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Arm Locking for Laser Interferometer Space Antenna

Permanent Link: http://ufdc.ufl.edu/UFE0042841/00001

Material Information

Title: Arm Locking for Laser Interferometer Space Antenna
Physical Description: 1 online resource (214 p.)
Language: english
Creator: YU,YINAN
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ARMLOCKING -- GENERALRELATIVITY -- GRAVITATIONALWAVE -- LASERINTERFEROMETRY -- LISA
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 Laser Interferometer Space Antenna (LISA) is a collaborative National Aeronautics and Space Administration (NASA)/European Space Agency (ESA) space mission to detect gravitational waves in the frequency region of 0.03 mHz to 1 Hz by means of laser interferometry. It will be the first space-based interferometric gravitational wave detector to be launched in 2020s. LISA will consist of three identical spacecraft arranged in a quasi-equilateral triangular constellation with 5 Gm on each side. Each spacecraft houses two drag-free proof masses that follow the geodesic motion. The Interferometric Measurement System (IMS) of LISA monitors changes in the proper distance between two proof masses on each respective spacecraft. Laser frequency stabilization is one of the most significant and difficult issues for the IMS of LISA. Arm locking as a proposed frequency stabilization technique, transfers the stability of the long arm lengths to the laser frequency. The arm locking sensor synthesizes an adequately filtered linear combination of the inter-spacecraft phase measurements to estimate the laser frequency noise, which can be used to control the laser frequency. Due to the large propagation delay during the light transmission, the arm locking controller needs to be carefully designed to retain enough phase margin. A potential problem for arm locking is that the Doppler shift of the return beam will cause a constant pulling in the master laser frequency if unaccounted for in the phase measurement. Until now all the benchtop experiments on arm locking verified only the basic single arm locking configuration with unrealistic short delay time and without any Doppler estimation error at the phasemeter. At the University of Florida we developed the hardware-based University of Florida LISA Interferometer Simulator (UFLIS) to study and verify laser frequency noise reduction and suppression techniques under realistic LISA-like conditions. These conditions include the variable Doppler shifts between the spacecraft, LISA-like signal travel times, far-end heterodyne phase-locking, realistic laser frequency and timing noise. In this dissertation we will systematically introduce the cutting edge of experimental studies of arm locking under these realistic conditions. We have built an analog/digital hybrid system to demonstrate the control system of various arm locking schemes and their incorporation with pre-stabilization subsystems. We measured the noise suppression in our experiments as well as the frequency pulling in the presence of Doppler frequency error. With the achievement of meeting the requirement, our pioneering work have sufficiently demonstrated the validity and feasibility of arm locking under LISA-like conditions.
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 YINAN YU.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Muller, Guido.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042841:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042841/00001

Material Information

Title: Arm Locking for Laser Interferometer Space Antenna
Physical Description: 1 online resource (214 p.)
Language: english
Creator: YU,YINAN
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ARMLOCKING -- GENERALRELATIVITY -- GRAVITATIONALWAVE -- LASERINTERFEROMETRY -- LISA
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 Laser Interferometer Space Antenna (LISA) is a collaborative National Aeronautics and Space Administration (NASA)/European Space Agency (ESA) space mission to detect gravitational waves in the frequency region of 0.03 mHz to 1 Hz by means of laser interferometry. It will be the first space-based interferometric gravitational wave detector to be launched in 2020s. LISA will consist of three identical spacecraft arranged in a quasi-equilateral triangular constellation with 5 Gm on each side. Each spacecraft houses two drag-free proof masses that follow the geodesic motion. The Interferometric Measurement System (IMS) of LISA monitors changes in the proper distance between two proof masses on each respective spacecraft. Laser frequency stabilization is one of the most significant and difficult issues for the IMS of LISA. Arm locking as a proposed frequency stabilization technique, transfers the stability of the long arm lengths to the laser frequency. The arm locking sensor synthesizes an adequately filtered linear combination of the inter-spacecraft phase measurements to estimate the laser frequency noise, which can be used to control the laser frequency. Due to the large propagation delay during the light transmission, the arm locking controller needs to be carefully designed to retain enough phase margin. A potential problem for arm locking is that the Doppler shift of the return beam will cause a constant pulling in the master laser frequency if unaccounted for in the phase measurement. Until now all the benchtop experiments on arm locking verified only the basic single arm locking configuration with unrealistic short delay time and without any Doppler estimation error at the phasemeter. At the University of Florida we developed the hardware-based University of Florida LISA Interferometer Simulator (UFLIS) to study and verify laser frequency noise reduction and suppression techniques under realistic LISA-like conditions. These conditions include the variable Doppler shifts between the spacecraft, LISA-like signal travel times, far-end heterodyne phase-locking, realistic laser frequency and timing noise. In this dissertation we will systematically introduce the cutting edge of experimental studies of arm locking under these realistic conditions. We have built an analog/digital hybrid system to demonstrate the control system of various arm locking schemes and their incorporation with pre-stabilization subsystems. We measured the noise suppression in our experiments as well as the frequency pulling in the presence of Doppler frequency error. With the achievement of meeting the requirement, our pioneering work have sufficiently demonstrated the validity and feasibility of arm locking under LISA-like conditions.
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 YINAN YU.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Muller, Guido.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042841:00001


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ARMLOCKINGFORLASERINTERFEROMETERSPACEANTENNA By YINANYU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011YinanYu 2

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Tomyparentswhoprovidedmeeverythinginmylifewithoutan yreservation 3

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ACKNOWLEDGMENTS Thisworkwouldnothavebeenpossibleifwerenotforthehelp frommanyUF graduatestudents,facultyandstaff.EspeciallyIwouldlik etoexpressmydeepest gratitudetomyPhDdegreesupervisor,Dr.GuidoMuellerwhoh asbeeneverlasting helpfulandhasofferedallkindsofassistance,supportand guidanceonmyresearchas wellasonthewritingofthisthesis. Iwouldalsoliketoshowmygratitudetomycommitteemembers ,Dr.Tanner,Dr. Whiting,Dr.KlimenkoandDr.Sarajedini,fortakingtheirprec ioustimetohelpmefulll myrequirementsforthePhDdegree.Ialsowishtogivespecial thankstomycurrent andformerlabcolleagues:VinzenzWand,ShawnMitryk,DylanSw eeneyandJoseph 'Pep'Sanjuan,fortheirsupportiveassistancesandproduct ivediscussions. Finally,Iwouldliketoexpressmyloveandgratitudetomyfa milyfortheirsupport andencouragement.Especially,Iowemypassionategratitud eandlovetomywife,also academicpartner,YiliangBao,forherheartfulandconsider ateassistancetobothmy lifeandcareer. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1GRAVITATIONALWAVEASTRONOMY ...................... 15 1.1Introduction ................................... 15 1.2GravitationalRadiationinGeneralRelativity ................. 19 1.2.1PropagationofGravitationalWaves .................. 19 1.2.2GenerationofGravitationalWaves .................. 21 1.2.3InteractionofGravitationalWaveswithTestMasses ........ 24 1.3SourcesofGravitationalWaves ........................ 26 1.3.1HighFrequencyRange ......................... 27 1.3.2LowFrequencyRange ......................... 28 1.3.2.1Galacticbinaries ....................... 29 1.3.2.2Coalescenceofmassiveblackholes ............ 33 1.3.2.3Extrememassratioinspirals ................ 34 1.3.2.4Cosmicgravitationalbackgroundradiation ......... 36 1.4DetectionofGravitationalWaves ....................... 37 1.4.1DetectionMethods ........................... 38 1.4.2InterferometricDetectors ........................ 41 1.4.2.1Principleandconguration ................. 42 1.4.2.2Noiselimitations ....................... 45 2LASERINTERFEROMETERSPACEANTENNA ................. 49 2.1Overview .................................... 49 2.1.1Sensitivity ................................ 50 2.1.2DisturbanceReductionSystem .................... 52 2.1.3InterferometricMeasurementSystem ................. 54 2.2NoiseCancellationforLISA .......................... 59 2.2.1TimeDelayInterferometry ....................... 59 2.2.1.1MichelsonX-combination .................. 60 2.2.1.2Sagnaccombination ..................... 62 2.2.1.3TDILimitations ........................ 63 2.2.2Pre-stabilization ............................. 64 2.2.2.1Pound-Drever-Hallfrequencystabilization ......... 65 2.2.2.2Mach-Zehnderfrequencystabilization ........... 67 5

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2.3ArmLocking .................................. 71 2.3.1Architecture ............................... 72 2.3.2ArmLockingSensors .......................... 74 2.3.2.1Singlearmlocking ...................... 74 2.3.2.2Commonarmlocking .................... 79 2.3.2.3Dualarmlocking ....................... 82 2.3.2.4Sagnacarmlocking ..................... 86 2.3.3FrequencyPulling ............................ 88 2.3.3.1Dopplerimpactonarmlocking ............... 88 2.3.3.2Frequencypullingrate .................... 90 2.3.3.3Modieddualarmlocking .................. 94 2.3.4IntegrationwithTunablePre-stabilizationReferenc es ........ 99 2.3.4.1ArmlockingwithPound-Drever-Hallstabilization ..... 99 2.3.4.2ArmlockingwithMach-Zehnderstabilization ....... 102 2.3.4.3Armlockingonly ....................... 102 2.3.5ArmLockingLimitations ........................ 103 2.3.5.1Limitedcontrollergaininfar-endPLL ........... 103 2.3.5.2Realisticnoisesourcesinarmlocking ........... 107 3UNIVERSITYOFFLORIDALISAINTERFEROMETERSIMULATOR ...... 115 3.1OpticalComponents .............................. 115 3.2ElectronicComponents ............................ 117 3.2.1DigitalSignalProcessingHardware .................. 117 3.2.2Phasemeter ............................... 118 3.2.2.1Design ............................ 119 3.2.2.2Performance ......................... 121 3.2.3ElectronicPhaseDelay ........................ 122 4EXPERIMENTALVERIFICATIONOFSINGLEARMLOCKING ......... 125 4.1Motivation .................................... 125 4.2PreliminaryTestWithNumericalControlOscillator(NCO) Tracking .... 127 4.2.1ExperimentalSetup .......................... 127 4.2.2DigitalFilterDesign ........................... 130 4.2.3MeasurementResults ......................... 132 4.2.4NoiseAnalysis ............................. 136 4.3SingleArmLockingIntegratedwithaTunableReference .......... 138 4.3.1HeterodynePhase-lockedLoop .................... 139 4.3.1.1Experimentalsetup ..................... 139 4.3.1.2Closed-loopdynamics .................... 141 4.3.1.3Resultsandanalysis ..................... 143 4.3.2PiezoelectricTransducer(PZT)ActuatedCavity ........... 146 4.3.2.1CharacterizationofthePZTcavity ............. 146 4.3.2.2Experimentalsetup ..................... 148 4.3.2.3Resultsandanalysis ..................... 150 6

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4.3.3Electro-opticModulator(EOM)SidebandLocking .......... 152 5EXPERIMENTALVERIFICATIONOFDUAL/MODIFIEDDUALARMLOCKING 154 5.1CommonArmLocking ............................. 154 5.1.1CommonArmLockingSensor ..................... 154 5.1.2PreliminaryTestwithNumericalControlOscillator(N CO)Tracking 156 5.2DualArmLocking ............................... 159 5.2.1DualArmLockingSensor ....................... 159 5.2.2PreliminaryTestwithNCOTracking .................. 161 5.2.3IntegrationwithPre-stabilizedLaser ................. 166 5.3ModiedDualArmLocking .......................... 168 5.3.1ModiedDualArmLockingSensor .................. 168 5.3.2PreliminaryTestwithNCOTracking .................. 170 5.3.3IntegrationwithPre-stabilizedLaser ................. 172 5.4ArmlockingIntegratedWithFar-endPhase-locking ............. 173 5.4.1TransponderNoiseFloor-TimeDomainSimulation ......... 175 5.4.2ExperimentalVerication-SimpleModel ............... 177 5.4.3ExperimentalVerication-FullModel ................. 179 6DOPPLERFREQUENCYERRORINARMLOCKING .............. 187 6.1DopplerFrequencyErrorinLISA ....................... 187 6.2InvestigationofFrequencyPullingonUFLIS ................. 189 6.2.1GenerationofDopplerFrequencyErrors ............... 190 6.2.2FrequencyPullinginSingleArmLocking ............... 191 6.2.3FrequencyPullinginDualandModiedDualArmLocking ..... 193 6.2.3.1Time-domainsimulationswithAC-coupledcontroll er ... 193 6.2.3.2Experimentswithmodieddualarmlockingsensor ... 196 7CONCLUSIONANDOUTLOOK .......................... 202 7.1ControlSystemofArmLocking ........................ 202 7.2NoiseLimitations ................................ 203 7.3Doppler-inducedFrequencyPulling ...................... 204 7.4Outlook ..................................... 205 REFERENCES ....................................... 207 BIOGRAPHICALSKETCH ................................ 214 7

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LISTOFTABLES Table page 2-1Parametersofthenoiseanalysisforarmlocking ................. 110 5-1Parametersintime-domainarmlockingsimulationswith transpondernoise .. 175 6-1ParametersinAC-coupleddualarmlockingsimulationsw ithDopplererrors .. 194 6-2ParametersoftheAC-coupledlterusedinsimulations ............. 194 8

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LISTOFFIGURES Figure page 1-1Inspiralofawhitedwarfbinarysystem ....................... 30 1-2Artist'sillustrationofanEMRI ............................ 35 1-3SchematicofabasicMichelsoninterferometer .................. 41 1-4StrainsensitivitiesofLIGOS6 ........................... 47 2-1HeliocentricorbitoftheLISAconstellation ..................... 50 2-2LISAsensitivitycurve ................................ 51 2-3InterferometricMeasurementSystemofLISA ................... 55 2-4InterferometersontheopticalbenchofLISA ................... 56 2-5MichelsonX-combinationandSagnaccombination ................ 61 2-6Pound-Drever-Hallfrequencystabilization ..................... 65 2-7Mach-ZehnderfrequencystabilizationforLISA .................. 68 2-8ArmlockingarchitectureintheLISAconstellation ................. 72 2-9Magnitudeandphaseresponsesofthesinglearmlockings ensor ....... 75 2-10Instabilityofsinglearmlockingwhenusinga 1 = s controller ........... 76 2-11Typicalvaluesof whichindicatesthenoisesuppression ............ 78 2-12Exampleoftheclosed-looptransferfunctionofsinglea rmlocking ....... 79 2-13Timedomainsimulationtodemonstrateinitialtransie ntsinsinglearmlocking 80 2-14Transferfunctionofthecommonarmlockingsensor ............... 81 2-15Transferfunctionsofthedifferentialarmandtheinte grateddifferentialarm .. 82 2-16Transferfunctionofthelinearcombination H ( s ) .................. 84 2-17Transferfunctionofthedualarmlockingsensor .................. 85 2-18TransferfunctionoftheSagnac-baseddualarmlockings ensor ......... 87 2-19GenericsinglearmlockingloopwithaDopplerfrequenc yerror ......... 90 2-20GenericdualarmlockingloopwithDopplerfrequencyer rors .......... 92 2-21Magnitudeandphaseresponsesofthemodieddualarmlo ckingsensor ... 95 9

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2-22Magnitudeandphaseresponsesofthedualandcommoncom ponents .... 96 2-23GenericmodieddualarmlockingloopwithDopplerfreq uencyerrors ..... 97 2-24Magnituderesponsesof H + ( s ) and H ( s ) formodieddualarmlocking .... 98 2-25PZTactuationcavityasatunablereferenceforarmlocki ng ........... 100 2-26Sidebandcavitylockasatunablereferenceforarmlocki ng ........... 101 2-27Heterodynephase-lockingasatunablereferenceforar mlocking ........ 101 2-28Armlockingwiththephase-lockedlaseratthefarspacec raft .......... 103 2-29Armlockingwithvariousnoisesources ...................... 108 2-30Noiseoorsincommonarmlocking ........................ 111 2-31Noiseoorsindualarmlocking ........................... 112 2-32Noiseoorsinmodieddualarmlocking ..................... 113 3-1Frequencynoiseofthecavitystabilizedlaser ................... 116 3-2ImplementationofthephasemeteronaPentekboard .............. 119 3-3Phasemeterperformancemeasuredwithsplit-and-subtra ctedVCOsignals .. 121 3-4TransferfunctionandnoiseooroftheEPDunittestedwith 1s delay ...... 124 4-1ExperimentalsetupofsinglearmlockingusinganNCOtotr acktheinputnoise 128 4-2SinglearmlockingwithNCOtracking-Laplacedomain ............. 129 4-3Magnituderesponseofcontrollerforsinglearmlocking ............. 131 4-4Magnituderesponseofthe 1 = p s lter ....................... 132 4-5Noisespectraofthesinglearmlockingtestathighfrequ encies ......... 132 4-6Timeseriesofthesinglearmlockingtestathighfrequen cies .......... 133 4-7Noisespectraofthesinglearmlockingtestatlowfreque ncies ......... 134 4-8Timeseriesofthesinglearmlockingtestatlowfrequenc ies ........... 134 4-9SinglearmlockingwithNCOtracking-Laplacedomainwith noisesources .. 135 4-10Comparisonwitha30-bitdigitizationnoiseoor .................. 137 4-11Comparisonoffrequencydriftratebetweena30-bitlt eranda32-bitlter ... 138 4-12Experimentalsetupofsinglearmlockingusingaphase-l ockedlaser ...... 140 10

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4-13SinglearmlockingwithheterodynePLL-Laplacedomain ............ 141 4-14NoisespectraofsinglearmlockingwithheterodynePLL ............. 144 4-15SinglearmlockingwithheterodynePLL-Laplacedomainwi thnoisesources 145 4-16ExperimentalsetupofsinglearmlockingusingaPZTactua tedcavity ..... 148 4-17SinglearmlockingwithaPZTcavity-Laplacedomain .............. 149 4-18NoisespectraofsinglearmlockingwithaPZTactuatedca vity ......... 151 4-19SinglearmlockingwithaPZTcavity-Laplacedomainwithn oisesources ... 152 5-1Designofthecommonarmlockingsensorandthesensortra nsferfunction .. 155 5-2ExperimentalsetupofcommonarmlockingusingNCOtracki ng ........ 156 5-3NoisespectraofcommonarmlockingtestwithNCOtrackin g .......... 157 5-4CommonarmlockingwithNCOtrackingintheLaplacedomai n ......... 158 5-5Lockacquisitionprocessincommonarmlocking ................. 159 5-6Implementationdiagramofthedualarmlockingsensor ............. 160 5-7Magnituderesponsesofthedualarmlockingsensor ............... 161 5-8PreliminaryexperimentalsetupofdualarmlockingwithN COtracking ..... 162 5-9NoisespectraofdualarmlockingwithNCOtracking ............... 163 5-10Noisespectraof48-bitdualarmlockingwithNCOtracki ng ........... 164 5-11Lockacquisitionprocessindualarmlocking ................... 165 5-12Experimentalsetupofdualarmlockingusingaphase-loc kedlaser ....... 166 5-13NoisespectraofdualarmlockingwithheterodynePLL ............. 167 5-14Implementationdiagramofthemodieddualarmlocking sensor ........ 168 5-15Magnituderesponsesofthemodieddualarmlockingsen sor .......... 169 5-16ExperimentalsetupofmodieddualarmlockingwithNCOt racking ...... 170 5-17NoisespectraofthemodieddualarmlockingtestwithN COtracking ..... 171 5-18Magnituderesponsesof H + ( s ) and H ( s ) usedintheexperiment ........ 172 5-19Experimentalsetupofmodieddualarmlockingusingaph ase-lockedlaser .. 173 5-20NoisespectraofdualarmlockingwithheterodynePLL ............. 174 11

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5-21Timedomainsimulationofmodieddualarmlockingwith transpondernoise 177 5-22ExperimentalsetupofdualarmlockingwithFGsignalast hetranspondernoise 178 5-23NoisespectraofdualarmlockingwithFGsignalasthetr anspondernoise ... 180 5-24ExperimentalsetupofmodieddualarmlockingwithfarendPLL ....... 181 5-25Closed-loopdynamicsofmodieddualarmlockingwithf ar-endPLL ...... 182 5-26Noisespectraofthetranspondernoiseobservedinmodi eddualarmlocking 185 5-27Noisespectraofthestabilizedlaserfrequencyandthe transpondernoiseoor 186 6-1ExperimentalsetupofAC-coupledsinglearmlockingwith Dopplererror .... 192 6-2MeasurementresultsofAC-coupledsinglearmlockingwi thDopplererror ... 192 6-3PerformanceofdualarmlockingwithanAC-coupledcontr oller ......... 195 6-4Experimentalsetupofdual/modieddualarmlockingwith Dopplererrors ... 197 6-5Frequencypullingofdual/modieddualarmlockingwith Dopplererrors .... 197 6-6Performanceofmodieddualarmlockinginthepresenceo fDopplererrors .. 198 6-7Lineardriftinthelaserfrequencystabilizedbymodie ddualarmlocking ... 199 6-8Frequencypullinginlockacquisitionduetoastepfunct ioninDopplererrors 200 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ARMLOCKINGFORLASERINTERFEROMETERSPACEANTENNA By YinanYu May2011 Chair:GuidoMuellerMajor:Physics TheLaserInterferometerSpaceAntenna(LISA)isacollaborativ eNational AeronauticsandSpaceAdministration(NASA)/EuropeanSpaceAgency( ESA)space missiontodetectgravitationalwavesinthefrequencyregi onof 3 10 5 Hz to 1Hz by meansoflaserinterferometry.Itwillbetherstspace-bas edinterferometricgravitational wavedetectortobelaunchedin2020s.LISAwillconsistofthr eeidenticalspacecraft arrangedinaquasi-equilateraltriangularconstellation with 5Gm oneachside.Each spacecrafthousestwodrag-freeproofmassesthatfollowth egeodesicmotion.The InterferometricMeasurementSystem(IMS)ofLISAmonitorscha ngesintheproper distancebetweentwoproofmassesoneachrespectivespacec raft. Laserfrequencystabilizationisoneofthemostsignicant anddifcultissuesfor theIMSofLISA.Armlockingasaproposedfrequencystabilizati ontechnique,transfers thestabilityofthelongarmlengthstothelaserfrequency. Thearmlockingsensor synthesizesanadequatelylteredlinearcombinationofth einter-spacecraftphase measurementstoestimatethelaserfrequencynoise,whichc anbeusedtocontrolthe laserfrequency.Duetothelargepropagationdelayduringt helighttransmission,the armlockingcontrollerneedstobecarefullydesignedtoret ainenoughphasemargin. ApotentialproblemforarmlockingisthattheDopplershift ofthereturnbeamwill causeaconstantpullinginthemasterlaserfrequencyifuna ccountedforinthephase measurement.Untilnowallthebenchtopexperimentsonarml ockingveriedonlythe 13

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basicsinglearmlockingcongurationwithunrealisticsho rtdelaytimeandwithoutany Dopplerestimationerroratthephasemeter.AttheUniversit yofFloridawedeveloped thehardware-basedUniversityofFloridaLISAInterferomet erSimulator(UFLIS)tostudy andverifylaserfrequencynoisereductionandsuppression techniquesunderrealistic LISA-likeconditions.Theseconditionsincludethevariable Dopplershiftsbetweenthe spacecraft,LISA-likesignaltraveltimes,far-endheterody nephase-locking,realistic laserfrequencyandtimingnoise.Inthisdissertationwewi llsystematicallyintroduce thecuttingedgeofexperimentalstudiesofarmlockingunde rtheserealisticconditions. Wehavebuiltananalog/digitalhybridsystemtodemonstrat ethecontrolsystemof variousarmlockingschemesandtheirincorporationwithpr e-stabilizationsubsystems. Wemeasuredthenoisesuppressioninourexperimentsaswell asthefrequency pullinginthepresenceofDopplerfrequencyerror.Withthea chievementofmeeting therequirement,ourpioneeringworkhavesufcientlydemo nstratedthevalidityand feasibilityofarmlockingunderLISA-likeconditions. 14

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CHAPTER1 GRAVITATIONALWAVEASTRONOMY 1.1Introduction Sincethedawnofhumancivilization,thecuriositytotheuni versehasalways beenaprimarymotivationforhumantopursuescience.Einste inoncesaid,“Themost incomprehensiblethingabouttheworldisthatitiscompreh ensible.”Theobservational astronomyisundoubtedlyarststepinhuman'shistoricala ttemptstounderstandthe universe,thankstoelectromagneticwavesthatallowedAris totletoobservecelestial constellationsbynakedeyesandallowedGalileotoobserve Jupiterandorbitingmoons byopticaltelescopes.Startingfromthe19thcentury,theun iverseobservabletohuman hasbeensubstantiallyexpandedfromvisiblelighttocurre ntlythefullspectrumof electromagneticwaves.Avarietyofground-basedandspace -basedobservatories, coveringfromradiotoGammaray,havebeendevelopedtostud ytheevolutionofstars andgalaxies,aswellastheoriginoftheuniverse. Inadditiontothemagnicentachievementselectromagneti cobservationshave alreadyobtained,anotherdifferentobservationmethodst artedtograduallygrowin thesecondhalfofthe20thcentury.Gravity,asthedominant forceintheuniverse, resultsingravitationalwavesgeneratedfromaccelerated objects,whichispredicted byEinstein'sGeneralTheoryofRelativity.Inthistheory,g ravitationalwavesare oscillationsofspacetimegeometrypropagatingwiththesp eedoflight[ 1 ].Einstein neverthoughtgravitationalwavescouldpossiblybedetect edduetotheirextremely weakinteractionswithmatters.Nevertheless,JosephWebe rstartedtobuildthevery rstgravitationalwavedetectorsusingresonantmassesin 1960s[ 2 ].Alsoduringthis period,generalrelativitybegantodemonstrateitspoweri ntheresearchofastrophysics andcosmology.Relativisticgravity,aswellasgravitatio nalradiations,werefoundto playanimportantroleinvariousastronomicalsystems.The mostfamousexampleof gravitationalradiationsisthe13-yearobservationtothe binarypulsarPSR1913+16 15

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[ 3 4 ],whichrstindirectlyveriedtheexistenceofgravitati onalwaves,aswellasthe validityofgeneralrelativityinstronggravitationalel ds.ThesuccessofPSR1913+16 indicatesthatlikeelectromagneticwaves,gravitational wavescanalsobeusedas anobservationmethodtostudytheuniverse,whichisknowna sgravitationalwave astronomy. Actually,gravitationalwaveastronomywillrevealabundan tinformationabout astronomicalsystemsthatcouldnotbeachievedbytraditio nalelectromagneticwave astronomy[ 5 ].Forexample,thegravitationalradiationfromblackhole coalescence providestheonlypossibledirectobservationmethodtostu dyblackholephysics. Theelectromagneticradiationfromblackholes(knownasHa wkingradiation),ifany, wouldbeimpossibletodetectforcurrenttechnology.Alsodu etotheweakinteraction withmatters,gravitationalwavesarehardlyattenuatedor scatteredeversincethey aregenerated.Thisindicatesthattheycouldcarryinforma tionaboutsomeexotic phenomenathatelectromagneticdetectionscannotreach,s uchastheinteriorof supernovaexplosionsorphysicsoftheveryearlyuniverse. Althoughtheprospectofgravitationalwaveastronomyisexc iting,thedetectionof gravitationalwavesstillremainstobeachallengeforcurr enttechnologyduetotheir extremelyweakinteractionswithmasses.Forexample,thet ypicalupperlimitofthe gravitationalwavestraingeneratedbythecoalescenceofs tellarmassblackholesis nomorethan 10 23 whenarriveattheEarth.Themanifestationofgravitational wave strainsisanoscillatingchangeintherelativedistancebe tweentestmasses.Based onthisprinciple,laserinterferometerscanbenaturallyu sedtoconstructgravitational wavedetectorsviapreciseinterferometry.Theground-bas edLaserInterferometer GravitationalWaveObservatory(LIGO)isbasedonanequalarmedMichelson interferometerwitharmlengthsof 4km [ 6 ].Nevertheless,therequireddisplacement sensitivityisstillontheorderof 10 19 mHz 1 = 2 .Toreachtherequiredsensitivity, 16

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varioustechniqueshavebeendevelopedforLIGOandothergr ound-baseddetectorsto enhancetheirdetectionsensitivityandsuppressrandomno ises. LIGOandotherground-baseddetectorsareexpectedtodetec tgravitational wavesatfrequencieshigherthan 30Hz .Forfrequenciesbelow 30Hz ,the seismicnoisefromgroundvibrationsstartstodominatethe sensitivitycurverapidlyand becomesalimitationforcurrentground-baseddetectors.T hedetectionofgravitational wavesatlowerfrequenciesrequireseitheracompletelydif ferentdetectionmethod oranabsoluteisolationfromseismicnoisebyplacingthede tectorinspace.Laser InterferometerSpaceAntenna(LISA)isaproposedlarge-scaled laserinterferometer ontheheliocentricorbitinordertodetectgravitationalw avesatfrequenciesfrom 3 10 5 Hz to 1Hz [ 7 8 ].LISAconsistsofthreespacecraftforminganear-equilate ral trianglewitheachsidelength 5 10 9 m .Sincethearmlengthisfarlongerthanall ground-baseddetectors,thelengthchangeduetogravitati onalwavestrainswillbecome moresignicant,whichprovidesLISAasensitivityof 10 20 Hz 1 = 2 at 3mHz .Withthis sensitivity,LISAwillbeabletodetectlow-frequencygravi tationalwavesemittedfrom coalescencesofmassiveblackholes(MBH)outtoredshift z 20 [ 9 10 ].Otherprimary gravitationalwavesourcesLISAwilldetectincludeextreme massratioinspirals(EMRI) wherethedynamicsofatestmasscapturedinKerrgeometryca nbeexperimentally studied,resolvedandunresolvedGalacticcompactbinarie s,evengravitationalrelicsof BigBangknownasthestochasticbackgroundandmaybecosmicst ringspredicted insomeversionsofthestringtheory.Insummary,LISAwillpe rformaseriesof newscienticmeasurementstovariousgravitationalwaves ourcesintheuniverse withmultiplegoals,includingrelevantastrophysicalres earchinsuchasbinariesand galaxies,testsofgeneralrelativityinstronggravitatio nalelds,aswellasdiscoveriesof newphysicsandcosmology. LiketheMichelsoninterferometerusedinground-baseddet ectors,LISAwillalso measurethechangeindistancebetweentwofree-fallingpro ofmasses.However, 17

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thecongurationandoperationofLISAarequitedifferentfr omastandardMichelson interferometerinanumberofways.AsLISAconsistsofthreesp acecraftandeach ofthemorbitsthesunindependently,thearmlengthbetween twoproofmassesisa time-variablequantity,whichcannotbemadeidenticaltoe achothereveninprinciple. TimeDelayInterferometry(TDI)asapost-processingtechn iquesynthesizesequal arminterferometrytoextractgravitationalwavesignals[ 11 12 ].Nevertheless,the randomuctuationsinthelaserfrequencycannotbesuppres sedadequatelybyTDI withlimitedprecisionofthearmlengthknowledge.Tooverc omethisproblem,the pre-stabilizationofthelaserfrequencyisessential.Ina dditiontocommonmethods suchasPound-Drever-Halltechniqueusedforthelaserstab ilizationinground-based detectors,thearchitectureofLISAinherentlyprovidesaun iquefrequencystabilization techniqueknownasarmlocking.SincetheLISAarmlengthisave rystablereference intheLISAfrequencyband,armlockingtakesitasafrequency referencetostabilize thelaserfrequencyviathesynthesizationofanadequately lteredlinearcombination oftheinterferometrysignals.Themainsubjectofthisdiss ertationisaboutarmlocking, includingtheanalyticalperformance[ 13 ],numericalsimulationsinthetimedomainand especiallytheexperimentalvericationinlaboratory[ 14 15 ]. Thisdissertationisdividedintosevenchapters.Theremai ningpartofChapter 1introducesthetheoryofgravitationalradiation,gravit ationalwavesourcesand detections.InthispartIputemphasisonthegravitational wavesourcesforLISA. Chapter2coverstheoverviewofLISA,includingthescientic requirementsand payload,wherethetechniqueofarmlockingisthemainsubje cttobefocusedon. Chapter3toChapter6discusstheexperimentalverication sofarmlockingonthe hardwaresimulatorofLISAinterferometrydevelopedattheU niversityofFlorida.The nalpart,Chapter7,givestheconclusionsandoutlookonar mlockingandLISA interferometry. 18

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1.2GravitationalRadiationinGeneralRelativity 1.2.1PropagationofGravitationalWaves Gravitationalradiationisoneofthemostfundamentalphen omenapredictedby Einstein'sGeneralTheoryofRelativity.Thepropagationof gravitationalwavesin spacetimecanbedescribedasanapproximatesolutionoflin earizedEinsteineld equationsinthepresenceofaweakgravitationaleld. 1 TheEinsteineldequationscanbewrittenas R 1 2 g R = 8 G c 4 T (1–1) Weassumethegravitationaleldisweakenoughsuchthatthe metric g canbe decomposedintotheatMinkowskimetricplusasmalllinear perturbation,i.e., g = + h j h j 1. (1–2) Thisapproximationisknownasthetheoryoflinearizedgrav itywhereonlytherst orderoftheperturbation h istakenintoaccount.Inthistheorytheoverallspacetime metric g canbedescribedasaperturbationtensoreld h propagatingwithintheat Minkowskispacetime.ThistensoreldissymmetricandLore ntzinvariant. TosolvetheEinsteineldequationsusingthisapproximatio n,wecalculatethe Riemanncurvaturetensor R = 1 2 ( @ @ h + @ @ h @ @ h @ @ h ). (1–3) Wedenethetraceoftheperturbationas h = h andthetrace-reversemetric perturbationas h = h 1 2 h .BasedontheRiemanntensorcalculatedinEq. 1–3 1 Thetheoreticalintroductionofgeneralrelativityandgra vitationalwavesinthis chapterismainlyadaptedfromRef.[ 16 ]byMisner etal. andRef.[ 17 ]byMaggiore. 19

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theEinsteineldequationscanbereducedintothelinearize dform 2 h + @ @ h @ @ h @ @ h = 16 G c 4 T (1–4) DuetotheLorentzinvarianceof h ,wecouldexploitthegaugefreedomtochoose theLorenzgauge @ h =0. (1–5) InthisgaugethelastthreetermsontheleftsideofEq. 1–4 vanish.Therefore,we obtainthelinearizedEinsteineldequation 2 h = 16 G c 4 T (1–6) whichisessentiallyafour-dimensionwaveequationwithsi xindependentcomponentsof h Intheregionfarawayfromthesource,thewavepropagatione quationis 2 h =0. (1–7) Thegeneralsolutionofthisequationisgivenbythelinears uperpositionofthe followingeigenfunction h = A exp( ik x ), (1–8) wherethefour-dimensionwavevector k mustsatisfythecondition k k =0 .This indicatesthegravitationalwavetravelsthroughanullgeo desicwiththespeedoflight. SincetheLorenzgaugeisnotunique,itallowsustoaddmoreco nstraintsto eliminateadditionaldegreesoffreedombyintroducingthe transverse-tracelessgauge (TTgauge).Inthisgaugetheperturbationmetricisindepen dentofthetimecomponents andisbothtracelessandtransverse.Weusethenotationof h TT torepresentthemetric tensor h intheTTgauge,whichisgivenby h TT 0 =0, h TT =0, @ h TT =0. (1–9) 20

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UndertheconstraintsofboththeLorenzgaugeandtheTTgaug e,themetrictensor hasonlytwoindependentcomponents,correspondingtotwoo rthogonalmodesoflinear polarizationofgravitationalwaves.Ifweassumethegravi tationalwavetravelsinthez direction,themetrictensorcanbecompletelycharacteriz edbytheform h TT ( t z )= 0BBBB@ h + h 0 h h + 0 000 1CCCCA cos[ ( t z = c )], (1–10) wherethenotationsof h + and h areknownastheplusandthecrosspolarizations ofgravitationalwaves,respectively.Eq. 1–10 canalsobewrittenintheformofa spacetimeinterval: ds 2 = cdt 2 + dz 2 +[1+ h + cos( ( t z = c ))] dx 2 +[1 h + cos( ( t z = c ))] dy 2 +2 h cos( ( t z = c )) dxdy (1–11) 1.2.2GenerationofGravitationalWaves Intheregionneartheradiationsource,theenergy-momentu mtensor T doesnot vanish.Bysolvingthematter-coupledEinsteinequation,wec ouldobtaintherelation betweenthemotionofthesourceandgenerationofgravitati onalwaves.Generally,an inhomogeneouswaveequationsuchas 2 h = 16 G c 4 T (1–12) canbesolvedusingtheretardedGreen'sfunction,whichisa lsousedforsimilar problemsinelectromagnetism.Weconsidertheradiationfr omafour-dimensionpoint source (4) ( x x 0 ) andthentheGreen'sfunction G ( x x 0 ) isthesolutionforthewave equation: 2 x G ( x x 0 )= (4) ( x x 0 ) ,where 2 x representsthed'Alembertianoperator withderivativesrespectivetotheeldpoint x 21

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TheretardedGreen'sfunctioncanbeexpressedas G ( x x 0 )= 1 4 j x x 0 j ( x 0 ret x 0 0 ), (1–13) where x 0 0 = ct 0 x 0 ret = ct ret and t ret t j x x 0 j c iscalledtheretardedtime. ThusthesolutionforEq. 1–12 isgivenbytheintegration h ( t x )= 16 G c 4 Z G ( x x 0 ) T ( x 0 ) d 4 x 0 = 4 G c 4 Z 1 j x x 0 j T t j x x 0 j c x 0 d 3 x 0 (1–14) Iftheradiationsource T isnon-relativistic,whichmeansthetypicalvelocities insidethesourcearesignicantlysmallerthanthespeedof light,inthefareld approximationwehave h ( t x ) 4 G c 4 1 r Z T t r c x 0 d 3 x 0 (1–15) where r = j x x 0 j isthespatialdistance. WiththeLorenzgaugeconditionandtheconservationlawofth eenergy-momentum tensor @ T =0 ,theformulacanbefurtherreducedinto h ( t x )= 2 G c 4 1 r d 2 I dt 2 ( t r c ), (1–16) where I = Z x 0 x 0 T 00 ( t x 0 ) d 3 x 0 (1–17) isdenedasthequadrupolemomenttensoroftheenergydensi tyofthesource.Ifthe sourceisaperfectuidwitharestframeenergydensity ,thequadrupolemoment tensorissimplyequivalenttothemomentofinertiaoftheso urce.Eq. 1–16 isknown asthequadrupoleformula,whichindicatesthatthegravita tionalwavestraingenerated fromanon-relativisticsourceisproportionaltothesecon dderivativeofthequadrupole momenttensor.Incontrast,theleadingterminanelectroma gneticradiationisa time-changingdipolemoment. 22

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Itiseasytoshowthatforabinarysystemthatconsistsoftwo starswithmass M andorbitalradius R ,theorbitalfrequencyisgivenby f orb = 1 2 q GM 4 R 3 .Fromthe quadrupoleformula,onecandeducethatthegravitationalw avefrequencyofthisbinary systemistwiceoftheorbitalfrequency,i.e., f GW =2 f orb Theluminosity(i.e.,thetotalpower)ofthegravitational quadrupoleradiationis givenby L GW = G 5 c 5 d 3 I dt 3 d 3 I dt 3 = G 5 c 5 d 3 I dt 3 d 3 I dt 3 1 3 d 3 I dt 3 2 + (1–18) where I = I 1 3 I = R ( x 0 x 0 1 3 r 2 ) T 00 ( t x 0 ) d 3 x 0 isknownasthereduced quadrupolemoment,whichisthecounterpartofthereducedq uadrupolemomentin electromagnetism.Theanglebracketsrepresentanaverage overseveralcharacteristic wavelengthsofthesource. Nowwewillestimatehowmuchthepowerthegravitationalrad iationhas.First, thethirdtimederivativeofthereducedquadrupolemomenth asthesameorderof magnitudetothequantity MR 2 = T 3 ,where M istheacceleratedmass, R isthesizeof theradiationsystemand T isthetimeforthemasstomovefromonesidetotheother. Forabinarysystem T canbeconsideredastheorbitalperiod.Ontheotherhand,th e quantity MR 2 = T 3 canbewrittenas MR 2 T 3 = M ( R = T ) 2 T E kinetic T L internal (1–19) where E kinetic isthetranslationalkineticenergyoftheacceleratedmass whenmoving fromonesidetotheotherand L internal istheinternalluminosityorthepowerinsidethe systemowingfromonesidetotheother. 23

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Therefore,aroughestimationofEq. 1–18 isgivenbythesquareoftheinternal luminosity: L GW G c 5 L 2internal (1–20) Fromthisrelationonecanfurtherdeducehowlongthegravit ationalradiationwill needtoexhaustthetotalenergyofthesystem.Wewillexplai nthebinarysystemasan exampleinSection1.3.2.1.2.3InteractionofGravitationalWaveswithTestMasses Themotionofatestmassinacurvedspacetimeisdescribedby thegeodesic equation: d 2 x d 2 + dx d dx d =0. (1–21) Weconsidertwonearbygeodesics,whichareparametrizedby x ( ) and x ( )+ ( ) respectively.Ifwetakethedifferencebetweentheirgeode sicequationsand expandittotherstordersince j j isverysmall,ityields d 2 d 2 +2 dx d d d + @ dx d dx d =0. (1–22) Thisequationcanbewritteninasimplerformifweintroduce thecovariant derivative DV D dV d + V dx d ,whichisknownasthegeodesicdeviationequation: D 2 D 2 = R dx d dx d (1–23) Therefore,thegeodesicdeviationequationdescribesatid algravitationalforce applyingonnearbygeodesics.Tofurtherderivehowthetest massbehavesinthe presenceofthetidalforce,wewillneedtochooseareferenc eframe. IntheTTgaugegravitationalwavescanberepresentedassim pleasinEq. 1–10 Weconsidertwotestmasseswithacoordinateseparation i andiftheyareatrestat =0 ,wehave dx i = d =0 and dx 0 = d = c .Therefore,Eq. 1–22 isreducedto d 2 i d 2 = 2 c i0 d d + c 2 @ i00 (1–24) 24

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Moreover,intheTTgaugetheChristoffelsymbol i00 = 1 2 (2 @ 0 h 0 i @ i h 00 ) (1–25) alsovanishessince h 00 and h 0 i equalzerointheTTgauge.Theonlynon-vanishingterm is i0 j = 1 2 @ 0 h ij andthentheequationisfurthersimpliedinto d 2 i d 2 = h ij d i d (1–26) Therefore,ifthetwotestmassesdonothaveaninitialrelat ivevelocity d i d ,then wealsohave d 2 i d 2 =0 andtheircoordinatedistancealwaysremainsthesame.Inot her words,intheTTgaugethepositionoftestmassesdoesnotcha ngeduetogravitational waves.Thisconclusiondoesnotimplythattheinteractiono fgravitationalwaveswith testmassesiszero;instead,whatgravitationalwaveschan geistheproperdistance betweentestmassesasdemonstratedinEq. 1–11 .Ifweassumethatthecoordinates ofthetwotestmassesaregivenby (0,0,0,0) and (0, L ,0,0) ,fromEq. 1–11 theproper distanceisthengivenby s = L (1+ h + cos( t )) 1 = 2 L (1+ 1 2 h + cos( t )). (1–27) DuetotheinvarianceoftheTTgauge,gravitationalwavedet ectorsrequireamore convenientreferenceframetomeasurethepositionchangeo ftestmasses.Such areferenceframeknownastheproperdetectorframerequire sthetestmasstobe drag-free(atleastincertaindirections)anditsrelevant coordinateswillbechangedby gravitationalwaves.Inthisframethegeodesicdeviatione quationyields i = 1 2 h TT ij j (1–28) wherethesecondorderderivativeiswithrespecttothecoor dinatetimeratherthanthe propertime. 25

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StartingfromEq. 1–28 ,weconsideragravitationalwavepropagatingalongthez axisandthetestmassesarelocatedinthex-yplane.At z =0 ,the + polarizationcan bewrittenas h TT ab = h + sin t 0B@ 100 1 1CA a b =1,2. (1–29) SubstituteEq. 1–28 withEq. 1–29 andweobtainthecoordinatechangesonxand ydirections: x = h + 2 ( x 0 + x ) 2 sin t y = h + 2 ( y 0 + y ) 2 sin t (1–30) Ignorethenegligible x y termsontherightsideandintegratetwiceovertime: x = h + 2 x 0 sin t y = h + 2 y 0 sin t (1–31) Similarapproachcanbeperformedtoevaluatethecoordinate changesduetothe polarization: x = h 2 y 0 sin t y = h 2 x 0 sin t (1–32) Eq. 1–31 andEq. 1–32 describeshowgravitationalwavesdisplacetestmasses transverselywithrespecttotheirpropagationdirection. Thiskindoftidaldistortionsis themeasurementprinciplebehindallresonantandinterfer ometricgravitationalwave detectors. 1.3SourcesofGravitationalWaves Inthissectionwehaveareviewontypicalastrophysicalgra vitationalwave sourcesclassiedbytheirradiationfrequencies.Thefreq uencyrangeforobservable 26

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gravitationalwavesourcesstartsfrom 10kHz andextendsdownwardbyroughly22 ordersofmagnitude[ 18 ]. InSection1.2.2wealreadymentionedthatthefrequencyofgr avitationalwaves emittedfromabinarysystemistwiceoftheorbitalfrequenc y.Ingeneralcases,the gravitationalwavefrequencyofacertainsourceisofthesa meordertothenatural frequencyofthesystem,whichisgivenby f 0 = r G 4 (1–33) where isthemeandensityofthemass-energyinthesource.Itcanbe simply estimatedby = 3 M 4 R 3 ,where M isthemassand R istheradius.Therefore,a gravitationalwavesourcewithahighermeandensitywillra diatesatahigherfrequency. Foraneutronstarwitha 1.4 M massanda 10km radius,thegravitationalwave frequencyiscloseto 2kHz .Forawhitedwarfwitha 0.5 M massanda 10 4 km radius, thegravitationalwavefrequencyiscloseto 3mHz ,whichisintheLISAscienceband. Forblackholes,sincetheSchwarzschildradiusofablackhol eisrelatedtothemass: R =2 GM = c 2 ,thenaturalfrequencyisinverselyproportionaltothebla ckholemass.For a 10 M stellarmassblackholethefrequencyiscloseto 1kHz ,whilefora 2.5 10 6 M massiveblackholethefrequencyisaslowas 4mHz 1.3.1HighFrequencyRange Thegravitationalwavesourcesinthishighfrequencyrange willbedetectedby ground-basedinterferometricdetectorsaswellasresonan tbars.Thesesourcesinclude thecoalescenceofneutronstarandstellarmassblackholes ,gravitationalcollapseof supernovae,spinningpulsarsandstochasticradiationbac kgroundfromtheBigBang. Wewillintroducetheinspiralbinariesandthestochasticb ackgroundinthenextsection (Lowfrequencyrange)andthissectiononlyinvolvessource sthatareexclusivetothe highfrequencyrange. 27

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Oneparticularlyinterestinghigh-frequencysourceisspi nningpulsars.Toradiate gravitationalwaves,themass-energydistributionofthep ulsarmustbeasymmetric otherwisethequadrupolemomentistime-independent.Them echanismtoproducethis asymmetrycouldbea“lump”,wheretheasymmetryisstaticre lativetotheneutronstar, ora“wave”,wheretheasymmetryisinmotion.Theformercase scanpossiblybeseen inspheroid-shapedneutronstars[ 19 ],neutronstarswithamisalignmentofthemagnetic eldwiththerotationaxisorneutronstarswithaccretions .Sincethecontributionto gravitationalradiationsiscompletelyfromtheasymmetri cportion,thegravitational wavefrequencyofapulsarisalsotwiceofthespinningfrequ ency.Sincethespinning frequencyofapulsarisultrastable,thegravitationalwav esignalwillbeseeninavery narrowfrequencybin.Unlikeinspiralbinaries,thegravit ationalradiationofapulsaris probablynottheprimarymechanismtocauseittospindown[ 18 ]. Inaddition,thecorecollapseofasupernovathatformsanew neutronstar orblackholeisalsolikelytobeanimportantsourceinthehi ghfrequencyrange. Therotationalcorecollapseofasupernovaisasymmetric,w hichwasalready conrmedbyobservationstothesupernovaSN1987A[ 20 ].Thisasymmetrymay beassociatedwithatransientandnon-periodicburstsigna l.However,thewaveform andamplitudepredictionsofburstsignalsareverydifcul ttomodelanalyticallyandso farreliesentirelyonnumericalsimulations.Itisyetnotc learwhatfractionofthetotal mass-energywillbereleasedintheformofgravitationalra diation.Thecurrentbest estimationyieldsanupperlimitof 10 6 [ 18 ],whichmakesthegravitationalburstsinthe Virgoclusterundetectable.Incomparison,therateofcorec ollapsesupernovaeinour Galaxyisestimatedtobeoneperafewdecadesandthecorresp ondingburstsignal shouldbedetectableforcurrentdetectors.1.3.2LowFrequencyRange Inthissectionwewillfocusonthegravitationalwavesourc esintheLISAscience band,includinggalacticbinaries,coalescenceofmassive blackholes,extrememass 28

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ratioinspiralsandcosmicgravitationalradiationbackgr ound. 2 Thescienceobjective ofLISAistodetectandstudythegravitationalradiationsfr omtheseunprecedentedly observedsources.Basedonthedetectionresults,LISAwillco mprehensivelyreveal moreastrophysicalinformationonbinaries,blackholes,g alaxystructures,etc.LISAwill alsopreciselytestthevalidityofgeneralrelativityinve rystronggravitationaleldslike thevicinityofaKerrblackhole.Finally,LISAwillprobenew physicsandcosmologyby tracinggravitationalwavesfromtheveryearlyuniverse.1.3.2.1Galacticbinaries Theinspiralofcompactbinarysystemsisthebestunderstoo dgravitationalwave sourceandsofartheonlyobservationallyconrmedsource. InEinstein'stheory, gravitationalradiationsdamptheenergyfromtheorbitalm otionandcausetheorbitto graduallyshrink.Thisphenomenonhasalreadybeenconrme dintheobservation tothebinarypulsarPSR1913+16byHulseandTaylorsince1974[ 3 4 ].Asthe twoneutronstarsinspiralcloser,theorbitalfrequencyco ntinuestoincrease,which generatesachirpsignalintheformofgravitationalradiat ions.InPSR1913+16one memberisaradiopulsar,whichprovidesaveryaccuratecloc kfororbitalperiod measurements.Ontheotherhand,boththequadrupoleamplit udeandtheorbitaldecay rateonlydependonthemassesofthebinarysystem(seebelow ).Theorbitaldecay ratethereforecanbedirectlypredictedbygeneralrelativ ity,giventhattherequired parametersarealreadyobtainedbyotherobservations.Byco mparingthetheoretical andthemeasuredorbitaldecayrate,thepredictionofgener alrelativityissuccessfully veriedwithanobservationalerrorlessthan 1% Herewewillderivehowabinaryorbitevolvesinthepresence ofgravitational radiationbraking.Supposewehavetwostarsofmasses m 1 and m 2 onanellipticalorbit 2 Foracomprehensivereviewofgravitationalwavesourcesin theLISAscienceband, seeRef.[ 21 ]. 29

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Figure1-1.Artist'sillustrationofawhitedwarfbinarysys temintheinspiralphase, courtesyofNASA. withanangularvelocity andthesemi-majoraxisoftheellipticalorbitis a .Thenthe Kepler'slawyields 2 a 3 =( m 1 + m 2 ) G = MG (1–34) Fromthevirialtheorem,thekineticenergyofthebinarysys temisgivenby E kinetic = 1 2 E potential = G 2 m 1 m 2 a (1–35) BasedonEq. 1–20 ,theluminosityofthegravitationalradiationisproporti onaltothe squareoftheorbitingpower,whichisgivenbytheproductof thekineticenergyandthe angularvelocity: L GW G c 5 L 2internal G c 5 G 2 m 1 m 2 a 2 G 4 2 M 3 4 c 5 a 5 (1–36) where = m 1 m 2 = M isthereducedmass. Ifwefollowanexactcalculation,theresultisgivenby L GW = 32 G 4 2 M 3 5 c 5 a 5 f ( e ), (1–37) 30

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where f ( e ) isacorrectionfunctionduetotheeccentricity.Foracircu larorbitwith e =0 wehave f ( e )=1 Forbinarysystems,thedecayrateofthetotalenergyisequa ltothegravitational radiationpower,i.e., dE total dt = L GW (1–38) wherethetotalenergy E total justequalstothenegativeofthekineticenergy. Weareinterestedinthedecayrateoftheorbitalperiod dT = dt .Basedonthe Kepler'slaw,theperiodisassociatedtothekineticenergy viatherelation T =const ( E kinetic ) 3 = 2 .Bytakingderivativesoneachsidewehave T T = 3 2 E kinetic E kinetic (1–39) SubstitutewithEq. 1–38 andwehave T T = 96 5 G 3 M 2 c 5 a 4 f ( e ), (1–40) whichcanbefurtherwrittenas T T = 96 5 G 5 = 3 M 2 = 3 c 5 T 2 8 = 3 f ( e ). (1–41) Ifweassume f ( e )=1 ,thesolutionofthisdifferentialequationis T ( t )= T 8 = 3 0 8 3 kt 3 = 8 (1–42) where T 0 istheperiodat t =0 andtheconstant k isgivenby k = 96 5 c 5 (2 ) 8 = 3 ( G 3 = 5 M 2 = 5 ) 5 = 3 (1–43) Eq. 1–42 describeshowfasttheorbitalperiodofabinarysystemdecr easesover timeduetogravitationalradiations,whichcompletelydep endsontheconstant k Eq. 1–43 indicatesthatthespeedoforbitshrinkingonlydependsont hecombination 3 = 5 M 2 = 5 fromthebinarymasses.Thiscombinationisknownasthechir pmassofthe 31

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binarysystem: M chirp = 3 = 5 M 2 = 5 =( m 1 m 2 ) 3 = 5 ( m 1 + m 2 ) 1 = 5 (1–44) Inotherwords,thechirpmassofthebinarysystemdetermine showfastthe frequencyofthechirpsignalsweepsthespectrum.Byobservi ngthetime-varying frequencyofthegravitationalradiation,onecandirectly deducethechirpmassofthe binarysystem. Sincemoststarsintheuniversehaveastellarmassandalsoar ebinaries,white dwarfbinariesarenumerousintheGalaxy,muchmorethanneu tronstarorblackhole binaries.Althoughthegravitationalradiationfromwhited warfsisrelativelyweak,the closedistanceinourGalaxyensuresahighSNR 100 atrelativelyhighfrequencies ( > 1mHz ).Thesetwoconditionsmakethewhitedwarfbinariesthegua ranteed sourcesforLISA.LISAshouldbecapableofdetectingthousands ofthemindividually andmeasuringtheirphysicalparameterssuchasdistance,o rbitalperiodandspatial orientationprecisely[ 22 ].Inparticular,anumberofresolvedwhitedwarfbinariesw ith ashortorbitalperiod(afewminutes)areexpectedtobedete ctedsoonafterLISAis engaged.LISAcanusethemasinstrumentvericationsources bymeasuringand comparingtheirdistancesandphysicalparameters[ 23 ].Inaddition,LISAisalso expectedtodetectneutronstarandstellarmassblackholeb inariesthatareatin-band lowfrequencies[ 24 25 ]. Ontheotherhand,thepopulationofwhitedwarfbinariesint heGalaxyisvery large( 10 7 )atfrequenciesbelowa 1mHz [ 26 ].Thesegalacticbinariesforma confusion-dominatedforeground,whereasadiffusebackgr oundisgeneratedfrom extragalacticbinaries.Consequently,onlythebrightest andclosestsourcesamongthe confusionforegroundcanbeparticularlyresolvedbyLISA.Ne vertheless,thisforeground isstillofinterestforLISAmeasurementssinceitrepresent sstatisticalinformationsuch asthetotalnumberandgeometricaldistributionofthegala cticbinariesinthisfrequency region. 32

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1.3.2.2Coalescenceofmassiveblackholes Massiveblackholesareblackholesofmass 10 5 M to 10 9 M .Abundantevidences indicatethatalmosteverygalaxyhasamassiveblackholein itscenter.Theyare thoughttobethe“engine”topowerupactivegalacticnuclei (AGN)andquasars[ 27 ]. Duetothegiganticmass,thefrequencyofgravitationalrad iationsfromcoalescences ofmassiveblackholesismuchloweryettheamplitudeismuch moresignicantthan stellarmassblackholes.Thegravitationalradiationfrom coalescencesofmassiveblack holesisthestrongestsignalthatLISAisexpectedtodetect. Thedetectionrangecan reachasfaras z 20 andthesignalisevenwellabovethenoiseamplitudewithahi gh SNRof 10 2 to 10 3 .Alsocomparedtostars,thedimensionofgalaxiesisnotinsi gnicant relativetothedistancebetweeneachother,whichmakesthe galaxymergerrate actuallyratherhigh.Therefore,LISAshouldbeabletodetec tthemergersofmassive blackholeswithadecentdetectionrate,approximately1pe ryearatredshift z < 1 [ 28 ]. Thisestimationdoesnottakeintoaccounttheformationofg alaxiesviamergersofsmall protogalaxiesofmassupto 10 6 M .Iftheseprotogalaxiesalsocontainaseedblack holeofmass 10 4 M intheircenter,themergerrateforLISAmaybeashighasone thousandperyear. Sameasstellarmassblackholebinaries,thecoalescenceofm assiveblackhole binariescanbedividedintothreephases:inspiral,merger andringdown[ 29 30 ].The inspiralandringdownphasecanbeanalyticallymodeledbyt hepost-Newtonian(PN) approximationandtheperturbationtheory,whileafulldes criptionofthemergerphase requiresnumericalrelativity[ 31 ].Duringtheinspiralphasethetwoblackholesare separatedfarfromeachother( R 4 M )andspiraltogetherwithaninitialvelocity v = c 0.05 ,whichcanbeconsideredasanadiabaticevolution.Likegal acticbinaries, LISAwillbeabletomeasurethephysicalparametersduringth einspiralphasewith veryhighaccuracy.Duringtheemissionofgravitationalra diations,thetwoblackholes approacheachotherandtheorbiteventuallydecaystotheIn nermostStableCircular 33

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Orbit(ISCO).InSchwarzschildgeometry,theISCOislocatedat R ISCO =6 GM = c 2 ,where M = m 1 + m 2 isthetotalmassoftheblackholebinary. Whentheirdistancebecomesshorterthan r ISCO ,twoblackholesplungeinto eachotherandtheirhorizonsstarttomergeintoone,formin gadistortedblackhole. Duringthemergerphasethevelocityofblackholescanreach v = c 0.5 andthe post-Newtonianapproximationbreaksdown.However,numer icalrelativitycouldexploit thepreciseparametersobtainedintheinspiralphasetopre dictthemergerwaveform [ 32 ].Themergeoftwoblackholesradiatesatransientyetextre melyenergeticburst signal,whichfeaturesaveryhighSNRofthousandsformerger sof 10 6 M at z =1 Attheendofthemerger,themergedblackholestartstosettle downfroman excitedstate.Duringthisringdownphase,theexcitedblac kholeradiatesgravitational wavesinastyleofdampedoscillations.Eventuallyitwillbe stabilizedintoastableKerr blackhole,whichisentirelycharacterizedbythemassands pinangularmomentum, asrequiredbytheno-hairtheorem.Thisprocesscanbemodel edusingalinear perturbationtheoryinKerrspacetime.Theringdownwavefo rmisgivenbythe superpositionofawholesetofquasi-normalmodessolvedby theperturbationtheory [ 33 ].Bydetectingthedampedoscillationsfromtheblackholein theringdownphase, LISAshouldbeabletoconrmifamassiveblackholeisactuall yinthegalaxycenter andidentifyaKerrblackholefromitsmassandspinangularm omentum.Also,itisa testofgeneralrelativityinaextremelystronggravitatio naleld. 1.3.2.3Extrememassratioinspirals AparticularlyinterestingclassofsourceforLISAisknowna sextrememassratio inspirals(EMRIs),inwhichasmallcompactobjectiscapture dbyamassiveblack holeandinspiralsintoit[ 34 ].ThesmallcompactobjectinaEMRIcanbeawhite dwarf,neutronstarorstellarmassblackhole,whileLISAwil lprobablydetectEMRIs withstellarmassblackholesmostly.Thisismainlybecause blackholestendtobe concentratedtothegalacticcenterduetodynamicmasssegr egation.Alsocomparedto 34

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Figure1-2.Artist'sillustrationofanEMRIinthecenterofag alaxy,courtesyofNASA. EMRIswithwhitedwarfsorneutronstars,EMRIswithblackhole swillradiateobservable signalswithahigherSNR.TheformationmechanismofEMRIsisc onjecturedtobe two-bodyscattering,whereacompactobjectsufcientlycl osetothegalacticcenter happenstobecapturedbythemassiveblackholeandsubseque ntlydrivenontoa highlyeccentricorbit(closeto1).Asthecompactobjectorb itsthemassiveblackhole, gravitationalradiationshrinkstheorbitanddecreasesth eorbitaleccentricity,causing thecompactobjecttospiralinuntiltheitisnallydisrupt edbythetidalforce[ 35 ]. Inthelastyearsofinspiralbeforeplunge,EMRIwillberadia tingcontinuouslyat frequenciestowhichLISAissensitive( 3mHz ).Duetotheextrememassratio, theinspiralprocessisveryslow,whichmakesanindividual waveformobservablefor 10 5 cycles = yr [ 36 ].MostabundantEMRIsforLISAdetectionsarebelievedtocons ist ofastellarmassblackholeof 10 M andamassiveblackholeof 10 6 M .LISA iscapableofdetectingsucheventswithin z 1 andthenearesteventsinoneyear maybenofurtherawaythan z =0.1 .Providedthegenerationmechanismoftwo-body scattering,therateofEMRIformationsinourGalaxyisappro ximatelyoneper4million years[ 37 ].ThisgenerationratecorrespondstoaconservativeLISAde tectionrateof50 35

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peryearyetwithalargeuncertaintyduetotheweakconstrai ntsonstellarpopulations neargalacticnuclei[ 38 ]. EMRIsprovideanexcellentsignaltostudytheKerrspacetime .Althoughearly EMRIradiationsduetohighlyeccentricorbitsarewidelydis creteintimeandthereby unresolvableforLISA,theradiationinthelastyearsofinspi ralisbelievedtofaithfully encodetheinformationofthesurroundingKerrspacetime.T hemotionofthecompact objectintheKerrspacetimeisgeodesicinashorttimescale ,whileforalongertime scaletheparametersofthegeodesicmotionwilladiabatica llyprecessduetotheorbital circularization.Also,theorbitalplaneisexpectedtohave Lense-Thirringprecession ascribedtotheblackholespin.Bydetectingtheseintriguin gfeatures,LISAshouldbe abletopreciselymeasurethemassandspinangularmomentum ofthecentralblack hole[ 39 ].Iftheobservedgravitationalwavesignalsareuniquelyd eterminedbythe measuredmassandspin,asrequiredbytheno-hairtheorem,o necanfurtherconrm thatthecentralmassiveobjectisaKerrblackhole.Iftheyd onotsatisfytheno-hair theorem,onecandeducethatthecentralmassiveobjectisso methingelselikeaboson star[ 40 ]. AlthoughthetheoreticalmodelofEMRIisseeminglysimpledue totheextreme massratio,thehighorbitaleccentricitycausedbytwo-bod yscatteringcomplicates thewaveform.Sofar,theEMRIwaveformsstillcannotbecomple telycalculatedfrom theperturbationtheory.Themostcommonmethodexploitsnu mericalsolutionsof Einsteinequationsintheperturbationtheory,knownasTeuk olskyformalism[ 41 ].In thisformalismtheeldequationdescribestheperturbativ eeldsinaKerrmetricanda wholesetofTeukolsky-based(TB)waveformshavebeensolved extensively. 1.3.2.4Cosmicgravitationalbackgroundradiation Analogoustothecosmicmicrowavebackground(CMB),astochas ticbackground ofgravitationalwaveswithafrequencyrangeof 10 18 Hz tohighfrequenciesbeyond 10kHz wasproducedintheearlyuniverse.Theprimordialgravitat ionalwaves 36

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decoupledfrommatteratthePlancktimeof 10 43 s andtraveledthroughtheuniverse almostwithoutanyattenuationorscattering.Therefore,t hedetectionofthecosmic gravitationalwavebackgroundmightbeanexclusivewaytod irectlyprobethephysics intheveryearlyuniverse.Thestochasticbackgroundofgra vitationalwavesisusually describedintheirenergydensity,whichcanbeexpressedas afraction n gw ofthecritical energydensityoftheuniverse.Althoughthespecicvalueof theenergydensityis stilluncertain,anupperlimithasbeendeterminedbytheco nstraintsfromBigBang nucleosynthesis(BBN),aswellasobservationstotheanisotr opyoftheCMBandthe periodofpulsarsignals.TheobservationofCOBEtotheCMBin dicatesanupperlimit of n gw < 10 13 at 10 18 Hz [ 42 ]. Thestochasticbackgroundofgravitationalwavesmayhaveb eengeneratedfrom theamplicationofquantumuctuationsduringination,w hichtransferredenergyinto theuctuationsandconvertedthemintogravitationalwave s.Theconventionalination theorypredictsaatspectrumof n gw thatisindependentoffrequencies,whichmakes theenergydensityoftheination-inducedbackgroundfarb elowthesensitivityofLIGO, LISA( 10 10 at 1mHz )orpulsartiming.However,theactualenergydensitymight stillbemuchhigherthantheconventionalexpectationinso mevariationsoftheination theory[ 43 44 ].Besidesination,thereareothertwomechanismsthatmayp roduce stochasticgravitationalbackground:therst-orderelec troweakphasetransition[ 45 46 ] andcosmicstrings[ 47 ].Theoretically,thesetwomechanismsmayproducegravita tional waveswithanobservableenergydensityforLISAinitsscienc eband. 1.4DetectionofGravitationalWaves Theoriesandobservationshaveindicatedthatthegravitat ionalwavesourcesinour universearenumerousandfullofinformation,yetwestilll ackareliabletooltomeasure them.Withoutdirectdetectionsofgravitationalradiation fromastrophysicalsystems, gravitationalwaveastronomycannotbeconsideredabranch ofobservationalastronomy intherealsense.Startingfrom1960s,researchershavebeen endeavoringinorderto 37

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enhancegravitationalwaveastronomyfrom“thebeginningo fknowledge”to“thestage ofscience”.Inthissectionwewillhaveareviewonvariousd etectiontechniquesthat havebeendevelopedoratleastproposed.1.4.1DetectionMethods Thepioneeringdetectionmethod,knownastheresonantmass detectororthe“bar” detector,wasproposedandbuiltbyJosephWeberintheearly 1960s[ 2 ].Atypicalbar detectorconsistsofacylindermadeofaluminumwithalengt hof L 3m andaradius of R 30cm .Thesensitivityofthebardetectorisattributedtothesha rpresonanceof thecylinder.Bardetectorstypicallyhavearesonantfreque ncyof f 0 500Hz to 1.5kHz Ifthefrequencyofthegravitationalwaveisveryclose(wit hinafewHz)totheresonant frequencyofthebardetector,itwillbeabsorbedandexcite mechanicalvibrationsinthe cylinder.Intheory,ashortgravitationalwaveburstwitha strain h willdriveamechanical vibrationwithanamplitude L hL .Foradecentstrainamplitudeof 10 21 ,thevibration willhaveanamplitudeof 10 21 m .Toreadoutsuchatinyoscillation,aseriesof mechanicaloscillationamplierandresonanttransducers areimplementedtoamplify theoscillationandconvertitintoameasurableelectricsi gnal. Thesensitivityofawell-suspendedandisolatedbardetect orisprimarilylimited bythreeintrinsicsourcesofnoise[ 5 48 ].Therstnoisesourceisthethermalnoiseof theatomsinthecylinder.ComparedwithWeber'soriginalde signthatwasoperatedat roomtemperature,currentdetectorsarecooledtocryogeni ctemperatures( 100mK ) toreducethethermalnoise.Nevertheless,thethermalnois eatthistemperatureisstill toohighsuchthattheoscillatorisrequiredtohaveahighme chanicalqualityfactor Q 10 6 tofurthersuppressthethermalnoise.Thesecondnoisesour ceisintroduced bythereadoutchains,suchastheelectronicnoisefromthea mplierandtransducer, aswellasthereverseeffectofthetransducerthatconverts theelectricsignalinto anmechanicalforceapplyingonthecylinder,knownastheba ck-actionnoise.The readoutnoiselimitsthedetectorbandwidthwithinaveryna rrowrange( 1Hz )around 38

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theresonantfrequency.Thethirdkindofnoiseisaquantume ffectcomingfromthe zero-pointvibrations. Inadditiontobardetectors,electromagneticwavescanals obeusedtoprecisely measurethechangeinthecoordinatedistancebetweentestm asses.Inanelectromagnetic detectionthegravitationalwavemodulatesthepropagatio ntimeoftheelectromagnetic wavebydisturbingthemotionoftestmasses.Sofar,develope ddetectionmethods basedonelectromagneticdetectionsfallintothreecatego ries: Spacecraftranging Pulsartiming Interferometry Intherstmethod,thetestmassesareEarthandafree-fallin gspacecraftin geodesicmotions.Duringthedetectionacommunicationsig nalisemittedfrom thetransmitteronEarthandtravelstothespacecraft.Thesp acecraftisusuallya scienticprobeinaJupiterorSaturnmission,inordertoget alongtransmissiontime ( 2 4 10 3 s ).Atthespacecraftthesignalwillbecoherentlytransponde dbackand returnedtoEarth.Bymonitoringtheoutgoingandreturningti meofthecommunication signal,onecandeducetheeffectofgravitationalwavesont hesignalpropagation. Similarmeasurementsunderthesameprinciplecanbeperform edbytrackingthe Dopplershiftfrequencyaddedontothecommunicationsigna l[ 49 ],sincegravitational waveswillinducerelativemotionsbetweentestmasses.The fractionalchangeinthe communicationsignalfrequencyisgivenby = 1 2 cos2 [(1 cos ) h ( t ) 2cos h ( t l = c l cos = c ) (1+cos ) h ( t 2 l = c )], (1–45) where l istheEarth-spacecraftdistanceatangle tothepropagationdirectiononthe z-axis. istheanglebetweentheprinciplepolarizationvectorofth egravitationalwave andtheprojectionofthespacecraftpositiononthetransve rseplane.Thisequation indicatesthatanimpulsein h ( t ) willappearatthreedifferenttimesintheDopplershift 39

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frequency,knownasthethree-pulseresponse.Itshouldben otedthattimedelaysor Dopplershiftscausedbyotherclassicalandrelativistice ffects(e.g.,orbitalmotions, Shapirotimedelay,etc.)needtobemodeledandsubtractedou tfromthemeasurement data. Pulsartimingexploitstheextremeregularityofpulsarsign alsasatimingreference tomonitorthetimeofarrival(TOA)totheEarth.Thetimingdat afrompulsarscanbe analyzedtosearchforlow-frequencygravitationalwaves[ 50 ],especiallytheupperlimit oftheenergydensityofthecosmicstochasticbackground[ 51 ].Thefractionalchangein thepulsefrequencyduetogravitationalwavesisgivenby = 1 2 cos2 (1 cos )[ h ( t ) h ( t l = c l cos = c )], (1–46) where l istheEarth-pulsardistanceatangle tothepropagationdirectiononthez-axis. istheanglebetweentheprinciplepolarizationvectorofth egravitationalwaveandthe projectionofthepulsarpositiononthetransverseplane.I ncomparisontoEq. 1–45 animpulsein h ( t ) onlyappearsattwodifferenttimesinthefrequencychange, thereby knownasthetwo-pulseresponseduetotheone-waytransmiss ionfromthepulsarto theEarth. Indataanalysis,theactualTOAofradiopulsesiscomparedw ithatheoretical modelbasedonEq. 1–46 ,whichyieldsatimingresidual t .Intheory,pulsartimingis capableofdetectinggravitationalwavesontheorderofmag nitude h ( f ) t f ,where f isthegravitationalwavefrequency.Thisgivesastrainsen sitivityof 10 15 10 16 at 10 9 Hz .Inpracticalmeasurements,thedataoftimingresidualswi llbecorrelated betweendifferentpulsarswithanumberof N toenhancethedetectionsensitivity,which yields h ( f ) t f = p N .Inadditiontoinuencesoftheintrinsictimingjitterand local instrumentnoises,theradiopulsesalsoencounteraseries ofinterstellarmedium(ISM) effectsduringthepropagation,suchasdispersion,scatte ring,scintillationandrefraction, whichrequireadditionalmeasurementsandanalysistocorr ectthem. 40

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Figure1-3.SchematicofabasicMichelsoninterferometer Undercurrentunderstandingofgravitationalwavedetecti ons,themostpromising techniqueisinterferometry.Michelsoninterferometersa reinherentlyidealgravitational wavedetectorsascribedtotheapplicationonpreciselengt hmeasurements.Today, avarietyofthe1stgenerationinterferometricgravitatio nalwavedetectors(LIGO[ 6 ], VIRGO[ 52 ],GEO[ 53 ],TAMA[ 54 ])areinoperationandthe2ndgenerationdetectors (AdvancedLIGO[ 6 ],LCGT[ 55 ])arealreadyindevelopment[ 56 ].Moreover,the space-basedinterferometricdetectors,LISAandconceptua lizedDECIGO[ 57 ],mayalso belaunchedinthenearfuture.1.4.2InterferometricDetectors TheconceptofinterferometricdetectorsisassimpleasaMi chelsoninterferometer, asshowninFigure 1-3 .Amonochromaticlasersourceemitsabeamtothebeamsplitt er, whichseparatesthebeamwithequalprobabilityamplitudes .Thetwointerferometer armshavenearlyidenticallengthsandareperpendicularto eachother.Attheend ofeacharm,atotallyreectivemirrorisplacedtobounceth ebeamback.Afterthe round-triponeacharm,thetwobeamsrecombineatthebeamsp litterandarethen 41

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receivedbythephotodetector.Themirrorsinaground-base dinterferometerarenot free-fallingduetothesuspensiontobalancethegravity,b uttheystillcanbeusedastest massessinceweareonlyconcernedwiththehorizontalplane .Ifgravitationalwaves modulatethearmlengthsinto L x and L y ,thelaserpowerreceivedatthephotodetector isproportionalto j E PD j 2 = E 2 0 sin 2 [ k L ( L y L x )], (1–47) where E 0 isthemagnitudeoftheinputelectriceldand k L isthewavenumberof thelaser.Herewetakeintoaccountthe phaseshiftinthereectedbeamatthe beamsplitter.1.4.2.1Principleandconguration AspreviouslymentionedinSection1.2.3,theinteractionofg ravitationalwaves withinterferometricdetectorscanbeevaluatedineithert heTTframeortheproper detectorframe.Herewewillbrieyreviewthecalculationi ntheTTframe,wherethe coordinatesofmirrorsdonotchangeinthepresenceofgravi tationalwaves,whilethe properdistancedoes.Thelightround-triptraveltimeonea charmcanbecalculated startingfromthespacetimeintervalgivenbyEq. 1–11 orEq. 1–27 .Herewedirectly givetheresult: t 2 t 0 = 2 L x c + L x c h ( t 0 + L x = c )sinc L c (1–48) Assumingthattheinterferometerisonthex-yplaneandthegr avitationalwave h ( t )= h + cos( t ) onlyhasthepluspolarizationandispropagatinginthez-ax is, thisequationgivesthelightround-triptraveltimeinanar monthex-axis,wherethe armlengthis L x .Thelaserbeamisemittedfromthebeamsplitterat t 0 andreturns tothebeamsplitterat t 2 .Notethatwhenthefrequencyofgravitationalwavesisnot toohigh,i.e., L = c 0 ,thesincfunctionapproaches 1 andthelighttraveltime approaches (2 L x + L x h ( t )) = c .However,whenthefrequencyofgravitationalwaves iseithertoohigh( L = c 1 )oratmultiplesof c = L ,thesincfunctionequalsto zeroandtheinterferometerisinsensitivetosuchgravitat ionalwaves.Themultiples 42

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of c = L intheangularfrequencyarethereforecalledinterferomet ernulls.Sincethe armlengthofcurrentground-baseddetectorsisontheorder of km ,interferometer nullshavenoinuenceontheirdetectionperformance.Howe ver,thesensitivityof space-baseddetectorssuchasLISAandDECIGOwithamuchlonge rarmlengthmight becompromisedatthesefrequencies. Thelightround-triptraveltimeintheotherinterferomete rarmcanbeobtainedina similarform.Fromthelighttraveltimeswecanwritedownth ereturningelectricalelds atthebeamsplitterandthenthetotalelectriceldreceive datthephotodetectorisgiven by E PD ( t )= iE 0 e i L ( t 2 L = c ) sin[ 0 + ( t )], (1–49) where L ( L x + L y ) = 2 isdenedasthecommonarmlengthand 0 = k L ( L x L y ) is theintrinsicphasedifferenceduetothearmlengthmismatc h. ( t ) comesfromthe modulationsfromgravitationalwavesandisgivenby ( t )= h + k L L sinc L c cos[ ( t L = c )] h + k L L cos[ ( t L = c )].(for L = c 1) (1–50) Fromthepurposeofdetection, ( t ) shouldbeaslargeaspossible,whichrequires thedesignofinterferometerstobeoptimized.Inotherword s,inthephaseshiftthe factor k L L sinc L c = k L k sin L c (1–51) shouldreachthemaximum.Therefore,werequire L = c = n = 2 ,orthearmlength needstobeatleast = 4 ,where isthewavelengthofgravitationalwaves.Unfortunately thisisaproblemformostgravitationalwavesources:Forag ravitationalwaveat 100Hz ,theoptimizedarmlengthis 750km ,whichisanunrealisticrequirementfor currentground-baseddetectors. 43

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Thesolutionforground-baseddetectorsisanovercoupledF abry-Perotcavity.A Fabry-Perotcavitythatconsistsoftwolow-transmissivit ymirrorsiscapableoftrapping thephotonsinsideforacertainamountoftimeandthensubst antiallyincreasesthe effectivearmlength.Intheory,thestoragetime,whichist heaveragetimeforaphoton trappedinsidethecavity,isgivenby s L c F (1–52) where L isthecavitylengthand F isthenesseofthecavity.Comparedtoasimple Michelsoninterferometer,theeffectivearmlengthisenha ncedbyafactorof (2 = ) F .For ground-baseddetectorsthetypicalnesseis 10 2 .InthepresenceofaFabry-Perot cavityoneacharm,thephaseshiftduetogravitationalwave sisgivenby ( f ) h + 4 F k L L 1 p 1+( f = f p ) 2 (1–53) where f p =1 = (4 s ) iscalledthepolefrequencyoftheFabry-Perotcavity. Inpractice,aninterferometricdetectorwillencounterva riousrealisticissuesthat needtobeaddressed.First,laserbeamsmusthavespatialpr olesthatarenotideal planewavefrontsinthetransversedirection.Thebestchoi ceofaproleisaGaussian beam,whichcorrespondstothe TEM 00 modefromthespatialHermite-Gaussian solutionsoftheparaxialwaveequation.TheadvantageofaG aussianbeamisthe sphericalwavefrontthatcanbemode-matchedtosphericalshapedmirrors.For interferometricdetectors,thelaserisrequiredtobeover whelminglyoperatingatthe TEM 00 modeandtheotherhighermodesthatcausenoisesmustbemini mized.In additiontocarefulalignments,thelasereldisrstsenti ntoanotherFabry-Perotcavity knownasthemodecleanerbeforeentersthebeamsplitter,en suringthatonlythe TEM 00 modeisonresonanceandtransmittedtothemirrors.Asecond modecleaner isplacedbetweenthebeamsplitterandthephotodetectorto furtherlteroutthehigher modes.AnotherimportantissueistomakesureeveryFabry-Pe rotcavityisoperating 44

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onresonance.ThesolutionisknownasthePound-Drever-Hal l(PDH)techniquewhich locksthelasertotheresonantfrequencyofthecavityviafe edbackcontrol.Wewillsee moredetailsofthePDHtechniqueinSection2.2.2.1.1.4.2.2Noiselimitations Themainnoisesourcesofaninterferometricdetectorcanbe classiedintotwo categories:Therstclasscomesfromthequantizationofli ght,includingtheshot noiseandtheradiationpressurenoise.Theyjointlydeterm inethestandardquantum limit(SQL)ofinterferometricdetectors.Thesecondclassi nvolvesnoisesourcesfrom internalBrownianmotionsorotherexternaleffects,whichm anifestthemselvesasa displacementnoiseinthemeasurement. Shotnoise-Thelimitednumber N ofphotonsreceivedpersecondbyaphotodiode willcauseastandarddeviationgivenby p N inthenumberofphotons.The observedpoweructuationisthengivenby P shot = p N ~ L T = r ~ L T P (1–54) Thispoweructuationwillcoupleintothereadoutoftheint erferometeroutput. Whenrepresentedinthestrainsensitivity,itisgivenby S 1 = 2 shot ( f )= 1 8 F L s 4 ~ L c P bs p 1+( f = f p ) 2 (1–55) where P bs = CP 0 isthepowerenteringthebeamsplitterafterpowerrecyclin gand istheefciencyofthephotodiode.Therefore,inthefreque ncydomaintheshot noiseoorisatuntil f = f p andthenincreaseswitha f slope,dominatingthehigh frequencyrangeabove 100Hz .Also,increasingthelaserpowertoaccumulate morephotonspersecondwillhelptoreducetheshotnoise. Radiationpressurenoise-Ontheotherhand,atoolargenumb erofphotons mayalsocauseaproblem.Whenthephotonsstrikethemirrorth eywillexerta continuousradiationpressurebytransferringamomentum 2 ~ L = c .Theradiation pressureisthengivenby F =2 P = c whichcomeswithuctuationsduetothe variationsinthepower.FromEq. 1–54 ,theuncertaintyintheradiationpressureis thengivenby S 1 = 2 rad =2 r 2 ~ L P c 2 (1–56) 45

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Thisforceuctuationcanbeconvertedintoadisplacementn oiseinthemirrorvia theLaplacetransformof F = M x ,where M isthemassofthemirror.Whenthe radiationpressurenoisecouplesintothereadout,thestra insensitivityisgivenby S 1 = 2 rad ( f )= 16 p 2 F ML (2 f ) 2 r ~ 2 P bs L c 1 p 1+( f = f p ) 2 (1–57) Thisequationshowsthatinadditiontosquarerootofthelas erpower,theradiation pressurenoiseisalsoproportionaltothenesseofthearmc avity. Seismicnoise-Groundmechanicalvibrationsarethelimitfo rground-based detectorsinthefrequencyrangebelow 10Hz .Theseismicnoiserepresentedas adisplacementnoiseisapproximatelyontheorderof L seismic 10 7 1Hz f 2 mHz 1 = 2 (1–58) Themitigationofseismicnoiseinthemirrorisrealizedbys ophisticatedisolation andsuspensionsystems.Inthesuspensionsystem,asetof N pendulumsare placedincascadetolteroutthehighfrequencynoisewitha transferfunctionof ( f 2 0 = f 2 ) N ,where f 0 istheresonancefrequencyofthependulum. Gravitygradientnoise-ThelocalNewtoniangravitational eldcanvarydueto theenvironmentchange,suchasthetime-varyingtidalforc eoructuationsinthe massdensityfromnearsources.Sincethegravitationalforc ecannotbescreened, thegravitygradientnoisecannotbeattenuated.However,i tdecreasesrapidlyat highfrequenciesandisnotdominantatlowfrequenciesforc urrentground-based detectors.Gravitygradientnoisewillbetheultimatesens itivitylimitforthenext generationground-baseddetectorstopursue. Thermalnoise-Thermalvibrationsinthesuspendedpendulu mandthemirrorwill bothcoupleintothemeasurements.Thedominantnoiseinthe frequencyrange 10 10 2 Hz istheBrownianthermalnoiseofthemirrorcoating.Inadditi on,the thermaluctuationsinthesuspensionwillinducespurious motionsofthemirrorin bothhorizontalandverticaldirections.Thenormalmodeso fthesuspensionwire mayalsobepresentinthefrequencyspectrumasverynarrowp eaks. Asanexample,thesensitivitycurveofLIGOisillustratedin Figure 1-4 .The gureshowsthatatabout 100Hz thestrainsensitivityreachesitsmaximumof 3 10 23 Hz 1 = 2 .Athigherfrequenciesthesensitivitycurveincreasesmode rately witha f slopeduetotheshotnoiselimit.Forfrequenciesintherang efrom 40Hz to 100Hz thesensitivityismainlydominatedbythecoatingthermaln oiseonthetest mass.Forfrequenciesbelow 40Hz themostdrasticchangehappens:Theseismic 46

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Figure1-4.StrainsensitivitiesofLIGOS6,courtesyofLIGO. noisedominatesthelowfrequencyregionandsensitivitysh arplydrops.AdvancedLIGO willutilizemuchbetterisolationschemes;nevertheless, thecornerfrequencywherethe seismicnoisestartstodominatewillstillbearound 10Hz ,whichmakesthedetectionof low-frequencygravitationalwavesimpossibleforgroundbaseddetectors. Foranabsoluteisolationfromseismicnoise,theonlyfeasi blewayistoplacethe interferometerintospace.Therstspace-basedinterfero metricdetectorprojectis LaserInterferometerSpaceAntenna(LISA)[ 7 8 ],whichisaimedtodetectgravitational wavesbetween 3 10 5 Hz and 1Hz .Anotherproposedspace-basedinterferometric detector,Deci-hertzInterferometerGravitationalWaveO bservatory(DECIGO)[ 57 ],will detectgravitationalwavesmainlybetween 0.1Hz and 10Hz tobridgethefrequencygap betweenLISAandground-baseddetectors.DECIGOwillsynthes izetheconstellation designofLISA(SeethenextchapterfordetailsofLISA),yetwith muchshorterarm lengthsof 1000km suchthatLIGO-likeFabry-Perotinterferometerscanbeuse das armcavities.TheinterferometryusedinLISA,ontheotherhan d,isquitedifferentfrom 47

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standardMichelsoninterferometryusedinground-basedde tectors.Wewillfocusonthe LISAtechnologyinthenextchapter. 48

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CHAPTER2 LASERINTERFEROMETERSPACEANTENNA 2.1Overview TheLaserInterferometerSpaceAntenna(LISA)isacollaborativ eNASA/ESA spacemissiontodetectgravitationalwavesinthefrequenc yrangeof 3 10 5 Hz to 1Hz [ 58 ].Itwillbetherstspace-basedinterferometricgravitat ionalwavedetectorto belaunchedin2020s.Unliketheequal-armMichelsoninterf erometerconguredin standardground-basedinterferometricgravitationalwav edetectors,LISAconsistsof threeidenticalspacecrafttrailingtheearthbyabout 20 inindependentheliocentric orbits,arrangedinaquasi-staticequilateraltriangle,a sshowninFigure 2-1 .Thearm lengthbetweeneachtwospacecraftisapproximately 5Gm andvariesbyupto 1% overthe10-yearlifetimeofthemission.Thistime-depende ntvariationinarmlengthis mainlycausedbythedifferentgravitationalpullofEarthon theindividualspacecraft. ThismotionalsoDopplershiftsthefrequencyofthereceive dlasersusedtoperformthe interferometry. Eachspacecrafthousestwodrag-freeproofmassesthatfollo wthegeodesic motion.Ahousingaroundtheproofmassfunctionsasasensor todetecttherelative positionbetweentheproofmassandthespacecraft.TheDist urbanceReduction System(DRS)controlsthethrustersonthespacecraftandmini mizetheacceleration oftheproofmassduetoundesiredexternalforces.TheInter ferometricMeasurement System(IMS)ofLISAmonitorschangesintheseparationbetween twoproofmasseson eachrespectivespacecraft.Anymodulationontheseparatio ncausedbygravitational wavesandotherspuriousaccelerationsoftheproofmassesw illbemeasuredvia interferometrywiththedesiredsensitivity. LISAwilluse 1064nm laserswithanoutputpowerofapproximately 2W .The transmittedlasereldhasasignicantdiffractionlossdu etothelongarmlength, thereforeonlyaverysmallportionoflight( 100pW )willbereceivedbythephotodetector 49

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Figure2-1.TheheliocentricorbitoftheLISAconstellation .Theconstellationtrailsthe Earthby 20 andtheplaneofitisinclinedwithrespectoftheellipticby 60 Thearmlengthbetweeneachtwospacecraftisgenerally 5Gm onthefarspacecraft.Forthisreason,thelasereldemitte dfromthefarspacecraft transpondsthereceivedlaserphasethroughaheterodyneph ase-lockedloopwithan offsetfrequencyintherangeof 2 20MHz .Whenthelasereldfromthefarspacecraft istransmittedbacktothelocalspacecraftitwillbesuperi mposedwiththelocallaser eldtogenerateabeatsignal.Adigitalphasemeterwillmea surethephaseofthis interferometeroutputwithsufcientprecisions( cyclesHz 1 = 2 ),inordertoextractthe gravitationalwavesignalsduringthepost-processing[ 59 ]. 2.1.1Sensitivity ThesensitivityrequirementofLISAisusuallyrepresentedb ythelinearspectral density(LSD)ofthegravitationalwavestrainamplitude, p S h ( f ) ,whichisplotted inFigure 2-2 .Thestrainisproportionaltochangesintheseparationbet weentwo proofmasses,whichisalsoknownasthesinglelinkequivale ntpositionuncertainty L singlelink ( f ) .Mathematically,thestrainsensitivityisgivenby p S h ( f )= p 5 2 p 3 T ( f ) L singlelink ( f ) L (2–1) InEq. 2–1 p 5 representstheaveragedantennaresponseoverthewholesky 2 p 3 =1 = sin(60 ) istheprojectionoftheequilateraltriangularconstellat ionontoan 50

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Figure2-2.TheLISAsensitivitycurvedescribesthestraina mplitudespectraldensity versusfrequency.Theredcurverepresentstherequirement onthestrain sensitivity( 10 4 Hz to 1Hz )andthebluecurveistheextendedgoal ( 3 10 5 Hz to 1Hz ).AdaptedfromRef[ 60 ]: LISA:Unveilingahidden universe:Assessmentstudyreport equivalentL-shapedMichelsoninterferometer. T ( f ) istheinstrumenttransferfunction, whichconvertsthesinglelinkequivalentpositionuncerta intyintothestrainresponseto gravitationalwaveswithdifferentfrequencies,assuming thattheLISAconstellationwill formtheMichelsonX-combinationforTimeDelayInterferome try(TDI). RecalltheinterferometerresponseofaMichelsoninterfer ometertogravitational waves,aswediscussedinSection1.4.2.1.Theinstrumenttra nsferfunctionofLISA canbeunderstoodinaverysimilarway:Itisbasicallyatat lowfrequencies,which indicatestheinstrumentsensitivityisindependentofthe gravitationalwavefrequency ifitislowenough.Athighfrequencies f GW > c 2 L ,theinstrumentsensitivitygradually decreaseswithaslopeof f asthefrequencyofthegravitationalwaveincreases. Especiallyatfrequencieswherethearmlengthisexactlymul tiplesofhalfofthe gravitationalwavewavelength,theLISAarmhaszerorespons eforgravitationalwaves withnormalincidence.Attheseinterferometernullstheave ragedsensitivityfurther 51

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decreases(byaboutafactorof2)butnevergoesdowntozero[ 61 ],sincegravitational waveswiththesamefrequencywhilefromdifferentsourcesh avedifferentincident directionsontoLISA. BasedonthenoisebudgetdescribedinRef[ 60 ],thesinglelinkequivalentposition uncertainty L singlelink ( f ) islimitedatlowfrequenciesbyaccelerationnoise x DRS ( f ) causedbyspuriousresidualforcesthatactontheproofmass es.Athighfrequencies, itislimitedbysensingnoiseanddisplacementnoise x IMS ( f ) generatedinsidethe interferometricmeasurementsystem.Thecombinednoiseli mitationisthengivenbythe sumofthepowerspectraldensity(PSD)ofthesetwouncertaint ies,i.e., L singlelink ( f )= q x 2 DRS ( f )+ x 2 IMS ( f ). (2–2) Thespecicexpressionsoftheallocatedaccelerationnois eandthepathlength noiseare x DRS ( f ) 3fm = s 2 (2 f) 2 p Hz s 1+ f 8mHz 4 s 1+ 0.1mHz f (2–3) and x IMS ( f ) 18pm p Hz s 1+ 2.8mHz f 4 (2–4) wherethemagnitudesof 3fm = (s 2 p Hz) and 18pm = p Hz arerespectivelythe totalnoisebudgetsoftheDRSandtheIMS.Bothhavea 35% marginoverthetotal subsystemallocations.Asubstantialcontributiontothed isplacementnoiseintheIMS isattributedtotheshotnoiseatphotodiodes.Givenatypic al 100pW receivedpowerat photodiodes,theshotnoisebudgetisgivenby x shot = 2 r h P 7pm p Hz (2–5) 2.1.2DisturbanceReductionSystem Asagravitylaboratoryandinterferometricgravitationalw avedetector,LISAhasto containtestmasseswhosemotionispurelydeterminedbythe surroundingspacetime 52

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geometry,oratleastbefree-fallinginthedirectionsofin terest.Themainobjectiveof theDisturbanceReductionSystemistoprovidesuchadrag-fr eeenvironmentforthe proofmassesbyisolatingthemfromrandomexternalforces, aswellasmaintainthe spacecraftcenteredontheproofmassesviamicro-Newtonth rusters.Possiblerandom externalforcesthatmaycausespuriousaccelerationnoise includesolarradiation pressure,interplanetarymagneticelds,gravity,electr ostaticandmagneticdisturbances fromthespacecraft,time-variablegaspressureduetotemp eratureuctuations,etc. Tomaintaintheproofmasstofollowthegeodesicmotion,eac hspacecraftis loadedwithtwoGravitationalReferenceSensors(GRS's)[ 62 ],eachmountedinthe lineofsighttothetelescope.EachGRSprovidesahousingtha tcontainselectrodesto encloseaproofmassandmonitoritspositionindividually. Theelectrodesarearranged inacertainpatternsuchthatallpossiblerelevantdisplac ementsandrotationsofthe proofmasscanbereadoutbymeasuringthecapacitancebetwe entheproofmass andthehousing,knownascapacitivesensing[ 63 64 ].Themeasurementsensitivity ofcapacitivesensingisonthelevelof 1.8nmHz 1 = 2 fordisplacementsinsensitive directionsand 200nradHz 1 = 2 forrotations.ByapplyingACvoltagesontheelectrodes, theorientationandpositionoftheproofmassinthenon-sen sitivedirectionscanbe appropriatelycontrolled. TheproofmassofLISAisanalloycube( 73% goldand 27% platinum)withamass of 1.96kg anddimensionof 46mm .Thismixingratiowaschosentominimizethe magneticsuspectibilityoftheproofmassto j j < 2 10 5 [ 65 66 ].Nevertheless,the non-zeromagneticsuspectibilityrequiresstrictmagneti ccleanlinessofthespacecraft. Thesurfacesoftheproofmassesarecoatedwithagoldenlaye rtoincreasethe reectivityfortheshortarminterferometry. AsaprecursoroftheLISAmission,LISAPathnderwillbelaunch edinthe2010s [ 67 ].TheLISAPathndermissionwilltestthetechniqueofdragfreeoperations, includingtheidenticalGRSandsimilartestmassinterfero meters(seethenextsection), 53

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etc.Therequirementoftheresidualaccelerationnoiseoft hetestmassesonLISA Pathnderisrelaxedbyaboutafactorof10comparedtoLISA.2.1.3InterferometricMeasurementSystem TheInterferometricMeasurementSystemmonitorstheuctua tionsinthe separationbetweentwoproofmassesondifferentspacecraf t.TheIMSiscomposedof threesubsystems:theopticalsystem,thelasersystemandt hephasemeasurement system.Inthissectionwewillfocusontheopticalsystemof LISA,alsoknownas anopticalassemblyhousedbyeachspacecraft.Eachopticala ssemblyconsistsof apairofunitsandeachunitconsistsofanopticalbench,ate lescopeandaGRS. Thetelescopesarepointingtotheothertwospacecraftindi vidually.Theoptical benchisparalleltotheprimarymirrorofthetelescopesuch thattheopticalbenchis perpendiculartotheplaneoftheLISAconstellation,wherea stheGRSismounted behindtheopticalbench. Asthemainpartoftheopticalsystem,theopticalbenchdirec tsthelaserbeams tothedesiredspatialpositionsforinterferences.Thesep arationbetweentwoproof massesismonitoredthroughthecombinationofthreediffer entinterferencesgenerated ontheopticalbench,whichisillustratedinFigure 2-3 .Alltheinterferometricmeasurements inLISAareheterodynedetections,wherethetwolaserbeamsw ithclosefrequencies ( 2 20MHz )generateaheterodynebeatsignaltobemeasuredbyaphotod etector. ThephotodetectorLISAwillusearequadrantphotodetectors (QPD's)thatcan detecttheheterodynephaseaswellastheanglesofthewavef ront.Thisisknown asdifferentialphasesensingorwavefrontsensing[ 68 ].AsshowninFigure 2-4 : Alocallaserbeamistransmittedfromthelocallaserintoth eopticsthroughan opticalber. Anadjacentlaserbeamistransmittedfromtheadjacentbench throughanother opticalberknownastheback-linkber. Afarlaserbeamistransmittedfromthefarspacecraftthrou ghthetelescope. 54

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Figure2-3.TheInterferometricMeasurementSystemofLISA.Eac hspacecrafthouses twoopticalbenches,whichtransmit/receivelaserbeamsto /fromtheother twospacecraftviatelescopes,formingsixone-waylinks.I nthisdiagramone laser(the“red”laser)on SC 1 isusedasthemasterlaser,whichis pre-stabilizedtoanopticalcavity.Theothervelasersar ephase-locked eitherlocallytothemasterlaserviatheback-linkberoru singthebeat signalwiththefarlaserviaopticaltransponders.Thedist ancebetweentwo proofmassesoneacharmismonitoredbythreedistinctmeasu rements: onemeasurementbetweenopticalbenches(longarm)andtwomeasurementsbetweeneachproofmassanditsrespectiveopt icalbench (shortarm+reference). 55

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Figure2-4.InterferometersontheopticalbenchofLISA.Thei nterferometryusedin LISAisheterodynedetections,wheretwolaserswithclosefr equenciesare combinedtogenerateabeatsignalwithitsphasedetectedat the photodetector.Oneachopticalbenchthreeinterferometer sareconstructed usingthreephotodetectorsandtheirmeasurementsarecomb inedto measurethedistancebetweentwoproofmassesoneacharm.No tethat thisisjustanoversimplieddiagramtoillustratetheinte rferometers, whereastherearealsonon-interferometricbeamspropagat ingthroughthe opticalbench.Also,theGRSanditshousedproofmassareactu allynotin theplaneoftheopticalbench. Figure 2-3 showsthattheadjacentbeamandtheoriginallocalbeamgene ratea heterodyneinterferenceatthephotodiode PD R ,whichmeasurestherelativephase noiseofthetwolasersandtheadditionalnoiseintroducedb ytheback-linkber. Thismeasurementisknownasthereferencemeasurement.The adjacentbeamis alsobesuperimposedontothelocalbeamthatisreectedoff fromtheproofmass. Thisinterferencethatoccursatthephotodiode PD S ismonitoredastheshortarm measurement,whichcontainstheuctuationinformationof thedistancefromthe proofmasstotheopticalbench[ 69 ].Therefore,theshortarminterferometerofLISA isalsocalledthetestmassinterferometer,wherethereado utcanbesenttotheDRS toactivelycontrolthemotionoftheproofmassincorporati onwithcapacitivesensing. Thethirdkindofinterferencethatoccursbetweenthelocal beamandthereceived anddelayedfarbeamisrecordedbythephotodiode PD L .Thisbeatnotecarriesthe 56

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uctuationinformationoftheLISAlongarmlengthfromaloca lopticalbenchtoa faropticalbench,whichisknownasthelongarmmeasurement .Sincethelongarm interferometerisusedtodetectthelengthuctuationsdue togravitationalwaves,it isalsoreferredtoasthescienceinterferometer.Therefor e,wecanwritedownthe photodiodereadoutsofthesethreeinterferences: Referencemeasurement: R ( t )= l ( t ) a ( t )+ N a ( t ) Shortarmmeasurement: S ( t )= l ( t ) a ( t )+ 2 L pm ( t )+ N a ( t ) Longarmmeasurement: L ( t )= l ( t ) f ( t ji ( t ))+ N Trans ( t ji ( t ))+ h ji ( t ) Intheaboveexpressionsotheruncorrelatednoisesuchassh otnoiseintroducedat thephotodiodeandthetechnicalnoiseintroducedatthepha semeterADCsareignored. l ( t ) a ( t ) and f ( t ) arethephasesofthelocalbeam,theadjacentbeamandthefar beam,respectively. N a ( t ) containsthenoiseintheback-linkber. L pm ( t ) istherelative positionbetweentheproofmassandtheopticalbench.Theli ghttraveltimefrom SC j to SC i ji ( t ) isafunctionoftimeascribedtotherelativemotionbetween spacecraft.The transpondernoise, N Trans ( t ) ,primarilyincludesthephase-lockedloopnoisefromfar spacecraft. h ji ( t ) isthephaseuctuationduetotheincidentgravitationalwa vestrain appliedontheLISAlongarmlength(opticalbenchtoopticalb ench). Althoughthelaserphasenoisedominatesinallthreemeasure dphases,the linearcombinationofthereferencemeasurementandthesho rtarmmeasurement, S ( t ) R ( t ) ,eliminatesthelaserphasenoisein l ( t ) and a ( t ) ,aswellasthe additionalnoise N a ( t ) .Thustheonlyremainingterm 2 L pm ( t ) representstherelative positionoftheproofmass. However,amorecomplicatedversionofthisnoisecancellat ionschemeisused forthecaseoflongarminterferometry,wheretheexistence ofdelaytimeprevents thelaserphasenoisefrombeingautomaticallyeliminated. Duetothesignicant inequalitybetweenarmlengthwhichvariesovertime,thela serphasenoiseontwo longarmsbecomesuncorrelatedandnolongercancelsoutina standardMichelson 57

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combination,nottomentionthatthetranspondernoisesont wolongarmsare intrinsicallyuncorrelated.Thetechniquetosolvethispr oblemisknownasTimeDelay Interferometry[ 70 71 ].Asapost-processingalgorithm,TDIsynthesizesequal-ar med Michelsoninterferometrybytime-shiftingtheinterferom etrysignaldataproperlysuch thatthelaserphasenoise,clocktransfernoise,benchmoti on,etc.,arecanceledoutin certainlinearcombinations.MoredetailsofTDIwillbesee ninthenextsection. Inadditiontothethreebasicinterferometersdescribedab ove,theopticalbench alsoincludesthepoint-aheadanglemechanism(PAAM).ThePAAMe nsuresthatin thelongarminterferometrythedirectionofthetransmitte dbeamisnotchangedwhen receivedbythefarphotodetectorduetothetransverserela tivemotionofthespacecraft. Thetransversecomponentofthevelocityistime-dependent ;therefore,thePAAMis requiredtobeanon-boardtechniquetocorrectthetime-var yingdeviationangle.It isestimatedthatthistime-varyinganglewillaccuratelyt racktheorbitalmotionofthe constellation.Consequently,thecorrectionmechanismca nbeimplementedbefore launchandafeedbackcontrolloopisnotnecessary. Thetelescopeisanothercrucialcomponentoftheopticalas semblythatisused totransmitlasereldsinthelongarminterferometry.Thet elescopeforLISAwilluse anoff-axisCassegrainreectordesignwithamagnication of 80 x .Eachtelescopeis pointingtothecorrespondingfarspacecrafttogatherthei ncomingbeam( 100pW ) aswellastoexpandandcollimatetheoutgoingbeam( 1W ).Thelaserbeamfrom thelocalopticalbenchwillbefocusedontothesecondarymi rror,whereitismagnied andreectedtotheprimarymirror.Fromtheprimarymirrort hecollimatedbeamwillbe senttothefaropticalbench,whereareverseprocesswillbe performed.Thedesignof thetelescopedeterminesthepowerofthereceivedbeamandt herebytheshotnoise limitfortheIMS.Themechanicalstabilityofthepathlengt hbetweentheprimaryandthe secondarymirroriscriticalasthepathlengthnoisewilldi rectlyenterthereadoutofthe 58

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longarminterferometer.GiventhetotalIMSnoisebudgetof 18pmHz 1 = 2 ,therequired pathlengthstabilityforthetelescopeisnomorethan 1pmHz 1 = 2 2.2NoiseCancellationforLISA LISAwilluselaserswithawavelengthofprobablyaround 1 m .Thelaserwill havetohavefrequencyactuationcapabilitiessufcientto meettheLISAfrequency noiserequirements.Forquitealongtime,thelaserfrequen cycontrolhasbeen asignicantandcomplicatedtopicinstudiesonthelongarm interferometry.Ina laserinterferometricmeasurement,thelaserfrequencyno iseisrequiredtocausean equivalentnoisenohigherthanthequantitytobemeasured. IntheIMSnoisebudget, theallocatedpathlengthnoiseforthelaserfrequencyis 2pmHz 1 = 2 ,correspondingto alaserphasenoisebelow 1.2 10 5 cyclesHz 1 = 2 oralaserfrequencynoisebelow 5.6 10 6 HzHz 1 = 2 .SincethearmlengthsoftheLISAconstellationareinherentl y unequaltoeachotherandalsotime-varying,thelaserfrequ encynoisedoesnotcancel outinastandardMichelson-likeway.Incomparison,thetyp icalfrequencynoiseof afree-runninglaserisonthelevelof 10kHz = f HzHz 1 = 2 ,whichrequiresthelaser frequencynoisetobereducedbymorethan12ordersofmagnit udeat 3mHz Currentunderstandingofthelaserfrequencycontrolgives athree-levelapproachto mitigatethefrequencynoise[ 72 ].Thersttwostepsareactivestabilizationtechniques andthethirdstepisapost-processingtechnique.Thelaser frequencywillrstbe activelystabilizedtoalocalreferenceonspacecraft(pre -stabilization),thenstabilized totheLISAarm(armlocking)andnallygetcanceledouttothe requirementinthe post-processing(TDI).2.2.1TimeDelayInterferometry TheconceptofTimeDelayInterferometry(TDI)istosynthes izeinterferometry (TDIobservables)bylinearlycombiningphasemeasurement sofsinglelinksand appropriatelytime-shiftingthemeasurementdataduringp ost-processing[ 11 73 ]. Asanalgorithminthetimedomain,TDIiscapableofeliminati ngthelaserphase 59

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noise,differentialclocknoiseandspacecraftmotionsint helongarminterferometry whilemaintainingthephaseuctuationsduetogravitation alwavestrains.Thenoise suppressionperformanceofTDIisprimarilylimitedbythea rmlengthknowledge,which determinesthelighttraveltimeusedforaccuratelytime-s hiftingandfractionaldelaying themeasurementdata. TheTDIalgorithmthatassumesxedarmlengthbetweenspace craft,alsoknown astherstgenerationTDIor“TDI1.0”,onlyexploitsthepha semeasurementsdelayed bytherespectivepropagationdistancetoformTDIobservab les.Thesecondgeneration TDIor“TDI2.0”takesthetime-varyingarmlengthcausedbys pacecraftrelativemotions intoaccountbytime-shiftingthephasemeasurementsbymul tiplesofthepropagation distance[ 70 74 ].A“TDI3.0”algorithmwhichincludestherelativeacceler ations betweenspacecrafthasalsobeenconceptualized,butisunn ecessarytomeetthe LISAnoisebudget.Theindividualphasemeasurementcanbeli nearlycombinedto formvariousTDIobservables,suchasanunequal-armedMich elsoninterferometer (Xcombination)oraSagnacinterferometer( combination).Inthissectionwewill introducethebasicMichelsonX-combinationwithxedarmle ngthtoexplainthenoise cancellationinTDI.2.2.1.1MichelsonX-combination TheMichelsonX-combinationexploitsthephasemeasurement sonfourlinksto formanequal-armedMichelsoninterferometer.Itshouldbe notedthatthephase-locking onthefarspacecraft(orcorrelationbetweenlasers)isnot necessaryforanykindof TDIcombinations;however,usingopticaltransponderscou ldsimplifytheMichelson X-combinationsuchthatonlythetwophasereadoutson SC 1 arerequired.Asshownin Figure 2-5 (left), SC 1 isusedasthelocalspacecraftlocatedatthevertex.Sinceth etwo laserson SC 1 arephase-lockedtoeachotherviatheback-linkber,thelo calspacecraft canbeconsideredasthebeamsplitterthatgeneratestwoinphasecoherentbeamsin astandardMichelsoninterferometer.Thelasereld 1 ( t ) emittedfrom SC 1 propagates 60

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Figure2-5.(Left)TheMichelsonX-combinationinstaticTDI ,where SC 1 isusedasa beamsplittertogeneratetwocoherentbeamsandthelaserbe ams transmittedfrom SC 2 and SC 3 resemblethecoherentlightbackreected frommirrorsinaMichelsoninterferometer.Inpost-proces singthebeat signalsaredelayedbyavirtualround-triptraveltimeonth eotherarm,which isillustratedasthedashedline.(Right)TheSagnaccombina tioninstatic TDI,wheretwocoherentbeamsaretransmittedfrom SC 1 followinga clockwiseandcounterclockwiseclosedtrajectory.Duetot herotationofthe constellation,atraveltimedifferenceof 47 s isgenerated.Similar post-processingwillbeusedtocanceloutthelaserfrequen cynoise,clock noise,etc. throughthelongarm1-2and1-3individuallyandphase-lock sthefarlaseron SC 2 and SC 3 .Therefore,thelasereldtransmittedfrom SC 2 and SC 3 resemblesthecoherent lightbackreectedfrommirrorsinaMichelsoninterferome ter.Ifweassumethatthe far-endPLLyieldsaphasenoise PLLi ( t ), i =2,3 ,thephasemeasurementofthe heterodyneinterferometryon SC 1 isgivenby S i ( t )= 1 ( t ) 1 ( t 1 i i 1 )+ PLLi ( t i 1 )+ h 1 i ( t i 1 )+ h i 1 ( t ), (2–6) where ij istheone-waylighttraveltimefrom SC i to SC j h 1 i ( t ) and h i 1 ( t ) arephase perturbationsfromgravitationalwaves.Thelinearcombin ationofMichelsonXisgiven by X ( t )= S 2 ( t ) S 3 ( t ) S 2 ( t 13 31 )+ S 3 ( t 12 21 ). (2–7) 61

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Theadditionaltermscanbeinterpretedasthebeatsignal S i ( t ) on SC 1 delayed byavirtualround-triptraveltimeontheotherarm,whichis illustratedasthedashed lineinFigure 2-5 (left).IfwesubstituteEq. 2–7 withEq. 2–6 ,allphasenoiseterms arecanceledout.Inprinciple,thisalgorithmreconstruct sanequal-armedMichelson interferometerandextractthecommonphasenoiseoutofsee minglyuncorrelated phasemeasurements.Theresidualtranspondernoisefromth efar-endPLLwillalsobe measuredonthefarspacecraft.Itisthenappropriatelytim e-shiftedandsubtractedfrom theX-combination.Similarprocedureswillalsobemadeforth eremovalofclocknoise, benchmotion,back-linkbernoise,etc.Ontheotherhand,t hephaseperturbations fromgravitationalwaveswillbemaintainedinthepost-pro cessedsignal,yielding X ( t )= h 12 ( t 21 )+ h 21 ( t ) h 13 ( t 31 ) h 31 ( t ) h 12 ( t 21 13 31 ) h 21 ( t 13 31 ) + h 13 ( t 31 12 21 )+ h 31 ( t 12 21 ). (2–8) Brieyspeaking,thedeductionfromEq. 2–6 toEq. 2–8 demonstratesthebasic post-processingtechniquetoextractgravitationalwaves fromLISAreadouts. 2.2.1.2Sagnaccombination AnothernotableTDIobservableistheSagnaccombination,whi chusestheLISA signalstoformaSagnacinterferometer[ 75 ].AstandardSagnacinterferometerconsists ofabeamsplittertogeneratetwocoherentbeams,whichprop agatearoundaclosed loopinoppositedirectionsandthenarerecombinedatthebe amsplitter.InLISA,six linksfromthreespacecraftareusedtoconstructaSagnaccom bination.Asshown inFigure 2-5 (right),thetwocoherentbeamsaretransmittedfrom SC 1 followinga clockwiseandcounterclockwisetrajectory,respectively .SincetheLISAconstellationis notstaticduetotherotation,alightpathdifference L betweenthetwopropagationsin 62

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oppositedirectionsisgenerated.ThisisknownastheSagnac effect: L = L 12 + L 23 + L 31 ( L 13 + L 32 + L 21 )= 4 n A c (2–9) Inthisequation n istheangularvelocityoftherotationand A istheareaenclosed bythelightpath.TheangularvelocityoftheLISAconstellat ionisgivenby 1cycle = year correspondingto 2 10 7 rad = s .Giventhattheconstellationisinclinedby 60 with respecttotheecliptic,theparallelcomponentoftheangul arvelocityisonlyhalfofthe magnitude.Therefore,thelightpathdifferenceisapproxi mately L 14.4km ,whichis equivalenttoapropagationtimedifferenceofapproximate ly 47 s Ifweassumethephasenoisefromallphase-lockedloopsinth econstellationis zero,intheclockwisetrajectorythephasemeasurementsar egivenby 23 ( t )= 12 ( t 12 ) h 12 ( t ), 31 ( t )= 23 ( t 23 ) h 23 ( t ), S 13 ( t )= 13 ( t ) 31 ( t 31 )+ h 31 ( t ). (2–10) Thusweobtain S 13 ( t )= 13 ( t ) 12 ( t 12 23 31 )+ h 12 ( t 23 31 )+ h 23 ( t 31 )+ h 31 ( t ). (2–11) Byfollowingthesameprocedure,theotherphasemeasurement on SC 1 isgivenby S 12 ( t )= 12 ( t ) 13 ( t 13 32 21 )+ h 13 ( t 32 21 )+ h 32 ( t 21 )+ h 21 ( t ). (2–12) Sincethedifferencebetweentheclockwisepropagationtime CW = 12 + 23 + 31 andthecounterclockwisepropagationtime CCW = 13 + 32 + 21 is 47 s ,westillneed time-shiftingtocanceloutthearmlengthdifference.2.2.1.3TDILimitations TheTDIcapabilityofnoisesuppressionislimitedbysevera lrealisticeffects, primarilythearmlengthknowledgewhichdirectlydependso ntherangingaccuracy. 63

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Inadditiontotherangingaccuracy,limitingeffectsinclu dethevelocitycorrectionerror, analogchainerror 1 ,phasemeternoiseandscatteredlight,etc. TherangingsystemLISAwilluseispseudo-randomnoise(PRN)r anging,which phasemodulatesthecarrierofthefarlaserwithaPRNcodestr eam.Thetraveltime thencanbemeasuredviathecorrelationbetweenthelocalan dreceivedPRNcode. CurrentestimationofthePRNrangingerroris L 1m ( 1ns intermsoftimingerror). AllocatedbytheIMSnoisebudget,theresiduallaserfrequen cynoiseattheTDI outputisrequiredtobebelow 2pmHz 1 = 2 p 1+(2.8mHz = f) 4 .Witha 1m ranging accuracy,therequiredlaserfrequencynoiseattheTDIinpu tisgivenby pre TDI ( f ) < 282 s 1+ 2.8mHz f 4 HzHz 1 = 2 (2–13) Incomparisontothenoisesuppressionlimitedbyranging,o therlimitingeffectsyield morerelaxedrequirementsfor pre TDI ,whichindicatestherangingaccuracyonLISA armlengthisthedominantlimitationforTDI.Duetothelimi tedrangingaccuracy,TDI wouldnotmeetthe 2pm requirementonresiduallaserfrequencynoisebyitself,si ncea free-runninglasertypicallycarriesfrequencynoiseonth elevelof 10kHz = f HzHz 1 = 2 Thisfrequencynoiseisseveralordersofmagnitudebeyondt hestabilityspeciedin Eq. 2–13 ;therefore,thelaserfrequencymustbeactivelypre-stabi lizedtoachievethe requirement.2.2.2Pre-stabilization Thefrequencystabilizationplanistolockthelaserfreque ncytoastablereference. Onthelocalspacecraft,suchastablereferencecanbeeithe rahigh-nesseFabry-Perot cavitymadeofultra-lowexpansionglassoranatomicormole cularspectralline[ 76 ]. Themodulation-demodulationmethodusesanopticalcavity knownasPound-Drever-Hall 1 Theanalogchainconsistsofaphotoreceiver,apre-amplie randananti-aliasing ltersuccessivelyplacedbeforetheADCs. 64

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Figure2-6.TheexperimentalsetupofPound-Drever-Hallst abilization. (PDH)technique[ 77 ].Ithasbeenwellstudiedinlaboratory,givingafrequency noisesuppressionperformanceof 30Hz = p Hz at 3mHz [ 78 ].Besidesthewidely favoredPDHtechnique,anotherpre-stabilizationmethodis basedonaheterodyne Mach-Zehnderinterferometer,whichexploitsthepathleng thmismatchbetweentwo differentlasersasthefrequencyreference.Thismethodha sbeenusedintheLISA TechnologyPackage(LTP)forLISAPathnder[ 79 ].Notethatinallsixlasersonly onemasterlaserisneededtobepre-stabilizedusingoneoft hemethodsdescribed above,andtheothervelasersarestabilizedusingphase-l ockingeitherlocallytothe masterlaserviatheback-linkberorusingthebeatsignalw iththefarlaserviaoptical transponders.2.2.2.1Pound-Drever-Hallfrequencystabilization ThePound-Drever-Halltechniqueisthemostusedfrequency stabilizationmethodin opticalexperiments[ 80 ].Theopticalcavityusedinlaboratoryisnormallyahighnesse ( 10 4 )Fabry-Perotcavity,whichconsistsoftwolow-lossmirror sbondedtoarigid spacermadeoflow-expansionmaterialsuchasZerodurorULE. AsshowninFigure 2-6 ,thelaserbeamwithanominalfrequency istransmittedthroughanelectro-optical modulator(EOM)drivenbyalocaloscillatorwithanominalfr equency n (typically > 5MHz ,welllargerthantheFWHMofthecavity),producingtwosideb ands n 65

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Afterthepolarizingbeamsplitterandthequarter-waveplat e,thebeamisreectedfrom theopticalcavity.Weusethereectedbeamtofeedbackcont rolthelaserfrequency. Thereectivity T r ofanopticalcavityisafunctionofthelaserfrequency : T r ( ) E r ( ) E i ( ) = r 1 r 2 e i 1 r 1 r 2 e i (2–14) where =2 L = c isthephaseshiftduetoaround-tripinsidethecavity. r 1 r 2 t 1 and t 2 arethereectioncoefcientsandtransmissioncoefcient sofmirror1and2andwe assumealosslesscavitywith r 2 1 + t 2 1 =1 Ifthecarrierfrequencyofthebeamexactlymatchesupthere sonantfrequencyof thecavity L =2 nc = L ,thereectedcarrierwillhavezerophaseshift.Also,aslon gas n islargeenoughsuchthatthesidebandsarefarawayfromther esonantfrequency, theirphaseshiftwillbenegligible.Therefore,thereect edcarrierwillgeneratetwo heterodynebeatsignalswitheachsideband.Sincetheyare 180 outofphase,the superpositionofthemonthephotodiodewouldbezero. Ifthecarrierfrequencyisslightlyofftheresonantfreque ncy,aphaseshiftwillbe generatedinthereectedcarrier.Whentheoffset totheresonantfrequencyissmall, thereectivitytransferfunctionapproximatestotherst linearorderbytakingTaylor expansion: T r ( )= r 1 r 2 1 r 1 r 2 +2 i r 2 1 r 2 r 2 (1 r 1 r 2 ) 2 L c = r 1 r 2 1 r 1 r 2 2 i F FSR = r 1 r 2 1 r 1 r 2 i HWHM (2–15) Inotherwords,thecavityisabletodetectwhetherthecarri erfrequencyisabove orbelowtheresonantfrequencysincethesignofthephasesh iftdependsonthesign ofthefrequencyoffset.Alsoduetothephaseshift,theheter odynebeatsignalswill nolongercancelouteachother.Ifwedemodulatethereecte dbeamwiththesame 66

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oscillator n andlow-passlterthehighfrequencycomponent,azero-cro ssingDCsignal thatislinearwithrespecttothefrequencyoffsetcanbegen eratedtofeedbackcontrol thelaserfrequency. IntheLISAbandespeciallybelow 1mHz ,thefrequencystabilityofthePDH techniqueismainlylimitedbythetemperatureuctuations thatcausesspuriousnoises inthecavitylength.Therelativestability L = L ofthereferencingcavitylengthinour experimentsisaslowas 10 13 ,resultinginafrequencynoiseofabout 30HzHz 1 = 2 2.2.2.2Mach-Zehnderfrequencystabilization Mach-Zehnderfrequencystabilizationrequiresadedicate dheterodyneinterferometer withunequalarm,whichisusedasthefrequencyreference.I ntheLTPbaselinedesign theunequal-armedMach-Zehnderinterferometeriscongur edusingabeamsplitter andtwoacousto-opticmodulators(AOM)togenerateheterod yneinterferometrysignals at 1 2kHz [ 81 82 ].OntheLISAopticalbench,theintegrationofsuchaheterod yne interferometerbecomeseasierastheinterferometrysigna lbetweentwoadjacent lasersisalreadyincorporatedintothephasereadout.Comp aredwithstabilizationtoa Fabry-Perotcavity,Mach-Zehnderstabilizationsimplie stheopticalbenchsetup,where neitheranopticalcavitynoranEOMisnotnecessarytobeimpl emented.Also,this pre-stabilizationmethodallowstotunethelaserfrequenc yoverawiderangewithoutthe necessityofchangingthepathlengthmismatch. TheproposedinterferometersetupisillustratedbyFigure 2-7 .Betweenthe twoadjacentlasersonthesamespacecraft,onelasermarked astheslavelaseris phase-lockedtothemasterlaserwithafrequencyoffset( 2 20MHz )drivenbythelocal oscillator.Aswealreadyknow,theheterodyneinterferomet rysignalbetweenthemis measuredatthephotodiode PD R andthephasereadoutisgivenby R = S M (2–16) 67

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Figure2-7.TheinterferometersetupforMach-Zehnderfreq uencystabilization.Inthis setuptheslavelaserisheterodynephase-lockedtothemast erlaser.Their heterodynesignal R ismeasuredatthereferencephotodiode.Theother armoftheMach-Zehnderinterferometerhasanadditionalpa thlength L whichgeneratesapropagationdelayintheheterodynesigna l F .Theerror signalofMach-Zehnderstabilizationisgivenby F R ,whichexploitsthe pathlengthmismatchasareferencetostabilizethemasterl aser.Notethat themixersinthisdiagrammayrepresentphasemeters,eachd rivenbya commonlocaloscillator.Thephasemetersdemodulatethehe terodyne signalswiththesameoffsetfrequencyandmeasurethephase noiseineach ofthem. BasedonthedynamicsofthePLL,intheLaplacedomainthephase noise S ofthe phase-lockedslavelaserisgivenby S = 1 1+ G 1 0S + G 1 1+ G 1 M (2–17) Thersttermrepresentsthenoisesuppressionofthefree-r unningslavelaserand thesecondtermsindicatesthatthephasenoiseoftheslavel asertracksthephase noiseofthereferencedmasterlaser.Therefore,thephasem easurementat PD R canbe writtenas R = 1 1+ G 1 0S 1 1+ G 1 M (2–18) 68

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Weplaceanadditionalunequal-armedinterferometer F ontheopticalbench, whichtogetherwith R constitutesanunequal-armedMach-Zehnderinterferomete r.The phasemeasurementat PD F isgivenby F = S M e s (2–19) where = L = c and L isthearmlengthmismatchofthisinterferometer.Substitut eEq. 2–17 intothisequationandwehave F = 1 1+ G 1 0S + G 1 1+ G 1 e s M (2–20) Eq. 2–20 cannotbedirectlyusedasanerrorsignaltocontrolthemast erlaser frequencysinceitdependsonthephasenoiseofthefree-run ningslavelaserandthe transferfunctionofthePLLcontroller.However,ifwecombi nethephasemeasurements onbothtwophotodiodesbysubtractingthem,weget F R = 1 e s M (2–21) whichcanbeusedtodetectthephaseuctuationsofthemaste rlaser.Thissensor transferfunctiononlydependsonthearmlengthmismatch.Wi thahigh-gainfeedback controller G 2 ,thephasenoiseofthemasterlasercanbestabilizedto M lock = 1 1+(1 e s ) G 2 M (2–22) InSection2.3.2.1wewilllearnthatMach-Zehnderstabiliza tionisessentially abenchtop“mini”versionofsinglearmlocking,whichexplo itstheLISAarmasa referencetostabilizethelaserfrequency. Ifweapplyaphasemodulation mod atthesubtractorofthesensor,thisphase modulationwillbecoupledintothephasenoiseofthestabil izedmasterlaserasa reference: M lock = 1 1+(1 e s ) G 2 M + G 2 1+(1 e s ) G 2 mod (2–23) 69

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Thuswehave @ M lock @' mod = G 2 1+(1 e s ) G 2 (2–24) Itiseasytodemonstratethatinthehighgainlimit( G 2 1 ),Eq. 2–24 approximates to @ M lock @' mod 1 s (2–25) Theaboverelationholdsforfrequenciesmuchlowerthan 1 = ,whichisalwaysvalid inthelockingbandwidth( 20Hz ).Intermsoffrequency,itcanberepresentedas @ M lock @' mod 1 (2–26) whichindicatesthefrequencyofthemasterlasercanbecont inuouslytunedby introducingaphaseoffset mod atthesensor.Theactuationfactorfromthephase modulationtothefrequencytuningisgivenby 1 = ThenoiseperformanceofMach-Zehnderstabilizationislim itedbythepathlength stabilityoftheinterferometer,aswellassensingnoiseso urcesintroducedbythe photodiodesandphasemeters(shotnoise,digitizationnoi se,etc.)Ifweassumethe sensingnoiseisequivalentto 1pm = p Hz ,fora 0.5m pathlengthmismatchtheexpected stabilizedfrequencynoiseisapproximatelygivenby MZ =800 s 1+ 2.8mHz f 4 HzHz 1 = 2 (2–27) Althoughthisexpectedperformanceisinferiortothexedca vityanddoesnot meettherequirementforpre-stabilization,Mach-Zehnder stabilizationprovidesan intrinsicfrequencytunabilitythatthePDHstabilizationd oesnothave,whichmakes theincorporationofMach-Zehnderstabilizationwitharml ockingmucheasier.The incorporationcanbeachievedbyadjustingthephaseofthel ocaloscillatorthatisused todemodulatetheMach-Zehnderheterodynesignalthrougha rmlocking.Moredetails oftheintegrationofMach-Zehnderstabilizationandarmlo ckingwillbeseeninSection 70

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2.3.4.2.Inaddition,avariationofMach-Zehnderstabiliz ationthatusesaFabry-Perot cavitytoachievethearmlengthmismatchhasalsobeendevel oped.Duetothebetter stabilityofthecavitylength,thestabilizationperforma nceisexpectedtobesuperiorto thestandardMach-Zehndermethod,whichhasbeendemonstra tedinrelevantbenchtop experiments[ 83 ]. 2.3ArmLocking IncomparisontoMach-Zehnderstabilizationthatexploits thepathlengthto theadjacentopticalbenchasareference,theLISAlongarm(f romopticalbenchto opticalbench)isanotherpathlengthreferencewhichprovi desaevenbetterstability ( L = L 10 21 Hz 1 = 2 )intheLISAband.Techniquesthattransferthestabilityoft he longarmlengthstothelaserfrequencyaregenerallyknowna sarmlocking.Unlike Mach-Zehnderstabilizationthatrequiresanextrainterfe rencepathinthesetup,the armlockingarchitectureisintrinsicallyembeddedintheL ISAconstellation:Armlocking synthesizestheerrorsignalfromtheinter-spacecraftpha semeasurements,whichare alreadyavailableatthephotodetectors.Thisfeatureallo wsthatthecontrolsystemof armlockingcanbefullyimplementedinon-boarddataproces singandnoadditional resourceisneeded.Withconstellationphaselockingonfars pacecraft,thelocallaser beamhasaninterferencewithitsreplicadelayedbytheligh tround-triptraveltime 33s .Suchalongtimedelayisveryuncommoninordinarysensors,a sitmay causeexcessivephaseshiftatinterferometernullfrequen ciesandthenlimitthecontrol bandwidth.However,thepioneeringworkdonebySheard etal hasdemonstratedthat thebandwidthofarmlockingisnotnecessarilylimitedbyth elongtimedelayanda highgaincontrollerdesignisachievableifthephasemargi niscarefullyretained[ 84 ]. Morestudiesonarmlockinghaveanalyticallydemonstrated thattheinstantaneous phaseinformationcanberecoveredviaanadequatelinearco mbinationofphase measurements. 71

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Figure2-8.Thebaselinedesignofarmlocking.Thelaserel dofLaser1(L1)onthe local SC 1 issplitandsenttothefarSC2and SC 3 .ThelasereldsofLaser2 (L2)andLaser3(L3)arephase-lockedtotheincomingeldof L1and transmittedbackto SC 1 .On SC 1 thephase-lockedlasereldsofL2andL3 aredemodulatedbytheinstantaneouslasereldofL1.These twobeat signalsaremeasuredbytwophasemetersindividually.Armlo ckingtakesa linearcombinationofthesetwophasemeasurementstoestim atethephase noiseinformation,whichisthenusedtocontrolthefrequen cyofL1. 2.3.1Architecture ThebasicschematicofarmlockinginLISAisillustratedinFi gure 2-8 .Inthis diagram,onespacecraft SC 1 isusedasalocalormasterspacecraftonwhichthelaser ispre-stabilizedbylockingtoalocalfrequencyreference .Whenthelasereldfrom SC 1 istransmittedtoeitherofthetwofarspacecraft SC 2 and SC 3 ,thelaseronthemwillbe phase-lockedbythecontroller G 2,3 totheincominglaser.Ifweassumetheinitialphase 72

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noiseofthepre-stabilizedlaseron SC 1 is 0L 1 ( s ) intheLaplacedomain,thenthephase ofthelaserfrom SC 2,3 receivedbythephotodetectoron SC 1 willbegivenby Li ( s )= 1 1+ G i ( s ) 0Li ( s ) e s i 1 + G i ( s ) 1+ G i ( s ) 0L 1 ( s ) e s ( 1 i + i 1 ) + N(i) ( s ), (2–28) where i =2,3 and 0Li ( s ) istheinitialphasenoiseofthefree-runninglaser. N(i) ( s ) includesallotherkindsofnoiseinthephase-lockedloop,i ncludingshotnoise,clock noiseandspacecraftjitter.Weignorethistermasitismuch lowerthanthelaserphase noise,althoughitwillultimatelylimittheperformanceof armlocking(SeeSection2.3.5). Thebeatsignalsbetweenthelasereldsfrom SC 2,3 withthemasterlaseron SC 1 willbemeasuredbythephasemeteron SC 1 : 1 i ( s )= 0L 1 ( s ) Li ( s ) = 0L 1 ( s ) 1 G i ( s ) 1+ G i ( s ) e s i 1 1+ G i ( s ) 0Li ( s ) e s i 1 (2–29) where 2 = 12 + 21 and 3 = 13 + 31 aretheround-triplighttraveltimesonbotharms, respectively.SinceinLISAthephase-lockedloopcontroller yieldsthehighgainlimit,Eq. 2–29 canbesimpliedbyapproximations G i ( s ) 1+ G i ( s ) 1 and 1 1+ G i ( s ) 0 .FromEq. 2–29 we canderivethetransferfunctionofthetwoarms: P 1 i ( s )= 1 i ( s ) 0L 1 ( s ) 1 e s i i =2,3. (2–30) Thetwophasemeteroutputs 12 ( s ) and 13 ( s ) aremanipulatedtoconstructan armlockingsensorsignal.FollowingthenotationusedinRe f[ 13 85 ],thismanipulation canbedescribedbya 1 2 mappingvector S k .Ifwewritedownthetwointerferometer outputsasa 2 1 phasevector 1 ,thesensorsignalisthensimplygivenby err ( s )= S k ( s ) 1 ( s ), (2–31) where 1 ( s )= 264 12 ( s ) 13 ( s ) 375 73

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Whenthearmlockingloopisclosed,thephasenoiseattheloca llaseroutputis givenby L 1 ( s )= 0L 1 ( s ) G 1 ( s ) 1+ G 1 ( s ) H ( s ) S k 1 ( s ) = 0L 1 ( s ) 1+ G 1 ( s ) H ( s ) (2–32) where G 1 ( s ) isthearmlockingcontrollerand H ( s )= S k ( s ) 264 P 12 ( s ) P 13 ( s ) 375 (2–33) isthearmlockingsensor.Comparedwiththeinitialphaseno iseofthepre-stabilized laser,thestabilizedphasenoiseissuppressedbytheopenloopgain G 1 ( s ) H ( s ) 2.3.2ArmLockingSensors2.3.2.1Singlearmlocking Singlearmlockingistheoriginalideaofarmlockingpropose dbySheard etal. .In thesinglearmlockingcongurationonlytheinterferomete routputononearmwillbe usedandconsequentlywehave H S ( s )= P 12 ( s )=1 e s 2 .AsshowninFigure 2-9 ,the sensortransferfunctionhaszeromagnituderesponseatDCa swellasatfrequencies n = 2 knownasnulls.Thephaseresponsestartsfroma 90 phaseadvanceatDCand hasdiscontinuitiesatthesenullswherethephasejumpsfro m 90 to 90 .Thetransfer functionalsohavemultipleunitygainfrequencies,onboth sidesneareachnull. Theseunitygainfrequenciesplaceadditionalstabilityco nstraintsonthecontroller design.InpracticeanintegratorwithasinglepoleatDCisw idelyusedinfeedback controllersasitprovidesahighgainatDCandthegaindecli neswitha 1 = s slopeathigh frequencies,whichisrequiredbythelimitedbandwidth.Ho wever,thiskindofcontroller essentiallydoesnotworkforsinglearmlockingbecausethe 1 = s slopewouldbringinan additional 90 phaseshift.Astheloopgainincreases,thephasemarginwill befurther decreased,whichwilldestroythesystemstability.Thispr oblemisillustratedbyFigure 74

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10 -4 10 -3 10 -2 10 -1 10 0 0 1 2 Frequency (Hz)Magnitude 10 -4 10 -3 10 -2 10 -1 10 0 -100 0 100 Frequency (Hz)Phase shift (degree) Figure2-9.Themagnitudeandphaseresponseofthesinglear mlockingsensor.Note thatforthemagnituderesponsewearenotusingthelogarith micscaleinthe yaxis(aswedoinmostcases)justtoshowthesituationneari nterferometer nullsmoreclearly.Inthelogarithmicscalethemagnituder esponseversus theFourierfrequencyactuallyapproximatestoan f slopeatlow frequencies. 2-10 ,wheretheopen-looptransferfunctionisgivenbythesingl earmlockingsensor multipliedbya 1 = s slope.Withagainfactorof10,attheunitygainfrequenciest he phaseshiftisalreadyverycloseto 180 Toavoidlosingexcessivephaseinbandwidth,theproposedd esignofthe controllerhastohaveaslopelesssteepthan 1 = s tocompensatethephaselossat zerocrossings.Forinstance,a 1 = s 1 = 2 or 1 = s 2 = 3 slopewillonlybringina 45 or 60 phase shift,respectively.Weassumeagenericarmlockingcontro llerlterwiththetransfer function G ( f )= kf e i ,where 0 << 1 k isarealgainfactorand isthephase shiftintroducedbythelter.Asweknow,theclosed-looptra nsferfunctionofafeedback controlloopisdenedas TF CL ( f ) g out ( f ) f in ( f ) 1 1+ TF OL ( f ) (2–34) 75

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10 -4 10 -3 10 -2 10 -1 10 0 10 0 Frequency (Hz)Magnitude 10 -4 10 -3 10 -2 10 -1 10 0 -200 -100 0 Frequency (Hz)Phase shift (degree) Figure2-10.Anordinaryintegratorwithatransferfunction of 1 = s isnotsuitabletobea singlearmlockingcontrollerduetotheexcessivephaseshi ftplacedbythe sensornulls.Withagainfactorof10thesystemhasalreadybe come unstablesinceattheunitygainfrequenciesthephaseshift isverycloseto 180 Withthecontrollerlterdescribedabove,theopen-looptra nsferfunctionofthe singlearmlockingloopisgivenby TF OL ( f )= G 0 (1 e i 2 f ) e i kf (2–35) where G 0 istheopen-loopgainand = 2 33s istheround-triplighttraveltime. Thenoisesuppressionperformanceofanarmlockingloopisd eterminedbythe magnitudeoftheclosed-looptransferfunction j TF CL ( f ) j .When j TF CL ( f ) j < 1 wehave noisesuppressions,otherwisewehavenoiseamplications 76

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IfwesubstituteEq. 2–35 intoEq. 2–34 andevaluatethemagnitudeofthe closed-looptransferfunction j TF CL ( f ) j ,ityields j TF CL ( f ) j = kf e i kf e i + G 0 (1 e i 2 f ) = s k 2 f 2 k 2 f 2 +2 G 2 0 (1 cos2 f ) 2 kG 0 f (cos( +2 f ) cos ) = r k 2 f 2 k 2 f 2 + (2–36) FromtheresultofEq. 2–36 ,thenoisesuppressionperformancedependsonthevalue of =2 G 2 0 (1 cos2 f ) 2 kG 0 f (cos( +2 f ) cos ), (2–37) i.e., > 0 =0 < 0 correspondstothecaseofnoisesuppression,nosuppressio n andnoiseamplication,respectively. 2 Aspecicmodeltonumericallyevaluate isshowninFigure 2-11 .Inthissystem modelweassumetheloopgain G 0 =10 andthelterhasamagnituderesponse j G ( f ) j =1 = f 1 = 2 witha 45 phaseshift.Theplotshowsthatinthefrequencyregion f 1 = ,wehave 0 ,whichmeansasimple 1 = f slopedoesnotprovideanynoise suppressionatDC.Thiscanalsobeseenfromthetransferfun ctionofthesinglearm lockingsensor,whichhasa f slopestartingfromDC.Toovercomethis f slopeinthe sensorandobtainsomenoisesuppressionatDC,thecontroll ermusthaveatleasta 1 = f slopeinthefrequencyregion f 1 = Inthefrequencyregionwhere f iscomparableto n = isgenerallylargerthan zerobutwithafewexceptions.Atsomecertainfrequencieswh ichareverycloseto 2 Fromthepointofviewofmathematics,thevalueof isobviouslyrequiredtohavea lowerlimit: > k 2 f 2 ,otherwisetheexpressionofthemagnituderesponsewouldn ot bemeaningful.Inpractice,thevalueof generallymeetsthisrequirementintheLISA bandsincetheFourierfrequencyislow. 77

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10 -4 10 -3 10 -2 10 -1 10 0 0 200 400 Frequency (Hz) 10 -4 10 -3 10 -2 10 -1 10 0 -0.2 0 0.2 Frequency (Hz) Figure2-11.Anexampleofthevalueof thatindicatesthenoisesuppression,whichis givenbyEq. 2–37 .Largervalueof indicatesmorenoisesuppressions. Notethatinthismodelwedidnotincludeintegratorsatlowf requencies; thereforewedonothaveanysuppressionatDC.Thevalueof canbe negativeatcertainfrequenciesneareachnull,correspond ingtonoise amplicationsduetotheexcessivephaseloss. eachnull n = ,numericalcalculationshowsthat canbelessthanzero,causingnoise amplicationsatthesefrequencies.Thesenoiseamplicat ionscorrespondtomultiple peakswithniteheightsintheclosed-looptransferfuncti onasshowninFigure 2-12 Itisworthnotingthatnoiseamplicationsdonotoccurexac tlyatsensornulls n = Actuallywhen f = n = =0 andthecontrolloophasnonoisesuppressionsor amplications. AsshowninFigure 2-12 ,anidealsinglearm-lockingcontrollerwouldbedesigned inthefollowingmanner:startingfromDCthelterhasastee pslopeofatleast 1 = f to suppressthelowfrequencynoiseandstopsrightbeforethe rstnull 1 = ;startingfrom therstnulltheltertransferfunctionhasa 1 = f 1 = 2 or 1 = f 2 = 3 slopeinbandwidthand startstoattenoutbeyondthebandwidth. Althoughtheideaofsinglearmlockinghasbeentheoreticall yandexperimentally validatedbymultipleresearchers,theshortcomingsofsin glearmlockingmakeit 78

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10 -4 10 -3 10 -2 10 -1 10 0 10 -4 10 -3 10 -2 10 -1 10 0 10 1 Frequency (Hz)Noise suppression Figure2-12.Anexampleoftheclosed-looptransferfunction ofsinglearmlocking, wherethelighttraveltimeis 33s .Thecontrollerconsistsofalterwitha 1 = f 1 = 2 slopeandtwo-stageintegrators. ultimatelyinappropriateforrealisticLISA.Itisverynotic eablethattherstnullshows uprightinsidetheLISAband,whichinanycaseisnotanidealn oisesuppression performanceduetothenoiseamplicationsnearthenulls.M oreover,sincetherst nullisintheLISAband,thegainaroundthatregionmustbehig henoughtoprovide adecentnoisesuppression.Consideringthecontrollerwil lhaveaslopelesssteep than 1 = f ,anextremelylargebandwidthwouldbeneededinsinglearml ocking. Thethirddisadvantageofsinglearmlockingisthestart-up transientscausedbythe non-zeroinitialerrorsignal,whichcreatestherepeating 33secondsnoisewithdamped oscillations[ 86 87 ]. 2.3.2.2Commonarmlocking Undermostcircumstances,thelengthsoftwoLISAarmsaredif ferentbyarelatively smallamountexceptforsomecertainshortperiods.Forthis reason,commonarm locking,asthesimplestarmlockingcongurationthatutil izestheinterferometeroutputs onbothtwoarms,exploitsthearmlengthmismatchtopushthe rstnullofthesensor tohigherfrequenciesbeyondtheLISAband.AsMarkusHerzpoin tedoutusingthe 79

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0 500 1000 1500 2000 2500 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time (s)Phase variation (cycle) Figure2-13.Thetimedomainsimulationtodemonstratethei nitialtransientsinsingle armlocking.Thequasiperiodictransientis“frozen”insid etheoutputsignal witha 33s oscillationanddecayswithatimeconstantofapproximatel y 500s forthisspecicmodel. secondarmallowstoreducethestart-uptransients[ 88 ].Themappingvector S k ( s ) of commonarmlockingcongurationis [1,1] ,asitsimplycombinesthetwointerferometer outputsbyaddingthem.Thusthesensortransferfunctionof commonarmlockingis givenby H C ( s )= P + ( s )=[1,1] 264 P 12 ( s ) P 13 ( s ) 375 = P 12 ( s )+ P 13 ( s ) =2 e s 2 e s 3 (2–38) Ifwedene ( 2 + 3 ) = 2 and ( 2 3 ) = 2 > 0 ,thetransferfunctioncanalso berepresentedintheangularfrequencybytakingthetransf ormation s = i : P + ( )=2[1 cos( ) e i ]. (2–39) Thissensortransferfunctionresemblesthatofsinglearml ocking H S ( )=1 e i Thedifferenceisthattheround-triptimeononearmisrepla cedbytheaveraged 80

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10 -3 10 -2 10 -1 10 0 10 1 10 0 Frequency (Hz)Magnitude 10 -3 10 -2 10 -1 10 0 10 1 -100 0 100 Frequency (Hz)Phase shift (degree) Figure2-14.Themagnitudeandphaseresponseofthecommona rmlockingsensor. Theaveragedlighttraveltimeis 33s andthearmlengthmismatchis 1% Notethattherstsensornullislocatedat 1 = 6Hz andthemagnitude at 1 = =30mHz isnotzeroas cos(2 ( = )) isnotequalto 1 .Therefore, frequenciesatthemultiplesof 1 = (butnotatthemultiplesof 1 = )are referredtoas“localminima”ratherthansensornullsinthi sdissertation. Nevertheless,localminimastillcorrespondtolargephase shiftandthereby maycausenoiseamplicationsintheclosed-loop. round-triptimeonbotharms.Also,thetermthatrepresentst hedelayedphaseis multipliedwithacoefcient cos( ) .Duetothiscoefcient,themagnitudeofthe transferfunctiondoesnotdecreasetozerowhen =2 n = unless isalsointeger multiplesof 2 = .Themagnituderesponseandphaseshiftofthecommonarm lockingsensorisshowninFigure 2-14 ,whichdescribesthesituationof =33s and =0.16s ( 1% armlengthmismatch). Figure 2-14 alsoshowsthatthephaseshiftat 1 = 30mHz isstillcloseto 90 whichmeansthatthecontrollerisstillrequiredtomaintai narelativelylowgainto compensatetheexcessivephaseloss.Similartosinglearmlo cking,noiseamplications willalsooccurnearthefrequencieswithlargephaseshifti ncommonarmlocking.This characteristicindicatesthatdespiteoftheintroduction ofasecondarm,commonarm 81

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10 -3 10 -2 10 -1 10 0 10 1 10 0 Frequency (Hz)Magnitude 10 -3 10 -2 10 -1 10 0 10 1 -200 0 200 Frequency (Hz)Phase shift (degree) differential arm integrated differential arm scaled by 1/ D t Figure2-15.Themagnitudeandphaseresponseofthediffere ntialarm(green)andthe integrateddifferentialarmscaledby 1 = (red).Theaveragedlighttravel timeis 33s andthearmlengthmismatchis 1% .Theyonlycontainthe informationofthedelayedphase,thereforethemaximumoft heirphase shiftis 180 andtheycannotbesolelyusedasanarmlockingsensor. lockingisessentiallynotasignicantimprovementofsing learmlockingintermsofthe gainadvantageandsystemstability.2.3.2.3Dualarmlocking Dualarmlocking,alsoknownasdirectarmlocking,combines thephasemeasurements ontwodifferentarmsandthendirectlyprovideafeedbacksi gnalbyestimatingthe instantaneousphaseinformation.Theoriginalideaofdual armlockingisfromAndrew SuttonandDanielShaddock[ 85 ].Comparedwithsinglearmlockingandcommonarm locking,dualarmlockingexhibitsseveralremarkableimpr ovementssuchasnonoise amplicationintheLISAbandandsignicantlybetternoises uppressionperformance. InadditiontothecommonarmsensordescribedinEq.(2-15),w ecanalsodene thedifferentialarmsensor: P ( s )= P 12 ( s ) P 13 ( s )= e s 3 e s 2 (2–40) 82

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Again,whenrepresentedintheangularfrequencydomain,the differentialarm sensorisgivenby P =2 i sin( ) e i (2–41) wherethedenitionsof and arethesameasinthecommonarmsensor.We plotthemagnituderesponseandphaseshiftofthedifferent ialarmsensorinFigure 2-15 Thedifferentialarmsensoronlycontainsinformationofth epurelydelayedphase, thereforeitcannotbesolelyusedforfeedbackcontrol.How ever,thedifferentialarm signalcanbelinearlycombinedwiththecommonarmsignalin acertainwaysuchthat thedelayedterminthecommonarmsensorwillbeeliminated. IntheLaplacedomain, suchalinearcombinationisgivenby H ( s )= P + ( s )+ 1 s P ( s ). (2–42) Switchedbacktotheangularfrequency,thelinearcombinati onyields H ( )=2 2cos( ) e i + 1 2 i sin( ) e i i =2 1 ( cos( ) sinc( ) ) e i (2–43) Atlowfrequencies( 1 ),both cos( ) and sinc( ) areapproximately equalto 1 .Therefore,thetransferfunctionofthiscombinationisap proximatelygiven by H ( ) 2 ,i.e.,themagnituderesponseretainsalmostatnesswitha factorof2and thephaseshiftisalsoalmostentirelyzerointheLISAband.I notherwords,thislinear combinationextractstheinstantaneousinputphasefromth einterferometeroutputs. Duetotheintegrationofthedifferentialarmpath,thediff erentialarmsensordominates atlowfrequenciesandtheintegrateddifferentialarmreta insthezerophaseshiftinthis region.Theatmagnituderesponseandzerophaseshiftatlo wfrequenciesallowa moreexiblecontrollerdesign,comparedwiththenullsath ighfrequenciesthatlimitthe controllerslopeandthereforetheachievablegain. 83

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10 -3 10 -2 10 -1 10 0 10 1 10 0 Frequency (Hz)Magnitude 10 -3 10 -2 10 -1 10 0 10 1 -200 0 200 Frequency (Hz)Phase shift (degree) Figure2-16.Themagnitudeandphaseresponseofthelinearc ombination H ( s ) .The averagedlighttraveltimeis 33s andthearmlengthmismatchis 1% .For frequenciesfarbelow 1 = thefrequencyresponseisalmostat,sincethe informationoftheinstantaneousphaseinthisfrequencyre gionis recoveredthroughthelinearcombination.However,athigh frequenciesthe phaseshiftcanbeaslargeas 180 atfrequenciesof 1 = (2 ) ,duetothe excessivephaselossfromtheintegrateddifferentialarm. Figure 2-16 illustratesthemagnituderesponseandthephaseshiftofth islinear combination,wherewehave =33s and =0.16s .Itclearlyshowsthatathigh frequenciestheshapeofthetransferfunctionhasadrastic changecomparedwith thelowfrequencycase.Therstnullshowsupatthefrequenc y 1 = =6.1Hz and themaximumphaseshiftcanreachalmost 180 atfrequenciesslightlylessthan 3Hz duetotheexcessivephaselossfromtheintegrateddifferen tialarm.Therefore,sucha sensorwouldeasilyproducesomesysteminstabilityinthep resenceofamoderately highloopgain. AmethodproposedbySutton etal. tosolvethisstabilityproblemistoadda second-orderellipticallow-passlter E ( s ) inthedifferentialarmpath.Moregenerally, anylow-passlterthathasaunitygainatDCandapolenearth efrequencyof 1 = 4 is capableofattenuatingandphase-shiftingthedifferentia larmwithintheinstabilityregion. 84

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10 -3 10 -2 10 -1 10 0 10 1 10 0 Frequency (Hz)Magnitude 10 -3 10 -2 10 -1 10 0 10 1 -100 0 100 Frequency (Hz)Phase shift (degree) Figure2-17.Themagnitudeandphaseresponseofthedualarm lockingsensor.The averagedlighttraveltimeis 33s andthearmlengthmismatchis 1% .In thisdesignalow-passlterisaddedintothedifferentialp athtoattentuate thephaseshift.Thelow-passlterensuresthatthecommona rm dominatesatallfrequenciesabove 1 = suchthatthephaselossatsensor nullsisalleviatedbackto 90 .Thecontrolleristhenonlyrequiredto maintainenoughphasemarginat 1 = (2 ) ,whichiswellabovetheLISA band. Withoutaffectingtheoverallmagnituderesponse,thislte rensuresthatthecommon armdominatesatallfrequenciesabove 1 = suchthatthephaselossatsensornullsis alleviatedbackto 90 .Themagnitudeandphaseresponsesofthislinearcombinati on withtheltereddifferentialarmareshowninFigure 2-17 ,wherethelow-passlterhas asinglepoleat 1Hz .Therstimpulsenullofthisdualarmlockingsensorisat 1 = whileat n = 2 themagnituderesponsehasalocalminimum.Thephaserespon se showsthateitheratsensornullsoratlocalminima,thesens ortransferfunctionhasa phaseshiftofapproximately 90 .Sincethedualarmlockingsensordoesnothaveany localminimumintheLISAbandandathighfrequenciesbothsen sornullsandlocal minimacorrespondtoa 90 phaseshift,inthisdissertationwedonotdistinguishthe 85

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sensornullsandlocalminimafordualarmlocking.Inthefol lowingchaptersthesensor nullsofdualarmlockingaredenedatfrequenciesof n = (2 ) Takentogether,themappingvectorofdualarmlockingcanbe writtenas S k ( s )= [1+ E ( s ) s ,1 E ( s ) s ] andthedualarmlockingsensorisgivenby H D ( s )= P + ( s )+ E ( s ) s P ( s ) (2–44) orrepresentedintheangularfrequency: H D ( )=2 1 ( cos( ) E ( i )sinc( ) ) e i (2–45) Anothersignicantimprovementcomparedtosinglearmlocki ngisthefastdecaying initialtransients.Similartosinglearmlocking,theiniti altransientsindualarmlocking arealso 33s quasi-periodicoscillations,butwithamuchshortertimec onstant. InrealisticLISA,therealizationofdualarmlockingrequire stheknowledgeofthe time-variablearmlengthmismatch.Ifthe usedinthedualarmlockingconguration doesnotexactlymatchupthereal-timearmlengthmismatch, thesensortransfer functionwillbeslightlydistortedwithincreasedripples atmultiplesof 1 = .Thecurrent knowledgeisthatevenwithouton-boardrangingtheLISAarml engthwillbeknownto theaccuracyof 10km ,correspondingtoanerrorin 30 s .Thistinyerrorwill onlycauseanegligibleeffectonthedualarmlockingsensor andconsequentlythearm lockingperformance.2.3.2.4Sagnacarmlocking Sagnacarmlocking,whichresemblesthecongurationoftheSa gnaccombination forTDI,utilizestheentireLISAconstellationtoobtainthe twointerferometeroutputs. Sagnacarmlockingexploitsthelengthmismatchintheperime teroftheconstellation loopinsteadoftwoLISAarms.Onthemasterspacecraft,theou tgoingsignalfrom thepre-stabilizedlaseristransmittedaroundtheconstel lationloopcounter-clockwise andtheninterferedwithitself,whileanotheroutgoingsig nalfromtheotherlaser,which 86

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10 -4 10 -2 10 0 10 2 10 4 10 -2 10 0 Frequency (Hz)Magnitude 10 -4 10 -2 10 0 10 2 10 4 -100 -50 0 50 100 Frequency (Hz)Phase shift (degree) Figure2-18.ThemagnitudeandphaseresponseoftheSagnac-b aseddualarmlocking sensor.Theaveragedlighttraveltimeis 50s anddifferentialdelaytimeis 47 s .TheSagnacarmlockingsensorprovidesaveryniceperforman cein termsofthebandwidthasboththemagnitudeandphaseshifta reentirely atupto 100Hz isphase-lockedtothemasterlaserviathebacklinkber,is transmittedaroundthe loopclockwiseandthenalsointerferedwithitself.Whenthe phase-lockedloopnoise onallspacecraftisignored,thetwointerferometeroutput sintheSagnacarmlocking congurationaregivenby CCW ( t )= 0 ( t ) 0 ( t 12 ) 0 ( t 23 ) 0 ( t 31 ) CW ( t )= 0 ( t ) 0 ( t 13 ) 0 ( t 32 ) 0 ( t 21 ) (2–46) Therefore,theequivalentaverageddelaytimeofthesetwop hasemeasurements isapproximately 50s andtheequivalentdifferentialdelaytimeisapproximatel y 47 s .Comparedwiththevariabledifferentialdelaytimeindual armlocking,these twovaluesareveryconstantsincetherotationoftheconste llationiscontinuousand alwaysinonedirection.Bytakingthetwophasemeasurements ,aSagnac-baseddual armlockingsensorcanbeconguredwiththetransferfuncti onshowninFigure 2-18 87

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Therstnull,accordingtothedifferentlighttraveltime 47 s ,comesat 21kHz ,which allowsanextremelyhighgainperformance. Despitetheevidentadvantagesdescribedabove,Sagnacarml ockinghasnot beenconsideredseriouslybyresearchers.Themaindrawbac kisitscomplexity,which requiresallsixlinkstobeoperationalsimultaneouslytoc onstructthesensorsignal. Also,thenoiselimitationsandtheDopplerimpactofSagnac-b aseddualarmlockingare moresignicantthananyotherarmlockingcongurations.W ewillseemoredetailsof thisinthefollowingsections.2.3.3FrequencyPulling Sofar,thediscussionsaboveonarmlockingarelimitedtothe sensorconguration, suppressiongainandsystemstabilityunderveryidealized circumstances.Inreality, thesimplearchitectureofarmlockingmakeitencountersev eralrealisticissues.These knownissuesincludethefrequencypullingonthestabilize dlaserfrequencyduetothe Dopplerestimationerrorandvariousnoiselimitationssuc hasclocknoise,spacecraft jitterandshotnoisethatrestrictthenoisesuppressionpe rformance. 2.3.3.1Dopplerimpactonarmlocking Beforethediscussionontheissueoffrequencypulling,wer stlookataverybrief introductiononhowthephasemetermeasuresthephaseofint erferometrysignals.More detailsontheprincipleofthephasemeterwillbeseeninSect ion3.2.2.Tomeasurethe phasenoise ( t ) ofaninputbeatsignal S ( t )= A ( t )sin(2 L t + ( t )) ,thephasemeter needstoprovideaninternaloscillation S 0 ( t )=sin(2 0 t + 0 ( t )) whichthentracksthe inputsignalviaphase-locking.Therefore,thepresetfreq uencyofthelocaloscillation 0 needstobecloseenoughtothenominalfrequencyofthebeats ignal L .Thefeedback signalofthisphase-lockedloop,whenrepresentedasafreq uency,isgivenby f feedback ( t ) L 0 + d (t) dt (2–47) 88

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where d (t) dt isessentiallythefrequencynoiseoftheinputsignal.This frequencywillbe fedbacktotheinternaloscillationforphase-locking;als oitcanbeintegratedtogenerate thephaseuctuations.Thus,if L 6 = 0 ,alineardriftcausedbythefrequencyoffsetwill beseeninthemeasuredphasenoise. Thisfrequencyoffsetalsoexistsinthelongarmmeasuremen tinLISA:During thetransmissionofthelasereld,theheterodynefrequenc yofthemeasurementis modulatedbytheDopplershiftduetotherelativespacecraf tmotion.Thetime-variable relativevelocitiesineacharmwillbeupto 18m = s ,correspondingtoatime-variable Dopplershiftofupto 17MHz .Consequently,thefrequency L ofthelaserbeatsignal dependsontheknownoffset 0 usedatthefar-endphase-lockedloopandthenotso wellknownDopplershift D ( t ) .Thusthephaseoftheheterodynebeatsignalreceived bythephasemeterisgivenby L ( t )=2 ( 0 + D ( t )) t + l ( t ) l ( t ( t )). (2–48) Ideally,thephasemeterwouldberequiredtodemodulatethe beatsignalwiththe differencefrequencyofthetwolasers ( 0 + D ( t )) .Consequently,thepresetfrequency atthephasemeterwouldberequiredtobeupdatedinreal-tim eduetothetime-variable Dopplershift D ( t ) ,whichisunfortunatelydifculttoestimateataveryaccur atelevel. Actually,LISAwillnotestimatetheDopplerfrequencyinreal -timewhenarmlocking isinthesteadystateandatime-variableDopplerfrequency error D istherefore unavoidable.ItappearstobepossibletolimittheDopplerf requencyerrorto 20Hz per 200s averagingtimebyintroducingadditionalfunctionalityin on-boarddataprocessing. AmoredetaileddescriptionofDopplerfrequencyerrorsinL ISAwillbeseeninSection 6.1. TheDopplerfrequencyerrorwillmanifestitselfastherema iningfrequencyoffset inthephasemeasurementandthenwillbeintegratedupinthe armlockingcontroller, whichcausesafrequencypullinginthestabilizedlaserfre quency.Generallyalinear 89

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Figure2-19.AgenericsinglearmlockingloopwithaDoppler frequencyerrorinthe Laplacedomain.TheDopplerfrequencyerror D isaddedtotheerror signal e ( s ) duringthedemodulationofthephasemeter.ThisDopplererr or willbeseenbythecontroller G ( s ) andmaycauseafrequencydriftinthe stabilizedphase ( s ) driftinfrequencyismostlyharmlessinthedatapost-proce ssing.However,alarge frequencydriftintroducesadditionalrequirementsonthe tunabilityofthearmlocked laserandconsequentlyonthatofallsixlasersintheLISAcon stellation.Inthefollowing sectionwewillquantifythefrequencypullinginducedbyDo pplererrorsbyinvestigating thefrequencyresponseofagenericarmlockingloop.2.3.3.2Frequencypullingrate Figure 2-19 showsagenericsinglearmlockingloopwithaDopplerfreque ncyerror inputintheLaplacedomain.Thetransferfunctionsofthesi nglearmlockingsensor andcontrollerare H ( s ) and G ( s ) ,respectively.Forsinglearmlocking,theDopplererror isintroducedinthedemodulationatthephasemeterrightbe forethecontroller.Inthis diagram,theclosed-looptransferfunctionfromtheDopple rerrorinputtothestabilized lasernoiseoutputisgivenby TF D ( s )= G ( s ) 1+ H ( s ) G ( s ) (2–49) 90

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Inastandardcontrolloopwhenthecontrolleryieldsthehig hgainlimit G ( s ) !1 thetransferfunctionisreducedto TF D ( s ) 1 H 1 ( s ) 1 s (atverylowfrequencies) (2–50) ThisindicatesfortheDopplererrorinput,thefrequencyre sponseofaDC-coupled singlearmlockingloopisequivalenttoanintegrator,whic haccumulatestheDoppler errorovertimeandcausesadriftintheoutputfrequency.In thetimedomain,the instantaneousoutputfrequencyisgivenbytheconvolution betweentheinstantaneous Dopplererrorinputandthetransferfunctiondescribedabo ve.Overshorttimeintervals theDopplererrorisconstantandthelaserfrequencychange sby L ( t )= 1 Z D dt = D t (2–51) Onepotentialsolutionistoreducethecontrollergaingrad uallybelowthelower limit( 3 10 5 Hz )oftheLISAband,whichisknownasAC-coupling.Thisisbecau se theDopplershiftfrequencycausedbyspacecraftmotionsis averyslowoscillation at 10 8 Hz andthesamecaseisforDopplerfrequencyerrors. 3 Withsuchan AC-coupledcontrollertheDopplerfrequencyerrorwillnot coupleintothearmlocking loop.However,thetrade-offisthattheimpulseresponseof theAC-coupledcontroller toaDopplererrorwillshowupinthelockacquisitionandbec omeapartofinitial transients.Relevantexperimentalandnumericaldemonstr ationswillbeseeninChapter 6andhereweonlyevaluatethefrequencypullingrateinthes teadystate.Inthelow 3 Strictlyspeaking,theDopplerfrequencyerroroscillatesa tthesamefrequencyas theDopplershiftfrequencyonlyinthesteadystateofarmlo ckingbutnotduringthelock acquisition.SeeChapter6formoredetails. 91

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Figure2-20.AgenericdualarmlockingloopwithDopplerfre quencyerrorsinthe Laplacedomain.TwoindependentDopplerfrequencyerrorsa regenerated attwophasemetersonthelocalspacecraft.Whenthetwophasemeasurementsareaddedandsubtractedinthedualarmlockin gsensor, theDopplerfrequencyerrorsarealsoaddedandsubtracted. The differentialDopplererrorwillbeaccumulatedbytheinteg ratorinthe differentialpath. gainlimit( H ( s ) G ( s ) 0 atverylowfrequencies),wehave TF D ( s ) G ( s ) (atverylowfrequencies) (2–52) Thismeansinthelowgainlimit,theeffectonoutputfrequen cynoiseissimplythe inputDopplererrormultipliedwiththecontrollergainint helowfrequencyrange.Thus, inthepresenceofanAC-coupledcontrollertheDopplererro rwillnotbeaccumulatedby thecontrollerandconsequentlytheresidualeffectisnomo rethananinsignicantoffset addedtotheoutputlaserfrequency 4 Figure 2-20 showsthatindualarmlockingthetwoarmsgeneratetwoindep endent Dopplererrors D2 and D3 ,whichcanbecombinedtoconstructacommonanda 4 Theresidualeffectdependsonthespecicmagnituderespon seof G ( s ) atverylow frequencies.Hereforsimplicityweassumethat G ( s ) providesaconstantlylowgain downtoDC. 92

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differentialDopplererror.Wedene D+ = D2 + D3 D = D2 D3 (2–53) Theywillshowupinthecommonanddifferentialarmsensorsi ndependentlyand willbeampliedbytheclosed-loopgainineachpath: TF D+ ( s )= G ( s ) 1+ H D ( s ) G ( s ) (2–54) TF D ( s )= G ( s ) E ( s ) s [1+ H D ( s ) G ( s )] (2–55) Inthecommonarm, TF D+ ( s ) isapproximatelyequalto 1 = H D ( s ) inthehighgain limit.Sincethemagnitudeofthedualarmlockingsensorisa tatlowfrequencies, acommonDopplererrorwillonlycauseafrequencyoffsetind ualarmlocking.This characteristicsistobedistinguishedfromthefrequencyr esponseofacommonarm lockinglooptoacommonDopplererror.Inthatcasethearmlo ckingsensorisreplaced with H + ( s )= P + ( s ) .Consequently,inthehighgainlimitacommonDopplererror D+ = D2 + D3 willcausethefrequencypullingwithadriftrateof D+ = 2 Inthedifferentialarm,wehave E ( s ) 1 and H D ( s ) 2 atlowfrequencies andinthehighgainlimittheequationapproaches 1 = 2 s .Thisfrequencyresponse correspondstoafrequencydriftrategivenby d L dt D = D 2 (2–56) Sincethisdriftrateisinverselyproportionaltothediffer entialdelaytime,it becomesundesirablewhenthearmlengthmismatchisverysma ll,whichisoneof themaindisadvantagesofdualarmlocking.Toreducethisfr equencypulling,several modicationstothecontrollerhavebeenproposed,includi ngaddinghigh-passltersto suppressthecontrollergainbelowtheLISAband. 93

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2.3.3.3Modieddualarmlocking Modieddualarmlockingwasproposedbyourgroupand,indep endently,byKirk McKenzietosolvetheDoppler-inducedfrequencypullingpr oblemaswellastoenhance thenoisesuppressionperformance[ 13 ].Theconceptofmodieddualarmlockingis tomaintainthegainadvantagesofdualarmlockingandrecov erthefrequencypulling characteristicsofcommonarmlockingbyamplifyingthecom monarmsensorbelow 1 = andkeepingthedualarmsensorabove 1 = .Thiscombinationretainstheoverallat transferfunctionofthedualarmlockingsensorbelow 1 = 2 andsignicantlyreduces thefrequencypullingduetoaDopplererror. Themodieddualarmlockingsensorisconstructedbasedona linearcombination ofcommonarmlockingsensoranddualarmlockingsensor: H MD ( s )= F C ( s ) P + ( s )+ F D ( s ) H D ( s ), (2–57) where F C ( s ) and F D ( s ) aretwoltersdesignedinamannerthatthecommonarm sensordominatesatfrequenciesbelow 1 = andthedualarmlockingsensorgradually dominatesaboveit.Althoughthedesignof F C ( s ) and F D ( s ) isnotunique, F C ( s ) is essentiallyalow-passlterwithonepoleatDCand F D ( s ) isahigh-passlterwith acornerfrequencyaround .Bothltersrequireanappropriategaintosmooththe crossoverbetweenthetwosensors.Anexampleofthespecicd esignof F C ( s ) and F D ( s ) isprovidedbyKirkMcKenzie.Thelow-passlter F C ( s ) hasapoleatDCanda leadcompensatorwithazeroat 5 = 13 andapoleat 5 = 2 .Thehigh-passlter F D ( s ) hasfourzeroesatDC,twopolesat 7 = 10 11 = 20 andtwopolesat 1 = 90 .Here weprovideanotherrelativelysimplerdesignforourhardwa resimulations(SeeSection 5.3formoredetails).Thetransferfunctionofourmodiedd ualarmlockingsensoris illustratedinFigure 2-21 ,whereboththemagnituderesponseandthephaseshiftare maintainedintheLISAband.Wealsoplotthemagnitudeandpha seresponseofthe dualandcommoncomponentsthatconstitutethemodieddual armlockingsensorin 94

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 0 Frequency (Hz)Magnitude 10 -4 10 -3 10 -2 10 -1 10 0 10 1 -100 -50 0 50 100 Frequency (Hz)Phase shift (degree) Figure2-21.Themagnitudeandphaseresponseofthemodied dualarmlocking sensor.Theaveragedlighttraveltimeis 33s andthearmlengthmismatch is 1% .Themodieddualarmlockingsensorretainstheatnessoft he dualarmlockingsensortransferfunction.Atfrequenciesne ar 1 = =30mHz thetransferfunctionhassomedistortionscomingfromthe crossoverbetweenthecommonanddualarmcomponents.(SeeFi gure 2-22 ) Figure 2-22 .Thegureshowsthatthecommonarmcomponentdominatesfre quencies below 1 = whilethedualarmcomponentdominatesfrequenciesaboveit Eq. 2–57 canalsobewrittenasalinearcombinationofcommonarmsens orand differentialarmsensor: H MD ( s )= F C ( s ) P + ( s )+ F D ( s )[ P + ( s )+ E ( s ) s P ( s )] =[ F C ( s )+ F D ( s )] P + ( s )+ E ( s ) F D ( s ) s P ( s ) = H + ( s ) P + ( s )+ H ( s ) P ( s ). (2–58) Itiseasytoverifythatthemappingvectorofthemodieddua larmlockingsensoris givenby S k ( s )=[ H + + H H + H ] ,where H + ( s )= F C ( s )+ F D ( s ) and H ( s )= E ( s ) F D ( s ) s Actuallythisgeneralformalismcanalsobeadaptedtothecom monarmlockingand 95

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 0 Frequency (Hz)Magnitude 10 -4 10 -3 10 -2 10 -1 10 0 10 1 -200 -100 0 100 Frequency (Hz)Phase shift (degree) F D (s)H D (s) F C (s)P + (s) Figure2-22.Themagnitudeandphaseresponseofthedualand commonarm componentsthatconstitutethemodieddualarmlockingsen sorinFigure 2-21 .Thedualarmcomponentishigh-passlteredandthecommona rm componentislow-passltered.Therefore,thecommonarmco mponentwill dominatethefrequenciesbelow 1 = andthedualarmcomponentwill dominatethefrequenciesaboveit.Thephaseshiftofthecom monarm componentinthelowfrequencyregionneedstobeattenuatet ozeroto maintaintheoveralltransferfunction. dualarmlockingsensor,with H + ( s )=1 H ( s )=0 (common)and H + ( s )=1 H ( s )= E ( s ) s (dual). Figure 2-23 showstheDopplerfrequencyerrorsenterthecontrolsystem of modieddualarmlocking.Followingthetheprocedureinthe previoussection,inthe highgainlimittheclosed-looptransferfunctionstotheco mmonanddifferentialDoppler erroraregivenby TF D+ ( s )= G ( s ) H + ( s ) 1+ H MD ( s ) G ( s ) 1 2 H + ( s ), TF D ( s )= G ( s ) H ( s ) 1+ H MD ( s ) G ( s ) 1 2 H ( s ). (2–59) Sincethespecicexpressionsof H + and H dependonthecustomizeddesign oflters F C ( s ) and F D ( s ) ,thefrequencyresponsestotheDopplererrorscouldbe 96

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Figure2-23.AgenericmodieddualarmlockingloopwithDop plerfrequencyerrorsin theLaplacedomain.ComparedwithFigure 2-20 ,twoltersareplacedin thecommonanddifferentialpathindividually.Thehigh-pa sslterinthe differentialarmpatheliminatesthefrequencypullingdue tothedifferential Dopplererror,whilethelow-passlterdesigninthecommon armpath allowsthefrequencypullingduetothecommonDopplererror .Notethatit isalsopossibletohigh-passlterthecommonarmpathtoeli minatethe frequencypullingcompletely,whichisessentiallyequiva lenttoan AC-coupledcontroller. complicatedtocalculate.However,asweknowthatessentia lly F C ( s ) isalow-passlter withonepoleatDCand F D ( s ) isahigh-passlterwithacornerfrequencyaround 1 = thenoiselimitationcanberoughlyestimatedusingasimpli eddesignof F C = 1 s and F D = s s +2 = .Thuswehave H + ( s )= 1 s + s s +2 = H ( s )= 1 s s s +2 = = 1 1 s +2 = (2–60) Weplotthemagnituderesponsesof H + ( s ) and H ( s ) with =33s and =0.16s inFigure 2-24 .Themagnituderesponsesof H + ( s ) and H ( s ) featureaslopeof 1 = s and 1 = s ,respectively.Atlowfrequencies f 1 = H ( s ) attensoutandapproaches aconstantvalueof = (2 ) .Athighfrequencies f 1 = H + ( s ) attensoutand approaches 1 97

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10 -4 10 -3 10 -2 10 -1 10 0 10 -1 10 0 10 1 10 2 Frequency (Hz)Magnitude H + (s) H (s) 1/(s*0.16s) 1/(s*33s) Figure2-24.Themagnituderesponseof H + ( s ) and H ( s ) with =33s and =0.16s formodieddualarmlocking.Inthisdesignthe F D ( s ) onlyhasasingle zeroatDC,whichmakesthemagnituderesponseof H ( s ) atatlow frequencies.Amorecarefuldesignof F D ( s ) shouldprovideasteeperslope tosufcientlysuppressthe H ( s ) atlowfrequencies. Ifweonlyconsiderthesteadystate,theDopplerfrequencye rrorisanoscillation atfrequenciesbelowtheLISAband,wherethethemagnitudere sponsesof H + ( s ) and H ( s ) approximateto 1 = s and = (2 ) .Consequently,thedifferentialDopplererror D willnotcauseanyfrequencypullingbutaninsignicantfre quencyoffset.The dominantfrequencypullingrate,whichisnowcompletelyat tributedtothecommon Dopplererror D+ ,isgivenby d L dt MD = D+ 2 (2–61) Comparedwithdualarmlocking,thisfrequencypullingrate isinverselyproportional totheaveragedelaytime,whichisobviouslyasubstantiali mprovementwhenthearm lengthmismatchissmall.However,itshouldbenotedthatth ecommonDopplererror D+ mayalsobemuchlargerthanthedifferentialDopplererror D bytwoordersof magnitude,especiallyduringtheinitialestimationsofDo pplererrors.Thisfactindicates thatmodieddualarmlockingisnotalwayssuperiortoduala rmlockingintermsof 98

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frequencypulling.InrealisticLISA,itshouldbeallowedtos witchtotheproperarm lockingcongurationbetweenthemaccordingtothereal-ti mearmlengthmismatch. 2.3.4IntegrationwithTunablePre-stabilizationReferenc es BeforethefrequencypullingcausedbyDopplererrorsinarml ockingwasrealized, theintegrationofarmlockingwithpre-stabilizationsubs ystemswasalreadyconsidered becausethiscombinationwouldprovidemorenoisesuppress ionstorelaxthe burdenonTDI.Theincorporationofarmlockingwithapre-st abilizationsubsystem requiresthereferencetobetunablebecausethelockpoints forthelocalreference andthearmlockingreferencearenotgenerallythesame.For instance,astandard Pound-Drever-Hallsetupdoesnotprovidetunabilitybetwe entworesonantfrequencies ofthereferencecavity;therefore,somemodicationtothe PDHsetupmustbepresent tomaketheresonantfrequencyadjustable.Nowthediscover yoffrequencypulling increasestherequiredtuningrangeforthelocalreference .Areasonabletuningrange ofthereferenceshouldbewelllargerthantheexpectedfreq uencypullingrangeofthe lasertoavoidpossiblepre-stabilizationissues.Givenac avitypre-stabilizedlaserand 200s averagingestimationtime,thefrequencypullingonthesta bilizedlaserisonthe levelofafewMHz,whichmeansthetuningrangeofthetunable referenceshouldbeat leastattensofMHz. Severaltunabilityoptionswithasufcienttuningrangehav ebeensuggested forarmlockingwithdifferentpre-stabilizationschemes. Herewewilldiscussthree feasibleoptions:ArmlockingwithPound-Drever-Hallstabi lization,armlockingwith Mach-Zehnderstabilizationandarmlockingwithoutanypre -stabilization. 2.3.4.1ArmlockingwithPound-Drever-Hallstabilization ForPound-Drever-Hallstabilizationasimpleapproachist oreplacethexedoptical cavitywithaPZTtunablecavity,whichisactuatedbythearml ockingfeedbacksignal. AsshowninFigure 2-25 ,bytuningthelengthofthecavitythecentralfrequencyof thestabilizedlasercanbetunedandfurtherstabilizedbyt hearmlockingcontroller. 99

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Figure2-25.TheintegrationofarmlockingandPound-Dreve r-Halltechniqueusinga PZTactuatedcavityasatunablereference.Inthissetupthel aserislocked tothePZTactuatedcavityviathePDHmethod.Thearmlockingfe edback signalisusedtodrivethePZTtocontrolthecavitylengthand therebythe resonantfrequency. TheadvantageofusingaPZTcavityisthattheimplementation issimplesuchthatthe changetothepre-stabilizationsubsystemisminimized.Ab enchtopvericationofthis methodisdescribedinSection4.3.2,wheremoredetailswill beexplained. Asecondapproachisknownassidebandlocking,whichisillu stratedinFigure 2-26 .InthisapproachtheRFsidebandgeneratedbytheEOMislocke dtoaxed opticalcavityviastandardPound-Drever-Halltechniquea ndthelocaloscillatordriving theEOMistunedbythearmlockingfeedbacksignal.Therefore thetuningofthe modulation/demodulationRFsignalallowsthetuningofthe centralfrequencyofthe stabilizedlaser.Thismethodrequiresahighbandphasemod ulator/demodulatorto replacetheoriginallocaloscillator,butnomodicationi sneededfortheresonantcavity. RelevantexperimentsaredescribedinSection4.3.3. AsshowninFigure 2-27 ,itisalsopossibletokeepthePound-Drever-Hall stabilizationcompletelyuntouchedbyintroducinganauxi liarylaserthatisphase-locked tothecavitystabilizedlaserwithanfrequencyoffset.Simi lartothesidebandlocking, thetunabilityisaccomplishedviathearmlockingfeedback signaltuningtheheterodyne frequencyofthephase-lockedloop.Thiscongurationisea sytodemonstrateona 100

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Figure2-26.TheintegrationofarmlockingandPound-Dreve r-Halltechniqueusinga xedcavitylockedtothesideband.Inthissetupthesideban dfrequencyof thelaserislockedtothexedcavityviathePDHmethod.Thear mlocking feedbacksignalisusedtoadjustthemodulation/demodulat ionRFsignal thatdrivestheEOM. Figure2-27.TheintegrationofarmlockingandPound-Dreve r-Halltechniqueusingan auxiliaryphase-lockedlaser.Inthissetupanauxiliaryla serisheterodyne phase-lockedtothecavitystabilizedlaser.Thearmlockin gfeedbacksignal isusedtoadjustthelocaloscillatorthatdrivesthePLL. benchtopbutwouldrequireanadditionallaserinrealistic LISA.Theexperimental demonstrationisdescribedinSection4.3.1. Itisalsoworthnotingthatinallthesesetups,thearmlocki ngfeedbacksignalisnot necessarilyusedtoonlyadjusttheactuatorsinthepre-sta bilizationsetup;instead,itcan alsobedirectlyaddedtothefeedbacksignalfromthePDHcont rollertocorrectthelaser 101

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frequency.Inpractice,thearmlockingfeedbacksignalwil lbeelectronicallysplitupand entersthepre-stabilizationloopatthetwopointsdescrib edabove;thelaserfrequency thereforewillbestabilizedbytheircombination.Theadva ntageofthiscongurationis thattheperformanceofthearmlockingsystemisnotlimited bythebandwidthofthe pre-stabilizationloop.2.3.4.2ArmlockingwithMach-Zehnderstabilization InadditiontothestandardPound-Drever-Hallstabilizati on,theintegrationofarm lockingandotherkindsofpre-stabilizationsubsystemhas alsobeenconsidered.In particularwealreadymentionedtheintegrationofarmlock ingwithMach-Zehnder stabilizationinSection2.2.2.2.Unlikethewidelydiscret elock-pointsinthePDH stabilization,Mach-Zehnderstabilizationprovidesanin herentfrequencytunability describedinEq. 2–26 ,whichmakestheintegrationmuchsimpler.Intheintegrati onthe armlockingfeedbacksignalisnotonlyusedtoadjustthepha seofthelocaloscillator thatdemodulatestheheterodynebeatsignal,butitisalsod irectlyaddedtothefeedback signalintheMach-Zehnderstabilization.Thesecondadjus tmentensuresthatthe high-bandwidtharmlockingperformancecanberetainedand notaffectedbythelow bandwidthoftheMach-Zehndersetup.Withthecontrollerdes ignedinRef.[ 13 ],the expectedperformancemeetstheTDIcapabilityrequirement withamargingreaterthan 50at 1Hz and800at 3Hz 2.3.4.3Armlockingonly Itisalsopossibletohavearmlockingtakeovertheentirest abilizationofthe laserfrequencywithoutanytunablepre-stabilizationref erence.Insuchanoption thearmlockingfeedbacksignalisdirectlysentbacktothef requencyactuatorofthe lasercontroller;thereforetheopticalbenchsubsystemis totallyuntouched.Analytical studiesindicatethatevenwithoutanypre-stabilizations chemes,thedual/modied dualarmlockingwillprovidesufcientnoisesuppressions andmeettheTDIcapability requirementacrosstheentireLISAbandinmosttimeofayear[ 13 ].Theonlyperiod 102

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Figure2-28.Michelson-typedarmlockingcongurationwit hthephase-lockedlaserat thefarspacecraft. whenthearmlockingperformanceisinsufcientonlylastsf or 30min ,twiceayear. Thishappenswhenthearmlengthmismatchislessthan 12km .Inthenextsectionwe willanalyzethenoiselimitationsinarmlockingandidenti fywhentheywillbecomea problem.2.3.5ArmLockingLimitations Inthissectionweconsidertherealisticlimitationstothe noisesuppression performanceofarmlocking.Inreality,thestabilizedlase rfrequencynoiseisdominated byseveralexpectednoisesources.Oneofthenominalnoises ourcesisthetransponder noisewhichcomesfromthephase-lockingofthefarlaser.Th etranspondernoiseis adequatelyremovedinTDIalgorithms;however,armlocking doesnothaveanysimilar mechanismtotakecareofit.Wewillanalyticallyinvestiga tehowthetransponder noisewillaffectthearmlockingperformance.Also,theeffe ctsofothersignicantnoise sources,includingnoiseofultra-stableoscillators(USO' s),randomspacecraftmotion, shotnoiseandtechnicalnoisewillbediscussed.2.3.5.1Limitedcontrollergaininfar-endPLL Figure 2-28 illustratesthegenericarmlockingcongurationwiththei ncorporationof thephase-lockedlaseratthefarspacecraft,wherethelimi tedcontrollergainintroduces 103

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atranspondernoise.Intheanalysiswerstassumethefar-e ndPLLsareclosed whilethearmlockingloopisstillopen.BasedonEq. 2–29 ,inthepresenceofagain limitedfar-endPLL,thephasemeasurementofthelongarmint erferometryatthelocal spacecraft SC 1 isgivenby 1 i ( s )= 0L 1 1 G i 1+ G i e s i 1 1+ G i 0Li e s i 1 (2–62) Thersttermisdirectlyrelatedtotheinitialphasenoise 0L 1 ofthepre-stabilized locallaser.Thesecondtermisanadditionalrandomnoiseso urce(theresidualnoise unsuppressedbythefar-endPLL)since 0L 1 and 0Li areuncorrelated.Therefore,we canstilldenetheLISAarmtransferfunction,whichyieldst hersttermdividedbythe theinitialphasenoise 0L 1 : P 1 i ( s )=1 G i 1+ G i e s i (2–63) whichwouldbereducedtoEq. 2–30 when G i 1 .Thedenitionofthearmlocking sensor H ( s ) remainsthesameasEq. 2–33 ,butnowitinvolvesthePLLcontrollergain G i i =2,3 Whenthearmlockingloopisclosed,thestabilizedphasenois eofthelocallaseris givenby L 1 ( s )= 0L 1 G 1 1+ G 1 H S k 1 (2–64) where S k isthemappingvectorand 1 isa 2 1 phasevectordescribingthetwo interferometeroutputs.(SeeSection2.3.1) Nowwecalculatethestabilizedphasenoisedependingonspe cicmappingvectors individually.Forsinglearmlocking S k =[1,0] ,Eq 2–64 yields L 1 ( s )= 0L 1 G 1 1+ G 1 H 0L 1 1 G 2 1+ G 2 e s 2 1 1+ G 2 0L 2 e s 21 = 1 1+ G 1 H 0L 1 + G 1 1+ G 1 H 1 1+ G 2 0L 2 e s 21 (2–65) 104

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ThersttermofEq. 2–65 demonstratesthattheinitialphasenoise 0L 1 issuppressed bytheopen-loopgain G 1 H .Thesecondtermreectstheclosed-loopresponseof singlearmlockingtothetranspondernoise 1 1+ G 2 0L 2 .Essentiallyitisequivalenttoa Dopplerfrequencyerrorintroducedduringthelighttravel astheybothcorrespondtoa closed-loopresponseof G 1 1+ G 1 H exceptforatrivialphasedelay e s 21 .Inthehighgain limitofthearmlockingcontroller,theclosed-looprespon seapproximatesto 1 = H 1 = s whichincreasesthelevelof L 1 ( s ) bytheamountoftheaccumulatedtransponder noise. Inthecaseofdualarmlocking S k =[1+ E ( s ) s ,1 E ( s ) s ] ,ifweuse E ( s ) 1 inthe LISAfrequencyband,Eq 2–64 yields L 1 ( s )= 0L 1 G 1 1+ G 1 H [1+ 1 s ,1 1 s ] 264 0L 1 1 G 2 1+ G 2 e s 2 1 1+ G 2 0L 2 e s 21 0L 1 1 G 3 1+ G 3 e s 3 1 1+ G 3 0L 3 e s 31 375 = 1 1+ G 1 H 0L 1 1 1+ G 2 0L 2 e s 21 1+ 1 s 1 1+ G 3 0L 3 e s 31 1+ 1 s (2–66) Thersttermrepresentstheidealizednoisesuppression.T headditionaleffectsof thetranspondernoise,whicharedescribedbytheothertwot erms,canbewrittenas L 1 ( s )= 1 s 1 1+ G 3 0L 3 e s 31 1 1+ G 2 0L 2 e s 21 1 1+ G 3 0L 3 e s 31 + 1 1+ G 2 0L 2 e s 21 (2–67) TherstterminEq. 2–67 indicatesthatinthedifferentialarmpaththedifference betweenthetranspondernoisesontwoarmswillbeaccumulat edandscaledwith 1 = Thesecondtermindicatesthatinthecommonarmpaththesumo fthetransponder noiseswillbedirectlysubtracted,whichisnegligibleinc omparisontothedifferentialarm path. 105

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Inthepresenceofamodieddualarmlockingsensor S k =[ H + + H H + H ], H + ( s )= F C ( s )+ F D ( s ), H ( s )= E ( s ) F D ( s ) s ,thestabilizedphasenoiseisgivenby L 1 ( s )= 0L 1 G 1 1+ G 1 H [ H + + H H + H ] 264 0L 1 P 12 + L 2 0L 1 P 13 + L 3 375 (2–68) where P 1 i =1 G i 1+ G i e s i isthearmtransferfunctionand Li = 1 1+ G i 0Li e s i 1 isthe transpondernoisedelayedbythereturningtraveltime. Eq. 2–68 yields L 1 ( s )= 0L 1 G 1 1+ G 1 H ( H + P + + H P ) 0L 1 + G 1 1+ G 1 H [ H + ( L 2 + L 3 )+ H ( L 2 L 3 ) ] 1 1+ G 1 H 0L 1 + H + H ( L 2 + L 3 )+ H H ( L 2 L 3 ). (2–69) Thisindicatesthesumofthetranspondernoisesontwoarmsw illbemultiplied with H + H whilethedifferencebetweenthemwillbemultipliedwith H H ,whichisagain equivalenttothefrequencyresponsetotheDopplererrorsg ivenbyEq. 2–59 .Herewe stillusethetransferfunctionsof H + ( s ) and H ( s ) determinedbyEq. 2–60 toanalyze thetranspondernoiseoor. Sincethetranspondernoises L 2 and L 3 areuncorrelated,intermsofthenoise amplitudewehave L 2 + L 3 L 2 L 3 .Therefore,asshowninFigure 2-24 thedifferentialnoisefromthedifferentialarmpathwilld ominatemostoftheLISAband untilthecrossingof H + and H atabout 0.2mHz .Forfrequenciesbelowthecrossing thecommonnoisewilldominate.Comparedwithdualarmlocki ng,thenoiselimitation below 1 = hasbeendecreasedfrom 1 = s tothedominanttermbetween = (2 ) and 1 = s .InrealisticLISA,thelter F D canbedesignedtohaveaslopesteeperthan s such thatthemagnitudeof H willrollofffasteratlowfrequenciesandthecommonnoisew ill starttodominateatahigherfrequency. Inthissectionwehavedemonstratedtheanalyticaltranspo ndernoiselimitation solelycausedbythenitegainofthePLLcontroller.Inreali ty,thetranspondernoise 106

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ofthefar-endPLLisprimarilycontributedotherrealisticn oisesources,suchasthe frequencynoisethatenterstheultra-stableoscillatordr ivingtheheterodynePLL.The frequencynoiseoftheUSO,ormorefrequentlycalledclockno ise,willalsobecomea partofthetranspondernoiseandmayfurthercompromisethe armlockingperformance. Moredetailswillbediscussedinthenextsection.2.3.5.2Realisticnoisesourcesinarmlocking Thelimitationofarmlockingperformancedependsonsevera lrealisticnoise sources,aswellascertainarmlockingcongurationsthatd eterminehowthenoise sourceswillcontributetothestabilizedlaserfrequency. Basically,theformalismusedto evaluatetheeffectofthetranspondernoisecanbedirectly adaptedtoquantifythearm lockinglimitationcausedbythesenoisesources,onlywith differenterrorpoints.Sofar, signicantnoisesourcesthatmaybepresentinarmlockingi nclude: Clocknoise-Thephaseofabeatsignalismeasuredbycompari ngittoatiming reference(thelocalultra-stableoscillator).Therefore ,theacquiredphasevalue isalwaysrelativetothephasenoiseofthereferencingcloc k.Theclocknoiseis proportionaltothenominalfrequency n ofthemeasuredbeatsignal,i.e., clock ( f )=n 0 clock ( f ), (2–70) where 0 clock ( f ) arethefractionalfrequencyuctuations,correspondingt othe normalizedclockfrequencynoiseat 1Hz clockfrequency.Thefractionalfrequency uctuationisestimatedtobeapproximately 2.4 10 12 = p f Hz 1 = 2 .Theclock noisesfromphasemetersonthesamespacecraftarecorrelat ed,whiletheyare uncorrelatedondifferentspacecraft. Spacecraftmotion-LISAarmlength(benchtoptobenchtop)isa nexcellent referencetostabilizethelaserfrequency.However,thest abilityofthislength referenceisstilllimitedbytheDRSthatdragsthespacecra fttotrackthegeodesic motionoftheproofmass.Theconsequentlengthuncertainty isapproximately givenby L SC ( f )=2.5 10 9 p 1+( f = 0.3Hz) 4 mHz 1 = 2 (2–71) Thelengthuncertaintyofonearmincludesthespacecraftmo tionsoftwo spacecraftateachend.Thislimitedstabilityinthelength referencewillcause aphasenoiseinthephasemeasurement. Shotnoise-Limitednumber N ofphotonsreceivedpersecondbyphotodiodes. FromEq. 2–5 ,with 100pW lightreceivedatthephotodiodetheshotnoiseisgiven 107

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Figure2-29.Thecontrolsystemofarmlockingwithvariousn oisesourcesenteringthe loop.Inthediagram C21 representstheclocknoiseintroducedinthe phasemeasurementofthebeatsignal 21 ( t )= 2 ( t ) 1 ( t 12 ) .The samenotationisusedfortheshotnoise S andthephasemetertechnical noise T X21 representsthespacecraftmotionof SC 2 appliedontothe instantaneousphase 2 ( t ) andthedelayedphase 1 ( t 12 ) by shot = 1 2 p N = r ~ c 2 1 P =6.9 10 6 cyclesHz 1 = 2 (2–72) Technicalnoise-IncludingtheADCnoiseintheA/Dconversion ofthebeatsignal, aswellastheniteprecisionofintegerarithmetic,knowna sthedigitizationnoise inthephasemeters,armlockingsensorandcontroller.Adig italsignalwitha samplingfrequencyof f s andaprecisionofN-bitgenerallycarriesthedigitization noise dig = f clock 2 N p 6 f s (2–73) Inthepresenceofa 50MHz clockfrequencyanda 100kHz datarateina48-bit phasemeter,thearmlockingsensorwillsensethedigitizat ionnoisegivenby 2.3 10 10 HzHz 1 = 2 Figure 2-29 illustratesthearmlockingschematicincludingtherealis ticnoise sourcesdescribedabove.Firstweassumethefar-endphaselockedloopsarein operationwhilethearmlockingloopisnotclosedyet.Thust helaserphasenoise 108

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Li i =2,3 fromthefarspacecraftisgivenby Li ( s )= 1 1+ G i 0Li + G i 1+ G i 0L 1 e s 1 i + X1i e s 1 i + Xi1 Si1 Ci1 Ti (2–74) Inthisanalysisweignorethetranspondernoise 1 1+ G i 0Li andassumethehighgain limit G i 1+ G i 1 .Therefore,thephasenoiseofthelongarminterferometrys ignalonthe localspacecraftisgivenby 1 i ( s )= 0L 1 ( s ) ( Li ( s )+ Xi1 ) e s i 1 X1i + S1i + C1i + T1i 0L 1 ( s ) P 1 i X1i 1+ e s i 2 Xi1 e s i 1 + Si1 e s i 1 + S1i + Ci1 e s i 1 + C1i + Ti e s i 1 + T1i (2–75) FollowingthenotationusedinEq. 2–31 ,thephasenoisesofthetwointerferometer outputscanbewrittentogetherasa 2 1 phasevector 1 ( s )= 264 12 ( s ) 13 ( s ) 375 =[ N L + N C + N X + N S + N T ], (2–76) whereeachvectorisgivenby N L = 264 0L 1 ( s ) P 12 0L 1 ( s ) P 13 375 N C = 264 C12 + C21 e s 21 C13 + C31 e s 31 375 N T = 264 T12 + T2 e s 21 T13 + T3 e s 31 375 N X = 264 X12 ( 1+ e s 2 ) 2 X21 e s 21 X13 ( 1+ e s 3 ) 2 X31 e s 31 375 N S = 264 S12 + S21 e s 21 S13 + S31 e s 31 375 (2–77) Notethatfor N C N X N S and N T ,eachvectorelementconsistsoftwonoiseterms (e.g., C12 and C21 )thatcomefromtwouncorrelatedsources.Thecombinednois e amplitudeisgivenbythequadraturesumofthetwoterms. 109

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Table2-1.Parametersofthenoiseanalysisforarmlocking ParameterSymbolValueUnits Averagedelaytime 33s Differentialdelaytime 0.016s Nominalfrequencyonarm1-2 n 12 15MHz Nominalfrequencyonarm1-3 n 13 14MHz Fractionalfrequencyuctuation 0 clock 2.4 10 12 = p f Hz 1 = 2 Armlengthstability(SCtoSC) L SC 2.5 p 1+( f = 0.3Hz) 4 nmHz 1 = 2 Receivedpoweratphotodiodes P 100pW TDIcapability TDI 282 p 1+(2.8mHz = f ) 4 HzHz 1 = 2 Armlockingrequirement[ 60 ] AL 0.3 p 1+(2.8mHz = f ) 4 HzHz 1 = 2 Whenthearmlockingloopisclosed,weevaluateEq. 2–64 L 1 ( s )= 0L 1 G 1 1+ G 1 H S k 1 = 1 1+ G 1 H 0L 1 G 1 1+ G 1 H S k [ N C + N X + N S + N T ]. (2–78) Thesecondtermrepresentsthenoiselimitations.Hereweco nsiderthearmlocking performancelimitedbyvariousnoisesourcesbyassumingan innitecontrollergain. TheanalysisisbasedontheLISAparameterslistedinTable 2-1 underdifferentarm lockingcongurations,whereweneglectthetechnicalnois easitshouldbesmall comparedtotheotherthree.Thearmlengthmismatchisassum edtobeasshortas 0.1% ,correspondingtoa 0.016s differentialtimedelay,inordertodemonstrateacritical scenario. Forcommonarmlocking,thearmlockingperformanceisindep endentofthearm lengthmismatch.ThenoiselimitationsareplottedinFigur e 2-30 ,whichshowsthatfor frequenciesabove 20mHz andbelow 0.7mHz thecommonarmlockingperformance islimitedbyspuriousspacecraftmotionsandforfrequenci esinthemiddlerangeitis dominatedbytheclocknoise.Atfrequencies n = thespacecraftmotioncaussenoise peaksthatmayfailtomeetthearmlockingrequirement. Forthedualarmlockingconguration,thenoiselimitation sareplottedinFigure 2-31 .Thedominantclocknoiseandspacecraftmotionoorpreven tarmlockingfrom 110

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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 Frequency (Hz)Frequency noise (Hz Hz -1/2 ) clock spacecraft motion shot TDI capability Arm locking requirement Figure2-30.Thenoiseoorsincommonarmlockingareindepe ndentofthearmlength mismatch.Athighfrequenciesthedominantnoiseisspacecra ftmotions andatlowfrequenciesthedominantnoiseistheclocknoise. meetingtherequirementforfrequenciesbelowabout 10mHz .Analytically,theclock noiseoorisgivenby clock ( s )= 1 H [1+ 1 s ,1 1 s ] 264 C12 + C21 e s 21 C13 + C31 e s 31 375 = 1 H C12 + C21 e s 21 + C13 + C31 e s 31 + 1 s C12 + C21 e s 21 C13 C13 e s 13 (2–79) Notethattheoperationof“addition”or“subtraction”betw eentheclocknoises contributedbydifferentarmsshouldstillbetakenasaquad raturesum.Thisresult indicatesthatthedominanttermisthedifferentialarmpat h,whichintegratesthe differentialclocknoisebetweentwoarms.Duetothestruct uralsimilarity,theotherkinds ofnoisesourceswillbecoupledintothestabilizedfrequen cynoiseinthesamemanner. Therefore,thenoiseoorindualarmlockingisprimarilyco mposedoftheintegrationof 111

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10 -4 10 -3 10 -2 10 -1 10 0 10 -4 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)Frequency noise (Hz Hz -1/2 ) clock spacecraft motion shot TDI capability Arm locking requirement Figure2-31.Thenoiseoorsindualarmlockingaresensitiv etothearmlength mismatch.Hereweassumearelativelyshortarmlengthmisma tchof 0.1% andthenoiseoorsaresignicantlyhigherthanthecommona rmlocking situation.Asthenoiseoorindualarmlockingisinverselyp roportionalto thearmlengthmismatch,theperformanceofdualarmlocking is insufcienttomeettheTDIcapabilitywhenthearmlengthmi smatchisless thanabout 60km thedifferentialnoisesandinverselyproportionaltothea rmlengthmismatch.Ashorter armlengthmismatchcorrespondstoaworsenoisesuppressio nperformanceandour analysisindicatesthatforthedifferentialdelaytimeles sthan 0.2ms ( L < 60km )the dualarmlockingperformancewouldevenbeinsufcienttome ettherequirementfor TDI. Asimilarapproachcanbeusedtoevaluatethenoiseoorinth eSagnac-based dualarmlockingconguration.SinceintheSagnacconstellat iontherelaybeamswill pickupalltheclocknoises,spacecraftmotions,etc.thate nterthearmlockingcontrol systembothclockwiseandcounter-clockwise,thequadratu resumofthesenoiseswill beintegratedandmultipliedwith 1 = ,wheretheequivalentdifferentialdelaytimeis 112

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10 -4 10 -3 10 -2 10 -1 10 0 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)Frequency noise (Hz Hz -1/2 ) clock spacecraft motion shot TDI capability Arm locking requirement Figure2-32.Thenoiseoorsinmodieddualarmlockingarel essdependentonthe armlengthmismatchduetothelow-frequencylteringschem e.Thenoise oorsareeffectivelysuppressedbythehigh-passlteratf requencies below 1 = .Withamorecarefullydesignedhigh-passlter,thenoiseo ors willbefurtherreducedandasymptoticallyapproachthenoi seoors determinedinthecommonarmlockingconguration. 47 s .Sinceeverynoisesourcewillcontributetotheoverallnois eoortwiceand scalewithaveryshortdifferentialdelay,thenoiselimita tioninSagnac-baseddualarm lockingismuchhigherthananyotherarmlockingcongurati on. Theanalysisformodieddualarmlockingcanbeadaptedfrom Eq. 2–69 .Forthe modieddualarmlockingsensor [ H + + H H + H ] ,theclocknoiseoorisgivenby clock ( s )= H + H ( C12 + C21 e s 21 + C13 + C31 e s 31 )+ H H ( C12 + C21 e s 21 C13 C31 e s 31 ). (2–80) Asasimpledemonstration,westilltake H + ( s ) and H ( s ) givenbyEq. 2–60 as anexample.ThenoiselimitationsareplottedinFigure 2-32 .Comparedwiththenoise limitationindualarmlocking,thenoiselevelhasbeendecr easedbelowthearmlocking 113

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requirementfortheentireLISAband,whichiswellbelowtheT DIcapability.Notethat duetothesimplieddesignof F D ( s ) ,whichexhibitsanunoptimized s slope,thenoise oorsatlowfrequenciesarenotmaximallysuppressed.Inre alisticLISAabetternoise suppressionperformanceshouldbeexpectedwithamorecare fullydesigned F D ( s ) ifmodieddualarmlockingwillbeused.Ifthemagnitudeoft he F D ( s ) lterrollsoff fasteratlowfrequencies,thenoiseoorswillbecomelessd ependentonthearmlength mismatchandasymptoticallyapproachtheoordeterminedb y 1 = Attheendofthischapterletushaveabriefestimationregard ingtheLISA armlengthmismatch.Nowthatweknowashortarmlengthmisma tchiscriticalfor Doppler-inducedfrequencypullingandthearmlockingperf ormance,weareinterested inhowlongacriticalscenario,say j L = L j < 0.1% ,willlastintheorbitalperiodofone year.Asweknow,thearmlengthmismatchisroughlyasinusoid functionwithaperiod ofoneyearandthemaximummismatchisabout 1% ,i.e., L =0.01 L sinn t (2–81) where n=2 = 1yr 2 10 7 Hz Therefore,thearmlengthmismatchis L 5 10 7 m sin(2 10 7 Hz t) 10ms 1 t. (2–82) Thusweobtainthechangerateofarmlengthmismatchofappro ximately 10ms 1 Forthearmlengthmismatchbelow 0.1% ,thedurationisapproximately 2 5 10 6 m = 10ms 1 10 6 s ,whichislessthan12days,twiceperyear.Notethatthisisj ust aquickandroughestimationandamoredeliberatecalculati ondependsontheexact LISAorbits.Nevertheless,itstillgivesustheinformation thattheperformanceofarm lockingwillnotbesubstantiallydegradedbythearmlength variationsandthescenario willonlybecomecriticalduringashortamountoftimeevery year. 114

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CHAPTER3 UNIVERSITYOFFLORIDALISAINTERFEROMETERSIMULATOR TheLISAgroupattheUniversityofFloridahasdevelopedahar dwareinthe loopsimulatorforLISA'slongarminterferometrywhichissim ilartoLISAaspossible. Sofar,asynthesizedhardwareLISAinterferometryandmeasur ementsystem,the UniversityofFloridaLISAInterferometerSimulator(UFLIS), whichconsistsofavariety ofLISAcomponentmodels,hasbeendevelopedwithcontinuous modicationand enhancement.ThesecomponentmodelsincludeLISA-likecavit ypre-stabilization, digitalintegrated-circuitboardbasedsciencephasemete rs,LISA-likeinter-SCranging andlasercommunications,LISA-likevariablelighttravelti meandvariableDoppler shifts,analog/digitalhybridrealisticarmlocking,synt hesizedrealisticTDIwithfractional delayltering,generationofgravitational-wave-likesi gnals,etc. UFLISiscurrentlytheonlyexistinghardwaresystemthatca nrealisticallysimulate longarminterferometrybyprovidingLISA-likelighttravelt imeaswellasvariable Dopplershifts.Sincethethreeinterferometersareindepen denttoeachotherin theIMS,theabsenceoftheothertwointerferometerwillnot degradethevalidity ofourexperimentsbutwillalsothereducethecomplexity.A long-termgoalof UFLISistoinjectthegeneratedgravitational-wave-likes ignalsintothesimulator andmocktheLISAdatawithintherequirednoiselevel,andtos uccessfullyextractthe gravitational-wave-likesignalsbydevelopingproperLISA dataanalysismethods.In thischapterwewillintroducethecurrentresearchontheop tical/electroniccomponents relatedtothehardwaresimulationofarmlocking,whichare accomplishedbyourgroup attheUniversityofFlorida. 3.1OpticalComponents InthecurrentopticallayoutofLISAinterferometrybenchto p,three 1064nm Nd:YAG lasersarereferredtoas L i i =1,2,3 ,respectively.AnotherindependentNd:YAGlaser RL isareferencelaserwhichfunctionsasanopticalclocktoge neratebeatsignals 115

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10 -4 10 -2 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Frequency (Hz)Frequency noise (Hz/rtHz) RL L1 beat signal TDI capability Figure3-1.Frequencynoiseofthebeatsignalbetweentwoca vitystabilizedlaser RL and L 1 withfrequenciesbelow 20MHz witheachoftheotherthreelasers.Thereasonweuse beatsignalratherthanthedirectlaserfrequencyisthatas inglelaserfrequencyisin therangeof 300THz whichisnotpossibletobemeasuredbyanormalcommercial photodiode. Thelasers RL and L 1 areseparatelylockedtotheirownreferencehighnesse opticalcavityusingPound-Drever-Halltechniqueforprestabilization.Eachcavity consistsofaZerodurspacer.Therefore,thebeatsignalbet ween RL and L 1 provides theinputphasenoiseonthemasterspacecraftinTDIandarml ockingsetup.Thebeat signalsbetween RL and L 2 or L 3 canbeusedasreturningbeamsfromfarspacecraft with L 2 or L 3 phase-lockedtothemasterlaser L 1 Figure 3-1 showsthefrequencynoisespectrumofthebeatsignalbetwee n RL and L 1 atanominalfrequencyofapproximately 13MHz .Inthefrequencyrangefrom 1mHz to 1Hz ,thefrequencynoisestaysatalevelofapproximately 100 200Hz = p Hz whichisalreadybelowthe 282HzHz 1 = 2 pre-TDIfrequencynoiserequirement.Also, thebeatfrequencyexhibitsalongtermlineardriftbyupto 1MHz ,withatypicaldrift 116

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rateof 1Hz = s .Thismoderatefrequencydriftcanbetakenoutinthedatapr ocessing andessentiallyhasnosignicantinuenceonlasermeasure ments. 3.2ElectronicComponents ToreproducethemeasurementenvironmentinLISA'slongarmin terferometry, weselectedahigh-speeddigitalsignalprocessingsystem( referredtoasthe“Pentek board”)manufacturedbythePentekCorporation.Phasemeter s,aswellasmost importantelectroniccomponentsofUFLIS,areallimplemen tedonthissystem. 3.2.1DigitalSignalProcessingHardware ThePentekboardconsistsofthreeindividualelectronicbo ards:Model4205 motherboard,Model6256A/DdaughterboardandModel6228D/A daughterboard. Model4205containsthefollowingprimaryelements: A32-bitcentralmicroprocessor(CPU)witha 1GHz clockfrequency:Likeall CPUs,itcarriesouteachinstructionoftheprograminsequen ceandperformsthe basicarithmetical,logical,andinput/outputoperations ofthesystem. A 1GB synchronousdynamicrandomaccessmemory(SDRAM):Itisusedt o storedatawhilesynchronizedwiththesystem'sPCIbus(mean ingitwaitsfora clocksignalbeforerespondingtocontrolinputs). Four32-bitvelocityinterfacemodules(VIMs):VIMisaninter faceforthedata transferandcommunicationsbetweenthemotherboardandth edaughterboards. EachtwoVIMsareconnectedtoonedaughterboard,withonedata interfaceand onecontrol/statusinterface. Fourbi-directionalrst-inrst-outbuffers(BIFOs):After /Beforethedatais transferedthroughtheVIM,thedatawillrstbewritteninto theBIFObefore memoryaccess.EachBIFOisconnectedtooneVIM.Datatransferc analsobe realizeddirectlybetweendifferentVIMswithouttheproces sofwriting/readinginto thememory. Fourdirectmemoryaccess(DMA)controllers:DMAiscapableo freading/writing datafrom/intotheBIFOwhiledirectlyaccessingthememoryf orreadingand writingindependentlyoftheinterventionfromtheCPU. OneserialandoneEthernetport:Theseportsareforthecommu nicationbetween Model4205andcomputerstoupload/downloadcommandsandl es. 117

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TheModel6256A/DdaughterboardandtheModel6228D/Adaught erboardare connectedtotheModel4205motherboardviaVIMs.Theprimary elementsonModel 6256include: Four14-bitanalog-digitalconverters(ADCs):TheseADCscan beclockedatupto 105MHz .SincetheconnectorsoftheseADCsareAC-coupledwithahighpass lter,theADCscanonlydigitizeanRFsignalwiththecarrier frequencyhigher than 400kHz ,whichisidealforLISA-likeheterodynebeatsignals.Theful lscaleof theADCinputamplitudeis 4dBm TwoXilinxVirtex-2eldprogrammablegatearrays(FPGAs):Thed igitalsignal fromtheADCswillbesenttotheFPGAs,wherethelogicalfunctio nsand operationsindigitalsignalprocessingcanbeimplemented .FPGAcontains programmablelogiccomponentsknownaslogicblocks,andah ierarchyof recongurableinterconnectsthatallowtheblockstobewir edtogether,which allowsustoeasilyconguretheFPGAaccordingtotherequire ddesign.The digitalsignalontheFPGAyieldsxed-pointarithmetic,whi chlimitsthedata precisionwhileimprovesthecalculationefciency. TheprimaryelementsonModel6228include: Four16-bitdigital-analogconverters(DACs):TheseDACsc anbeclockedatupto 500MHz .SimilartotheADCs,theDACsarealsoAC-coupledsuchthatthe ycan onlygenerateRFsignalswiththecarrierfrequencyhighert han 400kHz .Thefull scaleoftheDACoutputamplitudeis 2dBm AXilinxVirtex-2eldprogrammablegatearray(FPGA):Model622 8alsoprovides asimilarFPGAthatcanbeprogrammedforsignalprocessing.T heFPGAcanbe directlyconnectedtotheinputsoftheDACs. Withthesethreemodels,thePentekboardcanrealizeavariet yoffunctionssuch asthedataacquisition,signalmeasurementanddigitallt ering.Inthefollowingsections wewilldiscusstwobasicapplicationsofthePentekboard,w hichhavebeenwidelyused inUFLIS.3.2.2Phasemeter PhasemeteristhemostfundamentalinstrumentinLISA'smeasur ementsystem. Itisessentialfortheprecisemeasurementofheterodynein terferometryphases atphotodiodes,whichistheprimaryquantitiestobemeasur edinIMS.Moreover, phasemetersplayanimportantroleindigitalcontrolloops inwhichthesensordetects 118

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Figure3-2.TheimplementationofthephasemeteronaPentek board.Thephasemeter isessentiallya“frequencymeter”thattracksthefrequenc yuctuationsof theinputsignal. thephase,e.g.,phase-lockingandarm-locking.Therefore ,phasemeterscanbe classiedintotwotypesbasedonthedifferentfunctions.T heformertypefocuses onthephasemeasurementintheLISAband,requiringhighprec isionswithalowdata rate( 3Hz ).Thelattertypefocusesonthephasedetectionindigitalc ontrolswherea highbandwidthandconsequentlyfastdatarate( 100kHz )willbeneeded. AsdiscussedinSection2.3.5,thephaseismeasuredwithrespe cttoaclock referencesuchthattheacquiredphasevalueisalwaysrelat ivetothephasenoiseofthe referencingclock.Therefore,allphasemeasurementsinev itablycarrytheclocknoise. 3.2.2.1Design GiventhatthedesiredsensitivityofIMSisapproximately 18pm = p Hz ,thescience phasemeterneedstobeabletomeasurethearmlengthchangew ithaprecisionof 1pm = p Hz .Therefore,thephasemeasurementwith 1064nm wavelengthisrequired tohaveanaccuracyof 1 cycles = p Hz [ 59 ]. ThephasemeterdevelopedattheUniversityofFloridaadapt sanarchitecture ofdigitalphase-lockingsimilartotheLISAphasemeter.Assh owninFigure 3-2 ,the phasemeterisimplementedontheModel6256A/Ddaughterboar d.Theclockfrequency 119

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tooperatethe14-bitADCis f clock =62.5MHz .Thedigitizedheterodynesignal,which carriesafrequency 0 + ( t ) isdemodulatedbya48-bitnumericalcontroloscillator (NCO)thatisphase-lockedtothemeasuredsignal.TheNCOfr equency m tracks thefrequencyofthemeasuredsignal,anditisgivenbythesu mofa16-bitpreset offsetfrequency o anda48-bittime-varyingfrequencycorrection corr .Thefrequency differencebetweentheNCOandthemeasuredsignalgivesthe PLLerror e .Basedon thephasemetermodelshowninFigure 3-2 ,wehave corr = e G m = o + corr e = 0 + ( t ) m (3–1) where G isthetransferfunctionofthePLLcontrollersatisfyingthe condition G 1 .We solve corr givenby corr = G 1+ G [ ( t )+( 0 o )] ( t )+( 0 o ). (3–2) Therefore,thefrequencyuctuationsofthemeasuredsigna lisfaithfullyreproduced by corr if 0 = o withinthephasemeterbandwidth.Ifthepresetoffsetfrequ encyis notexactlyequalto 0 ,aconstantoffsetwillbeaddedinto corr .Thisfrequencyoffset canbeeasilyremovedinpost-processingofphasemeasureme nts,butmightcausea frequencypullingissueinarmlocking. Thephasemeterisessentiallya“frequencymeter”thatdete ctsthefrequency uctuationsofthebeatsignal.Integratingthefrequency uctuationswillgeneratethe phaseinformation: ( t )= R t 0 ( t ) dt .Thisintegrationcanbedoneinpost-processing. Notethatthephasemeasurementcanalsobeachievedbyinteg ratingtheNCO frequency m .AstheNCOfrequencycontainstheconstantfrequencyoffset ,alinear driftinphasewillshowupintheresult. 120

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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 Frequency (Hz)Phase noise (cycle/rtHz) Chan 1 Chan 2 Residual 1 pm requirement Figure3-3.Thephasemeterperformancemeasuredwithsplit -and-subtractedVCO signals.ThenoiselimitationofthephasemeteristheADCnoi se,whichis contributedbythetemperatureuctuationsforlowRFfrequ enciesand timingjitterforhighRFfrequencies.AlthoughtheADCnoise oorprevents thephasemeterfrommeetingthe 1pm requirement,itdoesnotcausea signicantprobleminourarmlockingexperiments.However ,thenoise suppressionperformanceinourtestsmaybelimitedbytheADC noise, whichwillbeseeninChapter5. ThemainloopofthedigitalPLLisrunningat f clock = 128 488kHz asdownsampled byacascadedintegratingcomb(CIC)lterwithafactorof12 8.Thedatarateofthe frequencyuctuationscanbefurtherdownsampledto 61.0kHz or 14.9Hz ,depending onwhetherthefunctionofthephasemeterdataishigh-bandw idthtrackingorscience measurements.However,inourarmlockingsetupwedirectly usethe 488kHz frequencydatastreamwithoutanyadditionaldownsampling stagetominimizethe systemprocessingdelay(seethenextchapter).3.2.2.2Performance Thephasemeterischaracterizedbytakingmeasurementsofa voltagecontrolled oscillator(VCO)signal,asshowninFigure 3-3 .Inthissimplesetup,the 4MHz VCO signalissplitbyanelectronicsplitterandmeasuredbytwo phasemeters.Thetwo phasemetersareprogrammedonthesameFPGAbutconnectedtod ifferentADCs; 121

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therefore,theyhaveacommonclocknoisebutuncorrelatedAD Cnoises.Figure 3-3 showsthatthenoisespectrumoftheirdifferenceislimited bya 1 = p f ADCnoiseoor, whichisempiricallygivenby ADC ( f )= 3 10 7 p f 0 4MHz cycleHz 1 = 2 (3–3) where 0 isthenominalfrequencyoftheinputsignal. Inaddition,theniteprecisionofthexedintegersinthep hasemetercausesthe digitizationnoiseasfrequencynoisegivenby Dig ( f )= f clock 2 N p 6 f s (3–4) whichisalimited-bandwidthwhitenoiseintermsoffrequen cyuctuations.Forthehigh samplingfrequencyof 61.0kHz ,thedigitizationnoiseofthe48-bitphasemeterisabout 3.67 10 10 HzHz 1 = 2 .Forthelowsamplingfrequencyof 14.9Hz ,thedigitizationnoise isabout 2.35 10 8 HzHz 1 = 2 .Whenconvertedintophaseuctuations,thedigitization noiseoorismarginallybelowthe 1pm requirementcurve. Latestprogressonthephasemeterdevelopmentindicatesth attheADCnoise sourceisprimarilyattributedtothetempature-dependent phasedispersionintheRF transformerforlowRFfrequencies( < 8MHz )andtimingjitterforhighRFfrequencies ( > 8MHz ).Thephasemetersensitivityperformancehasbeensubstan tiallyimproved byreplacingtheRFtransformersandhasalreadymetthe 1pmHz 1 = 2 requirement. Nevertheless,itshouldbenotedthatthephasemetersusedi ncurrentEPDunits(see thenextsection)andarmlockingsetupdescribedinthisdis sertationhavenotadapted thisenhancement.3.2.3ElectronicPhaseDelay AchallengingissueinbenchtopexperimentsofLISAinterfer ometryisthe simulationoftheround-trippropagationbetweenspacecra ft.Thedifcultyofreproducing aLISA-like 33s delaylinecompromisesthevalidityofLISAinterferometrye xperiments 122

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ofTDIorarmlocking.Comparedwithunrealisticshortdelay linesviaverylongcables orbersusedinmostlaboratories[ 89 90 ],onedistinctivefeatureofUFLISisthe emulationofrealisticLISA-likedelaytimesandMHz-rangeDo pplershiftsviaelectronic delay.Suchanelectronicdelaysystembuiltwithhigh-bandw idthphasemetersiscalled anElectronicPhaseDelay(EPD)unit. TheEPDsystemisalsoimplementedonaPentekboard[ 91 92 ]clockedat 62.5MHz .AnEPDunitconsistsofthreemaincomponents,whichareimplem ented onthreepartsofthePentekboardrespectively.TheModel62 56A/Ddaughtercardis programmedwitha48-bitphasemeter. 1 Thephasemetermeasuresthefrequency uctuationofthedigitizedlaserbeatnotewithadatarateo f 61kHz andthensendthe datastreamtothememoryoftheModel4205motherboard.Them otherboardstores thefrequencyinformationinamemorybufferforacertainam ountoftime.Thehigh datarateinthephasemeterensuresthefrequencyinformati onwithinthearmlocking bandwidth( kHz )canbeproperlydelayedbytheEPDsystem.Afterthedelayinthe memorybuffer,thefrequencyinformationissenttotheMode l6228D/Adaughtercard, whereanNCOintegratesthesumofthefrequencyuctuationa ndthefrequency offsettoregeneratethedelayedcopyoftheinputlaserphas e.AftertheNCOoutput, thedigitizedsignalisconvertedbacktoananalogsignalby the 500MHz sampling frequencyD/Aconverter.Duringthisroutine,aMHz-rangeD opplerfrequencycanbe addeddynamicallytothenominalfrequencyofthedelayedsi gnalonthemotherboard. TheEPDunitiscapableofdelayingaMHz-rangesignalbyaslong as 280s .The performanceoftheEPDunitismeasuredusingasimplesetupwhe retheVCOsignalis electronicallysplitandonechannelisdelayedby 1s throughtheEPDunit.Wemeasure 1 Actually,theFPGAisprogrammedwithfouridenticalphasemet ersthatare connectedtofourdifferentADCsinordertooperatefour-cha nneldelayssimultaneously. Herewejustconsideraone-channelcaseforthepurposeofsi mplicity. 123

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10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 Frequency (Hz)Magnitude idealized measured 10 -3 10 -2 10 -1 10 0 10 1 10 -5 10 0 10 5 Frequency (Hz)Phase noise (cycles/rtHz)Linear Spectral Density Prompt phase Delay phase Differential phase (fractional delayed) Figure3-4.(Left)Themagnituderesponseofaninterferome terresponsewith 1s delay. (Right)ThephasenoisespectrumoftheEPDunit. thefrequency/phasenoiseofthedelayedsignalandtheprom ptsignalindividually.The interferometerresponseofaLISAarm,whichalsoyieldsasin glearmlockingsensor,is givenby P 12 ( s )=1 e s = 12 ( s ) 0 ( s ) (3–5) where 12 ( s ) isthedifference 0 ( t ) 0 ( t ) betweenchannel1and2representedin theLaplacedomain. ThemeasurementresultsareshowninFigure 3-4 .Theleftgureshowsthe magnituderesponseof P 12 ( s ) ,whichexhibitsinterferometernullsat n = andan s slope forfrequenciesfarbelow 1 = .TherightgureshowsthenoiseooroftheEPDunit.To evaluatetheadditionalnoiseinducedbytheEPDunit,wetimes hiftandfractionaldelay lterthedelayedphasesuchthatitstimeseriescouldbemax imallysynchronizedto thepromptphaseinpost-processing.ThustheEPDnoiseisgive nbydifferentialphase noisebetweenthetwosignals.ThemeasuredEPDnoiseisessent iallyonanequivalent levelastheADCnoiseinthephasemeter,sincetheprimarycon tributiontotheEPD noiseisfromthephasemeterADCs.Notethattheapparentincr ementoftheEPDnoise atfrequenciesabove 1Hz showninthegureisanartifactfromfractionaldelaylter ing, whileinrealitytheEPDnoisecontinuestodecreasewiththesa metrend. 124

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CHAPTER4 EXPERIMENTALVERIFICATIONOFSINGLEARMLOCKING 4.1Motivation Currently,UFLISistheonlyexistinghardwaresystemthatc ansimulatearmlocking experimentallyunderrealisticconditions,whichismainl yascribedtothelonglighttravel timeprovidedbytheEPDunit.Comparedwithprevioussinglear mlockingexperiments whichwerelimitedto s delaytimesproducedbylongcables[ 89 ]orbers,therst generationofEPD-basedarmlockinghardwaresimulationsbyI raThorpehasachieved signicantprogress[ 93 ].Nevertheless,therstgenerationarmlockingonUFLISis still anextremelysimpliedmodelasitexploitedonlyonesingle armtobuildthesensor anddidnotaddtheDopplershiftaswellasthenoiselimitati onsintothesetup.By summarizingthediscussionsinSection2.3,welistthefollo wingconditionsthatneedto bereectedinarealisticarmlockinghardwaresimulation. Duetothesignicantadvantagesofdual/modieddualarmlo cking,theerror signalofarmlockinghastobeacombinationoftwointerfero metrysignals,which meansweneedtohavetwodelaylines,twophasemetersandama ppingvector implementedonthehardwaresystem. Theround-triptimeoneacharmshouldbearound33secondsan d,ifpossible, time-dependent.Thearmlengthmismatchshouldbegenerate dinarealisticrange (nomorethan 1% ,butalsonotequaltozero). Thephasenoiseintroducedbythephase-lockedcontrollero nthefarspacecraft willcoupleintothesensorsuchthatitcannotbesuppressed bythearmlocking controller.Therefore,thePLLonthefar-endshouldbeimple mentedrealistically ratherthanbeassumedtohaveaninnitegaingivenbyanidea lopticaltransponder. ADopplershiftfrequencyappliedtotheheterodynefrequen cyofthebeatsignal needstobetakenintoaccount.Also,atime-dependentDopple rshiftfrequencyis desirable. BecauseitisimpracticaltomeasureandupdatetheDopplerfr equencyonevery datacycle,aDopplerfrequencyerrorwithinanacceptabler angeshouldbe introduced. Finallythereareotherimportantnoisesourcesweneedtoco nsider,includingthe clocknoiseinthephasemeasurement,thespacecraftjitter andtheshotnoiseat 125

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photodiodes.Thesenoisesourceswilllimitthenoisesuppr essionperformance. Andtheeffectofgravitationalwavesonarmlockingshouldal sobeconsideredin ordertoinvestigatetheimpactofarmlockingongravitatio nalwavedetections. TheultimategoalofarmlockingsimulationsonUFLISistore producetheabove conditions,suppressthelaserfrequencynoiseandincorpo rateitintoaTDIexperiment todemonstratetheprimaryfunctionsofIMS.Amethodicalpr oceduretobuildsuch anultimatearmlockinghardwaresystemistostartwiththeb asicsinglearmlocking loopandtaketherealisticconditionsintoaccountstepwis e.Thusinthischapterwewill describeaseriesofenhancedsinglearmlockingexperiment swhichfeatureDoppler shifts[ 14 ]andincorporationwithatunablereference. Inourexperimentalvericationofsinglearmlocking,weus eaMHzbeatsignal betweentwocavitystabilizedlasersastheinputnoisesour ce.Thisprovidesa frequencynoiseofapproximately 100 200HzHz 1 = 2 atfrequenciesabove 1mHz Thesinglearmlockingsensor,whichisessentiallytheinte rferometerresponseofa singlearm,isanelectronicmodelsameasthetransferfunct ioninFigure 3-4 .The realisticround-triptraveltimeontheLISAarmissimulated viatheEPDunit,where variableDopplerfrequencyshiftscanbeaddedtothenomina lfrequencyofthedelay signal.Thearmlockingcontrollerisimplementedonasecon dPentekboard,wherea phasemeterwillrstmeasuretheerrorsignalwithahighdat arate( 488kHz )andthen sendthedatatoafeedbacklter(Section4.2.2).Thefeedbac ksignalfromthecontroller outputwillbeusedtodrivetheactuator(atunablefrequenc yreference)tocontrol thelaserfrequency.TheactuatorcanbeeitheraPZTactuator totunetheresonant frequencyofthereferencecavity(Section4.3.2),oralocal oscillatorofaheterodyne phase-lockedloop(Section4.3.1). Amongthesecomponents,thesensortransferfunctiondeterm inesthecontroller design,becausetheslopeofthecontrollerlterwillneedt opreservesomeextraphase forfrequenciesabovetherstnullinthesensor.Tosimplif yourexperimentsetup,we setthedelaytimetobeapproximately 1s attheEPDunit,whichisstillmuchlonger 126

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thanthedelaytimesusedinpreviousarmlockingexperiment s.Thebenetisthatinthis caseweonlyneedtobeconcernedwiththelterdesigninthef requencyrangeabove 1Hz .Also,the 1s delaytimeiscomparabletothetypicaldifferentialdelayt imewhen thesecondLISAarmisusedtotestthedual/modieddualarmlo ckingcongurations, wherethesensornullsarelocatedatoraboveafewHz. 4.2PreliminaryTestWithNumericalControlOscillator(NCO) Tracking Beforeweactuallystarttobuildacompleterealisticarmloc kingsystemincorporated withpre-stabilizedlasers,itisreasonabletopreliminar ilyverifythenoisesuppression capabilityandclosed-loopstabilityofabasicsinglearml ockingloop.Suchabasic singlearmlockingloopdoesnotneedtobeincorporatedwith atunablereference, butsimplyreectstheclosed-loopdynamicsofthefeedback controlsystem.Also,the realisticnoiselimitationsaswellasDopplerfrequencyer rorsshouldbeinsignicant enoughsuchthattheyshouldonlycausenegligibleeffectso nthemeasurementresults. Astraightforwardmodeltotestabasicsinglearmlockinglo opisanelectronicmodel withaVCOsignaloralaserbeatsignalbetweentwocavitystab ilizedlasersastheinput noise.4.2.1ExperimentalSetup Thepreliminaryexperimentalsetupisessentiallyasingle armlockingloopfeaturing theEPDunittostabilizethefrequencyofanumericalcontrolo scillator(NCO)toa noisylaserbeatsignal.AsshowninFigure 4-1 ,theNCOsignaldrivenbythecontroller demodulatesthebeatsignalfromthephotodiodesuchthatit willtrytotracktheinput frequencynoisewhentheloopisclosed.Theoutputofthemix er,whichgivesthe residualfrequencynoise,isthensenttothesinglearmlock ingsensorandsplitintotwo parts.TherstpartisdelayedattheEPDunit,whichisimpleme ntedonaPentekboard witha 62.5MHz clockfrequency,byapproximately1secondastheround-tri ptravel time.Duringthedelayprocess,aDopplershiftwithaconsta ntfrequencyisaddedonto thenominalfrequencyofthedelayedsignal.Thisfrequency -shiftedanddelayedsignal 127

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Figure4-1.ThepreliminaryexperimentalsetupwheretheNC Otrackstheinputlaser beatsignalbymeansofsinglearmlocking.Thispurelyelect ronicroutine simulatesthecontrolsystemofsinglearmlocking.Thesing learmlocking sensor(EPD)andcontrollerareimplementedontwodifferentP entekboards synchronizedtoacommonclock.NotethattheDopplerfreque ncyof 3MHz isperfectlydemodulatedinthephasemeasurementofthecon troller. isthenmixedtogetherwiththenon-delayedsecondparttoge neratetheerrorsignalof singlearmlocking.Comparedwiththeheterodyneinterfero metryinrealisticLISA,this purelyelectronicroutinepreciselysimulatesthegenerat ionoftheerrorsignalcoupled withtheDopplershiftfrequency.Duetotheabsenceofthefa r-endPLL,thenominal frequencyofthemixedsignaldoesnotincludetheoffsetfre quencyofthePLL.However, thissimplicationwillnotaffectthepurposeofthispreli minarytest. Subsequently,themixedsignalissenttothecontrollerimpl ementedonasecond Pentekboardwiththesameclockgenerator.Thecontrollerc onsistsofafastdatarate phasemeterwithasamplingfrequencyof 62.5MHz = 128 448kHz ,adigitallter andanNCO.Thephasemeterextractsthefrequencyuctuatio nsoftheerrorsignal bycomparingapresetoffsetfrequencywiththenominalfreq uencyoftheerrorsignal. IfthepresetoffsetfrequencyperfectlymatchesuptheDopp lerfrequency,noDoppler frequencyerrorwillenterthefrequencyuctuationsofthe errorsignal.Afterthedigital 128

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Figure4-2.Theclosed-loopdynamicsofsinglearmlockingw ithNCOtrackinginthe Laplacedomain.Notethatweanalyzemostvariablesintheph aseexceptin thearmlockingcontroller,asthearmlockingltermanipul atesthe frequencyuctuationsratherthanthephase,whichissentf romthe phasemeter.Therefore,thetransferfunctionofthephasem eterisgivenby s representingaconversionfromphasetofrequency.Opposit ely,thetransfer functionoftheNCOis 1 = s ,whichintegratesthefrequencyuctuationsinto phaseuctuationstotracktheinputphasenoise. lter,thelteredfrequencyuctuationswillbeintegrate dbytheNCO,whichgenerates thedrivingsignaltotracktheinputnoise. Todescribetheprincipleofthisexperimentalsetup,wedra wFigure 4-2 that illustratestheclosed-loopdynamicsintheLaplacedomain .Thephasenoisesof theinputsignal,thetrackingNCOsignal,theresidualsign alandtheerrorsignalare representedby o ( s ) NCO ( s ) r ( s ) and e ( s ) ,respectively.Thetransferfunctionof thedigitallterisgivenby G ( s ) .Thephasemeteryieldsaconversionfromphaseto frequencysuchthatithasatransferfunctionof s .Ontheotherside,theNCOyieldsan inverseconversionfromfrequencybacktophasesuchthatit hasatransferfunctionof 1 = s .Therefore,therelationsbetweenthesephasenoisesaresi mplygivenby r ( s )= o ( s ) NCO ( s ), e ( s )= r ( s )(1 e s ), NCO ( s )= e ( s ) G ( s ). (4–1) 129

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Thusweobtainthefamiliardenitionoftheclosed-looptra nsferfunction r ( s ) o ( s ) = 1 1+(1 e s ) G ( s ) (4–2) whichagreeswithEq. 2–32 .Thetransferfunctionindicatesthatourpreliminary experimentalsetupissufcienttodemonstratethenoisesu ppressionandclosed-loop stabilityofthesinglearmlockingconguration.4.2.2DigitalFilterDesign AspreviouslymentionedinSection2.3.2.1,thetransferfunc tionofthesinglearm lockingcontrollerwillneedtoprovideaslopesteeperthan 1 = s atfrequenciesbelow 1 = andaleadcompensatorlesssteeperthan 1 = s atfrequenciesabove 1 = .This characteristicallowsustodesignaPIcontrollerthatconsi stsofacompensatorlter withaslopeof 1 = p s forfrequenciesabove 1 = and4-stageintegratorsforfrequencies below 1 = inparallel.Thecombinationofthecompensatorlterandin tegratorsyields themagnituderesponseofthecontrollershowninFigure 4-3 .Thearmlockingcontroller isalsoimplementedonthe6256ADCboard,receivingtheerror signalviathedirect connectiontothephasemeteroutput. Nowweconsiderthedesignandrealizationofthedigitalcom pensatorlter.The advantageofaninniteimpulseresponse(IIR)lteristhat ithashigherefciency andshorterlatencyduringthecalculationprocess,whichi sidealforthefeedback control.Also,digitalIIRlterscanbefastandaccuratelyc onvertedfromtheiranalog counterpartsinthes-domainbymeansofdiscretizationtec hniquessuchasthebilinear transform.Ontheotherhand,allthepolesofanIIRltermus tbewellinsidetheunit circleinthez-domainotherwisetheopen-loopstabilitywo uldbecompromised. The 1 = p s slopeofanIIRlterinthes-domaincanbeapproximatelyach ieved byplacingzerosandpolesalternativelywithafrequencysp acingratioof10,i.e., polesat 1Hz 10Hz 100Hz1kHz andzerosat 3Hz 30Hz 300Hz3kHz .Toconvert thecoefcientsintothez-domain,weperformthebilineart ransformwithasampling 130

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10 -4 10 -2 10 0 10 2 10 4 -60 -40 -20 0 20 40 60 Frequency (Hz)Magnitude (dB) compensator filter 4-stage integrators overall controller Figure4-3.ThemagnituderesponsesofthePIcontroller:The 4-stageintegrators(red) dominatethefrequenciesbelow 1 = 1Hz toprovidemoresuppression andthecompensatorlterwithaslopeof 1 = p s (blue)dominatesthe frequencybandfrom 1Hz to 10kHz toreducephaseloss.Forfrequencies beyondthebandwidth( > 10kHz )thecompensatorlterattensoutto maintaintheclosed-loopstability. frequencyof 448kHz ,whichisinheritedfromthedatarateofthephasemeteroutp ut. TheIIRlterisimplementedasatwo-stagesecond-order-se ction(SOS)directIIform lter.SincethelterisimplementedontheFPGAwhichyieldst hexed-pointarithmetic, thecoefcientsofthelterinthez-domainaswellastheinp utandoutputvaluesneed tobefurtherquantizedintoxed-pointnumbers,wherethei ssuesrelatedtothelimited precisionandthedynamicrangearise.Inpractice,theseis suesarenormallyresolvable andabalancebetweenthemcanbeobtainedthroughempirical attempts.Inthisdesign, thephasemetersends32-bitfrequencydatatothelterandd uringtheentiredata transferitiskepttobea32-bitxed-pointvaluedowntothe NCO. ThemagnituderesponseoftheimplementedIIRltershownin Figure 4-4 is measuredbydividingthelinearspectraldensitiesofthein putnoiseandtheoutput noise.ItisworthnotingthatunliketheEPDunit,theoutputof thecontrollerlterwillnot bestoredinthememoryandthenreadoutbytheDACboard.Inco ntrast,theoutput signalwillbypassthe4205motherboardanddirectlyreceiv edbytheDACboardviaVIM 131

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10 0 10 2 10 4 -60 -50 -40 -30 -20 -10 0 10 20 Frequency (Hz)Magnitude (dB) measured theoretical Figure4-4.Themagnituderesponseofthe 1 = p s lter,wherethemeasuredtransfer functionisveryconsistentwiththetheoreticaldesignfor frequenciesbelow 200Hz .Thehighfrequencynoise( > 1kHz )observedinthemeasurement isbeyondthearmlockingbandwidth. 10 -2 10 0 10 2 10 4 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) Input frequency noise of laser beat Residual frequency noise of laser-NCO beat 10 -2 10 0 10 2 10 4 10 -4 10 -3 10 -2 10 -1 10 0 10 1 Frequency (Hz)Noise suppression measured theoretical Figure4-5.(Left)Themeasurementresultofnoisesuppress ionperformanceathigh frequenciesshowsthatcomparedwiththeinitialfrequency noisespectrum ofthebeatsignalbetweentwocavitystabilizedlasers(blu e),theresidual frequencynoisespectrumofthelaser-NCObeat(red)issupp ressedbythe compensatorlter.(Right)Theclosed-looptransferfunct ionisgivenbythe ratiobetweentheinitialfrequencynoisespectrumandther esidual frequencynoisespectrum. transfer.Thisfeatureisenabledwhenthemotherboardisse ttotheVIM-to-VIMdata transfermodeinordertominimizethedataprocessingdelay 4.2.3MeasurementResults Afterthearmlockingstartsandentersthesteadystate,weme asurethephase uctuationsoftheinitialbeatsignal o ( s ) andtheresidualsignal r ( s ) usingtwo 132

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0 10 20 30 40 50 60 -400 -300 -200 -100 0 100 200 300 Time (s)Phase (cycles) Initial phase of the laser beat Residual phase of the laser-NCO beat 7 8 9 10 11 12 13 14 15 -10 -8 -6 -4 -2 0 2 4 6 Time (s)Phase (cycles) Figure4-6.(Left)A 1min timeseriesofphaseuctuationsoftheinitialbeatsignal (blue)andtheresiduallaser-NCObeat(red)withafastsamp lingfrequency of 61kHz .(Right)Aclose-upoftheresiduallaser-NCObeat.Thetime series haveadistinctoscillationatabout 1Hz ,aswellashigherharmonicsonthe topofit. phasemetersindividually.Herewehavetakentwokindsofph asemeasurementswith twodifferentphasemeterdatarates.Thephasemeterrstse ndsdataatthehighrateof 61.0kHz for 60s inordertoinvestigatethenoisesuppressionandthesystem bandwidth athighfrequencies.Thelinearspectraldensitiesofthefr equencynoisesareplottedin Figure 4-5 (left),whichshowsamoderatenoisesuppressionduetothec ompensator lterforfrequenciesabove 1Hz .Theratiobetweenthemyieldsthemagnitudeofthe closed-looptransferfunction,whichdisplaysmultiplepe aksatfrequenciesverycloseto 1Hz 2Hz 3Hz ,etc.Thedeviationfromthetheoreticaltransferfunction mainlyliesat frequenciesabove 500Hz ,whereaservobumpshowsupatapproximately 1kHz .The servodumpisascribedtotheexcessivephaseshiftduetothe propagationdelayindata transfer.Figure 4-6 plotsthetimeseriesofthephaseuctuationswithinthisti mespan. Notethatinthisplotthelineardriftduetoaconstantfrequ encyvalueisalreadyremoved fromthephasedata.Figure 4-6 (right)istheclose-upoftheresidualphaseuctuations, whichdistinctlyexhibitsanoscillationwithafrequencyv erycloseto 1Hz ,aswellas otherhigherharmonicsappliedonit. Inthesecondsetofphasemeasurements,thephasemeterruns atthelowdatarate of 14.9Hz for15hourstoinvestigatethenoisesuppressionperforman ceintheLISA 133

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10 -4 10 -2 10 0 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) Input frequency noise of laser beat Residual frequency noise of laser-NCO beat 32-bit digitization noise floor 10 -4 10 -2 10 0 10 -6 10 -4 10 -2 10 0 10 2 Frequency (Hz)Noise suppression measured idealized Figure4-7.(Left)Themeasurementresultofnoisesuppress ionperformanceatlow frequenciesshowsthatcomparedwiththeinitialfrequency noisespectrum ofthebeatsignalbetweentwocavitystabilizedlasers(blu e),theresidual frequencynoisespectrumofthelaser-NCObeat(red)issupp ressedbythe 4-stageintegratorsby5to6ordersofmagnitudefrom 0.1mHz to 10mHz (Right)Theclosed-looptransferfunctionisgivenbythera tiobetweenthe initialfrequencynoisespectrumandtheresidualfrequenc ynoisespectrum. Themeasurementhasshownthatthecontrolsystemofthissin glearm lockingloopisconsistentwiththetheoreticaldesign. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 6 Time (s)Phase fluctuations (cycle) Residual phase of the laser-NCO beat Initial phase of the laser beat 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 -6 -4 -2 0 2 4 6 Time (s)Phase fluctuations (cycle) Figure4-8.(Left)A 12.5h timeseriesofphaseuctuationsoftheinitialbeatsignal (blue)andtheresiduallaser-NCObeat(red)withaslowsamp lingfrequency of 14.9Hz .Inthistimespantheinitialphaseuctuatesbymorethan 3.5 10 6 cycles .(Right)Aclose-upoftheresiduallaser-NCObeat.The residualphaseuctuatesbyabout 10cycles andthegeneralprole resemblestheinitialphaseuctuations,indicatingthatt henoisesuppression atlowfrequenciesislimited. 134

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Figure4-9.SinglearmlockingwithNCOtracking-Laplacedom ainwithnoisesources. Thissinglearmlockingloophasfourprimaryerrorinputpoi nts.Notethat thenoisesourcesintroducedoutsidethearmlockingcontro lleraretreated asthephasenoiseandotherwisethefrequencynoise,inacco rdancewith theconventionusedinFigure 4-2 band.Thelinearspectraldensitiesoftheinitialfrequenc ynoiseandresidualfrequency noise,andthetimeseriesofthephaseuctuationsareshown inFigure 4-7 (left)and Figure 4-8 ,respectively.Theratiobetweenthetwospectrayieldsthe magnitudeof theclosed-looptransferfunction,whichindicates5to6or dersofmagnitudenoise suppressionfrom 0.1mHz to 10mHz asshowninFigure 4-7 (Right). Theclosed-looptransferfunctionshowninFigure 4-7 (Right)yieldsa s 3 slopefrom 10mHz to 0.1Hz duetothe4-stageintegratorsinthecontroller.Themeasur ement resultagreeswellwiththetheoreticaldesign,whichindic atesthatthenoisesuppression performanceisgainlimitedinthisfrequencyregion.Howev er,forfrequenciesbelow 10mHz theresultshowsanobviousdeviationastheclosed-looptra nsferfunction attensoutratherthancontinuedecreasingwitha s 3 slope.Thisbehaviorindicates thatinthisfrequencyregionthesuppressionperformancei slimitedbyadifferentnoise source.AsshowninFigure 4-7 (Left),thedeviationintheclosed-looptransferfunction iscausedbythe 1 = s slopeintheresidualfrequencynoisespectrum. 135

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4.2.4NoiseAnalysis Toanalyzethenoiselimitsanddeterminethedominatingnoi seoorinthesetup, weconsiderpossiblenoisesourcesinFigure 4-2 andthendrawFigure 4-9 .Thesenoise sourcesincludethe48-bitdigitizationnoise 48 dig andtheADCjitter ADC introduced attheEPDunitaswellasthearmlockingphasemeter,whicharer epresentedasphase noisesinthegure;anyDopplerfrequencymismatchintrodu cedatthearmlocking phasemeter,whichisalsorepresentedasaphasenoise D t ;the32-bitdigitization noise f 32 dig atthefeedbacklter,whichisrepresentedasafrequencyno ise;aswellas frequencynoise f NCO introducedbytheNCO. Amongthesenoisesources,itisobviousthatanyNCOnoise f NCO willbe suppressedbytheopen-loopgain.SincetheDopplershiftfre quencyisdesigned tobeperfectlyaccountedforbythephasemeterpresetfrequ ency,thephasenoise representingtheDopplerfrequencyerroralsovanishes.Th e48-bitdigitizationnoiseand theADCjitteraddedattheEPDunitwillbetrackedbytheresidua lsignal;however,they areatafairlylowlevelcomparedwiththe 1 = s slope.Actually,thecalculationindicates thatthe 1 = s slopeisaneffectofthe32-bitdigitizationnoiseaddedint hefeedbacklter, wherethe32-bitfrequencyuctuationissampledat 488kHz .Thuswedeterminethe digitizationnoise N Dig = f clock 2 N p 6 f s =8.50 10 6 HzHz 1 = 2 (4–3) Intermsoffrequencynoisethedigitizationnoiseiswhitet hroughtheentireband. Asthisfrequencynoisebecomesanadditionaltermintheerro rsignaltransferedfrom thephasemetertothelter,ithasanequivalenteffectasth eDopplerfrequencyerror enteringandbeingaccumulatedbytheclosed-loop.Therefo reinthepresenceofa DC-coupledcontroller,thedigitizationnoisemultiplied by 1 = s willappearinthespectrum oftheresidualfrequencynoiseandthelevelofthenoiseoo rscaleswith 1 = : Dig ( f )= N dig s = 1.35 10 6 f HzHz 1 = 2 (4–4) 136

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10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) free-running VCO frequency noise residual VCO-NCO beat frequency noise 30-bit digitization noise floor 32-bit digitization noise floor Figure4-10.Thedashedgreennoiseoorisgivenbythe30-bi tdigitizationnoise coupledintothesinglearmlockingloop.Comparedwiththe3 2-bit digitizationnoiseoor(dashedblack),thedashedgreeno orishigherbya factorof4sincetheprecisiongivenbythelowesttwobitsis lostinthe frequencyinformation.Themeasurementhasshownthatthen oise suppressionislimitedbythenewnoiseoorwhenthe30-bita rmlocking controllerisused. AsshowninFigure 4-7 ,thisnoiseooraccuratelymatchesupthe 1 = s slopein theresidualfrequencyspectrum.Thisnoiseoorcanfurthe rbeveriedbychanging thedelaytimeorchangingthexed-pointprecisionofthefr equencyuctuations.To demonstratetheinuenceoftheprecisiontothelevelofthe noiseoor,wetruncate the32-bitfrequencyuctuationinthelterto30-bitbyhar dcodingthevaluesofthe lowesttwobitstozero.WetakethemeasurementusingafreerunningVCOasthe inputnoiseandtheresultofthe30-bitlterisshowninFigu re 4-10 .Theresultexplicitly demonstratesthatthemeasured30-bitdigitizationnoise ooragreeswiththetheoretical predictionthatthenewnoiseoorincreasesbyafactorof4d uetothe2-bitprecision lossinthelter. Sincethedigitizationnoiseinthefeedbacklterisequival enttoaDoppler frequencyerroratthephasemeter,thedirectconsequenceo fthis 1 = s noiseoorin thetimedomainisalineardriftintheoutputfrequency.Her eahigherdigitizationnoise 137

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0 500 1000 1500 2000 2500 3000 3500 -20 0 20 40 60 80 100 120 Time (s)Frequency drift (Hz) filter with 32-bit frequency fluctuation filter with 30-bit frequency fluctuation Figure4-11.Comparisonoffrequencydriftratebetweena30 -bitlter( 28.6mHz = s )and a32-bitlter( 7.5mHz = s ).Thefrequencydriftrateofa30-bitlterislarger byafactorofabout4,scalingwiththermsamplitudeofthedi gitization noise. oorcorrespondstoafasterfrequencydrift,asillustrate dinFigure 4-11 .Thefrequency datainthisgurehasbeenlow-passlteredbyaveragingout thehighfrequencynoise (e.g.,usingaboxcarlter)suchthatthelineardriftcanbe explicitlyrevealed.The linearregressionapproachshowsthatforthe32-bitltert hefrequencydriftrateis approximately 7.5mHz = s andforthe30-bitlteritisapproximately 28.6mHz = s ,which isagainroughlyfourtimesthedriftrateofthe32-bitcase. GiventhatinrealisticLISA thelighttraveltimeononearmisapproximately33timeslar gerthanthe 1s delaytime inourexperiment,the32-bitdigitizationnoiseshouldnot havesignicanteffectsonthe noisesuppressionperformanceorfrequencypullingifsing learmlockingisused. 4.3SingleArmLockingIntegratedwithaTunableReference Inthissectionwemoveforwardtotheauthenticelectro-opt icalmodelswheresingle armlockingisusedtostabilizethelaserfrequency.Theinc orporationofarmlockingand pre-stabilizationsubsystemrequiresanadditionalactua tortotunethepre-stabilization reference.Thespecicapproacheshavebeenalreadyintrod ucedinSection2.3.4and sofar,allproposedmethodstointegratesinglearmlocking withPound-Drever-Hall techniquehavebeenexperimentallydemonstratedonUFLIS. Insteadofthepreliminary 138

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vericationofthenoisesuppressionperformanceandclose d-loopstabilityinthe previouselectronicmodel,wewillfocusonthecharacteriz ationofthetunablereferences andadditionalnoisesourcesintroducedbytheminthesemor erealisticsinglearm lockingdemonstrations.4.3.1HeterodynePhase-lockedLoop Theactuationmethodofheterodynephase-lockingrequires anauxiliarylaser tofaithfullyreproducethelaserfrequencynoiseofthecav itystabilizedlaser.For thisreasonitisnotaconciseoptionforrealisticLISA,yetit isasimpleapproachto demonstrateinbenchtopexperiments.4.3.1.1Experimentalsetup AsdescribedinFigure 2-27 ,theconceptofPLL-basedsinglearmlockingisto stabilizeanauxiliarylaserwhichisphase-lockedtotheca vitystabilizedlasersuch thatthelocaloscillatorofthePLLcanbetunedbythearmlock ingcontroller.The bench-topexperimentalsetupofthePLL-basedsinglearmloc kingresemblesthe setupinFigure 2-27 exceptthatweuseanothercavitystabilizedlaser RL togenerate abeatsignalagainstthephase-lockedlaser.Figure 4-12 illustratestheexperimental setupinwhichthereferencelaser RL andthemasterlaser L 1 arecavitystabilizedvia Pound-Drever-Halltechnique.Theslavelaser L 2 isphase-lockedto L 1 withafrequency offset,whichisdrivenbytheNCOinthearmlockingcontroll er.Therefore,thestability ofthecavitystabilizedlaser L 1 istransferedinto L 2 throughphase-lockingandthe frequencynoiseinthe RL L 2 beatsignalisapproximatelyequaltothefrequencynoise inthebeatsignal RL L 1 .Also,duetotheheterodynePLLthestabilityofthebeat signal RL L 2 istunable,whichmeansitcanbefurtherstabilizedbyamore stable reference:thearmlength(i.e.,thexeddelaytimeinourex periment).Comparedwith thepreliminarytestwithNCOtrackingtheinputnoise,thef requencyofthelaserbeat signal RL L 2 isdirectlysuppressedbythearmlockingopen-loopgain. 139

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Figure4-12.Theexperimentalsetupofsinglearmlockingex perimentusingan additionalheterodynephase-lockedlaser.Inthissetupth ereferencelaser RL andthemasterlaser L 1 arecavitystabilizedviaPound-Drever-Hall technique.Theslavelaser L 2 isphase-lockedto L 1 withafrequencyoffset, whichisdrivenbytheNCOinthearmlockingcontroller.Ther efore,within thePLLbandwidth L 2 faithfullyreproducesthelaserfrequencynoiseof L 1 bothreferencedtothe“opticalclock” RL .NotethatthePLLbandwidthis about 20kHz ,whichiswelllargerthanthearmlockingbandwidth ( 1kHz );thereforeitwillnotlimitourarmlockingperformancean dadirect feedbackofthearmlockingcontrolsignaltothelaserisnot necessary.The armlockingcontrollerisusedtoadjusttheNCOthatdrivest hePLLandthe frequencynoiseof L 2 canbefurthersuppressedbythearmlocking controllergain. Inourexperimentthenominalfrequencyofthe RL L 1 beatsignalisapproximately 13MHz .IfwesetthenominalfrequencyoftheNCOsignaltobe 8MHz andphase-lock L 2 to L 1 ,therewillbetwopossibilitiesforthenominalfrequencyo fthethe RL L 2 beat signal: 5MHz and 21MHz ,dependingonwhichvalueislargerbetweenthenominal frequenciesof L 1 and L 2 .Onlyoneofthetwobeatfrequencyvaluesisworkingfor thesetupastheothercongurationwouldyieldawrongsign. Inoursetuptheright congurationisa 5MHz RL L 2 beatsignal.TheEPDunitisconguredinthesame wayasfortheNCOtrackingsetup,providinga 1s delaytimeand 2MHz Dopplershift. 140

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Figure4-13.Theclosed-loopdynamicsofsinglearmlocking tunedbyaheterodynePLL intheLaplacedomain.Thecontrolsystemisbasedontwoloop s:PLLand singlearmlocking.Thestabilityofthearmlockingreferen ceistransferred intotheerrorsignalofthePLLviathedrivingNCO. 4.3.1.2Closed-loopdynamics Figure 4-13 illustratestheexperimentalsetuprepresentedintheLapl acedomain. Inthisdiagramweuse i torepresentthephaseofthelaser L i (Thelaserphaseofa free-running L 2 isgivenby 02 andthein-loopphaseofthephase-locked(andthenarm locked) L 2 isgivenby 2 .)and torepresentthephaseofthePLLfeedback,thephase oftheNCOsignal,etc.Notethatinthisdiagramthecongura tionoftheplusandminus signsassumesthenominalfrequenciesofthethreelasersyi eld n RL > n L 2 > n L 1 Notallpermutationsofthethreefrequenciesareallowedfo rthesystemstabilityand differentpermutationswillresultindifferentcongurat ionoftheplusandminussigns. Ifweconsiderthephase-lockedloop,therelationsaregive nby 2 = 02 PLL PLL =( 2 1 NCO ) G PLL (4–5) Hencewesolvethein-loopphaseof L 2 givenby 2 = 1 1+ G PLL 02 + G PLL 1+ G PLL ( 1 + NCO ). (4–6) 141

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Thersttermrepresentstheinitialnoisesuppressedbythe open-loopgainand thesecondtermrepresentsthereferencenoisethatthisPLLt racks,i.e.,theslavelaser L 2 tracksthemasterlaser L 1 .Aphasemodulation NCO fromtheNCOsignalisalso coupledintothePLLreference.Thetunabilityof NCO allowsthephaseof L 2 canbe furtherstabilized. Inthearmlockingloop,theNCOnoiseisgivenby NCO =( 0 2 )(1 e s ) G AL (4–7) SubstituteEq. 4–7 intoEq. 4–6 andwehave 2 = 1 1+ G PLL 02 + G PLL 1+ G PLL ( 1 +( 0 2 )(1 e s ) G AL ) = 1 1+ G PLL 02 + G PLL 1+ G PLL 1 + ( 0 2 )(1 e s ) 1+ G PLL G PLL G AL (4–8) Wecombinethetermsinvolving 2 totheleftandaddterms 0 1 1+ G PLL 0 G PLL 1+ G PLL 0 (=0) totheright: 1+ G PLL 1+ G PLL (1 e s ) G AL 2 = (4–9) 1 1+ G PLL 02 + G PLL 1+ G PLL 1 + 0 (1 e s ) 1+ G PLL G PLL G AL + 0 1 1+ G PLL 0 G PLL 1+ G PLL 0 (4–10) Thepurposeoftheseadditionaltermsissuchthattheabovee quationcanbe simpliedinto 1+ G PLL 1+ G PLL (1 e s ) G AL ( 2 0 )= 1 1+ G PLL ( 02 0 )+ G PLL 1+ G PLL ( 1 0 ), (4–11) where i 0 i =1,2 isthephaseofthebeatsignal,i.e.,thephaseof L 1,2 relativeto thereferencelaser RL .Fromthisequationweobtaintheequivalentopen-looptran sfer function TF OL = G PLL 1+ G PLL (1 e s ) G AL (4–12) 142

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Thersttermontherightindicatesthatthephasenoiseofaf ree-running L 2 relative tothereferencelaserisdoublesuppressedbytheopen-loop gainofphaselockingas wellasthatofarmlocking.Thesecondtermontheright,whic hwastrackedinphase locking,isnowsuppressedbytheopen-loopgainofarmlocki ng.Inthehighgainlimit ( G PLL 1 )ofthePLL, 1 1+ G PLL 0 and G PLL 1+ G PLL 1 ,thenEq. 4–12 approximatestothe open-looptransferfunctioninEq. 4–2 andEq. 4–11 isfurtherreducedinto 1+(1 e s ) G AL ( 2 0 )= 1 0 (4–13) Finally,theclosed-looptransferfunctionisgivenby TF CL = 1 1+(1 e s ) G AL = 2 0 1 0 (4–14) Thisrelationindicatesthephasenoiseof RL L 2 beatsignalissuppressedfrom thenoiselevelof RL L 1 bythearmlockingopen-loopgainandtheclosed-looptransf er functionoftheentiresystemcanbeapproximatelygivenbyt heratiobetweenthem.In otherwords,thePLL-basedsinglearmlockingsetupismathem aticallyequivalenttothe preliminaryexperimentofNCOtrackinginthepresenceofan idealPLL. 4.3.1.3Resultsandanalysis Wemeasurethefrequencynoisesofthe RL L 1 beatsignalandthe RL L 2 beatsignalastheinputnoiseandtheoutputnoise,respecti vely.Theirnoisespectra inthelowfrequencyregionareshowninFigure 4-14 .Asseeninthegure,thenoise spectrumofthe RL L 2 isalmostidenticaltotheresidualfrequencynoiseshownin Figure 4-7 (left),exceptforsomedifferenceintheoverallloopgain. Themeasured closed-looptransferfunction,whichisgivenbytheratiob etweenthetwospectra, indicatesthatthe RL L 2 beatsignalhasbeenstabilizedwith5to6ordersofmagnitud e. Inthelowfrequencyregionthenoisesuppressionisagainli mitedbythesame32-bit digitizationnoiseoordescribedinSection4.2.4. 143

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)Relative frequency noise (Hz/rtHz) Frequency noise of RL-L1 beat signal Frequency noise of RL-L2 beat signal 32-bit digitization noise floor 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 Linear Spectral Density Frequency (Hz)Noise suppression measured idealized Figure4-14.(Left)Thenoisespectraoftheinitial RL L 1 beatsignal(blue)andthe stabilized RL L 2 beatsignal(red).Thenoisesuppressionperformanceis almostidenticaltotheresultgivenbyFigure 4-7 .(Right)Theclosed-loop transferfunctiongivenbytheratiobetweenthetwoLSDsisal sothesame asthepreviousexperiment.Thedeviationfromthetheoreti caldesignagain comesfromthe32-bitdigitizationnoiseoor. Inadditiontothe32-bitdigitizationnoiseandADCnoisealr eadymentionedin Section4.2.4,thephase-lockedloopinthissetupalsointro ducesspuriousphase variations PLL intotheerrorsignal.Also,theuncertaintyinthelightpath lengthfrom thelasertothephotodioderesultsintheuncertaintyinthe laserphase;therefore,the L 1 L 2 beatsignalandthe RL L 2 beatsignalwillhaveuncorrelatedlightpathnoises LP12 and LP02 attributedtothedifferenceinthelightpathofeachlaser. Weneedto investigatetheinuenceofthesenoisesourcestothemeasu rementresultaswell.The noisemodelintheLaplacedomainisdrawnasFigure 4-15 Itisstraightforwardtoshowthatthephaseof L 2 alsotracksthePLLnoise PLL in thePLL: 2 = 1 1+ G PLL 02 + G PLL 1+ G PLL ( 1 + NCO + PLL ), (4–15) whichindicatesthatthePLLnoiseissimplycoupledintothep hasenoiseof L 1 Althoughitincreasesthenoiselevelof L 2 fromthepointofviewofthephase-locked loop,thephasenoiseofthe RL L 1 beatsignal,includingthisPLLnoise,iswell suppressedbythearmlockingopen-loopgain.Thisdemonstr atesthatanynoise 144

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Figure4-15.SinglearmlockingwithheterodynePLL-Laplaced omainwithnoise sources.InadditiontothenoisesourcesanalyzedinFigure 4-9 ,the heterodynePLLactuatoralsointroducesthePLLnoise PLL ,aswellas thelightpathnoise LP12 and LP02 introducedbythePLLwillbeadequatelysuppressedandunlik elycausealimitationin thearmlockingperformance. Thesamesituationisforthelightpathnoise LP12 and LP02 ,whichiscoupledinto the L 1 L 2 beatsignaland RL L 2 beatsignalindividually.Theexistenceofthelight noise LP12 and LP02 isnothingmorethananadditionaltermaddedin PLL and NCO i.e., PLL =( 2 1 + LP12 NCO ) G PLL NCO =( 0 2 + LP02 )(1 e s ) G AL (4–16) Inotherwords,theycanbeconsideredasbeingcoupledintot hephasenoise 1 and 0 ,respectively. LP12 ,asasmallportionof 1 ,willagainbesuppressedbythe armlockingopen-loopgain.However, LP02 coupledinthereferencelaserwillshowup inallthebeatsignalsandthenincreasesthestabilizednoi seof RL L 2 beatsignalby aninsignicantamount. 145

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4.3.2PiezoelectricTransducer(PZT)ActuatedCavity ThePZTactuatedcavityisastraightforwardconcepttoconti nuouslytunethe frequencyofacavitystabilizedlaserbyactivelymodulati ngthecavitylength.The advantageofaPZTcavityisthatthemodicationtotheoptica lcavitydoesnot signicantlycomplicatetheimplementationwhilestillco uldachieveconsiderabletuning range.AlthoughthePZTactuatorwillinevitablycompromiset hereferencestabilityof pre-stabilizationbyprobablyoneorderofmagnitude,thei ncorporationwitharmlocking iscapableofsuppressingthenoisesignicantlywellbelow therequirement. 4.3.2.1CharacterizationofthePZTcavity InaPZTactuatedcavity,aPZTactuatorisusedtochangethedis tancebetween thetwomirrorsformingthereferencecavity.Inourexperim entthePZTactuatoris placedbetweenthefusedsilicamirrorandtheZerodurspace rofthecavityusing hydroxidebonding.ThisadditionalPZTactuatorcompromise sthestabilityofthecavity length:Measurementshaveshownthatthefrequencynoiseof thebeatsignalbetween twolasers,whicharelockedtoastandardZerodurcavityand aPZTcavityindividually, isapproximatelyonthelevelof kHzHz 1 = 2 at 1mHz ,withanappliedvoltagefrom 0V to 100V .Withthevoltageincreasedfrom 100V to 200V ,thefrequencynoisewillalso increaseslightly.Thetypicalfrequencydriftrateoftheb eatsignalhasbeenobservedto bewithintherangeof 50 350Hz = s 1 ThefeedbacksignalarrivingatthePZTactuatorhastobeaDCs ignal.Since theNCOinthecontrollerisAC-coupled,weprogramanotherN COgeneratinga free-runningsignalwiththesamenominalfrequencyandthe nuseananalogmixerto generateaDCoutputbymeansofdemodulation.TheDC-couple derrorsignalisused todrivethePZTactuatedcavity.Whenthearmlockingloopiscl osed,thefrequencyof 1 ThehydroxidebondingofthePZTcavityaswellasthecharacte rization measurementsdescribedinthisparagrapharedonebyAlixPres ton. 146

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theerrorsignalisclosetoDC.Inthisregionasmallchangei nphasecanbeconsidered linearlyproportionaltoasmallchangeinthevoltageappli edonthePZTactuator,i.e., dV = Kd ,where K istheconversioncoefcientdeterminedbytheanalogmixer .Using thefullrangeoftheDAC,theconversioncoefcientwasfoun dtobeapproximately 3.2mV = degree Consequently,thePZTactuatorconvertsthechangeinvoltag eintoachangeinthe cavitylength.Thisconversionisalsoalmostlinearatfreq uenciesbelow 20kHz .The cavitylengthchangethenresultsinthechangeoftheresona ntfrequency.Asweknow, therelationbetweentheresonantfrequency n L ofareferencecavityanditsopticalpath length L isgivenby n L = 2 nc L (4–17) Bytakingtherst-orderderivativeonbothsides,weobtaint herelationbetweenthe changeintheresonantfrequencyandthechangeintheoptica llength: d n L = 2 nc L 2 dL (4–18) Thelinearconversionfromtheappliedvoltagetothechange intheresonant frequencycanbedescribedusingatuningcoefcient,which ismeasuredtobe 1.5MHz = V .AspreviouslymentionedinSection2.3.4,thetuningrangefo rthetunable referenceisrequiredtobeatleasttensofMHz.Ifthedesire dtuningcapabilityis designedtobe 100MHz ,theappliedvoltageisthenrequiredtohavearangeof 0 70V ,whichgenerallyrequiresalow-noiseDCampliertoamplif ythefeedbacksignal. IfwetaketheconversionsstartingfromtheNCOoutputtothe resonantfrequency changeasawhole,itisessentiallyaconversionfromthepha sechangetothe frequencychange.Thisconversionresemblesthefrequency actuatorinlasercontrollers, whichactuatethephasechangefromthecontrollersignalin tothelaserfrequency change.Sinceafrequencyactuatorisequivalenttoaphaseac tuatorwitha 1 = s factor, 147

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Figure4-16.Theexperimentalsetupofsinglearmlockingex perimentusingaPZT actuatedcavity.Inthissetupthereferencelaser RL islockedtoastandard cavityandthemainlaser L 1 islockedtothePZTcavity.Theirbeatsignalis rstdemodulatedbyafunctiongeneratorsignaltothedesir edfrequency rangeandthensentintothecontrolsystemofsinglearmlock ing.Asthe PZTactuatorneedsthefeedbacksignalatDC,asecondfree-ru nningNCO isusedtodemodulatetheNCOsignalthatisadjustedbythear mlocking controller.Notethatthecontrollerlterismultipliedwi th s inthetransfer functiontocompensatethephase-to-frequencyconversion inthePZT actuator. thePZTactuatedcavitycontainsanintrinsic 1 = s integratorthatmustbetakeninto accountinclosed-loopdynamics.4.3.2.2Experimentalsetup TheexperimentalvericationofPZT-actuatedsinglearmloc kingisillustrated byFigure 4-16 .Inthissetupthemainlaser L 1 isstabilizedtoaPZTactuatedcavity viaPound-Drever-Halltechnique.Anotherpre-stabilizedr eferencelaser RL ,whichis lockedtoastandardZerodurcavitywithaxedlength,isuse dtogeneratea 81MHz heterodynebeatsignalwith L 1 atthephotodiode.Thisbeatsignalisdemodulatedby a 72MHz functiongeneratorsignalto 9MHz ,whichissentintothearmlockingsensor with 1s delayandthenappropriatelylteredbythecontroller.Asme ntionedinthe 148

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Figure4-17.Theclosed-loopdynamicsofsinglearmlocking tunedbyaPZTactuated cavityintheLaplacedomain.Theclosed-loopdynamicsofth is experimentalsetupissimpleasthearmlockingfeedbacksig nalisdirectly sentbacktothelaserviathePZTactuator.ComparedwithFigu re 4-2 ,the maindifferenceistheadditionalgainstagesandtheintrin sicintegratorin thePZTactuation. previoussection,thecontrollerisconguredtohavetwopa rallelNCOsrunningatthe same 5MHz nominalfrequency.ThedifferencebetweenthemisthatoneN COreceives thelteredfrequencyuctuationsfromthearmlockingerro rsignal,whiletheother NCOisfree-runningtodemodulatetherstNCOsignal.Theou tputoftheanalogmixer carriestheinformationofthephaseuctuations,whichisa lreadyconvertedintoaDC voltage.Afterappropriatelyampliedto 70V ,thevoltageisappliedontothePZT actuatortomodulatetheopticallengthofthecavity.Themo dulationonthecavitylength isthenusedtocorrecttheresonantfrequencythat L 1 islockedto. Duetotheintrinsic 1 = s integratorinthePZTactuator,thecontrollerlterneedsto bemodiedbymultiplyingan s slopetomaintaintheclosed-loopstability.Thecontrolle r lterinthissetupisdesignedtohavea p s slopetoprovide 45 phaseadvance,which canberealizedbyswitchingthefrequenciesofzerosandpol esinthe 1 = p s lter. TheexperimentmodelintheLaplacedomainisshowninFigure 4-17 .Inthismodel thecontrolleroutputisfollowedbyaseriesofgainstages, where K 0 representsthe 149

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ampliergain, K 1 representstheconversionfromthephasetothevoltageatth eanalog mixer, K 2 representstheconversionfromthevoltagetotheresonantf requencyatthe PZTactuator.Toproperlyclosetheloop,another 1 = s stageisrequiredtoconvertthe frequencybacktothephase.Therefore,therelationstodes cribethismodelaregivenby 1 = 01 PZT PZT =( 1 0 )(1 e s ) K 0 K 1 K 2 G s (4–19) Ifwedenetheopen-looptransferfunction TF OL =(1 e s ) K 0 K 1 K 2 G s (4–20) Thestabilizedphasenoiseof L 1 isthengivenby 1 = 1 1+ TF OL 01 + TF OL 1+ TF OL 0 (4–21) Sincewearemoreconcernedwiththephasenoiseofthe RL L 1 beatsignal, Eq. 4–21 isequivalentto TF CL = 1 0 01 0 = 1 1+ TF OL (4–22) 4.3.2.3Resultsandanalysis ThemeasurementresultsofthefrequencynoiseareshowninF igure 4-18 Accordingtothenoisespectra,althoughthefrequencynoise ofthebeatsignal stabilizedbyafree-runningPZTcavitydoesnotmeetthestri ngent 30HzHz 1 = 2 pre-stabilizationrequirement,singlearmlockingiscapa bleofappropriatelytuning thePZTcavitysuchthatthenoisespectrumissignicantlyre ducedby6or7orders ofmagnitude.Notethateveninthegainlimitedregion,thei nitialfrequencynoise andthestabilizedfrequencynoisearenotcorrelatedsince theywerenotmeasured simultaneously.Nevertheless,wemodeledtheopen-looptr ansferfunctionofthesetup givenbyEq. 4–20 andtheidealizedstabilizednoisespectrum,representedb ythegreen 150

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10 -3 10 -2 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) Frequency noise of free-running PZT cavity Frequency noise of arm locking stabilized PZT cavity 32-bit digitization noise floor Idealized frequency noise under arm locking LISA pre-stabilization requirement Figure4-18.Noisespectraofsinglearmlockingexperiment usingaPZTactuated cavity.Theinitialnoisespectrum(blue)isapproximately 200HzHz 1 = 2 and itwasnotmeasuredsimultaneouslywiththestabilizednois espectrum (red),whichisagainlimitedbythe32-bitdigitizationnoi seoor.Thegreen curveyieldstheidealizedfrequencynoiseinthepresenceo fthe closed-loopgain,ignoringthenoiselimitations.Thereda ndgreencurves areconsistentathighfrequencies(gainlimitedregion)an dthedeviation comesinbelow 0.1Hz curve,isevaluatedbytheinitialfrequencynoisemultipli edbythemagnitudeofthe modeledclosed-looptransferfunction.Thefrequencyregi onwheretheidealizednoise spectrumagreeswiththemeasuredoneisobviouslygainlimi ted,whilethedeviation indicatesthenoiselimitedregionwhichisdominatedbythe digitizationnoiseoor. Figure 4-19 illustratesthenoisesourcesgeneratedinthesetup.Theno isesources thatareintroducedaftertheNCOoutputcanbevariousandco mplicated,includingthe electronicnoiseintheamplier,thermalnoiseinthePZTact uatorandintheoptical cavity,etc.However,wecanroughlygeneralizethesevario usnoisesasaphase uctuation PZT introducedintotheNCOoutput.Thentheclosed-looptransf erfunction fromthiserrorpointtotheoutputnoise 1 isgivenby TF PZT ( s )= K 0 K 1 K 2 = s 1+ K 0 K 1 K 2 (1 e s ) G ( s ) = s (4–23) 151

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Figure4-19.SinglearmlockingwithaPZTactuatedcavity-Lap lacedomainwithnoise sources.ComparedwithFigure 4-9 ,thePZTactuationintroduces additionalnoisesourcessuchastheelectronicnoise,ther malnoise,etc. Thesenoisesourcescanbegeneralizedasthephasenoise PZT introducedaftertheNCOoutput,andareconsequentlysuppr essedbythe closed-loopgain. Atlowfrequencies,itiseasytoshowthatthistransferfunct ionapproximatesto 1 (1 e s ) G ( s ) suchthatanyphasenoiseintroducedaftertheNCOoutputiss tillwell suppressedbytheintegratorgain.4.3.3Electro-opticModulator(EOM)SidebandLocking Inadditiontothetwotunablereferencesdescribedabove,t heexperimental vericationofsinglearmlockingwithEOMsidebandlockingh asalsobeendemonstrated onUFLIS.TheexperimentwasperformedbyJefferyLivas etal. atGoddardSpaceFlight Center(GSFC)[ 94 ],incorporatingthearmlockingcomponentsonUFLISandthe pre-stabilizationsystemincludingthebroadbandEOMprovi dedbyGSFC.Inthe setupasidebandgeneratedattheEOMislockedtothexedleng thcavityusingthe standardPDHtechnique.Tunabilityisachievedbychangingt hefrequencyofthephase modulationsignalappliedontheEOMandthelocaloscillator ,whichbothneedtobe 152

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broadband.Intheexperiment,thetunablerangeoftheentir econtrolsystemislimited bythetunabilityofthelocaloscillatortoafewtensofMHz. Thisexperimenthasdemonstratedadditionalnoisesuppres sioncanbeachieved bytheincorporationofsinglearmlockingandsidebandlock ingasthetunable pre-stabilizationreference,whichonlyrequiresminimal modicationstothePDH cavitystabilizationanddoesnotexplicitlydegradetheno isesuppressionperformanceof it. 153

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CHAPTER5 EXPERIMENTALVERIFICATIONOFDUAL/MODIFIEDDUALARMLOCKING 5.1CommonArmLocking Inthischapterwediscusstheexperimentalvericationofa dvancedarmlocking congurationsthatutilizetheinterferometeroutputsonb othtwoLISAlongarms, includingcommonarmlocking,dualarmlockingandmodiedd ualarmlocking[ 15 ]. Amongthesecongurations,althoughcommonarmlockingtake salinearcombination ofthephasemetermeasurementsonbotharms,itstillresemb lessinglearmlockingina varietyofwayssuchthatithasneverbeenconsideredpromis ing.Nevertheless,wewill stillrstdemonstratethevalidityofcommonarmlockingwi threlativelyshorttimedelays onUFLIS.Thereasonisthatasthesimplesttwo-armcongura tion,thecommonarm lockingexperimentcanbeconsideredasapreliminarytestf orthefollow-onarmlocking experimentswithmoreelaboratedlinearcombinations.5.1.1CommonArmLockingSensor BasedonthediscussioninSection2.3.2.2,thecommonarmsens orisessentially givenbythesumoftwolongarminterferometryoutputsontwo differentarms.Inthe realisticarchitecture,thephasenoiseofthelongarminte rferometryoneacharmis fastsampledandthenmeasuredbythecontrollerphasemeter individually.Amapping vector,whichisanadderforthecaseofcommonarmlocking,w illbeimplemented rightbehindthephasemeterstogenerateansensor/errorsi gnal.Foroursimulationon UFLIS,thelongarminterferometryisrepresentedbyelectr onicmixingoftheprompt andEPD-delayedphase.Togeneratedifferentiallighttravel timebetweentwoLISA arms,theEPDunitwillneedtoexploitthebottomVIMonthePente kboardwherethe delaybuffercanbespeciedwithadifferentarraylength. AsshowninFigure 5-1 (left),thecommonarmsensorofUFLISisimplemented ontwoPentekboardswhichfunctionrespectivelyasfronten dandbackend.Thefront endincludestwoEPDunitsthatsimulatetheround-triptravel timeontwodifferent 154

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 10 1 Frequency (Hz)Magnitude Figure5-1.(Left)Thedesignofthecommonarmlockingsenso rontheFPGA.The commonarmsensorisimplementedontwoPentekboards.DSPboa rd1 functionsastheround-tripdelaylinesontwoarmsandDSPboa rd2 functionsasthephasemeasurementsandthemappingvectoro nthelocal spacecraft.Thephasemeterssendthefrequencydatawithaf astrateof 488kHz .(Right)Themeasuredcommonarmsensortransferfunction. The averageddelaytimeis 5s andthedifferentialdelaytimeis 0.1s .Notethat therstsensornullisat 1 = =10Hz ,whichishigherthantheNyquist frequencyofthismeasurement.Themeasuredtransferfunct iononlyhave localminimaat n = armsindividually.Weuseanvoltagecontrolledoscillator asanoisyoscillatorforinitial experiments.TheVCOsignaliselectronicallysplitintotwo armsandthesignaloneach armissplitagaintogenerateapromptsignalandadelayedan dDopplershiftedsignal viatheEPDunit.OntheotherPentekboardthebackendstartswi thtwophasemeters thatmeasuresthephasedifferenceoneacharmindividually .Identicaltothesinglearm lockingcase,thephasemeterdesignisoptimizedinawaytha ttheysenddatawitha fastrateof 488kHz tominimizethedatatransferdelay.Themappingvectorcalc ulates thesumofthetwophasemeasurementsandthensendthe32-bit errorsignalintothe followingcontrollerlter. 1 Forthepurposeofdemonstration,thedelayprovidedbytheEPD unitsaresetto berelativelyshorttimessuchas 2s or 5s .Thereasonissimplythatthedesignofthe 1 Thephasemetermeasuresthefrequencyuctuationsandsend outthedatawith aprecisionof48-bit.Thenthe48-bitdataistruncatedinth eVHDLcodeandonlyhigh 32-bithasbeenkeptbeforetheyarereceivedbythemappingv ector. 155

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Figure5-2.Theexperimentalsetupofcommonarmlockingusi nganNCOtotrackthe inputnoise.ThebasiccongurationresemblesthesetupinF igure 4-1 .The delaytimesare 2.1s and 1.9s controllerlterforsinglearmlockingexperimentscanbed irectlyadaptedforcommon armlockingwithsimilardelaytimes,whileacommonarmlock ingdemonstrationwith realistic 33s delaywouldonlybecompatiblewithatotallydifferentseto fzerosand polestobedesignedinthecontrollerlter.Figure 5-1 (right)showsthemeasurement ofthemagnituderesponseofacommonarmlockingsensorwith delaytimes 5.1s and 4.9s ( =5s and =0.1s ).Themagnituderesponsehasnullsstartingfrom 1 = ( )=10Hz ,whichisalreadybeyondtheLISAband;however,thereareals olocal minimaatmultiplesof 1 = =0.2Hz whereeachphaseshiftisstillcloseto = 2 with variousdegrees.5.1.2PreliminaryTestwithNumericalControlOscillator(N CO)Tracking Theexperimentalsetuptodemonstratethecontrolsystemof commonarmlocking isshowninFigure 5-2 ,whichisidenticaltothesetupdescribedinFigure 4-1 exceptfor thereplacementwiththecommonarmsensor.Inthisexperime ntthedelaytimesare settobe 2.1s and 1.9s .ByfollowingsimilarproceduresdiscussedinSection4.2.1, the residualphasenoise r ( s ) isgivenby r ( s )= o ( s ) G ( s ) 1+ H C ( s ) G ( s ) S k 1 ( s ), (5–1) 156

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) Free-running VCO VCO-NCO beat signal 32-bit digitization noise 2 s 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -8 10 -6 10 -4 10 -2 10 0 Frequency (Hz)Noise suppression measured idealized Figure5-3.(Left)Thenoisespectraofthefree-runningVCOs ignal(blue)andthe VCO-NCObeatsignal(red).Thenoisespectrumoftheresidual frequency noiseislimitedbythe32-bitdigitizationnoiseoor,whic hisgivenbythe integratedquadraturesumofthe32-bitdigitizationnoise fromthetwo independentphasemeter.(Right)Theclosed-looptransfer functionisgiven bytheratiobetweenthetwonoisespectra. Thentheclosed-looptransferfunctionisgivenby r ( s ) o ( s ) = 1 1+(2 e s 2 e s 3 ) G ( s ) (5–2) Themeasurementresultsoftheoriginalnoiseandresidualn oise,aswellasthe closed-looptransferfunctiongivenbytheratiobetweenth em,areshowninFigure 5-3 .Duetothelocalminimaat n = inthesensormagnituderesponse,theresidual frequencyspectrumstillexhibitsperiodicpeaks,althoug hnotnoiseenhancements,at thesefrequenciesandat 1 = =0.5Hz thepeakreachesthehighest. Inthelowfrequencyregion,thenoisesuppressionisagainl imitedbythe32-bit digitizationnoiseoor.TheexperimentmodelintheLaplac edomainisshownin Figure 5-4 ,whichdepictsagenericarmlockingcongurationwithNCOt racking.In thisdiagramtheblocklabeledwith“mappingvector”repres entsanadderforcommon armlocking.NotethattheADCnoiseandthedigitizationnois eintroducedondifferent armsisuncorrelatedtoeachother.Basedonthediscussionin Section2.3.5.2,the armlockingsensorwilldetectthecommonnoisebetweentwoa rms,whichisgivenby thequadraturesumofthem.Sincethedigitizationnoisesint woarmsareuncorrelated buthavethesameamplitudeinoursetup,thequadraturesumi sgivenby p 2 N dig .The 157

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Figure5-4.Theclosed-loopdynamicsofcommonarmlockingw ithNCOtrackinginthe Laplacedomain.Thisdiagramcanrepresentthecontrolsyst emofany two-arm-basedarmlockingconguration,wheretheblockla beledwith mappingvectordeterminesthespecicarmlockingtype.Not ethatthesame kindnoisesources(digitizationnoise,ADCnoise)betweent hetwochannels areuncorrelatedandtheircommonnoiseisgivenbythequadr aturesumof them. magnituderesponseofthecommonarmlockinglooptothecomm ondigitizationnoise scaleswith 1 = ,thereforewehave Dig ( f )= p 2 N dig s = 9.53 10 7 f HzHz 1 = 2 (5–3) Comparedwiththesinglearmlockingexperiments,thedigit izationnoiseoorhas beendecreasedbyafactorof p 2 duetothelongeraverageddelaytime.Forrealistic 33s lighttraveltime,thedigitizationnoiseoorcanbefurthe rdecreasedforsingle orcommonarmlockingandhenceshouldnotcausesignicanti mpactonthenoise suppressionperformance. 158

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0 50 100 150 200 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time (s)Frequency variation (Hz) Figure5-5.Thelockacquisitionprocessobservedinthecom monarmlocking experiment.Thestart-uptransientscausedbyanon-zeroin itialerrorsignal appearinthetimeserieswithaperiodof 2s .Thetransientsdecaywitha timeconstantlessthan 50s ;however,thisobservedtimeconstantisbased onanaveragedlighttraveltimeof 2s .Fora 33s round-triptraveltime,the timeconstantwouldincreaseproportionallyandbecomecom parabletothe singlearmlockingcase. Analogoustosinglearmlocking,theperiodicimpulsepeaksa realsovisiblein thetimeseriesandthemostdistinguishableoscillationyi eldsaperiodof 2s .Figure 5-5 illustratesthemeasurementsintherstcoupleofminutes, wherethestart-up transientswithaperiodof 2s appearinthelockacquisitionprocessanddecaywitha timeconstantlessthan 50s .Notethatthetimescaleusedinourexperimentisshorter thantherealisticLISAsituationbyafactorofabout16.Ther efore,theexpectedtime constantforarealistic 33s delayisonthelevelofmorethan 500s ,whichiscomparable tothesinglearmlockingmodeldescribedinFigure 2-13 5.2DualArmLocking 5.2.1DualArmLockingSensor Asanenhancedversionofcommonarmlocking,dualarmlocking recoversthe instantaneousphasenoisebycombingthecommonphasenoise andtheintegrated differentialphasenoiseontwointerferometeroutputs.To realizesuchalinear 159

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Figure5-6.Theimplementationdiagramofthedualarmlocki ngsensor.Incomparison toFigure 5-1 ,themappingvectorprovidesthedifferentialpathwhereth e differentialfrequencynoisebetweentwochannelsisinteg ratedandscaled by 1 = .Thelow-passlter E ( s ) isalsoimplementedtoattenuatethe excessivephaselossinthedifferentialpath. combination,wemodifythemappingvectoronthebasisofcom monarmsensor.As showninFigure 5-6 ,theupdatedmappingvectorcalculatesthecommonanddiffe rence betweenthetwophasemeasurements.Inthedifferentialpat h,thedifferentialfrequency noiseisintegratedandscaledby 1 = .Alow-passlterwithasinglepoleat 1Hz is placedinthedifferentialpathtoattenuatetheexcessivep haselossintheopen-loop transferfunctionofthedualarmlockingsensor.Depending onwhicharmislonger, anadderorsubtractorsubsequentlycombinesthetwopathst ogeneratethedualarm lockingsensorsignal.Inallthefollowingexperimentswea lwaysassume 2 > 3 such thatthedifferentialarmiskepttobeaddedtothecommonarm Tomakesureeachcomponentinthedualarmlockingsensorwor ksappropriately, werstmeasuredthemagnituderesponsesofthecommonarm,d ifferentialarmand integrateddifferentialarmindependently.Thedelaytime saresettobe 5.1s and 4.9s ( =5s and =0.1s )andthemeasurementresultsareshowninFigure 5-7 (left). Theentiretransferfunctionofdualarmlockingsensor,giv enbythecombinationof commonarmandintegrateddifferentialarm,ismeasuredand displayedinFigure 5-7 (right).Themeasuredtransferfunctionagreesverywellwi ththetheoreticaltransfer function,whichretainsaatmagnituderesponseatlowfreq uencies.Themeasurement 160

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -3 10 -2 10 -1 10 0 10 1 Frequency (Hz)Magnitude common differential integrated differential (scaled with 1/ D t ) 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -3 10 -2 10 -1 10 0 10 1 Frequency (Hz)Magnitude measured theoretical Figure5-7.(Left)ThemeasurementresultsonUFLISdisplay thetransferfunctionsof commonarm,differentialarmandintegrateddifferentiala rmusingtwoEPD unitswithdelaytime 5.1s and 4.9s .Theintegrateddifferentialarmis alreadyscaledwith 1 = .(Right)Themeasurementresultofthedualarm lockingsensortransferfunction,whichisalinearcombina tionofthethree transferfunctionsontheleft.Thelow-passlter E ( s ) isalsoincludedinthe measuredtransferfunction.Therstnullislocatedat 5Hz resultsalsoexplicitlyshowtherstnullat 1 = 2 =5Hz ,whichiswellbeyondtheLISA band. 2 Sincethesensornullsofdualarmlockingonlydependonthedi fferentiallight traveltime,thesensortransferfunctionwillessentially remainthesameiftheaveraged delaytimeischangedto 33s 5.2.2PreliminaryTestwithNCOTracking Tovalidatethefeasibilityofthedualarmlockingsensor,w eperformedhardware simulationsfeaturingthissensortostudyitsnoisesuppre ssionperformance.Inour preliminaryexperimentweconguredadualarmlockingloop withthesameNCO trackingscheme.Figure 5-8 illustratestheexperimentalsetupinwhichthefrequency ofa 2MHz NCOsignalisstabilizedtoafree-running 7MHz VCOsignal.ThetwoEPD unitsprovide 33.25s and 32.75s delaytimes(armlengthmismatch 1.5% )aswellas 2MHz and 3MHz Dopplershifts,respectively.Thissensorsignalisltere dbythe feedbackcontrollertodrivetheNCOsignaltrackingtheinp utphasenoise.Sincethe 2 Strictlyspeaking,thisisanon-zerolocalminimumanditisn otexactlylocatedat 5Hz 161

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Figure5-8.Thepreliminaryexperimentalvalidationofthe dualarmlockingsensoron UFLISistostabilizethefrequencyofanNCOsignaltotrackt heinput frequencynoiseofaVCOsignal.ThetwoEPDunitsprovide 33.25s and 32.75s delaytimesaswellas 2MHz and 3MHz Dopplershifts, respectively.Theanalogmixersmixthepromptanddelayeds ignals.The mixeroutputsarethensenttothephasemetersandthemappin gvectoron thesecondDSPboardtogeneratethesensorsignalofdualarml ocking. differentialdelaytime =0.25s ,thecompensatorlterdesignedinSection4.2.2still satisestherequirementofclosed-loopstability. Thelinearspectraldensitiesofthefree-runningVCOsignal andtheVCO-NCObeat signalareshowninFigure 5-9 (Left).Theratiobetweenthetwospectrumyieldsthe closed-looptransferfunction,whichindicates6ordersof magnitudenoisesuppression below 1mHz asshowninFigure 5-9 (Right).Figure 5-9 (Right)alsodemonstrates somedeviationfromtheidealizedclosed-looptransferfun ctionforfrequenciesbelow 30mHz .Theoreticalcalculationindicatesthatthe 1 = f 32-bitdigitizationnoiseoor, whichnowscaleswith 1 = ,stilldominatesthisfrequencyrange.Therefore,the32-b it controllerlterinthepresenceofa 0.25s armlengthmismatchresultsinanoiseoor givenby Dig ( f )= N dig s = 5.41 10 6 f Hz = p Hz. (5–4) Thisresultindicatesthatasmallerarmlengthmismatchwil lcauseahigher 1 = f digitizationnoiseoor.InrealisticLISAthiskindofdigit izationnoisewithinthedualarm lockingsensor/controllermaylimitthenoisesuppression performanceundercertain 162

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/rtHz) Free-running VCO signal Residual VCO-NCO beat signal 32-bit digitization noise ( D t = 0.25 s) 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 Frequency (Hz)Noise suppression measured idealized Figure5-9.(Left)Themeasurementresultofnoisesuppress ionperformanceshowsa noisesuppressionof 6 ordersofmagnitudebelow 1mHz .Intheresidual frequencynoisecurvethe 1 = s slopestartingfromaround 10mHz towardDC isduetothe32-bitdigitizationnoiseintegratedbythedif ferentialarm.For thisreason,thisnoiselimitationisproportionaltothefa ctorof 1 = .(Right) Theclosed-looptransferfunctionisgivenbytheratiobetw eentheinitial frequencynoisespectrumandtheresidualfrequencynoises pectrum.The deviationfromtheidealizedclosed-looptransferfunctio ninthelow frequencyrangeismainlyduetothedigitizationnoiseoor circumstancesofveryshortarmlengthmismatch.Foradiffe rentiallighttraveltimeless than 1ms ( 300km ),theresidualnoisewouldfailtomeetthe 0.3Hz = p Hz requirement, aswellasexhibitrelativelyfastfrequencydrift.Thispot entialissueindicatesthatthe controllerwitha32-bitxed-pointprecisionissomewhati nadequateespeciallyforthe dualarmlockingconguration.Ifweenhancetheprecisiono fthesensor/controllerupto 48-bit,thecorrespondingdigitizationnoiseis N Dig 48 = f clock 2 48 p 6 f s =1.30 10 10 Hz = p Hz. (5–5) Althoughweareabletoincreasetheintegerprecisionofthea rmlockingsensor/controller, theprecisionintheVIMtransferfromthecontrolleroutputt otheNCOisstilllimitedto 32-bitduetothehardwarespecication.However,aslongas wemaintainthe48-bit datauntilitissenttotheVIMtransferatthecontrolleroutp ut,theadditional32-bit digitizationnoiseintroducedbytheVIMtransferwillbesup pressedbytheopen-loop gainandwillnotcauseanissue.Theresultsusingthesameex perimentalsetupare showninFigure 5-10 .ThedelaytimesgeneratedattheEPDunitsare 33.025s and 163

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10 -4 10 -2 10 0 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Relative frequency noise (Hz/sqrtHz) VCO-NCO residual signal Free-running VCO signal 48-bit digitization noise ( D t = 0.025 s) ADC noise @ 2MHz & 3MHz Figure5-10.Inthismeasurementweusethesameexperimenta lsetupexceptthatthe xed-pointprecisioninthecontrollerlterhasbeenimpro vedto48-bit. ComparedtoFigure 5-9 ,thedigitizationnoiseoorhasbeendecreasedby afactorof 2 16 andtherebythenoisesuppressionperformancehasbeen enhanced.However,itdoesnothitthe48-bitdigitizationn oiseoorbutis limitedbytheADCnoiseinthephasemeasurements.TheADCnois ein thephasemeterisa p f slopeandisintegratedbythearmlockingloopin thesamemannerasthedigitizationnoise. 32.975s ,respectively.Comparedwiththe32-bitcase,thedigitiza tionnoiseoorhas beendecreasedbyafactorof 2 16 duetothehigherprecision.However,theresult indicatesthattheresidualnoisespectrumiswellabovethe 48-bitdigitizationnoiseand limitedbyadifferentnoiseoorfeaturinga 1 = f 1 = 2 slope. Theclosed-loopresponsetothetechnicalnoise(48-bitdig itizationnoise,ADC noise)introducedattheEPDunitsis H ( s ) G ( s ) = (1+ H ( s ) G ( s )) 1 .Asaresult,these noiseswillonlyincreasetheresidualfrequencynoisebyan insignicantamount.In contrast,theADCnoiseatthecontrollerphasemeterswillbe integratedbythetransfer functionofthedifferentialarm.Basedontheintroductioni nSection3.2.2.3,theADC frequencynoiseatthephasemeter,whendigitizinga 3MHz and 2MHz RFsignal,is 164

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0 100 200 300 400 500 600 700 800 -150 -100 -50 0 50 100 150 200 250 Time (s)Frequency variations (Hz) Figure5-11.Thelockacquisitionprocessobservedinduala rmlocking.Inthetime seriestheinitialtransientsoscillatewithaperiodof 33s anddecaymuch fasterthanthesingleorcommonarmlockingcase.Thetimeco nstantin thismodelislessthan 100s approximatelygivenby N ADC@3MHz ( f )=10 6 p f Hz = p Hz, N ADC@2MHz ( f )=1.5 10 6 p f Hz = p Hz. (5–6) SincethetwoADCsondifferentchannelsareuncorrelated,the abovetwonoises arecombinedinquadraturewhenenterthearmlockingcontro ller.Theclosed-loop responsetothecombinedADCnoiseisdominatedbythediffere ntialarm,whichcauses a 1 = f slopescaledwith 1 = : ADC ( f )= p N ADC@3MHz ( f ) 2 + N ADC@2MHz ( f ) 2 s = 1.15 10 5 p f Hz = p Hz. (5–7) AsshowninFigure 5-10 ,this 1 = p f noiseooragreeswiththeresidualnoise spectrum. Anotherdistinctivefeatureofdualarmlockingisthatthein itialtransientsdecaywith arelativelyshorttimeconstant.AsshowninFigure 5-11 ,theinitialtransientsexhibita dampedoscillationwithaperiodof 33s .Afterabout 400s the 33s oscillationbecomes 165

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Figure5-12.Theexperimentalsetupoffrequencynoisemiti gationofthecavity stabilizedlaserusingdualarmlocking.Theexperimentalp rincipleisthe sameasdescribedinFigure 4-12 ,wherethesinglearmlockingsensoris replacedwiththedualarmlockingsensordesignedinthepre vioussection. TheaverageddelaytimebetweenthetwoEPDunitsisrealistic 33s .We measuredthestabilizedfrequencynoiseofthe RL L 2 beatsignalwith multipledifferentialdelaytimes. invisibleinthetimeseriesofthestabilizedfrequency,wh ichismuchshorterthanthe requireddurationofmorethan 2000s insingleorcommonarmlocking. 5.2.3IntegrationwithPre-stabilizedLaser Theapplicationofdualarmlockingonthefrequencynoisemi tigationofpre-stabilized lasershasbeenperformedbymeansofanauxiliaryphase-loc kedlaser.Thisexperiment forthersttimedemonstratesthatarmlockingiscapableof sufcientlyreducingthe frequencynoiseofacavitystabilizedlaserinthepresence ofrealistic 33s lighttravel times.Theexperimentalsetupresemblesthesetupillustra tedbyFigure 4-12 ,whilethe armlockingcongurationhasbeenexpandedintotwoarms.Ass howninFigure 5-12 thefrequencynoiseof L 2 tracksthe L 1 asaphase-lockingreference.Thephase-locking reference,whichisdrivenbyaNCO,isfurtherphase-modula tedviathedualarm lockingsetup. 166

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 10 6 Frequency (Hz)Relative frequency noise (Hz/rtHz) RL L1 beat signal RL L2 beat signal ( D t = 0.25 s) RL L2 beat signal ( D t = 0.025 s) ADC noise floor ( D t = 0.25 s) ADC noise floor ( D t = 0.025 s) Figure5-13.Thenoisespectraofthecavitystabilizedbeat signal RL L 1 (blue)and thebeatsignal RL L 2 furtherstabilizedbydualarmlocking.Comparedto the RL L 1 beatsignal,thefrequencynoiseof RL L 2 issuppressedby7 to8ordersofmagnitudesinthenoiselimitedregion,wheret hedominant noiseoorcomesfromthephasemeterADCs.Alsoweinvestigate the frequencynoisespectraof RL L 2 inthepresenceofvariousdifferential delaytimes.Thesemeasurementssimulatetheinverselypro portional changeofthelimitingnoiseoorinaccordancewiththechan geofthearm lengthmismatchindualarmlocking. ThemeasurementresultsareshowninFigure 5-13 ,whichincludesthefrequency noisespectraunderthecaseswithtwodifferentdifferenti aldelaytimes.Startingfrom thefrequenciesaroundabout 10mHz ,thearmlockingperformancegraduallychanges fromgainlimitedtonoiselimited.Forthedifferentialdel aytimethatequals 0.25s ,the frequencynoiseofthe RL L 2 beatsignalinthenoiselimitedregionissuppressed below 10 4 HzHz 1 = 2 .Incomparisontothecavitystabilized RL L 1 frequencynoise, theclosed-loopgainofdualarmlockingprovidesanoisesup pressionwith7to8orders ofmagnitudesinthenoiselimitedregion.LiketheNCOtrack ingmeasurement,the limitingnoiseoorexhibitsaslopeof 1 = f 1 = 2 ,whichmatchesuptheexpectedADCnoise oorgivenbyEq. 5–7 .Whenthedifferentialdelaytimedecreasesbyafactorof10, the ADCnoiseooralsoincreasesbyafactorof10accordingly. 167

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Figure5-14.Theimplementationofthemodieddualarmlock ingsensoronUFLIS expandstheoriginaldualarmlockingdesignbylinearlycom biningthe commonarmanddualarmsensorsignals.Thelow-passlter F C ( s ) tobe multipliedwiththecommonarmsensorissimpliedintoanin tegratorwith ascalingfactorof 1 = .Thehigh-passlter F D ( s ) hasazeroatDCanda poleataround 1 = 30mHz tosmooththecrossoverbetweenthetwo transferfunction.Notethatthesamplingfrequencyof F D ( s ) is downsampledbyafactorof32tomaintaintheopen-loopstabi lity. 5.3ModiedDualArmLocking 5.3.1ModiedDualArmLockingSensor AsintroducedinSection2.3.3.3,themappingvectorofamodi eddualarm lockingsensorconsistsofalow-passlter F C ( s ) toamplifythecommonarmanda high-passlter F D ( s ) tosuppressthedifferentialarmatlowfrequencies.Inspit eof thetwoadditionallters,Figure 5-14 showsthatthemodieddualarmlockingsensor isdesignedinasimilarwaytothedualarmlockingsensor.Ou rdesignfollowsthe simplieddesigndescribedinSection2.3.5,wherethelow-p asslter F C ( s ) isreduced intoanintegratorwithascalingfactorof 1 = forthepurposeofsimplicityandefciency. Thehigh-passlter F D ( s ) hasazeroatDCandapolearound 1 = 30mHz todecreasethelowfrequencygainaswellastosmooththecro ssoverbetweenthe commonarmandthedualarm. 3 Comparedwiththedatatransferrateof 488kHz ,the 3 Inthespecicdesign,thepoleof F D ( s ) isactuallyatapproximately 10mHz to smooththecrossover.ComparedwithEq. 2–60 ,thiswillincreasethenoiselimitationsin 168

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -3 10 -2 10 -1 10 0 10 1 MagnitudeFrequency (Hz) H D (s)F D (s) P + (s)F C (s) 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -1 10 0 10 1 Frequency (Hz)Magnitude measured theoretical Figure5-15.Anexampleofamodieddualarmlockingsensorwi thtimedelaysof 2.025s and 1.975s .(Left)Thetransferfunctionmeasurementsofthedual armandcommonarmcomponentsinthemodieddualarmlocking sensor. Theoriginallyatmagnituderesponseinthedualarmcompon entisltered bythehigh-passlter F D ( s ) suchthatitstartstorolldownwitha s slopeto zeroatDC.Thecommonarmcomponentismultipliedby 1 = s andscaledby thefactorof 1 = .(Right)Thecombinationofthosetwocomponentsyields theoveralltransferfunctionofthemodieddualarmlockin gsensor,where theoverallmagnituderesponsestillretainsatnessyetth ecommonarm nowdominatesatlowfrequencies. poleisatasignicantlylowerfrequencysuchthatthehighpasslterwouldbefairly unstablewithsuchahighsamplingrate. 4 Forthisreason,thesamplingfrequencyof thehigh-passlterisdownsampledbyafactorof32tomainta intheopen-loopstability. Theoveralltransferfunctionofthemodieddualarmlockin gsensorisgivenby H MD ( s )= 1 s P + ( s )+ F D ( s ) H D ( s ). (5–8) Wemeasuredthetransferfunctionofthemodieddualarmloc kingsensorandan examplewithtimedelaysof 2.025s and 1.975s isshowninFigure 5-15 .Theleftgure showsthatthecommonarmdominatesthefrequenciesbelow 1 = andthedualarm modieddualarmlockingbyafactorofabout 3 andinthefollowinganalysisweneedto keepthisfactorinmind. 4 Ifweperformaz-transformtoconvertthelterwitha 30mHz cornerfrequencyinto thez-domainusingasamplingfrequencyof 488kHz ,thepolewouldbeveryclosetothe edgeoftheunitcircleonthez-plane. 169

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Figure5-16.Thepreliminaryexperimentalvalidationofth emodieddualarmlocking sensoronUFLISresemblesthepreviousdualarmlockingexpe riment.The modieddualarmlockingsensorextractstheinstantaneous phase informationoftheinputlasernoiseandsendittothearmloc kingcontroller, whichdrivesan 8MHz NCOsignaltotrackthe 13MHz laserbeatnote. Thedelaytimeoneacharmis 33.025s and 32.975s ,respectively. dominatesthefrequenciesaboveit.Theirsmoothcrossover allowstheshapeofthe combinedtransferfunctiontobemaintainedatintheLISAba nd. 5.3.2PreliminaryTestwithNCOTracking Thenoisesuppressionperformanceofthismodieddualarml ockingsensoris validatedinasimilarexperimentsetupwithdelaytime 33.025s and 32.975s ,asshown inFigure 5-16 .Theinputnoisesourceisa 13MHz beatsignalbetweentwoZerodur cavitystabilizedlasers RL and L 1 .ThemeasurementresultsareshowninFigure 5-17 Comparedwiththeresidualfrequencynoisespectrumobtain edviadualarmlocking underthesamearmlengthmismatchcondition,thenoisesupp ressionperformancehas beensubstantiallyimprovedtoapproximately 10 4 HzHz 1 = 2 at 3mHz Theenhancementinthenoisesuppressionperformanceisatt ributedtothe mitigationofthenoiselimitations.Sameastheexperimenti ntheprevioussection, theprimarynoisesourcesinthiscontrolsystemincludethe 48-bitdigitizationnoise insidethearmlockingsensor/controllerandtheADCnoiseat thephasemeters.The noiseamplitudeisspeciedbyEq. 5–5 andEq. 5–6 ,respectively.Thefrequency 170

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10 -4 10 -2 10 0 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Frequency noise (Hz/rtHz) RL L1 beat signal Laser NCO beat signal ADC noise floor 0.025 s Figure5-17.Themeasurementresultshowsthemodiedduala rmlockingsignicantly improvedthenoisesuppressionperformancecomparedwitht heoriginal dualarmlockingexperiment.The 1 = p f slopeADCnoiseoorobservedin dualarmlockingexperimentshasbeensurpassed.Thenalno ise limitationinamodieddualarmlockingconguration,asde scribedinEq. 5–9 ,isacombinedeffectofthedigitization/ADCnoisecoupledi nthe commonarmandinthedifferentialarm. responseofthemodieddualarmlockingsystemtotheinputn oiseisdescribedbyEq. 2–80 ,i.e., Dig = ADC ( f )= H + H MD N Dig = ADC ( f )+ H H MD N Dig = ADC ( f ). (5–9) Whetherthecommonarmcomponentorthedifferentialarmcomp onentdominates dependsonthelterdesignandthearmlengthmismatch.Inou rsetupthetransfer functionof H + ( s ) and H s isspeciedbyEq. 2–60 .Undertheconditionofa 0.025s differentialdelaytime,themagnituderesponsesareplott edinFigure 5-18 .Dueto thesimplieddesignofthelter F D ( s ) andtherelativelyshort ,thedifferentialarm componentdominatesacrosstheentireLISAband.Thusforfre quenciesbelow 10mHz (alsothenoiselimitedregion),theADCnoiseandthe48-bitd igitizationnoisewillbe ampliedbythedifferentialarmwithafactorof 3 = (2 H MD ( s )) 318 ,wherethe factorof3comesfromthepolefrequencyof F D ( s ) at 10mHz .Forfrequencieswell 171

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10 -4 10 -3 10 -2 10 -1 10 0 10 -1 10 0 10 1 10 2 10 3 Frequency (Hz)Magnitude H + (s) H (s) 1/(s*33s) 1/(s*0.025s) Figure5-18.Themagnituderesponsesof H + ( s ) and H ( s ) forthemodieddualarm lockingexperiment.Theexpressionsof H + ( s ) and H ( s ) arespeciedby Eq. 2–60 ,exceptthatthepoleof F D ( s ) isat 10mHz ratherthan 30mHz causingthemagnitudeof H ( s ) increasesbyafactorof3intheatregion. Duetotherelativelyshortdifferentialdelay,thediffere ntialarmdominates acrosstheentireLISAband. below 1 = ,thedominantnoisesourceistheproportionallyscaledADCn oiseoor, whichisgivenby ADC ( f )= N ADC ( f ) 2 H MD =5.73 10 4 p f HzHz 1 = 2 (5–10) Inthenoiselimitedregion,thenoisespectrumoftheLaserNCObeatsignalis consistentwiththeADCnoiseoorgivenbyEq. 5–10 .Inthismeasurementwehave demonstratedthatunderthesameconditionofarmlengthmis match,modieddualarm lockinghasabetternoisesuppressionperformancethandua larmlocking.Nowthatthe noiseoorinthecontrolsystemofourmodieddualarmlocki ngexperimentshasbeen identiedandveriedquantitatively,wewouldexpectthes amenoiseoorwhenthis controlsystemisappliedontothefrequencynoisemitigati onofcavitystabilizedlasers. 5.3.3IntegrationwithPre-stabilizedLaser Theincorporationofmodieddualarmlockingandacavityst abilizedlaser wasexperimentallydemonstratedusingthesetupillustrat edbyFigure 5-19 .The 172

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Figure5-19.Experimentalsetupofmodieddualarmlockingu singanauxiliary phase-lockedlaser.Theexperimentalprincipleisthesame asdescribedin Figure 5-12 ,wherethedualarmlockingsensorisreplacedwiththe modieddualarmlockingsensordesignedintheprevioussec tion.The averageddelaytimebetweenthetwoEPDunitsis 33s andthedifferential delaytimeis 0.025s (armlengthmismatch 0.15% ).Wemeasuredthe stabilizedfrequencynoiseofthe RL L 2 beatsignal. experimentalsetupisessentiallythesameasinFigure 5-12 ,wherethefrequencynoise ofthe RL L 1 beatsignalisfaithfullyreproducedbytheheterodynephas e-locked RL L 2 beatsignal.Thefrequencynoiseof RL L 2 canbethereforefurthersuppressed viathephasemodulationfromthearmlockingfeedbacksigna l.Thedelaytimeon eacharmis 33.025s and 32.975s ,respectively.Themeasurementresultsareshown inFigure 5-20 .Inthismeasurementthefrequencynoiseofthe RL L 1 beatsignalhas beensuppressedbyabout7ordersofmagnitudeat 3mHz .Thenoiselimitedregion startsfrom 20mHz ,wherethesameADCnoiseoorwitha f 1 = 2 slopedominatesthe frequencynoiseofthe RL L 2 beatsignal.Themeasuredclosed-looptransferfunction isconsistentwiththeidealizedmodeldownto 20mHz 5.4ArmlockingIntegratedWithFar-endPhase-locking Uptonowwehavecharacterizedvariousarmlockingcongura tionswithLISA-like delaytimeandDopplershifts.Wehavealsodemonstratedthe incorporationofarm lockingwithcavitystabilizationsystemsonourEPD-basedel ectro-opticalmodeland 173

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10 -4 10 -2 10 0 10 -6 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Frequency noise (Hz/rtHz) RL-L1 beat signal RL-L2 beat signal 10 -4 10 -2 10 0 10 -8 10 -6 10 -4 10 -2 10 0 10 2 Frequency (Hz)Noise suppression idealized measured Figure5-20.(Left)Thenoisespectraofthe RL L 1 beatsignal(blue)andthe RL L 2 beatsignal(red).Inthismeasurementwehaveachievedabou t7ordersof magnitudenoisesuppressionat 3mHz .Thenoiselimitationstartingfrom 10mHz isthesameADCnoiseoorwitha f 1 = 2 slopeasseeninFigure 5-17 .Thefrequencynoisefrom 0.1Hz to 1Hz isslightlyabovethe 0.3HzHz 1 = 2 requirement,whichcanbealleviatedbyincreasingthe integratorgaininthatregion.(Right)Themeasuredtransf erfunction agreeswiththeidealizedmodelverywelldowntoabout 10mHz .Below 10mHz thecontrolsystembecomesnoiselimited,whichcausesthe deviationfromtheidealizedmodel. achievedsubstantialnoisesuppressionperformance.Inth issectionwetakeone stepforwardtotheaimofrealisticarmlockinghardwaresim ulationbyintroducingthe transpondernoiseatfarspacecraft.Wewillimplementthep hase-lockedloopatthe far-endrealisticallyratherthanassumethattheopticalt ransponderfunctionsperfectly withaninnitefeedbackgain. Moreover,inSection2.3.5wehavederivedthatotherrealist icnoisesources(clock noise,spacecraftjitter,etc.)wouldcoupleintothestabi lizedlasernoiseinaccordance withthesametransferfunctionasthetranspondernoise.Th erefore,thetransponder noiselimitationobservedinthestabilizedlasernoisecan beequivalentlyconsideredas amanifestationofanyadditionalnoisesourcepresentedin armlocking.Ourexperiment hasforthersttimedemonstratedthearmlockingperforman ceinthepresenceofthe realistictranspondernoise. 174

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Table5-1.Parametersintime-domainarmlockingsimulatio nswithtranspondernoise ParameterSymbolValueUnits Samplingfrequency f s 10 Hz Pre-stabilizedfrequencynoise 0 1 30 p 1+(2.8mHz = f ) 4 HzHz 1 = 2 Free-runningfrequencynoise 0 2 10kHz = f HzHz 1 = 2 Averageddelaytime 33s Differentialdelaytime 0.025s Dopplerfrequencyerroronarm1-2 D2 0Hz Dopplerfrequencyerroronarm1-3 D3 0Hz Transpondernoiseonarm1-2 2 PLL 10 2 f HzHz 1 = 2 5.4.1TransponderNoiseFloor-TimeDomainSimulation Theadditionalfar-endPLLisnothingmorethanintroducinga nadditionalnoise sourcebetweenthetwodelaylines,whichcanbeeitherthetr anspondernoiseorany otherkindofrealisticnoisesources.Thenoiselimitation derivedfromthetransponder noiseisgivenby trans = H + H ( trans2 + trans3 )+ H H ( trans2 trans3 ), (5–11) wheretheindependenttranspondernoisesoneacharm, trans2 and trans3 ,are uncorrelated.Thecommonanddifferencenoisebetweenthem yieldthequadrature sum,whichiscomparabletothetranspondernoiseonasingle arm.Therefore,we couldapplytheadditionaltranspondernoiseononlyonearm withoutlossofquantitative validity.Fordualarmlockingwehave H + =1 and H =1 = s andformodieddual armlockingwehave H + = F C + F D and H = EF D = s Thevericationofthetranspondernoiselimitationswasin itiallyperformedby time-domainsimulationsusingMATLABSIMULINKpackages.The softwaresimulations replicateLISA-likeconditionsaswellasthesensor/control lerdesignthathavebeen builtinourbenchtopexperiments.Theltersusedinthetim e-domainsimulationare designedintheLaplacedomain.Therefore,somedeviationi nthefrequencyresponse betweenthesoftwareltersandthehardwareltersshouldb eexpected.Tabel 5-1 lists theparametersthatareusedinoursimulations. 175

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Inthesimulationtheoutgoingtraveltimeandthereturntra veltimeareequalon botharms,i.e.,for =0.025s wehave 12 = 21 =16.5125s and 13 = 31 =16.4875s TheLaplacedomainlters,including E ( s ) F C ( s ) F D ( s ) andthe f 1 = 2 controllerlter, havethesamezerosandpolesastheirquantizedz-domaincou nterpartsinthe hardwareimplementation.Hereweassumethetransponderno iseisaband-limited bluenoise(an f slope)intermsoffrequencyuctuations.Notethatthisisj ustan over-simpliedmodelforthepurposeofquickdemonstratio n,whiletherealisticnoiseof ananalogPLLexhibitsa f 1 = 5 slopeinourmeasurement(seenextsection). ThesimulationresultsareshowninFigure 5-21 usingmodieddualarmlocking. Inthisgure,thearmlockingstabilizedfrequencynoiseis determinedbythreedifferent limitations:Forfrequenciesabove 0.1Hz ,thecontrolsystemisgainlimitedandthe stabilizedfrequencynoiseagreeswiththegainlimitedcas ewithoutthetransponder noise.Forfrequenciesbelow 0.1Hz thenoisesuppressionperformanceislimitedbythe transpondernoiseoor,whichisgivenbytwosegments:Inth eregionfrom 30mHz to 0.1Hz ,thetranspondernoiseoorisdominatedbythedifferentia larm,whichyieldsa noiseoor Trans ( f )= 10 2 f 2 s =3.2 10 2 HzHz 1 = 2 (5–12) Inadualarmlockingcongurationthisnoiseoorwoulddomi natedownthrough lowerfrequencies.Howeverformodieddualarmlocking,fo rfrequenciesbelow 30mHz thehigh-passlter F D effectivelysuppressesthemagnitudeofthedifferentiala rm,which yieldsanoiseoorwithalowermagnitude: Trans ( f )=10 2 f 2 2 =1.05 f HzHz 1 = 2 (5–13) Thesimulatedfrequencynoisespectrumofthestabilizedla sersignal(thered curve)agreeswiththeanalyticallimitationsinallfreque ncyregions.Thissimulation numericallydemonstratesthatthetranspondernoiseoori namodieddualarmlocking congurationislesssensitivetothearmlengthmismatch. 176

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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 Frequency (Hz)Frequency noise (Hz/rtHz) Pre-stabilized laser frequency noise Arm locking stabilized laser frequency noise (limited by open-loop gain) Transponder noise floor ( dn (f) 33/(2 p 0.025)) Transponder noise floor ( dn (f) 1/s/0.025) Arm locking stabilized laser frequency noise (limited by transponder noise) Figure5-21.Timedomainsimulationofmodieddualarmlock inglimitedbythe transpondernoiseoor.Thefrequencynoiseofthestabiliz edlaser(red)is determinedbydifferentmechanismindifferentfrequencyr egions.For frequenciesabove 0.1Hz thecontrolsystemisgainlimitedandthered curvefollowsthepurplecurve(thegainlimitedcase).Forf requencies between 0.1Hz and 30mHz ,thefrequencynoiseofstabilizedlaseris limitedbythetranspondernoiseoordeterminedindualarm locking(the yellowcurve)withatransferfunctionof 1 = s .Forfrequenciesbelow 0.1Hz ,thehigh-passlterstartstosuppressthemagnitudeofthe differentialarmsensor,whichalthoughstilldominatesth roughtherestof themeasurementband.Inthisregiontheredcurvefollowsth egreen curve,whichisgivenbytheintroducedtranspondernoisemu ltipliedwith thedifferentialarmgainof = (2 H MD ) 5.4.2ExperimentalVerication-SimpleModel Tosimulateanadditionalnoisesourcefromfarspacecrafto nourhardwaresystem, weneedtodividethedelaytimeattheEPDunitsequallyanddela ytheinputelectronic signalusingtwosplitdelaylinesincascade.Analogously,t hearmlengthchangedue totheLISAspacecraftmotionisinsignicantoverashorttim eintervalof 33s ,which makestheoutgoingtimeandthereturntimearealmostthesam e.Attheoutputofthe rstdelayline,thedelayedsignalcanbeusedtophase-lock anotherlaserthatfunctions 177

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Figure5-22.Experimentalsetupofdualarmlockingwithfunc tiongeneratorsignalas thetranspondernoise.Inthissetupthedelaylineonthearm 1-2isequally dividedintotwo,representingtheoutgoingtraveltime 12 andthereturn traveltime 21 .Afunctiongeneratorsignalisusedtodemodulatethesigna l delayedby 12 suchthatthenoiseofthefunctiongeneratorsignalenters theentireround-tripdelayline.Itshouldbenotedthatint hephase measurementthefrequencyofthefunctiongeneratorsignal hastobe exactlyaccountedfortoavoidthefrequencypulling. asafarspacecraftlaser.Thentheoutputsignalofthephase -lockedlaserissenttothe seconddelaylinetogeneratethereturningbeam. Asimpliedexperimentalvericationofdualarmlockingwi ththetranspondernoise wasperformedusingtheNCOtrackingconguration.Sincethe transpondernoisecan besimpliedintoanarbitrarynoisesourceenteringbetwee ntheoutgoingdelayand thereturndelay,weuseafunctiongeneratorsignaltosimul atethisadditionalnoise source.AsshowninFigure 5-22 ,thedelaylineonthearm1-2isequallydividedinto twodelaytimes,representingtheoutgoingtraveltime 12 andthereturntraveltime 21 TheDopplershiftfrequencyaddedtothenominalfrequencyo fthe“outgoingbeam” is +2MHz ,generatinga 7MHz Doppler-shiftedsignal.Weplaceafunctiongenerator producinga 3MHz signaltodemodulatethe 7MHz signalsuchthatthenoiseofthe functiongeneratorsignalenterstheentireround-tripdel aylineviatheanalogmixer.Itis 178

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worthnotingthatlikethelocaloscillatorthatdrivesthef ar-endphase-lockinginrealistic LISA,theoffsetfrequencyofthefunctiongeneratorsignalha stobetakenintoaccount inthephasemeasurementofthelongarminterferometry.Tha tis,thefunctiongenerator hastobesynchronizedtothemasterclockanditsnominalfre quencyhastomatchup thecorresponding16-bitintegervalue(e.g., 3MHz correspondsto 2.999305725MHz ) toavoidgeneratinganunwarrantedDopplerfrequencyerror .Thedemodulated 4MHz signalisthensenttotheseconddelayline,whichsimulates thereturnbeambeing transferredbacktothelocalspacecraft. Thefunctiongeneratornoiseismeasuredtobeapproximatel y 10 4 f 3 = 4 HzHz 1 = 2 viafractionaldelayltering.Withadifferentialdelaytim eof 0.25s and 0.025s ,the expectedtranspondernoiseoorisgivenby 3.18 10 5 f 1 = 4 HzHz 1 = 2 and 3.18 10 4 f 1 = 4 HzHz 1 = 2 ,respectively.Weplotthemeasuredresidualfrequencynoi se spectruminFigure 5-23 .Themeasurementssufcientlydemonstratethatindualarm lockingtheperformancelimitationduetothetranspondern oiseinverselyscaleswith thearmlengthmismatchandthenoiseamplitudeisconsisten twithouranalytical expectations.5.4.3ExperimentalVerication-FullModel Thecompleteexperimentalsetuptoverifythetranspondern oiseoorinamodied dualarmlockingcongurationisillustratedbyFigure 5-24 .Inthissetupwestillusean auxiliaryphase-lockedlaser L 2 toobtainthetunabilityofthepre-stabilizationreferenc e. Astheoutgoingbeamfromthe“localspacecraft”,thebeatsig nal RL L 2 iselectronically split.Thenthesplitsignalstraveltocorresponding“fars pacecraft”individually,which issimulatedbytheEPDunits.Similartothesimplemodelinthep revioussection, weallocatethe 33.025s round-triptimeequallyintotwodelaylines,representing the outgoingtraveltime 12 andthereturntraveltime 21 .Oncethe RL L 2 beatsignalis delayedby 12 andarrivesatthe“farspacecraft”,itsdelayedandDoppler -shiftedversion (a 3MHz NCOsignal)isusedtoheterodynephase-locka“farlaser”re presentedby 179

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10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Frequency noise (Hz/rtHz) Free-running VCO VCO-NCO beat (0.25 s) VCO-NCO beat (0.025 s) FG noise floor (0.25 s) FG noise floor (0.025 s) Figure5-23.NoisespectraofdualarmlockingwithFGsignal asthetranspondernoise. Inthisguretheredandpurplecurvesaretheresidualfrequ encynoisein thepresenceofthedifferentialdelaytime 0.25s and 0.025s ,respectively. Thegreenandyellowcurvesareobtainedfromthemeasuredfu nction generatornoisemultipliedwiththetransferfunction 1 = s .Our measurementshaveveriedthatthelimitationoftheresidu alfrequency noiseisconsistentwiththetranspondernoiseoor. the RL L 3 beatsignal.The 5MHz RL L 3 beatsignaltracksthefrequencynoise of RL L 2 andalsocarriesanuncorrelatedresidualnoiseduetothen itegainofthe PLLcontroller.The 2MHz offsetfrequencyofthePLL,whichisdrivenbyafunction generator,issynchronizedtothemasterclockandroundedt oitscorresponding16-bit value.Also,thefrequencynoiseofthefunctiongeneratorsi gnalentersthePLLand becomesapartofthetranspondernoise.Thisprocessresemb lestheclocknoisethat entersthefar-endPLLduringthephasemeasurementatfarspa cecraft.Thenwedelay thephase-locked RL L 3 beatsignalbythereturntraveltime 21 andformthelongarm interferometrywiththe“localbeam” RL L 2 beatsignal. Figure 5-25 illustratestheexperimentalsetuprepresentedintheLapl acedomain. FollowingthenotationusedinFigure 4-13 ,weuse i torepresentthephaseofthelaser 180

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Figure5-24.Experimentalsetupofmodieddualarmlockingw ithfar-endPLL.This setupexploitsanauxiliaryphase-lockedlaser L 2 toobtainthetunabilityof thepre-stabilizationreference.Thedelaylinetosimulat etheLISAarm1-2 isdividedequallyintotwo,representingtheoutgoingandr eturntraveltime individually.Anotherbeatsignal RL L 3 isphase-lockedtothe RL L 2 delayedbytheoutgoingtraveltimewithanoffsetfrequency of 2MHz and thendelayedbythereturntraveltimetohavetheheterodyne interference withthepromptbeam.Inthissetupthefunctiongeneratorde modulating theheterodynefrequencyofthePLLhasanequivalentfunctio nasthe clockonfarspacecraft,wheretheclocknoiseentersthepha se measurementofthefar-endPLLinasimilarway. L i andanupperindexof 0 toindicatetheout-of-loopphase.Inthisdiagram G 0 isthe PLLgainforthephase-locking L 2 to L 1 ,while G 1 and G 2 followthenotationconvention torepresentthearmlockinggainandthePLLgainonthelocals pacecraftandthefar spacecraft,respectively.Intheanalysisweonlyconsider thetranspondernoiseasitis theonlydominantprimarynoisesourceinthissetupandnegl ectanyothersecondary noiseincludingthedigitizationnoise,theADCnoise,theli ghtpathnoise,etc. IdenticaltoEq. 4–6 ,thephaseoftheslavelaser L 2 isgivenby 2 = 1 1+ G 0 02 + G 0 1+ G 0 ( 1 + NCO ). (5–14) 181

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Figure5-25.Closed-loopdynamicsofmodieddualarmlocki ngwiththefar-endPLL. Inthisdiagram G 0 isthegainofthelocalPLLactuator, G 1 isthegainof armlockingcontrollerand G 2 isthegainofthefar-endPLL.Theprimary noisesourcesinthissetupisthenitegainof G 2 causingtheresidual frequencynoiseinthefar-endPLL,whichtheclocknoise CL fromthe“far spacecraft”(thefunctiongeneratornoise)alsocontribut esto.Comparedto thetranspondernoiseinthesetup,theothersecondarynois esourcessuch asthedigitizationnoiseorADCnoisecanbeneglected. TheNCOnoiseisgivenby NCO = S k 264 12 13 375 G 1 (5–15) where 13 =( 0 2 )(1 e s 3 ) isthephasemeasurementonthearmwithzero transpondernoise. Ontheotherhand,thephasemeasurementwiththetransponde rnoiseaddedinis givenby 12 =( 0 2 ) ( 0 3 ) e s 21 (5–16) 182

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Sincethe RL L 3 beatsignalisphase-lockedtothedelayed RL L 2 beatsignal,we have 0 3 = 1 1+ G 2 ( 0 03 )+ G 2 1+ G 2 ( 0 2 ) e s 12 + CL (5–17) where CL isthephasenoiseofthefunctiongeneratorthatdrivesthePL L.Oncethe functiongeneratorissynchronizedtothemasterclock,ith asanequivalenteffectasthe clocknoisefromthefarspacecraft SC 2 SubstituteEq. 5–17 intoEq. 5–16 ,weobtainthephasemeasurementonarm1-2: 12 =( 0 2 ) 1 1+ G 2 ( 0 03 ) G 2 1+ G 2 ( 0 2 ) e s 12 + CL e s 21 = 1 G 2 1+ G 2 e s 2 ( 0 2 ) 1 1+ G 2 ( 0 03 ) e s 21 G 2 1+ G 2 CL e s 21 (5–18) ThersttermofEq. 5–18 isthephaseofthelocalbeammultipliedbythetransfer functionoftheround-triponarm1-2.Wedenethetranspond ernoiseonarm1-2: Trans = 1 1+ G 2 ( 0 03 ) e s 21 + G 2 1+ G 2 CL e s 21 (5–19) Eq. 5–19 indicatesthatthetranspondernoiseinvolvestheintrinsi cnoiseofthe far-endPLLduetothelimitedgainaswellastheclocknoisefr omfarspacecraft. Forthepurposeofsymmetricalformalism,wealsowritethep hasemeasurement onarm1-3inthefollowingform: 13 = 1 G 3 1+ G 3 e s 3 ( 0 2 ), (5–20) wherethePLLgain G 3 isassumedtobeinnite. 183

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Ifweconsideragenericarmlockingcongurationwithamapp ingvector S k = [ H + + H H + H ] ,thephaseoftheNCOsignalisthengivenby NCO = [ ( H + + H ) 12 +( H + H ) 13 ] G 1 = [ H + ( 12 + 13 )+ H ( 12 13 ) ] G 1 = [ ( H + P + + H P )( 0 2 )+( H + + H ) Trans ] G 1 =[ H ( 0 2 )+( H + + H ) Trans ] G 1 (5–21) where H = H + P + + H P isthesensorofanarbitraryarmlockingconguration. SubstituteEq. 5–21 intoEq. 5–14 and 2 becomes 2 = 1 1+ G 0 02 + G 0 1+ G 0 1 +[ H ( 0 2 )+( H + + H ) Trans ] G 1 (5–22) FollowingthesameprocedureinEq. 4–9 ,wecombinethetermsinvolving 2 tothe leftandaddterms 0 1 1+ G 0 0 G 0 1+ G 0 0 (=0) totheright: 1+ G 0 1+ G 0 HG 1 2 = 1 1+ G 0 02 + G 0 1+ G 0 1 + G 0 1+ G 0 HG 1 0 + G 0 1+ G 0 ( H + + H ) G 1 Trans + 0 1 1+ G 0 0 G 0 1+ G 0 0 (5–23) whichcanbesimpliedinto 1+ G 0 1+ G 0 HG 1 ( 2 0 )= 1 1+ G 0 ( 02 0 )+ G 0 1+ G 0 ( 1 0 )+ G 0 1+ G 0 ( H + + H ) G 1 Trans (5–24) ThersttwotermsontherightsideofEq. 5–24 representthenoisesuppression, whilethethirdtermdeterminesthenoiselimitation.Thetr anspondernoiseoorisgiven by Trans = 1 1+ G 0 ( H + + H ) G 1 1+ G 0 1+ G 0 HG 1 Trans H + + H H Trans ( G 0 G 1 1). (5–25) 184

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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)Frequency noise (Hz/rtHz) Phase-locked loop noise Function generator noise Combined transponder noise Figure5-26.Noisespectraofthetranspondernoiseobserve dinmodieddualarm locking.Themeasurementhasidentiedthattheprimarycon tributionto thetranspondernoisecomesfromthegainlimitedPLLandthef unction generatornoiseiswellbelowit. TheresultofEq. 5–25 isconsistentwithEq. 5–11 ,provingthevalidityofthis experimentalsetuptodemonstratethetranspondernoiseo or. Withacertaincontrollergain,thetranspondernoiseismeas uredtobeapproximately a 3 10 4 f 1 = 5 HzHz 1 = 2 slope,asshowninFigure 5-26 .Theprimarycontributionin thetranspondernoisecomesfromthephase-lockedloop,whi lethefunctiongenerator noiseismuchlower.AsthePLLgaindecreases,thetransponder noiseamplitudewill proportionallyincrease. ThemeasurementresultsareillustratedinFigure 5-27 .Asthedifferentialdelay timeequals 0.025s ,theexpectedtranspondernoiseoorbelow 30mHz isapproximately givenby 0.20 f 1 = 5 HzHz 1 = 2 ,whichlimitsthenoisesuppressionperformanceinthat region.Ourmeasurementhasoriginallydemonstratedthatt hearmlockingstabilized frequencynoiseagreeswiththisexpectednoiseoorandsti llsufcientlymeettheLISA requirementinthepresenceoftranspondernoise.Forfrequ enciesaround 30mHz thearmlockingperformanceisstillgainlimitedandthetra nspondernoiseoorgiven 185

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Frequency noise (Hz/rtHz) RL L1 beat signal RL L2 beat signal Transponder noise floor RL L2 beat signal (lower PLL gain) RL L2 beat signal (higher PLL gain) Figure5-27.Noisespectraofthestabilizedlaserfrequenc yandthetranspondernoise oor.Inthisgurethestabilizedfrequencynoiseisrepres entedbythered curve,whichislimitedbythetranspondernoiseoor.Theye llowcurve representthetranspondernoiseoorgivenbythecombinedn oisein Figure 5-26 multipliedwiththedifferentialarmgainof = (2 H MD ) .The redandyellowcurvesagreewitheachotherinthenoiselimit edregion. Notethatthefar-endPLLwassettogainlimitedsuchthatthea mplitudeof thePLLnoisecanbecontinuouslymanipulatedbyadjustingth eloopgain. AsthePLLnoiseincreasesordecreases,thetranspondernoise oorinthe stabilizedlaserfrequencyalsotracksthischange,whichi sshownbythe green(ahigherPLLgain)andpurple(alowerPLLgain)curves. by 1 = s isstillbelowthe RL L 2 frequencynoise.Wealsohavedemonstratedthat asthePLLgainchanges,thecorrespondingtranspondernoise willtrackthechange accordinglywithinacertainrange.Themeasurementsindic atethatthetransponder noiseoorinverselyscaleswiththePLLgainasexpected. 186

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CHAPTER6 DOPPLERFREQUENCYERRORINARMLOCKING InthischapterwewillinvestigatetheimpactofDopplerfre quencyerrorsonarm locking.TheresearchofDopplerfrequencyerrorscanbediv idedintotwocategories: Theon-boardestimationofDopplerfrequencyerrorsandthe techniquestorestrict theinducedfrequencypulling.InrealisticLISA,theDoppler shiftfrequencyisa time-variablequantityduetothechangingrelativeveloci tybetweenspacecraft. Therefore,theinitialDopplerfrequency,aswellasitsrs ttimederivativewillbe estimated.TheerrorinthisestimationwillcauseaDoppler frequencyerrorinthephase measurementsonthelocalspacecraft.SincetheestimatedDo pplerfrequencywillnot beupdatedwhilearmlockingisinoperation,theconsequent Dopplerfrequencyerroris alsoafunctionoftime.Inthischapterwewillfocusmoreont hefrequencypullingand associatedremedyschemes,ratherthanthesimulationofDo pplerfrequencyerrors. NotethattheDoppler-inducedfrequencypullingalsoinclu desthesituationsinlock acquisitionandinthesteadystate,andhereweprimarilyin vestigatethelattersituation asithasalong-termeffectonthelaserfrequency. 6.1DopplerFrequencyErrorinLISA TheLISAorbitsaredesignedtohavea1-yearperiodoscillati onsofthearm lengths,interioranglesandrelativevelocities,whichme anstheDopplerfrequency variationalsooscillateswitha1-yearperiod.Therefore, theDopplerfrequency correspondstoanoscillationat 3.17 10 8 Hz .Sincetherelativevelocitybetween spacecraftcanreachupto v SC =18m = s ,thecorrespondinground-tripDopplershift frequency D tothelaserfrequency 0 yieldstherelation D 0 = v SC c (6–1) ThustheDopplerfrequencycanbeuptoapproximately 17MHz .Asthisfrequency isestimatedandthensubtractedinthephasemeasurements, aDopplerfrequencyerror 187

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willshowupinthephasedataandeventuallydrivethefreque ncyofthemasterlaser.In realisticLISA,theDopplerfrequencyaswellasitstimederiv atives(theDopplerchange rate)willbeinitiallymeasuredandthensubtractedbefore armlockingisengaged.The estimationoftheDopplerfrequency D canberealizedbyvarioustechniques:For example,theon-boardrangingcanmeasuretherelativevelo citybetweenthespacecraft ortheaveragedlaserbeatfrequencyallowstoestimatetheD opplerfrequencydirectly bycomparingtheincomingbeatfrequencyandtheoutgoingbe atfrequency. However,howlargetheDopplerfrequencyerror D willbeisstilluncertain.Ifwe expandthetime-varyingDopplerfrequencyerrortotheseco ndorderoftimederivative inthetimedomainandassumethattheDopplerchangerate(th erstandsecond derivatives)areconstantasthevariationisverysmall,th eDopplerfrequencyerror D asafunctionoftimeisgivenby D ( t ) D0 +_ D t + 1 2 D t 2 (6–2) TheperformanceofDopplerestimationsdependsontheiniti allaserfrequency noise ~ 0 ( f ) beforearmlockingandtheestimationdurationtime T ,yetisindependent ofarmlockingcongurations,providedthattheround-trip traveltime isconstantly 33s .Intheinitialphasemeasurement,thecontributiontothev arianceofthemeasured frequency, 2 ,primarilycomesfromtheDopplerfrequencyerror.Thusthe ycanbe consideredasequivalenttoeachother.AccordingtoRef[ 13 ],thevarianceisgivenby 2 =4 Z 1 0 df ~ 0 ( f ) 2 sinc 2 ( fT )sin 2 ( f ), (6–3) wherethe sinc functioncomesfromtheLaplacetransformoftheaveragingt ime T and the sin functioncomesfromtheinterferometerresponseononearm. Foranaveragingestimationtimeof 100s ,theinitialDopplerfrequencyerror wouldbeapproximately 400Hz withaMach-Zehnderpre-stabilizedlaser;whilefora free-runninglasertheerrorcouldbeashighas 10 5 Hz 188

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Oncethearmlockingisengaged,theDopplerfrequencyerror willenterthe controllerwhichcausesthelaserfrequencytoramp.During thislockacquisition process,theDopplerfrequencyerrorcouldgeneratearelat ivelylargeinitialtransient (severalhundredMHz)imposedontothelaserfrequency,whi chisgivenbytheproduct oftheerrorandthestepresponseofthecontrolsystem.Whena rmlockingentersits steadystate,theDopplerfrequencyestimationswillnotbe updateduntilthenexttime thearmlockingloopisneededtobeunlocked(e.g.,whenchan gingtheheterodyne frequencyorswitchingthearmlockingconguration).After daysormonths,theDoppler frequencyestimationswillbemadeveryaccurateduetothel ongaveragingtimeand therebytheinitialtransientsoccurringinlockacquisiti onwillbesignicantlysmallerthan thersttime. Thusinthesteadystate,theresidualDopplerfrequencyerr orprimarilycomesfrom thetime-varyingDopplerfrequencythattrackstheorbital motionofLISAspacecraft. Dependingondifferentarmlockingcongurations,thefreq uencypullingrateisderived inSection2.3.3.Theexpectedfrequencypullinginthestead ystateislimitedwithin 10MHz peaktopeak,whichisevenlesssignicantthantheintrinsi cdriftofthelaser frequency,providedthatthearmlockingcongurationcanb eexploitedappropriately basedonthearmlengthmismatch. 6.2InvestigationofFrequencyPullingonUFLIS InthissectionwewillstudytheDoppler-inducedfrequency pullingrateinthesteady state,inthepresenceofvariousarmlockingcongurations .Sofar,allofthearmlocking benchtopexperimentssimulatedanidealsituationwithout anyDopplershiftorwhen theDopplerfrequencywasperfectlyaccountedforinthepha semeasurements.Inthe followingtime-domainsimulationsandbenchtopexperimen ts[ 95 ],wewillgenerate constantandstepDopplerfrequencyerrorsinphasemeasure ments,whichisadequate toobservethefrequencypullinginthesteadystate.Wewill alsousethemodieddual 189

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armlockingsensortoexperimentallydemonstrateitscapab ilitytoconciselyreducethe frequencypullingrate. Inadditiontothemodieddualarmlockingsensor,anAC-cou pledcontrollerwithits magnitudesuppressedbelowtheLISAbandisalsocapableofre ducingthefrequency pullingrate.Suchalowfrequencylteringmechanismcansti llbeincludedinthearm lockingcontrollerdesignasanadditionalconstraint,eve nifamodieddualarmlocking sensorisused.Theadvantageisthatitcanfurtherreduceth eresidualfrequency pullingrateandensurethefrequencypullingratewillneve rreachtoohighincase thesystemisswitchedtootherarmlockingcongurations.H owever,anAC-coupled controllermaybringalimitationtotheachievablegain,wh ichmakesthecontrolsystem ofarmlockinggainlimitedwhenthearmlengthmismatchisla rge.Also,thephase marginatthezero-crossingbelowtheLISAbandneedstobecar efullyretainedforthe purposeoftheclosed-loopstability.Bothofthetwoissuesr equireadeliberatecontroller design.6.2.1GenerationofDopplerFrequencyErrors SincetheDopplershiftfrequencygeneratedbycurrentEPDunit sisaconstant valueandcannotbecontinuouslytuned,theDopplerfrequen cyerrorcanonlybe generatedasaconstant.InChapter3wealreadymentionedth atthephasemeteroffset frequencyisa16-bitxed-pointintegernumberwhiletheDo pplerfrequencyspecied onthemotherboardisaoating-pointnumber.Andtoperfectl yaccountfortheDoppler frequency,weroundedtheoating-pointDopplerfrequency tothecorresponding16-bit integervalue.Therefore,ifwegeneratetheDopplerfreque ncyerrorbychangingthe phasemeteroffsetfrequency,itcanonlybediscretelyintr oducedwithafrequency resolution 62.5MHz = 2 16 954Hz .WithsuchalargeDopplererror,thearmlocking lterswouldberequiredtohaveamuchlargerdynamicranget otrackthefastfrequency pulling.TogenerateasmallerDopplerfrequencyerror,weh avetodecreasethe frequencyresolutionbyincreasingtheprecisionofthepha semeteroffsetfrequency. 190

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Incomparison,theEPDdelaylineismoreexibletochangefort hegenerationofa Dopplerfrequencyerrorwithessentiallyanarbitraryvalu e.Currently,aconstantDoppler frequencyerrorcanbesimulatedinthreesimplewaysonUFLI S: 1. DirectlymodifytheDopplerfrequencyvalueintheCprogram byaddingan additionalconstanttotheoriginal16-bitnumbersuchthat theDopplerfrequency errorisgivenbytheconstant.Theadvantageofthismethodi sthataLISA-like time-varyingDopplerfrequencycanalsobesimulatedbased onthismethod, wheretheadditionalvaluebecomesafrequencymodulationw itha1-yearperiod. 2. ChangetheclockfrequencyfortheEPDunit.SinceintheEPDunitth eDoppler frequencyscaleswiththeclockfrequencyof 62.5MHz ,theDopplerfrequency error D isgivenby ( f clock = f clock ) D .Thismethodisstraightforwardasneither softwareorhardwareisneededtobemodied.Alsoifweapplya frequency modulationontheclockfrequency,atime-varyingDopplerf requencyisessentially achievable.However,thefrequencyofthefrequencymodula tioncannotbemade aslowas 10 8 Hz duetothehardwarelimit.Anotherissueofthismethodisthat theDopplerfrequencyerrorgeneratedontwoarmsbecomecor related. 3. UsethesetupillustratedbyFigure 5-22 ,wheretheDopplershiftfrequencyis appliedinthemiddleoftwodelaylines.Thesignalsourceto providetheDoppler frequency,suchasafunctiongenerator,issuggestedtobes ynchronizedtothe masterclock(thoughnotrequiredinthissituation).Simila rtothesecondmethod, atime-varyingDopplerfrequencycanalsoberealizedbyfre quency-modulating thesignalfrequency,althoughtheperiodofthefrequencym odulationislimitedby theparameterofthefunctiongenerator. InthefollowingexperimentswemainlyemployMethod2and3. 6.2.2FrequencyPullinginSingleArmLocking Inthissectionwewillrstdiscussapioneeringexperiment inwhichaconstant Dopplererrorisintroducedintooursinglearmlockingloop andadesignedAC-coupled controllerisdedicatedtoeliminatethefrequencypulling .Figure 6-1 illustratesthe experimentalconguration,inwhicha 6MHz freerunningVCOsignalisdemodulated witha 9MHz NCOsignal.Thebeatsignalof 3MHz isthensplitintoapromptsignal andasignalwhichisdelayedby 1s aswellasshiftedby 4MHz .Thephasemeterwill demodulatetheerrorsignalwithan16-bitoffsetfrequency 191

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Figure6-1.ThepreliminaryexperimentalsetupofAC-coupl edsinglearmlockingwith Dopplererror.A 5Hz constantDopplerfrequencyerrorisgeneratedby shiftingtheclockfrequencyoftheEPDunitby 80Hz .Theclocksthatdrive theEPDunitandthearmlockingcontrolleraresynchronizedto accurately controltheDopplerfrequency.AnAC-coupledcontrollerwit haat magnituderesponseatDCisusedtoeliminatethefrequencyp ulling. 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 -2 -1.5 -1 -0.5 0 0.5 1 x 10 4 Time (s)Frequency Variations (Hz) Free-running VCO Stabilized VCO with AC-coupling Stabilized VCO with DC-coupling 10 -4 10 -2 10 0 10 2 10 -3 10 0 10 3 10 6 10 9 Frequency (Hz)Phase Noise (cycles/sqrtHz) Free-running VCO Stabilized VCO with AC-coupling Stabilized VCO with DC-coupling Figure6-2.(Left)Timeseriesofafree-runningVCOsignal,a stabilizedVCOsignalwith theAC-coupledcontrollerandastabilizedVCOsignalwithas tandard DC-coupledcontroller.TheAC-coupledcontrolleriscapab leofremovingthe frequencypullingthatshowsupinthesituationwiththeDCcoupled controller.(Right)Linearspectraldensitiesofthephase noiseofthethree timeseries.AlthoughtheDopplererrordoesnotcausefreque ncypulling withtheAC-coupledcontroller,thecontrollergainissign icantlylimitedand thenoisesuppressionperformanceiscompromised. 192

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InthebasicexperimentdescribedinSection4.2,thisoffset frequencymatchesthe Dopplerfrequencyperfectly,whilehereweintroduceacons tantDopplerfrequencyerror of 5Hz bychangingtheclockfrequencyoftheEPDunit.Inthepresence ofastandard DC-coupledcontroller,thefrequencypullingrateistheng ivenby D = =5Hz = s whichisdemonstratedbythestabilizedtimeseries(thegre encurveinFigure 6-2 (left)).Incontrast,theredcurverepresentsthetimeseri esstabilizedbyanAC-coupled controller.Thetimeseriesoffrequencyuctuationsexhib itthefrequencypullingin lockacquisition,whichisgivenbytheconvolutionbetween theDopplererrorand theAC-coupledcontrollertransferfunctionatlowfrequen cies.SincetheAC-coupled controllerisdesignedtohaveaconstantlowgainatDC,acon stantfrequencyoffset ( 9kHz )isaddedtotheoutputfrequencythroughthelockacquisiti onprocess.After theinitialfrequencypullingisrelaxedandthesystemente rsthesteadystate,the stabilizedfrequencystaysatandtheresidualfrequencyp ullingrateisstrictlyzero. Figure 6-2 (right)showsthelinearspectraldensitiesofthephaseuc tuationsofthe threebeatsignals.TheDC-coupledcontrollerhasthebesti n-bandperformancebutthe frequencyisrampingdownwiththeexpectedrate.Notethatt hisAC-coupledcontroller isjustforthepurposeofdemonstration:Weuseafairlyhigh lowunitygainfrequency comparedtoLISAtoperformthemeasurementinareasonableti me.Thislimitsthe noisesuppressioninourexperimentscomparedtothenoises uppressionexpectedin LISA.6.2.3FrequencyPullinginDualandModiedDualArmLocking6.2.3.1Time-domainsimulationswithAC-coupledcontroll er TheDopplerimpactbecomescriticalinadualarmlockingcon gurationwhen thearmlengthmismatchissmall.Herewerepresentatime-do mainsimulationof anAC-coupleddualarmlockingcongurationwithDopplerer rors.Thistime-domain simulationisrunningonaMatlabSimulinkmodelwithoating pointarithmetic.Indual armlockingthecriticalparameteristhedifferentialDopp lerfrequencyerrorbetween 193

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Table6-1.ParametersinAC-coupleddualarmlockingsimula tionswithDopplererrors ParameterSymbolValueUnits Samplingfrequency f s 10 Hz Free-runningfrequencynoise 0 1 10kHz = f HzHz 1 = 2 Averagedelaytime 33s Differentialdelaytime 0.255s Dopplerfrequencyerroronarm1-2 D2 150kHz Dopplerfrequencyerroronarm1-3 D3 149kHz Loopgain G 0 100 Table6-2.ParametersoftheAC-coupledlterusedinsimula tions zeros(Hz)poles(Hz) 10 7 5 10 6 10 7 5 10 6 10 6 3 10 3 5 10 5 3 10 3 5 10 5 3 10 3 0.13 10 3 0.33 10 3 twoarms,whereasmallerarmlengthmismatchresultsinasma llerdifferentialDoppler error.Inoursimulationweassumeafree-runninglaserwith alargeDopplerfrequency erroroneacharm,whilethedifferencebetweenthemis 1kHz .Theparametersused arelistinTable 6-1 andtheAC-coupledcontrollerisdesignedwiththezerosand poles listedinTable 6-2 TheblackcurveinFigure 6-3 (right)istheopen-looptransferfunctionwhichisgiven bythedualarmlockingsensor(thebluecurve)multipliedby theAC-coupledcontroller (theredcurve)withaloopgainfactor G 0 .TheAC-coupledcontrollerconsistsofa f 1 = 2 slopelow-passlterforthehighfrequencysuppressionand ahighpassltertoreduce thefrequencypulling.Theopen-looptransferfunctionpro videsabandwidthofaround 42kHz andaunitygainfrequencyof 15 Hz atlowfrequencies.Themaximumofthe open-loopgainisapproximately 100dB at 3mHz ;thenthegainstartstodecreasewith aslopeof f 2 duetotheAC-coupling.NotethattheslopeoftheAC-coupled lteratthe zero-crossingyieldsan s slopetoavoidexcessivephaseshift. 194

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0 2 4 6 8 10 x 10 5 -1.5 -1 -0.5 0 0.5 1 x 10 6 Time (s)Laser Frequency (Hz)0.04841 Hz/s 10 -8 10 -6 10 -4 10 -2 10 0 -200 -150 -100 -50 0 50 100 150 200 Frequency (Hz)Magnitude response (dB) OLTF(s) G(s) G eff (s) H eff (s) H(s) TF D+ (s) TF D(s) Figure6-3.(Left)Simulatedtimeseriesoftheoutputfreque ncychangeduetoa 1kHz DopplererrorwithanAC-coupledcontroller.(Right)Model edfrequency responsesofagenericdualarmlockinglooptocommonanddif ferential DopplererrorsinthepresenceofanAC-coupledcontroller. ThefrequencyresponsetoDopplererrorscanbemoreeasilye xplainedbydening aneffectivesensor H e ( s )= P ( s ) andeffectivecontroller G e ( s )= G 0 G ( s ) = ( s ) inthedifferentialpath.InFigure 6-3 (right)theyellowcurveandthecyancurve demonstratetheeffectivecontroller G e andtheeffectivesensor H e inthedifferential armrespectively.Theclosed-loopfrequencyresponsetoth edifferentialDopplererroris illustratedbythegreencurveinthegure.Dependingonthe gainleveloftheopen-loop transferfunction H e ( s ) G e ( s ) ,theclosed-loopfrequencyresponseconsistsofthree segmentsindifferentfrequencyregions: TF D ( s )= 8>>>>>><>>>>>>: 1 H e ( s ) 1 2 s when f f UG G e(s) 1+ H e ( s ) G e(s) when f f UG G e(s) when f f UG (6–4) where f UG isthelowerUGFofthehighpasslter. Wearemoreinterestedintheregionwheretheclosed-loopfr equencyresponseis approximatelyequaltotheeffectivecontroller,i.e., TF D ( s )= G e(s) = G 0 G ( s ) = ( s ) The 1 = s slopeinthefrequencyresponseindicatesaDopplererrorin ducedpullingstill occursintheoutputfrequency.However,comparedwiththel argegainfactorof 1 = (2 ) intheDC-couplingsituation,thegainfactornowissignic antlyreducedduetothe 195

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lowgainoftheAC-coupledcontroller.Ifwerecallthattheg ainfactorof 1 = s isdirectly proportionaltothedriftrateofthefrequencypulling,iti sstraightforwardtoderivethe reduceddriftrategivenby d L dt AC coupled =2 G 0 G ( s ) d L dt DC coupled (6–5) Thisrelationindicatesthatthereduceddriftrateisalsod irectlyproportionalto theloopgain.Ourtime-domainsimulationisbasedonalonga rmlengthmismatchof 2 =0.51s andaninaccuratedifferentialDopplerestimationof D =1kHz .Without theAC-couplingschemethedriftratewouldbe 1kHz = 0.51s 1961Hz = s forahighgain DC-coupledcontroller.Figure 6-3 (left)showsthetimeseriesofthevariationofthelaser frequencyusingourAC-coupledcontroller.Thefrequencyp ullinginlockacquisition yieldsamaximalfrequencyoffsetofapproximately 1.2MHz .Inthesteadystatethedrift ratehasbeenreducedto 0.04841Hz = s ,whichisexactlywhatweexpectbasedonthe frequencydomainanalysispresentedabove.Notethatwered ucedthelaserfrequency noiseinthissimulationtozerotobeabletodeterminethedr iftratewiththisaccuracy. 6.2.3.2Experimentswithmodieddualarmlockingsensor Inthefollowingexperimentswewillverifythefrequencypu llingrateofdual/modied dualarmlockinginthesteadystate.Theexperimentalsetup isillustratedbyFigure 6-4 ,wheretheDopplerfrequencyerrorsaregeneratedbyshifti ngtheclockfrequency. Theblocklabeledwithmappingvectorrepresentsthedualor modieddualarmlocking mappingvectordesignedinChapter5.TheDopplerfrequency errorsgeneratedinthe phasemeasurementsare 6.4Hz and 9.6Hz ,respectively.Therefore,with =33s and =0.025s ,theexpectedfrequencypullingratesare d L dt Dual = D 2 =64Hz = s, d L dt ModiedDual = D+ 2 =0.24Hz = s. (6–6) 196

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Figure6-4.Experimentalsetupofdual/modieddualarmlock ingwithDoppler frequencyerrors,whicharegeneratedfromtheclockfreque ncyoffsetof 200Hz .TheDopplerfrequencyerrorsinthephasemeasurementsare 6.4Hz and 9.6Hz ,respectively.Thecontrollerinthissetupisstillhigh-g ain DC-coupledtoobservethefrequencypullingrate. 0 100 200 300 400 500 600 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 Time (s)Frequency fluctuation (Hz) Modified dual arm locking (0.245 Hz/s) Dual arm locking(64.2 Hz/s) Figure6-5.Observedfrequencypullingofdual/modieddua larmlockingwithDoppler frequencyerrors.Foradualarmlockingandamodieddualar mlocking conguration,theobservedfrequencypullingrateis 64.2Hz = s and 0.245Hz = s ,respectively.Thesemeasurementresultsareconsistentw iththe theoreticalpredictionsanddemonstratethatthemodiedd ualarmlocking sensoriscapableofalleviatingtheDopplerissuesubstant iallywhenthearm lengthmismatchisnotverylarge. 197

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0 500 1000 1500 2000 2500 3000 3500 4000 -200 0 200 400 600 800 1000 Time (s)Frequency fluctuation (Hz) Free-running frequency fluctuations Frequency fluctuations stabilized by MDAL 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 -4 10 -2 10 0 10 2 10 4 Frequency (Hz)Frequency noise (Hz/rtHz) Free-running VCO noise Residual noise stabilized by MDAL Figure6-6.(Left)Thefrequencyuctuationsofthefree-ru nningVCOsignalandthe residualsignalstabilizedbymodieddualarmlocking.Com paredtothe frequencypullingofdualarmlockinginFigure 6-5 ,wherethestabilized frequencyhasbeenpulledbymorethan 35kHz in 600s ,themodieddual armlockingsensorrestrictsthefrequencypullingwithina rangeoflessthan 900Hz inonehour.(Right)Thenoisespectraofthetwotimeseries. Here wecanseethatthenoisesuppressionperformanceisessenti allynot affectedbythefrequencypullingafterthelineardriftisr emovedin post-processing. Theobservedfrequencypullingforbothtwocasesisshownin Figure 6-5 .Ina durationof 600s ,theoutputfrequencyhasdriftedbymorethan 35kHz whendualarm lockingisused.Incontrast,themodieddualarmlockingco ngurationlimitsthe1-hour frequencypullingwithinarangeoflessthan 900Hz ,asshowninFigure 6-6 (left).Note thatthisfrequencypullingrateisevensmallerthanatypic aldriftrateofcavitystabilized lasers.Fromthefrequencydatawehaveobtainedthefrequen cypullingratesare 64.2Hz = s and 0.245Hz = s .Themeasuredfrequencypullingratesmatchuptheexpected valuesandhasdemonstratedthatthemodieddualarmlockin gsensoriscapableof alleviatingtheDopplerissuesubstantiallywhenthearmle ngthmismatchisnotvery large.Figure 6-6 (right)showsthelinearspectraldensityoftheresidualfr equency noise,whichisessentiallynotaffectedbythelinearfrequ encydrift. 1 1 TheLSDoftheresidualfrequencynoiseislimitedbyprecisio nlossinthe E ( s ) lter, whichisanolddesign,andisinanycaseindependentofthefr equencypullingrate. 198

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0 500 1000 1500 1000 1500 2000 2500 3000 Time (s)Frequency fluctuation (Hz) cavitystabilized laser MDAL stabilized laser Figure6-7.Observedfrequencypullingofmodieddualarml ockingwithDopplererrors inthecavitystabilizedlaser.TheDopplerfrequencyerror ononearmisset tobe 10Hz whilezeroontheotherarm.Themeasuredfrequencypulling rateiscomparabletotothespontaneousdriftrateofcavity stabilizedlasers. Similarlinearfrequencydriftshavealsobeenobservedonst abilizedlasers.The experimentalsetupisessentiallyahybridofthesetupsinF igure 5-19 andFigure 5-22 : Themasterlaserisstabilizedviaamodieddualarmlocking loopthatisactuatedby aheterodynePLL.Weplaceasynchronizedfunctiongenerator inthemiddleofthe twodelaylinesononearmtosimulatetheDopplershift.Thec onvenienceofthissetup isthattheDopplershiftfrequencycanbepreciselytunedto generateacontinuously varyingDopplererrorononearm,whilesimultaneouslytheD opplerfrequencyonthe otherarmwillnotbeaffected. Figure 6-7 illustratesthelineardriftinthelaserfrequencystabili zedbyamodied dualarmlockingloopwith =33s and =0.025s .InthissetupwesettheDoppler frequencyerrorononearmtobe 10Hz ,whiletheDopplerfrequencyisperfectly accountedforontheotherarm.Therefore,thefrequencypul lingrateshouldbegiven by D+ = 2 = 0.15Hz = s andtheobservedfrequencypullingrateisconsistentwith thetheoreticalvalue.Forcomparisons,theintrinsicfreq uencydriftrateinthiscavity stabilizedlaserismeasuredtobeapproximately 0.18Hz = s andifdualarmlockingis 199

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100 200 300 400 500 600 700 800 900 -4000 -3000 -2000 -1000 0 1000 2000 Time (s)Frequency fluctuation (Hz) cavity stabilized MDAL stabilized MDAL simulation Figure6-8.Observedfrequencypullinginlockacquisition duetoastepfunctionin Dopplererrors.AfterthestepDopplererrorenterstheloop, thelaser frequencyhasbeenpulledby 3kHz inlockacquisitionandthendriftswith aconstantrateinthesteadystate.Atime-domainsimulatio nwiththesame conditionswasperformedandthetimeseries(thecyancurve )haveshown consistencywithourmeasurements. used,thefrequencypullingratewouldbecome 200Hz = s .Ourmeasurementhas demonstratedthatmodieddualarmlockingiscapableofred ucingthefrequency pullingratetoanegligiblelevelthatiscomparabletothes pontaneousdriftofcavity stabilizedlasers,providedthatareasonableDoppleresti mationcouldbegiven. Inaddition,wehavealsostudiedtheresponsivebehaviorof thestabilizedlaser frequencytoastepfunctionintheDopplererror.Usingthes ameexperimentalsetup, westartthearmlockingloopwithoutanyDopplererror.After thelaserfrequencyis settleddown,weinitiateastepfunctionintheDopplerfreq uencyby 10Hz andobserve thefrequencypullingintherelockduration.Thetimeserie softhelaserfrequencyis showninFigure 6-8 ,wherewecanseethelaserfrequencyhasbeenpulledby 3kHz andre-enteredthesteadystatewithaconstantdriftrateof 0.152Hz = s inabout 2min 200

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Toverifythislockacquisitionprocess,wehaverunatime-d omainsimulationwiththe sameconditionsandtheresultisgivenbythecyancurveinth egure.Wecanseethat inmostregionsthenumericalsimulationagreeswiththemea surementverywell. 2 The deviationthathappensduring 350s 400s ismainlyascribedtothedifferenceinthe lterdesignbetweenthesimulationandhardwaremodels,su chastheerrorcausedby bilineartransforminthetime-domain. Still,torealisticallysimulatethetime-variableDoppler frequencyerror,asimulatorto generatetime-variableDopplershiftfrequenciesthattra ckthespacecraftorbitalmotions mustbeimplementedontheEPDunit.However,anoscillationfr equencyof 10 8 Hz istoolowforlaboratoryexperimentstoeffectivelysimula te.Tocircumventthisproblem, theplanistorescaletheorbitalfrequencyofLISAandconseq uentlythedelaytimes, aswellaszerosandpolesinarmlockinglters,toarelative lyshort“lab-reasonable” timescale,whichrequirestheredesignoftheentirearmloc kingsystem.Currently, anewversionoftheEPDunitfeaturingsuchtime-varyingDoppl erfrequenciesisin development.Consequently,anadaptedarmlockingsystemw illbebuilttostudythe armlockingperformancewithrealistictime-variableDopp lererrors. 2 Thetimeseriesofthestabilizedlaserfrequencyshowsspur iouspeaks,whichare highlycorrelatedtotheunwarrantedjumpsintheinputlase rfrequency.Ourarmlocking loopiscapableofattenuatingfrequencyjumpswithinarang eof 1kHz 201

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CHAPTER7 CONCLUSIONANDOUTLOOK WetesteddifferentarmlockingschemesforLISAusingourLISA interferometer testbed.TheEPD-basedelectro-opticalarmlockinghardware simulationscan effectivelyandfaithfullyreproducerealisticLISA-likeco nditionssuchas 16s light traveltimeandvariableMHzDopplershifts,whicharevital forthevalidityofarmlocking experiments.Inparticular,thedualandmodieddualarmlo ckingcongurationsare linearcombinationsofLISAinter-spacecraftphasemeasure mentsoptimizedforthe noiseperformanceandminimizationoftheDoppler-induced frequencypullingissue. Inthisdissertationexperimentalvericationstoexamine themunderrealisticLISA-like conditionsarepresentedforthersttime. 7.1ControlSystemofArmLocking OnUFLISwehaveimplementedfourbasicarmlockingcongura tionsand demonstratedtheircontrolsysteminaseriesofvericatio nmeasurements.Inour experimentsalaserbeatsignalwithaheterodynefrequency intherangeof 2 20MHz isreceivedandconvertedintoanelectronicsignalattheph otodiode.Thelongarm interferometeroutputissimulatedviatheanalogmixingof theelectronicsignaland itsdelayedreplica.Bydemodulatingtheinterferometerout puts,thephasemeters extractthefrequencyuctuationsfromeachheterodynebea tsignal.Adedicated mappingvectorisusedtosynthesizeandmanipulatethefreq uencyuctuationsto formanadequatelinearcombination,knownastheerrorsign alofarmlocking.The measurementshaveshownthattheopen-looptransferfuncti onsofthesearmlocking sensorsagreeverywellwiththetheoreticaldesign. ThearmlockingcontrolleronUFLISwasdesignedandbuiltin parallelpaths:A compensatorlteryieldsa 1 = p f slopeatfrequenciesabove 1Hz toalleviatethelarge phaseshiftduetotheround-tripdelays.Four-stagecascad edintegratorsprovide sufcientnoisesuppressionsintheLISAband.Throughtheco mbinationofthesetwo 202

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controlpaths,arobustcontrolsystemofarmlockinghavebe enachievedwithboth ahighbandwidthandasatisfactorynoisesuppressionperfo rmance.Consequently, wepresentthersteverexperimentalproofofthesearmlock ingschemesinthe presenceofrealisticLISA-likelongtimedelaysandDopplers hifts,withunprecedented noisesuppressionperformances.Inourexperimentsweachi evedupto6ordersof magnitudenoisesuppressionviasinglearmlockingandupto 8ordersofmagnitudevia dual/modieddualarmlockingintheLISAband. Wealsohavestudiedtheincorporationofarmlockingwithca vitystabilizedlasers. ThecombinationofarmlockingandthePDHstabilizationtech niquewillprovidemore noisesuppressions;however,itrequiresthereferenceoft hePDHstabilizationtobe frequencytunable.Therefore,wepresentamodicationtot hestandardPDHsetup knownasthePZTactuator,bywhichtheresonantfrequencyoft hecavitycanbe adjustedinaccordancewiththearmlockingfeedbacksignal .Wealsopresentasecond methodtointroduceanauxiliarylaseroffsetphase-locked tothemasterlaser.Theslave laserreproducesthenoisepropertyofthemasterlaseraswe llasobtainsthefrequency tunabilitythroughtheoffsetfrequencyofthelocaloscill ator.Thesemeasurementshave demonstratedthatarmlockingcanbeeasilyreconciledwith thecavitystabilization withoutexplicitlydegradingthenoisesuppressionperfor manceofeitherofthem. 7.2NoiseLimitations Dependingonschemeusedandactuallighttraveltimes,arml ockingcanbegain limitedorlimitedbyseveralexternalnoisesources.Thepr imarynoisesourcesinarm lockingincludeclocknoise,spacecraftmotion,shotnoise ,technicalnoiseandpossibly far-endPLLnoise.Inourexperimentsweanalyzedtheimpacto fvariousnoisesources onthenoisesuppressionperformanceunderdifferentcircu mstances. Wealsostudiedtheinuenceofdigitalizationnoiseinthed igitalarmlocking sensorsandcontrollersontheperformanceofarmlocking.T heseresultscanbe usedtodesignthedigitalarmlockingsystemforLISA.TheADCno iseinphase 203

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measurementsisanotherprimarynoisesourcethatlimitsth earmlockingperformance inourexperiments.InrealisticLISAtheADCnoiseperformanc einthephasemeterwill beoptimizedtomeetthe 1pmHz 1 = 2 requirementandshouldnotcauseaproblem. Nevertheless,ourmeasurementshaveveriedthatthelimit ingADCnoiseoor isconsistentwiththetheoreticalpredictionsintheconte xtofUFLIS.Moreover,by investigatingtheADCnoiselimitationwepresenttheexperi mentalproofthatwiththe samearmlengthmismatch,thenoiseperformanceofmodiedd ualarmlockingis substantiallysuperiortothatofdualarmlocking. Amorerealisticdemonstrationofmodieddualarmlockingw iththefar-end transpondernoiseisalsopresentedinthisdissertation.I ntheexperimentweimplemented ananalogphase-lockedloopatthefar-endtophaselockthel ocalbeatsignaltothe incomingdelayedsignal.Thetranspondernoiselimitation observedinthestabilized lasernoisecanbeequivalentlyconsideredasamanifestati onofanynoisesource presentedonthefarspacecraftastheyallcoupleintothear mlockingcontrolsystemin thesamefashion,i.e., = H + H ( 2 + 3 )+ H H ( 2 3 ), (7–1) where i i =2,3 isanytranspondernoiseintroducedonthefarspacecraft SC 2,3 Wemeasuredthetranspondernoiselimitinthestabilizedla serandthePLLnoiseatthe far-endindependentlyandfoundtheminaccordancewithEq. 7–1 .Theexperimentalso revealsthatinthepresenceofanon-negligibletransponde rnoiseandarelativelyshort armlengthmismatch( =0.025s ),ourmodieddualarmlockingcongurationwith cavitystabilizationstillsufcientlymeetstheTDIcapab ilitywithamarginofmorethan 25,000 at 3mHz 7.3Doppler-inducedFrequencyPulling TheimpactofDopplershiftsonarmlockingwasrstlyidenti edfromthearm lockingexperimentsonUFLISandisaverycriticalissuetha twouldultimatelyaffect 204

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thefeasibilityofarmlocking.Morein-depthresearchonth eDopplerimpactgavea betterunderstandingoftheDopplerfrequencyerror,frequ encypullingandhelped toidentifysolutionstoovercomeit.Thearmlockingcontro lsystemimplementedon UFLISprovidesanidealenvironmenttotesttheeffectsoffr equencypullingandrelevant remedies. ByintroducingaDopplerfrequencyerrorinthephasemeasure ment,wehave demonstratedhowthefrequencypullingdependsondifferen tarmlockingsensors. Theseexperimentsarethersteverexperimentalproofsofa rmlockinginarealistic andnon-staticLISAconstellation.Inparticular,anAC-cou pledcontrollerforsingle armlockinganddualarmlockinghasbeendevelopedinhardwa reandsoftware respectively,conrmingthatthefrequencypullingcanbes ubstantiallyreducedoreven eliminated.Wealsocomparedthedistinctfrequencypullin gratesofdualarmlocking andmodieddualarmlockinginresponsetosameDopplerfreq uencyerrors.Our experimentindicatesthatthemodieddualarmlockingsens oriscapableofalleviating thefrequencypullingrateinthesteadystatetoanegligibl elevel,whichiscomparableto thetypicaldriftrateofacavitystabilizedlaser. Wealsohavestartedtheinvestigationoftime-variableDop plerfrequencyerrorsin hardwaresimulations.Theobservedfrequencypullinginlo ckacquisitionduetoastep functionintheDopplererroragreesverywellwiththetimed omainsimulationunderthe sameconditions. 7.4Outlook Withtheeverlastingendeavorongravitationalwavedetecti onsinthepast50years, gravitationalwaveastronomyisexpectedtoopenanewwindo wtotheuniverseinthis decade,withthehelpofthenextgenerationground-basedde tectors,suchasadvanced LIGO,VIRGOandLCGT,andthespace-basedprojectLISA.Fromgra vitational wavedetectionsoncompactbinaries,blackholesandBigBangr elics,entirelynew 205

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informationonastrophysicsandcosmologywillberevealed andconsequentlyextend ourunderstandingoftheuniverse. Thelow-frequencyregionthatLISAwillprobeisfullofinter estinggravitationalwave sources,manyofwhichstillhaveneverbeenseen.Forcoales cencesofmassiveblack holes,aswellasprimordialgravitationalwavesgenerated intheextremelyshorttime afterBigBang,aspacegravitationalwaveobservatorysuchas LISAmightbetheonly accessiblemethodtostudythem.LISAwillpreciselymeasure thousandsofbinary systemsinourGalaxyandyieldnewinsightsintotheastroph ysicsofbinarystars;LISA willalsoprovideaveryrsttestofgeneralrelativityinex tremelystronggravitational eldsonthevicinityofblackholesbylisteningtotheinspi ralsignals. Fromthepointofviewofinstrumentation,LISAisalsoatechn icallychallenging missionduetotherequiredhighsensitivity.Thenoiserequ irementsforlaserinterferometry anddrag-freecontrolareunprecedentedlystringentforcu rrenttechnology.Thelaser frequencystabilizationwasonceconsideredadifcultiss uethatmaycauseTimeDelay Interferometrytofail.Nowmultiplefrequencycontrolopt ionshavebeenproposedto solvethisproblemandtheselectionismorebasedonthecons iderationofcomplexity, cost,etc.Armlockingasoneoftheproposedlaserfrequencys tabilizationtechniques, entirelyexploitsLISA'slongarminterferometrytogenerate anerrorsignal.Thissimple architecturesubstantiallysavesonimplementationresou rces.Inaddition,theintegration ofarmlockingandotherpre-stabilizationsubsystemscans uppressthelaserfrequency noisebelowthepre-TDIrequirementof 1m rangingcapabilitywithanabundantmargin, whichwillfurtherrelaxthedependenceonranging.Theissu eofDoppler-induced frequencypullingslightlycomplicatesthedesignoftheco ntrolsystemofarmlocking andadditionalon-boarddataprocessingtoestimatetheDop plerfrequencyisneeded. Nevertheless,bothanalysisandexperimentshaveshowntha tmodieddualarm lockingsolvesthisDopplerproblem.Welookforwardabette runderstandingofLISA technology,aswellasagravitationalwaveuniversethatLI SAwillultimatelydiscover. 206

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BIOGRAPHICALSKETCH YinanYuwasborninTianjinintheeastofChina.Hespentmost ofhischildhood andadolescenceinTianjinandgraduatefromNankaiUnivers itywithabachelor's degreeinsciencein2005.Inhissenioryearhedecidedtoapp lyforgraduateschool topursuehisPhDdegreeaswellastoexpandhiscareerofscien ticresearch.He wasacceptedbytheDepartmentofPhysicsattheUniversityof Floridawithteaching assistantshipin2005.Inthesummerof2006hejoinedDr.Mue ller'sresearchgroup andbeganworkingwithhimonthebenchtopexperimentsofLas erInterferometerSpace Antenna(LISA)instrumentationsforgravitationalwavedetec tions.Theirworkhas demonstratedthevalidityandfeasibilityoflaserfrequen cystabilizationbymeansofarm locking.Afterveyearsofresearch,hegraduatedfromtheUn iversityofFloridawitha DoctorateofPhilosophyinphysics. 214