelation coefcientmeasurementswecandeneanelectronictempera tureenvironmentstability requirement.3.4.2.4Clock-ADCTimingJitter ADCtimingjitterisdenedasthetime-changingdelaybetwe enthewell-dened risingedgeoftheclockandtheactualsampletriggeringoft heADCinputsignal.This timingjittervaluecanhaveafrequencydependence, ~ t ( ) ,whichscalesproportionately withthebeat-frequency, f in = h d = dt i ,tophasenoise[ 67 ]: ~ ( )= f in ~ t Jit ( ). (3–27) Thisfrequencycouplingcanandwillbeexploitedtoestimat etheADCtimingjitter. 54

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3.4.2.5DemodulationNoiseCoupling Nowthatwehaveseenhoweachoftheseindividualnoisesourc esscalewith theirinputcharacteristics,weevaluatehowthesemeasure mentswillcoupleinto thedemodulatedphasemeterreadout.First,asanexampleof howthisreadoutis performed,weconsideraninputsignalwithsomephaseandam plitudevariationwhere A ( t ), ( t ) << 1 ,while in and' A 'areconstant,givenby: x ( t )= A [1+ A ( t )]sin( in t + ( t )). (3–28) Ifwedemodulatethiswiththein-phaseandquadraturecompo nentsoftheinputatthe samefrequency,suchthat o = in ,weobtainfollowingterms: y In phase ( t )= x ( t ) sin( o t ) (3–29) = A [1+ A ( t )] 2 [cos( ( t )) cos(( in + o ) t + ( t ))], y Quad ( t )= x ( t ) cos( o t ) (3–30) = A [1+ A ( t )] 2 [sin( ( t ))+sin(( in + o ) t + ( t ))]. Afterlow-passlteringtoremovethesum, ( in + o ) term,wecanwritetheresultasa functionoftheamplitudeandphaseerroras: y In phase ( t )= A [1+ A ( t )] 2 [[cos( ( t ))], (3–31) y Quad ( t )= A [1+ A ( t )] 2 [[sin( ( t ))] (3–32) Basedonthedenitionsofthein-phaseandquadraturecompo nents,onecanexactly reconstructtheamplitude A Read Out ( t )=2 p y In phase ( t ) 2 + y Quad ( t ) 2 = A [1+ A ( t )] (3–33) 55

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andphase, Read Out ( t )= tan 1 y Quad ( t ) y In phase ( t ) = ( t ). (3–34) Ontheotherhand,usingthesmallvalueassumption, ( t ) << 1 ,wecanmake theapproximations, cos( ( t ))=1 and sin( ( t ))= ( t ) ,suchthattheamplitude andphaseread-outssmplify.Usingthein-phasecomponento nlywecouldwritethe amplitudeas: A Read Out ( t ) 2 y In phase ( t )= A [1+ A ( t )]cos( ( t )). (3–35) Usingthesameargumentwecanwritetwoapproximationsfort hephase,givenby: ( t ) y Quad ( t ) y In phase ( t ) =tan( ( t )), (3–36) or ( t ) 2 A y Quad ( t )=(1+ A ( t ))sin( ( t )). (3–37) Theseapproximationsareusefulinreducingtheloadonthed igitalprocessingdevices but,aswecanseefromthesecalculations,theyalsotendtoc oupletheamplitudeand phasenoisesourcestogetherwhichmustbeconsideredforLI SA-likehigh-precision phasemeasurements. Thedetailsofthephasemeterphasetrackingandread-outbe comemore complicatedwhenweallowfor A ( t ) and ( t ) tochangearbitrarily,butthismodel willgiveusanideaofhowun-desirednoisetermswillcouple intothedemodulated phasemeasurement.Atthispointwecanseethat,dependingo nthetypeofreadout approximationused,theseerrortermswillcoupleindiffer ently.Ourfocuswillbeon ( 3–37 )since,aswewillseeinChapter 5.2.1 ,thisisthephasemeterread-outscheme weusetomeasuretheinputphase.Alreadywecanseehowtheam plitudenoise couplesintothephase( G ( t ) in( 3–25 )forexample).Unfortunately,thisfailstogivea 56

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completeunderstandingofthenoisecouplingsincethisrs torderapproximationfailsto considerhowhigh-frequencytermscoupleintothemeasurem entthroughtheclocking andfeed-backtrackingdemodulationprocesses. Theclock-samplingdemodulationwillaliashigh-frequenc ytermsintothedigital ADCoutputsuchthattheinputphasetermsat nf Clk f in willsuperimposewithagain of1directlyintothe f in frequencybin.Followingstandardelectronicsprocedures ,RF transformersareusedtoACcoupletheinputandlow-passant i-aliasingltersareused tosuppressanyinformationatfrequenciesgreaterthan f in EventhoughtheinputsignalisACcoupled,wewillincludean offseterrorinthe extendedmeasurement-demodulationtreatmentsuchthatwe nowwritetheinputas: x ( t )= A O ( t )+ A [1+ A ( t )]sin( in t + ( t )). (3–38) Again,demodulatingwiththequadraturecomponentweobtai n: y Quad ( t )= A [1+ A ( t )] 2 [sin( ( t ))+sin(2 o t + ( t ))]+ A O ( t )cos( o t ). (3–39) wherewehaveset o = in = o Takingadeviationfromtheprevioustreatmentweexpandout theseresultsto explicitlyshowtheDC, o ,and 2 o terms: y Quad ( t )= A [1+ A ( t )] 2 [ sin( ( t ))+sin(2 o t )cos( ( t ))+cos(2 o t )sin( ( t ))) ] + A O ( t )[cos( o t )]. (3–40) 57

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Finallyapplyingthe A A o << 1 approximation,low-passlteringtheterms whichhavenonoisecoupling,applying( 3–37 ),andexpandingexplicitly,weobtain: Est ( t )= 2 A y Quad ( t ) (3–41) = ( t ) 1+ A ( t ) A + ( t ) 1+ A ( t ) A cos(2 o t ) (3–42) + ( t )cos(2 o t )+ A ( t ) A sin(2 o t )+ A O ( t ) A cos( o t ). Fromthiswecanreadoffthenoise-couplingofeachterm.The termweare interestedin, ( t ) ,isscaledby 1+ A ( t ) = A .Generally,since A ( t ) = A << 1 ,thiseffect issmall,butmaybealimitingnoisesourcewhenattemptingt oformdifferentialphase cancellationcombinations.Although,sincethistermscal esinverselywiththeamplitude, itcanbereduced,dependingonthesource,byincreasingthe signalpower.Ifthe amplitudenoisecomesfromuctuationsinthelaserpower,f orexample,increasingthe signalpowertendstoalsoincreasethenoisesuchthat A ( t ) = A remainsconstant.Next wenoticeboththe ~ (2 o ) and ~ A (2 o ) termsoftheinputwave-formcoupleintothe phasemeasurement.Finally,wenoticethattheoffseterror ~ A O ( o ) ,termcouples directlyintothemeasurement.Thistermwouldbeofanerror like V Gnd ( t ) in( 3–25 ). Wecanmitigatethe 2 o termsbyanti-aliaslow-passlteringtheinputsignalat frequenciesgreaterthan o .Unfortunately,ifthesetermsaregeneratedbytheADC aftertheanti-aliasingltertheywill,again,coupleinto thephasedata.Takingthisinto account,extensivelow-frequencytestingoftheADCsmustb eperformedtoensure theseerrorsdonotlimitthephasemeasurementsensitivity .Wewillseehowthe UF-phasemeter'sADCsaretestedandcharacterizedinChapt er 5.3 3.4.3HeterodyneTime-DelayInterferometry Untilnow,wehavenotspecicallystatedhowtheone-wayint erferometry observables, sr b sr ,and s sr ,aremeasuredandmitigated.Herewederivetheterms on SC 1 buttheprocedureusedcanbeemployedtoderivetheobservab lesontheother 58

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Figure3-3.Diagramoftheinterferometrymeasurementsyst em:Thismodelshowsa moredetaileddepictionoftheindividualsatallitesshown inFigure 3-2 .Here weseethebeampathandheterodynedlasereldsresultingfr omthetwo locallasersandthetwolasereldsbeingtransmittedfromt hefarSC.The LISAobservables, sr b sr ,and s sr andhowtheyaregeneratedwiththe heterodynedlasereldsisshownexplicitly. 59

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SC.Forthe sr and b sr observables,weconsiderthetwolocal SC 1 lasers,Laser 21 andLaser 31 ,whichtransmittheireldsto SC 2 and SC 3 ,respectively,havingalaser frequencies, 21 and 31 = 21 + ,andlaserphases, 21 ( t ) and 31 ( t ) ,suchthatwe cancouldwritethelasereldsas: 21 ( t )= E 21 ( t ) e i ( 21 t + 21 ( t )) 31 ( t )= E 31 ( t ) e i (( 21 + ) t + 31 ( t )) (3–43) Theselasernoisesourcesareusedtotrackthemajorphaseno isecouplingterms throughtheinterferometeralthough,fromhereon,wewilli gnoretheamplitudeterms andtake E 21 ( t )= E 31 ( t )= E 3.4.3.1FiberNoise Theindependentlasereldsarepassedthroughaberbank-l inkaddingboth common-modeandnon-reciprocalphasenoisesuchthatthee ldsafterpassing throughtheback-linkberscanbewrittenas: 21, b ( t )= Ee i ( 21 t + 21 ( t )+ b C ( t )+ b ,21 ( t )) (3–44) 31, b ( t )= Ee i ( 31 t + 31 ( t )+ b C ( t )+ b ,31 ( t )) (3–45) where i 1, b aretheindependent,counter-propagatingbernoiseterms and b C are thecommonbernoiseterms. Superimposingtheselasereldsfromeithersideoftheber ontoaphoto-detector andACcouplingthePDoutput,weobtainphoto-currentsgive nby: PD 21 ( t )= p ( 21 ( t )+ 31, b ( t ))( 21 ( t )+ 31, b ( t )) / cos( t + 31 ( t ) 21 ( t )+ b C ( t )+ b ,31 ( t )) (3–46) PD 31 ( t )= p ( 31 ( t )+ 21, b ( t ))( 31 ( t )+ 21, b ( t )) / cos( t + 31 ( t ) 21 ( t ) b C ( t ) b ,21 ( t )) (3–47) 60

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Demodulatingthesesignalswith sin ( t ) andmeasuringthephasetermsweobtain twophaseread-outsgivenby: 21 ( t )= 31 ( t ) 21 ( t )+ b C ( t )+ b ,31 ( t )+ f f CLK CLK ( t ) (3–48) 31 ( t )= 31 ( t ) 21 ( t ) b C ( t ) b ,21 ( t )+ f f CLK CLK ( t ) (3–49) withtheexpecteddifferentiallaserphase, 31 ( t ) 21 ( t ) ,counter-propagatingber noiseterms,andclocknoiseterms.Thecommonbernoiseter mscoupleintoeachPD withtheoppositephasesuchthatthesumoftheseobservable sgivesameasureofthe differentialcounter-propagatingphasenoiseterms: Sum = 21 + 31 =2 31 ( t ) 21 ( t )+ f f CLK CLK ( t ) + b ,31 b ,21 (3–50) whilethedifferenceofthesetermsgivesameasureofthecom monmodeberterms: Dif = 21 31 =2 b C ( t )+ b ,31 + b ,21 (3–51) Theindividualcounter-propagatingphasenoisetermsmust besmallerthan 1 cycle sincethereisnowaytodifferentiatebetweenthisandthein dependentlasernoise terms.Inthenextsectionwewillseehowthesedifferential laserphaseandcommon modebertermscancelwhenevaluatingtheproof-masstospa cecraftdistance. 3.4.3.2Spacecraft/Proof-MassMotion Nowthatwehaveameasureofthebernoise,weusethesamelas ereldsto getameasureoftheindividualSCtoproof-massdistance.If wedenethedistance betweentheSCbeam-splitterandtheproof-massas d ( t ) wecancovertthistolaser phasenoiseafterreectingofftheproofmassbyre-writing theberterms, 21, b ( t ) and 21, b ( t ) ,as: 21, mass ( t )= Ee i ( 21 t + 21 ( t )+ b C ( t )+ b ,21 ( t )+ d 31 ) (3–52) 31, mass ( t )= Ee i ( 31 t + 31 ( t )+ b C ( t )+ b ,31 ( t )+ d 21 ) (3–53) 61

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Demodulatingthesewiththelocalbench'slasereld,weobt ainthe b sr signals: PD b 21 ( t )= p ( 21 ( t )+ 31, mass ( t ))( 21 ( t )+ 31, mass ( t )) / cos( t + 31 ( t ) 21 ( t )+ b C ( t )+ b ,31 ( t )+ d 2 ) (3–54) PD b 31 ( t )= p ( 31 ( t )+ 21, mass ( t ))( 31 ( t )+ 21, mass ( t )) / cos( t + 31 ( t ) 21 ( t ) b C ( t ) b ,21 ( t ) d 3 ). (3–55) Measuringthebeatnotephasewithphasemeters,wecanwrite the b sr observablesas: b 21 ( t )= 31 ( t ) 21 ( t )+ b C ( t )+ b ,31 ( t )+ d 2 + f f CLK CLK ( t ) (3–56) b 31 ( t )= 31 ( t ) 21 ( t ) b C ( t ) b ,21 ( t ) d 3 + f f CLK CLK ( t ) (3–57) Linearcombinationsofthesefoursignals, 21 ( t ) 31 ( t ) b 21 ( t ) ,and b 31 ( t ) provide uswithalocalmeasureoftheSCtoproofmassdistanceandthe differentiallaser phaseterms: d 2 = [ b 21 21 ] (3–58) d 3 = [ 31 b 31 ] (3–59) 31 21 = 21 + 31 2 + b ,21 b ,31 2 f f CLK CLK ( t ) (3–60) WenoticeintheseequationsthattheSC-to-proof-massdist ances, d 2 and d 3 ,are independentofanybernoiseterms.Meanwhile,thediffere ntiallaserphasetermsare limitedbytheindividualcounter-propagatingberterms b ,31 and b ,21 .The d terms areindependentofclocknoisesincetheyaremeasuredonthe samespacecraftatthe sameheterodyneoffsetfrequency, ,andthecommonclocknoisecancelsinthese differentialmeasurements.3.4.3.3Inter-SpacecraftMotion Nowthatwehaveamethodofevaluatingthelocal-SCtolocalproof-massdistance wemustmeasuretheinter-SCarm-lengthtermstobeabletoco nstructthecomplete 62

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differentialinter-proof-massinterferometer.Inthisde scriptionwewillfocusona singleone-wayinterSClinkbetween SC 1 and SC 2 whereLaser 12 ,havingalaser frequency 12 ( t ) ,istransmittedalongArm-3'acquiringatime-delay, 3 0 ( t ) ,andthen isheterodynedwithLaser 21 ,havingalaserfrequency 21 ( t ) .Inter-SClengthchanges atvelocitiesof 0 20 m = s forLISA 4 ,causerelativisticDopplershiftsandtime-domain phasescalingofthelaserelds. Tocalculatetheeldtransformation,werstdenethemeas uredphaseasthe integrationofthelaserbeatfrequencywithrespecttoaret ardedtime, t r = t x = c ,such thatbothpositionandtimechangesaffectthemeasuredphas e: ( t )= Z t 0 ( t r ) dt r (3–61) Taking SC 2 asthestationaryframeofreferencewedenetheinter-SCli ghttravel timefrom SC 2 to SC 1 as 3 0 ( t )= 3 0 (0)+ 3 0 t where 3 0 = v 3 0 = c = d ( t ) = dt .Apositive velocityreferstoanincreasinginter-SCdistancesuchtha t SC 1 ismovingawayfromthe SC 2 .Inthestationary SC 2 framewecanwritethelasereldgeneratedby Laser 21 as: 12; ; ( x t )= Ee i ( t x c ) 12 ( t x c ) (3–62) andthemeasuredphaseevaluatedatx=0: 12 ( t )= Z t 0 12 0 c d (3–63) Wecantransformthistothemoving SC 1 referenceframethrougharelativistic Lorentztransformationas: 12;3 0 ( x 0 t 0 )= Ee i r (1 ) t 0 x 0 c 12 ( r (1 )( t 0 x 0 c ))] (3–64) 4 Othermissionsmayhavedifferentialvelocitieswhichares ignicantlylarger[ 56 ] 63

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where,forthisarm, = 3 0 and r = r 3 0 =1 = p 1 2 and t r = r (1 )( t 0 x 0 = c ) Fromthisequation,weseetherelativisticDopplershifttr ansforming 12 r (1 ) 12 Integratingthebeatnotefrequencyinthismovingframe: ( t 0 )= r (1 ) Z t 0 0 12 r (1 ) L q (0) c d (3–65) suchthatwecanwritethemeasuredphaserelationship: 0 ( t 0 ) ; q = ( r q (1 q )( t 0 q (0))) (3–66) where q (0)= L q (0) = c .InmostLISA-likecases,weignorethesmallfactorsof 2 < 10 12 andso,willdropthe r factorsintheanalysisfromthispointforward.Noticethe' ;' notationusedtotransformthelasereldbetweenmovingfra mesin( 3–64 )and( 3–66 ). Ingeneral sr ; q ( t )= sr ((1 q )( t q (0))) wherewehavetaken t (1 q )( t q (0)) WewillusethissamenotationthroughtherestoftheTDIanal ysis. TheeldfromthefarSCisreectedthroughatransmittingte lescope,transmittedto thelocalSCacquiringamulti-secondtime-delay 5 ,andcapturedbythelocalreceiving telescope. Heterodyningtheselasereldsweobtainaphoto-currentgi venby: PDs 21 ( t )= p ( 21 ( t )+ 12;3 0 ( t ))( 21 ( t )+ 12;3 0 ( t )) (3–67) / cos (( 21 (1+ 3 0 ) 12 ) t ( 21 ( t ) 12 ((1 3 0 )( t 3 0 (0)))) (3–68) wherewenoticethatthesignofthephaseinformationdepend sontherelative magnitudeofthelocallaserfrequencyincomparisonwithth einter-SCDoppler shiftedlaser'sfrequency.Althoughthismaybeeasilycorr ectedinpost-processing, therelativelaserfrequenciesandinter-SCDopplershifts willchangethesignonboth 5 =5 Gm = c =16.7 sforLISA 64

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theobservableandclocknoisecorrectionstermsintheTDIc ombinationsandmustbe keptinmindwhenformingthepost-processedTDIcombinatio ns. GeneralizingtoincludetheclocknoiseandGWtermswecanwr itethe s 21 inter-SC termas: s 21 = 21 ( t ) 12 ((1 3 0 )( t 3 0 (0)))+ h 21 ( t )+ f 21 f Clk Clk :1 ( t ), (3–69) wherewehaveintroducedtheGWstrainmodulationof h ( t )= L ( t ) = L o ,andalocal SC i clocknoiseterm, Clk : i ( t ) ,scaledbythePDbeatnotefrequency, f 21 .Wehavealsomade assumptionsaboutthesignofthedifferentialmeasuredpha sebasedontherelative laserandDopplerfrequenciesforevaluatingtheTDIcombin ations. 3.4.3.4BasicTDI-CombinationsandConsiderations Atthispoint,wehaveformedandevaluatedalltheobservabl esrequiredto completelyre-constructthedifferentialproof-massinte rferometer.Thesesignals, sr b sr ,and s sr ,canbeevaluatedforeachofthe3SCand6inter-SClaserlink sresulting in18differentobservables.Whenanyoftheseobservablesa remeasuredwitha phasemeter,clocknoisetermsgivenby ( f in = f Clk ) Clk : i ( t ) whereisthephasenoiseof clocklocatedon SC i ,areaddedtothedigitalsignalsandmustbeaccountedforwi ththe inter-SCside-bandclocknoisetransfers[ 68 – 70 ].Theseobservablescanalsobeused tophase-locktheadjacentopticalbenchlasersortheadjac entinter-SClasersintoorder totransferthelaserstabilityandobtaincommon-modelase rnoisecancellationtowithin theaccuracyofthephaselockloops(PLLs)[ 71 ]. ThefocusofthisdissertationistoformtheTDIcombination sbasedonthe inter-SClaserlinks, s sr .Tosimplifytheanalysis,fromthispointonwewillmake someassumptionsaboutthemeasuredobservables.First,as sumingthelocal-SC tolocal-proof-massmotioncanbemeasuredthroughheterod yneinterferometryand thattheaccelerationnoiseoftheproof-masscanbeeffecti velyreducedwiththeDRS, itisreasonabletoimaginetheproof-massesasbeingmounte ddirectlytotheSC, 65

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thussetting d sr =0 .Next,assumingthenon-reciprocalber-backlinknoiseca nbe reducedbeyondthe 1 cycle requirement,wewillhaveareasonablemeasureofthe adjacentbench-topdifferentiallasernoiseandcanignore the sr signals,referencingall measurementstoasingle Laser i on SC i withalaserphase i ( t ) Inthisspeciccasewecanwriteacompletebasis 6 forall6inter-SCinterferometry measurementsincludingclocknoisetermsas: s 21 ( t )= 1 ( t ) 2 ((1 3 0 )( t 3 0 (0)))+ h 21 ( t )+ f 21 f Clk Clk :1 ( t ), (3–70) s 31 ( t )= 1 ( t ) 3 ((1 2 )( t 2 (0)))+ h 31 ( t )+ f 31 f Clk Clk :1 ( t ), s 12 ( t )= 2 ( 12 t ) 1 ((1 3 )( 12 t 3 (0)))+ h 12 ( 12 t )+ f 12 f Clk Clk :2 ( 12 t ), s 32 ( t )= 2 ( 12 t ) 3 ((1 1 0 )( 12 t 1 0 (0)))+ h 32 ( 12 t )+ f 32 f Clk Clk :2 ( 12 t ), s 13 ( t )= 3 ( 13 t ) 1 ((1 2 0 )( 13 t 2 0 (0)))+ h 13 ( 13 t )+ f 13 f Clk Clk :3 ( 13 t ), s 23 ( t )= 3 ( 13 t ) 2 ((1 1 )( 13 t 1 (0)))+ h 23 ( 13 t )+ f 23 f Clk Clk :3 ( 13 t ). wherewehavedifferentiatedbetweenthedifferentclockph asenoiseterms, Clk : i ,and absoluteclockfrequencyoffsets 7 : ij = f Clk : j f Clk : i (3–71) Theclockphasenoisetermsareremovedwiththeinter-SCclo cknoisetransferswhen formingtheTDIcombinationswhiletheclockfrequencyoffs etsareaccountedforby time-scaledinterpolation[ 72 73 ]ofthe'far'SCsignalsbytheinverseclockratio, ij 6 AssumingSC-1isourstationaryframeofreferenceandClock -1isourabsolute clockfrequencyreference. 7 TheclockfrequencyoffsetsfromclocksondifferentSCcaus eanerrorinthe referencetoanabsolutetimewhichisdenedbythefrequenc yoftheclockon SC 1 66

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beforeformingtheTDIcombinationswithrespecttotheloca lreferenceclock,inthe aboveexamplecase, f Clk :1 Twousefulcombinationsforfurtheranalysiswillbethesin gle-armround-trip sensorsignals sr andthecounter-propagatingSagnacsignals sr ,asdiagrammedin Figure 4-2 whichprovideuswithameasureoftheconstellationinterfe rometrydynamics referredtoasinglelasersource. Thelocalsensorsignalscanbederivedfromthe s sr datastreamsbytime-scaling thefarSCmeasurementsbytherespectiveinverseclockrati o ( ij 1 ) ,thentime-shifting andtime-scalingtheresultbythereturntrip'sdelaytrans formation, s sr ; q 21 ( t )= s 21 ( t )+ s 12;3 0 ( 12 1 t ) (3–72) = 1 2;3 0 + h 21 + f 21 f Clk Clk :1 +( 2 1;3 + h 12 + f 12 f Clk Clk :2 ) ;3 0 = 1 1 ((1 3 )(((1 3 0 )( t 3 0 (0))) 3 (0))) (3–73) + h 21 ( t )+ h 12 ((1 3 0 )( t 3 0 (0))) + f 21 f Clk Clk :1 ( t )+ f 12 f Clk Clk :2 ((1 3 0 )( t 3 0 (0))) and,forArm-3, 31 ( t )= s 31 ( t )+ s 13;2 ( 13 1 t ) (3–74) = 1 3;2 + h 31 + f 31 f Clk Clk :1 +( 3 1;2 0 + h 13 + f 13 f Clk Clk :3 ) ;2 = 1 1 ((1 2 0 )(((1 2 )( t 2 (0))) 2 0 (0))) (3–75) + h 31 ( t )+ h 13 ((1 2 )( t 2 (0))) + f 31 f Clk Clk :1 ( t )+ f 13 f Clk Clk :3 ((1 2 )( t 2 (0))) whichcancelthefarSC'slaserphasenoiseandreferenceall phasemodulationstoa singlelasersource, 1 ( t ) .Herewecanexplicitlyseetheclock-noisecouplingwhich 67

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entersintothesensorsignals'data-streamswiththeform: Clk s 1 ( t )= f s 1 f Clk :1 Clk :1 ( t )+ f 1 s f Clk Clk : s ((1 q )( t q (0))). (3–76) AswewillseeinCh. 4.1.2 andCh. 4.3 ,thesetermscanbeaccountedforbythe clock-noisetransfersin( 4–3 ). Anothersetofusefulcombinationsare(Ch. 4.3.2.1 )thecounter-propagating round-tripSagnacsignals: 21 ( t )= s 21 + s 32 ( 12 1 t ) ;3 0 + s 13 ( 13 1 t ) ;1 0 3 0 (3–77) = 1 1;2 0 1 0 3 0 +[ h 21 + h 32;3 0 + h 13;1 0 3 0 ] (3–78) and 31 ( t )= s 31 + s 23 ( 13 1 t ) ;2 + s 12 ( 12 1 t ) ;12 (3–79) = 1 1;312 +[ h 31 + h 23;2 + h 12;12 ]. (3–80) Finally,thelastcombinationofinterestisthefullysymme tricSagnaccombination: = s 31;1 + s 12;2 + s 23;3 ( s 21;1 0 + s 32;2 0 + s 13;3 0 ) (3–81) =[( 1 3;2 ) ;1 +( 2 1;3 ) ;2 +( 3 2;1 ) ;3 ] (3–82) [( 1 2;3 0 ) ;1 0 +( 2 3;1 0 ) ;2 0 +( 3 1;2 0 ) ;3 0 ] whichisindependentoflasernoiseforanon-rotating 8 constellation. Wehaveignoredtheclocknoisetermsforthesakeofsimplici tysincetheseSagnac combinationsarenotthefocusofthisdissertation. 8 ( q ( t )= q 0 ( t )) 68

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CHAPTER4 TIMEDELAYINTERFEROMETERY Nowthatwehavedevelopedamethodofcharacterizingthesig nalswithwhichto measuretheinter-proof-massnoisetermsthequestionente rs:Howdowecombine thesesignalstocancelthelaserphasetermsandextractthe GWmodulations? 4.1LaserNoiseCancellation TheabilitytocancelthelasernoisetheIMSislargelybased ontheaccuracywith whichwecanmeasurethetime-dependentdistance 1 betweenthespace-craftand interpolatethephasesignalstoaccountforthisdistance. Togetanideaofwhythisis thecase,let'sconsidertwolaserphasesignals, ( t ) and ((1 )( t ))+ g ( t ) .These signalsareindependentlymeasuredondifferentSCproduci ngtwodigitallysampled signals: x 1 [ n ]= ( f s n )+ f in 1 f s Clk 1 ( f s n ), (4–1) x 2 [ n ]= ((1 )( n ( f s + f s ) )) (4–2) + h ( n ( f s + f s ))+ f in 2 f s + f s Clk 2 ( n ( f s + f s )). The (1 ) factorisintroducedtoaccountforthelaserphasetime-sca lingasa resultofSCmotionwhilethe f s factoraccountsforthesmalldifferenceintheabsolute clockfrequencies.Althoughthis f s factorswillbesmall, 2 theycancauseaccumulated errorsinthephasecorrectionsiftheyarenotaccountedfor 3 Thesecanberelated tothe ij factorsdenedin( 3–71 )with f Clki = f s and f Clkj = f s + f s .Theability toextract g ( t ) fromthesesignalsdependsonfourthings:(1)theinitialla ser-phase 1 Orequivalently,thelaserlighttraveltime-delay: ( t )= L ( t ) = c 2 f s 50 MHz, f s 1 10 Hzdependingonclocktolerances. 3 Relativistictime-dilationcausesthesameeffectbutitis small, 2 < 10 12 ,andcan simplyincorporatedintotheclockfrequencyerrorterm. 69

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spectraldensity, ~ ( ) ,(2)theabilitytomeasurethedifferentialclocknoiseter ms,(3)the accuracyofourknowledgeofthetimedelay, ,and(4)theabilitytoperformfractional delaylteringandinterpolationtotime-shiftandtime-sc alethesesignalstoaccountfor the f s ,and (1 ) factors. 4.1.1LaserNoise Obviously,theinitiallasernoiselevel, ~ ( ) ,playsalargerollinourabilityto extractthe g ( t ) terms 4 fromtheabovesignals.Thefrequencyreferencesproposed tostabilizethelasersinLISAincludeopticalcavities,Ma ch-Zanderinterferometers, molecularresonators,andarmlocking.Opticalcavityrefe rencesandtheassociated lockingmethodsarewelltestedandrobustbuttheEOMsandca vitiesusedasa frequencyreferenceaddtoweightoftheSC.Mach-Zanderint erferometerstwellwith theLISAinterferometrybasedesignbut,incomparisontoca vities,lacklow-frequency stability[ 34 ].Molecularhyper-neresonanceisbenecialinproviding anabsolutelaser frequencyreferencebutisgenerallycomplicatedinimplem entation.Arm-lockinghas thegreatestadvantageforLISAsinceitrequiresnoadditio nalhardwareandcanbe implementedcompletelythroughdigitalsignalreadoutsan dcontrolswhicharealready intheLISAdesign.Atthesametime,arm-lockingisatagreat dis-advantagesinceit hasonlybeentestedthroughelectronicsimulationsandapo ssibleriskoffailurewhen implementedinLISA.(Chapter. 6.3.1 ) 4.1.2ClockNoiseTransfers ThelasereldsoneachSCaremodulatedwithup-converted 5 clocknoise side-bands[ 74 ]suchthattheside-bandbeat-notesbetweenlasereldsfro madjacent SC, s sr ,alsoproducedifferentialclocknoiseterms.Giventhatth eclocksignalsare up-convertedbyafactor, G up ,andthattheside-bandbeatnoteproducedwhentheclock 4 Generally, h ( ) < 100 Cycles = p Hz 5 From50MHzto2GHz, G up 40 70

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from SC 2 istransmittedto SC 1 ismeasuredwithaphasemeteratafrequencyof f Clk weobtain: Clk Sidebands ( t )= G up ( Clk 1 + Clk 2;3 ) + f Clk f Clk 1 Clk 1 (4–3) = G up 1+ f Clk G up f Clk 1 Clk 1 + Clk 2;3 (4–4) G up ( Clk 1 + Clk 2;3 ) (4–5) Aswe'llseeinChapter. 4.3 ,thesearethesametermswhichshowupintheTDI combinationsandareusedtocorrectforthisclocknoisecou pling. 6 4.1.3RangingErrors Beforeweoutlinethemethodsofinter-SCranginginChapter 4.4 ,wederivea generalrelationshipbetweentherangingerrorandtheinpu tlasernoise.Giventhat wehaveameasurementofsome ( t ) aswellassometime-delayedmeasurement delayed ( t )= ( t )+ g ( t ) ,wetime-shiftandsubtractthesesignalstoextract g ( t ) Assumingwehavesometime-shiftingerror, + ,whenwefractional-delayinterpolate thetime-delayedsignal, delayed ( t ) ,andsubtractitfromtheinputsignalweestimatethe noisecancellation.UsingtheTaylorapproximationwecanw rite, X Err ( t ) ( t + + ), ~ X Err [ e i t e i ( t + ) ] ~ j ~ X Err j' j ~ j where istheangularFourier-transformationfrequency.Nowweca nestimatea simpliedbutreasonablemeasureoftherelationshipbetwe enlasernoisecancellation 6 Formoreinformationontheclocknoisetransfersandcorrec tions,see[ 68 – 70 75 ]. 71

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andtherangingprecisionasafunctionoffrequencyfor !<< 1 withtherelationship: X Err ( )= ( ) ( ). (4–6) Thus,givenacertainrangingcapabilitywecanderivethere quiredlaserprestabilization, orvisa-versa,givenalaserpre-stabilizationlevelwecan derivetherequiredranging accuracytomeettheinterferometrysensitivityspecicat ions.ThecompleteTDI derivationincludescomplexphasefactors, ( ) ,whichmayrangeinmagnitudefrom0 to4dependingonthelocationofthearm-lengthdependent sr zeros(Chapter. 4.4.2.1 ). 4.1.4FractionalDelayFilteringandInterpolation Dependingonthenalizeddesign,LISAwillproducesatelli tetoEarthdata-streams oftheTDIobservablesata f data =3 10Hz datarate.Integersampleshiftsofthese data-setswillresultinashiftingerrorof 0.33 0.1s = s ,respectively.But,giventhe expectedlaserpre-stabilizationlevel,LISAwillhavetot ime-shiftandtime-scalethese data-streamstoa3.3nsaccuracy 7 .Thus,assuminga10Hzdata-rate,LISArequiresa fractionalshiftingaccuracyof = T data =3.3ns = 0.1s=3.3 10 8 Fractionaldelayinterpolation[ 72 ]isusedtointerpolatethedata-setsandapply thefractionalshift[ 73 ].InLISAapplications,theLagrangelterisideallysuite dfor data-interpolationduetotheconstantlow-frequencyphas eloss 8 .Generallywecan writetheinterpolateddata-set, s ( n D ) whereDisthefractionalshift D = = T data ,as afunctionoftheinputdata-set,theLagrangianlterwindo w,andthesinc(x)functionas: s ( n D )= ( N 1) 2 X k = ( N 1) 2 s ( n + k ) w ( k ) sinc ( D k ) (4–7) 7 L =1meter, =3.3 ns 8 constantgroupdelay 72

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where t D = D +( N 1) = 2 w ( n )= N sin ( t d ) B ( t d N ) B N 1, n + N 1 2 (4–8) and[ 76 ] B ( n k )= ( n )( k ) ( n + k ) (4–9) Inthefollowingdescription,wedefaulttoalterlengthof N=51whichprovidesus withashiftingaccuracyofgreaterthan 10 12 [ 73 ].Generalizingthistoincorporate time-scalingaswellastime-shifting,wecanwritethetime -changingtimedelayasa functionoftheinteger, n D ,andfractionalshift, D ( t ) as: ( t )= (0)+ t = n D T data + D ( t ) T data (4–10) Inthismannerweadvancethefractionaldelay,D,asafuncti onofthedata-sample,n, foreachiterationoftheloop: D ( n )= D (0)+ n .TheMATLABcoderequiredandused toperformthedatalteringandinterpolationisoutlinedi nAppendix A 4.2LaserPre-stabilization Basedontheexpectedrangingaccuracy,thecurrentLISAdes ignrequiresalaser frequencystabilityof: Pre Stabilization ( f )= 280 Hz p Hz s 1+ 2.8 mHz f 4 (4–11) Thiscouldbeachievedthroughasinglestabilizationmetho dorbysomecombinationofthesemethods.Thecostsandbenetsofeachofthese methodsareoutlined 73

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wellintheLISAfrequencycontrolwhitepaper[ 34 ].Thefollowingdiscussionwillfocus ontwomethods:(1)Pound-Drever-Hallcavitystabilizatio n 9 and(2)arm-locking 10 4.2.1Pound-Drever-HallLocking Figure4-1.ModelofthePDHlockingscheme:AmodelofthePou nd,Drever,Halllaser frequencylockingtechniqueisdepicted.Thelasereldism odulatedwith side-bandsandalignedthroughapolarizingbeamsplitteri ncidentonthe cavity.Thebackreectedeldisusedtocontrolthelaserfr equencyafter demodulatingtheside-bands. TheapplicationofPound-Drever-Hall(PDH)laserstabiliz ation[ 50 ]involves stabilizingthefrequencyofalasertothelengthofanultra -lowexpansion(ULE)cavity suchthatanintegermultipleofthelaser'swavelength, =1064 nm ,equalsthelengthof thecavity: N = L (4–12) 9 Thisisusedasthepre-stabilizedinputfortheTDIsimulati onsaswellasastable referenceagainstwhichtocompareotherlaserstabilityme asurements 10 Theseexperimentsmutuallyprovidedaproof-of-conceptfo rtheUFLISelectronics alongwiththevericationofthearm-lockingstabilizatio nmethodsthemselves. 74

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Thisgivesusarelationshipbetweenthelaserfrequency, f + f ( t ) ,andthecavitylength, L + L ( t ) givenby[ 52 ]: f ( ) f = L ( ) L (4–13) transferringthelengthstabilityofthecavitytothefrequ encystabilityofthelaser. ThePDHfrequencystabilizationisaccomplishedbymodulat ingthelaser phaseusinganelectro-opticalmodulatorwithMHzside-ban ds[ 51 ]asshownin Figure 4-1 .Thephaserelationshipoftheseoff-resonanceside-bands reected fromthecavityarescaledbyacomplexreectioncoefcient whichisafunction ofthecavitiesmirrors'reectionandtransmissioncoefc ients.Demodulatingthe PDsignalagainstthemodulationgeneratingoscillatorusi nganelectronicmixer cancelsthecommonfrequencynoiseofthelocaloscillator 11 andresultsinan errorsignalwhichisproportionaltothephaseoffsetoflas ercarrierphasetothe cavitiesresonantlength.Thiserrorsignalcanthenbeused withtheappropriate proportional-integrating-differentiating(PID)ornit e-impulse-response(FIR)control electronicstofeedbacktothepiezo-electrictransducer( PZT)andtemperaturecontrols ofthelaseroutputfrequency.Formoreinformationonthisl ockingtechniqueandit's applicationsinLISA,referto[ 34 51 52 ]. 4.2.2ArmLocking ArmlockingisalaserstabilizationtechniqueproposedbyS heard,et.al.[ 53 ]which exploitsthelong-termstabilityoftheLISAarm-lengths 12 asareferenceagainstwhich tostabilizethelong-termlaserfrequency.Utilizingthes ensorsignalsfrom( 3–72 )and ( 3–74 ),wehavearst-ordermeasureofthechangeinthelaserfreq uencyoverthe 11 Thiscouldbeavoltagecontrolledoscillator,afunctionge nerator,oranyrelatively stableMHzoscillator 12 5 0.1Gm overoneyear. 75

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individualround-triparm-lengthdelayswithatransferfu nctiongivenby: T AL : s ( s )= L ( s 1 ( t )) L ( 1 ( t )) (4–14) = 1 ( s ) 1 e s (4–15) wherethefunctionalvariable,'s,'isthecomplexLaplacef requency s = + i while s 1 referstotheround-tripdelayto SC s .Alteredoutputofasinglearmsensorsignal 13 [ 53 77 78 ]orsomelteredlinearcombinationofthetwoindividualar msignals 14 [ 79 – 81 ]canbeusedtocontrolthelocal, 1 ( t ) ,laserphasewiththecompleteopen-loop transferfunctiongivenby: T AL : Comp ( s )= T Sum ( s ) A sum ( s )+ T Dif ( s ) A dif ( s ). (4–16) where A sum ( t )= 21 ( t )+ 31 ( t ), (4–17) A dif ( t )= 21 ( t ) 31 ( t ) (4–18) resultinginalasernoisesuppressiongivenbytheclosedlo optransferfunction: T Closed = 1 1+ T AL : Comp ( s ) (4–19) Theimplementationofarm-lockingrequiresreal-timecons tructionofthesensorsignals aswellassomemethodofactuatingthelaserfrequencyeithe rthroughaPZT-mounted cavity[ 82 ],offsetphase-lockedlasers[ 78 ],orside-bandlocking[ 83 ].Specialcare mustbetakentoestimatetheDopplershiftssinceintegrati onofDopplererrorscause 13 Singlearm-locking 14 Dual-modiedarmlocking 76

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afrequency-pullingeffectwhichlimitsthenoisesuppress ionandcausesthelaser frequenciestodriftasafunctionoftheintegratedDoppler error[ 84 ]. TheelectroniccomponentsdescribedinChapter. 6 incombinationwithThe UniversityofFloridaLaserInterferometrySimulator(UFL IS)benchtopwereusedby theauthorandotherstoperformthersthardwareimplement edproof-of-conceptsingle arm-lockingexperiment(Chapter. 6.3.1 )revealingtheyet-to-be-discoveredfrequency pullingeffects[ 41 77 84 ].Expansionstothesimulatorhavealsoprovendualand modiedarm-lockingcongurationsalongwithfrequencypu llingeffects.[ 54 81 ]. 4.3TDITheory Priortonow,wehavesimplymentionedthatwemustformparti culartime-shifted andtime-compressedlinearcombinationstoaccountforthe time-changingunequal arm-lengthsandcancelthedominantlaserphasenoisebutwe havenotdened thesespeciccombinations.Theselinearcombinationsfal lintotwomajorcategories: (1)TDI-Sagnaccombinationsand(2)TDI-Xcombinations.Th ethreeTDI-Sagnac combinationsareconstructedbycompletingthelasertrans ferchainaroundcounter propagatingdirectionsoftheLISAconstellation.Theseco mbinationsaresignicantly lesssensitivetogravitationalwaves,thusprovidinganes timationofnon-lasernoise sourcessuchasPDnoiseorscatteredlighteffects.TheTDIXcombinationsare constructedbycompletingthelasertransferchainalongin dividualarmsofthe interferometercancelingthelaserphasenoiseandextract ingtheGWsignals.As previouslystated,theabilitytoformthesecombinationsd ependsdirectlyonthe accuracyofthemeasuredarm-lengths(Chapter. 4.1.3 ),butatthesametime,these combinationscanbeexploitedtoestimatethearm-lengths( Chapter. 4.4.2.1 ). 4.3.1TDICombinations WebeginouranalysisbyconstructingthetwomajorTDIcombi nationswhich areshowndiagrammaticallyinFigure 4-2 .TheTDI-Xcombinationisrepresentative ofaMichelson-typeinterferometerwhiletheTDIcombinationisrepresentativeof 77

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aSagnac-typeinterferometer.TheTDIformulationshaveno tbeenconsistentwithin theliterature[ 55 ]suchthatinthisdescription,wewillusethesamenotation asthose denedinChapter. 3.4.3 TheTDI-analysisandcombinationsuseddependonthedynami csoftheorbits.The differentTDIgenerationsarederivedbasedontheorbitald ynamicsapproximationused. The TDI X 0.0 combinationassumesthearm-lengthsareconstant, d q ( t ) = dt =0 non-rotating q (0)= q 0 (0) ,andequal, q (0)= q (0) 15 .The TDI X 1.0 combination assumesthearm-lengthsareconstant, d q ( t ) = dt =0 andnon-rotating q (0)= q 0 (0) butunequal, q (0) 6 = q (0) .The TDI X 1.5 combinationassumesthearm-lengthsare constant, d q ( t ) = dt =0 ,butrotating, q (0) 6 = q 0 (0) ,andunequal, q (0) 6 = q (0) .Finally, the TDI X 2.0 combinationassumesthearm-lengthsarenon-constant, d q ( t ) = dt 6 =0 rotating, q (0) 6 = q 0 (0) ,andunequal, q (0) 6 = q (0) .TheseareoutlinedinTable 4-1 Thus,the TDI X 2.0 combinationshouldcompletelyaccountforthelasernoisec oupling giventhelineararm-lengthrateofchangewehaveconsidere dintheprevioussections. Assumingwehaveacontinuousdata-streamoftheobservable sforSCacceleration termstohaveaneffectonthedatacombinations,thiscouldb eaccountedforwith furtherexpansionoftheTDIcombinations.Thisisusuallyu n-necessarygiventhe likely-hoodofa'unbroken'data-set,butnone-the-less,w ewillseeinChapter. 4.4 thatthiscanbeaccountedfor,giventheLISA-orbitaldynam ics,withsegmented data-analysis. 15 The q referstothe'opposite'armofthe q -armvs. q -armMichelson-type interferometer. 78

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Table4-1.OrbitaldynamicsapproximationsforTDIgenerat ions GenerationMichelsonArm-LengthCounter-PropagatingDel ayDelayDynamics TDI0.0 q ( t )= q ( t ) q (0)= q 0 (0) d q ( t ) = dt =0 TDI1.0 q ( t ) 6 = q ( t ) q (0)= q 0 (0) d q ( t ) = dt =0 TDI1.5 q ( t ) 6 = q ( t ) q (0) 6 = q 0 (0) d q ( t ) = dt =0 TDI2.0 q ( t ) 6 = q ( t ) q (0) 6 = q 0 (0) d q ( t ) = dt = q TDI3.0+ q ( t ) 6 = q ( t ) q (0) 6 = q 0 (0) d q ( t ) = dt = q ( t ) Figure4-2.DiagrammaticmodelsoftheTDI-XandSagnaccomb inations:Herewe presentthegeometricrepresentationoftheLISAconstella tionsinparallelto thosedevelopedin[ 85 ].Thelasertransferchaininthesediagramatic representationsshowhowtotime-shifttheobservablestof ormtheTDI combinationsandensurethatthelasernoisesourceswillca ncelwhenthe lasertransferchainisaclosedloop. 4.3.2SagnacCombinations4.3.2.1TDISix-PulseCombinations Therstordersix-pulsecombination,named areconstructedfromthe sr combinations( 3–77 and 3–79 )fromeachoftherespectiveSCandtaketheform: 1.0 = 21 31 (4–20) = 1 1;2 0 1 0 3 0 [ 1 1;312 ] +[ h 21 + h 32;3 0 + h 13;1 0 3 0 ] [ h 31 + h 23;2 + h 12;12 ] = 1;312 1;2 0 1 0 3 0 +[ h 21 + h 32;3 0 + h 13;1 0 3 0 ] [ h 31 + h 23;2 + h 12;12 ]. (4–21) 79

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Generally 16 ,thesecombinationscancelthelaserphasetermsandleaves ix h sr GW termsinthe TDI 1.0 approximation.Theyarenamedbasedonthefactthata delta-functiongravitationalwaveinputisreplicatedsix timesintheresultingcombination. Ifweextendthistothe TDI 2.0 approximationbyaccountingfortime-changing delaysandun-equalcounter-propagatinglighttraveltime -delays,wewouldhavetotrace thelaserchainbacktothestartingpointresultingina12 s sr -termexpressionanda twelve-pulseGWresponse.4.3.2.2TDISymmetric-SagnacCombination TherstorderSangnaccombinationwrittenas: 1.0 = s 31;1 + s 12;2 + s 23;3 ( s 21;1 0 + s 32;2 0 + s 13;3 0 ) (4–22) isfreeoflaserphasenoiseinanon-rotatingconstellation andisordersofmagnitude lesssensitivetoGWsignals.Thiscouldbeusedtodiscrimin atebetweeninstrument noisesandGWstochasticbackgroundsignals[ 86 87 ].Thiscombinationisshown geometricallyinFigure 4-2 ;onenotesthattherotationoftheconstellationinthe geometricrepresentationcausesanopenlasertransfercha inforarotatingconstellation whichdestroysthecommonmodelaserphasecancellationthe Sangac-1.5combinations[ 85 88 ].Again,thisisaccountedforbythemodiedSangaccombina tionwhich re-tracesthetime-delaypathresultingina12-term TDI 2.0 function. 4.3.3MichelsonX-combinations TheMichelsonXcombinations 17 formthreeinterferometerscombinationswhich collectivelyformabasisinthe2-dimensionalplaneofthec onstellationforthe h + and 16 Ignoringclock-noiseterms 17 UsingotherSCasourframeofreference,wecanobtainthe'Y' and'Z' combinations 80

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h strain.Inthefollowingsections,weconstructtheMichels onXcombinationsfromthe round-tripsingle-armsensorsignals, sr ,denedin( 3–72 )and( 3–74 ). 4.3.3.1TDIX 0.0 Inthespecialcasewherethetotalroundtripdelay-timesar eequal ([ q (0)+ q 0 (0)]=[ q (0)+ q 0 (0)]) andthedifferentialSCvelocitiesarezero ( d q ( t ) = dt = q =0) thedifferenceofthesensorsignals,( 3–72 )and( 3–74 ), X 0.0 = 21 31 (4–23) generatesthestandardequal-armMichelsoninterferomete routput,independentoflaser phasenoise.Expandingexplicitly, X 0.0 = 1 1 ( t 3 0 (0) 3 (0))+ h 21 ( t )+ h 12 ( t 3 0 (0)) (4–24) + f 21 f Clk Clk 1 ( t )+ f 12 f Clk Clk 2 ( t 3 0 (0)) [ 1 1 ( t 2 (0) 2 0 (0))+ h 31 ( t )+ h 13 ( t 2 (0)) + f 31 f Clk Clk 1 ( t )+ f 13 f Clk Clk 3 ( t 2 (0))] whichreducesinthisspecialcaseto: X 0.0 = h 21 ( t )+ h 12 ( t 3 0 (0)) h 31 ( t ) h 13 ( t 2 (0)) (4–25) + f 21 f Clk Clk 1 ( t )+ f 12 f Clk Clk 2 ( t 3 0 (0)) f 31 f Clk Clk 1 ( t ) f 13 f Clk Clk 3 ( t 2 (0)) whereweseethefour-pulseGWresponseandclocknoisecoupl ing.Maintainingthe =0 assumption,thelasereldsarenotDopplershiftedandtheb eatnotefrequencies onoppositeSCwillbeequalsuchthatwecanfurthersimplify thiscombinationandwrite 81

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itas: X 0.0 = h 21 ( t )+ h 12 ( t 3 0 (0)) h 31 ( t ) h 13 ( t 2 (0)) (4–26) + f 21 f Clk [ Clk 1 ( t )+ Clk 2 ( t 3 0 (0))] f 31 f Clk [ Clk 1 ( t )+ Clk 3 ( t 2 (0))]. Wenoticethattheseclocknoisetermsarethesameclock-noi setransferterms describedin( 4–3 ).Rescalingtheclock-noisetransfersidebandmeasuremen tsand subtractingthemfromthe TDI X 0.0 combination,weobtainaMichelsonGWoutput freeofanyothernoisesources: X 0.0 = h 21 ( t )+ h 12 ( t 3 0 (0)) h 31 ( t ) h 13 ( t 2 (0)) (4–27) Nowthatwehaveshownhowtheclock-noisetermsareaccounte dfor,wewillignore themintherestoftheTDIdescription.LighteldDopplersh iftsfrominter-SCmotion changethebeatnotetoclockfrequencyratio, ( f sr f Dop ) = f Clk ,butcanbesubtracted fromthesensorsignals sr beforeanyofthefollowingcombinationsareformed.That said,wewillrevisithowthesetermscoupleintotheUFLIS-T DIsimulationsdescribedin Chapter 7 Unfortunately,despiteallthiswork,the TDI X 0.0 combinationisrarelyareasonable laserphasecancellationtechniquesincetheLISAarm-leng thsarealmostalways un-equal.None-the-less,thisservesasarstorderexampl eofhowheterodyneGW interferometryisperformed.Theconsiderationsbeyondth isshowthetechniquesto accountfororbitaldynamicsandchangesinthelight-trave ltime-delaysbetweentheSC. 4.3.3.2TDIX 1.0 TheTDIX 1.0 combination[ 89 ],writtenas X 1.0 = 21 31 21;2 0 2 + 31;33 0 (4–28) 82

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replicatestheequal-armphasedelaysandcancelsthecommo nlaserphasenoiseinthe casewhere ([ 2 + 2 0 ] 6 =[ 3 + 3 0 ]) and 2 3 0 asshowngeometricallyinFigure 4-2 Expandingexplicitly,butstillallowingfornon-zero -values,wecanwrite: X 1.0 =[ 1 1;33 0 ] [ 1 1;2 0 2 ] [ 1 1;33 0 ] ;2 0 2 +[ 1 1;2 0 2 ] ;33 0 (4–29) +[ h 21 + h 12;3 0 ] [ h 31 + h 13;2 ] [ h 21 + h 12;3 0 ] ;2 0 2 +[ h 31 + h 13;2 ] ;33 0 =[ 1;33 0 2 0 2 1;2 0 233 0 ] (4–30) +[[ h 21 h 21;2 0 2 ]+[ h 12;3 0 h 12;3 0 2 0 2 ]] [[ h 31 h 31;33 0 ]+[ h 13;2 h 13;233 0 ]]. Fromtheexpansionwenoticethe TDI X 1.0 eight-pulseGWresponse.Failingto accountfortheSC-motionandtime-changingdelays,theTDI X 1.0 combinationislimited bythe [ 1;33 0 2 0 2 1;2 0 233 0 ] termswhichdonotcancelcompletelysincethetime-delay transformationsareperformedinadifferentorder.Calcul atingthedelayerrorasaresult ofthetransformationorderusingtheleading (1 ) terms 18 wecanwrite: = (4[1 2 ] 4[1 3 ])=4 [ 3 2 ] (4–31) where isthemeanone-waydelaytime 19 .Evaluatingthistheresultin( 4–6 ),we obtain[ 90 ]: ~ X 1.0 > 4 j 2 3 j ~ 1 (4–32) where ~ 1 isthetime-differentiatedlaserphasespectrum.Givenorb italcharacteristics wherethislimitislargeenoughtorestraintheIMSsensitiv ity,wemustfurtherexpandto thegeneral TDI X 2.0 combinationtoaccountforthisresidualnoise. 18 Wemaintaintheassumptionthat << 1 (1 ) 1 ,andapproximate (1 ) n =1 19 16.7 s 83

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4.3.3.3TDIX 2.0 TheTDIX 2 combination,writtenas[ 91 92 ], X 2.0 = 21 31 21;2 0 2 + 31;33 0 (4–33) 21;33 0 2 0 2 + 31;2 0 233 0 + 21;2 0 22 0 233 0 31;33 0 33 0 2 0 2 isusedtocancelthe, [ 1;33 0 2 0 2 1;2 0 233 0 ] ,laserphasenoisetermsleftinthe TDI X 1.0 combination.Thisproducesa16-pulseresponsetogravitat ionalwaves,writtenexplicitly as: X 2.0 =[[ h 21 + h 12;3 0 ] [ h 31 + h 13;2 ] [ h 21 + h 12;3 0 ] ;2 0 2 +[ h 31 + h 13;2 ] ;33 0 ] (4–34) [[ h 21 + h 12;3 0 ] ;33 0 2 0 2 [ h 31 + h 13;2 ] ;2 0 233 0 [ h 21 + h 12;3 0 ] ;2 0 22 0 233 0 +[ h 31 + h 13;2 ] ;33 0 33 0 2 0 2 ] but,giventhecorrecttime-delays,cancelsallthelaserph asenoiseandaccounts fortheindependentlinear-time-delays, q ( t ) ,assuming d 2 ( t ) = dt 2 =0 .Annual changesin ,orSCaccelerationterms( d 2 ( t ) = dt 2 6 =0 ),maybeaccountedforwith thefurtherexpansionoftheseTDIcombinationsalthough,t hisisunnecessaryas arguedinfollowingsection(Chapter. 4.3.4 ).TheMATLABcoderequiredtoformthese data-combinationsusingthe s sr data-setsisprovidedinAppendix B 4.3.4LISAOrbitalDynamicsandTDIDataAnalysis Inthefollowingexperiments, isassumedtobeconstantandwewillfocuson theTDIX 2.0 velocitycorrections.Thus,inordertoutilizetheTDI-ran gingmethods outlinedinChapter. 4.4.2.1 forLISA-TDIdata-analysis,the valuewillhaveto beadjustedtoavoidtheacceleration-dependentaccumulat ederror.Althougha continuousmeasureandcorrectiontothe valuesarepossible,thiscansimplybe accomplishedbysegmentingthedata-analysis,intheworst caseLISA-likescenario, every p T year = ( )= p 3.3ns 3.15 10 7 s = (66ns = s )= 708s[ 55 ].Giventhe time-framefortheaccelerationeffectstocoupleintothed ata-analysis,eveninthis 84

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worst-casescenario,itiseasiertoadjustthe -valueinthedata-analysisthanitisto forma TDI X 3.0 ,32-pulseGWresponse,combination. 4.4Ranging AswehavestatedinChapter. 4.1.3 andaswecanseeexplicitlyfromthe sr and TDI X 1.0 = 2.0 combinations,thelasernoisecancellationdirectlydepen dsonourability toestimatetheone-wayinter-SCarm-lengthsandformthese linearcombinations.From ( 4–6 )wecancalculatetherangingaccuracyneededtocancelthe 280Hz = p Hz laser pre-stabilizationinputnoise. Generally,thecancellationofthelocal 1 ( t ) laserphasenoisefromthefar s 1 s signalsdependsontheaccuracyoftheout-goingdelaytimes 3 ( t ) and 2 0 ( t ) ,while thecancellationofthefarlaserphasenoisefromthelocal s s 1 signalsdependsonthe accuracyofthein-comingdelaytimes, 3 0 ( t ) and 2 ( t ) .Assumingthelasersonseparate SCareindependentlystabilizedsuchthat 1 6 = 2 6 = 3 ,theneachoftheone-way inter-SCtime-delayfunctions, q ( t )= q (0)+ q t ,mustbeevaluatedtoa1meter (3.3ns)accuracy. Exploitingthephase-lockingtechniquesdescribedin[ 71 ],wecanphase-lockthe farlasers, 2 and 3 ,tothedelayed 1 eldfromthemasterSCandtransferthestability ofthemasterlasertothesefarlasers.Thisresultsinexpre ssionsforthefarSC'slaser phasenoisegivenby: 2 ( t )= 1;3 3 ( t )= 1;2 0 (4–35) towithinthetrackingaccuracyofthePLL.Inthisspecialca se,theone-waydelays areaccountedforbycontrollingthefarsensorsignalswith aPLLsuchthat, s 1 s = PLL Error 0 .Takingtheexpressionsfor 2 and 3 describedin( 4–35 ),andevaluating themin s 1 ( 3–72 and 3–74 ),weseethattheseexpressionssimplifyto: 21 ( t )= s 21 ( t ), 31 ( t )= s 31 ( t ). (4–36) 85

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Thus,sincethe s 1 s termsdonoteffectthenoisecancellationinthisphase-loc ked conguration,wecanreducethedelayconstraintstotwo,ro und-tripdelaysgivenby: 2 0 2 ( t )=[(1 2 )((1 2 0 )( t 2 0 (0)) 2 (0))] (4–37) 33 0 ( t )=[(1 3 0 )((1 3 )( t 3 (0)) 2 0 (0))]. Meanwhile,theconstraintsontheindividualone-waydelay saregreatlyreduced resultingfromtherelatively-stable 20 PLLnoise.Thesereducedrangingconstraints resultfromthefactthatthe'input'PLLnoiseusedin( 4–6 ),allowsforlargevaluesof whilestillmeetingtheX-combination'sIMSsensitivity. Twomethods,Pseudo-randomNoise(PRN)ranging[ 75 93 94 ]andTDI-Ranging [ 95 ]havebeenproposedtomeasuretheinter-SCarm-lengths(li ght-traveltimes).The PRNrangingmethod,describedinthenextchapterinvolvest heuseofadditionaloptical componentstomodulatethelasercarrierswithPRN-codes;t hecross-correlationofthe sixinter-SCrangingcodeswiththelocalcopiesofeachofth esePRNcodesprovides anindependent,real-timemeasureoftheone-wayinter-SCd istances.TDI-Ranging, ontheotherhand,requiresnoadditionalcomponentsanddet erminestheinter-SC rangingvaluesinpost-processingbyexploitingthelaserc ancellationcharacteristicsof theTDI-combinations.Providedwiththeeaseofimplementa tionoftheTDI-Ranging technique,theTDIexperimentsdescribedinChapter. 7 willattempttoexperimentally developandimproveupontheTDI-Rangingmethodsproposedi n[ 95 ]. 4.4.1Pseudo-randomNoise(PRN)CodeRanging Pseudo-randomnoise(PRN)codecross-correlationtechniq uesarewellunderstood inter-devicedistancetrackingmethodswhichhavebeendev elopedandveriedforuse inglobalpositioningsatellites(GPS).Theapplicationof thesemethodsinLISAinvolves 20 1 Hz = p Hz 1mHz = p Hz 86

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modulatingoneofsixlocalcodes(oneforeachlasersource) ontothecarrierlaser eld.Aftertheinter-SC/intra-SClasereldtransmission anddetection,thePRNcodes fromtheadjacentlasereldiscross-correlatedwiththelo calPRNcodes.ThesixPRN codeshavethedesignedpropertysuchthattheydonotcorrel atewitheach-other,but doperiodicallycorrelatewiththemselvesdependingonthe lengthofthePRNcode.A positivecross-correlationofthePRNcodeisusedinconjun ctionwithadelay-lock-loop toactivelytrackthe'incoming'codefromtheadjacentbenc htopandproducean inter-SCdelayread-out.Assumingthemeasureddelayscorr ectlyaccountforelectronic andspecialrelativisticdelayterms,thesedelayread-out scanbedirectlyusedtoform sr and TDI X 1.0 = 2.0 combinationsinrealtime. Thisrangingmethodhasbothbenetsandcomplications.The real-timePRNdelay measurementsallowustoformthe sr and TDI X 1.0 = 2.0 combinationsinreal-timeon theSC.The sr termscanbeusedforarm-lockingasdescribedinChapter. 6.3.1 .The TDI X 1.0 = 2.0 combinationscanbeformedon-boardfromthe s sr observablesandsent intheirpre-constructedTDI-formtoEarthratherthanhavi ngtotransmitthe18individual sr b sr ,and s sr observablesignals. Ontheotherhand,aswe'llseecomparativelyinthenextsect ion,thisranging methodaddsunnecessarycomplicationstotheLISAdesign.E lectro-opticalmodulators (EOMs),whichmightintroduceadditionalnoiseterms,must beusedtomodulatethe laser-eldwiththePRNcodesbeforetheinter-SCtransmiss ion,addingweightand complexitytotheLISAdesign.Also,thereisnoguaranteeth atthedelaysmeasured bythePRNrangingmethodsareequaltothoseneededtoformth eTDIcombinations sincethelasernoisecancellationintheTDI-combinations dependsonthetime-delay fromthelasereld'sgenerationtoobservables'detection 21 TDI-ranging,ontheother 21 IncludingPD,ADC,andphasemeterphasedelayresponses. 87

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hand,doesnotrequireanadditionalEOManddeterminestheo ne-waydelays,including electronicphasedelays,inpost-processing.4.4.2Time-delayInterferometryRanging(TDIR) Time-delayinterferometryranging[ 95 ]isamethodofdeterminingtheLISAsignal traveltimesinpost-processingbyminimizingthetotalroo tmeansquared(RMS)power intheLISAmeasurementfrequencyband.Takingthemeasured s sr ,combinations,we formthetheoreticalTDI-2.0combinationsusinginitiales timatesofthesixvariable, 22 lighttraveltimedelays.Thetime-delayparametersaresca nnedoverthepossible values 23 untilthetotalRMSpowerinthe TDI X 2.0 isminimized.Theinitialdelayoffset parameters, q (0) ,canvaryby0.66secondsandmustbemeasuredtoanaccuracy oflessthan 3.3 nanoseconds( 4–6 ).Thus,assumingnopreviousknowledgeofthe time-delayvalues,wehavealargeparameter-spaceoverwhi chtoscanwitheachofthe fourtime-delayoffsetparameterstakingoneof 2 10 8 possiblevalues. 24 Unlesssome trackingmethodisdeveloped,thisparametersearchcouldb ecomputationallyintensive. Although[ 95 ]provesthecapabilitiesandlimitationsoftheRMSminimiz ation rangingmethod,itdoesnotprovideamethodofactivelydete rminingthetimedependent delayparameters.Inaddition,theRMSminimizationmethod 'sdelayparameter calculationintroduceserrorsinthetime-delayvaluescau sedbylow-frequency gravitationalwavesignals.Inthisexperiment,wewillpre sentandemployanewmethod ofTDI-rangingbymodulatingthelasereldwitharangingto neatafrequencyoutside oftheLISAmeasurementband,inthiscase, 1 1.5 Hz .Althoughthismethodhasbeen usedforspacecraftrangingonactiveprojects[ 96 ]andhasbeenconsideredforuseon LISA[ 97 ],theauthorisnotawareofaformalanalysisintheliteratu re.Inthefollowing 22 Weassumethat q = q 0 and q (0) 6 = q 0 (0) 23 (16.33 s < q (0) < 17 s ) ( 66 ns = s << 66 ns = s ) 24 0.66 s = 3.3 ns =2 10 8 88

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sections,weattempttopresentthisformalanalysisoftheT DIranging-toneapplication andrealization,includingatime-delaysearchalgorithmd evelopedbytheauthor. 4.4.2.1TDIRangingTone TheTDI-rangingtonescanbeplacedoneachofthesixLISAlas ereldsbyadding a f Tone Hz sinusoidalmodulationtothepiezo-electrictransducerwh ichcontrolsthe laserfrequency.Thisissignicantlymorestraight-forwa rdthantheEOMsrequiredto implementthePRN-rangingmethod.Ontheotherhand,therea l-timemeasureofthe arm-lengthsprovidedbythePRNrangingmethodhastheadvan tageofproducingthe sensorandTDIvariablesinreal-timewhichisbenecialtow ardstheimplementationof arm-lockingandreducingthesatellite-to-Earthdatatran smissiondemands. ExtendingtheRMSminimizationconcept,theapplicationof arangingtone effectivelyincreasesthelasernoisebyintroducingmores ignalpoweratthespecied rangingtonefrequencies.Asshownby( 4–6 ),thisresultsinabetterestimationofthe rangingerror, ,thantheinherentlasernoisecancellationwouldprovide. Inaddition, becauseweareonlyinterestedinthepowerminimizationnea rtherangingtone frequency,thisrangingmethodshouldnotbeaffectedbylow frequencygravitational waves. Therangingtoneisoptimallymodulatedontothelaserelda tafrequencymid-way betweenthefrequency-domainzerosoftheinter-SCsensors ignals, sr ,toavoid inherenttonecancellationalongasinglearm 25 : N q (0)+ q 0 (0) < f Tone < N +1 q (0)+ q 0 (0) (4–38) Also,toavoidconfusionbetweentheindividuallasermodul ations,eachoftheLaser sr eldsshouldbemodulatedatadifferentfrequency, f Tone sr .Usingthereverseargument oftherangingrequirementderivation,thecancellationof thelocal 1 ( t ) tonefromthefar 25 ThiscanbeseengraphicallyinFigure 7-4 89

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s 1 s signalsconstrainstheout-goingdelaytimes, 3 ( t ) and 2 0 ( t ) ,whilethecancellation ofthefarlasertonesfromthelocal s s 1 signalsconstrainsthein-comingdelaytimes, 3 0 ( t ) and 2 ( t ) ,resultinginallfourone-waytime-delayfunctions. RevisitingtheLISA-modelwherethefarlasersarephase-lo ckedtothedelayed masterlasereld,( 4–35 ),thedelayedlocallasermodulationistransmittedbackto to localSC.Thisresultsintworound-triptime-delayfunctio nconstraints,( 4–37 ),usingthe cancellationofthelocaltonesintheTDIcombinations.Ins piteofthis,thephase-locked laserscouldstillbemodulatedwiththeirownrangingtones restoringallfour,one-way constraints.4.4.2.2TDIRangingParameterSearchAlgorithm Themethodusedtodeterminethesix-parameterswhichdene thefourone-way delay-functionsisoutlinedinFigure 4-3 .Therststepinvolvesprovidinganestimate ofthetime-delayfunctions.Theestimatedoesnotneedtobe accurate;anyestimation errorwillbecorrectedbytheconvergentpropertiesofthes earchalgorithm.The estimatesarethenusedtotime-scalethe s sr signalsbytheappropriatefactorsas denedbythe TDI X 2.0 combination 26 .Thetime-scaleddatasetisthenbrokeninto 'N'sectionswhichareindividuallyusedtodeterminethefo urtime-delayoffsets, q (0) foreachsection. Aswe'vepreviouslystated,thespanofpossiblevaluesofth etime-delayoffsetsin comparisonwiththerequiredtime-delayrangingaccuracyr esultsinalargeparameter spaceoverwhichthesevaluescanvary.Bruteforcescanning andcomputingtheTDI combinationforall 10 32 valuesiseffectivelyimpossible.Instead,webeginbyeval uating theTDIcombinationforeach-timedelayintherangeofpossi blevaluesfrom16.2sto 17.2swithaprecisionof0.1s.Inthisspeciccase,thedela y-segmentationresults 26 TheTDIX 2 combinationisusedinfavorofthetheTDIX 1 combinationbecauseof thepossibleinherentconstraintsoftheTDIX 1 :( 4–32 ) 90

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in10possibledelaysvaluesforeachofthefourdimensionso ftheone-waydelay times,resultingin 10 4 possibledelaypermutations.TheRMSpoweroftheTDIX 2.0 combinationinthe f Tone Hz 1mHz frequencybandiscalculatedforeachofthe 10 4 possiblevalues.Thesetofdelayswhichcontainstheminimu mRMSpowerisusedas anewreferencepoint.Thetime-delayparameterspacearoun dthisminimum( 0.1 s ) is,brokenintomoreprecisedelayvalues( 0.01 s ).TheRMSpowerminimizationscan inthisnewparameterspaceisperformedto,again,determin ethedelayparameters toabetterprecision.Theprocessisrepeatedimprovingthe precisionbyafactorof 10eachtimeuntilthefourdelay-timeoffsetsareconstrain edtoa 1 ps precisionand therangingtoneisdominatedbyinstrumentnoisesources. 27 Thedelaysegmentation precision,inthiscaseafactorof10foreachiteration,isc hosentoavoidthepossibility ofconvergingonalocalminimaandobtainingthewrongdelay -timeestimate.Amore effectivemethodmightincludetheevaluationofasurfaceg radientwhichconverges ontheRMS-minimizeddelaytimeswhichhasbeenttedtothet ime-delaygrid throughMonte-Carloanalysis;thismethodcouldaccelerat ethedata-analysis,assist inconrmingtheresultisnotalocalminima,andensurethat theactualoptimized delay-timeshavebeendetermined. Oncethefourtime-delayoffsetsarecalculatedforeachoft heNdata-segments,a linearregressionoftheoffsetsisperformedtoobtainafun ctionaldenitionofallfourof theone-waytimedelays: 2 ( t )=(1 2 )( t 2 (0)), 3 0 ( t )=(1 3 )( t 3 0 (0)), (4–39) 2 0 ( t )=(1 2 )( t 2 0 (0)), 3 ( t )=(1 3 )( t 3 (0)). 27 Althoughwescanthedelayparameterstoa 1 ps precision,theactualdelayerroris determinedbasedontherangingtonecancellationwhichisl imitedbyinstrumentnoise sources. 91

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Thecalculatedslopeprovidesamoreaccurateevaluationof -valuesforthissectionof datathantheoriginalestimation.Assumingtheranging-to necancellationandtime-delay offsetcalculationwaslimitedby( 4–32 ),theimprovementinthe estimationincreases theprecisionofthetime-delayoffsetsintheseconditerat ionoftheranging-tone cancellationalgorithm.Generally,afterthreeiteration stheone-waydelayfunctions areoptimallyevaluated.Therstiterationdeterminesthe values.Thesecond iterationdeterminesthedelay-offsets.Thethirditerati onoptimizesthevaluesover theentiredata-set.Oncethefourone-waydelayfunctionsa rederived,theyareused tocalculatetheTDIX 2 combinationfortheentiredata-set.Inaddition,thevaria nceon thelinear-regressionprovidesuswithameansofdetermini ngtherangingprecisionin comparisonwiththeprecisiondenedbytherangingtonecan cellationusing( 4–6 ).The processissimpliedtoafour-parameter( 2 3 2 0 2 (0), 33 0 (0) )estimationwhenthefar laserareslavephaselockedtothemasterlocallaser. 92

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Figure4-3.Flowchartoftheranging-toneminimizationpro cess:Theprocessdepictedbythisowchartminimizesthe rangingtoneandmaximallyconstrainsthesixvariableligh ttraveltimedelays.Theresultsofthisprocessfor thedifferentexperimentalcongurationsarepresentedin Table 7-1 andTable 7-2 93

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CHAPTER5 THELISAPHASEMETER 5.1DigitialSignalProcessing(DSP)System Thephasemeterread-outs,digitalcontrolsystems,andele ctronicinter-SCdelay simulationcomponentsweredevelopedusingprogrammableD SPequipmentacquired fromPentek,Inc.ThePenteksystemisconstructedfromthre ebasiccomponents.The 6256 1 isafour-channel,14-bit,analogtodigitalsignalconvert er(ADC)samplingcard containingaeldprogrammablegatearray(FPGA)capableof high-speedreal-time xed-pointdata-processing.Meanwhile,the6228 2 isafour-channel,16-bit,digitalto analogconverter(DAC)read-outcardwiththesameFPGAcapa bilities.Thesetwo daughtercardsareconnectedthroughthemasterprocessing andcontrolcard,the 4205. 3 The4205handlesthedatatransfersbetweenthedaughtercar dsandcanbe controlledthroughaserialbaseduserinterface.The4205' sprocessorcanperform oatingpointcalculationsonthemeasureddataandstoreth eresultsin1GByteof synchronousdynamicrandom-accessmemory(SDRAM)orsendt hedatathroughan Ethernettransfertoanexternaldata-storagecomputer.Bo ththeinputADCsandoutput DACscanbeexternallyclockedeitherbythesamesourceorby twoindependentclock sources. 5.2 CyclePhaseMeasurements Thephasemeter,programmedtothe6256'sFPGA,isdesignedt omeasurethe phaseofa 2 20MHz PDbeatnotesignalwithanaccuracyof 1 cycle = p Hz (Table 3-1 ). Thisdeviceisused,notonlytomeasureLISA-likescienceob servables,butalsoto generatethephasedatafortheelectronicsimulationofint er-SCeldtransmissiondelay 1 Model6256Dual/Quad105MHzA/Dw/Virtex-IIProFPGA-VIM-2 2 Model62284-Ch.D/A,DigitalUp-converter&FPGAVIM-2Modu le 3 Model4205VIM/PMCCarrierandMPC7457PowerPCVMEBoard 94

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andphasemodulationeffects.Itcanalsobeusedtogenerate high-speedfrequency read-outsforreal-timedigitalcontrolsystemssuchasarm -locking.Inthefollowing sectionsweoutlinethedesign,characteristics,noisesou rces,andperformanceofthe UniversityofFloridaphasemeter. n r n n n rnn n r n n r r r r r Figure5-1.AmodeloftheLISAphasemeter:Hereweshowhowad igitaloffsetphase lockloopislockedtothesampledinputsignal.Meanwhile,t hefeedback frequencysignal, f fb ( t ) ,andthemultiplierdemodulatedoutputs, Q ( t ) and I ( t ) ,arerecordedtoreconstructthephaseandamplitudeofthei nputsignal. Thedetailsandtheoreticalanalysisofthephasemeteroper ationcanbe foundinthetext(Chapter 5.2.1 ) 5.2.1Design Thephasemeterdesignismodeledoffofastandardoffsetpha selockloop(PLL) trackingcontroller.Thename'phasemeter'ismisleadings incethePMcoreactually recordsa64-bitfrequency-proportionalfeedbacksignali nadigitalPLLwhichisthen integratedinpost-processingtogeneratethephase.Assho wninFigure 5-1 ,aninput signal, in ( t )= A in ( t ) sin ( in ( t )) ,issampledwitha14-bitaccuracyatasampling 95

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frequency, f s 4 Thedigitizedbeatnoteisthenmixedwithsineandcosinecom ponents ofadigitalfeedbacksignalwithaconstantamplitude, A fb ,andatimevaryingfeedback phase, fb ( t )= R dt ( f fb ( t )+ f oset ) ,givenauser-denedoffsetfrequency, f oset .The feedbacksinusoids, A fb sin ( fb ( t )) and A fb cos ( fb ( t )) ,aregeneratedwithnumerically controlledoscillators(NCOs)usinga n LUT =28 -bitlook-uptable(LUT).Once thefeedbackandinputsignalsaremultiplied, 5 thesignalislteredwithaNstage cascaded-integrating-comb(CIC)lter[ 98 ].Thedataisdown-sampledbyafactorofR andlow-passlteredbytheCIC-transferfunctionsuchasth oseshowninFigure 5-2 ThetransferfunctionoftheCIClterintheLaplacedomaint akestheform: G ( s )= R 1 N (1+ z R ) N (1 z 1 ) N wherez=e s f s (5–1) AnexampleoftheCICtransferfunctionsfor f s =50.0 MHz, R fb =16 ,and N = 2,4,8 stageltersareplottedinFigure 5-2 f s ischosenatafrequencywithalower bounddenedbytheNyquistsamplingfrequency, f s > 2 f Ny ,where f Ny isthesignals largestfrequencycomponentofinterest,andwithanupperb ounddenedbythe timingconstraintsoftheFPGAandADCs.Alargerdown-sampl ingfactor,R,reduces thetrackingloop'supdaterate,reducingthetimingrequir ementsoftheFPGA,but alsoresultsinanincreaseofthePLL'sin-loopphasedelay, reducingthetracking bandwidth.Thenumberofstages,N,denesbywhatfactorthe high-frequencydatais suppressedbeforedown-samplingandaliasingthehigh-fre quencyinformationintothe measurement.WithregardstotheLISAmission,anicebenet ofusingtheCIClterfor thisdown-samplingprocessisthatthe'zeros'ofthetransf erfunctionarealiasedtoDC, signicantlysuppressinganyout-of-bandinformationbef oreitisaliasedintotheLISA sciencedatafrequencyband(DC-1Hz).Thisisdiscussedfur therinChapter 5.2.2.2 4 f s =40 100 MHz 5 The14-bitADCoutandthe28-bitLUTsinusoidresultina n Q =42 -bitI/Qprecision 96

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 7 10 -40 10 -20 10 0 Frequency (Hz)Suppression Magnitude 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 7 0 100 200 300 400 Frequency (Hz)Radians 2 Stage 4 Stage 8 Stage 2 Stage 4 Stage 8 Stage Figure5-2.CICltertransferfunctions:Themagnitudeand phaseresponseofthe N =2,4,8 stageCICltersusingadata-rate, f s =50 MHz ,anda down-samplingfactor, R =16 ,isplotted.Alargernumberofstages, althoughincreasingthesuppressionofhigher-orderalias edfrequency bands,alsocausesanincreaseinthelter'sphasedelay. SwitchingourfocustothePM'sPLLtrackingloop,ifweassum ethat f in = d in ( t ) = dt f oset ,andsuppressthesumtermofthedemodulatedsignalwitha CIClow-passlter,itresultsinthesineandcosineofthedi fferencephase, err ( t )= in ( t ) fb ( t ) .TheCIClteralsoreducesthetrackingloop'sdatarateto f Core = f s = R fb Asaresultofthesine/cosinemultiplicationandlteringw eobtaintwosignalswhichare proportionaltothein-phaseandquadraturecomponentsoft hefeed-backerrorsignal, Q ( t ) / G ( s )[ A in ( t ) sin ( in ( t )) A fb cos ( fb ( t ))]= A fb A in ( t ) 2 sin ( err ( t )) (5–2) I ( t ) / G ( s )[ A in ( t ) sin ( in ( t )) A fb sin ( fb ( t ))]= A fb A in ( t ) 2 cos ( err ( t )). (5–3) Using Q ( t ) astheerrorsignalforthePLL,itislteredusingxed-poin tbit-shiftersand cascadedaccumulatorswhichareconstructedtoproducethe feed-backcontroller,H(s), 97

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givenby: H ( s )= 2 N 1 +2 N 2 f s s +2 N 3 f 2 s s 2 (5–4) The N 1,2,3 valuesarethebit-shiftingfactorsintheP-I-I 2 feed-backlter,where N 1 < N 2 < N 3 6 denesthegainandfrequency-zerosoftheP-I-I 2 lter.Thisltergenerates thephase-proportionalfeed-backfrequencysignal, f fb ( t ) ,witha60-bitprecision 7 .The user-denedoffsetfrequency, f oset ,isaddedtothisfeed-backsignal,integratedwitha xed-pointaccumulator 8 togeneratethefeed-backphase, fb ( t ) ,andusedtodrivethe NCO/LUTmentionedabove.Assumingthesystemis'locked'th efeedbacksignalshould tracktheinputsignalsuchthat,withinthebandwidthofthe controller: Q ( t ) 0 fb ( t ) in ( t ). (5–5) Sinceweareinterestedinthephaseuctuations,thefreque ncyfeedbacksignal, f fb ( t ) ,willbeintegratedtogeneratethein-bandphaseoutput, in band ( t ) .Usingthe feed-forward,G(s),andfeed-back,H(s),transferfunctio nsincombinationwithbasic controltheory[ 99 ],wecancalculatetheexpectedclosed-loopin-bandandout -of-band transferfunctionsbasedonthefeed-forwardandfeed-back transferfunctionsasdened above: fb ( s ) in ( s ) = A fb A in H ( s ) G ( s ) s f s + A in H ( s ) G ( s ) # 1 (5–6) Q ( s ) in ( s ) = A fb A in G ( s ) s f s + A in H ( s ) G ( s ) # 0. (5–7) 6 Formostdesigns, N 1 11, N 2 18, N 3 28 7 n fb =n Q + N 3 =42+28=70 bits,whicharethentruncatedto60bits. 8 T Accum ( s )= f s = s 98

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Thetime-changinginputamplitudecoupling,analyzedinCh apter 3.4.2.5 ,is assumedasbeingconstantforthisfeed-backanalysissucht hatamplitudeuctuations oftheinputwillhavelittleaffectonthetransferfunction .Fromthedenitionsin( 5–6 ) and( 5–7 ),wecaninterprettheseasthein-bandandout-of-bandfreq uencyinformation basedontheseclosed-looptransferfunctions: in band ( t )= fb ( t ) out of band ( t )= atan Q ( t ) I ( t ) = err ( t ). (5–8) Thus,wecanreconstructtheentireinputsignal'sphaseinf ormationbysimplyadding thesesignals: Out ( t )= in band ( t )+ out of band ( t ) in ( t ) (5–9) Wecanalsocalculatetheinputamplitude: A Out ( t )= p I ( t ) 2 + Q ( t ) 2 A in ( t ) (5–10) Theexpectedbandwidth,dependingontheamplitudeofthein putsignalandthe timingdelaysoftheFPGA,shouldbeatleastafewkHzsuchtha tintheLISAfrequency band,thephaseinformationoftheinputsignaliscompletel ycontainedinthein-band informationand,thus,wecanignoretheout-of-bandinform ation: f out of band ( t ) 0 (5–11) Out ( t )= fb ( t ) in ( t ) (5–12) Thiseliminatestheneedforthesinecomponentofthemixerm ultiplicationandreduces theconstraintsontheFPGAdesignifamplitudemeasurement sarenotneeded. Togeneratethephasemeterdata,the n fb =60 -bitfeedbacksignals, f fb ( t ) foreachofthefourADCinputsandPMoutputsarepackedintoa seriesof32-bit values,transferedthroughtheDSPsystemtothe4205,andco mmunicatedtoa 99

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data-storagecomputerthroughanEthernetconnection.Thi sprovidesuswitha frequencyquantizationprecisionof f = f s = 2 n fb (Figure 5-3 ). Inthefollowingchaptersthesamplingrateanddown-sampli ngfactorvarybased ontheexperimentbeingperformedandthecomputationaldem andsontheFPGA. Theprimaryclockratesusedare40MHz,62.5MHz,and100MHz 9 .Togetanidea ofthefrequencyprecision,taking f s =50.0 MHz ,weproducefrequencydatawitha quantizationprecisionof 50.0MHz = 2 60 =54pHz .Thisisunnecessarilyprecise,but sincetheFPGAandprocessorperformcomputationsbasedon3 2-bit'words,'andthe 32-bitprecision, 62.5MHz = 2 60 =14.5mHz ,isnotsensitiveenoughtomeettheLISA requirements,wearerequiredtousetwo'words'perdata-po intwhichallowsforup-to 64-bitxedpointfrequencyoutputs. 10 ThisisexploredmoreinChapter 5.2.2.1 5.2.2PhasemeterReadouts Nowthatwehavepresentedthedesignandcapabilitiesofthe PMcore,the60-bit, f Core = f s = R fb frequencyoutputdatacanbeusedforthreepurposes:(1)tor ecord LISA-likesciencedataofPDobservables,(2)togenerateth ephase/frequencydata forsimulatingtheinter-SCphase/frequencydelays,(3)fo rhigh-speeddigitalfeedback controlsystemssuchasarm-lockingandphase-locking.Mai ntainingtheLISAprecision throughoutthesystem,wekeepthe60-bitfrequencyprecisi onforallpossibleLISA-like usagesofthePMcore.Theread-outrate,ontheotherhand,va riesdependingon theapplication.LISAsciencemeasurementscallfor3-10Hz PDphasereadouts[ 29 ]. Simulationsoftheinter-SCdelaysrequirearelativelyhig hdata-ratetomaintainthe phasedata-rateforaccuratedata-interpolation(Chapter 6.2.2.1 )andtopreventaliasing intothemeasurementbandwhenelectronicallyreplicating theLISA-likelasereld 9 LISAisexpectedtousea50MHzultrastableoscillator(USO) clock[ 29 ]. 10 4bitsareusedtolabelthefrequencyvaluewithanindexingn umberbasedonthe associatedADCchannel. 100

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delays.PMreadoutsfordigitalcontrolsystemssuchasarmlockingusethemaximum alloweddata-ratetominimizethephasedelaysappliedbyth edigitalcontrollerwhich reducethetrackingband-width. Toachievethedesireddata-rateforLISA-sciencemeasurem entsandelectronic phasedelay(EPD)simulations,thePMfrequencyreadoutdat aisdown-sampledagain byCIClters.TheCICltersdown-samplethedatatoafast, f PLL 400kHz ,medium, f EPD 50kHz ,orslow, f LISA 10Hz ,readoutrate,whichvaryslightlydependingonthe down-samplingfactor,R,andtheclockrate, f s .Theimportantpointhereistoconsider thealiasingeffectsoftheCIClterandthequantizationno iseofthefrequencyread-out. 5.2.2.1PhaseQuantizationNoise Wecancalculatethephasemeters'quantizationphasenoise basedonthe bit-resolutionandreadoutrateofthefrequencyinformati onfromthesamearguments usedtoderive,( 3–24 ),andiswrittenexplicitlyas: Dig ( )= 1 f s 2 n fb p 6 f data Cycles p Hz (5–13) where n fb isthebitprecisionofthefrequencyfeedbackdataand f data istheCIC down-sampledoutputdatarate.Toseehowthiscomparestoth eLISAphasemeter precisionrequirements,thelow-frequencyphasequantiza tionlimit,scaledtoaclock rateof f s =50 MHz ,isplottedfortwodifferentread-outrates, 3 Hz and 10 Hz ,andthree differentbitprecisions,47,48,and49bits,inFigure 5-3 .Assumingwehavea 3 Hz data-rate,werequire49-bitsscaledtothe 50 MHz clock.Usinga 10 Hz data-ratewecan reducethistoa48-bitprecision.Wealsoplotthequantizat ionprecisionforthe 10 Hz 60-bitUF-LISAsciencephasemeter.Despitethatthe60-bit PMprecisionisfarbeyond theLISArequirements,wemaintainthishighbit-precision simplyasaresultoftheDSP system'scapabilities. 101

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -10 10 -8 10 -6 10 -4 Frequency (Hz)Cycles/rt(Hz) 47-bits, 10Hz 48-bits, 10Hz 49-bits, 10Hz 60-bits, 10Hz 47-bits, 3Hz 48-bits, 3Hz 49-bits, 3Hz LISA Req Figure5-3.TheoreticalPMreadoutdigitizationlimitatio ns:Thefrequencybit-precision anddataread-outratecoupletolimitthedigitization-lim itedPM measurementsensitivitythroughquantizationnoise( 5–13 ).Althoughthis specieshowmanybitsareneededinthePMtrackingloop,the data precisioncouldbereducedforsatellite-to-Earthtransmi ssiononceithas beenreadoutofthePMtrackingloopbyremovingthefrequenc yoffsetand retainingonlythedynamicbits. 5.2.2.2CICDownsamplingandAliasing TheCIClterisusedtodown-samplethedatabecauseofthere lativeeaseof programmingonanFPGAsinceitconsistsofcascadeddiffere ntiatorsandintegrators whichmaybeconstructedusingxed-pointaccumulatorsand subtracters.Inaddition, asshowninFigure 5-4 ,aliasedtermshaveaninnitesuppressionatallfrequenci es whicharealiasedtoDC.Despitethisfeature,wemustensure thattheCICaliased phase-noiseissuppressedbeyondtheLISArequirementinth eLISAmeasurement band,upto 1 Hz .Asanexample,wetakethepre-stabilizedlaserinputnoise of 280 Hz = p Hz andplotthemagnitudeofthepass-bandandrstaliasedfreq uency bandfora 10 Hz dataratedown-sampledwith N =2 and N =6 stageCICltersin Figure 5-4 .Wecanseethat,giventhisdata-rateanda6-stageCICdownsamplinglter, weobtainarst-aliasedbandsuppressionwhichmeettheLIS Arequirements.This isveriedbyanexperimentalmeasurementasshowninFigure 5-14 .Aniteimpulse 102

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response(FIR)lterwithaatpass-bandandsignicantsup pressioninthestop-band topreventaliasingcouldbeusedtoimprovetheperformance nearthesamplingrate andreducethedatarateto 3Hz butthisiscomputationallydemandingforaxed-point FPGA. 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)Cycles/rt(Hz) Pre-stabilized Laser (280 Hz/rt(Hz)) 2-Stage Pass-band 2-Stage 1st-aliased band 2-Stage Limit 4-Stage Pass-band 4-Stage 1st-aliased band 4-Stage Limit 6-Stage Pass-band 6-Stage 1st-aliased band 6-Stage Limit LISA phasemeter Requirement Figure5-4.PhasemeternoisecausedbyCIClteraliasing:T hedown-samplinglters usedtoreadoutthephasemeterdatamustensurethatthenois einthe aliasedfrequencybandsaresufcientlysuppressedbefore beingfoldedinto themeasurementband.Therstaliasedbandofthe N =2,4,6 stageCIC ltersusedtodown-samplethePMcoredataintheUF-phaseme terare plottedasafunctionoftheexpectedinputlasernoise.Usin ga 10 Hz data-rate,theCIC-lterdown-samplingltersrequireatl eastN=6-stagesto meettheLISArequirements. 5.2.3PhasemeterTest-Measurements Webeginthevericationandnoiseanalysisprocessbyperfo rmingsoftware simulationsinMATLAB-Simulinkusingthexed-pointXilin x-DSPtoolkitandhardware 103

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simulationswithavoltagecontrolledoscillator(VCO)tes tinput. 11 Inboththesoftware andhardwaresimulations,werecordthe60-bitfeedback, f fb ( t ) ,andthe30-bit quadratureerror, Q ( t ) ,andin-phase, I ( t ) ,signals. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 -0.8 -0.6 -0.4 -0.2 0 Phase (Radians)In-Band Phase 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 -0.2 -0.1 0 0.1 0.2 Phase (Radians)Out-of-Band Phase 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 -0.2 0 0.2 0.4 0.6 Time (sec)Phase (Radians)Corrected Phase Input Phase Measured Phase Figure5-5.SoftwarevericationofPMperformance:Theinband(top)andout-of-band (middle)phaseinformationisplottedforasimulatedlaser -likenoiseinputfor 2 ms .Thebottomcurveshowstheinputphasecomparedagainstthe summedin-bandandout-of-bandphaseinformation.Theinit ialtracking transientsareseenatthebeginningofthesimulationandth ecurvesmatch asexpectedoncethephasemeterhaslockedontotheinputsig nal. Thetime-seriesoutputofasoftwaresimulationisshowninF igure 5-5 whilethe spectralresultsofahardwareexperimentusingaVCOinputs ourceisshownin Figure 5-6 .Fromthehardwareexperimentwecanseethattheout-of-ban dphase erroriswellbelowtheLISArequirement,justifying( 5–11 ).The'difference'termsrefer tothesubtractionoftwodifferentADCs,butwhichsampleth esameinputsourcebeing 11 OurVCOshavesimilarnoisecharacteristicsasthepre-stab ilizedlaserbeatnotes andisusedasatestinputformanyvericationmeasurements asshowninFigure 6-2 104

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electronicallysplit,effectivelyshowingthecombinedun -commonADCandphasemeter phasenoiselimitation.Thiswillbediscussedextensively inthefollowingchapters.At thispoint,itsimplyveriesthatbothADC'sandphasemeter saremeasuringthesame signalstoa0.1 cycle = p Hz accuracy,atleastinthehighfrequencyrange. 10 0 10 1 10 2 10 3 10 4 10 5 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 Frequency (Hz)Cycles/sqrt(Hz)Linear Spectral Density In-band Phase Out-of-Band Phase Total Phase In-band Difference Out-of-band Difference 1pm LISA Requirement Figure5-6.HardwarevericationofPMperformance:Thehig h-frequencynoise spectrumofthehardwaretestedphasemeterusingaVCOinput sourceis plotted.Thein-bandphase(bluecurve)matchestheVCOinpu tnoisetoa levelof 0.1 cycles = p Hz (cyancurve)whenperformingadifferentialADC measurement(Figure. 5-9 ).Theout-of-banderror(greencurve)immediately dropsbelowtheLISArequirementandcontinuestotrackthei nputnoisetoa betterprecisionatlowerfrequenciesduetotheP-I-I 2 feedbacktransfer function(Figure. 5-7 ).Thus,includingtheout-of-banderrorinthemeasured phasedoeslittletoimprovethephaseprecision(purplecur ve). Dividingthespectraofthefrequencyfeedbacksignal, ~ f fb ( ) ,bythequadrature errorspectra, ~ Q ( ) ,asshowninFigure 5-7 ,weobtaintheexpectedfeedbacktransfer functionbasedontheprogrammedFPGAcontrollerdesign.Mo reonthedesignand vericationaspectsofthephasemetercanbefoundbyrefere ncingthegroundwork experimentsperformedbyIraThorpe[ 41 ]. 105

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10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)|H(s)| Theory Simulation Hardware Test Figure5-7.VericationofthePMfeedbacktransferfunctio n:Dividingthespectraofthe feedbackfrequencysignalbytheout-of-banderrorsignal, weobtainthe expectedfeedbacktransferfunction( 5–6 )forboththesoftwareand hardwarevericationtests. 5.2.4PhasemeterNoiseModel Takingintoconsiderationphasemeter,ADC,andclocknoise sources,wecanwrite thesampledphasemetersignalas: PM i = in + PM + ADC + f in f s Clk (5–14) where PM = CIC o + Quant i (5–15) Clk = CLK o + CLK i (5–16) ADC = ADC o + ADC i (5–17) 106

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sothatwemaydifferentiatebetweennoisetermswhichareco mmonbetweenall thechannels, x o ,andtermswhichareindependentlyappliedtothe'i-th'cha nnel, x i .TheCICnoiseislabeledwithan'o'sinceitcorrelateswith theinputnoise.The quantizationnoise,ontheotherhand,isappliedtoeachcha nnelindependently.In Chapter 5.3 ,wewillfurtherinvestigatetheADCnoiseterms.Todisting uishbetweenthe clocknoiseandADCnoiseterms,wewilldenetheclocknoise sourcesasanysignal whichscalesproportionatelywiththeinputfrequency,suc hastimingjitter,whileADC noisesourcesaredenedasabsolute,frequency-independe ntadditivenoisesignal. Inanattempttomodelthesedifferentnoisesources,adiagr amofwheretheyare introducedinthemeasurementprocessisshowninFigure 5-8 5.2.5DifferentialandEntangledMeasurements Todistinguishbetweentheseterms,wetakethreetypesofme asurements,as showninFigure 5-9 ,witheachattemptingtoprobeadifferentnoisesourceasou tlinedin thefollowingsections.5.2.5.1DigitallySplitDifferentialNoise Therstmeasurementusesa1MHzVCOtestinputwhichissampl edatarate of f s =62.5MHzwithasingleADC.TheADCsampleddataispassedtot wodifferent phasemetercoreswithtwodifferentPLLoffsetfrequencies .Thisresultsintwo data-streamswhichareonlylimitedbythephasemetercore' smeasurementand digitizationprecision, Quant i ,in( 5–14 ).Themeasurementsaretakenatfourdifferent ratesthenplottedtogethertospanthe 100 Hz to 10 kHz frequencyrangeasshownin Figure 5-10 .The'quantizationnoise'limitisatthe 100 pcycle = p Hz levelforfrequencies above 10 mHz andequalsthePMphasequantizationlimitatfrequenciesbe low 10 mHz Usingtheamplitude-phasequantizationnoiselevelasgive nby( 3–24 )andconsidering the f in loop =62.5 = 128 MHz in-loopdatarate,wecansolvefortheADC-noise-free 107

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Figure5-8.Phasemeternoisemodel:AmodeloftheADCinputs andclockdistribution isdepicted.Commonandindependentnoisetermsareaddedto theADC inputandclockinputsignalsintorepresentthe Clk and ADC termsin 5–14 .TheADCitselfhasagroundreferencewhichproducesanaddi tive phasenoiseandindependentvoltagereferencesfortheADCcomparators whichrepresentthe V Gnd ( t ) and G ( t ) ( 3–25 )termsrespectivelyand discussedinChapter 3.4.2.3 .Thesetermsarealsorepresentativeofthe A O ( t ) and A ( t ) ( 3–38 )termsinChapter 3.4.2.5 108

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Figure5-9.ExperimentalmodelsofADCnoiseestimationmea surements:The quantizationnoiseexperiment(left)maintainsthecommon ADCnoiseterms butdistinguishesbetweenindependentPMcorequantizatio nnoiseterms. Thedifferentialnoiseexperiment(middle)distinguishes between independentADCnoisetermsbutcancelscommonADCnoiseter msas explainedinChapter 5.2.5.2 .Theentangledphasemeasurement(right) providesameasureofthecommon-modeADCnoiseterms(Chapter 5.2.5.3 ). effectivein-loopphasequantization, n fb : x ADCAmp Quant ( )= 2 1 n fb p 6 f in loop 100 pcycles p Hz (5–18) resultinginaneffectivephasemeteramplitudequantizati onprecisionof n fb =23.5 -bits. Again,thesenoisesources,beingwellbelowtheLISAmeasur ementrequirements, arenotofconcerntous.Although,thedigitizationprecisi oncalculationscanbeusefulin determiningtheamountofdatathatmustbetransmittedtoEa rthfordata-processingof thesciencePMread-outs.5.2.5.2ElectronicallySplitDifferentialNoise Thenextmeasurementsaretakentoprobetheun-commonclock ,ADC,and phasemeternoisesources.Splittingademodulated1MHzVCO outputusingan electronicradio-frequency(RF)transformersplitter,we measurethesignalphase usingtwodifferentADCs,sampledwiththesameclocksource .Inthiscase,the commonnoisesources, CIC o ADC o ,and Clk o cancelfromthemeasurement whiletheun-commonsources, ADC i and Clk i ,donot.Weexpectthe Clk i tobe 109

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dominatedbytiming-jittertermswhile ADC i mightintroducephaseandamplitude noisetermsoutlinedinChapter 3.4.2.5 .Acompletespectrumofthe1MHzinput andthetwo-channeldifferentialoutputisplottedinFigur e 5-10 .Themeasurement precisionislimitedatlowfrequencies,below10Hz,byacom binationoftiming-jitterand ADC/transformerdispersion.Theprecisionislimitedathi ghfrequencies,above10Hz, bythecouplingoftheamplitude-samplingnoisetermsdescr ibedinChapter 3.4.2.3 Thisisveriedbyvaryingthesignalfrequencyandamplitud e,thencomparingthePM measurementsensitivitywiththechangesinthemeasuremen tcharacteristics. Figure5-10.PM/ADCQuantizationanddifferentialnoise:H ereweplottheresultsofthe quantizationanddifferentialnoisemeasurementsmodeled inFigure 5-9 from 100 Hz to 10 kHz .Thequantizationnoiseshowsanin-loopwhite n fb =23.5 -bitquantizationphasenoiseforfrequenciesabove 10 mHz the expected60-bitfrequencyquantizationPMread-outlimita tionfor frequenciesabove 10 mHz .Thedifferentialmeasurementislimitedbythe n V =13.78 -biteffectiveamplitudequantizationnoiselevelforfreq uencies above 10 Hz andatimingjitterlimitednoisegivenby( 5–19 )forfrequencies below 10 Hz 110

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Inordertoprobethetimingjitterterms,wetakelong-term, low-frequencymeasurementsofinputbeatnotefrequenciesrangingfrom1.0MHzto7 9.5MHz.Themeasurementat 79.5 MHzisaliasedintothemeasurement-bandbythe f s =60MHzclock, producingameasurableoscillationat19.5MHz.Theresults ,Figure 5-11 ,showthat themeasuredphasenoisedecreasesasthemeasurementfrequ encyisdecreased from79.5MHz,to12.0MHz,to8.0MHz.Basedonthenoiselevel andscalingofthese signals,weestimateatimingjitternoiseof, t Jit ( )= 40 p f fs p Hz (5–19) Conrmingthisestimatenumericallyweobtainaphasenoise of: Jit ( )= f in t (0.1Hz) (5–20) =(79.5 10 6 ) t (0.1Hz)= 31.8 p f Cycles p Hz =(12.0 10 6 ) t (0.1Hz)= 4.8 p f Cycles p Hz =(8.00 10 6 ) t (0.1Hz)= 3.2 p f Cycles p Hz =(4.00 10 6 ) t (0.1Hz)= 1.6 p f Cycles p Hz allofwhichmatchwiththemeasuredphasenoiseforfrequenc ieslargerthan8MHz. Unexpectedly,lowerfrequencies(4.0,2.0,and1.0MHz)res ultinahigherphasenoise precisionandareducedmeasurementsensitivity.Theincre asednoiseattheselower frequenciesiscausedbytemperaturecorrelatedphasedisp ersionintroducedbythe RFtransformersusedtoACcoupletheADCsignalinput;seeCh apter 5.4.1 fora descriptionoftheexperimentsandmethodsusedtocorrectf orthisnoise. Inanattempttoevaluate( 3–25 ),wevarytheinputamplitudeofademodulated 10MHzVCOinput.Usingtwopeak-to-peakinputamplitudes,2 00mVand600mV,we observeafactorof3improvementinthemeasurementsensiti vityathighfrequencies whenthesignalisnotlimitedbytimingjitter.Thetracking bandwidthforthisPMdesign 111

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 Frequency (Hz)Cycles/sqrt(Hz) ADC Noise 79.5MHz ADC Noise 12MHz ADC Noise 8MHz ADC Noise 4MHz ADC Noise 2MHz ADC Noise 1MHz 1 pm Level Figure5-11.ADCtimingjitternoiselimitations:Thephase noiseduetothetimingjitter scalesinverselywiththe f in MHz inputfrequency( 3–27 )andmatchesthe theoreticallycalculatedjittervaluesfor f in > 8, MHz for t Jit ( ) ( 5–19 ).The phasemeasurementforinputfrequencieswith f in < 8 MHz arelimitedby temperaturedependentdispersionduetotheRFtransformer s (Chapter 5.4.1 ). alsoincreasesfrom 3 kHz to 10 kHz .Theamplitudenoisecouplingallowsusto evaluatethe G ( ) V Bias factorin( 3–25 ).Basedonthemaximumpeak-to-peakinput amplitudeof2000mV,wecanestimatethisamplitude-to-pha senoisecouplingas: Amp ( )= 2000 mV V Pk Pk : in ( mV ) 7.5 nCycles p Hz (5–21) Notethattheamplitudenoiseisnotaresultofaliasedampli tudenoisefrom 2 (Chapter 3.4.2.5 )sincethesewouldnotscalewiththesignalpower.Wehaveal so accountedfortheout-of-bandphaseterms( tan 1 ( Q = I ) )inthesemeasurements,which indicatesthatthisisnotaresultofvariationsinthesigna l'samplitudeasdescribedby Chapter 3.4.2.5 .Usingtheinverseargumentusedtoderive( 3–26 ),wecancalculatethe 112

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effectivequantizationprecisionof n V =13.78 whichisnearlyequaltothefull14-bit ADCprecision. Wewillhaveanindependentmeasureofthisnoisesourceusin gadifferent experimentalsetupinChapter 5.3 10 0 10 1 10 2 10 3 10 4 10 5 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Frequency (Hz)Cycles/sqrt(Hz) Input ADC Noise Only (High) ADC Noise Only (Low) Expected Amplitude Noise (200 mV Pk-Pk) Expected Amplitude Noise (600 mV Pk-Pk) Expected Timing Jitter Figure5-12.ADCamplitudenoiselimitations:Theamplitud enoisescalesinverselywith theinputamplitudeandindicatesthatthenoiseoorisduet otheADC quantizationbit-limitation.Usingthisassumption,wen daneffective numberofbits n V =13.78 whichisnearlyequaltothespecied14-bit ADCprecision. Atthispoint,weclaimthemeasurementislimitedbyacombin ationoftiming jitterandRFtransformerphasedispersioninthelow-frequ encyrangeandlimitedby amplitudenoiseinthehigh-frequencyrange.SincetheADC' s,althoughindependent integratedcircuits,areonthesameprintedcircuitboardt heymayhavesomecommon temperaturedependentphasedelayorcommonamplitude-vol tagereferencenoise. Inordertodifferentiatebetweenthesecommonterms,wetak eanentangledphase measurementinvolving3differentsignalswhichcombineto canceltheinputandclock noisesourcesbutleavethecommonADCandphasemeternoises ources. 113

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5.2.5.3EntangledPhaseNoise Finally,weperformwhatiscommonlyknownasanentangledph asemeasurement. ThismeasurementinvolvesmixingthreeindependentVCOsig nals( VCO 1 ( t ) VCO 2 ( t ) VCO 3 ( t ) )withfrequencies( f VCO 1 f VCO 1 f VCO 3 )andmeasuringthedifferentialnoise betweeneachoftheVCOswiththreeADCsandPMstoobtainthre esignals( A ( t ) B ( t ) C ( t ) )givenby: A ( t )= VCO 1 ( t ) VCO 2 ( t )+ f VCO 1 f VCO 2 f Clk Clk ( t )+ ADC A ( t )+ PM A ( t ) (5–22) B ( t )= VCO 2 ( t ) VCO 3 ( t )+ f VCO 2 f VCO 3 f Clk Clk ( t )+ ADC B ( t )+ PM B ( t ) (5–23) C ( t )= VCO 1 ( t ) VCO 3 ( t )+ f VCO 1 f VCO 3 f Clk Clk ( t )+ ADC C ( t )+ PM C ( t ) (5–24) Takingalinearcombinationofthedifferentialnoisemeasu rements,namely: Ent ( t )= A ( t )+ B ( t ) C ( t ) (5–25) wecanceltheVCOandclocknoisetermsinthenalcombinatio nbutareleftwithnoise duetotheindividualphasemeterandADCs. TheresultsplottedinFigure 5-13 ,showuptoafactorof10increasednoiseatlow frequenciesduetoacommonADCnoisewhichcancelsinthe2-c hanneldifferential measurement.Theadditionallow-frequencyerrorincompar isontothedifferential measurementresultsfromcommon-modetemperaturedepende ntphasedispersion oftheRFtransformers(Chapter 5.4.1 ).Wealsonoticeanincreasednoiselevelnear thesamplingratebecauseofthealiasinglimitationsofthe CICdown-samplinglter describedinChapter 5.2.2.2 .VerifyingtheCIClteraliasingerror,weperformthesame measurementwitha2-stage,4-stage,and6-stage,CICdownsamplinglters.Aswe canseeinFigure 5-14 ,the2-stageCICdown-samplinglterwillnotmeettheLISA requirementsatfrequenciesabove10mHz. 114

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/sqrt(Hz) LISA Requirement 1pm LISA Requirement 10pm Input VCO Electronically Split Entangled Phase Figure5-13.Entangledphasemeasurementresults:Theenta ngledphase measurementsshowanincreasednoiseatlowfrequenciesbyu ptoa factorof10duetotheindependentRFtransformerphasedisp ersionterms (Chapter 5.4.1 ).Themeasurementcouplesinaliasednoisenearthe samplingrateduetotheCICdown-samplinglters(Chapter 5.2.2.2 ). 5.2.6ApplicationsinLISAandLIGO AlthoughtheLISAphasemeterdesignwasmotivatedbythenee dtomeasure thephaseofLISA'sheterodynebeatnotestoa1 Cycleaccuracy,thephasemeter hasapplicationsinmanyheterodyneandhomodyneinterfero metryschemes.LISA willusethePMtomeasurenotonlytheLISAobservables,buta lsotheclock-noise sidebandsandinter-SCdatacommunicationsignals.ThePMf requencyreadouts arerequiredtoperformarm-lockingcontrolsandisalsoapp licableinMach-Zehnder laserpre-stabilizationanddigitalheterodynephase-loc king.Thehigh-frequencyPM sensitivitycanalsoevaluatethermalcoatingnoiseforLIG O[ 100 ]. 5.3ADCNoiseEstimation WeemployanovelADCphase-noisemeasurementtechniquetoo btainabetter understandingofthenoisecouplingaddedbytheADCs.Using theclockastheinput 115

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10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/sqrt(Hz) Input Noise 2-Stage CIC 4-Stage CIC 6-Stage CIC LISA Requirement Figure5-14.N-stageCIClteraliasingandentangledphase measurements:The entangledphasemeasurementsareperformedusingaPM-core which down-samplesthemeasuredPMdatawitha N =2,4,6 stageCIClter.As describedinChapter 5.2.2.2 andshowninFigure 5-4 ,theCIClter down-samplinglteraliasesnoiseintothemeasurementban d,limitingthe phaseprecisionofthemeasuredinput. sourceasshowninFigure 5-15 ,weadjustthelengthofthecablefeedingtheinput signal, L ,tochangethephaserelationshipbetweentheinputsignala ndtheclock source. Whenthereisanoffsetphaseof0betweentheinputandtheclo cktrigger,we sampletheclockitselfatit'szerocrossing.Inthiscase,w earesensitivetothelinear combinationoftheground-reference, V Gnd ( t ) in( 3–25 ),timingjitternoise, ~ t Jit ( ) in ( 3–27 ),andotherun-accounted-forwhite-noiseterms, Thermal ( t ) .Assumingthesignal hasaslopeof =2 f Clk V Clk ,measuredin [ V = s ] with V Clk beingtheclockamplitude,we canwritethesampledoutputofthismeasurementas: V =0 ( t )= V Gnd ( t )+ ( f Clk ) t Jit ( t )+ Thermal ( t ). (5–26) 116

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Figure5-15.ModelforestimatingADCphaseandamplitudeno ise:Splittingaclock sourceandadjustingthephaserelationshipbetweentheclo ckandthe ADCinput,weeffectivelysampletheclockwithitselfcance lingthe commonclocknoise.Whenthephaserelationshipbetweenthe clockand theinput, =0 ,wesampleatthezero-crossingwhichcouplesintiming jittertermswithaproportionalityfactorgivenbythe'slo pe', .Whenthe phaserelationshipbetweentheclockandtheinput, = = 2 ,wesample thepeakoftheclockoscillation;differentialmeasuremen tswillcancelthe commonclockamplitudenoiseandbelimitedbytheADCamplit udenoise factor, G ( t ) ( 3–25 ). 117

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Whenthereisanoffsetphaseof = 2 betweentheinputandclocktrigger, wesamplethesignal'speak.Inthiscase,wearesensitiveto theADCamplitude uctuations,the G ( t ) V Bias o termin( 3–25 ),theclockamplitudeuctuations, V Clk ( t ) ,and thegroundoffsetterm, V Gnd ( t ) ,asinthezero-crossingmeasurement.Assumingthe clockoscillatorsourcehasanamplitudeof V Clk ( t ) ,wecannowwritethesampledoutput as: V = = 2 ( t )= V Gnd ( t )+ V Clk ( t ) G ( t ) V Bias o (5–27) Splittingtheclockinputsignalandperformingthesamepre scriptionforasecond ADC,wecancomparethetwoADCmeasurementstocancelcommon noisetermssuch astheclock'samplitudeuctuations, V Clk ( t ) ,orcommon V Gnd ( t ) terms. Thefollowingzero-crossingandpeak-samplingmeasuremen tsusea 62.5 MHz clockinputwhichhasaamplitudeof V Clk =0.45 V .Thephasemetercoreisreplaced withCICdown-samplinglterstoproducethesampledvoltag eADCoutputata 14.9 Hz down-sampledrate.Atthesametime,wemeasurethetemperat ureoftheADC'stosee ifthereisanythermalnoisecorrelationsinthemeasuremen ts. Theresultsofthezero-crossingmeasurement,asshowninth eleftcolumnof Figure 5-16 ,showastrongcorrelationbetweenthetwoindividualADCme asurements (redandbluecurveintop-left,pinkinbottom-left)whichc ancelsinthedifference betweenthesignals(blackcurveintop-left).Thisindivid ualADCnoiseisstrongly temperaturecorrelatedatfrequenciesatandbelow1mHzand indicatescommonmode phasedispersion.Subtractingthesignalsweobtainavolta genoiseestimatedby(green curveintop-leftofFigureg:ADCPhaseAmpNoise): ~ V =0 ( )=5 10 6 r 1 Hz f Volts p Hz (5–28) 118

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Figure5-16.CommonsourceADCphaseandamplitudenoiseres ults:The zero-crossingmeasurements(leftcolumn)showastrongADC noise correlationwhich,whendifferentialmeasurementsareper formed,cancelto revealthethermalandtimingjitternoisesources(top-lef t)asexplicitly showninFigure 5-17 .TheindividualADCnoisesourcescorrelatestrongly withthetemperatureatfrequenciesbelow 2 mHz indicatingthecouplingof temperaturedependentRFdispersion.Theindividualpeaksampling measurements(rightcolumn)alsoshowastronglow-frequen cycorrelation withthetemperature.Thelossofcorrelationbetweenthein dividualADC samplesbelow 2 mHz (bottom-left),indicatesthatthesamplingprecisionis dominatedbyvoltagebiasreferencenoise, V Bias o G ( t ) in( 3–25 ). 119

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Scalingthisby ,weobtainatimingjitterof: ~ t ( )= ~ V =0 ( ) (5–29) =5 10 6 r 1 Hz f Volts p Hz 1 2 f Clk V Clk s Volt =5 10 6 r 1 Hz f Volts p Hz 1 2 62.5 MHz 0.45 Volts 28 r 1 Hz f fs p Hz (5–30) Thisissmaller,yet,comparabletothetimingjittermeasur edinChapter 5.2.5.2 .Ifthisis actuallythetimingjittertermitshouldscalewithrespect totheclockfrequencythrough the dependence.Toprobethisdependence,wetakethesamemeasu rementat 50MHz,75MHz,and100MHz.Theresults(Figure 5-17 ),showthesamenoiselevel forthe50MHzand75MHzsignalsasweobservehere.Thisindic atesthatthenoise sourcewearemeasuringisnottimingjitter,butrather,aph aseerrorwhichislikelydue totheinternalelectronicthermalnoisebackground, Thermal ( t ) .The100MHzsignal showsaslightincreaseinthephasenoiseerrorindicatingt hatwehavenowmade largeenoughtodominateoverthethermalbackgroundtermsi n( 5–26 ).Giventhis assumption,were-evaluatethejitterlevel: ~ t ( )= ~ V =0 ( ) (5–31) =9 10 6 r 1 Hz f Volts p Hz 1 2 f Clk V Clk s Volt =9 10 6 r 1 Hz f Volts p Hz 1 2 100.0 MHz 0.45 Volts 32 r 1 Hz f fs p Hz (5–32) whichisclosertotheestimategivenby( 5–19 ).Scalingthis100MHzzero-crossing, voltagenoisemeasurementtoa10MHzinputsignalandcompar ingitagainsta10MHz measurement,theresultsmatchthemeasuredjitterasshown inFigure 5-18 120

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Figure5-17.Zero-crossingtimingjitterestimate:Thedif ferentialzero-crossingtiming jittermeasurementsprovideuswiththeabilitytodirectly probetheADC timingjitterandit'sfrequencydependence.Theinputfreq uency dependenceof allowsustodifferentiatebetween Thermal ( t ) and ( f Clk ) t Jit ( t ) in( 5–26 ).The50and75MHzmeasurementsshowthesame noiselevelandaredominatedby Thermal ( t ) .The100MHzmeasurement increases ( f Clk ) toallowustoprobe t Jit ( t ) ( 5–32 ). Movingontothepeak-samplingmeasurementsasshowninther ightcolumnof Figure 5-16 ,weobservesomecommonamplitudenoiseatfrequenciesbelo w1mHz andatfrequenciesabove10mHz.Theamplitudenoiseatthese frequenciesislikely duetocommonmodeuctuationsintheinputsignalorinthevo ltagebiasreference, G ( t ) V bias ,although,withoutanindependentmeasureoftheinputampl itude,thereis nowaytodifferentiatebetweentheseterms.Thedifferenti alADCnoiseshowssome 121

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -8 10 -7 10 -6 10 -5 10 -4 Frequency (Hz)Cycles/rt(Hz) Zero-Crossing:50 MHz Zero-Crossing:75 MHz Zero-Crossing:100 MHz Differential ADC, 10.0 MHz 10MHz*40 fs/sqrt(f) Figure5-18.ThetimingjitterestimateobtainedfromFigur e 5-17 isscaledand compareddirectlytoanexperimentalPMmeasurementofa10M Hzinput signal.Theresultsindicatethatwehaveobtainedanaccura teestimateof theADCtimingjittergivenby 30 40 = p ffs = p Hz whichwilllimitthe LISAmeasurementprecisionsuchthatthePMsandADCsdonotm eetthe 1 cycle = p Hz measurementrequirementbyuptoanorderofmagnitude nearthe 3 mHz cornerfrequency. temperaturecorrelationatfrequenciesbetween2and20mHz .Thisismostprobably duetotemperaturedependentnoiseinthevoltagebias, G ( t ) V bias ,sinceitisnotlikely thattheamplitudeoftheclocksourcecorrelateswiththete mperatureoftheADCs.That said,thesenoisesourcesonthisabsolutevoltagescaleare smallerthanthetimingjitter sourcesandarescaledbytheinverseamplitudebeforecoupl ingintothemeasured phaseasexplainedinChapter 3.4.2.5 .Although,becauseofthe'white'natureofthe voltagenoiseinthis1-20mHzregionof 5 10 5 Volts = p Hz ,itindicatesthatwe shouldbeabouttocalculatetheeffectivenumberofbitsint hislow-frequencyrange. Assuming( 3–24 )stillholds,wecalculatetheeffectivenumberofbitsfrom j V Bias j j V In j 2 1 n p 6 f s = 5 10 5 V Full Scale Volts p Hz (5–33) 0.5Volts 0.225Volts 2 1 n p 6 14.9Hz = 5 10 5 1.0 Volts Volts p Hz (5–34) Thesolutiongivesabit-resolutionof n=13.2 and,thus,itseemsthatthermal uctuationshavedecreasedtheeffectivenumberofbitsint hisfrequencyrange.These 122

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low-frequencyamplitude-phaseuctuationtermswillbeco meapparentoncethetiming jittertermshavebeenextractedinthenextchapter. Nowthatweknowthattimingjitteristhelimitingnoisesour ceofthesemeasurements,weattempttoestimateandcorrectfortheseterms.Al so,wemustfurther investigateandcorrectnon-jitterlimitationsofthephas emeterinthe1-8MHzfrequency range,ofwhich,thesezero-crossingandpeakmeasurements donotprobe.Thatsaid, thecorrelationbetweentheADCinputphaseandthetemperat ureatlowfrequencies inFigure 5-16 (bottomleft)indicateatemperaturedependentphaserespo nseofthe ADCs.Iftheslopeofthisphase-responseislargeenough,it mightover-comeand dominatethetimingjitterterms. 5.4TimingJitterExtraction Timingjittercorrectionisbasedofftheconceptthat,alth oughthetimingjitterterms areADCindependent,asecondmodulation,addedtotheADCin putsignal,canbe usedusedasareferencetomeasureandcorrectforthetiming jitter.Asshownin Figure 5-19 ,asecondoscillatorisaddedusinganRFtransformer-split ter.Thesummed outputissplitagainintotwodifferentADCs.Thephaseofth einputandtherangingtone oneachADCismeasuredwithit'sownphasemeter.Measuringt heinputsignals,we obtainPMoutputswhichtaketheform: In : ADC 1 ( t )= In ( t )+ f In f Clk ( Clk ( t )+ f Clk t ADC 1 ) (5–35) In : ADC 2 ( t )= In ( t )+ f In f Clk ( Clk ( t )+ f Clk t ADC 2 ) (5–36) Repeatingthisfortheaddedreference-tonesignals,weobt ain: Tone : ADC 1 ( t )= Tone ( t )+ f Tone f Clk ( Clk ( t )+ f Clk t ADC 1 ) (5–37) Tone : ADC 2 ( t )= Tone ( t )+ f Tone f Clk ( Clk ( t )+ f Clk t ADC 2 ) (5–38) Usingthethese,weareabletoprobethedifferentialtiming jittertermstocorrectfor theinputsignal'stimingjitter.Notethatahigherfrequen cyreferencetoneisdesirable 123

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sincethetimingjitterscaleswiththisfrequency.Infact, thejittertonecanbeplacedata frequencyabovethesamplingrate;measuringthealiasedos cillationresultsinstronger jittercouplingsasshowninFigure 5-11 Subtractingthetwoinputsignalsandthetworeferencesign als,weareableto evaluatethedifferentialtimingjitterateachfrequency, f in and f Tone : ( t )= f In [ t ADC 1 t ADC 2 ] (5–39) Tone ( t )= f Tone [ t ADC 1 t ADC 2 ] (5–40) Rescalingandsubtractingthesesignalswecancelthediffe rentialADCtimingjitter andobtainacalibratedresultwhichcancelsallthenoiseso urceswehaveconsidered: Calib = ( t )+ f In f Tone Tone ( t ) (5–41) Notethatthisdifferentialtimingjittercorrectionmetho dwillnotgiveameasureof theabsolute,individualtimingjittertermsfromwhichwec anobtainatruemeasurement of In ( t ) .Afewpossiblemethodsofperformingthisabsolutejitterc orrectionare discussedinChapter 5.4.2 Aseriesofinitialtestsperformedwellwhentherangington eandinputsignal wererelativelyequalinfrequencyasshownintherightcolu mnofFigure 5-20 nearlyobtainingthe 1 cycle = p Hz requirement.Thetemperaturemeasurementsand correlationsfortheADCnoiseestimationofChapter 5.3 arealsousedheretoseeifthe measuredphasecorrelateswiththeADCtemperature.Thetop -rightplotofFigure 5-20 showednocorrelationwitheitherinputsignalorthediffer entialtimingjitterindicating thattemperatureeffectsplaylittleroleinthenoisecoupl ingatthesehighfrequencies. Whenthetimingjittercorrectionisattemptedusingalow-f requency(4MHz)inputsignal andahigh-frequency(15MHz)tone,weobtainnonoticeablei mprovementinthephase measurementprecision. 124

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Figure5-19.Timingjittercalibrationmodel:Hereweshowa modelofhowtheinput signal,inthiscase,aVCO,isaddedtoacalibrationsignal[oscillator/functiongenerator(FG)]withaninverselydr ivensplitter.Thetwo summedoscillationsaresplitandsampledwithtwoindepend entADCsbut withthesameclocksource.ThePMcoresmeasurethephaseoft he VCO-inputandcalibrationsignalswhichareeachcontainth etimingjitterof theirrespectiveADCsscaledbythemagnitudeofthemeasure dinput frequencyasdescribedinChapter 5.4 Correlatingeachofthesewiththemeasuredtemperature,we seeastrong correlationbetweenthelow-frequencyinputandthemeasur edtemperatureasshown inthetop-leftplotofFigure 5-20 .Thisindicatesthatthereissomephaselosswhich changesasafunctionoftemperatureattheselowfrequencie s.Thislow-input-frequency limithadalreadybeenindicatedtosomeextentbytheresult splottedinFigure 5-11 .To seehowstrongthetemperaturecouplingeffectis,weperfor mthetimingjitterextraction forthreedifferentfrequenciesacrossthe2-20MHzLISAinp ut-frequencyrange(2MHz, 8MHz,and16MHz)andattempttocalibratethemagainsta(19M Hz)oscillatortone. TheresultsareshowninthetopplotofFigure 5-22 5.4.1PhaseDispersionMitigation AdetailedlookintothePentekdesignindicatesthattheRFt ransformersused toACcoupletheinputsignal,theADT4-5WT,haveastronginp utreturnlossand, 125

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Figure5-20.Temperaturedependentphasedispersion:Plac ingtheinputoscillationat 4MHzandthejittercalibrationinputat15MHz,thetimingji tterextractionis limitedbyjitter-uncorrelatednoiseintheinputoscillat ion.Thedifferential phasenoiseofthis4MHzinputsignalcorrelatesstronglywi thtemperature oftheRFtransformerswhichintroducedispersiontermstha tarenot accountedforinthetimingjitterextraction.Placingthei nputoscillationand jittercalibrationtoneclosetoeach-other,14and15MHzre spectively,we reducethetemperaturedependentdispersionandthetiming jitter extractionworksasexplainedinChapter 5.4 thus,asteepphasedependenceattheselowfrequenciesassh ownintheleftcolumn ofFigure 5-21 .Temperaturevariationschangethetransferfunctionresp onseand phaselossfortheseRFtransformerswhichcoupleintothelo w-frequencymeasured phase(Figure 5-11 ).Mini-circuits,theintegrated-circuitproductioncomp anywhich makestheADT4-5WTalsoproducesanothermodel,theADT1-6T ,whichhasa signicantlymoreconstantinputreturnloss,andthus,asm allerphasechangein theLISA-frequencyrange.Aftertherisksandcostswherewe ighed,wedecidedto 126

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replacetheRFtransformersonthePentekboardstobettersu itthe 2 20 MHz LISA inputfrequencyrangeofinterest. Figure5-21.ThestockADT4-5WThasabandwidthbetween300k Hzand500MHz. ThereplacementADT1-6T'sbandwidthisfrom30kHzto125MHz providingabettersuitedACcouplertomakePMmeasurements at LISA-likefrequencies.[ 101 102 ] Thankfully,changinginRFtransformersdidnotdamagetheP entek'selectronics andimprovedthelow-frequencyperformanceasshownbythei mprovedtimingjitter resultinFigure 5-22 .Atthispoint,wecanseethatprovidedwiththereferenceto ne's measureofthetimingjitter,weshouldbeabletocorrectfor thetimingjitterofthe measuredinputphaseacrosstheentire 2 20 MHz LISAinput-frequencyrange. 5.4.2AbsoluteTimingJitterExtraction Performingabsolutetimingjitterextractionissomewhatm oredifcultsinceweneed thesamereferencephasenoise, Tone ( t ) ,attwodifferentfrequencies, f Tone 1 and f Tone 2 127

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Figure5-22.ReplacementRFtransformerdispersionmitiga tion:Repeatingthetiming jitterextractiontestswiththeold(ADT4-5WT)andnew(ADT 1-6T)RF transformers,wendtheRFtransformerphasedispersionha sbeen correctedandthetimingjitterextractionhasbeensignic antlyimproved. ThejitterextractionresultsusingthenewADT1-6TRFtrans formersmeet theLISAmeasurementrequirementof 1 cycle = p Hz forallfrequenciesin the2-20MHzrange.Oncethetimingjitterhasbeenremoved,t hephase precisionislimitedbythesignal-amplitudedependentqua ntizationnoise oor;Chapter 5-12 for f > 20 mHz ,( 5–33 )for f < 20 mHz ,overthe frequenciesforwhichtheRFtransformerdispersionnolong erlimitsthe results. suchthatwecanformthecombination: In Calib ( t )= In : ADC 1 ( t ) f In f Tone 1 f Tone 2 [ Tone 1: ADC 1 ( t ) Tone 2: ADC 1 ( t ) ] (5–42) = In ( t )+ f In f Clk ( Clk ( t )+ f Clk t ADC 1 ) f In f Tone 1 f Tone 2 f Tone 1 f Tone 2 f Clk ( Clk ( t )+ f Clk t ADC 1 ) In ( t ) 128

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Thiswouldbeaccomplishedbyup-convertingordown-conver tingacommonsignal witha 1 cycle = p Hz accuracy,orrather,anaccuracybetterthanthetimingjitt ernoise limitation,whichmayormaynotbefeasiblebasedonthenois echaracteristicsofthe electronicsandmethodsusedtoperformthefrequency-conv ersion. 5.5PhasemeterPerformanceReview Compilingtheresultsofallofthesemeasurements,wehavea well-denedmodel ofthelimitingnoisesourcesinthePMmeasurementprocess. Theregionsofwhichover thesetermscoupleisshowninFigure 5-23 .Thefrequencyrangefrom 10 Hz to 10 kHz isdominatedbywhiteamplitudenoisewhichcouplesintothe phasefromacombination ofvariationsintheinputbeatnote'samplitude(Chapter 5.2.5.2 )andquantizationnoise (Chapter 3.4.2.5 ).Increasingthesignalvoltagedecreasesthequantizatio nnoiseand reducesthemeasuredphasenoiseproportionately. TheLISAmeasurementband, 0.1 mHz to 1 Hz ,isdominatedbytimingjitterand temperaturedependentphasedispersion.Thetimingjitter scalesproportionatelywith adecreaseinthesignalfrequencyandcanbecorrectedforby themethodsmentioned inChapter 5.4 andChapter 5.4.2 .Oncethetimingjitterhasbeenremoved,thelimiting noisesourcesissuppressedtotheamplitudenoiselimitwit ha 1 = p f icker-typenoise sourcescouplinginatfrequenciesbelow 10 mHz ThephasedispersionfromtheRFtransformersplaysalarger ollinlimitingthe phasemeterphasesensitivity.Internalorexternalheatin gofanylteredelement, includingphoto-detectors,transformer-splittersorACc ouplers,andmixers,willhave atransferfunctionwithsomedenedphaseresponse.Ifthep haseresponseisnot atoverthe2to20MHzLISAheterodynebeatnotefrequencyba nd,thetemperature dependentelectronicswillproduceatime-changingphasel osswhichcouplesinto measuredphase.Thedominateheatingsourceinthe1to100mH zfrequencyrange intheseexperimentsresultedfrominternalheatingoftheA DCsuponpowerupmore sothanenvironmentaltemperaturechanges.Thus,despitet hequiettemperature 129

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Figure5-23.Phasemeternoisemodel,estimation,andlimit s:Theprimaryphase precisionlimitingeffectshavebeenidentiedandplotted incomparison withthe 1 cycle = p Hz phasemeterrequirement.Thefrequencyrange above 10 Hz isdominatedbyamplitude-quantizationnoiseandscales inverselywiththeinputamplitudefrom2Voltspeak-to-pea kto200mVolts peak-to-peak.The 40 = p ffs = p Hz timingjitterdominatesthefrequencies below 1 Hz andscalesproportionatelywiththe2-20MHzinputfrequenc y. Thisassumesthatanytemperature-correlatedlow-frequen cyphase dispersion,whichscalesbasedontheslopeofthetransferf unctions' phaseresponse,hasbeenmitigated;thatsaid,aconstantph aseresponse inthe2-20MHzrangeisdesirable. environmentoftheLISAmission,thesenoisesourceswillst illbepresentfromelectronic heatingandcooling. 130

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/sqrt(Hz) LISA Requirement 1pm LISA Requirement 10pm Input VCO Noise 10 MHz, Electronically Split Entangled Phase Noise Jitter Calibrated Phase Noise Figure5-24.ComparisonofthePMnoisecharacteristics:Th e10MHzdifferentialphase (Chapter 5.2.5.2 ),entangledphase(Chapter 5.2.5.3 ),andtimingjitter extraction(Chapter 5.4 )resultsareplottedtogetherforadirectcomparison ofPMexperimentalresultsandnoiseperformance. 131

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CHAPTER6 THEUNIVERSITYOFFLORIDALISAINTERFEROMETRYSIMULATOR 6.1TheLISALaserTest-bench TheUFLISexperimentalbenchtopaspicturedinFigure 6-1 consistsoffour controllableNd:YAG,1064nmlasers,Laser Ref ,Laser 1 ,Laser 2 ,andLaser 3 ,which generatelasereldswithphasesignalsgivenby Ref ( t ) 1 ( t ) 2 ( t ) ,and 3 ( t ) .We interpreteachindividualLaser i asthelaseron SC i producingthelasereld ( t ) as describedinChapter 3.4.3.4 .HeterodynedPDsignalsbetweenLaser Ref andtheother threelasereldsproducedifferentialmeasurementsofthe laserphases: 1 R = 1 Ref 2 R = 2 Ref 3 R = 3 Ref (6–1) suchthatcombinationsofthesesignalswillcancelthecomm onLaser Ref noise.Thus, thereferencelaserisusedasaglobalreferencewithwhicht heotherthreelaserelds canbemeasured.ReproducingtheexpectedLISApre-stabili zedlasernoise,we PDHlockLaser Ref andLaser 1 toaULEcavity.Thefollowingsectionreportsonthe phasemetermeasurementsofthispre-stabilizedlasernois e.Thenextsectiondescribes themethodsusedtosimulatetheinter-SCelectronicdelayi ncludingthemulti-second time-changinglaserphasedelay,MHzlasereldDopplershi fts,and CycleGW modulations.Thelastsectiondescribeshowthesethreecom ponentsarecombinedto performadvancedarm-lockingandTDIsimulations. ReferenceCavityStabilization TheUniversityofFloridalaserbenchtopPDHlockstwolaser s, Laser Ref (orRL) and Laser 1 (orL1),totwodifferentcavities, 26.0cm and 22.5cm inlengthrespectively, resultinginalaserbeat-notesbetweenthetwolaserswitha frequencystabilitygivenby therelationship: f RL = L 1 ( )= p f RL ( ) 2 + f L 1 ( ) 2 (6–2) 132

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Figure6-1.UFLISlaserbenchtop:Thelaserbenchtop,pictu redhere,PDH pre-stabilizesthelasersandformsallthenecessaryPDsig nalstoperform theLISAinterferometrysimulations.Thefourlasers,thre erepresentingthe individualSCandoneactingasasopticalclock,canbeseeni nthebottom leftcorner.ThevacuumtankontherightcontainstheULEcav itiesand providesthenecessarytemperatureandpressureshielding requiredto transferthecavitystabilitytothelaserfrequency.Theph asemeterdata acquisitionandEPDdataprocessingDSPelectronicscanbes eeninthe backgroundnearthetop-leftoftheimage. 133

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Thestabilityoftheselasersdependshighlyonthefeedback electronicswho'sgaincan beadjustedtoachievea 30 300Hz = p Hz stabilityandarelimitedbythetemperature stabilityofthecavitylength[ 52 ].Phasemetermeasurementsofthefree-running laserstabilityincomparisonwiththecavitystabilizedla serstabilityareplottedin Figure 6-2 fromwhichweobserveanimprovementinthelaserphasenoise by4orders ofmagnitudeat10mHz.Also,asacomparison,weplottheVCOn oiselevelusedfor manyofthepreviousphasemetervericationmeasurements. InLISA,thelaserscan beindividuallylockedtotheirownreferencecavityoronem astercavitystabilizedlaser canbetransferedtotheotherlasersbyphase-lockingthedi fferentialheterodynedlaser elds.Thus,fromthispointon,weassumethatLaser 1 isthemasterstabilizedeldon SC 1 andthattheotherlasereldsarephase-lockedasdescribed in[ 71 ]. 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/sqrt(Hz)Linear Spectral Density Free Running Laser Cavity Stabilized Laser VCO Figure6-2.Measurementsofcommonlyusedsources:Hereweh aveplottedthe phasemetermeasurementsofthespectralnoiseofthefree-r unning,cavity stabilized,andVCOnoisesourcesusedintheUFLISsimulati ons.Cavity pre-stabilizationprovidesuswith4orderofmagnitudefre e-runningnoise suppressionat10mHz.TheVCOprovidessimilarnoisecharac teristicsto thecavitystabilizedlasersbelow10Hzandisusedasatesti nputformany simulations. 134

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6.2TheElectronicPhaseDelay(EPD)Unit Theelectronicphasedelay(EPD)unitisconstructedoutofa llthreePentek componentsdescribedinChapter 5.1 .Generallydescribed,thePentek-6256phasemeterfront-endproducesfrequencyinformationata10-100 kHzdataratewhichisthen passedtothePentek-4205.Thefrequencyinformationissto redonthePentek-4205in abufferwho'slengthisbasedonthedata-rateandthedesire dholdtime. 1 Oncethe delayisappliedthefrequencyinformationisinterpolated tointroducethetime-changing delay.TheinterpolateddataisaddedtoalargeMHzDopplerm odulationandsmallmHz GWmodulation.Theprocessedfrequencyinformationisthen senttothePentek-6228 whereitisintegratedtogeneratethephaseandusedtodrive anumericallycontrolled oscillator(NCO).TheNCOsignalisreconstructedusingthe Pentek-6228'sDACs.The detailsofthedesignandnoisearedescribedinthefollowin gsections. 6.2.1Design AdiagrammaticmodelofasinglechanneloftheEPDunitissho wninFigure 6-3 ThePentek-6256has4ADC'sandtwoFPGA'swhichareconnecte dtofourindependentphasemetersproducing60-bitfrequencydataataCICdo wn-sampledrateof f EPD =10 400 kHz dependingontheapplicationandclockrate.Assumingthe EPDmeasuresasignal, Asin (2 f in ( t ) t ) ,withaclockgivenby Asin (2 f Clk ( t ) t ) ,the phasemeteroutputtakestheform: f EPD [ n ]= f in [ n ]+ f in f Clk f Clk [ n ] (6–3) with n = t = f EPD duetothedown-sampledEPDdata-rate. Thisdataispackedinto16,32-bitpacketswhichtakethefor mshowninFigure 6-4 Thelasttwobitsofeachpacketareusedasatagwithwhichtoe nsureproper data-communication.Data-spacefor60bitsofDopplerand6 0bitsofGWinformation 1 delaytime( ) 135

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n r Figure6-3.Modeloftheelectronicphasedelay(EPD)unit:H erewehavedepicteda modelofEPDunit,whichisconstructedusingtheDSPsystems describedin Chapter 5.1 .TheinputsignalismeasuredwithaPMcoreandthefrequency dataisstoredinmemoryforapre-denedstoragetimerepres entingthe inter-SClighttraveltimedelay.Thedelayedfrequencydat aislinearly interpolatedtosimulateatime-changingdelay(notpictur ed)andaddedtoa Doppleroffset/modulationandthesimulatedGWsignal.Anu merically controlledoscillator(NCO)reconstructsthedelayedfreq uencyinformation andanDACregeneratesthesignalwiththesameclockusedtos amplethe input. isleftopeninthepackingschemewhichisbepopulatedwithd ataonthePentek-4205 oncethefrequencydatahasbeenreadfromthe6256.Dependin gonthedesireddelay, ,thestoragespaceforthe = f EPD data-pointsmustbereservedforeachchannel. Afterthefrequencydataisstoredfortheproperamountofti me,itislinearly interpolatedtoapplyatime-changingdelay.Itisimportan tthattheinterpolationtake therightform;LISAwillhavealinearchangeinthephaserat herthanalinearchange inthefrequency.Asaresult,assumingwearetryingtointer polatethedatasuchthat t t (1 ) ,afactorof (1 ) mustbeintroducedtoaccountforthetime-integration natureofthefrequencytophaseconversion.Thatis,whereo nemightnormallyuse: f [ n (1+ )]=(1 n ) f [ n ]+ f [ n +1] (6–4) 136

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Figure6-4.EPDunitdata-packingscheme:The60-bitfreque ncy,Doppler,andGW modulationdataofthetwoindependentEPDchannelsismulti plexedintoa 16pointdata-streamof32-bitvalues.Numericalagsaread dedtothe most-signicant-bitsofthefrequencydatatoensurethatt hedataiscorrectly interpretedanddata-pointsarenotmissed. 137

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toapplyalinearinterpolationbetween f [ n ] and f [ n +1] ,wewilluse: f [ n (1+ )]=(1 )(1 n ) f [ n ]+ f [ n +1] (6–5) toaccountforthefrequencytophaseconversion. Oncethelinearinterpolationisappliedthefrequencydata ispackedwitha1 Hzto 1mHzGWmodulationsignalanda 20 MHzDopplersignal.Itispossibletomodulate theDopplersignaltoaccountforthebreathingoftheLISAco nstellationbutinthe followingexperiments,wewillalwaysuseaconstantDopple rshift.TheGWsignal canbearbitrarilygeneratedusingtheon-boardprocessoro rtheGWdatacanbe uploadedtotheEPDunitfromthedata-processingcomputerb eforethedelayprocess isbegun.OncetheGW, f GW ( t ) ,andDoppler, f Dop ,informationispackedalongwith thetime-interpolatedfrequencydata,again,asshowninFi gure 6-4 ,theinformationis addedtogether,integratedtogeneratethephaseinformati on,andgeneratedusingthe NCO/DACoutputdescribedabove. Zero-DelayEPDUnit(FrequencyControlFiltering) BypassingthePentek4205data-storageand,instead,lter ingandtransmittingthe PMgenerateddatadirectlytotheNCO,wecandesignhigh-spe edfrequencycontrol ltersforheterodynelocking,phaselocking,andarm-lock ingexperiments.The60-bit phasemeterfrequencydata, f PM [ n ] ,ispassedthroughadigitalxedpointniteimpulse response(FIR)lter;theltereddatacanbereconstructed byanNCOwithlessthana 1msdelay 2 andusedtocontrolthelaserfrequency.Formoredetailonth eseltering andlockingmethods,referto[ 54 ]andChapter 6.3.1 2 Optimalperformanceresultsina 0.06 ms lterdelay.Arstorderapproximation resultsina 17 kHz feed-backbandwidth. 138

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6.2.2Verication TheEPDunit'snoisecanbewrittenintermsoftheinput-freq uencydata, f in ( t ) ,the delay, ,andthesampling( Clk ADC ( t ) )andregeneration( Clk DAC ( t ) )clocks 3 as: EPD ( t )= in ( t ( t ))+ h GW ( t ) (6–6) + f in f Dop f Clk DAC Clk ( t ) f in f Clk ADC Clk ( t ( t )) fromwhichwecanseetheclocknoisetermscouplingintothem easurement.Interestingly,theseclocknoisetermstakethesameformasthosedes cribedin( 3–70 )and ( 4–3 ).Inthisway,theEPDunitproducesthedifferentialclockn oisetermswhichcancel in( 3–70 )andwouldnormallyneedtobeaccountedforbyclocknoisetr ansfers. TestingtheEPDperformance,weuseaVCOsignalasaphasenoi sesourceand measuretheoriginalVCOandandEPD-delayedVCOsignalswit hphasemeters.This producestwosignalsoftheform: VCO PM ( t )= VCO ( t ) f VCO f Clk PM Clk ( t ), (6–7) VCO EPD PM ( t )= VCO ( t ( t ))+ h GW ( t )+ f VCO f Dop f Clk DAC Clk ( t ) f VCO f Clk ADC Clk ( t ( t )) f VCO f Dop f Clk PM Clk ( t ). (6–8) Timeshiftingthephasemetermeasurementandsubtractingi tfromtheEPDmeasurementweobtainadifferentialresultgivenby: EPD ( t )= VCO EPD PM ( t ) VCO PM ( t ) (6–9) = h GW ( t )+ f VCO f Dop f Clk ( DAC Clk ( t ) PM Clk ( t )) f VCO f Clk ( ADC Clk ( t ( t )) PM Clk ( t ( t ))). (6–10) 3 Althoughthesearegenerallythesamesource,theymayhaven on-commonterms whichdonotcancel. 139

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IfweassumethattheclockusedtodrivetheEPDinputADCando utputDACarebased onthesamesource, EPD Clk ( t ) ,wecanwritetheexpectednoiselevelas: ~ EPD ( t )= ~h GW ( t )+ f VCO 1 e 2 i f Dop f Clk [ ~ EPD Clk ( ) ~ PM Clk ( )]. (6–11) Thus,wecoupleinthedifferentialclocknoisescaledbythe armtransferfunction, 1 e 2 i .Takingmeasurementsofthisformwithindependentclockso urces, phase-lockedclocksources,andsplitclocksources,andco mparingthenoiselevels tothedifferentiallysubtractedEPDmeasurementsasshown inFigure 6-5 wesee thatthephasenoiseofthemeasurement(solidlines)equals theclocknoise(dotted lines)scaledbythearmtransferfunction, 1 e 2 i ,with =16.6 s,asexpectedby ( 6–11 ).Obviouslywewillusethesplitclocksourcetodrivebotht heEPDunitsandPM measurementstoreducethisclocknoisecoupling. Usinga4MHzVCOinputsourceandasplitclocksourcewetakea long-term EPDmeasurementtoseetheover-allnoiseleveloftheEPDuni t.Theresults,withand withouta 7.2 mHz GWmodulationareplottedinFigure 6-6 andcomparedagainstthe phasemeternoiselevel.Thus,itisreasonabletosaythatth eEPDunitreproducesthe inter-SClighttraveltimedelaytowithina10 Cycle/ p Hz accuracy 4 AlthoughtheEPDunitmeetstherequirement,wemustcheckto seehowtheclock noisesourcescoupleintoaheterodyneddifferentialarmme asurement.Toreproduce thesensorsignalsdescribedin( 3–72 )or( 3–74 )wetaketheEPDdelayedsignal,( 6–6 ), andelectronicallymixitwiththesameVCOusedastheinputt otheEPDunit.The 4 Theminimizedtime-delayisfoundusingthemethodsdescrib edinChapters. 4.4.2.1 and 4.4.2.2 140

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10 -3 10 -2 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 10 2 Frequency (Hz)Cycles/sqrt(Hz)Linear Spectral Density Input Delay Unit (Same Clock) Clock Splitter Noise Delay Unit (Unlocked Clock) Clock Noise Delay Unit (Locked Clock) Clock PLL Noise LISA 12pm Requirement Figure6-5.NoiselimitationsoftheEPDunit:Thedifferent ialclocknoisebetweenthe EPDunitandthephasemeterusedtoverifytheEPDnoiselimit sthe measuredperformancederivedin( 6–11 )for f Dop =0 .Theclocknoise (dottedlines)forthethreedifferentcases,(1)free-runn ing,unlockedclocks (2)phaselockedclocks,and(3)electronicallysplitclock sareestimatedby takingadifferentialmeasurementofthesame1MHzVCOsourc eusing differentclocksources.TheEPDnoisescaleswiththisdiff erentialclock noisemultipliedbythedelayed-comparison(orsensor)tra nsferfunction, 1 e 2 i ,asexpected. mixedoutputofthepromptanddelayedsignalstakestheform 5 : Arm ( t )= VCO ( t ) EPD ( t ) (6–12) = VCO ( t ) VCO ( t ( t )) h GW ( t ) f VCO f Dop f Clk DAC Clk ( t )+ f VCO f Clk ADC Clk ( t ( t )) 5 WeassumetheDopplershiftisnegativeinthiscase. 141

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/sqrt(Hz) VCO Input EPD Unit Noise Performance EPD Unit with GW Modulation 1pm Requirement 12pm Requirement Figure6-6.EPDUnit'sphase-noisetransmissionaccuracy: Whenusingthesameclock sourcefortheEPDsampling,EPDregenerationandPMmeasure ment systems,thecommonclocknoisetermscanceltowithintheac curacyofthe clockdistributionandtheEPDunitresultsintheplottedtr ansmission replicationaccuracy.A 7.2 mHz GWmodulationisaddedtooneofthe measurementstoverifythatthe h GW ( t ) termsin( 6–11 )donotcancel. Measuringthiswithaphasemeter: Arm PM ( t )= VCO ( t ) VCO ( t ( t )) h GW ( t ) (6–13) f VCO f Dop f Clk DAC Clk ( t )+ f VCO f Clk ADC Clk ( t ( t )) f Dop f Clk PM Clk ( t ). which,reducesinthecasewhere PM Clk ( t ) = DAC Clk ( t ) = ADC Clk ( t ) to: Arm PM ( t )= VCO ( t ) VCO ( t ( t )) h GW ( t ) (6–14) f VCO f Clk ( DAC Clk ( t )+ ADC Clk ( t ( t ))). wherewehaveleft DAC Clk ( t ) and ADC Clk ( t ) shownexplicitlywithwhichwecan comparethesedirectlyto( 3–72 )and( 3–74 ).Again,thismeasuredEPDunitsignal 142

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acquiresthesameinputandclocknoisescalingastheclockn oisecorrectedinter-SC observables,( 3–70 ). 6.2.2.1Time-changingTimeDelay Toverifythecapabilitiesoftheelectronicreplicationof thetime-changingdelay, wetakethepromptmeasurementofaVCOsignalandcompareitw iththeanEPDprocessedsignalasdescribedintheprevioussection.Thee xperimentaldelayisset to ( t )=16.6 s +100 tns = s .Ifwesimplyminimizethenoisewithoutconsidering thetime-changingdelay,wewillobtaintheaveragedelayfo rthemeasurementas thecalculatedoffsetdelaytime.Thedifferencebetweenth esignalsasplottedin Figure 6-7 showsthattheofthedifferenceisin-phasewiththeprompts ignalatthe beginningofthemeasurementandout-of-phasewiththeprom ptsignalneartheend ofthemeasurement.Thisobviouslylimitsthenoisecancell ationcapabilitiesasshown explicitlyinFigure 6-8 Accountingforthetime-changingdelay,weperformathefra ctionaldelayinterpolationoftheinputsignalusingthemethodsdenedinChapter 4.1.4 and 4.4.2.2 .Once thepromptdatahasbeencorrectlyinterpolatedwesubtract itfromtheEPDprocessed signal.Comparingtheresultswiththebase-lineEPDnoisep erformanceasshownin Figure 6-8 ,weseethatoncethetime-changingdelayhasbeenaccounted for,thenoise performancematchestheEPDnoiselevel.6.2.2.2GravitationalWaveInjection TotesttheGWinjectionaccuracyweinjectafrequencymodul ationwitha power-functionincreasingfrequencyenvelopedbyaGaussi anamplitudemodulation givenby: f GW ( t )= sin 2 (10 3 Hz)t (1 t 3.7 10 4 s ) 3 = 8 e ( t t o ) 2 2 10 7 s 2 (6–15) Granted,thisisnotarealisticGWmodulationsourcebut,no netheless,itprovides atestoftheabilitytoreconstructthesignalsinjectedbyt heEPDunit.UsingaVCO 143

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Figure6-7.Timeseriesoftheinterpolateddelaydifferenc e:Thetime-seriesoftheinput signalsubtractedfromthetime-delayinterpolatedsignal showsthat,atrst, thedifferentialphaseleadstheinputphaseduetoasmallpo sitivetiming errorbuttowardstheendofthemeasurementthedifferentia lphaselagsthe inputphaseduetoasmallnegativetimingerrorcausinganap parentphase inversion.Oncethetime-changingdelayhasbeenaccounted forthe differentialphasecancelstotheEPDprecisionoor. sourcewetime-shifttheEPDdelayedsignalandsubtractitf romthepost-processed time-shiftedmeasurementofthepromptsignal.Thetime-se riesoftheinjectedinput modulation,themeasuredGWmodulation,andtheirdifferen ceisplottedinFigure 6-9 Fromthespectrumofthesemeasurements,weseethatthenois eleveloftheGW injectedEPDunitmatchesthebase-lineEPDnoiselevelonce theexpectedGWsignal hasbeenremovedfromtheEPDprocessedmeasurement. AnotherGWsourceofconcernisthelow-frequencybinarycon fusionnoise background.Estimatesofthisbackgroundnoisevary[ 46 103 ]suchthatwewill generallyestimatethebinaryconfusionnoiselevelas: ~ h GW Background ( )= .01 s 2 +.001 mHz p Hz (6–16) ThisisplottedexplicitlyinFigure 8-1 144

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Figure6-8.Interpolateddelaydifferentialmeasurements andcorrections:The time-changingdelaycausesaspectralnoiseoorwhichscal eswiththe delayerror = T = 2 where T isthemeasurementtimeasdenedby ( 4–6 ).Ifwetime-scalethemeasuredinputsignalusingfraction aldelay interpolationandsubtractitfromtheEPDprocessedtime-c hangingdelay signal,weaccountfortheadditionaldelayerrorandcancel thenoiseto withintheEPDunit'snoiseoorprecision. 6.3UFLISSimulations Usingthelaserbench-top,EPDunit,controllters,andpha semetermeasurements, weareabletoperformmanyLISA-liketestsoftheinterferom etrysystemincluding advancedarm-lockingandTDIsimulations.Althoughamorer enedandin-depth analysisofthearm-lockingtestsandresultscanbefoundin [ 54 ],wepresentarsttest oftheUFLISelectronicsbyperformingasingle-armarm-loc kingexperimentasoutlined inChapter 4.2.2 .Nextweoutlinethetime-delayinterferometrysimulation sandsetup theTDIcharacteristicswhicharetobeexploredinthenextc hapter. 6.3.1Arm-LockingStabilization TheUFLISelectroniccomponentswererstusedtotestthesi nglearm-locking methodsbeingpresentedin[ 53 77 ].Sinceamuchmorein-depthanalysishas beenpresentedsincethesesimulationswereperformedwewi llbrieydiscussthe 145

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Figure6-9.Timeseriesofanarbitrarygravitationalwave: Thegravitationalwave modulationdenedby( 6–15 )isinjectedintotheEPDunitandextractedwith adifferentialinputmeasurementusingtheappropriatetim e-shifttocancel thenoise.Theextractedsignalandthetheoreticallyinjec tedsignalagree well;subtractingthemeasuredGWsignalfromthetheoretic alsignal,we completelycanceltheinjectedGWsignaltowithintheEPDun itprecision. experimentalsetup,theresultsobtained,andtheun-antic ipatedlimitations.Thesingle arm-lockingexperimentispresentedatthetopofFigure 6-11 .ThePDHpre-stabilized Laser L 1 isheterodynedwithLaser PL whichisoffsetphase-lockedtoLaser RL .The frequencyoff-setinthephase-lockisdrivenbyanoscillat ormodulatedwiththe arm-lockingcontrolsignal.Thearm-lockingcontrolsigna lisgeneratedbytakingthe Laser L 1 /Laser PL beat-noteandformingthesensorsignalsasdenedin( 3–72 )using anEPDunitwithaDoppleroffset, f Dop ,andtimedelayof =1 s.Thissensorsignal islteredwiththefrequencyfeed-backcontrollerdescrib edinChapter 6.2.1 .The feed-backlterisdenedintermsofthefrequencyas: T AL ( f )= h a 4 f 4 + a 3 f 3 + a 2 f 2 + a 1 f i + a 0 p f (6–17) The f 1 = 2 providestheanadditionalphaseadvanceintheregionofthe locking frequency.Thisisbecausethesensorsignal'sphaseswings between and with 146

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Figure6-10.Spectralcorrectionofanarbitrarygravitati onalwave:TheEPDunitwitha VCOinputisinjectedwiththeGWmodulationdenedby( 6–15 );theEPD outputandinputaremeasuredwithphasemeterandthenfract ionaldelay lteredtocancelthecommonVCOnoise.ThemeasuredGWsigna l (green)matchesthespectraldensityoftheinjectedwave-f orm(red) exactly.SubtractingouttheGWmodulation(cyan)wearelef twithanoise levelwhichequalstheEPDprecision(purple).TheGWmodula tedcurve (cyan)islow-passlteredsothattheGWmodulationisappar entinthe time-series(Figure. 6-9 ) respecttotheinputlaserphaseatfrequenciesabovethers tzero, 1 = .Meanwhile,the low-frequencyintegratorskickinatfrequenciesbelowthe rstzero, 1 = ,andscalewith thecoefcientssuchthat a 4 < a 3 < a 2 < a 1 tomaintainlockingstability.Theresults, showninFigure 6-11 ,showanadditionalveordersofmagnitudenoisesuppressi onof thelaserfrequencynoiseat10mHzandsignicantlyreduced frequencyuctuationsin thetime-series.Thelow-frequencystabilizationlimitis denedbytheaccuracyofthe controlelectronics[ 54 ]. AdvancedArm-LockingControllers Theprimarycomplicationwithachievingthelockingcondit iondescribedinthe previouschapterwastheuser-denedDopplerandphasemete rfrequencies.Unless thesevalueswereexactlyequaltheexperimentencountered anintegrationofthe 147

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Figure6-11.Long-armhardware-basedsingle-arm-locking experiment:Themodelat thetopdepictstheexperimentalsetupofthersthardwareb ased heterodynesingle-arm-lockingexperiment.TheRL/L1beat noteprovidesa measureofthe'input'noisewhilethePL(phase-locked)/RL beatnoteis stabilizedaccordingtotheproposedarm-lockingtechniqu es[ 53 ].The low-frequencynoiseissuppressedby5ordersofmagnitudea tfrequencies below10mHz.Themeasurementislimitedbyquantizationnoi seinthe EPDunit.Referto[ 54 ]formoreinformationonthedetailsand advancementstothissimulationinperformingLISA-likete stsofthe arm-lockingcapabilities.Thesesimulationsprovidedar sttestofthe UFLIS/DSPsystem'scapabilitiestoperformLISA-likesimu lations. frequencyerrorwhichcausedsysteminstability.Thisindi catedafrequencypullingeffect whichhassincebeendescribedin[ 79 ]and[ 84 ]. Sincethesinglearm-lockingexperimentwasperformed,mor ein-depthexperiments havebeenconductedbytheauthorandcolleagues.Adetailed presentationofthe singlearm-lockingexperimentandresultscanbefoundin[ 78 ].Thesamearm-locking methodswereusedincombinationwithaPDHside-bandlockin gschemetoprovethe capabilitiesoflaserfrequencycontrolinLISA[ 83 ].Thesecomponentswerealsousedto 148

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Table6-1.TDIexperimentalcharacteristics:Five40000se xperimentsareperformed withincreasinglymorecomplicated,yetmoreLISA-like,ch aracteristics.The transponderexperimentprovidesuswithabaselinemeasure ofthe experimentalsetup'snoiseperformance.The'dynamic'exp eriments demonstratetheabilitytodetermineandaccountforthetim e-changing delay-times.The'LISA-like'experimentprovestheabilit ytoremove independentSCnoisesourcesandconstrainone-waydelayti mes.Finally, theconfusionnoiseexperimentveriesthattheTDI-rangin gcapabilitywillnot belimitedbylow-frequencyLISAnoisesources. SimulationName s 1 r Inter-SCVelocityInjected (ns/s)GWSignal StaticTransponder s 1 r 0 2 =0 6.227mHz (2-Way) 3 =0 Binary DynamicTransponder s 1 r 0 2 = 100 6.227mHz (2-Way) 3 =+150 Binary StaticLISA-like s 1 r = PLLr 2 =0 6.227mHz (4-Way) (1.0 = f ) mCycle = p Hz 3 =0 Binary DynamicLISA-like s 1 r = PLLr 2 = 100 6.227mHz (4-Way) (1.0 = f ) mCycle = p Hz 3 =+150 Binary Confusion-Noise s 1 r = PLLr 2 = 100 6.227mHz (4-Way) (1.0 = f ) mCycle = p Hz 3 =+150 Binary+CN verifythedualanddual-modiedarm-lockingschemesprese ntedin[ 79 ]andvalidated in[ 84 ]and[ 81 ]. 6.3.2TDISimulationOutline TheTDImeasurementsthefollowingsectionsaregeneratedu singthe2-4sensor observables, s 1 r and/or s s 1 ,denedin( 3–70 )overa10-12hoursimulations.Theve TDIexperimentswewillperformareoutlinedinTable. 6-1 .Eachstagewillintroducea newnoisesourceintothemeasurementsuchthatwecandeciph erthenoisecouplings basedontheresultsofeachsimulation. Webeginwitha'statictransponder'measurementinwhichwe willuseconstant,but unequal, 2 0 2 6 = 33 0 6 ,arm-lengthstotesttheTDIX 1.0 combinations.Thetransponder 6 [ 2 2 0 16.55s] 6 =[ 3 0 3 16.75s] 149

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measurements,asshowninFigure 7-1 ,willsimulateLISAinanoise-lessfar-SC conguration,asthoughthe'outgoing'masterlasereldsa rereectedperfectlyoff thefarSCandwhere s 1 r =0 .TherangingmethodsdescribedinChapter 4.4.2.2 are usedtocalculatetheround-tripdelaytime 7 andcancelthelaserphasenoise.Next,we introducetime-changingarm-lengthswiththe 2 and 3 valuesoutlinedinTable. 6-1 89 tovalidatethecapabilitiesoftheTDIX 2.0 combinationsthrough'dynamic'simulations. Next,weintroducethenoisecouplingfromthefarSCsignals s 1 r ,byphaselocking thelasersonthefarSCtothedelayed'out-going'lasereld sandmeasuringthe resultingPDsensorsignalsasshownintheexperimentalmod el(Figure 7-7 ) 10 Again,werepeatthe'static'and'dynamic'simulations,in cludingtheadditionallaser phasenoisesignals, s 1 r ,inthecombinations;meanwhile,weareabletotestthe ranging-accuraciesoftheone-waydelaytimesusingthemet hodsdescribedin Chapter 4.4.2.2 .Finally,weintroducealow-frequencyconfusion-noiseba ckground signaltoinvestigatethelimitations,ifany,ofthisaddit ionallow-frequencynoiseonthe rangingaccuracies. 7 Theone-waydelaytimesareun-denedinthesesimulations. 8 2 = 100.0 ns = sns = s ( v 2 = 30.0 m = s ) ; 3 =+150.0 ns = sns = s ( v 2 =45.0 m = s ) 9 j 2 3 j =250.0 ns = s 10 ThefarSCsensorsignals, s 1 r ,areusedastheinputsignalsforphase-lockingthe farSC'slaserssuchthatthesensorsignalisequaltothePLL 'serrorsignal, PLLr ,as showninFigure 7-7 andTable. 6-1 150

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CHAPTER7 TIME-DELAYINTERFEROMETERYSIMULATONS Thefollowingchapterpresentthedetailsandresultsofthe experimentsdescribed inChapter 6.3.2 .Inallofthesemeasurementswewillinjecta 6.277 mHz GWbinary withastrainamplitudeof 4.35 10 20 m = m 1 tosimulatetheRXJ0806.3+1527binary describedinthevericationbinarytable(Table 2-1 ).Althoughthisstrainamplitudeisa factorof100largerthanwhatisexpectedinaactualLISA-li kedetection,itisusedto provetheTDIcombinations'abilitytocancelthelaserphas enoisewithoutcancelingthe modulatedGWsignals. 7.1TransponderTDISimulations Werstestablishabaselinerangingandmeasurementprecis ionwithstatic arm-lengths( =0 )andwithoutnoisebeingintroducedbythe'far'SC.Referri ngto theexperimentalmodelshowninFigure 7-1 ,wetaketheLaser Ref /Laser 1 beatnoteand useitastheinputfortheTDIsimulator.ThePDbeatnoteisel ectronicallymixedwitha frequencymodulation f Mod ( t )= Asin ([ o +500 sin (2 pi t )] t ) toapplytherangingtone 2 andtoshiftthelaserbeatnotetoaPMmeasurablefrequency, inthiscase, PD o =7MHz.Thismixedoutputiselectronicallysplit,measured withaphasemeter,and processedbyEPDunitstosimulatethe'out-going'inter-SC lasereldtransfer.Constant EPDdelaysof 2 (0) 2 0 (0) 16.75 s and 3 (0) 3 0 (0)=16.55 s areprogrammedto theEPDunitswhileDopplershiftsof f Dop 2 =+3 MHz and f Dop 3 = 2 MHz areappliedto thefrequencysignals.Thedelayed'out-going'signalsare directlyconnectedtoanother EPDunittosimulatethe'returning'inter-SClasereldtra nsfer.Thisnoise-lesstransfer betweentheEPDunits,inaLISAmodelforexample,willbehav easthoughthelaser 1 ( t )=2sin(2 (6.227 mHz ) t ) Hz GW ( t ) 4 ( t ) = (2 (6.227 mHz ))=204.4 cyc = Hz h GW ( t )= GW ( t ) = L =4.3 10 20 Hz 1 2 f mod =1 Hz, A mod =500 Hz 151

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Figure7-1.ModeloftheTDI-Transponderexperimentalbenc htop:ThisTDI-Transponder modelshowshowtheindividualpre-stabilizedlasereldsa recombinedto formtheinputnoisesignalandhowthissignalisprocessedb ytheEPD unitstoreplicateaLISA-likeinter-SClaserlinkandprodu cethe s sr observables[].TheinputPDbeatnoteisdemodulatedtoaPMmeasurable frequencyusinganoscillatorwitha1Hzphasemodulation.T he PM-read-outsignalsaredelayedbythe 2 0 ( t ) and 3 ( t ) inter-SClighttravel times,addedwithGWmodulations,andregeneratedwithaDop pleroffset. Inthesetranspondermeasurementsthedelayedsignalsarei mmediately injectedintothe'return'tripEPDunits 2 ( t ) and 3 0 ( t ) asthoughthelaser eldswerereectedoffmirrorsatthefarSC.GWmodulations andDoppler offsetsareaddedtothereturn-tripEPDunitsalso.Theroun d-tripdelay signalsareelectronicallymixedwiththelocal Laser Ref = Laser 1 beatnoteto producethe s 21 and s 31 sensorsignals. eldswerereectedoffmirrorsonthefarSCsuchthatthefar sensorobservablesare signal-free, s 1 r =0 ,intheTDIcombinations.Finally,the'return'eldEPDsim ulated outputisheterodyned 3 withthelocallaserphase,inthiscasetheLaser Ref /Laser 1 3 Electronicallymixed 152

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beatnote,toproducethelocal s 21 s 31 PDobservables.ThePDsignalsaremeasuredfor 40000sbyphasemeterswiththe60-bitprecisionata14.9Hzd ata-rate. Wethenintroducethetime-changingarm-lengthsdenedinT able 6-1 suchthat theinter-SCtimedelaystaketheformdenedin( 4–39 ).Despitethisdenition,as wedescribedinChapter 4.4 ,withoutanynoisebeingintroducedatthefarSCwewill onlybeabletoconstraintheround-tripdelaytimesfoundin ,( 3–72 )and( 3–74 )dened explicitlyas: 2 0 2 ( t )=[(1 2 )((1 2 0 )( t 2 0 (0)) 2 (0))] (7–1) 33 0 ( t )=[(1 3 0 )((1 3 )( t 3 (0)) 2 0 (0))]. (7–2) Fortunately,atthesametime,thismeansweonlyhavetoscan overthe2-dimensional, 2 0 2 33 0 ,basis.WewillalsoverifytheTDI1.0limitationsdenedin Chapter 4.3.3.2 7.1.1Static-ArmTransponderSimulation UtilizingthePDmeasurementsof s 21 and s 31 ,the 40000 s data-setisbrokeninto 40, 1000 s segments.TheiterativeformationoftheTDIcombinationfo reachsegment minimizesattemptstominimizetherangingtoneandcalcula tethetime-delayfunctions throughlinearregressionasdescribedinChapter 4.4.2.2 andshowninFigure 4-3 .The rstandonlyiterationproducesaslopeerror(constrainto nthearm-lengthvelocities), of j 2 j < 50 fs = s andavariance(round-triprangingaccuracy)oflessthan 0.6 ns ( 0.18 m )asshowninTable 7-1 4 .Inthisexperiment,wenotethattheTDIX 1 combination'sranging-toneminimizationproducesthesam eestimationtowithinthe measurementerroroftheround-triplighttraveltimeasthe TDIX 2 combination.If 6 =0 thedifferentTDIcombinationswouldproducedifferentest imationssincetheranging toneminimizationusingtheTDIX 1 combinationwouldbelimitedby( 4–32 )andwill 4 RefertoFigure 7-4 toseetheresultsoftheranging-tonecancellationnearthe 1 Hz frequencybin. 153

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/ Hz Arm-2 Sensor ( D 2 (t)) Arm-3 Sensor ( D 3 (t)) Phasemeter Precision TDI 1.0 Combination TDI 2.0 Combination Ranging Tone Pre-Stabilization Requirement 6.227 mHz Binary Ranging Limit (0.6 ns) IMS Requirement PM Requirement Figure7-2.Rawstatictransponderexperimentalresults:T hesensorsignals ( s 21 ( t )= 21 ( t ) s 31 ( t )= 31 ( t ) )areusedwiththerangingvaluesfoundin Table 7-1 togeneratetherawspectraofTDIX 1 andTDIX 2 combinations. Thephasemetermeasurementlimitationforthe7MHzinputbe atnote frequencyisalsoplottedforcomparison.TheTDI-combinat ionsdropbelow themeasurementlimitationduetotheinherentTDItransfer functionsbutthe GWsignalsarealsosuppressedbytheTDItransferfunction. The 6.277 mHz gravitationalwaveisonlyrevealedoncethelasernoisehas beenremoved. Theexpectedranginglimitationisalmost2ordersofmagnit udebelowthe TDIcombinationsindicatingthatalltheinputlasernoiseh asbeen suppressedbeyondthemeasurementsensitivity. estimatethelight-traveltime-delayasthemeandelayfora particulardata-segment. Thus,sinceeachoftheseTDI-rangingmethodsproducethesa meresult,eveninthe casewhere =0 ,wewillusetheTDIX 2.0 combinationsforallTDI-rangingestimations. Usingtheranging-calculatedvalues(Table 7-1 )weformtheTDIX 1 andTDIX 2 combinationsfortheentiredata-set.Thelinearspectrald ensityoftheTDIcombinations, asplottedinFigure 7-2 ,showthelasernoisecancellationandrevealthephasemodulatedGWbinaryat 6.277 mHz .Theexpectedranginglimitationbasedonthe 0.6 ns variancecalculatedaboveandevaluatedwith( 4–6 )iswellbelowthemeasurement sensitivity.Wecanalsoseethedifferencebetweenthe8-pu lseand16-pulseresponse 154

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transferfunctionsoftheTDIX 1 andTDIX 2 combinations.Rescalingbytheaverage inverseTDIresponsetransferfunctionsgivenby: T 1.0 ( s )= j 1 e stau 2 0 2 j + j 1 e stau 33 0 j 2 (7–3) T 2.0 ( s )= T 1.0 ( s ) 1 e stau 2 0 233 0 (7–4) weseetheTDIX 1 andTDIX 2 combinationsresultinidenticaleffectivesensitivities (Figure 7-3 ),asexpectedforastaticinterferometer.Fromthisplotwe canalsoseethat bothmeasurementsarelimitedbytheEPDunit'sphase-trans missionprecisionaswas previouslyplottedinFigure 6-6 .Fromthispointforwardwewillinterpretthesestatic TDIX 1 andTDIX 2 combinationstorepresentthebaselineprecisionoftheUFLISA-TDI simulator.Wecalculatetheeffectivelasersuppressionma gnitudebydividingtheinput spectrumbytheinputnoisebythere-scaledTDIspectra;fro mFigure 7-6 weobtain greaterthan 10 10 noisecancellationat1mHz. 7.1.2Dynamic-ArmTransponderSimulation Expandingthesimulationcharacteristics,weincludethet ime-dependentarmlengthsintotheexperimentwiththe -valuesdenedinTable 6-1 resultinginan expectedTDIX 1 limitationasdenedby( 4–32 )with j j =250 ns = s .Initiallyassuming =0 ,the 40000 s measurementsofthe s 21 and s 31 signalsarebrokeninto40, 1000 s segmentsandprocessedusingthemethodsdescribedinChapt er 4.4.2.2 .These datasegmentsareusedtominimizetherangingtoneandcalcu latetheround-trip time-delayoffsets, 2 0 2 (0) and 33 0 (0) ,foreachsegmentasshowninFigure 4-3 .The linearregressionofthesetime-delayoffsetsprovidesar st-ordermeasureofthe to anaccuracyof 100 ps = s asshowninTable 7-1 .Theprocessalsocalculatesarstorder measureoftheround-triptimedelaywitharangingprecisio nof < 7.5 s( 1.7 km )but duetotheincorrect =0 assumption,thesevaluestendtoequaltheaveragedelay forthedata-segment.Aseconditerationimprovesthe accuracyto 80 fs = s andthe rangingprecisionto < 5.9 ns ( 1.8 m )providingamuchmoreaccuratemeasureof 155

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Table7-1.TransponderTDI-rangingestimation:Themethod outlinedinFigure 4-3 andtheexperimentsdescribedin capters 7.1.1 and 7.1.2 areusedtocalculateestimatesoftheinter-SCdelayfuncti ons,( 4–37 ),andranging errors.TheestimatesverifytheTDI-rangingcapabilities andareusedtogeneratetheTDI-combinationsfor Figures 7-3 and 7-5 .The'return'delaytimestendtobelongerthanthe'outgoin g'delaytimesby250msasa resultofinternaldelayswithintheDSPsystem'sEPDunits. Iteration (Slope) 22 0 (0), 33 0 (0) (Offset) 22 0 33 0 ( ) TDI1.0 Transponder 1(TDI1.0) 2 2 = 44.5 fs = s 20.9 fs = s 22 0 (0)=33.55204887148 s 0.23 ns 22 0 =0.54 ns 2 3 = 46.3 fs = s 12.5 fs = s 33 0 (0)=33.15222859583 s 0.14 ns 33 0 =0.32 ns 1(TDI2.0) 2 2 = 41.0 fs = s 21.2 fs = s 22 0 (0)=33.55204887151 s 0.24 ns 22 0 =0.55 ns 2 3 = 46.3 fs = s 12.7 fs = s 33 0 (0)=33.15222859579 s 0.14 ns 33 0 =0.33 ns TDI2.0 Transponder 1 2 2 = 200.247 ns = s 100 ps = s 22 0 (0)=33.5518847 s 2.3 s 22 0 =7.5 s 2 3 =+300.056 ns = s 95 ps = s 33 0 (0)=33.1525027 s 2.2 s 33 0 =7.0 s 2 2 2 = 199.9998668 ns = s 80 fs = s 22 0 (0)=33.5519484187 s 1.8 ns 22 0 =5.9 ns 2 3 =+300.0001130 ns = s 23 fs = s 33 0 (0)=33.1523897572 s 0.51 ns 33 0 =1.7 ns 3 2 2 = 200.0000058 ns = s 8.9 fs = s 22 0 (0)=33.55194832884 s 0.20 ns 22 0 =0.65 ns 2 3 =+300.0001361 ns = s 4.5 fs = s 33 0 (0)=33.15238977691 s 0.10 ns 33 0 =0.33 ns 156

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/ Hz Pre-Stabilized Laser Noise f 1 (t) Simulator Baseline (EPD Noise) Corrected TDI 1.0 Combination Corrected TDI 2.0 Combination PM Requirement IMS Requirement Pre-Stabilization Requirement TDI 1.0/2.0 Ranging Limit (0.6 ns) Figure7-3.Correctedstatictransponderexperimentalres ults:Theinputsignal, 1 ( t ) ,is plottedalongwiththecorrectedTDIX 1 andTDIX 2 spectrawhichhavebeen scaledbytheinverseTDItransferfunctions( 7–3 ).BothTDIspectramatch exactlyoncetheinverseTDI-combinationtransferfunctio nhasbeen accountedfor.Theincreasedlaserphasenoisearound100mH zisaresult ofanover-coupledtemperaturetrackinginthePDHcontroll erwhichmay resultinaranging-limitedsensitivitylossnear100mHz.T heEPDunit's phasetransmissionaccuracyisplottedandisidentiedast heprimaryTDI simulationlimitingnoisesource.ThisstaticTDI-transpo nderresultwillactas thesimulator's'baseline'performancefortherestoftheC hapter.Theresults meetthe 18 pm = p Hz IMSrequirement( 3–5 ). thetime-dependentarm-lengths, 2 0 2 ( t ) and 33 0 ( t ) .Thenaliterationoptimizesthe precisionto 8.9 fs = s andtherangingprecisiontolessthan 0.65 ns ( 0.2 m ).(Table 7-1 ) Applyingthecalculatedround-tripfunctionalvaluesfrom thethirditerationofthe rangingprocedure,theentiredata-setisusedtoproduceth elinearpowerspectral densityfortheTDIX 1 andTDIX 2 combinationsasplottedinFigure 7-5 .TheTDIX 1 combinationislimited,asanticipated,by( 4–32 )with 16.6 sand j j =250 ns = s TheTDIX 2 combination'scorrectiontermsaccountforthisdynamicar m-length limitationandremovethevelocitydependentlaserphaseno iseresultinginasensitivity equaltotheexperiment'sbaselinenoise.Thisresultmeets theIMSrequirementdened 157

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0.95 1 1.05 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 Baseline Frequency (Hz)Cycles/ Hz 0.95 1 1.05 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 TDI 2.0 Transponder Frequency (Hz) 1.45 1.5 1.55 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 TDI 1.0 Phase-Locked Frequency (Hz) 0.95 1 1.05 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz) TDI 2.0 Phase-Locked f 1 ( w ) D 2 ( w ) D 3 ( w ) Simulator Baseline TDI-X 2 ( w ) PLL Noise Figure7-4.Rangingtonecancellationspectralresults:Th erangingtonemodulated inputareplottedalongwiththeresultingTDIX 2 combinationsforthefour primaryexperiments(Table 6-1 ).Thedecreasedaccuracyofthe 2 armas comparedtothe 3 arminTable 7-1 islikelyduetotheproximityofthe nearestarmzero.NotethattherangingtonefortheTDI1.0wi th phase-lockedlaserssimulationisat1.5Hzinsteadof1Hzwh ichresultsina reducedone-wayrangingaccuracy(Table 7-2 )fromthereducedPLLnoise. bytheLISAmissionconceptdesign.(Table 3-1 )The 0.65 ns( 0.2m)rangingprecision, ascalculatedinTable 7-1 andplottedinFigure 7-5 ,isnotexpectedtobealimitingnoise source. TheTDIX 1 'snoisesuppression(Figure 7-6 )equalsthetheoreticalinter-SC velocitydependentlimit( 4–32 )whiletheTDIX 2 'snoisesuppressionequalsthatof thesimulator'sbaselinesuppressioncharacteristics. 7.2LISA-like(Master-SlavePhaseLockedLaser)TDISimula tions Atthispoint,weincludethephase-lockingandtransmissio nofthe Laser 2 = 3 signals onthefarspacecraftasshowninFigure 7-7 .Thus,thisexperimentwillgenerateand measureallfour s sr 5 beatnoteobservablesasdenedin( 3–70 ).Thetwofarsensor 5 Twolocal-SCmeasurements: s 21 s 31 ;Twofar-SCmeasurements: s 12 s 13 158

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/ Hz Pre-Stabilized Laser Noise f 1 (t) TDI 1.0 Combination TDI 1.0 Limit ( Db =250 ns/s) Simulator Baseline Corrected TDI 2.0 Combination TDI 1.0 TDI 2.0 PM Requirement Pre-Stabilization Requirement Ranging Limit (0.65 ns) IMS Requirement Figure7-5.Dynamictransponderexperimentalresults:The suppressionoftheTDI X 1 combinationislimitedbythearm-lengthtime-dependencea sdenedby ( 4–32 )with =250 ns = s .TheTDI X 2 combinationremovestheadditional time-dependent-coupledlasernoiseandrevealsthe 6.277 mHz GWsignal. Aswiththestaticcase(Figure 7-3 ),theEPDunit'sphasetransmission accuracyistheprimarylimitingnoisesourceintheTDIcomb inations although,somesensitivitylossmayoccurduetoalimitedra ngingcapability around100mHz(Figure 7-9 signals, s 12 s 13 ,areusedfor,andareequalto,thephasenoiseofthePLLlase r-locking errorsignal.Meanwhile,theheterodynedPDoutputreprese ntsthelocallaserphase andisusedastheinputforthe'return'eldsimulationofin ter-SClighttransmission. ThesesignalsareusedtoconstructtheTDIcombinationsand minimizetheranging toneusingtheiterativeprocessdescribedinChapter 4.4.2.2 andshowninFigure 4-3 througha4-dimensionalsweepoftheindividuallight-trav eltimes, 2 ( t ) 2 0 ( t ) 3 ( t ) 3 0 ( t ) 7.2.1Static-ArmLISA-likeSimulation Thereadershouldnotethatthesemeasurementsweretakenea rlierthanboth thosepresentedinChapters 7.1.1 7.1.2 ,and 7.2.2 .Asaresult,therangingtonewas 159

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10 -4 10 -3 10 -2 10 -1 10 0 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 Frequency (Hz)Suppression Magnitude (|X( w )|/|s sr ( w )|) TDI 1.0 Db =0 TDI 2.0 Db =0 TDI 1.0 Db =250ns/s TDI 2.0 Db =250ns/s Expected TDI 1.0 Limit ( Db = 250ns/s, t = 16.6s) Figure7-6.TDIlaserphasenoisesuppression:Theachieved lasernoisesuppression magnitudesforthetranspondersimulationsofthestatican ddynamicTDIX 1 andTDIX 2 combinationsareplotted.TheTDIX 1 combination'snoise suppressionequalsthetheorizedlimit( 4–32 )inthedynamic-armsimulation. TheTDIX 1 andTDIX 2 resultinthesimilarlasersuppressioncharacteristics inthestatic-armsimulation.Thedynamic-armTDIX 2 combination'slaser noisesuppressionequalsthatofthestaticcase,verifying thatthevelocity coupledlasernoisehasbeencompletelyaccountedforandre moved. placedatadifferentfrequency (1.5 Hz ) andtheEPDdata-ratewashigher.Despite thesechanges,theeffectsareonlynoticeablethroughther educedPLLranging accuracy.Thesesameresultshavealsobeenpublishedinpee r-reviewedliterature [ 104 ]. Theoptimizedoutcomeoftherangingprocessisfoundaftera singleiteration andresultsinameasureof 2 toanaccuracybetterthan 70 fs = s andaround-trip rangingprecisionoflessthan 5.0 ns ( 1.5 m )asshowninTable 7-2 .Applyinga linearregressiontothecalculatedone-waydelaytimeswe ndaone-wayranging errorof 100 s ( 30 km ).Theoutgoingandreturndelaytimesareun-equalby 160

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Figure7-7.ModeloftheLISA-likeTDIexperimentalbenchto p:ThisTDI-PLLmodel showshowtheindividualpre-stabilizedlasereldsarecom binedtoformthe inputnoisesignalandhowthissignalisprocessedbytheEPD unitsand usedtophase-lockthe'far'SClasers,thusreplicatingall fourLISA-like inter-SClaserlinkandproduceallfour s sr observables.Thesetupisthe sameasisdenedinFigure 7-1 expectthatthe'transmitted'EPDoutputs areusedtophase-lockthefar Laser 2 and Laser 3 .ThePLLerrorsignalsare exactlyequaltothefarSCsensorsignals, s 12 and s 13 .The Laser Ref = Laser 2 = 3 beatnotesareusedastheinputsignalsforthe'return'eld lighttransmission simulation. 250 0.1 ms ,provingtheTDI X 1.5 combination's 6 abilitytoextractindividualphase errorsdespiteun-equaldelaysalongasinglearm ( q (0) 6 = q 0 (0)) Usingthecalculatedone-wayfunctionaltime-delayswefor mtheTDI X 1.0 and TDI X 2.0 combinations;re-scalingtheTDIspectraldensitybythein versearmtransfer functionsin( 7–3 ),weploteachTDIcombinations'phasenoiseinFigure 7-8 resulting inthesamemeasuredsensitivityforbothcombinations.Thi snoiselevelislikelydue 6 Table 4-1 161

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/ Hz Pre-Stabilized Laser Noise f 1 (t) Total PLL Noise TDI 1.0 Combination Simulator Baseline Corrected TDI 2.0 Combination TDI 1.0 Laser Ranging Limit (3 ns) One-Way Ranging Limit (1 ms) 'Far' PLLs Figure7-8.StaticLISA-likelaserexperimentalresults:T heTDIspectralresultsand comparisonswiththeinputlaserphasenoiseandPLLnoiseof the static-armTDIsimulationwithphase-lockedlasersisplot ted.Thecorrected TDI-2.0combinationusingthevaluesfromtheTDI-ranginge stimation (Table 7-2 )matcheswiththebaselineperformancefromFigure 7-3 .The laserandPLLnoisesuppressionasaresultoftheestimatedr anging precisionisnotexpectedtobealimitingnoisesourcewhich isveriedby cross-correlatingtheTDIcombinationwiththeinputsigna lsasshownin Figure 7-9 .IfthePLLnoiseisnotaccountedfor,theTDIcombinationis limitedbytheseadditionalPLLnoisesources(notpictured ). tothecouplingofmultipleun-correlatedEPDclocknoiseso urces,denedin( 6–11 ), intothemeasurement.Basedonthepoorcross-correlations betweentheinputnoise andthethePLLnoisesourceswiththeresultingTDIcombinat ionsweconcludethat therangingprecisionisnotalimitingnoisesourceandthat alltheknownsourceshave beenremovedfromthenalcombination.Wealsonotethatthe expectedranging precisionusingtheround-tripandone-wayvariancesalong with( 4–6 )iswellbelowthe TDIcombinationasplottedinFigure 7-8 162

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10 -3 10 -2 10 -1 10 0 0 0.2 0.4 0.6 0.8 1 Frequency (Hz)C xy (Magnitude Squared Coherence)TDI Coherence Functions f 1 vs. TDI-X 2 (Static Transponder) f 1 vs. TDI-X 2 (Static LISA-like) f PLL2 vs. TDI-X 2 (Static LISA-like) f PLL3 vs. TDI-X 2 (Static LISA-like) Figure7-9.Cross-correlatedTDIcombinations-StaticTra nsponderandLISA-like:The magnitudesquaredcross-correlationofthemeasuredlaser phasenoiseand PLLnoiseshowapoorcorrelationwiththeTDIX 2 combinationfor frequenciesbelowforallfrequenciesinthestaticTDI-PLL simulation(4-way delayestimate:Chapter 7.2.1 ).Thelasernoisecorrelateswiththecalculated TDIX 2 combinationforfrequenciesabove100mHzinthestatic TDI-transpondermeasurement(2-waydelayestimate:Chapt er 7.1.1 ).The poorcorrelationoftheinputnoisesourceswiththeTDIcomb inationsverify thattheinputnoiseshavebeensufcientlyremovedusingth eTDI-ranging arm-lengthestimate. 7.2.2Dynamic-ArmLISA-likeSimulation Afterthreeiterationsoftherangingprocess,theoptimize destimationofthe inter-SCdelaytimesresultsinameasureof 2 toanaccuracybetterthan 70 fs = s andaround-triprangingprecisionof 5.0 ns ( 1.5 m )asshowninTable 7-2 .The constraint'sontheone-waydelaytimesthroughtheresidua lPLLnoiseremovalcan notbeapplieduntiltheprecisionoftheround-tripdelayti mesareaccurateenough toremovetheinputlasernoisefromtheTDIcombinationstor evealtheresidualPLL noisesintheTDIcombinations.Thus,itrequiresatleaston eiterationoftheranging processuntiltheone-waydelaytimesbegintobeconstraine d.Atthesametime, theaccuracyoftheconstrainedone-waydelaytimeswillbes ignicantlyreduced incomparisontotheconstraintsontheround-tripdelaytim esresultingfromthe comparativelyreducednoisesuppressionmagnitudeoftheP LLnoise( 4–6 ).Applying 163

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alinearregressiontothecalculatedone-waydelaytimesaf terthethirditerationwend aone-wayrangingerrorofabout 100 s ( 30 km ).Again,theoutgoingandreturn delaytimesareun-equalby 250 0.1 ms ,provingthecapabilitytoextractindividual SCphaseerrorwithun-equaldelaysalongeacharm ( q (0) 6 = q 0 (0)) Applyingtheoptimizedone-wayfunctionalvaluesfromther angingprocedure, outlinedinTable 7-2 ,wegeneratetheTDIX 1 andTDIX 2 combinations(Figure 7-10 ). Rescalingbytheinversearmresponsefunction,theTDIX 1 combinationmeetsthe expectedlimitation( 4–32 ).TheTDIX 2 combinationmeetstheLISAIMSrequirement towithinafactorof4andis,again,limitedbyacombination ofmultipleEPDclock noisecouplingsourcesresultinginareducedsensitivitya bovethesimulator'sbaseline performance.Again,thepoorcorrelationoftheinputnoise sourceswiththenal combination,Figure 7-11 ,leadsustoconcludethattherangingprecisionsarenota limitingnoisesourceandthatallknownnoisesourceshaveb eenremovedfromthenal combination.Furthersupportingthisclaim,theexpectedr angingprecisionusingthe round-tripandone-wayvariancesalongwith( 4–6 )iswellbelowtheTDIcombinationas plottedinFigure 7-10 BinaryConfusionNoiseInjection Finally,weintroducethelowfrequencysimulated'confusi onnoise'intothe measurementtoensurethattheselow-frequencytermsdonot limittheranging precision.Theconfusionnoisebackgroundgivenby: h Background ( ) .01 s 2 +.001 mHz p Hz (7–5) where's'isthecomplexLaplacefrequency,and 6.277 mHz mono-chromaticbinary withasingle-armstrainamplitudeof 3.5 10 20 aresimultaneouslyinjected.The injectedbackgroundnoiselevelissettomatchmostestimat ionsinthe0.1mHz to1.5mHzbutdoesnotrolloffasquicklyatfrequenciesabov e1.5mHzasmost confusionnoiseestimatesdo(Figure 8-1 )[ 46 103 ].Thisismodiedfromtheexpected 164

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Table7-2.LISA-likeTDI-rangingestimation:Themethodou tlinedinFigure 4-3 andtheexperimentsdescribedin capters 7.2.1 and 7.2.2 areusedtocalculateestimatesoftheinter-SCdelayfuncti ons,( 4–37 and 4–39 ),and theassociatedrangingerrors.TheestimatesverifytheTDI -rangingcapabilitiesandareusedtogeneratethe TDI-combinationsforFigures 7-8 and 7-10 Iteration (Slope) 22 0 (0), 33 0 (0) (Offset) 22 0 33 0 : 2 2 0 3 3 0 ( ) TDI1.0PLL 1(TDI1.0) 2 2 = 66.1 fs = s 45.4 fs = s 22 0 (0)=33.5508007898 s 1.01 ns 22 0 =3.2 ns (Round-trip) 2 3 =58.7 fs = s 37.4 fs = s 33 0 (0)=33.2152647154 s 0.83 ns 33 0 =2.6 ns 1(TDI1.0) 2 = 4.07 ns = s 13.6 ns = s 2 (0)=16.694054656 s 301 s 2 0 = 2 =1.12 ms (One-Way) 2 0 (0)=16.8567461 s 301 s 3 =26.1 ns = s 15.9 ns = s 3 (0)=16.51799231 s 352 s 3 = 3 0 =0.95 ms 3 0 (0)=16.6972724 s 352 s TDI2.0PLL 1 2 2 = 199.984 ns = s 12 ps = s 22 0 (0)=33.59821021 s 0.28 s 22 0 =0.895 s 2 3 =+300.052 ns = s 7.8 ps = s 33 0 (0)=33.21476669 s 0.18 s 33 0 =0.568 s 2 2 2 = 200.000015 ns = s 71 fs = s 22 0 (0)=33.5982645303 s 1.6 ns 22 0 =5.2 ns 2 3 =+300.000013 ns = s 26 fs = s 33 0 (0)=33.2146434958 s 0.58 ns 33 0 =1.9 ns 3(TDI-2.0) 2 2 = 200.000028 ns = s 68 fs = s 22 0 (0)=33.5982645401 s 1.5 ns 22 0 =5.0 ns (Round-trip) 2 3 =+300.000020 ns = s 25 fs = s 33 0 (0)=33.2146435166 s 0.58 ns 33 0 =1.9 ns 3(TDI-2.0) 2 = 103.3 ns = s 1.4 ns = s 2 (0)=16.68021 s 31 s 2 = 2 0 =99 s (One-Way) 2 0 (0)=16.91805 s 31 s 3 =+152.24 ns = s 1.4 ns = s 3 (0)=16.48824 s 32 s 3 = 3 0 =105 s 3 0 (0)=16.72640 s 32 s ConfusionNoise TDI2.0 3(Round-Trip) 2 2 = 199.999991 ns = s 50 fs = s 22 0 (0)=33.6012734891 s 1.1 ns 22 0 =3.7 ns 2 3 =+300.000137 ns = s 22 fs = s 33 0 (0)=33.2100302983 s 0.49 ns 33 0 =1.6 ns 3(One-Way) 2 = 96.81 ns = s 2.3 ns = s 2 (0)=16.73546 s 53 s 2 = 2 0 =169 s 2 0 (0)=16.86582 s 53 s 3 =+149.439 ns = s 1.4 ns = s 3 (0)=16.53994 s 32 s 3 = 3 0 =102 s 3 0 (0)=16.67009 s 32 s 165

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10 -4 10 -3 10 -2 10 -1 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz)Cycles/ Hz Pre-Stabilized Laser Noise f 1 (t) Total PLL Noise TDI 1.0 Combination TDI 1.0 Limit ( Db =250 ns/s) Simulator Baseline Corrected TDI 2.0 Combination Laser Ranging Limit (5 ns) One-Way Ranging Limit (100 m s) TDI 2.0 'Far' PLLs Pre-Stabilized Laser Figure7-10.DynamicLISA-likeexperimentalresults:Thes uppressionoftheTDI X 1 combinationislimitedbytheexpectedarm-lengthtime-dep endence denedby( 4–32 )with =250 ns = s .TheTDI X 2 combinationremoves theinputlasernoise,the'far'PLLresidualphasenoise,an dthe time-dependentcoupledlasernoisetorevealthe 6.277 mHz GWsignal. Thesensitivitylimitationcomes,mostlikely,asaresulto fmultiple unaccountedforEPDnoiseasaresultofthetime-changingde lay.Thisis determinedbasedonacomparativelyimprovedperformanceu singthe static-PLLsimulationandthefactthatthetime-changingd elayisthemajor differencebetweenthesemeasurements. situationtoensurethattheadditionlow-frequencynoised oesnotlimittheranging precision.ThemonocromaticGWphasemodulationduetotheR XJ0806.3+1527 binaryisafactorof100smallerthantheexpectedstrainamp litudeaveragedover a1yearmeasurement.Thatsaid,theGWphasemodulationampl itudeissimply programmedtobeout-of-phasewheninjectedintotheindivi dualarmssimulatorand withnoconsiderationfortheGWpolarizationordetectoror ientationoverthecourseof theyear. Again,theoptimizedrangingestimationplacesboundsonth e accuracybetter than 50 fs = s andaround-triprangingprecisionof 3.7 ns ( 1.1 m ).Thus,this 166

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10 -3 10 -2 10 -1 10 0 0 0.2 0.4 0.6 0.8 1 Frequency (Hz)C xy (Magnitude Squared Coherence)TDI Coherence Functions f 1 vs. TDI-X 2 (Dynamic Transponder) f 1 vs. TDI-X 2 (Dynamic LISA-like) f PLL2 vs. TDI-X 2 (Dynamic LISA-like) f PLL3 vs. TDI-X 2 (Dynamic LISA-like) Figure7-11.Cross-correlatedTDIcombinations-DynamicT ransponderandLISA-like: Themagnitudesquaredcross-correlationofthemeasuredla serphase noiseandPLLnoiseshowapoorcorrelationwiththeTDIX 2 combination forfrequenciesbelowforallfrequenciesinthedynamicTDI -PLLsimulation (4-waydelayestimate:Chapter 7.2.2 ).Aswiththestaticcase(Figure 7-9 ), thelasernoisecorrelateswiththecalculatedTDIX 2 combinationfor frequenciesabove100mHzinthedynamicTDI-transponderme asurement (2-waydelayestimate:Chapter 7.1.2 ).Thepoorcorrelationoftheinput noisesourceswiththeTDIcombinationsverifythattheinpu tnoiseshave beensufcientlyremovedusingtheTDI-rangingarm-length estimate. confusionnoiseresultachievesarangingprecisiononthes ameorderasthesimulator's phase-lockedperformance,provingthatlow-frequencynoi sehaslittletonoeffectonthe rangingtonecancellationorthemeasuredarm-lengths. Finally,oncethelong-termresidualquadraticphasedrift shavebeenremoved fromthestrainmeasurement,thetime-domaincomparisonsb etweentheTDIX 2 outputsofthephase-lockedandconfusionnoiseexperiment sareplottedinFigure 7-12 toshowtheadditionallow-frequencynoise.TheseTDI X 2 noisespectrawithand withouttheconfusionnoiseareplotted,Figure 8-1 ,inparticular,intermsoftheactual LISAlengthstrainin cycles = p Hz alongwiththelow-frequencyaccelerationnoise, mid-frequencyshot-noise,andscaledbythehigh-frequenc ysensitivityloss.We alsoplotthetheoretical'1-year'averagedstrainofafewa ctualgravitationalwave sourcesandtheexpectedGWconfusionbackground[ 103 ]incomparisonwiththe 167

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 -4 -3 -2 -1 0 1 2 3 4 x 10 -3 time (s)Cycles Phase-Locked (Monochromatic Binary) Phase-Locked (Monochromatic Binary + Confusion Noise) Figure7-12.Confusionnoisetime-seriescomparison:Thet imeseriesofthedynamic LISA-likesimulationswithandwithoutaconfusionnoiseba ckground (Chapters 7.2.2 and 7.2.2 )areplottedexplicitlyafterlow-passlteringthe aliasednoisenearthesampling-frequency.Theaddedlow-f requency confusionnoiseisreadilyapparentwhencomparingtheresu ltsofthe measured 6.277 mHz binarywiththebinaryplusconfusionnoise( 7–5 ). injectedbackground( 7–5 )andGWsignal.Thedeviationinthespectralamplitude ( 7 mCyc = p Hz )iscausedbyspreadingoftheGWintoapproximately3freque ncy bins 7 7 204 cycles = Hz p 10000 s =20.4 mCyc = p Hz 3 7 mCyc = p Hz 168

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CHAPTER8 CONCLUSION InthisdissertationwehavepresentedhowLISAandotherspa ce-basedinterferometerswillachievethestrainsensitivitiesrequiredtome asureastronomicallyinteresting GWsourcesusingadvancedstabilizationtechniquessuchas cavitylockingandarm lockingaswellasthroughtheapplicationofpost-processi ngtime-delayinterferometry combinations.WehaveconsideredhowtheLISA-IMSscienceo bservablesare formedandhowthedifferentialproof-masslengthisconstr uctedfromtheseindividual heterodynePDobservables.Wehavepresentedthedesign,ph aseprecision,andlikely low-frequencylimitingnoisesourcesofLISA-likephaseme tersandhowtheyareusedin heterodyneinterferometry.Amethodofusingthephasemete rstoelectronicallysimulate theinter-SClasereldtransmissioncharacteristicsincl udingthetime-changingdelay, Dopplershift,andGWmodulationwasdeveloped,tested,and usedtoperformsomeof thersthardwarebasedarm-lockingexperiments. Time-delayinterferometrysimulationsusing cycle phasemeters,multi-second delayEPDunits,andtheUFLISlaserbenchtopweredeveloped andcomparedto thenoisesourcesexpectedinLISA.Thesimulationstestedt heabilityoftime-delay interferometrycombinationstocancelthelaserphasenois eandextracttheGW informationwith,both,static(constantarm-length)andd ynamic(time-changing arm-length)LISA-likecharacteristics.Wehavealsoprese ntedasimplebutpowerful methodofestimatingthetime-dependentarm-lengthsusing aTDI-rangingreference tonemodulatedontothelasereldatfrequenciesabovetheG Wfrequencymeasurement band.Thepost-processingformationoftheTDIX 2 combinationwhichoptimallycancels therangingtoneprovidesafunctionalconstraintonthetim e-changinginter-SCdelays toanaccuracybetterthanthatrequiredtoremovethelaserp hasenoisefromtheTDI combinations. 169

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10 -4 10 -3 10 -2 10 -1 10 0 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Frequency (Hz)Cycles/ Hz Simulator Baseline Phase-Locked 4-Way TDI 2.0 Confusion Noise 4-Way TDI 2.0 Injected Confusion Noise Limit Theoretical Confusion Noise Limit RX J0806 V407 Vul ES Cet AM CVn Figure8-1.CompiledresultsandcomparisonwithTDIforLIS A:Inthisgurewehave compiledalltheresultsoftheTDIsimulationsandattemptt omakeadirect comparisonwiththeexpectedLISAstrainsensitivity.Theb aselinespectral noiseoftheUFLISsimulator(grey-blue)fromtheTDI-Trans ponder (Chapter. 7.1.1 )measurementsisplotted.ThevelocitycorrectedTDIX 2.0 spectrumofthedynamicarmTDIsimulationwith(cyan)andwi thout(blue) theinjectedbinaryconfusionnoise(dotted-magenta)ispl ottedin comparisonwiththeIMSsensitivityrequirement.Thethree TDIsimulations arescaledbythefrequencyfactorsof( 3–2 )toaccountforthe high-frequencyGW-sensitivitylossexpectedinLISA.TheD RSandIMS requirementareroot-squaresummedandscaledby( 3–2 )toproducethe effectivesingle-linkLISAsensitivitywith(black)andwi thout(dotted-black) the 2 p 5 = 3 sensitivityfactor.Anestimateoftheconfusionnoiselimi tis plotted(dotted-red)alongwiththefourbrightestverica tionbinariesrescaled froma1-yearaveragedstrainsensitivitytonoisespectrai n cycles = p Hz .The strainmagnitudeofthe 1 year averagedRX-J0806binaryandthe 10000 s EPDinjectedGWhaveamplitudessuchthattheyresultinsimi larLSD amplitudesinthisgure.Theresultsoftheseexperimentsv erifytheabilityto accountfora 0.4 s ( 0.1 Gm )un-equalarm-lengthmis-matchand 75 m = s velocitycoupledlasernoisebyTDI-rangingtheinter-SCdi stancetoan accuracyof 5 ns ( 1.5 m )andformingtheTDIX 2 combinationtoextractthe GWinformationtowithinafactorof5ofthesingle-linkLISA sensitivity. RX-J0806 4 10 22 = Hz p 1 year =2.2 10 18 = p Hz =10.5 mCyc = p Hz EPD-GW 4.3 10 20 = Hz p 10000 s =4.3 10 18 = p Hz =20.5 mCyc = p Hz 170

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Theexperimentalresultsshowthatmorethan10ordersofmag nitudeoflaser frequencynoisecanbecanceledusingappropriatelytime-s hifteddatastreamswith theTDI X 2 datacombination.Wehavealsoveriedthattheabilitytoca ncelthelaser frequencynoiseusingaTDI X 1 datacombinationislimitedbytherelativevelocities betweenthespacecraft.Meanwhile,wehavealsodemonstrat edthattheTDIX 2 combinationaccountsforthisinter-SCvelocitylimitedTD I X 1 combination.The simulationshavealsoveriedtheremovaloftheresidualph aselockloopnoiseadded atthefarspacecraftandhaveshowedthatthePLLnoiseextra ctionispossiblewith relaxedone-wayrangingrequirementsdueascomparedtothe round-triprequirements. TheresultsoftheseTDIexperimentsdemonstratedtheabili tytoaccountfora 0.4 s ( 0.1 Gm )un-equalarm-lengthmis-matchand 75 m = s velocitycoupledlasernoiseby TDI-rangingtheinter-SCdistancetoanaccuracyof 5 ns ( 1.5 m )byformingtheTDIX 2 combinationtoextracttheGWinformationtowithinafactor of5ofthesingle-linkLISA sensitivity. IntheprocessofdevelopingtheUFLIS-TDIsimulator,wehav edevelopedand testeddataanalysistoolswhichusetherawphasemeterdata streamstoextract thelight-traveltimefunctionandgeneratetheTDIX 2 datastreams.Wehavealso addedaconfusionnoiseGW-backgroundtotheTDIsimulation sandveriedthatthis low-frequencybackgrounddoesnotinterferewithourrangi ngcapabilities. Thecombinationoftheseexperiments,validatingthemulti -secondtime-changing inter-SCcharacteristics,andtheexperimentsperformedi n[ 105 ],verifyingtheoptical noisecouplings,representtheessentialcharacteristics ofaLISA-likeinterferometry measurementsystemandthepost-processingcancellationo flaserphasenoiseusing theappropriateTDIcombinations.FutureUFLISexperiment sshouldincludereal, LISA-likeGWsignalsusingdata-setsgeneratedwithLISA-t oolslikeSyntheticLISA [ 55 ].Theuseofindependentclocksourcesandthevericationt hatclocknoisesources canbeaccountedforintheTDIcombinationswouldalsobeuse ful.Finally,simulations 171

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withthreeindependentlystabilizedlasersmightalsobeva luabletowardsverifyingthe constrainsontheone-wayrangingcapabilities. 172

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APPENDIXA TIMEVARYINGFRACTIONALDELAYINTERPOLATIONFUNCTION function[xOut,tOut]=TVFDfilter(x,fs,beta,Tau,N)%[xOut,tOut]=TVFDfilter(x,fs,beta,Tau,N)%%Computesatime-stretched/compressedandtime-shiftedd ata-set %withanN-pointLagrangefractiondelayfilterasdefinedi n %"Post-processedtime-delayinterferometryforLISA"byS haddocket.al. %%INPUTS:%x=timeseriesinput[amplitude]%fs=samplingfrequency[Hz]%beta=fractionalshiftrate(ins/s)%Tau=absolutetime-shiftingvalueinseconds%N=interpolationlength(numberofpointstousein%FractionalDelayFilter)%%OUTPUTS:%xOut=outputamplitudevector%tOut=outputtimevector%%IraThorpe,ShawnMitryk%Updated2-18-12L=length(x);M=(N-1)/2;k=(-(N-1)/2):1:((N-1)/2);offset=floor(Tau*fs);Dfrac=-(Tau-(offset/fs))*fs;xOutShift(1:L-N+1)=0; fori=1:L-N+1 DTaufrac=beta*i;td=(N-1)/2+Dfrac-DTaufrac;b1=gamma(td+1)/(gamma(N+1)*gamma(td-N+1));b2=gamma(N)./(gamma(k+(N-1)/2+1).*gamma(N-k-(N-1)/2 )); w=((pi*N)/(sin(pi*td)))*b1*b2;h=sinc(Dfrac-DTaufrac-k).*w;xOutShift(i)=sum(h.*(x(i:i+N-1)')); end 173

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xOut=xOutShift(N+M+1:L-N+1)';tOut=(0:length(xOut)-1)'/fs; Anup-to-dateversionofthesefunctionsandtheirapplicat ioninproducingthe resultsdemonstratedintheseexperimentscanbeobtainedf rom[ 106 ]. 174

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APPENDIXB TDI2.0COMBINATIONFUNCTION function[TDI2Out,TDI1Out]=TDIComb(fs,s21,s31,s12,s1 3,tau33, tau13,beta3o,tau22,tau12,beta2o) %[TDI2Out,TDI1Out]=TDIComb(fs,s21,s31,s12,s13,tau33 %tau13,beta3,tau22,tau12,beta2)%%ComputestheTDI1.0andTDI2.0Combinationsbasedonthe6variable %arm-lengthdelays.Basedon"Datacombinationsaccountin gforLISA %spacecraftmotion"byShaddocket.al.%%INPUTS:%fs=Samplefrequency%%s21=Sensorsignal(sending:SC2receiving:SC1)%s31=Sensorsignal(sending:SC3receiving:SC1)%s12=Sensorsignal(sending:SC1receiving:SC2)%s13=Sensorsignal(sending:SC1receiving:SC3)%%tau33=Round-tripdelayfromSC1throughSC2(Arm3)%tau13=One-waydelaytimefromSC1toSC2(Arm3)%beta3=Arm3timecompressionfactor%beta=(velocitybetweenSC1&SC2/speedoflight)%tau22=Round-tripdelayfromSC1throughSC3(Arm2)%tau12=One-waydelaytimefromSC1toSC3(Arm2)%beta2=Arm2timecompressionfactor%%OUTPUTS:%TDI2Out=TDI2.0Combination%TDI1Out=TDI1.0Combination%%ShawnMitryk%Updated2-18-12L1=length(s12);N=51;M=(N-1)/2;alpha2=(1-2*beta2o);alpha3=(1-2*beta3o);gamma2=(1-beta2o);gamma3=(1-beta3o); 175

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beta3=1-alpha3;beta2=1-alpha2;%TDI2.0shiftvaluesTau23=(tau22)*alpha3+(tau33)beta23=1-(alpha2*alpha3)Tau32=(tau33)*alpha2+(tau22)beta32=1-(alpha3*alpha2)Tau232=alpha2^2*tau33+alpha2*tau22+tau22beta232=1-(alpha2^2*alpha3)Tau323=alpha3^2*tau22+alpha3*tau33+tau33beta323=1-(alpha3^2*alpha2)%Calculatethesensorsignalsfromtheone-waysignals[chan12.data,chan12.t]=TVFDfilter(s12,fs,beta2o,tau 12,N); [chan13.data,chan13.t]=TVFDfilter(s13,fs,beta3o,tau 13,N); chan21.data=s21(2*N:L1-M);chan31.data=s31(2*N:L1-M);chan22.data=chan21.data+chan12.data;chan33.data=chan31.data+chan13.data;%FormtheTDICombinationsbasedonthe6-variabledelaysL2=length(chan22.data);[chan2_2s.data,chan2_2s.t] =TVFDfilter(chan22.data,fs,beta2,tau22,N); [chan3_3s.data,chan3_3s.t] =TVFDfilter(chan33.data,fs,beta3,tau33,N); [chan22_23s.data,chan2_23s.t] =TVFDfilter(chan22.data,fs,beta23,Tau23,N); [chan33_32s.data,chan3_32s.t] =TVFDfilter(chan33.data,fs,beta32,Tau32,N); [chan22_232s.data,chan2_232s.t] =TVFDfilter(chan22.data,fs,beta232,Tau232,N); 176

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[chan33_323s.data,chan3_323s.t] =TVFDfilter(chan33.data,fs,beta323,Tau323,N); chan22a.data=chan22.data(2*N:L2-M)';chan33a.data=chan33.data(2*N:L2-M)';chan22_2a.data=chan22_2s.data;chan33_3a.data=chan33_3s.data;chan22_23a.data=chan22_23s.data;chan33_32a.data=chan33_32s.data;chan22_232a.data=chan22_232s.data;chan33_323a.data=chan33_323s.data;tdi1a.data=chan22a.data+chan33a.data -chan22_2a.data-chan33_3a.data; tdi1b.data=-chan22_23a.data-chan33_32a.data +chan22_232a.data+chan33_323a.data; tdi2.data=tdi1a.data+tdi1b.data;TDI1Out=tdi1a.data;TDI2Out=tdi2.data; Anup-to-dateversionofthesefunctionsandtheirapplicat ioninproducingthe resultsdemonstratedintheseexperimentscanbeobtainedf rom[ 106 ]. 177

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BIOGRAPHICALSKETCH ShawnMitrykwasborninWillingboro,NJinthesummerof1983 .Growingup inclosecontactwithalargefamily,Shawnwasalwayssurrou ndedbypeoplewho weremorethanhappytosatisfyhiscuriosity.Aftermovingt oFlorida,Shawnbecame interestedinmanyaspectsofboththeartsandsciences,and eventually,foundhisniche inphysics.Afterobtaininganengineeringbackgroundeduc ationfromtheUniversity ofFloridain2006,ShawncontinuedtoobtainaPhD.inexperi mentalphysicswiththe LaserInterferometerSpaceAntennaProjectinMayof2012. 186