Citation
Digital Heterodyne Laser Frequency Stabilization for Space-Based Gravitational Wave Detectors and Measuring Coating Brownian Noise at Cryogenic Temperatures

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Title:
Digital Heterodyne Laser Frequency Stabilization for Space-Based Gravitational Wave Detectors and Measuring Coating Brownian Noise at Cryogenic Temperatures
Creator:
Eichholz, Johannes Michael
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (212 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
MUELLER,GUIDO
Committee Co-Chair:
SIKIVIE,PIERRE
Committee Members:
TANNER,DAVID B
WHITING,BERNARD F
SAWYER,WALLACE GREGORY
Graduation Date:
12/18/2015

Subjects

Subjects / Keywords:
Cryogenics ( jstor )
Interferometers ( jstor )
Laser interferometer gravitational wave observatory ( jstor )
Laser interferometer space antenna ( jstor )
Lasers ( jstor )
Noise measurement ( jstor )
Noise reduction ( jstor )
Noise temperature ( jstor )
Signals ( jstor )
Thermal noise ( jstor )
Physics -- Dissertations, Academic -- UF
cryogenic -- frequency-stabilization -- gravitational-waves -- heterodyne -- thermal-noise
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
The Laser Interferometric Gravitational-Wave Observatory project LIGO operates two facilities in the US that house kilometer-scale laser interferometers, in which differential length fluctuations are monitored in an attempt to detect gravitational waves. The propagation of these minute space-time perturbations, which are generated in observable magnitude only by violent astronomic events such as supernovae or binary mergers of massive objects like neutron stars and black holes, follows directly from the linearized Einstein equations. The LIGO sensitivity limit at and around 100Hz is heavily impacted by Brownian noise in the reflective coatings of its inertial test masses, which is a consequence of the fundamental fluctuation-dissipation theorem. Direct measurements of the mechanical loss in optical coatings that causes this noise across LIGO-relevant frequencies are rare, and many predictions rely on loss values that have been interpolated from observations at much higher frequencies. The main science objective of the work presented in this dissertation was the construction of a thermal noise test bed at the University of Florida to participate in the quest to find better coatings. A novel frequency stabilization method that we developed for potential implementation in the Laser Interferometer Space Antenna space mission LISA also finds an application in the THermal noise Optical Resonator experiment THOR. For the assessment of coating Brownian noise we expanded some of the core phasemeter technology concepts of LISA to the LIGO band. Since the application of cryogenics to next generation gravitational wave detectors is becoming a very probable prospect method to lower thermal noise, it is important to understand the losses in cryogenically cooled mirrors. Because direct measurements of coating thermal noise at cryogenic temperatures have not yet been obtained, a second apparatus similar to THOR was constructed with the capability to cryogenically cool its test mirrors. Preliminary measurements performed with CryoTHOR conclude the presented work. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: MUELLER,GUIDO.
Local:
Co-adviser: SIKIVIE,PIERRE.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-06-30
Statement of Responsibility:
by Johannes Michael Eichholz.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
6/30/2016
Classification:
LD1780 2015 ( lcc )

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DIGITALHETERODYNELASERFREQUENCYSTABILIZATIONFORSPACE-BASEDGRAVITATIONALWAVEDETECTORSANDMEASURINGCOATINGBROWNIANNOISEATCRYOGENICTEMPERATURESByJOHANNESEICHHOLZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015JohannesEichholz

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Tomymother,whoenduredmanyhardshipstoopenpathsinlifeformeandmybrother

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ACKNOWLEDGMENTSImayhavenevercometoGainesvilleifitwasn'tforKendallAckley,whobrightenseverydayofmylife,helpedmebuildahomeawayfromhome,andprovidedimmeasurablesupportduringmytimeingradschool.SpecialthanksgotomymotherLuise,forgivingmethefreedomtogomyownwayearlyoninmylifeandalwayssupportingmeinmydecisions,andmybrotherJimmy,forbeingasourceofinspirationandconstantlyremindingmeoftheimportanceofbeingtruetomyself.OfthemanyfriendsandcolleaguesthathaveenrichedmytimeatUF,IwanttothankinparticularMichaelHartman,forsharingthepainandjoyofbuildinganexperimentfromthegroundup,RichOttensforhisvaluedexperiencewithcryogenicsystems,PaulFulda,foralwaysgladlyhelpingwithhisexpertiseinoptics,andGiacomoCianiforensuringthatresultsareproperlyunderstoodbeforerushingtothenextstep.Myco-workersandcross-disciplineintramuralteammatesAaronSpector,RyanGoetz,andBobbyBondimprovedthelifeonandocampusinmanyaspects.Ifurtherthankmyformerco-workersShawnMitryk,whobroadenedmyknowledgeofFPGAsandunderstandingofdigitalsystems,andPepSanjuan,forbeingashiningexampleofadiligentworker.Myprofessionalcareerwouldneverhavetakenthispathifitwasn'tformysupervisorGuidoMueller,whorecruitedmeoutoftheAEIandguaranteedasmoothtransitionbetweencontinents.Thebalancebetweenhisguidanceandencouragementtoworkindependentlypreparedmewellfortheroadthatliesahead.IalsothankmycommitteemembersDavidTanner,BernardWhiting,GregSawyer,andPierreSikiviefortheiroccasionalsupportandrefreshingadvice.MarcLink,BillMalphurs,EdStorchandJohnVanleerfromthephysicsmachineshopdeservealotofcreditfortheirmuchappreciatedworkandadvicewithdesigns.Lastly,mythanksgotoNSFandNASA,forprovidingthefundingthatenabledmetoconcentrateonresearchforthevastmajorityofmytimeingradschool. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 16 1.1TheGravitationalUniverse .......................... 16 1.2NotetotheReader ............................... 19 2GRAVITATIONALWAVES ............................. 21 2.1SpecialRelativity ................................ 21 2.2GeneralRelativity ............................... 22 2.2.1CovariantDerivative .......................... 23 2.2.2MeasuringCurvature .......................... 25 2.2.3TheEinsteinTensor ........................... 27 2.2.4Einstein'sEquations .......................... 27 2.2.5SpacetimeExamplesandBlackHoles ................. 29 2.3TestsofGeneralRelativity ........................... 30 2.3.1SolarSystemDynamics:PerihelionAdvance ............. 30 2.3.2GravitationalLensingofLight ..................... 30 2.3.3GravitationalRedshift ......................... 31 2.3.4FrameDragging ............................. 31 2.3.5ShapiroDelay .............................. 31 2.3.6TheHulse-TaylorBinaryPulsar .................... 31 2.4PropagationofMetricPerturbations ..................... 32 2.4.1WaveEquation ............................. 32 2.4.2PolarizationsofGravitationalWaves ................. 33 2.5SourcesofGravitationalRadiation ...................... 34 2.5.1Core-CollapseSupernovae ....................... 35 2.5.2CompactBinaryCoalescences ..................... 36 2.5.3Extreme-Mass-RatioInspirals ..................... 36 2.5.4Super-MassiveBlackHoleMergers ................... 37 2.5.5PrimordialGravitationalWaves .................... 37 3INTERFEROMETRICGRAVITATIONALWAVEDETECTORS ........ 38 3.1GravitationalWavesandLaserInterferometry ................ 39 3.2LaserInterferometerGravitational-WaveObservatory ............ 41 5

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3.2.1AdvancedLIGOInterferometerTopology ............... 42 3.2.2AdvancedLIGOInstrumentSensitivity ................ 43 3.2.3CryogenicsandGravitationalWaveDetectors ............ 46 3.2.4CoordinationwithOtherGround-BasedDetectors .......... 47 3.3LaserInterferometerSpaceAntenna ...................... 48 3.3.1Time-DelayInterferometry ....................... 52 3.3.2OpticalRanging ............................. 53 3.3.3LaserFrequencyNoiseinLISA ..................... 54 4BROWNIANDISPLACEMENTNOISEINMIRRORS .............. 55 4.1Fluctuation-DissipationTheorem ....................... 56 4.1.1InternalDampinginSolidMaterials .................. 58 4.1.2LevinForcesandFormalism ...................... 60 4.2CoatingandSubstrateBrownianNoise .................... 62 4.2.1AveragedCoatingLossAngle ..................... 63 4.2.2ClassicationintoParallelandPerpendicularLosses ......... 63 4.2.3ClassicationofLossesforBulkandShearModes .......... 64 4.3CoatingNoiseatCryogenicTemperatures .................. 66 4.3.1TemperatureDependenceofMaterialParameters .......... 67 4.3.2ACryogenicThermalNoiseTestBed ................. 67 5EXTENDEDFREQUENCYRANGELISA-TYPEPHASEMETER ....... 69 5.1HeterodyneInterferometry ........................... 70 5.1.1BeatSignalGeneration ......................... 70 5.1.2VariantsofHeterodyneInterferometry ................. 72 5.2TrackingPhasemeterConcept ......................... 73 5.2.1PhaseLockLoops ............................ 74 5.2.2LinearizedModel ............................ 76 5.2.3IQreadout ................................ 78 5.3DesignConsiderations ............................. 79 5.3.1FieldProgrammableGateArrays ................... 80 5.4UniversityofFloridaLISAInterferometrySimulator ............. 81 5.4.1UFLISPhasemeter ........................... 82 5.5CompactPhasemeterSolution ......................... 82 5.5.1AX3065HardwareSpecications .................... 84 5.5.2FPGADesignWorkow ........................ 86 5.5.3InterfacingviaPCIBus ......................... 87 5.5.4DataLoggingusingDMATransfer .................. 87 5.6Phasemeterz-DomainModel .......................... 89 5.6.1CICFilters ................................ 90 5.6.2PMFeedback .............................. 92 5.7InstrumentalNoiseLimitations ........................ 94 5.7.1QuantizationNoise ........................... 94 5.7.2TimingJitter .............................. 95 6

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5.8LISABandFrequencyPerformance ...................... 98 5.8.1DierentialNoise ............................ 98 5.8.2EntangledPhasePerformance ..................... 100 5.9HighBandwidthPhasemeter .......................... 102 5.9.1TruncationOptimization ........................ 103 5.9.2InternalBandwidthEnhancement ................... 104 5.9.3LIGOBandInstrumentalNoise .................... 107 6DIGITALHETERODYNELASERFREQUENCYSTABILIZATION ...... 109 6.1OpticalCavities ................................. 110 6.1.1MirrorReectionandTransmissionCoecients ........... 110 6.1.2OpticalCavityTransferFunctions ................... 114 6.2StabilizationMethod .............................. 117 6.2.1ErrorSignalExtraction ......................... 117 6.2.2DemonstrationPhase .......................... 121 6.2.3HeterodyneStabilizationControlTheory ............... 124 6.3ControllerImplementationinPhasemeterHardware ............. 127 6.3.1DigitalLaserController ......................... 128 6.3.2FastPhasemeterIndirectDemodulation ................ 130 6.3.3DualHeterodyneStabilization ..................... 132 6.4DigitalHeterodyneStabilizationforLISA .................. 135 6.4.1CavityParameters ............................ 136 6.4.2HeterodyneStabilizationSetupforLISA ............... 140 6.4.3StabilizationResults .......................... 142 7CRYOGENICTHERMALNOISETESTBED ................... 145 7.1CryoTHOROverview .............................. 146 7.1.1MeasurementPrinciple ......................... 146 7.1.2CryogenicTankUpgrade ........................ 148 7.1.3TestBenchSuspensions ......................... 150 7.1.4ActiveSeismicIsolation ......................... 151 7.1.5TestCavityMounting .......................... 154 7.1.6FlexibleThermalLinks ......................... 156 7.2CavityAssembly ................................ 157 7.2.1MirrorContacting ............................ 158 7.2.2CavityMounts .............................. 160 7.3InterferometricComponents .......................... 161 7.3.1InjectionBench ............................. 162 7.3.2DiagnosticBench ............................ 163 7.3.3TestBench ................................ 164 7.4ExperimentCharacterization .......................... 167 7.4.1OpticalCavitiesRevisited ....................... 167 7.4.2ReferenceSystemNoiseFloor ..................... 169 7.4.3CoolingCapacity ............................ 171 7

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7.5CryoTHOROperation ............................. 173 7.5.1HeterodyneSidebandLocking ..................... 174 7.5.2RoomTemperatureMeasurements ................... 175 7.5.3FrequencyNoiseofaCryogenicallyCooledTestCavity ....... 177 8CONCLUSIONS ................................... 180 8.1FastStreamingLISAPhasemeterforLIGOBand .............. 180 8.1.1VersatileUsage ............................. 180 8.2HeterodyneLaserFrequencyStabilization .................. 181 8.2.1ApplicabilitytoLISA .......................... 181 8.3CryogenicThermalNoiseTestBed ...................... 183 8.3.1NecessaryImprovements ........................ 183 APPENDIX AESSENTIALSOFSIGNALPROCESSINGANDNOISESPECTRA ...... 185 A.1LaplaceandFourierTransformations ..................... 185 A.2Wiener-KhinchinTheoremandSpectralDensities .............. 186 A.3LinearSystems ................................. 188 A.4FrequencyandPhaseNoise .......................... 189 BGAUSSIANBEAMS ................................. 190 B.1HelmholtzEquationandParaxialApproximation .............. 190 B.2GaussianBeamParameters .......................... 192 B.3Hermite-GaussModes ............................. 193 B.4MatrixFormalismforGaussianBeams .................... 195 COPTICALCAVITIES ................................ 197 C.1StabilityCriterion ................................ 197 C.2SupportedLaserModes ............................ 199 REFERENCES ....................................... 202 BIOGRAPHICALSKETCH ................................ 212 8

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LISTOFTABLES Table page 4-1CoecientsforthebulkandsheardecompositionofcoatingBrowniannoise. .. 65 5-1AcromagAX3065specications. ........................... 84 5-2InternalPLLsignaltruncation ............................ 104 5-3PhasedetectorIIRltercoecients ......................... 105 6-1Ultra-low-expansionspacermaterialproperties ................... 137 6-2Referencecavityassemblyparameters ........................ 139 7-1Stacisfeetgainsettings ............................... 153 7-2CryoTHORcavitycongurations .......................... 160 7-3Comparisonofphotodetectors ............................ 163 7-4CryoTHORmeasuredcavityproperties ....................... 168 9

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LISTOFFIGURES Figure page 2-1Paralleltransportofvectorsonasphere ...................... 24 2-2Riemanntensorillustration ............................. 26 2-3Polarizationsofgravitationalwaves ......................... 33 2-4Frequencybandsofgravitationalradiation ..................... 35 3-1Michelsoninterferometerintestmassring ..................... 40 3-2TheLIGOtwinobservatories ............................ 41 3-3AdvancedLIGOtopology .............................. 42 3-4AdvancedLIGOnoisebudget ............................ 44 3-5Gravitationalwaveobservatoriesontheglobe ................... 48 3-6LISAorbits ...................................... 49 3-7LISAarmsplit .................................... 50 3-8LISAdesignsensitivity ................................ 51 3-9Time-delayinterferometry .............................. 53 5-1Beatsignalcreation .................................. 71 5-2Twovariantsofheterodyneinterferometry ..................... 72 5-3ConceptualPLLsketch ................................ 74 5-4PLLLaplacemodel .................................. 76 5-5PhasemeterIQreadout ................................ 79 5-6FPGAfunctionality .................................. 81 5-7Penteksystemmodularassembly .......................... 83 5-8AcromagAX3065componentoverview ....................... 85 5-9AX3065coredesignworkow ............................ 86 5-10AX3065PMstatecontrol .............................. 87 5-11AX3065dataformat ................................. 88 5-12AX3065datastreamingprocess ........................... 89 10

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5-13CIClterarchitecture ................................ 91 5-14PMz-domainmodel ................................. 92 5-15DigitalPLLloopgains ................................ 93 5-16AX3065timingjittersetup .............................. 96 5-17AX3065timingjitter ................................. 97 5-18AX3065dierentialPMperformance ........................ 99 5-19AX3065entangledphasemeasurementsetup .................... 100 5-20AX3065entangledphaseperformance ........................ 101 5-21Quantizationnoiseinjectionmethod ........................ 103 5-22Quantizationnoisemodel .............................. 105 5-23In-loopIIRlter ................................... 106 5-24In-loopPLLIIRandCICltertransferfunctions ................. 106 5-25RevisedAX3065PMloopgains ........................... 107 5-26PMLIGObanddierentialperformance ...................... 108 6-1Dielectricmirror ................................... 111 6-2Cavitygeometryandeldreference ......................... 114 6-3Cavityresonancebehavior .............................. 116 6-4Basicheterodynesetup ................................ 118 6-5Heterodynestabilizationerrorsignal ........................ 120 6-6Heterodynestabilizationdemonstrationsetup ................... 121 6-7HSdemonstrationcavity ............................... 122 6-8HSdemonstrationthermalshield .......................... 122 6-9Heterodynestabilizationdemonstrationphaseresult ................ 123 6-10Heterodynelockopticalgain ............................. 126 6-11AX3065D/Aconversionprotocol .......................... 128 6-12AX3065digitalPI-controller ............................. 130 6-13AX3065indirectdemodulationmethod ....................... 131 11

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6-14HSdualservoarchitecture .............................. 133 6-15Potentialchannelcrosstalkinindirectheterodynelocks .............. 135 6-16Opticalgainversusmirrortransmissivities ..................... 138 6-17DualHSsetupforLISAbandmeasurements .................... 140 6-18HSdemonstrationexperiment ............................ 141 6-19DualHSfrequencynoiseintheLISAband ..................... 143 7-1THORprinciplesetup ................................ 147 7-2CryoTHORinitialinventory ............................. 149 7-3CryoTHORhardwaremodications ......................... 150 7-4CryoTHORtestbenchsuspensions ......................... 151 7-5CryoTHORactiveseismicisolationsystem ..................... 152 7-6Stacissystemsuppressionofaccelerationnoise ................... 154 7-7Cryogenictestcavitymountingsolution ...................... 155 7-8Flexiblethermallinks ................................ 157 7-9CryoTHORassembledreferencecavity ....................... 159 7-10Cavitymountingsolutions .............................. 161 7-11CryoTHORinjectionbench ............................. 162 7-12Photodetectordierentialphasenoisemeasurement ................ 164 7-13PhotodetectorNoiseInvestigation .......................... 165 7-14SchematicviewofcompleteCryoTHORopticslayout ............... 166 7-15CryoTHORreferencesystemperformance ..................... 170 7-16CryoTHORcooldownimpressions .......................... 172 7-17CryoTHORcooldowntemperature ......................... 173 7-18Osetheterodynesidebandlocking ......................... 174 7-19Osetheterodynesidebandlockingperformance .................. 175 7-20Testcavityfrequencynoise .............................. 177 7-21Cryogenicallycooledtestcavityfrequencynoise .................. 178 12

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8-1PossibleopticalbenchimplementationofHeterodyneStabilization ........ 182 B-1Gaussianbeamparameters .............................. 192 B-2HermiteGaussmodes.ThenomenclaturefortheorthogonalmodesusesthenumberofnodesofthespatialdistributionintheTransverse-Electro-Magnetic(TEM)modes.Toprow,lefttoright:TEM-00,TEM-10,TEM-01.Bottomrow,lefttoright:TEM-20,TEM-11,TEM-02. ..................... 194 C-1Twomirrorresonator ................................. 198 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDIGITALHETERODYNELASERFREQUENCYSTABILIZATIONFORSPACE-BASEDGRAVITATIONALWAVEDETECTORSANDMEASURINGCOATINGBROWNIANNOISEATCRYOGENICTEMPERATURESByJohannesEichholzDecember2015Chair:GuidoMuellerMajor:PhysicsTheLaserInterferometricGravitational-WaveObservatoryprojectLIGOoperatestwofacilitiesintheUSthathousekilometer-scalelaserinterferometers,inwhichdierentiallengthuctuationsaremonitoredinanattempttodetectgravitationalwaves.Thepropaga-tionoftheseminutespace-timeperturbations,whicharegeneratedinobservablemagnitudeonlybyviolentastronomiceventssuchassupernovaeorbinarymergersofmassiveobjectslikeneutronstarsandblackholes,followsdirectlyfromthelinearizedEinsteinequations.TheLIGOsensitivitylimitatandaround100HzisheavilyimpactedbyBrowniannoiseinthereectivecoatingsofitsinertialtestmasses,whichisaconsequenceofthefundamentaluctuation-dissipationtheorem.DirectmeasurementsofthemechanicallossinopticalcoatingsthatcausesthisnoiseacrossLIGO-relevantfrequenciesarerare,andmanypredic-tionsrelyonlossvaluesthathavebeeninterpolatedfromobservationsathigherfrequencies.Sincetheapplicationofcryogenicstonextgenerationgravitationalwavedetectorsisbeco-mingaveryprobableprospectmethodtolowerthermalnoise,itisimportanttounderstandandcharacterizethelossesincryogenicallycooledmirrors.ThemainscienceobjectiveoftheworkpresentedinthisdissertationwastheconstructionofathermalnoisetestbedattheUniversityofFloridatoparticipateinthequesttondbettercoatingsandbepreparedforthecryogenicrouteoffuturedetectors.AnovelfrequencystabilizationmethodthatwedevelopedforpotentialimplementationintheLaserInterferometerSpaceAntennaspace 14

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missionLISAalsondsanapplicationintheCryogenicThermalNoiseOpticalResonatorexperimentCryoTHOR.FortheassessmentofcoatingBrowniannoiseweexpandedsomeofthecorephasemetertechnologyconceptsofLISAtotheLIGOband.Theexperimentaldetails,referencesystemcharacterization,andpreliminarymeasurementsperformedwithCryoTHORconcludethepresentedwork. 15

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CHAPTER1INTRODUCTIONGravitationisthemosttangibleofthefourfundamentalforces,beingtheonethatwemostconsciouslyexperienceeveryday.Yet,itisprobablyalsothemostmysteriousamongstthem,asitexistsoutsidethestandardmodelofparticlephysics,whichsuccessfullyunitedthedescriptionofelectromagnetism,theweak,andthestrongnuclearforcewithinasingletheory.Thequantizationofelds,whichstandsattheheartofparticlephysics,failstodeliverre-normalizableexpressionsinthecaseofgeneralrelativity(GR),whichisthebestdescriptionofgravitywehavetodate.Asaconsequence,thetheoriesstandsidebyside,andeachonitsownprovidespredictionsthatmatchtheobservedphysicalrealityexcellently.Therefore,astechnologyadvances,itbecomespossible(andnecessary)toputtheexistingtheoriestothetest,eitherwithincreasinglyprecisemeasurements,orinpreviouslyuntouchedlimitsontheenergyscale.Ontheparticlephysicssidemankindhasadvancedtoarticiallycreatehighenergyregimesinparticleaccelerators,andengageinactiveandtargetedsearchesfornewphysics.TheequationE=mc2isoneofphysics'mostprominentformulas,andonewaytoreaditisthatmassisanextremelycompactformofenergy,andthatevenweakgravitationaleectsinvolvehugeamountsofenergy.Asaresult,whatcanbeconsideredhighenergyforparticlephysics,isbutamerefractionofwhatwouldgeneratestronggravitationaleects.Wearethereforelimitedtopassiveobservationsofstrongeldgravity,andunfortunately(orfortunately),truestrongeldsourcesarenotfoundinoursolarsystem,norourrelativegalacticneighborhood. 1.1TheGravitationalUniverseInthehistoryofastronomy,electromagnetic(EM)radiation(invariousbandsandatvariousenergiesandfrequencies)hasforalongtimebeentheonlymediumforobservationsoftheuniverse,andonly30yearsagoneutrinodetectorsbegancomplementingthepicturefromtheparticleside.Thegravitationaleldisbyfarnotvoidofinformationaboutoursurroundings,butthemediatorshavealwaysbeenphotons.Inrecentyearsimportant 16

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gravitationalobservationsweremade,whichhaverequiredare-thinkingoftheconstituentsoftheuniverse.Theanalysisofgalaxyrotationcurvesshowedthatouterregionstendtorotatetoofastfortheaccountablemasstoholdthegalaxytogether.Thenecessaryadditionaldarkmattercannotsimplybeconcentratedinthecenter,butmustbespreadacrossthegalaxy,andparticleswithinthescopeofthestandardmodelhavebeenidentiedasdarkmattercandidates.Furthermore,resolvingtheredshiftdistributionoffaroutgalaxieswithspectralmethodshasunveiledthattheexpansionoftheuniverseisaccelerating,whichhassproutedthenotionofdarkenergydrivingthisexpansion.Lookingatsourceswithhighredshiftandthecosmicmicrowavebackground(CMB)allowsustotracethehistoryoftheuniversealmostallthewaybacktothebigbang,andshowsthattheremusthavebeenaneraofination,whichisnotaninherentfeatureofgeneralrelativity(GR),butmustbearticiallytackedon.GRisbyfarnottheonlytheoryofgravitythatisabletoproducepredictionsthatmatchtheobservedphysicalreality,butitisthesimplestinthatithastheminimumnumberoffreeparameters.WithinGRandothermetrictheoriesofgravityarisesthephenomenonofgravitationalwaves(GWs),whicharealsoapotentialcarrierofinformation,andtheirobservationwouldopenacompletelynewwindowontheuniverse.Theweakcouplingofmatterandgravitatio-naleldcanprovetobeablessingandacurseatthesametime,asitmakesitverydiculttodetectGWs,butalsoguaranteesthattheyarenotattenuatedbymatterinitsway.Comparedtophotons,thiscanbeanadvantage,becausemattercloudssuchasinterstellardustandnebulaarecommonlyopaquetoEMradiationandshieldwhatishiddenbehindthemfromourview.Furthermore,everythingintheuniverseinteractsgravitationally,suchthatobjectsthatdonotemitphotons(e.g.blackholes),orexoticmatterthatdoesnotcoupletotheEMeldmaystillbeobservedviaGWs.ThetopologyofGWobservatoriesisradicallydierentfromopticaltelescopes,asweshallsee,andtheyresembleantennasmorethandirectionaltelescopes.Whiletheyarecapableofproducingscienceontheirown,theyieldwouldnaturallyreceiveanimmenseboostfromcoincidentEMandGWobservations. 17

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GWdetectorsarecomplexprecisionmeasurementdevicesthatattempttoobserveGWsfromthestraintheyexertonspace-timeastheypasstheinstrument.Theirexperimentalhistorybeganintheearly1960'swiththedesignandconstructionoftherstresonantbardetectors(RBDs)undertheleadofJosephWeber[ 1 ].PassingGWsexcitemechanicalresonancesinmassivesuspendedbarsthataresensedwithpiezo-electrictransducersattachedtotheirsurface.Webersoonreportedpossibleobservationsofgravitationalwavesduetostatisticallysignicanteventrates[ 2 ],andaftergatheringdataoverseveralyears,hefoundanincreasednumberofGWscomingfromthedirectionofthegalacticcenter[ 3 ].TechnologyadvancesacrossgenerationsofRBDshaveincreasedthesensitivityofthismeasurementprinciple,andmodernRBDsareoperatedbymanyresearchgroupsacrosstheglobe.However,tothisdaynonewereabletoconrmWeber'soriginalndings,andtherehasyettobeaconrmedobservationofGWs.Aninterferometricsensingschemewithinertialtestmasses(TMs)wasdevelopedaroundthesametimeasRBDs,andfeasibilitystudiesoflargescaleinterferometricGWdetectors(IGWDs)weresetintomotionintheearly1970s[ 4 ].Theconstructionofprototypeinter-ferometersbegansoonthereafter,andtheirsizewasgraduallyincreasedformorerealisticassessments.Therstfull-scaleIGWDswithastrophysicallyrelevantsensitivitieshoweverdidn'tgoonlineuntiltheearly2000s,andhaveonlynowcompletedtheirrstmajorupgradestage.DespitetheimpressivesensitivitiesthatmodernIGWDsachieve,theirsciencegainhassofarbeenlimitedtoevaluatingthelackofobservations,resultinginconstrainingparametersinastrophysicalmodels[ 5 ],re-evaluatingpopulationestimates[ 6 ],andinonecasevetoingthelocalizationestimateforaGamma-Ray-Burst(GRB)[ 7 ].TheresultingdrivetoimprovethedetectorstobettersensitivitiesandnallyobserveGWsdirectlyhasfacedexperimentalistswithmany(sometimesunexpected)formsofinstrumentalnoise.AchallengeforRBDshasalwaysbeenthethermalnoiseofthesuspendedrod,whichoriginatesintheexcitationofitsinternalmodesduetothethermalenergyithasstored,andisthereasonforthecryogenicoperationofallmodernRBDs.Similarly,thermalnoise 18

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inthereectivecoatingsoftheTMsisexpectedtolimitthesensitivityofadvancedIGWDsintheircurrentcongurations.Toimprovethedetectorperformancewithfutureupgrades,coatingresearchisangledatbothndingalternativereectorsolutionswithlowerintrinsicloss,whichisthecauseforthermalnoise,andcryogenicallycoolingtheTMs.Lossreductioneortsfocusonthemechanicalpropertiesofthemirrors,andinferthecorrespondingopticalnoiseinIGWDswithmodelsthathavenotrigorouslybeentested.Tocomplementthisresearch,attheUniversityofFloridaweareconstructinganopticalthermalnoisetestbedforinterferometermirrors,whichistheoverlyingsciencethemeofthisdissertation.PlanstolaunchaspacemissionthatcansenseGWsonafrequencyscalethatisnotaccessibletodetectorsonEarthduetoenvironmentalnoisearecurrentlyledbytheEuropeanSpaceAgency(ESA)withatentativelaunchdatein2034.Thetechnologydemandsforthisendeavorareradicallydierentfromthoseofground-baseddetectors,andhardwaredevelopmentandtechnologydemonstrationarebeingpushedonmultiplefrontsbyvariousgroupsworld-wide.Ofthemanychallengessuchaninterferometricmissionwillfaceisthefrequencystabilityofthelasersthroughoutthescienceband,whichisvitaltothesuccesstothemission.Inexperimentsleadinguptothethermalnoisetestbedwedevelopedanewtakeonpossiblefrequencystabilizationschemesfortheproposeddetector.Weintegrateditscontrolsintoacompacthardwareimplementationofthephasemeasurementsystemthatisfavoredinthescopeofthemission,whichwealsoadaptedasthemainreadoutdeviceforthermalnoiseinducedfrequencyuctuationsinheterodyneRFsignals. 1.2NotetotheReaderAtitsrootthepresentedworkismotivatedbythedrivetoputgeneralrelativitytothetest,andbeyondthattopromotegravitationalwavedetectorstomainstreamastronomyinstruments.However,thetopicofGRisvast,andChapter 2 isthereforeintendedtobeabriefbutmostlyself-containedintroductiontogeneralrelativity,withafocusongravitationalwaves.Chapter 3 thenexploreshowgravitationalwavesinteractwiththephotoneldsinlarge-scaleMichelsoninterferometers,makingtheirbroadbanddetectionpossible.The 19

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NSF-fundedLIGOproject,whichsuppliedthemajorityofthefundingforthepresentedwork,andtheproposedspace-basedLISAdetector,forwhichNASAusedtobeamajorcontributoralongwithESAandalsoprovidedsomefundingforthiswork,arediscussedwithsomedetail.Inparticular,LISA'sfrequencycontrolschemeandLIGO'ssensitivitylimitationbycoatingBrowniannoiseareaddressed,sincetheywerethemotivationbehindthiswork.Chapter 4 exploresthenatureofthecoatingnoiseasaconsequenceofthefundamentaluctuationdissipationtheorem,anddiscussespossibilitiestoreduceitinfuturedetectordesigns.Theexperimentalpartisnotpresentedinastrictchronologicalorder,asmuchoftheworkprogressedinparallel.Itisratherstructuredbyexperimentaltopic,startingwiththeLISAphasemeterconceptinChapter 5 thatwepushedtowardshigherfrequenciesduetotheneedtoresolvefrequencyuctuationscausedbycoatingBrowniannoise.Chapter 6 explainsthefrequencystabilizationmethodwedevelopedtopotentiallysimplifythefrequencycontrolschemeinLISA,andalsondsapplicationinthethermalnoiseexperiments.Finally,Chapter 7 presentsthethermalnoiseexperimentCryoTHORinfull,startingwithanexplanationofthemeasurementconceptandanassessmentofindividualmechanicalandopticalcomponents.Aftercharacterizingsomeofthekeyfunctionalityintheexperiment,preliminarymeasurementsofaBrowniannoisetestcavityareobtained.InChapter 8 theworkandresultsofthisthesisaresummarizedandplacedincontextwiththestatusofgravitationalwavedetection. 20

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CHAPTER2GRAVITATIONALWAVESOnehundredyearsagoAlbertEinsteinpresentedhisrstworkonwhatwouldbecomewidelyknownasthegeneraltheoryofrelativity(GR)andrevolutionizedourunderstandingofgravitation[ 8 ].Untilthen,theprevalentunderstandingofthenatureofgravitation,whichwentbacktoSirIsaacNewton,hadbeenforittobeaforceduetoagradientinthegravitationalpotential(~r).Foragivenmassdensitydistribution(~r),(~r)wouldbedeterminedfrom r2(~r)=4G(~r);(2{1)whereG=6:67410)]TJ /F5 7.97 Tf 6.59 0 Td[(11m3kg)]TJ /F5 7.97 Tf 6.58 0 Td[(1s)]TJ /F5 7.97 Tf 6.59 0 Td[(2isthecouplingconstantbetweenmassandgravitationaleld.Incontrast,GRexplainsthemutualattractionbetweenmassiveobjectsasageometriceect.Anyapparentaccelerationisaresultofthemotiononastraightpaththroughacurvedspace,inwhichthemeaningofdistancehasbeenredenedtoincludetime.WithintheframeworkofGRacceleratedmassesarethesourceofpropagatingdistortionsofspace-time,whichareknownasgravitationalwaves(GWs).InthischapterweexaminetheequationsthatgovernthedynamicsofGWs.Afterarecaptureofspecialrelativity,weextenditsconceptstoGRandtakethenecessarystepstoobtaintheEinsteinequations.WebrieyaddressmethodsoftestingGR,andtheninvestigatesmallperturbationsofstaticspace-times.ThiswillleadtoawaveequationforGWs,withtwoindependentpolarizationsforplane-wavesolutions. 2.1SpecialRelativityAtanypointinhistoryprobingthelawsofphysicsinregimesthatarenotaccessibletolaboratorysciencehasrequiredmankindtoresorttoastronomicalobservations.TherstevidenceforthenitenessofthespeedoflightccamefromthelunareclipsesofJupiter'smoonIoin1675[ 9 ],andlaterarstrealmeasurementemergedfromthediscoveryoflightaberrationinanattempttodeterminestellarparallaxesin1727[ 10 ].Therst 21

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laboratory-scalemeasurementswereperformedinthemid-19thcenturyandinvolvedme-chanicallyeclipsedanddisplacedlightbeams[ 11 ].Therepeatedlackofsuccessinmeasuringadierenceinthespeedofcounter-propagatinglightbeams,forwhichthemostprominentexperimentisthatofMichelsonandMorleyin1887[ 12 ],togetherwithMaxwell'sformulationofthetheoryofelectrodynamics,eventuallyledEinsteintodevelopwhatwouldbecomeknownasthetheoryofspecialrelativity(SR).Adecadebeforeheexpandedhistheorytoalsoexplaingravitation,Einstein'sSRwasbuiltonthepremisethatcisaconstantofnature[ 13 ].Everyinertialobserver,regardlessoftheirstateofmotion,willconcludethesamevalueforcuponameasurementinwhicheverdirectiontheychoose.Thetimetandthethreespatialdimensionsx,y,andzarewovenintoa4-dimensionalspace-timefabric,inwhichthedistancedsbetweentwoeventsthatareseparatedbytheinnitesimaldisplacementdx=(cdt;dx;dy;dz)isgivenby ds2=)]TJ /F3 11.955 Tf 9.29 0 Td[(c2dt2+dx2+dy2+dz2:(2{2)AmorecompactwaytoexpressthisusestheMinkowskimetric=diag()]TJ /F1 11.955 Tf 9.3 0 Td[(1;1;1;1)foratspace-times.AdaptingtheEinsteinsummationconvention,inwhicheverydoublyoccurringindexinaproductimpliestheautomaticsummationoversaidindex,wewrite ds2=dxdx:(2{3)Lighttravelsonpathswithds=0,andtheinvarianceofEq.( 2{3 )undercoordinatetransformationsestablishestheconstancyofc. 2.2GeneralRelativityWiththedevelopmentofGREinsteinloosenedtheconceptoftheconstant,diagonalMinkowskimetricandwritesEq.( 2{3 )as ds2=gdxdx;(2{4) 22

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wherehasbeenreplacedwiththemoregeneralmetrictensorg.Cross-termsbetweenthedierentdxareallowedtoappearinEq.( 2{4 ),andthecomponentsofgmayvarywithlocationx.Constraintsongarethatitissymmetric,g=g,andthatitislocallyateverywhere,i.e.atanypointitispossibletondasetofcoordinatesexinwhichglocallyreducestotheatmetriceg=[ 14 ,p.145].Thepresenceofmass,ormoregenerallyenergy,couplestothe10independentcomponentsofg.Thistreatmentexplainsgravitationalaccelerationastheresultofmovingalonglocallystraighttrajectories(geodesics)throughadynamic,curvedspace-time. 2.2.1CovariantDerivativeThenotionofrelativityimpliesthatpreferredreferenceframesdonotexist,andconse-quentlythattheremustbewaytoformulatethetheorythatdoesnotrelyonaparticularchoiceofcoordinates.Eq.( 2{4 )wasarstexampleforsuchcovariantexpressions,whichareform-invariantundercoordinatetransformations.Itdenesthelengthofavector,andimpliesascalarproduct(;)betweenvectorsandinthespace-timeg, (;)==g:(2{5)Thespatialaspectofcurvaturecanbeunderstoodasanembeddingofthecurvedspace-timeasamanifoldintoahigherdimensionalspace[ 15 ,p.18].Themostrelatableexampleforthisisthetwo-dimensionalsurfaceofasphereinathree-dimensionalspace.Whenavector,whosedirectionalityisconstrainedtothesurface(space-time)ismovedacrossthespherebymeansofparalleltransport,whichholdstheangle(denedbythescalarproduct)withrespecttothetangentvectorofthetranslationconstant,theirnalorientationonthespherewilldependonthepaththeytook[ 14 ,p.153].Thispropertyofcurvedspace,whichisillustratedinFigure 2-1 ,distinguishesitfromatspace.Theobservationthatthischangeinorientationofavector~,whichiswritteninthebasis~eas ~=~e;(2{6) 23

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Figure2-1. Paralleltransportofvectorsonasphere.ThenalorientationofavectorthatisparalleltransportedfrompointAtopointCdependsonthepathittakes. occursevenforconstantcomponentsimpliesthatthebasisvectors~ethemselvesmustbesubjecttochangeunderthetransportoperation.Toaccountforthischangeunderinnitesimaltranslations,weletthecovariantderivativeractonthebasisaswell,andobtain r~=r(~e)=(@)~e+(@~e)=(@+)]TJ /F8 7.97 Tf 7.31 4.94 Td[()~e:(2{7)Inthelaststepwehaveexpressedtheobject@~einthebasisofthe~ethemselveswiththecoecients)]TJ /F8 7.97 Tf 89.52 4.34 Td[(,whicharecalledtheChristoelsymbols[ 14 ,p.127].Toobtainthecorrectingtermforthecovariantderivativeofcovectors,weusetheproductrulefordierentiationandexplicitlywrite r()=(r)+(r)=@();(2{8)wherethelastequalityholdsbecauseisascalar.TheadditionaltermsinvolvingChristoelsymbolsmustthereforecancel,andwecanidentify r=@)]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 7.32 4.93 Td[(:(2{9) 24

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Thedenitionofthecovariantderivativeiseasilyextendedtogeneraltensorsbyaddingacontractionwithanappropriate)]TJ /F8 7.97 Tf 196.7 4.34 Td[(foreveryfreeordinaryindex,andsubtractingacorresponding)]TJ /F8 7.97 Tf 96.04 4.34 Td[(foreverycovariantindex[ 14 ,p.131].Inparticular,thecovariantderivativeofthemetrictensoris rg=@g)]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 7.32 4.93 Td[(g)]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 7.31 4.93 Td[(g;(2{10)andisrequiredtovanishidenticallyifanglesandlengthsasdenedviathescalarproductinEq.( 2{5 )aretobepreservedundertheparalleltransportoperationthatbelongsto)]TJ /F8 7.97 Tf 386.24 4.34 Td[(.WecanpermutetheindicesinEq.( 2{10 )twiceandformalinearcombinationoftheresultingequations.Addingthepermutation(!!!)toEq.( 2{10 ),whilesubtractingthe(!!!)term,becauseofthesymmetryof)]TJ /F8 7.97 Tf 192.49 4.34 Td[(initslowerindicesweobtaintheexplicitexpression )]TJ /F8 7.97 Tf 7.32 4.93 Td[(=1 2g(@g+@g)]TJ /F3 11.955 Tf 11.96 0 Td[(@g)(2{11)for)]TJ /F8 7.97 Tf 25.2 4.34 Td[(intermsofg. 2.2.2MeasuringCurvatureParalleltransportonasphericalshellalreadyservedasanexampleforthegeometriceectsoftranslationsincurvedspace.Wenowuseittoderiveageneraltensorthatfullycharacterizesthecurvatureofspace-times.Westartwithanarbitraryvectorandpickadirection~einwhichwetransportitbyasmalldistanceb.Next,wetransportitbyainthedirectionof~e,whichtorstordertakesto !+rb!+rb+ra+rrab:(2{12)Ifwehadstartedmovingalong~erst,followedby~e,wewouldhavegottenasimilarexpressionwiththeordersofandswappedinthelastterm.ThisprocedureisillustratedinFigure 2-2 ,andmovinginreversealongthesecondpathcausesanetchangeinof =(rr)-222(rr)ab:(2{13) 25

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Figure2-2. Riemanntensorillustration.Alternateroutesfortheinnitesimaltranslationsbyaandbswaptheorderofdisplacementsalong~eand~e,whichaectstheorderingofthesecondderivativeterms.CovariantderivativesintroduceadditionalChristoelsymbolterms,anddonotcommutelikeordinarypartialderivatives. Itlandsusbackatthestartingpoint,withachangeinvalueofexclusivelyduetothelocalcurvature.Whenevaluatingthecovariantderivativesinthiscommutator,alltermswithpartialderivativesofcancel,andweareleftwithanexpressionoftheChristoelsymbolsthatislinearin.Initsexplicitform [r;r]=R=[@)]TJ /F8 7.97 Tf 7.31 4.93 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(@)]TJ /F8 7.97 Tf 7.31 4.93 Td[(+)]TJ /F8 7.97 Tf 19.08 4.93 Td[()]TJ /F8 7.97 Tf 7.32 4.93 Td[()]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 7.31 4.93 Td[()]TJ /F8 7.97 Tf 7.31 4.93 Td[(];(2{14)weidentifytheRiemanntensorRofg,whichencodesallinformationaboutlocalcurvature.Ithas4indicesandtherefore4444=256components,butusingvariousidentitiesandsymmetriesonecanshowthatthereareonly20independentdegreesoffreedom[ 14 ,p.160].TheBianchiidentitiesfortheRiemanntensor rR+rR+rR=0;(2{15)whereR=gR,canbeproveninlocallyatspace-timeusingvarioussymmetriesoftheRiemanntensorandthengeneralizedtothiscovariantexpression[ 14 ,p.164]. 26

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2.2.3TheEinsteinTensorThetwice-contractedBianchiidentitiesareobtainedfromEq.( 2{15 )bytracingtheindexwithandwith,followedbysimplifyingusingsymmetriesandEq.( 2{10 )[ 14 ,p.165].Theresultis gg[rR+rR+rR]=gr[2R)]TJ /F3 11.955 Tf 11.95 0 Td[(gR]=0;(2{16)wheretheRiccitensorRisthecontractionoftheRiemanntensorofitsrstindexwithitsthird, R=R;(2{17)whichremovesanytrace-freecomponentsofRandleavesasymmetrictensorwith10independentcomponents.ThescalarRiccicurvatureR=gRisthetraceoftheRiccitensor.ThelastequalityinEq.( 2{16 )denesatensorwhosecovariantderivativevanishes,whichwewriteexplicitlyas G=R)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2gR:(2{18)ThisistheEinsteintensor,whichisthecentralobjectofGR.ThecommonlyusedformofEq.( 2{16 )is rG=rggG=0;(2{19)whichisacompactexpressionforasystemofnonlinear,coupledpartialdierentialequationsing. 2.2.4Einstein'sEquationsThesymmetricenergy-momentumtensorTisdenedastheuxofthecomponentofthefourmomentumvectorp=(E=c;px;py;pz)acrossasurfaceofconstantx.Assuch,itisgenerallycommontodenetheTofaperfectliquid,inamomentarilyco-movingrestframe(MCRF)andgeneralizeitviacoordinatetransformations[ 15 ,p.39].ThepropertiesofaperfectliquidinaGRnotionarethatinastateofrestthereisnotransportofenergy,T0i=Ti0=0fori=1;2;3,andthatitisnotsubjecttoshearforces,Tij=0fori6=j. 27

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Explicitly,wewrite T=0BBBBBBB@0000P0000P0000P1CCCCCCCA;(2{20)wherethe00-componentistheenergydensity(generalized(~r)fromEq.( 2{1 )),andPisthepressureduetotherandommotionoftheuidelements[ 15 ,p.41].ThegeneralTfollowsbythetransportationalonganarbitraryfourvectoreldu T=(+P)uu+Pg:(2{21)ThevanishingofthecovariantderivativeofT rT=0(2{22)isequivalenttothelawsofenergyandmomentumconservation[ 14 ,p.99].SinceEqs.( 2{19 )and( 2{22 )arebothidenticallyzero,wecanequallywrite rG=rT:(2{23)Thischoiceisofcoursenotarbitrary,andthephysicalmotivationbehindEq.( 2{23 )isthatanequationthatcorrectlydescribesgravitymustreducetoEq.( 2{1 )intheclassicallimit,andthereforeinvolveasecondorderdierentialoperatoronthemetricwithasourcetermthatcontainsT.Moreover,theequationmustbecoordinatechoice-invariant,suchthatGandTareclearcandidates.Einstein'shypothesiswasthatthetwotensorsareequaluptoafactork,whichmustbechosentomatchtheobservedphysicalreality.ItsvalueisrecoveredfromEq.( 2{1 ),Newton'slawforgravitationintheweakeldlimit,andtheEinsteinequationsbecome G=8G c4T:(2{24) 28

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TheabbreviatednotationsomewhathidesthecomplicatednatureofG.Itstandsforasystemofnonlinearpartialdierentialequationsforthemetrictensorg,andTisitsinhomogeneoussourceterm.Eq.( 2{24 )isthemissinglinkthattellsspacehowtocurveinthepresenceofmatter. 2.2.5SpacetimeExamplesandBlackHolesWecancreatedierentscenariosforTandinsomecasessolveforganalytically.Asanexample,asphericallysymmetricinEq.( 2{21 ),whichamountstoatotalmassMwithinalimitedregionofanotherwiseemptyspacetime,hastheuniqueexteriorSchwarzschildsolution ds2=)]TJ /F9 11.955 Tf 11.29 13.27 Td[(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(rS rc2dt2+1)]TJ /F3 11.955 Tf 13.16 8.09 Td[(rS r)]TJ /F5 7.97 Tf 6.59 0 Td[(1dr2+r2d2;(2{25)whichwasfoundverysoonaftertheformulationofGR[ 14 ,p.263].TheSchwarzschildradius rS=2GM c2(2{26)representsacriticallengthscaleforspatialexpanseofthebody.IfallofMweretobecontainedwithinasphereofradiusrS,theresultingextremecurvatureattheboundaryrSwouldcausespacetimeitselftofalltowardstheinsideofthesphereatthespeedoflight.Notevenphotonscanescapethisconnedregionofspace,whichiswhysuchanobjecthasbeengiventhenameblackhole(BH).TheKerrmetricisageneralizationoftheSchwarzschildsolutionthatadditionallyallowsfortherotationofMwithanangularmomentumJ=aMc.Itreadsds2=)]TJ /F3 11.955 Tf 11.95 0 Td[(a2sin2 2dt2)]TJ /F1 11.955 Tf 11.96 0 Td[(2arSrsin2 2dtd++(r2+a2)2)]TJ /F3 11.955 Tf 11.95 0 Td[(a2sin2 2sin2d2+2 2dr2+2d2; (2{27)where2=r2+a2cos2,and=r2)]TJ /F3 11.955 Tf 11.49 0 Td[(rSr+a2arecalledBoyer{Lindquistcoordinates[ 14 ,p.309].Asconsistencybetweenthesolutionsrequires,anangularmomentumofJ=0 29

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producestheSchwarzschildmetric.SimilartoEq.( 2{25 )itistheuniquesolutionforitsscenario,andiscapableofrepresentingspinningblackholes. 2.3TestsofGeneralRelativityInthehundredyearssinceitsformulationGRhasenduredandsurvivedeverysingletestthathasbeenthrownatit.SeveralfeaturesofSRhavebroughtittofameinpopularscience,suchasthelengthcontractionandtimedilationbetweenobserverswithrelativespeednearthespeedoflight,andnaturallythesecarryoverintoGR.ButGRasatheoryofgravityneedstoproveitselfingravitationalobservations,andpreferablyinsituationswhereNewtoniangravityfailstoprovidesucientexplanations. 2.3.1SolarSystemDynamics:PerihelionAdvanceTheorbitofMercuryshowsananomalyintheadvanceofitsperihelion,whichisthethepointinitsorbitwhereitisclosesttothesun.Anundisturbedellipticalorbitinatwo-bodyproblemwouldhaveaxedorientationinspaceandtheperihelionwouldneverchange.Otherobjectsinthesolarsystem,mostnotablytheheavyplanets,disturbthismotionandcauseMercury'sorbittoprecess.Thereisadiscrepancyof43arcsecondspercenturybetweentheobservedvalueandthemulti-bodyNewtonianprediction[ 15 ,p.89].GRoerscorrectivetermsthatmatchtheobservationexactly[ 16 ]. 2.3.2GravitationalLensingofLightPhotonsdonothaverestmass(ordonotrest,onecouldsay),andwouldbeunaectedbygravitationaleldsinapurelypotential-basedtheory.ObservationshowevershowthatthepathsthatEMradiationtakesbendaroundgatheringsofmass(forexamplethesunorgalaxies),andevenifweattributeatinybitofmasstotheparticles,theobserveddeection(whichisindependentofmass)istwicethatoftheNewtonianvalue[ 15 ,p.91].InGR'sgeometricinterpretationofgravitythephotonscanremainrestmass-lessandstillcurvearoundheavyobjects. 30

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2.3.3GravitationalRedshiftTimeprogressesslowerthedeepertheobserverissittinginagravitationalwell.Anotherconsequenceisthatphotonsarerequiredtoloseenergywhenclimbingoutofthewell,andbecauseE=htheyhavetoberedshiftedtowardslowerfrequencies.ThesedaystestsoftheGRpredictionreachaprecisionof710)]TJ /F5 7.97 Tf 6.59 0 Td[(9[ 17 ]usingmatterwavesinatomicinterferometers. 2.3.4FrameDraggingRotatingmass,whichisasourceofcurvaturewithanintrinsicspincausesadragonthespace-timearoundit.SinceEarthisonacosmicscalenotverymassive,anddoesnotrotateveryquickly,theeectistiny,butstillmeasurable.GravityProbeB,aspacemissionthatusesrapidlyspinningnear-perfectspheresasdirectionalreferences,wasabletoresolvetheeecttoagreementwithin5%withGR[ 18 ]. 2.3.5ShapiroDelayInadditiontothegravitationallensingthatphotonsexperiencewhentheypassinproximityofcompactobjects,theyneedtocoveraneversoslightlylongerdistancegoingthroughthewellcomparedtojustemptyspace[ 19 ].Inbinarysystems,whereapulsardirectsitsjetinthedirectionofEarthandisaccompaniedbyaheavyobjectsuchasawhitedwarforanotherneutronstar,thevariationinarrivaltimeofitspulsesonEarthwhentheyhadtopassclosetothecompanionstarconrmtheadditionaldelaypredictionofGR[ 20 ]. 2.3.6TheHulse-TaylorBinaryPulsarGRpredictsthatbinarystarsystemsstirupthespace-timearoundthem,towhichtheyloseorbitalenergyintheprocess,causingthemtoslowlyspiraltowardseachotherwiththeireventualcollision.ThediscoveryofthebinaryneutronstarsystemPSRB1913+16byHulseandTaylorwasanimmensecredibilityboostforGR.ThefortunatehappenstancethatthejetofthepulsarisoccasionallyeclipsedbyanotherNS,coupledwiththefactthatpulsararrivaltimeshavefarbetterlong-termstabilitythaneventhebestclockswehaveatourdisposal,enabledextremelyprecisemeasurementsoftheslowchangeinorbitalperiod[ 21 ].Theseinturnallowedforanestimationoftherateatwhichorbitalenergyislost,which 31

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isinagreementwithGRtoanamazingaccuracy.Thisisalsothebestindicationfortheexistenceofgravitationalwaveswehavetodate. 2.4PropagationofMetricPerturbationsThenextquestiontoaskishowperturbationsofanotherwisestaticgpropagateoutwardsthroughemptyspace.Werestrictouranalysistosmalllinearperturbationshofanotherwisestaticbackgroundmetric,whichweinvestigateinlocallyatcoordinates,andwritethemetrictensoras g=+h:(2{28)TheEinsteinequationforthislinearizedgwillturnintoanequationofmotionforh. 2.4.1WaveEquationTherststepistocalculatethelinearizedChristoelsymbolsfrom( 2{11 )forthegasdenedinEq.( 2{28 ).Termsthatarequadraticinhorofhigherorderaredropped,andusingEq.( 2{14 )weobtaintheexplicitexpression R=1 2(@@h+@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h):(2{29)ThecontractiontotheRiccitensorisperformedagainst,leadingto R=1 2(@@h+@@h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h);(2{30)andnallytheEinsteintensor G=1 2(@@h+@@h)]TJ /F3 11.955 Tf 11.95 0 Td[(@@h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h)]TJ /F3 11.955 Tf 11.95 0 Td[((@@h)]TJ /F3 11.955 Tf 11.96 0 Td[(@@h));(2{31)whereh=histhetraceoftheperturbation.Wedenethetrace-reversedhas h=h)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2h;(2{32)andfurtherimposetheLorentz-gaugecondition[ 14 ,p.193] @h=0(2{33) 32

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Figure2-3. Polarizationsofgravitationalwaves.Thetwoindependentdegreesoffreedomh+andhforapassingGWstretchandsqueezespace-timeintransverseorthogonaldirections.Duetotheirquadrupolenaturetheycanbetransformedintoeachotherbyanin-planerotationby45. uponh.ThisremovesmostofthetermsinEq.( 2{31 )andleavesonlytheexpression )]TJ /F1 11.955 Tf 13.15 8.08 Td[(1 2@@h=)]TJ /F1 11.955 Tf 10.49 8.08 Td[(1 2)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(@2t+c2r2h=0;(2{34)forh.ThisisatypicalwaveequationforhandshowsthatGWstravelatthespeedoflightinGR,justlikephotons. 2.4.2PolarizationsofGravitationalWavesAsalineardierentialequation,Eq.( 2{34 )allowsitssolutionstobedecomposedassuperpositionofplanewaves.Wethereforewrite h=RfAexp(ikx)g(2{35)astherealpartofacomplexplanewavewithfour-vectork=(!;~k)andtensoramplitudeA.ThegaugeconditionEq.( 2{33 )hastheconsequence Ak=0(2{36)forthewave'stensoramplitude,meaningthatitscomponentsareperpendicular(inthefour-dimensionalsenseestablishedbyEq.( 2{5 ))tok.Thereisfurthergaugefreedom 33

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availabletorestrictA,andacommonchoiceistox A=A=0;andA0=0;(2{37)leadingtowhatiscalledthetransversetraceless(TT)gauge[ 14 ,p.204].Theoriginal10degreesoffreedomforhreduceto6duetothesecondpartofEq.( 2{37 ),andwithoutlossofgeneralitywithk=(!;0;0;k)wecanchoosethez-directionforthepropagationoftheplanewave.LookingatEq.( 2{36 ),thisforcesallcomponentsA3tobezero,reducingthenumberoffreeparametersfurtherto3.Lastly,Eq.( 2{37 )alsorequiresAtobetrace-free,leavingonly2degreesoffreedomthatcannotbegaugedaway.Theperturbationultimatelytakestheform h(TT)=0BBBBBBB@00000h+h00h)]TJ /F3 11.955 Tf 9.3 0 Td[(h+000001CCCCCCCAei(kz)]TJ /F8 7.97 Tf 6.59 0 Td[(!t);(2{38)andwecallh+andhthepolarizationsoftheGW.Figure 2-3 illustratestheirgeometricalinterpretationandquadrupolecharacter.Lookingataringoftestparticlesinthexy-plane,thepassingwavewillperiodicallywarpthering,stretchingitalongonedirectionandsqueezingitalongtheother.Arotationby45degreesswapstherolesofthepolarizations. 2.5SourcesofGravitationalRadiationTheconservationlawsforenergyandmomentuminferthatgravitationalradiationistoleadingordergeneratedbythetime-dependentmassquadrupolemomentofadistributionofmass[ 15 ,p.77].Sphericallysymmetricheavyobjects,eveniftheirshapebecomesoblateduetorotation,haveavanishingquadrupolemoment,andthereforedonotemitGWs.WecanhoweverthinkofvariousastrophysicalscenarioswhichlikelyinvolveasymmetriesthatwouldindeedgenerateGWs.Figure 2-4 providesanimpressionofthetypesofsourcesandtheircharacteristicfrequencyranges. 34

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Figure2-4. Frequencybandsofgravitationalradiation.Thefrequencyrangegenerallyscalesinverselywiththeinvolvedmasses.ThesensitivitiesofselectedGWdetectors,whicharediscussedinmoredetailinChapter 3 ,havebeenaddedtothegraph.Source: http://rhcole.com/apps/GWplotter/ and[ 22 ] 2.5.1Core-CollapseSupernovaeAstarisalargegatheringofmassinanequilibriumstatebetweenthegravitationalattractionoftheentiretyofitsparticleswiththeoutwardsdirectedphotonpressurethatisgeneratedbythefusionprocessinitscore.Eventuallyallofitsfuelisusedup,thephotonpressuresubsides,andisnolongerabletopreventthegravitationalcollapse.Wehavesofarnotbeenabletoprovideagap-lessexplanationofwhathappensduringtheensuingsupernova,whichleavesacompactremnantobjectsuchasadwarfstar,neutronstar,orevenablackholebehind.ThemassiveamountsofmatterejectedandEMradiationemittedmaskwhatishappeninginthecoreduringthecollapse,butanyasymmetriesduringthisprocesswillradiateawayasGWs[ 23 ].Sincethewaveformsofsupernovaecannotbemodeledanalytically,wehavetorelyonGWdetectorstopickupthemupasspontaneousburstsof 35

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gravitationalradiation[ 24 ].Thereceivedsignals,andinparticularacoincidentdetectionoftheiropticalcounterpartwithtelescopes,willhelpshedlightonthegravitationalaspectofthecollapse. 2.5.2CompactBinaryCoalescencesTheobservationsofHulseandTaylorconrmedthatgravitationalenergyisradiatedawayinbinarysystems,forwhichtheemissionofGWsistheonlylossmechanismthatGRhastooer.Objectsinbinarysystemswillgetclosertoeachotherastheyslowlylosetheirorbitalenergyinanadiabaticprocess,whichmeansthattheirorbitalvelocity(andthereforefrequency)hastospeedup.Theinspiralprogressesuntilthebinarypartnersreachhighlyrelativisticvelocitieswhentheirseparationisoftheorderoftheinner-moststablecircularorbit(ISCO)anditbecomesimpossibleforthemtospeedupanyfurther.Duringtheensuingmergerphasetheyplungeintoeachotherandmaybounceandrecoil,butonceasingleblobhasformeditwillringdownandradiatethelastofitsasymmetriesaway.Thesecompactbinarycoalescences(CBCs)areexpectedtobetheprimesourceforGWs[ 25 ]sincetheirwaveformscanbemodeledwithpost-Newtonianandnumericalrelativisticmethods[ 26 ]anddugfordeepinthedetectordatastreams.PredictionsoftheadvanceoftheorbitalphasecanbeturnedintoatestofGRbycomparingwaveformsagainstatemplatebank. 2.5.3Extreme-Mass-RatioInspiralsBinarysystemsinwhichoneofthepartnersisamassiveblackhole(MBH)ofmorethanhundredsofthousandsofsolarmasses,whiletheotheroneisastellarmasscompactobject(Whitedwarf,neutronstar,blackhole)areexcellentlaboratoriesfortestingGR.BecausethetidaleectsforsuchMBHsaremuchgentleratthehorizonthanforlessmassiveBHs,thesmallerobjectcanorbittheMBHmanytimesincloseproximitytothehorizonwithoutbeingtidallydisrupted,mappingalargeportionofthespace-timearoundit[ 27 ].Thiswouldallowforprecisiontestsofthemetricblackholesolutions. 36

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2.5.4Super-MassiveBlackHoleMergersOurMilkyWaygalaxyrevolvesaroundtheSuper-massiveblackhole(SMBH)Sagitta-rius-A,whichwasfoundbytracingthecurvedtrajectoriesofnearbystarsaroundalimited,darkregionofspaceatitscenter,whichmustthereforebeincrediblydenseandmassive[ 28 ].ThecollisionoftwoSMBHsisoneofthegravitationallymostviolenteventsimaginable,andthesheeramountofreleasedenergywillcarrygravitationalwavestouswithobservablemagnitudefromthefarreachesofspacewithhighredshifts,providinguswithsignalsfromtrulystrongeldgravitationalregimes[ 29 ],inwhichdeviationsfromGRwouldbemostnotable. 2.5.5PrimordialGravitationalWavesThedescriptionofthetimelineoftheuniverse,startingwiththebigbanguntilthestatethatweobservetoday,inageneralrelativisticframeworkhastointroduceaphaseofrapidinationduringwhichinhomogeneitiesinthegravitationaleldduetoquantumuctuationswouldhavebeenblownupandcouldpersisttothisdayasuctuationsoftheradiativegravitationaleld.WhiletheEMcosmicmicrowavebackground(CMB)datesbacktoabout300,000yearsafterthebigbang,inationoccurredwithin10)]TJ /F5 7.97 Tf 6.59 0 Td[(36sto10)]TJ /F5 7.97 Tf 6.59 0 Td[(32sofit,suchthatprimordialGWswouldenableustolookbackmuchfurtherintime.Detectorsmaybeabletosensetheseso-calledprimordialGWsasastatisticalbackgroundandhelpconstraintheoriesabouttheoriginoftheuniverse[ 30 ]. 37

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CHAPTER3INTERFEROMETRICGRAVITATIONALWAVEDETECTORSTheemergenceofGWsfromtheeldequationsofGRwasdiscoverednotlongafterGRhadbeenformulated,andwasrecognizedearlyon[ 31 ].However,duetothevast,backthennotwell-understoodgauge-freedominGR,thephysicalityofGWsremaineddebateduntilthe1950s,withevenEinsteinhimselfdoubtingtheirexistence[ 32 ].ThedevelopmentoftheplanewaveapproachbyBondi[ 33 ]helpedsettletheargumentabouttheirphysicalreality.CalculationsofthestraincausedbyapassingGWfromrealisticsourcescanbediscou-ragingatrstlook.ForanorderofmagnitudeestimateitsucestocombineaNewtonianapproach,forexampleasinstructedbySchutz[ 34 ],withEinstein'sworkontheenergeticsofGWs[ 35 ].Thedrivingmechanismisinthiscasethetime-varyingquadrupolemomentofagivenmassdistribution,whichcausesanapproximatestrain h=2GM Rc2GM rc2;(3{1)whereMisthetotalmassofthegravitatingsystem,Risthecharacteristicradiusofitsspatialexpanse,andristhedistancetotheobserver[ 36 ].Forthestandardexampleofacompactbinarycoalescenceoftwoneutronstars,eachwithmassMNS=1:4Msun,separatedbyR=90kmatadistanceof15Mpc,weobtainh810)]TJ /F5 7.97 Tf 6.59 0 Td[(22.AshasbeendiscussedinSection 2.3 ,amultitudeofweak-eldimplicationsofGRhavebeentestedtodate.Whatisstillmissing,however,isatruestrong-eldtestthatgoesbeyondthelossofkineticorbitalenergyinbinarysystems.TheobservationofGWswouldprovetobeinvaluabletoolforstrong-eldregimetests,astheorbitalevolutionleavesanimprintinphase,polarization,andamplitudedevelopmentofthewave.InthischapterwefocusonthelargeMichelsonlaserinterferometerapproachtodetectingGWs.Weillustratethecouplingofphotoneldsintheinterferometertopassinggravitationalwaves,andpresenttheAdvancedLIGOopticallayout.WebrieyaddressitsvariousGWcoupling-enhancingstrategiesandspectralsensitivity,andthendiscusshowcryogenically 38

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coolingkeyopticscanimprovetheinstrumentsensitivity.Lastly,theLISAdesignforaGWobservatoryinspaceisexplained,withemphasisplacedonthelasernoisedrivendesignrequirements. 3.1GravitationalWavesandLaserInterferometryThestretchingandsqueezingofinnitesimaldistancesintheoriginalexpressionEq.( 2{4 )canbeintegrateduptoachangeinthemacroscopicseparationbetweeninertialobjectsandturnedintoanobservableforameasurementdevice.Weplugintheresultsoftheplane-waveapproachEq.( 2{38 )andobtain ds2=)]TJ /F3 11.955 Tf 9.29 0 Td[(c2dt2+[1+h+(t)]dx2+[1)]TJ /F3 11.955 Tf 11.96 0 Td[(h+(t)]dy2+2h(t)dxdy+dz2(3{2)fortheseparationdsintheperturbedspace-time.Sinceds=0forphotons,thischangeinseparationcanbeimprintedintothephaseofanelectromagneticwavewiththedesignofasuitableantenna.LookingatFigure 3-1 ,itisclearthatduetotheleadingquadrupolenatureofGWs,thegeometryofaMichelsoninterferometeriswellsuitedfortheirdetection.Lightthattravelsinthexdirectioncoversthedistancedxinthetime dt=p 1+h+(t)dx c(1+h+(t)=2)dx c:(3{3)TotravelthenominaldistanceLxthroughthexarm,reectoamirrorandtravelbackitwilltaketheround-triptime Tx=2 cLxZ01+h+(t)=2dx=2Lx c+h+(t)Lx c:(3{4)WeneglectthephaseevolutionoftheGWduringtheround-trip,whichisagoodapproxi-mationifthewavelengthoftheGWislongcomparedtothebaselineoftheinterferometer.Thesametreatment,againassumingquasi-stationarystretching,producesaround-triptime 39

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Figure3-1. Michelsoninterferometerintestmassring.TheperpendiculararmsetuprespondsdierentlytothetwopossiblepolarizationsofGWs.Whiletheimpactofh+onthedetectoroutputismaximized,itisentirelyblindtoh. Tyalongtheydirectionof Ty=2 cLyZ01)]TJ /F3 11.955 Tf 11.95 0 Td[(h+(t)=2dx=2Ly c)]TJ /F3 11.955 Tf 11.95 0 Td[(h+(t)Ly c:(3{5)Monochromaticlightofangularfrequency!intheinterferometeraccumulatesthephase!Tiduringitstripthrougharmi,soifthexandybeamsarerecombinedatatthebeamsplitter,theywillhaveaphasedierenceof x(t))]TJ /F3 11.955 Tf 11.96 0 Td[(y(t)=!(Tx)]TJ /F3 11.955 Tf 11.95 0 Td[(Ty)=kL+h+(t)! c2L;(3{6)withL=Lx)]TJ /F3 11.955 Tf 12.46 0 Td[(Ly.Themacroscopicdierenceinlengthproducesastaticphaseoset,andtheGWinducedlengthchangesaddcoherentlyinthephasedierence.Therecombinedbeamsproduceadierentialarm(DARM)photodetectorsignal DARM(t)/cos2kL+h+(t)! c2L(3{7) 40

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ALIGOHanfordobservatory BLIGOLivingstonobservatoryFigure3-2. TheLIGOtwinobservatories.LocatedinHanford,WAandLivingston,LA,theobservatoriesareseparatedbyadistanceofjustover3,000kmandroughlymatchorientationstoenhancetheoddsforcoincidentdetectionofGWsinbothdetectors. attheinterferometeroutput.WhileL=0removesanynon-GWinducedsignalfromthedetectoroutput,italsosuppressestheGWsignaltorstorderintheseriesexpansionofthecosine.ThereforetheinterferometermustbeeitherdetunedfromL=0forDC-readout[ 37 ],orhavesidebandsaddedthatco-travelwiththecarrierbutdonotinterferedestructivelyintheoutput[ 38 ]. 3.2LaserInterferometerGravitational-WaveObservatoryTomaximizethesensitivitytospace-timestrain,thedisplacementnoiseinthereadoutofthetestmasspositionmustbebalancedagainstthetotallengthoftheinterferometerarms.TheNationalScienceFoundation(NSF)approvedthetwinLIGOdetectorsin1990,andsitesinHanford,WA(Figure 3-2A )andLivingston,LA(Figure 3-2B )werechosenfortheirconstructionwithanarmlengthof4km.Theprojecttime-lineforesawtheconstructionofInitialLIGOtounderstandtheoperationandhandlingoflarge-scaleinterferometers,followedbyseveralsmall-scaleupgradesleadingtoEnhancedLIGO,whichculminatedinthesciencerunsS5andS6[ 39 ].TheupgradeprocesstoAdvancedLIGObeganin2010,tookveyearstocomplete,andtherstobservationrunoftheadvanceddetectorerawasinitiatedinSeptemberof2015. 41

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Figure3-3. AdvancedLIGOtopology.AdvancedLIGOiscomprisedofseveralsubsystems,whosedevelopmentwasspreadacrossmultiplereasearchgroups. 3.2.1AdvancedLIGOInterferometerTopologyThelayoutoftheAdvancedLIGOdesignisshowninFigure 3-3 .Thepre-stabilizedhigh-powerlaser(PSL)isthesourceofthe130Wsinglemodecontinuouswavelasereldthatisthesciencecarrierintheinterferometer[ 40 ],whoselastpreparationstageistheinputmodecleaner(IMC)cavity,beforeitissenttothemaininterferometer(IFO).Startingwiththerstdesign,LIGOdeployedarmcavities,whichareformedbetweentheinputtestmass(ITM),locatedclosetothebeamsplitteratthebeginningofthearm,andtheendtestmass(ETM).Thearmcavitiesaretunedsuchthatthecarriereldresonates,increasingtheeectiveinteractiontimeofphotoneldandGWandboostingthephaseimprint. 42

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TheinterferometeristunedsuchthatconstructiveinterferenceinthesymmetricIFOportreectsthemajorityofthelightintheIFObacktowardstheIMC.Theplacementofthepowerrecyclingmirror(PRM)intheinputpathtransformstheeectivedistancetothetwoITMsintothepowerrecyclingcavity(PRC).IttrapsthecarrierlightintheIFOandprovidesafurtherincreasetothepowerstoredinthearmcavities,whichnowseeastrongerinputeld[ 41 ].Asignalrecyclingmirror(SRM)isplacedinthepathofthedetectoroutput,whichformsacavityinwhichthedierentialarmGWsignalscanresonate,beingsentbackintotheIFO.Signalrecyclingcanbetunedtoincreasethedetectorsensitivityatthegeneralexpenseofdecreasedbandwidthofitsresponse,andviceversa[ 42 ].TheimplementationofbothpowerrecyclingandsignalrecyclingmakesLIGOadual-recycled,cavity-enhancedMichelsoninterferometer[ 43 ]. 3.2.2AdvancedLIGOInstrumentSensitivityThebiggestchallengefromtheexperimentalsideofchasingsensitivityextremeswithGWdetectorsistotrackdownnoisesourcesanddevelopcopingstrategies.MuchofthenoisethatisultimatelyexpectedtolimitAdvancedLIGO'ssensitivitywasidentiedinpreviousdetectorcongurations,andresearcheortwasputintondingwaystoloweritintheupgradeddetectors.TheimpactofthevariousformsofnoiseforAdvancedLIGO,showninthedesignsensitivitycurveinFigure 3-4 ,isextrapolatedfromestablishedmodelsandknownparameters.Thesources{sometechnical,somefundamentalinnature{canbegroupedintothreegeneralcategories.EnvironmentalnoiseisincidentalnoisegeneratedbythesurroundingsofthedetectorthatcouplestotheGWsensitivechannels.Seismicnoise,causedforexamplebyhumanactivitiesortectonicplates,isoneofthemostobviousnoisemechanisms,asgroundshakingdirectlyaectsthepositionsofthemirrors.Themotionispre-lteredbyseveralactiveandpassivestages,andthetestmassesthemselveshaveaquadruplecombinationsuspension[ 44 ].Thisnoiseincreasessteeplytowardssub-Hzfrequenciesandbecomesinherentlydiculttolter,becausepassiveisolationwithharmonicdegreesoffreedomdoesnotrejectmotion 43

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Figure3-4. AdvancedLIGOnoisebudget.Atlowfrequenciesseismicandsuspensionthermalnoiselimitthedetectorsensitivity,whileinthebucketatintermediatefrequenciescoatingBrowniannoisepresentsahardlimit.Quantumnoiseseverelyimpactsthesensitivityacrosstheentirespectrum. belowtheresonancefrequencies,andbecauseaccelerationsensorsforactivefeedbackdonothavetherequiredsensitivityattheselowfrequencies.Newtoniannoiseoriginatesinchangesofthelocalgravitationaleld,introducingatime-varyinggravitationalpullonthetestmasses.Displacementsoflargeamountsofmass,suchascloudsinmotion,ormovingobjectsnearbythedetectorcoupledirectlytothetestmassesviagravitationitself,anditisthereforeimpossibletoshieldthem.Togetherwithseismicnoise,NewtoniannoisepresentsahardlowerfrequencywallforplanetaryGWdetectors.Quantumnoiseisatermthatengulfstwomanifestationsofthequantizedinteractionoftheelectromagneticeldwithmatter.Thequantummechanicalinterpretationexplains 44

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itasaformofcountingnoisewithPoissonianprobabilitydistribution,whiletheclassicalviewpointattributesanuncertaintytotheamplitudeandphaseoftheeldvectors.Thespectrumofthecorrespondingphaseandamplitudenoiseisfrequencyindependent(white).Ononehand,thiscausesphotonicshotnoise(SN)inthedetectorreadout,whichisshapedbythethearmcavitiestoappearproportionaltofabovethecavitypole.Ontheother,whenreectingoamirror,thephotonsofalaserbeamsuerelasticcollisions,inwhichtheytransfertwicetheirmomentumh=tothemirror.Theresultisanetpushingforceproportionaltotheinstantaneousnumberofreectedphotons(orclassicallytheincidentpower),andbecauseoftheelductuationsthiscausesradiationpressurenoise(RPN).Itsubjectsthemirrorstoawhiteforcenoise,whichneedstobeintegratedtwicetoyielddisplacementnoise,givingRPNanoverall1=f2slope.ItisimportanttonotethattheinputlasereldissplitcoherentlyintotheIFOarms.Thephaseandamplitudeuctuationsofthecarriereldinthearmsarethereforefullycorrelated,andanyradiationpressurevariationsorphaseuctuationsarecommontobotharmsanddonotcoupleintotheanti-symmetricportofthedetector.However,thequantummechanicalvacuumstate(withnophotons)exhibitszeropointuctuationsthatcoupleintothearmcavitiesthroughthebackportoftheIFO.Inthedetectoroutputuctuationsofthecarriereldareout-of-phase,butthevacuumuctuationsarein-phase,andthereforealtertherecombinedarmeldsdierently.IncreasingthelaserpowerreducesthephaseuncertaintyduetoSN,butincreasestheRPNatthesametime.Asimilartradeocanbeachievedwiththeuseofsqueezedvacuumstates[ 45 ]thatareinjectedintothebackportinsteadofnormalvacuum.Contrarytolaserpowertuning,however,byusingfrequencydependentsqueezingthesqueezingellipsecouldberotatedtoenablebroadbandsuppressionofquantumnoise[ 46 ].Thermalnoiseisaconglomerationofnoisesourcesthathavetheiroriginintheuctuationdissipationtheorem(FDT),towhichwededicatetheentireChapter 4 .ThequintessenceoftheFDTisthatanycomplexsystematatemperatureTissubjectto 45

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uctuationsinitsobservablesduetoitsmoleculesnotbeingatrest.Theexactcouplingmechanismscanbefoundfromthedissipativelossesintheinterrogateddegreeoffreedom.CoatingBrowniannoiseistheeectivetestmassdisplacementnoiseduetothemechanicallossintheircoatings,andtherelatedsubstrateBrowniannoiseiscausedbythemechanicallossinthesubstrate.Suspensionthermalnoise,asthenamesuggests,arisesfrommechanicallossinthesuspensionbers.Coatingthermo-opticnoiseisacombinationofthermo-elasticnoiseandthermo-refractivenoise,whichoriginateinspontaneousuctuationsofthetemperatureitselfthatdrivethermalexpansionandchangesintheindexofrefraction.Sincepropagationthroughthesubstrateisnotpartofthecavitythereisnorefractivenoise,butthermalexpansioncanstilldisplaceanddistortthereectingsurfaceasawhole. 3.2.3CryogenicsandGravitationalWaveDetectorsThemechanicallossesinthetestmassesandsuspensionsdeterminethecouplingstrengthbetweenthethermalbathofthesystemandmirrordisplacementnoise.WhileminimizingthemiskeytoimprovingthesensitivityofGWdetectors,amoreheads-onapproachistoreducetheamountofthermalenergyintherstplaceandcryogenicallycoolcriticalin-terferometercomponents.Naturally,thisisadicultendeavor,becausethetestmassesarerequiredtobeinert,andanyphysicalcontactwouldshortthenalsuspensionstages.Besidesstandardradiativecoolingeorts,heatcanbeextractedthroughthesuspensionbersthemselves[ 47 ],andpossiblybynear-eldradiativeheattransfer[ 48 ].Adrasticreductioninoperatingtemperaturecanbeaccompaniedbychangesintheinternalstructureofthetestmasses,suchthatthemechanicallossesandmaterialstiness,bothofwhichimpactthermalnoise,aresubjecttochangewithtemperature.FortheroadmaptowardsthenextgenerationofGWdetectorsthisneedstobewellunderstoodandinvestigatedtomakemajordesigndecisionsforlaserwavelength,substratesandcoatings,whicharethedrivingmotivesforamajorityoftheworkpresentedinthisthesis. 46

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3.2.4CoordinationwithOtherGround-BasedDetectorsAmapwithexistingandproposedGWobservatoriesisshowninFigure 3-5 .TheGEO600detectorinHannover,GermanyisaBritish-Germancollaboration.Its600minterferometerarmsarefoldedonce,doublingtheround-triptime,andinitscurrentstateitdoesnotemployarmcavities.WhileitsshorterlengthisadisadvantageforitsstrainsensitivityandshiftsitssensitivitybandtowardsslightlyhigherfrequenciesrelativetotheLIGOband,ithasbeenontheforefrontofimplementingadvancedinterferometertechniquessuchasdualrecyclingormonolithicmirrorsuspensions[ 49 ].TheFrench-ItalianVIRGOdetectorinCascina,ItalyisclosertotheLIGOdesignwith3kmarms.TheAdvancedVIRGOupgradeschedulewasosetwithrespecttoAdvancedLIGOtocomplementdeve-lopmenteortsandreducetimestretcheswithoutonlinedetectors.VIRGOwastherstdetectortopioneertestmasssuspensionwithquadruplestagedesigns.ConstructioneortsinJapanarefocusedonbuildingtherstundergroundGWobservatory.LocatedinthevacatedKamiokaminenearHidaintheGifuprefecture,KAGRAwillalsobetherstdetectortousecryogenicallycooledtestmassestocombatthermalnoise[ 50 ].TheroadtowardsKAGRAwaspavedbythesmaller-scaleexperimentaldetectorsTAMA300andCLIO.Indiahascommittedtotheconstructionofthenextlarge-scaleGWobservatoryLIGO-INDIA(orINDIGO).ItwilluseobsoleteandspareLIGOcomponentsthatareavailablefrompreviousupgradesorwereacquiredforcontingency.Theout-of-planecompo-nentthatLIGO-INDIAoerswillpositivelyaectthenetworksensitivitybeyondjustbeingonemoredetector[ 51 ].Spreadingthedetectorsacrosstheentireplanetisimportantformultiplereasons.Firstly,itdecouplesthelocalnoisesourcestheindividualdetectorsarefacing,bothhumanorenvironmentalinorigin(trains,woodworking,strongwinds,cloudmotion,etc.).Thismakesthecoincidentdetectionsindierentdetectorsmoreconvincingandraisesthenetworksignal-to-noiseratio(SNR)ofGWsignals[ 52 ].Secondly,thevastparameterspaceofGWeventsmakesitimpossibletopinpointtheoriginofawaveformwithonlyasingledetector. 47

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Figure3-5. Gravitationalwaveobservatoriesontheglobe.TheLIGOtwindetectorsinthecontinentalUShavethelargestbaselinewith4km,whileVIRGOsitsat3kmandGEO600at600m(foldedto1200m).ThecryogenicundergrounddetectorKAGRAiscurrentlyfarintoitsconstructionschedule,andINDIGOisproposeddetectorinIndia. CoherenttriggersforGWeventsnotonlyprovideamorepreciseestimateofthewaveform,possiblycomplementingthepolarizationinformation,butalsoenabletriangulationviati-mingdierences[ 53 ]. 3.3LaserInterferometerSpaceAntennaAmultitudeofsourcesatlowfrequenciesarehiddenfromground-basedinterferometricGWdetectorsduetotheseismicandNewtoniannoisewall.Longerlivedsourcesandthemoremassiveinspiralspectrumcanonlybeaccessibletoaspace-basedinterferometricdetector.ThemissionconceptfortheLaserInterferometerSpaceAntenna(LISA)hasbeenindevelopmentsincethe1990s[ 54 ].InitiatedasacollaborationbetweentheNationalAeronauticsandSpaceAdministration(NASA)andtheEuropeanSpaceAgency(ESA), 48

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Figure3-6. LISAorbits.Theindividualorbitsofthethreespacecraftaresynchronizedtomaintainastable,near-equilateralconstellationthatistiltedby60againsttheecliptic.Source: https://commons.wikimedia.org/ unfortunatebudgetdecisionshaverequiredastep-by-stepretreatofNASAfromthemissiontobecomeaprospectivejuniorpartner.TheprojecthassincebeenrenamedtoevolvedLISA(eLISA),andthetentativelaunchdatehasbeenpushedbackuntil2034attheearliest.Majordesignrevisionsarestillongoing,andwillbeforquitesometime,butthegeneralinterferometricreadoutschemewillremainintact,andmostlybeaectedbythechangeinarmlengthsduetocuttingcosts.Adjustmentstotherequirementsfortheinterferometricreadoutwillbenecessaryuponachangeinbaseline,butsincearmnewdesignhasnotbeenselected,weputthepresentedworkintoperspectivewiththeoriginalLISArequirements.Thevacuuminspaceallowsforaverylargeinterferometerbaselinewithtestmassesplacedonseparatespacecraft.ThelackofrigidinterferometerarmsrequiresthespacecrafttobeonindividualorbitsthatmaintainastableIFOgeometry.TheLISAdesignteamoriginallyfavoredtheconstellationshowninFigure 3-6 ,withthreeidenticalspacecraftin 49

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Figure3-7. LISAarmsplit.Thetotaldistancefromproofmasstoproofmassisdividedintothreeseparatemeasurements.Thespace-craftkeepthemshieldedfromexternalinuenceswhilethedistancesbetweentheopticalbenchesandtheproofmassesareevaluatedseparately. anapproximatelyequilateraltriangularcongurationseparatedfromeachotherby5millionkilometers[ 55 ].Theslightlyout-of-planeorbitscausethetriangletotrailtheEarthonitspatharoundthesuntiltedby60degreesagainsttheecliptic.Withinthespacecraftaregold-platedplatinumcubetestmasses,shieldedfromoutsideinuenceslikesolarwindsandspuriouselectromagneticradiation.LikeLIGO,LISAuses1064nmlasers,whichduetoGaussianbeampropagationspreadtoadiameterofabout30kmfromtheinitial40cm,whichisthesizeoftheprimarytelescopemirrors.Asaresult,onlyasmallfractionofthetotalavailablelightpowerontheorderofseveral100pWiscapturedfortheinterferometricmeasurement.Asimpleback-reectiontoformaconventionalMichelsoninterferometerwoulddecreasethereturnedlightpowertoanundetectablelevel.Instead,alocallaseronthefarspacecraftisusedasanopticaltranspondertorecoveritsphaseinaheterodyneinterferometricmeasurement.ThisprocedurewillbediscussedinmoredetailinChapter 5 .Ultra-stableopticalbenchesserveasthereferencepointforthetransponderlasers,splittingthedistancemeasurementsbetweeneachpairoftestmassesintothreesections,asillustratedinFigure 3-7 .Becauseofthelowlightpowers,shotnoiseplaysacrucialroleforthedetectorsensitivityofLISAaswell.TheLISAdesignsensitivityisplottedinFigure 3-8 foranintegrationtime 50

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Figure3-8. LISAdesignsensitivity.Newtoniannoisefromsolarsystemdynamicsisexpectedtolimitthedetectorperformance,andphotonicshotnoiseduetothelowrecoveredlightpowerspreventsLISAfromsensingfrequencieshigherthatafewHz.Source: http://www.astro.cardi.ac.uk/ of1yearwithaselectionofsourcesitwillbeabletodetect.Athigherfrequencieslasercarriershotnoise(notvacuumasforLIGO)shapesthesensitivitycurvemodulatedbythearmresponses,andatlowfrequenciesitisNewtoniannoisefromsolarsystemdynamicsthatlimitsLISA.Sinceagainthisisagravitationaleect,itisimpossibletoshieldthetestmassesfromit,andduetothehugeseparationtherearedierentialeectsthatleakintothescienceband.Theoriginalsensitivityrequirementforthecombineddierentialarmlengthnoisemeasurementwasgivenby ex(f)<1210)]TJ /F5 7.97 Tf 6.59 0 Td[(12s 1+3mHz f4m p Hz10)]TJ /F5 7.97 Tf 6.59 0 Td[(4Hz
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whichsplitinto10pm=p Hzfortheinter-spacecraftportionand1pm=p Hzforthetwolocalinterferometers,whichtheinterferometricphasemeasurementsystem(PMS)needstobeabletoresolve.There-designofeLISAwithshorterarmswouldrequireanincreasedaccuracyforequalstrainsensitivity,butalsoincreasetheamountoflightpoweravailableforthemeasurement,whichprovidesastrongersignal.ThedevelopmentofacompactimplementationofPMShardwarethatcanbeusedfordigitallaserfrequencycontrolanditsimplementationintoaLIGO-orientedthermalnoiseexperimentisamajorpartofthepresentedwork. 3.3.1Time-DelayInterferometryThedierentialchangeingravitationalpotentialhastheadditionaleectthatthetriangularconstellationisbreathing,andthearmlengthsgraduallyvaryoverthecourseoftheyearontheorderof1%[ 56 ].ThisliesverymuchoutsideofLISA'sscienceband,butitrequirescompensationforthedriftsinthedatastream.Mostimportantly,itcreatesanunequalarmlengthinterferometerthatcoupleslaserfrequencynoiseuctuationsintothephasemeasurements.Tocompensate,LISAutilizestime-delayinterferometry(TDI),whichsynthesizesanequalarminterferometerindatapost-processing[ 57 ].Thedatastreamsfortheinterfero-meterarmsaredelayedbytheround-triptimethroughtherespectiveotherarm,suchthatthecombinedpropagationtimesoftheoriginallaserphasematchinthenalexpressions,asillustratedinFigure 3-9 .SinceLISAisaversatileinterferometerwithmanyindividualphasedatastreams,avarietyofbasicTDIcombinationscanbeconstructedthatcancellasernoise[ 58 ],whichspananentireTDIspaceofnoisecancelingcombinations.TherstgenerationofTDIaddressesonlystaticarmlengthmismatches,butthroughconsecutivedelaycombinationslaterTDIgenerationsareabletotakerelativemotionandaccelerationintoaccount[ 59 ]. 52

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Figure3-9. Time-delayinterferometry.Themeasuredphasetimeseriesinthearmsaredelayedinpost-processingbytheroundtriptimethroughtherespectiveotherarm,synthesizinganequalarmlengthinterferometer. 3.3.2OpticalRangingForthesuccessfulimplementationofitsnoisecancelingcombinations,theTDIpipelinerequiresapreciseestimateoftheabsolutearmlengths.Anyresidualdeviationfromtheactualvaluewillremainasalengthmismatchandcouplelaserfrequencynoiseintothephasedata.TriangulationwiththeDeepSpaceNetwork[ 60 ]hasanaccuracyofseveralkilometers,andneedstobecomplementedwithinter-spacecraftranging.AproposedactiverangingschemeborrowsitsoperatingprinciplefromtheGlobalPositioningSystem(GPS),whichbroadcastspseudo-randomnoise(PRN)binarycodesfromsatellitesaroundtheglobeonradiocarrierwaves.Similarly,inLISAeachlaserlinkcanusefastbinaryphaseshiftkeyingfarabovethemeasurementbandtotransmitaPRNcodetothefarspacecraft,whereitstraveltimeisdetermined,whichyieldsanestimatefortheabsolutearmlength.Accuraciesbelow1mweredemonstratedinweak-lightscenarios[ 61 , 62 ]. 53

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3.3.3LaserFrequencyNoiseinLISAThefrequencystabilityofacoherentlightsourcecanbeessentialforthesuccessofaninterferometricexperimentwithamacroscopicopticalpathlengthdierenceLbetweenphasesamplingpoints.Insuchascenarioonesignalwillarrivelatecomparedtotheotherbythepropagationtimet=L=c,wherec2:998108m=sisthespeedoflight(invacuum).Ifthefrequencyofthesignalchangesbyduringt,torstorderaphasedierence=2tappearsbetweensamplesthatwouldbeinterpretedasavariationL==2oftheopticalpathlength.Thefrequencynoisee(f)ofasignalindicateshowmuchitsfrequencyuctuatesonagiventimescale.Accordingly,e(f)isresponsibleforthereadoutlengthnoise ex(f)=e(f)L c(3{9)inaninterferometricmeasurement.SpecicallyfortheLISAmissionthedierenceinarmlengthcanbeashighas1%ofthespacecraftseparation,whichaccountstoabout50,000km.TheuseofTDIsynthesizesanequal-armlengthinterferometerwithranginginformation,butaresidualuncertaintyremainsthatisestimatedtobeontheorderofL=1m.Ifthisisthecasethepicometerprecisioninex(f)fromEq.( 3{8 )thatLISArequirestoreachitsdesignsensitivitywouldneedtheabsolutelaserfrequencynoisetoobey e(f)
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CHAPTER4BROWNIANDISPLACEMENTNOISEINMIRRORSBytheendofthe19thcenturytheideathatmatterconsistsofatomsandmolecules,andthatthetemperatureofasubstanceisintimatelyrelatedwiththedynamicsofitsconstituentshadgainedplentyofmomentum.Thechaoticmotionofparticlesimmersedinsupposedlyundisturbedwaterpockets,whichhadrstbeenobservedin1827bybotanistRobertBrownunderamicroscope[ 63 , 64 ],wasregardedasstrongsupportforthistheory.EinsteinwasabletoprovideanexplanationforthisBrownianmotioninoneofhisannusmirabilispapers[ 65 ].Fromamoleculardynamicalpointofviewhebuiltamodelforthechaoticmotionofparticlesinahostgasorliquidbasedonrandomcollisionswiththeirsurroundingthermallyagitatedparticles.Inparallel,Langevinhaddevelopedanequivalentbuteasierapproach,whichledtothesameequationsforthediusionparameter[ 66 ].Einsteinwasfurtherabletoderiveamethodofdeterminingthesizeofmoleculesfromtheobservationofparticlerandomwalks.Threeyearslaterin1908Perrinpublishedhismeticulouswork,whichiswidelyregardedastherstdeniteproofforthegraininessofmatter,onthissubjectinmultiplearticles[ 67 ],andwasconsecutivelyawardedtheNobelprize.Inthelightofscientichistory,thetermsBrownianandthermalareoftenusedinterchangeably.Thethermalexcitationofthemicroscopicdegreesoffreedomofamacroscopicsystemisnotexclusivetogasesandliquids,butisalsopresentinsolids.Forobjectslikeinterferometermirrorsorsuspensionwires,itrepresentsaveryrealsourceofdisplacementnoisethatmaycompromisetheprecisionmeasurementsrequiredforGWdetection.Inparticular,theBrowniandisplacementnoiseofthereectivetestmasscoatingsisexpectedtobeoneofthelimitingfactorsofAdvancedLIGO.BecausematerialresearchandtheprospectofcryogenicsfortheLIGOtestmasseswouldstronglyprotfromalab-scalethermalnoisetestbed,weconstructedtheCryogenicThermalNoiseOpticalResonator(CryoTHOR)experimentattheUniversityofFlorida(UF).Thepreviousaccomplishmentsofthepresentedworkstrongly 55

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cateredtowardstheirapplicationinCryoTHOR,whichisthenalsciencethemeforthisthesis.WewillrstdiscusstheFluctuation-DissipationTheorem(FDT)andseehowthermalnoiseisadirectconsequencewheneverenergeticlossesarepresent.AsanapplicationoftheFDT,wederiveanexpressionforthethermaldisplacementnoiseduetomechanicallossinthecoatingandsubstrateofamirror,anddiscussdierentapproachestoquantifythemechanicallossofthesystemandconstructtheprobeddegreeoffreedom.Boththedirectscalingofthermalnoisewiththetemperatureofasystemandtheindirectdependenceviamaterialparameterscanhelpreducingthermalnoise,whichwediscussastheconclusionofthischapter. 4.1Fluctuation-DissipationTheoremTherstversionoftheFDTwastheNyquist-JohnsonTheorem,whichwasempiricallydeterminedbyJ.B.Johnson[ 68 ]andtheoreticallyexplainedbyH.Nyquist[ 69 ]in1928.Thethermalexcitationofthechargecarriersinaconductorspontaneouslyevokesgradientsinthechargedistributionandgeneratesanelectromotiveforce(EMF)E(!),whichinturncandriveanelectriccurrent I(!)=E(!)=ZN(!)(4{1)throughacircuitwithnetworkimpedanceZN(!).TheydeterminedandconrmedthattheaccompanyinguctuationhI2iinthiscurrentisgivenby hI2i=2kBT 1Z0R(!)jYN(!)j2d!;(4{2)wherekB=1:3810)]TJ /F5 7.97 Tf 6.58 0 Td[(23m2kgs)]TJ /F5 7.97 Tf 6.58 0 Td[(2K)]TJ /F5 7.97 Tf 6.58 0 Td[(1isBoltzmann'sconstantandTisthetemperatureoftheconductorinKelvins.ThedissipationenterstheequationwithR(!)=RfZ(!)g,whichistherealpartoftheEMF-generatingconductor'selectricalimpedanceZ(!),andYN(!)=1=ZN(!)istheoveralltransferadmittanceoftheelectricnetworkusedforthe 56

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currentmeasurement.ThevoltageorcurrentnoisethatfollowsfromEq.( 4{2 ),whichshouldnotbeconfusedwithelectronicshotnoise,iswidelyknownasJohnsonnoise.ThetheoremwasgeneralizedbyCallenandWeltonin1951forgeneralimpedancesZ(!)thatareassociatedwiththedissipationofenergyinanykindofphysicalsystem,notjustelectricalcomponents.UsinglinearperturbationtheorytheycalculatedthedissipationRV=jZVj2inageneralquantumsystemduetoanexternalperturbation^V.Similarly,theyderivedthequantummechanicaluctuationofthesameperturbationoperator^VusingBoltzmannstatistics,andfoundthatthetwoexpressionsareequaluptoafactor,intheclassicallimityielding hV2i=2 kBT1Z0RV(!)d!;(4{3)whereRV(!)isnowtherealpartofthegeneralizedimpedanceZV(!)[ 70 ]IncomparisontoEq.( 4{2 )thenetworkadmittanceYN(!)hasvanishedbecausetheuctuationinEq.( 4{3 )isnowstatedforthegeneralizedEMFitself,andisnotassessedviaaderivedgeneralizedcurrent.Whenmeasuring^V,givenamodelforthedissipationifitwasinsteadappliedtothesystemasaperturbation,Eq.( 4{3 )tellsthenoiseintheobservedvalueduethelonefactthatthesystemisconnectedtoathermalbathattemperatureT.CallenandGreenereformulatedthisversionoftheFDTayearlatertobestatedintermsofthegeneralizedadmittanceYV(!)=1=ZV(!)inwhichcaseitbecomes SV(!)=4kBT !2
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4.1.1InternalDampinginSolidMaterialsWhenasolidbodyisperturbed,itiscustomarytodecomposetheperturbationintothenormalmodesofthebody,becausetheirtimeevolutiondecouples,andeachindividualnormalmodecanbetreatedasaharmonicoscillatorwitheectivemassmandspringconstantk.Uponperturbation,thereisageneralizedforcethatstrivestoreturnthebodytoitsequilibriumconguration,whichislaggingthedynamicsoftheperturbationbyafrequencydependentphaseangle(!)1[ 72 ].Inotherwords,ifsuchanormalmodemovesasx(t)=x0cos(!t),itwillbeaccompaniedbyarestoringforceFx(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(kx0cos)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(!t)]TJ /F3 11.955 Tf 9.74 0 Td[((!),andbecausetheyareslightlyoutofphase,theworkdonebyFx(t) Wdiss=hFx(t)_x(t)i=)]TJ /F3 11.955 Tf 9.3 0 Td[(kx02!ZTcos)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(!t)]TJ /F3 11.955 Tf 11.96 0 Td[((!)sin(!t))]TJ /F3 11.955 Tf 21.91 0 Td[(!(!)1 2kx20(4{5)isdissipatedintothesystem.Sincethetotalenergyintheoscillatingmotionisequaltotheenergystoredinthemaximumelasticdeformationinthenormalmode, U=1 2kx02;(4{6)duringeachcyclewithdurationT=2=!thefraction2(!)ofthestoredenergyisdissipated,whichhasgained(!)thenamelossangle.Moregenerally,thephaselagofaharmonicoscillatorcanbemodeledusingacomplexspringconstant[ 73 ],suchthatFxisgivenby Fx=)]TJ /F1 11.955 Tf 9.62 3.16 Td[(^kx=)]TJ /F3 11.955 Tf 9.3 0 Td[(kei(!)x)]TJ /F3 11.955 Tf 21.92 0 Td[(k1+i(!)x:(4{7)IfwedrivethisxwithanexternalforceF(t),weobtaintheequationofmotion mx(t)+k[1+i(!)]x(t)=F(t);(4{8)forwhichaformalsolutioncanbefoundintheFourierdomain, ex(!)=1=m !02)]TJ /F3 11.955 Tf 11.96 0 Td[(!2+i(!)!02eF(!)=rx(!)eF(!);(4{9) 58

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where!0=p k=mistheresonancefrequencyoftheordinaryharmonicoscillatorwithspringconstantkandmassm.ThepowerspectrumSx(!)=jex(!)j2ofx(t)drivenbyF(t)isobtainedbymultiplyingEq.( 4{9 )withitscomplexconjugate, Sx(!)=jrx(!)j2jeF(!)j2=1=(m!02)2 (1)]TJ /F3 11.955 Tf 11.96 0 Td[(!2=!02)2+(!)2jeF(!)j2;(4{10)andwendatransferfunctionfromdrivingforcetopowerspectrumthathasasharppeakat!=!0for(!0)1withfullwidthathalfmaximum(FWHM)!=(!0)!0.ForthisresonancewendtheQ-Factor Q=!0 !=1 (!0);(4{11)whichistheinverseofthelossangleattheresonance.ToapplytheFDTweneedtondtheadmittanceY(!)=ev(!)=eF(!)forthissystemanddetermineitsrealpart.Sinceev(!)=i!ex(!)inFourierspace,weuseEq.( 4{9 )anddetermine Y(!)=i!=m !02)]TJ /F3 11.955 Tf 11.96 0 Td[(!2+i(!)!02;(4{12)fromwhichweobtain
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WerewriteEq.( 4{13 )as
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normalmodes,theirinuenceonSx(f)fadesasthelengthscaleoftheirspatialvariationbecomessmallcomparedtotheexpanseofthebeam,whichhasanaveraging,orspatiallow-passlteringeect,andthereconstructionconverges.Unfortunatelythisprocedurecanbecomecomputationallyverydemandingforarbitrarybeamsizesandidealizedhalf-innitetestmasses,asthenumberofnormalmodestobeconsideredgrowsaccordingly.InasimpliedapproachdevelopedbyLevin[ 75 ]weignorethefactthatx(t)isnotageneralizedharmonicdegreeoffreedom.InthiscasetheperturbationthatcorrespondstotheforceF(t)=F0sin(!t)fromSection 4.1.1 becomesapressure P(~r;t)=F(t)f(~r)(4{18)withthesameweightingfunctionasthelaserprole.Thekeytothesimpliedapproachisthattheloss(f)istreatedasageneralpropertyofthebody,notjustitsnormalmodes,andthedissipatedpoweristhefraction2f(f)ofthedynamicallydepositedenergyinthesystem,establishingthecouplingmechanismthatgeneratestheuctuations.Thisprocedurecanbeseenasageneralrecipeforthecalculationofthethermalnoiseinanyobservableofanysystemthathasassociatedlosses,notjustmirrordisplacementnoiseduetodeformations.Foraspecicgeneralizedcoordinateofinterest,acorrespondingvirtualLevinforceneedstobeappliedtothesystem,andtheamountofenergystoredintheelasticdeformationhastobecalculated.Thelossanglefunction(f)willthenscaletheobtainedstaticexpressiontothedissipatedpowerthatcausestheuctuationsintheobservable.Itisveryimportanttokeepinmindthatthevirtualnatureoftheoscillatorypressurethatisappliedtoasystemforthemodelingofthedissipationhasnothingtodowithanyforcesthatmaybeexertedduringtheactualmeasurement.Inparticular,theinterrogatingbeamonatestmassinaGWdetectormayexertaphotonicpressureonthetestmasswiththesameprolef(~r)weuseinP(~r;t),butthevirtualpressureisexclusivelyusedtomodel 61

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theback-reactionoftheinternaldissipationintheinterferometermirrorontothepositionreadout,andhasnophysicalrealityotherwise. 4.2CoatingandSubstrateBrownianNoiseTheoscillatorexampleillustratedthatfortheFDTwecancalculate
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incomplete.Thereectivecoatingisathinlayerwithmechanicalpropertiesquitedierentinsomeregardsfromthoseofthebulksubstratebody. 4.2.1AveragedCoatingLossAngleEmpiricallythelossinthedielectriccoatingsorotherreectivestructures(crystallinecoatingsorgratings)isfargreaterthanthesubstrateloss.Inaddition,theenergydensityishighestwherethecoatingislocated,asitisthesourceofthevirtualpressureinthemodel.ThesurfacedensityUofthedepositedenergyatthecoatinginterfaceis U=F02(1+s)(1)]TJ /F1 11.955 Tf 11.96 0 Td[(2s) Ysw2;(4{22)andacoatingwiththicknessdandlossanglecdissipatesthefraction2fcofUd[ 76 , 77 ].WeaccountforthisbyextendingtheexpressionforthetotaldissipatedpowerforuseinEq.( 4{16 )to Wdiss=2fUs+Ud Uc:(4{23)Itiscustomarytomergetheexpressionsforcoatingandsubstratethermalnoisebecauseoftheirsimilarappearance,despitethemnotbeingcorrelated.TheresultistheBrowniannoise SBNx(f)=2kBT 3=21)]TJ /F3 11.955 Tf 11.95 0 Td[(s2 Yswfs+2 p 1)]TJ /F1 11.955 Tf 11.95 0 Td[(2s 1)]TJ /F3 11.955 Tf 11.96 0 Td[(sd wc;(4{24)whichisledbytheoriginalsubstratenoisefromEq.( 4{21 )[ 78 ].Thecoatingnoiseexperiencesadilutionfactord=w,whichsomewhatcompensatesthehigherlossinthecoating,butexceedsthesubstratenoisebyatleastanorderofmagnitude,dependingonthebeamsize.Dielectric(andcrystalline)coatingsareheterogeneousmaterials,whichiswhycrepresentstheaveragelossinthecoating.Animportantpartofmodelingcoatinglossistoappropriatelyconstructtheaveragelossfromtheindividuallayerproperties. 4.2.2ClassicationintoParallelandPerpendicularLossesDuetotheanisotropyinthecoatingstructure,itispossiblethattheenergeticlossesaredierentdependingonifthedeformationisperpendicularorparalleltothelayer.This 63

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canbeaccountedforbywriting Wdiss=2f)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Uss+Ukk+U??;(4{25)whereitisassumedthattheenergyislostwithinthelayersandnottofrictionattheinterfacesbetweenthem.TheexpressionsforUkandU?willinvolvecoatingaswellassubstratematerialparameters,wheretheYoung'smodulusandPoissonratioofthedierentmaterialsusedinthecoatingmaydier,butareusuallysimilarandcanbeaveragedtoYcandc.TheaccordingnoisetermreadsSCBx(f)=2kBT 3=21)]TJ /F3 11.955 Tf 11.95 0 Td[(s2 Yswfs+1 p d w1 YsYc(1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2s)hYc2(1+s)2(1)]TJ /F1 11.955 Tf 11.95 0 Td[(2s)2k+YsYcc(1+s)(1+c)(1)]TJ /F1 11.955 Tf 11.96 0 Td[(2s)(k)]TJ /F3 11.955 Tf 11.96 0 Td[(?)++Y2s(1+c)2(1)]TJ /F1 11.955 Tf 11.95 0 Td[(2c)?i; (4{26)andreducestoEq.( 4{24 )ifk=?isassumed[ 79 ].Thistreatmentisowedexclusivelytothegeometryofthecoating,andwouldnotbeneededfortheanalysisofindividuallayers. 4.2.3ClassicationofLossesforBulkandShearModesBecauseanincidentbeamisnotreectedrightatthesurfaceofthecoating,buthasacertainpenetrationdepth,aholisticapproachtocalculatecoatingBrowniannoiseneedstoconsiderlightpropagationinsidethecoating[ 80 ]anditsphotoelasticandrefractiveeectsontheopticalpathlength.Becausetheyaecttherecombinationphaseofthepartialreectionsinthecoatingstacks,theycancauseadditionalphaseandamplitudenoise[ 81 ].Similartothespatialweightingwiththeinterrogatinglaserintensitythatleadstothew-dependencyinEqs.( 4{24 )and( 4{26 ),thedeeperalayersitswithinthecoating,thelessofaninuenceitcanhaveontheadditionalnoiseterms.ThediscussionofcoatingBrowniannoisebyHong[ 82 ]buildsonthisapproachandfurthermoredissectsthelossesintobulkandsheardeformationswiththerespectiveelasticmoduliandassociatedlossangles ^K=K(1+iB)^=(1+iS):(4{27) 64

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Table4-1. CoecientsforthebulkandsheardecompositionofcoatingBrowniannoise. Thickness(j)Surfaceheight(zS) BulkCBj=q 1+j 2DBj=1)]TJ /F8 7.97 Tf 6.58 0 Td[(s)]TJ /F5 7.97 Tf 6.59 0 Td[(2s2 p 2(1+j)Yj YsShearACSAj=p 1)]TJ /F1 11.955 Tf 11.96 0 Td[(2jDSAj=)]TJ /F5 7.97 Tf 10.49 4.71 Td[(1)]TJ /F8 7.97 Tf 6.58 0 Td[(s)]TJ /F5 7.97 Tf 6.59 0 Td[(2s2 2p 1)]TJ /F5 7.97 Tf 6.59 0 Td[(2jYj YsShearB(none)DSBj=p 3(1)]TJ /F8 7.97 Tf 6.59 0 Td[(j)(1s)]TJ /F5 7.97 Tf 6.59 0 Td[(2s2) 2p 1)]TJ /F5 7.97 Tf 6.59 0 Td[(2j(1+j)Yj Ys Usingelastictheory,theenergythatisstoredinthecoatingistheintegral Ucoating=UB+US=ZcoatingK 22+ijijdV;(4{28)whereisthebulkdeformation,andijisthesheartensor.Wemerelystatetheresultoftheirfulldiscussion,whichusestheabbreviationsforthecoecientsthataregiveninTable 4-1 .TheircalculationforthecoatingBrownianphasenoiseofthereectedlightisS=XjZzjzj+1dz j1)-222(=j(z) 2CBj+DBj2SBj++XjZzjzj+1dz j1)-222(=j(z) 2CSAj+DSAj2SSj+XjDSBj2lj jSSj; (4{29)andtheamplitudenoiseis S=XjZzjzj+1dz j(CBj
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willmostlyresorttoEq.( 4{24 ),sinceanopticaldirectmeasurementcanonlyresolveanaverageloss. 4.3CoatingNoiseatCryogenicTemperaturesLookingatEq.( 4{24 ),dierentcopingstrategiescanbedevelopedtoreducetheCBNinGWdetectors.WhilethespotsizeoftheGaussianbeamshasbeenpushedtoalimitwhereanadditionalincreasewouldleadtoproblemswithdiractionlosses(clippingthewingsoftheGaussianbyaniteaperture),theuseofhigherordermodescouldreduceCBNbecausetheydistributetheiravailableeldamplitudemoreevenlyacrosstheprobedsurface,enhancingthespatialltering[ 83 ].Thedierenceinindexofrefractionofthecoatingmaterialsdirectlyaectsthenecessarylayerthicknessforatargettransmissivity,howeverforcurrentGWdetectorswithawavelengthof1064nmtheredoesnotseemtobemuchroomforimprovement,andnear-optimalsolutionshavealreadybeenfound.Abaselinechangeforthewavelengthwouldenablethesearchforentirelydierentcoatings,where1550nmisaparticularlyattractivewavelength,becauseSiliconwithacomparablyhighnof3.48[ 84 ]couldbeusedasarefractor.Lastly,thereisthepossibilityofreducingthemechanicallossinthecoatingmaterials,butofcoursethisneedstobebalancedagainstthechangeinopticalandothermechanicalproperties.ThescienticprospectoftheapplicationofcryogenicsinGWinterferometryisextremelyenticing,despitetheengineeringeortthatisattachedtoit.Whilethegeneralscalingofthermalnoisewithp Tpromisesonlymarginalimprovementsevenforadrasticdecreaseintemperature,oneneedstokeepinmindthattheobservedstrainhofaGWscalesinverselywiththedistancetothesource,suchthatevenanimprovementofafactorof2resultsinanincreaseoftheobservablevolumebyafactorof8,whichdirectlyaectseventratesanddetectionSNRs.Moreover,theindirectdependencythroughmaterialparameters,whicharelikelytochangeacrossordersofmagnitudeintemperature,canbeturnedintoanadvantagewiththeselectionoftherightmaterialsortheintelligentdesignofoptical 66

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coatings.SignicantresearcheortconcentratesagainonSiliconduetoastarkdecreaseofitslossangletowardslowertemperatures[ 85 ]. 4.3.1TemperatureDependenceofMaterialParametersThemajorityofthemechanicallossofthestandardamorphousTantala-Silicacoatings(c410)]TJ /F5 7.97 Tf 6.59 0 Td[(4)hasbeentracedtooriginateintheTantalalayers.IncurrentGWdetectorstheyhavebeenreplacedwithTitania-dopedTantala-Silica,whichwerefoundtohaveadecreasedmechanicallossbyaboutafactorof4[ 86 ]withsimilarotherproperties.Whencooledtocryogenictemperatures,theirlosshasbeenfoundtoslightlyincrease(moresigni-cantlybelow100K)[ 87 ],theyarethereforenotastrongcandidateforacryogenicdetector.Themechanicallossofbulkfusedsilicajumpsupwardsbyuptothreeordersofmagnitudewhencooledto100K,andbyyetanotheronetowards40K[ 88 ].Incontrast,weseeadecreaseinlossinbothSapphireandSiliconbyonetotwoordersofmagnitude[ 89 ],makingthemsubstratematerialsofchoiceforcryogenicdetectors,andindeedtheprototypecryogenicKAGRAdetectorwillusesapphiremirrorswithTantala-Silicacoatings[ 90 ].Theirmechanicallossdoesnotchangesignicantlywhencooled,thereforethemajorityofthecryogenicgaininthiscasecomesfromthesuppressionofCBNwithp T. 4.3.2ACryogenicThermalNoiseTestBedThechoiceofamirrorcongurationforthetestmassesisatoughone,ashiddendependenciescanspoiltheeortsconcentratedonreducingCBN.Othernoisesourcesmayarise,orthelossmaybehaveinunforeseenways.AttheUniversityofFloridawethereforesetouttoconstructathermalnoisetestbedwiththegoaltodirectlyobservethedisplacementnoiseinGWdetector-relevantmirrorcoatings.Duetothestrongcaseforcryogenicsthecoolingcapabilitywasacentralobjectiveforitsconstruction.VitalinitialstepsforthisendeavorweretheimplementationofalaserstabilizationsystemthatreliablytransferstheCBN-inducedlengthnoiseinanopticalcavitytothefrequencyofaprobinglaser,andareferenceandreadoutsystemthatisabletoresolvetheselengthuctuations.OurworkontheLISAPMandtheHSdemonstrationexperimentcatered 67

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towardstheseneedsandprovidedanaturalprogressiontowardstheconstructionoftheCryogenicThermalNoiseOpticalResonatorexperimentCryoTHOR.Withtheinterferometricconceptinplace,oneofthemostpressingproblemsbecametheprincipallymutuallyexclusiveimplementationofsuspensionstokeeptheexperimentisolatedfromoutsideinuences,andtheneedforagoodthermalcontactforeectivecoolingoftheinvestigatedmirrors.ThereadoutsystemandfrequencystabilizationschemewillbediscussedinthecontextoftheoriginalLISAexperiments,andinChapter 7 weclosethecircleandpresentCryoTHOR. 68

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CHAPTER5EXTENDEDFREQUENCYRANGELISA-TYPEPHASEMETERTheaccuratemeasurementofsignalphasesiskeytoprecisioninterferometry,asthevitalinformationaboutopticalpathlengthvariations`iscontainedinthepropagationphase =2` :(5{1)Theabilitytoresolvedistancevariationsscalesinverselywiththeusedlaserwavelength,whichissubjecttotechnicalavailability(bothintermsofoscillatorsandlow-noisephotodetectors)andstabilityrequirements.Travelingadistance`throughamediumwithrefractiveindexn(),amonochromaticlasereldofwavelengthaccumulatesthephase +=2 )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(n()+n)]TJ /F3 11.955 Tf 10.96 -9.68 Td[(`+`;(5{2)andbothvariationsninn()and`in`canbemonitoredbyobservingovertime(andnotnecessarilydistinguished).Opticalandnearinfrared(NIR)lasersoscillateatseveralhundredsofTHz,whichisfartoofastforphotodetectorsandreadoutelectronicstofollow,andasecondlasereldisneededtodemodulatetheirphaseinformation.Thischapterfocusesonthelaserphasedemodulationusingheterodynetechniques.TheLISAapproachtomeasuringlaserphasesviabeatnotetrackingisexplained,andphasemeterprototypingeortsarediscussed.Thedevelopmentofaphasemeter(PM)usingintegratedcomputerdesktophardwarethatcanbeusedforfullydigitallaserphasedemodulationandfrequencyactuationinaLISAschemerepresentstherstmajorpracticalpartofthisthesis.WefurthermaximizedthePMbandwidthanddatatransferratestobeabletomeasurethefrequencyuctuationsoflaserbeatsintheLIGObandforthecoatingthermalnoiseexperimentthatispresentedinChapter 7 .Theprogrammingprocessandnalizeddesignarediscussedanditstechnicallimitationsforthephasereadoutareexplored. 69

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5.1HeterodyneInterferometryAsuperpositionwithasecondlasereldtranslatesthephasevariationsfromphysicalprocessesofinterest,whichhappenonamuchslowertimescalebutexistasmodulationsoftheextremelyfastoscillatinglaserphase,toelectronicallyprocessablefrequencies.TheMichelsontopologydiscussedinSection 3.1 ,inwhichthetwointerferedeldsstemfromthesamelaser,andthereforeoscillateatnominallythesamefrequency,isoneexampleforaho-modyneinterferometer.TheirinherentlyxedphaserelationrequiresasinglephotodetectortomeasurethedierentialphaseevolutionfromtheslopeoftheMichelsonfringe,orusingsidebandtechniques[ 91 ].Iftherelevantphaseinformationispresentinalone,weakcarriereld,asitisthecaseforthefractionoflaserlightcapturedbytheprimarytelescopesintheLISAmission,itneedstoberecoveredbyinterferencewithalocallyseeded,stronglasereldtomitigatetechnicalnoiseinthephasereadoutprocess.Aphotodetectorconvertsthesuperpositionofthetwoeldsintoasignalthatoscillatesattheirdierencefrequencyandcarriestheirdierencephase.SincepassingGWsmodulateonlythepropagatinglaserfromthefarspacecraft,thisdierencephasewillcontainitsGWimprints.However,thetwoeldsinaheterodynereadoutscenariodonotshareaxedphaserelation,andareferencephaseneedstoberecordedinordertoisolatetheGW,whichiswhyalllaserlinksinLISAneedtobebi-directional. 5.1.1BeatSignalGenerationInaconcreteexample,asuperpositionE(t;x;y;z)oftwomonochromaticplane-wavelasereldsthatpropagatetogetherwiththesame(andthereforeneglected)polarizationinthez-directionreads E(t;x;y;z)=E1(x;y)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i(!1t)]TJ /F8 7.97 Tf 6.58 0 Td[(k1z+1)+E2(x;y)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i(!2t)]TJ /F8 7.97 Tf 6.59 0 Td[(k2z+2);(5{3)whereEi(x;y)isthetransverseamplitudeprole,!iistheangularfrequency,ki=!i=cisthewavenumber,andiisthephaseoflaseri. 70

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Figure5-1. Beatsignalcreation.Asuperpositionoftwoelds(indicatedasorangeandblue)thatisincidentonaPDgeneratesanoscillatingsignalattheirdierencefrequency.Anydierentialphasevariations(indicatedbyarrows)aretranslatedtophasevariationsoftheinterferencesignal. Whenincidentonaphotodiode,thephoto-currentdensitygeneratedintheabsorptionprocessisproportionaltotheintensityI(t;x;y;z)=jE(t;x;y;z)j2oftheincidentlight[ 92 ].Theresponsivity(x;y)ofthephotodiodeandthePD'stransimpedancegainRTIconvertthephoto-currenttoavoltage V(z;t)=ZZRTI(x;y)jE(x;y;z;t)j2dxdy:(5{4)Assuminghomogeneousresponsivity(x;y)=0acrossthephotodiode,andneglectingthebandwidthlimitinRTIduetocapacitancesinthecircuit,aPDthatisilluminatedbytheeld( 5{3 )generatesthevoltage V(z;t)=0RTIhP1+P2+2p P1P2cos)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(!t+(z)i;(5{5)whichhasatime-dependenttermthatoscillatesatthedierenceangularfrequency !=!1)]TJ /F3 11.955 Tf 11.96 0 Td[(!2(5{6)andiscommonlycalledthebeat,beatnote,orbeatsignalofthelasers.Thecontrastfactor0<<1dependsonthespatialoverlapbetweentheEi(x;y),andisassumedtobeunity(perfectoverlap)fortheremainderofthisthesis.TheopticaldemodulationprocessisillustratedinFigure 5-1 . (z)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F1 11.955 Tf 11.95 0 Td[((k1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2)z(5{7) 71

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Figure5-2. Twovariantsofheterodyneinterferometry.Inthelocalscenariolaser2isusedexclusivelytogeneratebeatsatPDlocations,whereastheglobalinterferencesendsthesuperpositionofthelasersintotheinterferometer.Thersthasamuchstrongerinterferometricphaseresponse,whilethesecondislesssensitivetoauxiliarypathlengthnoise. isthedierencephase,intowhichthepath-dependentpropagationphasesk1zandk2zhavebeenabsorbed.ItisthisterminEq.( 5{7 )thatcontainstherelevantinformationaboutopticalpathlengthvariations.Interferometricphasesthatwouldcausesignalleveluctuationsintheoutputofahomodyneinterferometerhavebeentranslatedtoappearasphaseuctuationsofasignalthatoscillatesat!. 5.1.2VariantsofHeterodyneInterferometryIncontrasttohomodyneinterferometery,thephase2oflaser2withrespecttolaser1isnotknown,suchthatanyheterodynemeasurementrequiresatleasttwophotodetectors.AdedicatedPDsamplesthesuperpositionatareferencepointintheinterferometricsetup,whichestablishesareferencephase.Followingthephaseevolutionofaheterodynesignalwithrespecttothisreferencephaserevealslengthuctuationsintheinterferometer.Therearetwopossiblescenariosforheterodyneinterferometry,whichareillustratedinFigure 5-2 .Thebeamscanbelocallyinterferedbuttravelseparatepathsotherwise,ortheycantheycanbegloballyinterferedandpropagatethroughtheinterferometeroncommonpaths.Therstcasehasamuchmoresensitivephaseresponsetopathlengthuctuations,becausetheaccumulatedphaseisk1zasopposedto(k1)]TJ /F3 11.955 Tf 12.96 0 Td[(k2)z.Itisthereforepossibletoeitheraccentuatethephaseimprintoflengthvariations(forexamplethegravitationalwaveinteractionandtestmasspositionreadout)ormarginalizeitifdierentialphasenoisebetweenbeatsignalsistobekeptlow. 72

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5.2TrackingPhasemeterConceptOneofthexpointsinthedesignchoicesfortheLISAmissionhasbeentousealaserwavelengthof=1064nm.WithEq.( 5{1 )thedisplacementnoiserequirementfromEq.( 3{8 )thendictatesthatthephaseofthepropagatingbeamsneedstobedeterminedwithresidualphasenoise e(f)<210)]TJ /F5 7.97 Tf 6.58 0 Td[(6s 1+3mHz f4rad p Hz10)]TJ /F5 7.97 Tf 6.59 0 Td[(4Hz1HzincaseofLISA,anoiseoorofatleast1Hz=p Hzistobeexpected,andthephasesensitivitye(f)=e(f)=f(seeAppendix A )becomesevenworsethroughoutthemeasurementband.Countingzero-crossingsbecomesfurthermoreproblematicifasignalisweakandburiedinnoise,orifmultiplesignalsatdierentfrequenciesarepresent.Alternatively,thebeatcanbedemodulatedagainstanoscillatorofthesamenominalfrequency,whichprovidesaphase-resolvedreadoutwithalocaloscillator(LO)phaseasareference.ArstfrequencyestimatefortheLOcancomefromazero-crossingcountoranFFTbinpowercheck[ 93 ],andsetitcloseenoughtothebeatfrequencyforreasonableloggingratesofseveralHz. 73

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Figure5-3. ConceptualPLLsketch.ThebeatnoteX(t)ismultipliedwithalocaloscillatorY(t).Low-passlteringtheresultyieldsthequadratureQ(t),whichservesasanerrorsignaltothefeedbackservo,andadjuststheLOfrequencytotrackX(t).Withinthecontrollerbandwidth,allphasevariationsinX(t)aroundthefrequency!arecopiedtoY(t). BecauseofrelativeDoppler-shifts,inLISAthebeatfrequenciescanswingbyseveraltensofMHz,andtheLOsneedtotrackthebeatstoavoidcycleslipswhilemaintainingaxedsamplingrate.TheLOphasethenneedstobeloggedalongwiththephasedierencebetweenbeatandLO,otherwisethetruephaseprogressionofthebeatisnotrepresentedinthedata.TheLISAphasemeasurementsystem(PMS)needstofeaturephase-resolvedsignalreadoutatthemicro-cyclelevelandbeabletofocusonasinglefrequencycomponentoutofapotentiallycomplexsignal. 5.2.1PhaseLockLoopsThecorearchitectureoftheLISAtypePMrevolvesaroundthephase-lockedloop(PLL)principle,whichisacommontopicinsignalprocessing.APLLcangenerallybeusedasatranspondertorecoverthephaseof(potentiallyweak)oscillatingsignals,boostingitspowerandrejectingnoiseoutsidethefeedbackbandwidth.Let X(t)=AXsin)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(!t+X(t)(5{10)betheACportionofabeatnoteoftwolasereldswithamplitudeAX,inwhichweattributeanominalconstantangularfrequency!tothebeatnoteandidentifyalldeviationsofits 74

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phasefrom!ttochangesinX(t).WetreattheLO Y(t)=AYcos)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(!t+Y(t);(5{11)inthesameway,assumingthatitreadilyoscillatesat!withamplitudeAYandphasedeviationY(t).Makinguseofthetrigonometricidentity sincos=1 2hsin)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(+sin)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(+i;(5{12)wetake=!t+X(t)tobethephaseofthelaserbeat,and=!t+Y(t)thephaseoftheLO,andobtain X(t)Y(t)=`AXAY 2hsin)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(X(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Y(t)+sin)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(2!t+X(t)+Y(t)i:(5{13)Foranalogfrequencymixingthefactor`dependsonthepowerconversioneciencyoftheusedmixerandthepowerinthesecondaryoscillator,withtypicalvaluesintherangefrom:001to:5V=V2,whilefordigitalmixingitisunity(withAXandAYscaledtoareferencevoltage).The!ttermsinbothsignalscancelintherstterm,leavingonlythephasedierence(t)=X(t))]TJ /F3 11.955 Tf 12.83 0 Td[(Y(t).Forradiofrequency(RF)mixing,thesecondtermisoscillatingveryfastcomparedtobandwidthofinterestintherstat2!t,andisusuallyremovedwithappropriatelow-pass(LP)lteringbeforefurtherprocessing,leaving X(t)Y(t)LP=CAXAY 2sin)]TJ /F1 11.955 Tf 5.48 -9.69 Td[((t)=Q(t):(5{14)DrivingtheerrorsignalQ(t)tozerowithfeedbacktoY(t)establishesthephase-lockwith=X)]TJ /F3 11.955 Tf 11.95 0 Td[(Y=0andisequivalenttocopyingtheincomingsignalphasetotheLO.TheoriginalsignalphasecanbereconstructedfromthephaseofthelocaloscillatorwithresidualcorrectionsfromQ(t),whichconstitutestheoperatingprincipleoftheLISAPLL-basedphasemeter[ 94 ].TheinitiallockacquisitionrequiresthePLLoscillatorfrequencytobewithinthefeedbackbandwidthfromthesignalfrequency,orelsethechangesin(t)willbetoorapidfortheservotocatchthelock. 75

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Figure5-4. PLLLaplacemodel.TheerrorsignalquadratureeQ(s)issuppressedbytheservogainGS(s)andseedsthefrequencyactuationeF(s)forthesecondaryoscillatorY,whichcopiestheinputcharacteristiceX(s)ontoeY(s). 5.2.2LinearizedModelInthepresenceoffeedbackofQ(t)tothePLLoscillatorphase,(t)willbecomesmallenoughtoapproximatethesineinEq.( 5{14 )byitsargument, Q(t)CAXAY 2(t)=GPDGLP(t)=GPDGLP)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(X(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Y(t);(5{15)inwhichcaseQ(t)becomesproportionalto(t)withancombinedphasedetectorandlow-passgainofGPDGLP.UnderthispremisewecanbuildalinearizedmodelforthephasemeterPLLandillustratethemeasurementprincipleoftheLISAphasemeter.AsketchofthePLLmodelwiththenomenclatureofsignalsandtransferfunctionsisshowninFigure 5-4 .WetransformtheerrorsignaltotheLaplacedomainwiththevariables=+i!,whichisalinearoperationonEq.( 5{15 ),andobtain eQ(s)=GPD(s)GLP(s)eX(s))]TJ /F9 11.955 Tf 13.07 3.16 Td[(eY(s);(5{16)wherewehaveacknowledgedthatthedemodulationprocesswithsubsequentlow-passlteringcangiveGPD(s)GLP(s)afrequencydependency.InanalogcircuitstheroleoftheLOislledbyavoltagecontrolledoscillator(VCO),whosefrequencyrespondslinearlytoappliedvoltages.WemodeltheLOthereforeto 76

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interpretitsinputF(t)ascorrectionstothenominaloscillationfrequency.Inthelinearizedmodelitsoutputisaphase,whichestablishesanintegratorinitstransferfunction GY(s)=2 s:(5{17)Becauseofthisfrequencytophaseconversion,aPLLcanbestablebysimplyusingeE(s)astheinputforGY(s),howevermore(orless)gainmaybeneededtomeetperformancerequirementsortoguaranteeastablelockoperation.Therefore,beforebeingpassedtotheoscillator,theerrorsignalisrenedbytheservogainGS(s),whichistheonlytrulycustomizablecomponentinthismodel.FromacontroltheorystandpointtheplantG(s)ofthePLListheproductofallindividualtransferfunctionsofcontrolloopcomponents, G(s)=GY(s)GS(s)GLP(s)GPD(s);(5{18)whichisalsocalledtheopenloopgain(OLG)ofthefeedbackloop,sinceitrepresentsthefeed-throughgainoff(s)toarrivebackatitsstartingpointaseY(s).Moreverbosely,itis eY(s)=G(s))]TJ 6.6 -6.53 Td[(eX(s))]TJ /F9 11.955 Tf 13.07 3.16 Td[(eY(s);(5{19)andwecansolveforeY(s)tobe eY(s)=G(s) 1+G(s)eX(s)=H(s)eX(s);(5{20)wherewedenedtheclosedloopgain(CLG)H(s)ofthesystem.InthehighgainlimitwithG(s)1,wecanapproximateH(s)1,witharesidualerrorbetweeneX(s)andeY(s)thatisgivenby eX(s))]TJ /F9 11.955 Tf 13.08 3.16 Td[(eY(s)=1 1+G(s)eX(s)=E(s)eX(s)(5{21)andthereforesuppressedbytheOLGinthelooperrorfunctionE(s).ReadingtheoutputF(t)oftheloopservo,thedataacquisition(DAQ)samplesthefrequencyuctuationsoftheinputsignal,andthecorrespondingphaseuctuationscanbe 77

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restoredvia Y(t)=2tZt0F(t)dt:(5{22)BecauseoftheRFheterodynefrequenciestheLOphaseisrapidlyincreasing(onaveragewith!t),whichmakesF(t)thepreferredreadoutquantity,asithasanominalaverageanddoesnotfeaturecyclewrapslikeY(t).TheexpectedlevelofinputnoiseeX(s)andtheallowedresidualphasenoisefmax(s)dictatehowmuchgainisneededinG(s)tobeabletoneglecteQ(s),providingtherequirement jG(s)j>eX(s) fmax(s);(5{23)accordingtowhichGS(s)needstobedesigned.H(s)fromEq.( 5{20 )thenequalsunitytowithintherequiredprecision,andusingY(t)asanestimateforX(t)isjustied. 5.2.3IQreadoutThehighgainlimitcannotbestretchedtoarbitrarilyhighfrequenciesbecauseoffeedbackinstabilityifthephasemarginattheunitygainfrequency(UGF)becomestoosmall.Iftheservocannotprovidesucientfeedbackbandwidthtocoverthescienticallyinterestingfrequencies,samplingonlyY(t)isnolongersucientforthefullphasereconstruction.DeviationsofY(t)fromX(t)appearinQ(t),whichneedstobeloggedaswellinthiscasetoobtainthenecessarycorrectionstoY(t).However,thesignalamplitudeAXmaybesubjecttosmalluctuationsovertime,whichchangesthephasedetectorgaininQ(t)andwouldspoileortstoinvertthesineinEq.( 5{14 ).IntheLISAmissionthiscouldforexamplehappenduetovariationsinthereceivedlightpowerortheoverlapoftheincomingbeamwiththelocallaser.ThesignalQ(t)canberidofamplitudescalingusingatrigonometricrelationsimilartoEq.( 5{13 )thatreads sinsin=1 2hcos)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F1 11.955 Tf 11.95 0 Td[(cos)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(+i;(5{24) 78

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Figure5-5. PhasemeterIQReadout.ThefeedbacktotheQquadratureisusedforthesimultaneousdemodulationoftheout-of-phaseIquadrature.Usingthetwotheimpactofvariationsinthesignalamplitudecanberemovedfromthephaseestimate. fromwhichwe{afterlow-passltering{obtainthequantityI(t)=AXAY 2cos)]TJ /F1 11.955 Tf 5.48 -9.68 Td[((t).UsingbothquadraturesofthebeatsignalinanIQ-readoutscheme,thephasecorrectiontoY(t)canberecoveredas (t)=arctan CAXAY 2sin)]TJ /F1 11.955 Tf 5.48 -9.68 Td[((t) CAXAY 2cos)]TJ /F1 11.955 Tf 5.48 -9.69 Td[((t)!=arctanQ(t) I(t);(5{25)inwhichanyscalingwithsignalamplitudehasbeeneliminated.SincetheLISAsciencebandreachesuptoonlyafewHz,thereisnoshortageofgaininfeedbackloops,andIQreadoutwillusuallynotbenecessary. 5.3DesignConsiderationsInthemissionbaselineacompletelydigitalreadoutschemeusingField-ProgrammableGateArrays(FPGAs)hasbeenchosen[ 95 ].Thelaserbeatnotesaredirectlydigitized,andsignaldemodulationisexclusivelyhandleddigitally.Onceasignalhasbeendigitized,itfacesnofurtherelectronicnoisesources,andhighoscillatortuningbandwidthsanddynamicfrequencyrangescanberealizedwithindigitalcircuitsatfastclockrates. 79

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ThePLL-basedPMcannottrackasignalthroughazero-crossingofitsfrequency.BecauseoftheDoppler-shiftinducedswinginthelaserfrequencies,theyneedtobesucientlyosetfromeachother,sincetheinterferometricphasesmustbemeasuredcoherentlywithoutinterruptions.Ontheotherhand,low-noisePDsbecomeanissuetowardshigherRFfrequencies,andlocalpathlengthnoise(ontheinterferometricbenchandinthecables)scaleswiththesignalfrequency,suchthatitshouldnotbechosentoohigh.Moreover,introductorydigitalsignalprocessingtextbooksintroducetheNyquisttheoremearlyon,whichstatesthatinordertoresolvefrequenciesofinterestuptoanfmaxinasignal,ithastobesampledatafrequencyfsatleasttwicethevalueoffmax.Thefasterelectroniccircuitsareclockedandoperateat,themorecarefultheyneedtobedesigned,thelessrobusttheygenerallyget,andthemoreheattheygenerate,whichcancauseproblemsinspacemissions.Thedesigncompromisehasbeenthatthelaserfrequencieswillbechosenandactuatedsuchthatthebeatswillvarybetween2and20MHz[ 96 ].TheywillthenhavetobedigitizedatatmoderateRFfrequenciesofatleast40MHz. 5.3.1FieldProgrammableGateArraysPMprototypingisbasedonFPGAs,programmablemicrochipsnormallyusedforcustominterfacingandstreamliningcalculationsinreal-timesystems.Conventionalintegratedcircuitsandmicrochipsarehard-printedintosemiconductorwafers,whileFPGAsrepresentanintermediatestepbetweentheirsimulateddesignandphysicalimplementation.AnFPGAisanensembleoflogicgates,whichareinterconnectedbyarecongurablenetworkofconnections,asillustratedinFigure 5-6 .FPGAswererstintroducedtothemarketin1985,andsimilartomicrochipstheyhavesinceexperiencedanever-growingincreaseincomplexityandminiaturization.Becauseoftheircustomizability,FPGAscannotpossiblyachievethecomplexityanddensityofprintedmicrochips,butmodernFPGAscompensateforthisbyintegratingmi-croprocessorsandothersubcomponentsintotheirlayout.Havingon-the-yrecongurable 80

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Figure5-6. FPGAFunctionality.Theleftpictureshowsthearrayofcompactlogicblocks(CLBs)thatarethecentralprocessingunitsofanFPGA.CongureablesignalnetworksinterconnecttheCLBstoemulatethedigitallogicthatisspeciedatthetop-leveldesign.Physicalports(pinsonthechip)andintegratedRAMenableinterfacingwithexternaldigitallogicandlimiteddatastorage.TherightpictureshowsatypicalXilinxFPGACLB(thesmallestcomputingunit),whichconsistsof4x4bit-wiseinputsforlogicoperationsthataremultiplexedintoasingleoutput. digitalcircuitryisanimmenseadvantageforthePMdevelopment,andsincespace-qualiedFPGAsareavailable,PMprototypingeortsrevolvearoundanFPGA-basedapproach.SpecializedprogramminglanguageshavebeendevelopedforFPGAsfromotherhard-warelanguagesthathadbeeninuseforsimulatingdigitalcircuits.ThetwomostusedlanguagesareVerilogandVHDL,whichisshortforVeryHighSpeedHardwareDescriptionLanguage.Theyoeraverbose,user-readablegate-leveldescriptionofthelogiccircuitrywithlimitedsupportformulti-bitdatatypes.Theend-productofanFPGAdesignpipelineisacompiledgate-levelmachinecode,whichisstreamedtotheFPGAviaaspecializedinterfaceandallowsforitsreprogramming. 5.4UniversityofFloridaLISAInterferometrySimulatorPMdevelopmentattheUniversityofFloridawasinitiallyfocussedonsupplyingaprocessingunitfortheUniversityofFloridaLISAInterferometrySimulator(UFLIS)[ 97 ] 81

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withaPMthatmeetsthemissioncyclerequirementfromEq.( 3{10 ).UFLISwasusedsuccessfullytodemonstrateexperimentallytheuseofTDI[ 98 , 99 ],arm-locking[ 100 , 101 ],andGHz-sidebandclocksynchronization[ 102 ]onanopticaltestbenchwithreal-timeelec-tronicphasedelaysofupto32seconds,whichistheround-triptimethrougha5millionkmlonginterferometerarm. 5.4.1UFLISPhasemeterUFLISwasimplementedusingamodularassemblyofPentekcomponentswithXilinxVirtex-IIfamilyFPGAs,whichisillustratedinFigure 5-7 .Themainmodule(model4205)hastwoexpansionslotsfordaughtercards,andintheUFLIScongurationthosewerepopulatedbyanA/Dmodule(model6256)andaD/Amodule(model6228).The6256features2setsof214-bitADCs(AD6645)each,whichdigitizethebeatnotesatarateofupto100MHz.EachsetofADCsisconnectedtoitsownFPGA(XC2VP20),wherethePMlogicislocated.Throughavelocityinterfacemezzanine(VIM)thephaseinformationthatthePMchannelsgenerateisstreamedtothecarriermodule.AsecondaryFPGA(XC2V1000)interfaceswith1GBoflocalmemorytostoreitforlaterreadoutorfurtherprocessing.AsecondVIM-XC2V1000combinationcancalluponthestoredphaseinformationandsendittotheD/AmodulewithfourDAC5686digital-to-analogconverters(DACs).Thelaserbeatnotescouldberecreatedfromlocalmemoryafterasettimehadpassed.Thebeatoftwolaserswasthenosetphase-lockedtotherecreatedsignal,re-imprintingtheoriginalbeatnoteastheirphasedierence.Thecapabilitytoelectronicallydelayopticalsignalsforupto32secondsmadeUFLISavaluabletoolfortestingLISAinterferometrytechniques. 5.5CompactPhasemeterSolutionThePenteksystemwasusedastheprocessingunitforUFLISbecauseithadsignicantfunctionalityoverlapwithLISAphasemeterhardwareneeds,suchasfastdigitizationratesandrobustFPGAimplementation,butitwasnottailoredtowardsLISAapplications.Specically,thedirectactuationoflaserfrequencieswasproblematic,becausetheintended 82

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Figure5-7. Penteksystemmodularassembly.TheA/Dmodule6256digitizesthebeatnotesandhoststhePMlogic.Theacquiredphaseinformationisstreamedtothe4205hostmodule,whichstoresitinmemoryandlatersendsittoanauxiliarycomputerviaEthernetfordatalogging.Additionally,theD/Amodule6228hasaccesstothestoredphaseinformationandcanrecreatetheoriginallysampledbeatnotesafterasettimehaspassed. useofthe6228werefastRFapplications,forwhichtheDACshadbuilt-in,hardwiredsignalup-conversionandhigh-passlteredoutput.DigitallaserfrequencyactuationwouldthereforeonlybepossiblebymodulatingthefeedbackontoanRFsignal,andthenexternallydemodulatingit.Furthermore,themultipleFPGAsandBUSinterfaceshadtobepassedbetweentheA/DconversionandD/Aback-conversionaddedconsiderablephaselossforapotentiallaserfrequencycontrolservo.Afterthesuccessfularm-lockingandTDIexperiments,thenextstepwastomorefullyimplementthelaserfrequencycontrolinthephasemeterfunctionality.ThedemonstrationexperimentdidnotneedthecomplexityofUFLIS,whereprogram-mingthemultipleFPGAs,boardcontrollogic,andDAQsystemoccursonseveralhierarchylevelsandrequiresprogramswrittenindierentlanguages(VHDL,C,Labview)tointerfacewitheachotherwithoutastandardizedexchangeprotocol.WethereforedecidedtoacquireacompacthardwaresolutionthatoersthebarenecessitiesforrunningaLISAphasemeterwithback-endDC-coupledD/Aconversion. 83

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Table5-1. AcromagAX3065specications. PropertyValue A/Dchannels4(22)dierentialA/Dinputrange1VA/Dresolution14bitsA/Dmax.samplingrate64MHzA/Dclockjitter1psrmsmax.A/Dvoltagenoise0.13mVrmsD/Achannels2(12)D/Aoutputrange5VD/Aresolution16bitsD/Amax.samplingrate900kHzD/Avoltagenoise12nV=p HzFPGA13milliongatesSDRAM8MBdouble-sidedDAQInterfacePCI 5.5.1AX3065HardwareSpecicationsThePMC-AX3065modulemanufacturedbyAcromagisanFPGAcardwithA/D,D/A,andlimitedon-boarddatastoragecapability.Thesolutionwepurchasedismountedonanon-intelligentPCIAXmodule,whichactsasaPCI-standardmountandbusconvertertointegratetheAX3065intoadesktopcomputer.AsingleXilinxVirtex2FPGAislocatedonAXx065familycards,ofwhichtheAX3065hasthelargestavailablemodelwith3millionlogicgates.Relevanthardwarespecicationsfromthedatasheet[ 103 ]andtheuser'smanualoftheAX3065arecompiledinTable 5-1 ,andaschematicoverviewofthefeaturedcomponentsisshowninFigure 5-8 .BydefaulttheAX3065isclockedbyanintegrated64MHzcrystaloscillator,anddataexchangeswiththehostdesktopcomputeraredrivenbythe66MHzPCIstandard.Theinternalclockcannotbesynchronizedwithotherexternalclocks,butthecardhasabacksideportforanexternalclock,whichcanbesettooverridethecrystal,inwhichcaseitdrivesboththeA/Dsamplingandtheintra-FPGAclocknets.ThedataacquisitionisperformedbytwodualchanneldierentialADCchips(AD9248),whichgivethecardatotaloffourDC-coupledsamplingchannels.Thisistheminimum 84

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Figure5-8. AX3065componentoverview.Thecorefeatureisthe3millionlogicgateVirtex-2FPGA.Twodual-channelADCsandonedual-channelDACICconstitutethecard'sanaloginterface.CommunicationwiththehostcomputeranddataexchangeisdoneusingacompactPCIcontrollerchipand8MBofonboardRAM. numberneededfordierentialwavefrontsensingwithquadrantphotodetectors[ 104 ],whichatthetimewasapossiblefutureapplicationforthephasemeter.TheA/DresolutionoftheAD9248is14bits,anditsmaximumsamplingrateis64MHz.EachchannelofeachchiphasitsownbustotheFPGA,suchthatthereisnomultiplexinginvolved,andsignalsamplesofallfourchannelsarriveattheFPGAinparallelatthefullsamplingfrequency.Theinputvoltagerangeishard-codedtospanfrom-1Vto1V,andasignalthatexceedstheselimitsthrowsaagthatisalsopassedtotheFPGAtoindicatefaultydata.TheAX3065has8MBofrandomaccessmemory(RAM),inwhichdatacanbestoredtemporarilybeforeitispassedtothehostcomputer.TheRAMisdual-ported,andhasbeenwiredsuchthattheFPGAcaninterfacewithonesideviaadedicatedbus,whiletheothersideisconnectedtothebusthatissharedbyFPGAandPCIchip.Thisallowsforthestreamingofdatawhilesimultaneouslystoringvalues.TheD/AconversionishandledbyadualchannelAD5547chip.Ithasaprecisionof16bitsandcanoutputvoltagesrangingfrom-5Vto5V.Thetwochannelsreceivetheirdatathroughasinglemultiplexedbuswithanupperlimitfortheupdateratestatedtobe900kHz.Updatingthevoltageoutputofasinglechanneltakesatleast4clockcycles. 85

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Figure5-9. AX3065coredesignworkow.AtoolboxforSimulinkcontainsbasiclogicandcommoncomplexDSPoperationswhichisusedtoprogramthePMcoreandassociatedfunctionality.Asingletop-levelVHDLlemergesthePMcorewithcommunicationroutinesanddistributessignalinternally.Therouteddesigncontainsthecompleteimplementationofthedigitallogicandthewiringtoexternalsignals,fromwhichtheprogrammingleisgenerated. 5.5.2FPGADesignWorkowThedevelopmentplatformweusefortheAX3065istheproprietaryXilinxISEsoftware,whichincludesavarietyofcompilersanddiagnostictoolsforcircuitanalysis.TheAX3065PMiscodedinVHDL,andtherearetwomajorstepsalongthecompilationprocessthatyieldthenalFPGAprogrammingle.Theprogramisrstsynthesizedtoalogicgate-levelnetlist,whichprovidesablueprintforthegatedistributionandsignalroutingduringtheimplementationphase.Ifthecompilationissuccessful,thedesignwasabletobetintotheFPGA,andafullgateandsignalroutinglayoutistheresult.ItisthenconvertedintoaprogramminglethatcontainsthedatastreamforconguringtheFPGA.ThisworkowisoutlinedinFigure 5-9 .AlthoughitispossibletocodetheentirePMinVHDL,fortheprototypingprocessitprovedmoreecienttouseassociatedsoftwaretocreateexternalnetlists,whichcanbeimportedintotheFPGAdesignasblackboxcomponents.Specically,XilinxreleasedtheDSPToolssoftwarebundle,whichcontainsatoolboxfortheSimulinkenvironmentthatisapartofMatlab.Itsuppliesavarietyofcommondigitalsignalprocessing(DSP)functionsascompactblocksforgraphicalprogramming.BuildingthePMusingSimulinkhastheadvantageofbeingabletosimulatesub-processesinthePMwithitsfullanalyticcapability, 86

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Figure5-10. AX3065PMstatecontrol.Shownisanexemplarybinarystring(thecontrolstringforthePMchannel1,whichoccupiesthebyteaddresses00x8018through00x8021),whosersttwobitsaectthebehaviorofthecorrespondingPMchannel(lockingon/oandautomaticfrequencyacquisitionon/o).Thelower16bitsofthis32bitstringdeneamanualchannelosetfrequencyiftheautomaticacquisitionisturnedo. whichhelpsresolvetiminganddatahandlingissuesbeforeevenstartingthecompilationprocess. 5.5.3InterfacingviaPCIBusAdynamicdriverlibrary(DLL)issuppliedbyAcromagforinterfacingwithAX-seriescardsviathePCIbus.ImportantcallablefunctionsfromtheDLLforthePMareforexamplePCIAX ReadDwordandPCIAX WriteDword,whichcanaccessanemulatedaddressspaceinsidetheFPGA,wherecontrolregistersforthePMstatelogiccanbeplacedandrequested,andnumericalvaluescanbestored,asillustratedinFigure 5-10 (forexampleforsignalgainsorPMtargetfrequencies).TheFPGAdesignleisstreamedtotheFPGAbycallingthePCIAX DirectConfigurefunction,andthedriver'sdirectmemoryaccess(DMA)routinesallowtheAX3065'sPCIchiptotemporarilytakecontrolofthemotherboard'sPCIbusanddirectlyaccessthesystemmemoryformoreecientandfastertransferofdata.AllDLLfunctionscanbecalledfromNationalInstrument'sLabviewsoftware,whichweuseasafrontendgraphicaluserinterface(GUI). 5.5.4DataLoggingusingDMATransferOurinitialAX3065PMversionsusedthePCIAX ReadDwordfunctiontoextractthechanneldataviatheFPGA'sinternaladdressspace,butthisprocessiscomparablyslow,andloggingdatafromfourPMchannelssimultaneouslybecomeslimitedtoonlyafewreadoutoperationspersecond.WhilethisrateisnominallysucientforLISA,fordiagnostic 87

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Figure5-11. AX3065dataformat.Eachdatasegmentconsistsoften32-bitbinarystrings,ledbyasynchronizingID,adatapointcounter,andfollowedbythe64-bitindividualchanneldata,splitintohighandlowbits.Thedriveronlysupports32-bit-wisedatatransfer,butinthenalbinarylethehighandlowbitsareconcatenatedintolittle-endian64-bitformat. purposesandtousethePMintheLIGObandweimplementedamoreecientDMAstreamingmethod.TheAX3065has8megabytesofRAM,inwhichwepre-allocatetwodistinctmemoryblockstobelledwithPMdata.ThesetupoftheDMAtransferswiththePCIAXdrivercreatestwoequallysizedblocksinthesystemRAMthatmirrorthecardmemory.Thephase/frequencyestimatesforallfourchannelsbecomeavailableatthesametime,andastrobesignalgeneratedbytheSimulinkcoretellstheFPGA'sRAMaccessroutinesthatfreshdatahasarrived.Startingwithaleadingsynchronizingstring,andfollowedbyarunningcountervalueforthenumberIDofthesamplestofollow,the64bitdatastringsfromeachchannelaresplitintotheirhighandlowbits(sinceonly32bitscanbewrittenatatime),andconsecutivelystoredinthecardRAM(seeFigure 5-11 ).ThePMkeepswritingtoaRAMblockuntilitisfull,andthenrequestsaDMAtransfer,duringwhichthecomputerhandsoverthecontrolofitsmainbustotheAX3065,andcannotuseituntilthecardreleasesit.WitheveryDMAtransferthecorrespondingblockofcardRAMiscopiedtothecomputermemory,fromwhereitisloggedtodiskinbinaryformataftereachsuccessfulush,whilethecardRAMblockisvacatedtobeoverwritten.BecausethecardRAMisdual-ported,thePMchannelscanproceedtodumptheirdataintotherespectiveothermemoryblockevenwhiletheDMAtransferisongoing,whichguaranteestheuninterruptedstreamingofphasemeterdata(seeFigure 5-12 ).Logginga 88

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Figure5-12. AX3065datastreamingprocess.ThePMwritescontinuouslytoconsecutiveaddressvaluesuntiltheblockitiscurrentlyaccessingllsup.WhilethePMcontinueswritingtotherespectiveotherblock,aDMAtransferisinitiatedthatcopiestheentireblockintoabuerinthedesktopmemory.Fromthereitiswrittentole.Thisreadoutmethodminimizestimeoccupancyofthelocalbusandenablesfaststreamingrates. single64-bitnumber(8bytes)fromeachchannelatareadoutfrequencyfreadgeneratesdataatarateof 424bytesfread=32bytesfreadbytes s:(5{26)TogetherwiththeIDstringandthecountervaluethismeansthatatanfreadof30kHzittakesabout8secondstollthecardmemory,forcingablockreadouttobeatleastevery4seconds.Toobtainmorefrequentreal-timeupdatesofthechannelreadings(fordisplaypurposesonly)wesetthebuersizetokeepthenumberofDMAtransfersatabouttwotimesasecondatanydesiredreadoutfrequency. 5.6Phasemeterz-DomainModelInSection 5.2.2 wediscussedthelinearizedmodelforthePLLintheLaplacedomain,whichiscoinedtowardsanalogcircuits.Fordigitalcircuitswithdiscretesamplesadescriptioninthez-domainismoreappropriate.ThetransitionfromtheLaplacedomaintothez-domainisachievedwiththesubstitution z=es=fs;(5{27) 89

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whichisamappingofcomplexfrequenciestonumbersinthecomplexplane.Inparticular,realfrequencieswith=0aremappedontotheunitcircle.Thereisaninherentredundancyinthisdescriptionsincefrequenciesfoutsidetheinterval()]TJ /F3 11.955 Tf 9.3 0 Td[(fs=2;fs=2)donothaveauniquecomplexvalueassignedtothem,whichisamanifestationoftheNyquisttheorem.Signalsthatoscillatefasterthanhalfthesamplingfrequencycannotberesolvedbythedigitallogic,butratheraliasintolowerfrequencybands,whichmakeslow-passlteringbeforedigitizationandsamplingratechangesinsurmountable.Functionsofsbecomeexpressionsofzunderthetransformation,andsignalmanipu-lationshaveacorrespondenceinthez-domain.Whileordinaryanalogpropagationdelayscauseaphasedelay2fforananalogsignal,delaysfordigitalsignalsareregistersthatstoreavalueforasetamountofclockcyclesbeforepassingiton.Thephasedelayofaregisteris2f=fs,anditstransferfunctionis Greg(z)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2f=fs=z)]TJ /F5 7.97 Tf 6.59 0 Td[(1:(5{28)Thistransferstootheroperations,suchasthediscreteintegration Gint(z)=1+z)]TJ /F5 7.97 Tf 6.59 0 Td[(1+z)]TJ /F5 7.97 Tf 6.58 0 Td[(2+z)]TJ /F5 7.97 Tf 6.59 0 Td[(3+:::=1Xk=0z)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1;(5{29)wherethegeometricserieswasusedtoobtaintheclosedexpression.Itconvergesonthecomplexunitcircleunlessz=1(thezerofrequencycomponent),whichisconsistentwiththeintegrationintheLaplacedomain.Thediscretedierentiationis Gdi(z)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1;(5{30)andwecanusetheaboveinfomationtobuildaz-domainmodelofthePMchannels. 5.6.1CICFiltersDown-samplingofthephasemeterdataisessentialduetotheveryhighsamplingfrequenciesofthelaserbeatscomparedtotherelevantLISAsignalbandwidth.CascadedIntegratorComb(CIC)ltersarediscreteltersthatexcelatdown-samplingdataacross 90

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Figure5-13. CIClterarchitecture.Nintegratorstagesheavilyattenuatehighfrequencynoise,suchthatthedown-samplingfoldslittlenoiseintothelowerfrequencyband.Theequallymanydierentiatorsre-normalizetheintegralgain. manyordersofmagnitudeinfrequencybecausetheyexclusivelyfoldtruelterzerostoDCandthereforeminimizealiasingtothelowestfrequencies[ 105 ].TheanatomyofaCIClter,whichconsistsofpairedcombinationsofintegratorsanddierentiators,isillustratedinFigure 5-13 .Eachintegratorstageaddsapoleatzerofrequency,attenuatinghigherfrequenciesinthesignal.Becauseofthissuppressionthedown-samplingoperationfoldsverylittlenoiseintothelowerfrequencies,andthedierentiatingcombstagesre-normalizetheDCgain.Filtersthatinvolvefeedbackcanbecomeunstableduetobit-precisionissues,butbecauseCICltersrelyonlyonadditiveoperations,itiseconomictorunthematfullinternalprecision,whichiswhytheyareinherentlystable.ThetransferfunctionofanN-stageCIClterthatdown-samplesthedatabyafactorRis GCIC(z)=1 RN1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(R 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1N;(5{31)wherethegainofR)]TJ /F8 7.97 Tf 6.58 0 Td[(NisoptionallyappliedtonormalizethelterDCgaintounity.WeimplementedCICltersinthePMchannelsaspartofthephasedetectionprocess,andfurthermorefortheheavy-dutydown-samplingofthephaseincrementregisterdata. 91

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Figure5-14. PMz-domainmodel.Similartotheanalogcase,thephasedetectorandNCOarermcomponentsintheloopgain,andtheservoGS(s)isusedtoshapethefeedbacksignalandgiveitsucientgaintoguaranteelockstabilityanddierentialnoisemitigation. 5.6.2PMFeedbackInthelinearmodelthediscretephasedetectorisaproportionalgainwithregisteredoutput.ThediscretetimeseriesforQistherefore Q[n]=A 2hX[n)]TJ /F1 11.955 Tf 11.96 0 Td[(1])]TJ /F3 11.955 Tf 11.96 0 Td[(Y[n)]TJ /F1 11.955 Tf 11.95 0 Td[(1]i;(5{32)andifweincludethePM'snormalized2-stagedown-sampling(R=16)CIClterinitsassociatedtransferfunctionweobtain GPD(z)=A 2z)]TJ /F5 7.97 Tf 6.58 0 Td[(12)]TJ /F5 7.97 Tf 6.59 0 Td[(81)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(R 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(12:(5{33)ThePMaimstomeasurelaserbeatsdownto2MHz,thereforewesetthetheinternaldown-samplingfactorRto16,suchthatthePMsuppliessignalfrequencyestimatesat4MHz.LookingbackattheoscillatorinEq.( 5{17 ),whichwasdeterminedtobeanintegrator,thedigitalequivalenttransferfunctionofanumbercontrolledoscillator(NCO)is GNCO(z)=2z)]TJ /F5 7.97 Tf 6.59 0 Td[(3 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1;(5{34)whereatwocycledelayisincludedforthelook-upprocessbytheSimulinkblock.Because 92

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Figure5-15. DigitalPLLloopgains.ForafullscaleinputsignaltheOLGhasaunitygainfrequencyofabout63kHzwithaphaselossof120.TheCLGissuretotransferphasevariationsidenticallyuptoroughlyonekHz,wherethedierentialphaseerrorsuppressioncrosses60dB. ofthisintegratortheservoGS(z)mayonlyprovideproportionalgainattheUGF,orelsethephaselosswouldbecometoolarge.Sincelargelowfrequencygainisneeded(toavoidhavingtoimplementIQreadout),twointegratorswereaddedtotheservo,whichstartdominatingthefeedbackbelowseveralkHz.TheoveralltransferfunctionoftheServois GS(z)=2)]TJ /F5 7.97 Tf 6.58 0 Td[(5z)]TJ /F5 7.97 Tf 6.58 0 Td[(2R1+2)]TJ /F5 7.97 Tf 6.59 0 Td[(6z)]TJ /F8 7.97 Tf 6.58 0 Td[(R 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(R1+2)]TJ /F5 7.97 Tf 6.59 0 Td[(8z)]TJ /F8 7.97 Tf 6.59 0 Td[(R 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(R;(5{35)wheretheproportionalattenuationof2)]TJ /F5 7.97 Tf 6.59 0 Td[(5isneededtolowertheUGFdowntowherethecombinedphaselossduetopropagationdelayandtheNCOintegratorleavesenoughmarginforastablelockoperation.Asbefore,wecanwritetheOLGofthePMas G(z)=GPD(z)GS(z)GNCO(z):(5{36) 93

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WeplotitalongwithitsderivedlooptransferfunctionsinFigure 5-15 asfunctionsoftherealfrequencyf.ThemassivephaselossfromthetripleintegratorsisrecoveredtowardstheUGFof63kHz,andatreadoutfrequenciesintheLISAsciencebandthedierentialphaseerrorbetweenNCOandlaserbeatissuppressedbymorethan240dBofgain. 5.7InstrumentalNoiseLimitationsInadditiontotheintrinsicelectronicnoiseofanalogcomponentsgeneralfactorssuchasboardcharacteristicsandsignalroutingcanhaveanimpactonthenoisecharacteristicsofanalogcomponents.OnebigadvantageofthedigitalnatureoftheLISAPMapproachisthatthereisnointernalsourceofelectronicnoisebeyondthepointofdigitization,whichsimpliesthemodelingofthePM.However,thedigitizationprocessitselfproducesquantizationnoise,andvariationsinsampletimingleadtophasenoiseinthereadout,whichwefoundtolimittheperformanceoftheAX3065PMimplementation. 5.7.1QuantizationNoiseDigitizingananalogsignal,whichcanobviouslyvarysmoothlyinvalue,withtheinevitablyniteprecisionofadigitalcircuit,oneintroducesanerrorbetweentheactualsignallevelandthenumberthatrepresentsit.Thiserrorcouplesasquantizationnoise(QN),ortruncationnoiseintoconsecutivesignaloperations.ForrandomtimeseriesandsignalsthatarelargecomparedtotheprecisionofthesmallestquantizationunitwecanapproximateQNbythewhitepowerspectrum SQN(f)=ALSB2 6fs;(5{37)whereALSBistheresolutionoftheleast-signicantbit(LSB)andfsisthesamplingfrequen-cy[ 106 ].Intuitively,thenoiseislowerthebettertheresolutionofthedigitizerbecomes,butfastersamplingalsoreducesQN,becausetheavailabilityofmoresamplestendstoaverageoutthequantizationerror(whichisassumedtobenotcorrelatedbetweensamples).TheQNintroducedviaEq.( 5{37 )isnotexclusivetothedigitizationofanalogsignals,buteverytimetheprecisionofadigitalsignaloperationislimited,noisewillbeintroduced 94

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basedonthetruncatedlengthandthecurrentsamplingfrequency.Specicallyconsecutivemultiplicationstendtorapidlyincreasethenumbersofbitsrequiredtomaintainthefullprecision,whichmakessignaltruncationanecessity.TheADCQNrepresentsatechnicallimitationofthePM,anditsinternaltruncationnoiseneedstobecloselywatchedwhenmakingchangestothecode.The14-bitADCsignalsareinterpretedbytheFPGAassignednumbersintwo'scomplementformatthatarenormalizedto1.Thismeansthat214evenlyspaceddistinctvaluescanbegeneratedbetween)]TJ /F1 11.955 Tf 9.3 0 Td[(1and1)]TJ /F1 11.955 Tf 12.21 0 Td[(2)]TJ /F5 7.97 Tf 6.59 0 Td[(13.Thecorrespondingbinaryresolutionof2)]TJ /F8 7.97 Tf 6.59 0 Td[(Q(whereQistheeectivenumberofbits)istheALSBfromEq.( 5{37 ),andwouldideallybe2)]TJ /F5 7.97 Tf 6.58 0 Td[(13.However,fromthedatasheetoftheAD9248welearnthattheroot-mean-square(rms)errorofthedigitizationprocessis1.05bits,whichdecreasestheeectivebitresolutiontoQ=13)]TJ /F1 11.955 Tf 11.68 0 Td[(1:05=11:95.Furthermore,thequantizationerrorneedstobeassessedrelativetotheactualamplitudeofthedigitizedsignal(ratherthanthefullrangeofpossibleADCoutputvalues)ifitdoesnotllthedynamicrangeoftheADC.Thephasenoise(normalizedtocyclesratherthanradians)thatresultsfromADCQNistherefore eQN(f)=p SQN(f) A;(5{38)andusingtheabovevalueforQweobtaintheinstrumentalnoiseoor eQN(f)=2)]TJ /F5 7.97 Tf 6.59 0 Td[(11:95=A p 664MHz=1:2910)]TJ /F5 7.97 Tf 6.59 0 Td[(8 Acycles p Hz(5{39)foranyphasemeasuredbytheAX3065PM.InacomparativemeasurementbetweendierentPMchannelsthislevelisseeninuncorrelatedfashionbyeveryindividualchannel. 5.7.2TimingJitterSincethePMessentiallymeasuresthevariationinarrivaltimesofsignals,itcanonlyeverbeasgoodastheclockthatdrivesthesamplingofthesignals(whichiswhytheclocksondierentLISAspacecraftneedtobesynchronizedviathelaserlinks).Phasenoiseofthe 95

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Figure5-16. AX3065timingjittersetup.TheclockthattriggerstheADCtosampleisitselfsampled.Therelativetimingisadjustedsuchthattheexactzerocrossingissampled.Clocknoiseiscommon-modetobothsignallines,suchthatdeviationsfromzeroofthesampleddatacanbetracedtovariationsinADCtiming.Knowingtheclockslopeallowstoconvertthemeasuredinputuctuationstotimingjitter. sampleclock(CN)eclock(f)scalesdirectlyintothephasemeasurementvia eCN(f)=fbeat fseclock(f)(5{40)andworsenswithhigherbeatfrequenciesfbeat.ThisimpactstheaccuracyofabsolutephaseandfrequencyestimatesofthePM,howeverfordierentialmeasurementsthisinputclocknoiseiscommonmodetoallPMchannelsthatshareasamplingclock.Ontheotherhand,theexacttimeintervalbetweenaclockedgethattriggersthesamplinginanADCchannelandthemomentintimewhentheactualsampleistakenisalsosubjecttovariations.Thiscanforexamplebedrivenbytemperatureuctuationsorexistastechnicalnoise.Thecorrespondingtimingjitter(TJ)uctuationseTJ(f)aectPMestimatesinasimilarmannerasclocknoisevia eTJ(f)=fbeateTJ(f);(5{41) 96

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Figure5-17. AX3065timingjitter.Apowerlawwasttedtotheobservedsampletimingnoisebetween1mHzand100mHztoprovideareferencelevelinphasenoiseplotsandaddedtothegure.Between10mHzand100mHztheslopeofthetimingjitterisclosetop fwithalevelof2:4410)]TJ /F5 7.97 Tf 6.59 0 Td[(13s=p Hzat10mHz. andincontrasttothecommon-modeclocknoisethiswillbeuncorrelatednoisebetweenPMchannels.TheAX3065documentationsuggestsa1psrmsvariationinthesampletiming.ForMHzsignalstheresultingrmsphasenoiseexceedsmicro-cyclesandmaycorruptthephasedata.ExperiencewithUFLISshowedthattheTJnoiseisnotwhite,butgrowstowardslowfrequencies.WeadaptedamethodtocharacterizethetimingjitterindependentofPMmeasurementsdescribedin[ 107 ],wherethedrivingclockitselfissampledbyanADC.ThesetupforthismeasurementisillustratedinFigure 5-16 .Thephaseofthesampledclockinstanceisadjustedsuchthatthesamplingistriggeredonthezerocrossing.Variationsinthesampletimingwillresultinnon-zerovaluesbeingreturnedbytheADC,andbyexploringtheslope 97

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oftheclockedgewecanconvertthespectrumofdigitizedvaluestotimingjitter.Clocknoiseitselfiscommon-modetosampleclockandsampledclock,anddoesnotcoupleintothismeasurement.Figure 5-17 showsthespectralshapeofthetimingjitternoisethatwedeterminedwiththismethod.Atintermediatefrequenciesbetween1mHzand100mHz(lowerfrequencycomponentshavetoofewdatapointstobereliable,andhigherfrequenciesexperiencearoll-ofromthedown-samplinglter)wettedthepowerlaw e(f)2:3210)]TJ /F5 7.97 Tf 6.59 0 Td[(14s p Hz1Hz f0:54(5{42)tothecurve,whichweregardasasoftlimitforthephasemeterperformance.Wedeterminedthiscurvetobecharacteristicofthetimingjitter,butoverthecourseofthelongmeasure-mentsnecessarytoaccessitslowfrequencycomponents,weobservedinherentdriftsoftheaveragetimingawayfromthezero-crossing,whichmadeitdiculttoisolatethetimingjitter.AlthoughweplacedallsplittersandasmuchofthecablesaspossibleinatemperatureinsulatedStyrofoambox,itishardtocontrolthetemperatureuctuationinsidethedesktopcomputer,wheretheAX3065islocated.Givenalongenoughtimethetimingjitternoisecancertainlybecharacterizedasstationary,butwedoacknowledgethatsmallvariationsofthein-situjitterfromEq.( 5{42 )arepossible. 5.8LISABandFrequencyPerformanceTheperformanceandphasedelityoftheAX3065PMcanbeassessedwithcomparativemeasurementsbetweenphysicallydierentchannels.Thelevelofagreementbetweenchannelsthattrackthesamesignalprovidesinsighttowhatleveltheabsoluteestimatescanbetrusted.Therststepisadierentialmeasurementbetweentwochannels,buttheconceptcanbeextendedtoanentangledphasemeasurementacrossmultiplePMchannels. 5.8.1DierentialNoiseAdierentialmeasurementbetweentwochannelsistheeasiestwaytoquantifythePMperformance.AsinglesignalissplitintotwoPMchannels,andeachchannelproducesfrequencyestimatesindependently.Thedierenceoftheirrespectivereadingsshouldideally 98

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Figure5-18. AX3065dierentialPMperformance.WeconductedconsecutivemeasurementsusingthePMatallsupportedreadoutspeedsandplotthephasenoiseascompositecurves,spanningfrequenciesfrom0.1mHzto10kHz.TheresultshowsthatthePMislimitedbytimingjitterandADCquantizationnoise,withacrossoveratabout1Hz.Thedisplayednoisewasassessedinadierentialmeasurementofa1MHzfunctiongeneratorsignalwithphasenoiserelativetothephasemeterclockgivenbythebluecurve. yieldzero,andanydeviationsindicatethepresenceofdierentialnoisethatinuencestheabsoluteaccuracy.ThedierentphysicalPMchannelsareidenticalintheirdigitalarchitecture,thereforedigitizingasinglesignalandfeedingittoseparatePMchannelsproducesexactzerosinthedierentialPMoutput.AttheveryleasttheanalogsignalmustbesplitanddigitizedbyseparateADCs.ThedierentialphasenoiseperformanceoftheAX3065PMacrosstheLISAandLIGObandisshownasacompositephasenoisecurveinFigure 5-18 .Toobtainthisplot,a1MHzfunctiongeneratorsignalwassplitintotwooftheAX3065A/Dchannels.IndependentPMchannelsproceededtoextractthephaseinformationfromeachdigitizedsignal,andthedisplayeddierentialnoisewasassessedasthelinearspectraldensityoftheirdierencetime 99

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Figure5-19. AX3065entangledphasemeasurementsetup.Twosignalsoffrequenciesf1andf2aresplitandtrackedbydedicatedPMchannels.Amixercreatesasignalatthedierencefrequencythatestablishesaxedphaserelationbetweenthesignals.AthirdPMchannelmeasuresthedierencesignal,andthecombinationoffrequencyestimates^f1)]TJ /F1 11.955 Tf 14.5 3.15 Td[(^f1)]TJ /F5 7.97 Tf 6.58 0 Td[(2)]TJ /F1 11.955 Tf 14.51 3.15 Td[(^f2shouldcanceltozero. series.WeaddedtheQNandscaledTJlevelstotheplot,showingthatthePMisonlylimitedbythenoiseintheADCs.Theplotfurthershowstheinputphasenoiselevelofthe1MHzfunctiongeneratorsignalthatseededthedierentialmeasurement,illustratingthemassivesuppressionofinputnoiseduetothelargefeedbackgaininthePMchannels. 5.8.2EntangledPhasePerformanceAmorestringentperformancestatementforthePMisthenoiseoorinamulti-channelmeasurement.Becausethetwosignalsinatwo-waydierentialmeasurementarenominallythesameandthePMchannelshaveidenticalarchitecture,thereispotentialforcommon-mo-derejectionofnoisesources(forexampleduetoaliasingorfeedbackbehavior).Wethereforeextendedthedierentialmeasurementprinciple,spreadingitacrossthreechannelsinanentangledphasemeasurementofthreedistinctheterodynefrequencies.WiththesetupshowninFigure 5-19 weusedtwoRFsignalsof14MHzand19MHztoseeda5MHzsignalthatestablishesaxedphaserelationbetweenthem.ThethreesignalsaredistributedontodierentA/DchannelsandeachismeasuredwithadedicatedPMinstance.Theykeyisthatjustasinthetwo-waydierentialmeasurementthenominalfrequencyestimatesproducedbythePMchannelsshouldsubtracttozero,andthatdeviationsinthisnull-measurementhaveimplicationsfortheinstrumentalnoise. 100

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Figure5-20. AX3065entangledphaseperformance.Theobservednoiseinthethree-wayentangledphasemeasurementisclosetotheexpectedtimingjitterlimit.Whilethelevelsaresimilar,theslopesappeartobeslightlydierent,whichmaybeduethemeasurementnowspanningmorethanasingleADCchip,andnon-stationaryenvironmentalparameters. Figure 5-20 showstheresultofthisthree-wayentangledphasemeasurement.Asbefore,weincludedtheexpectedphasenoiseduetotimingjitterfromEq.( 5{42 )withascalingfactorofq (5MHz)2+(14MHz)2+(19MHz)2andtheLISAmicrocyclerequirement.Theobservednoisespectrumoorisinroughagreementwiththetimingjitterexpectations.ItisapparentthattheAX3065PMisunabletomeettheLISArequirementfortheentirerangeofbeatfrequenciesupto20MHz.However,theprimaryobjectiveforitsdevelopmentwastoincorporatelaserfeedbackcapabilities,whichweexploreinthenextchapter.Therearetechniquestoreducetheimpactoftimingjitterfromphasemeasurements.Becausethejitterisatimingoset,itseectiscoherentacrossoverlayedsignalsthataresampledby 101

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thesameADC.Asaresult,LISAPMhardwareprototypesaddultra-stablepilottonestoallinterferometricsignallines.ThepresenceofacoherentsignalinallPMchannelsenablesdedicatedadditionalphasemeterchannelstoresolvetheerrorintimingandcorrectthephaseestimates[ 94 ]. 5.9HighBandwidthPhasemeterThemeasurementprincipleoftheLISAPMprototypeprotsfromtheextremelyhighgainvaluesthatcanbeobtainedatlowfrequencies.BydesignitevaluatesthefrequencystabilityofasignalagainstaclockreferencefromthefeedbacktoitsNCO.ForthethermalnoiseexperimentthatrepresentsamajorpartofthisdissertationandisdiscussedindetailinChapter 7 weneedtodeterminetheabsolutefrequencynoiseoflaserbeatnotesintheLIGObandwithaninstrumentalnoiseoor ereq(f)<10)]TJ /F5 7.97 Tf 6.58 0 Td[(1s 1Hz fHz p Hz1Hz
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Figure5-21. Quantizationnoiseinjectionmethod.Indicatedaretheinjectednoisetermsandthetruncationwidths. 5.9.1TruncationOptimizationLISAprototypingeortsbyothergroupspointedoutthatthebitnumbersofsignalsinthePMcanbedrasticallyreducedatcertainkeypoints,becausetheresultingtruncationnoiseissuppressedviafeedbackatthereadoutpoint[ 93 ],aprocedurewhichiscallednoiseshaping.WeappliedthisknowledgetotheAX3065PMduetotheprospectthatreducingthesignalcomplexitymayfreegateresourcesandenableahigherinternalbandwidth.Basedonthez-domainPLLmodelthatwasintroducedinSection 5.6 ,weinjectedquantizationnoiseaccordingtoEq.( 5{37 )atkeypointsintheloop,andanalyzedthecorrespondingdegradationofthePMperformance.AswehaveseeninSection 5.7 ,theinitialA/DconversionintroducesarmnoiseoorforthePM,whichsetsatargetlevelforthetruncationofothersignals.Wedistinguishthein-looptruncation,whichcanbesubjecttotruncationerrorsuppressionduetofeedback,fromthereadouttruncationinthedown-samplinglters.UsingtheinjectionmodeldisplayedinFigure 5-21 ,wesetthetruncatedbitsindicatedbyP,Q,R,S,andTsuchthatthecorrespondingintroducederrorsleaveamarginofaboutanorderofmagnitudetoeeAD.ForthecalculationoftheOLGinEq.( 5{18 )itdoesnotmatterwhichpointintheloopservesasthestartingpoint,asallcomponentshavetobepassedexactlyonce.Consequently,eachtruncationnoisetermexperiencesthesameH(s)fromEq.( 5{20 ),andtondtheintroducedreadoutnoisewetraceittothePIR,passingthe 103

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Table5-2. InternalPLLsignaltruncation.InreferencetoFigure 5-21 thenalizedtruncationchoicesareshown. ComponentNoiseTermSymbolBitLength PhaseDetectoreePDQ16Low-passFiltereePDR16PIControllereePIS16PhaseAccumulatoreePAT13LookupTableeeLUP14CICReadouteeCICU48 gaincomponentsinreverseorder.Asanexample,thereadoutnoiseenPD(f)introducedbytruncatingtheoutputofthephasedetectorreads enPD(f)=H(s)eePD(f) GPDGLUGPA;(5{44)andothersareobtainedaccordingly.AsforthetruncationofthePIRvalueitself,itisofspecialimportancethatitisthetruncatedsignalthatgetspassedtothelow-passltersforreadout,astheun-truncatedsignalisnotsubjecttothenoisesuppression.BecausetheintegratorsinthearchitectureoftheCICdecimatorsthatdown-samplethePIRinformationprovideasimpleformoffeedback,noiseshapingcanbeappliedeventonotin-loopreadoutsignals[ 108 ].ThebandwidthofthePMwouldhowevernotbenetfromthis,suchthatwesimplytruncateatthelteroutput.InTable 5-2 theoptimizedchoicesforthebitlengthsarelisted,andFigure 5-22 showsthecorrespondingsimulatedPMreadoutnoiseimpact.CuttingbitsafterthePIcontrolleratthereadoutpointitselfexperiencesthemostdrasticsuppressionfromnoiseshaping.BecausethePMgeneratesfrequencyestimates,thereadouttruncationhasadierentspectralshapethanthein-looptruncations,whicharemostlyatinphase(exceptfortheheavilysuppressedPInoise). 5.9.2InternalBandwidthEnhancementTheCIClterusedinthephasedetectorwasthemainculpritforthelimitationofthefeedbackbandwidth.Notonlydoesthedecreasedsamplingrateafterthelteraddphase 104

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Figure5-22. Quantizationnoisemodel.ShownarethenoiselevelsduetointernalsignaltruncationinthePMandtheirrelativeinuenceontheinstrumentalnoisefortheLISAmissionrequirementandthethermalnoiseexperimentTHOR. Table5-3. PhasedetectorIIRltercoecients. FilterCoecientCoecientValueRequiredBinaryPrecision a1-1.515625S8 6a20.65625U5 5b10.625U3 3b20.75U2 2 losswitheachregister,thelteritselfcausedaseveredelay.Wethereforesubstituteditwithalow-passlterthatrunsatthefullsamplingrate.Thechoicefellontheminimalisticinniteimpulseresponse(IIR)lterarchitecturethatisshowninFigure 5-23 .Asopposedtoniteimpulseresponse(FIR)lters,IIRsinvolvesignalfeedbackthatcanoerbettersuppressionandlowercutofrequenciesforagivenltercomplexity(numberofmultiplicationsandadditions)atthedangerofbecomingunstable.InourcampaigntosimplifythePMlogic,wedesignedanIIRthatsuceswithlownumericalprecisionofthegainfactors,whilestillprovidingstablefeedback. 105

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Figure5-23. In-loopIIRlter.Theshownsecondorderlterisalinearcombinationusingfeed-backandfeed-forwardcoecients.Thez)]TJ /F5 7.97 Tf 6.59 0 Td[(1indicateregisteroperations,whichweminimizedtoincreasetheloopbandwidth. Figure5-24. In-loopPLLIIRandCICltertransferfunctions.TheIIRlterprovidesasofterroll-o,butthe45phaselossoccursataboutafactor4higher. WeimplementedasymmetricIIRlterofthetypeshowninFigure 5-23 .Table 5-3 containsthenalchoiceofcoecientsandtheirrequirednumericalprecision.Withinthelterblocksthemselvesthereisnotruncationofsignals.Figure 5-24 comparesthetransferfunctionofthenewIIRlterwiththatofthepreviousCIClter,showingthatthephaselosswasreducedbyaboutafactorof4. 106

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Figure5-25. RevisedAX3065PMloopgains.Theunitygainfrequencyhasbeenraisedto400kHz,wherethephaselossis120.Inthisimplementationthefeedbackbecomesunstableiftheinputsignalllslessthan10%oftheADC'sdynamicrangeduetothesecondintegratortakingover. WeplottherevisedloopgainfunctionsinFigure 5-25 ,whichshowsthatwereabletopushtheUGFofthePLLfeedbackto400kHz.Asaresult,wecutthesecondintegratorinthePIcontrollerfromthefeedback,becausetheorderofmagnitudeincreaseinbandwidthresultedinadditionalgainofabout20dBatlowerfrequencies.WiththeminimalIIRlow-passlterfrequenciesabove9MHzareattenuatedbymorethanafactorof100. 5.9.3LIGOBandInstrumentalNoiseAftertheimprovementstobandwidthandinternalsignalprecision,wetestedthephasemeteragainindierentialchannelmeasurementstoassertthattheinstrumentalnoisewasindeedunaected,asoutnoisemodel,suggested.Spanningarangeofbeatfrequenciesfrom750kHzto24MHzinincrementsoffactorsof2(motivatedbythetimingjitterscalingwiththebeatfrequency),weplotthedierentialphasenoiseinFigure 5-26 . 107

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Figure5-26. PMLIGObanddierentialperformance.Foravarietyoflog-spacedfrequencieswedeterminedtheinstrumentalnoiseoftheAX3065PMintherevisedarchitecturewithasplit-channeldierentialmeasurement.ThelimitationsarestilltheatADCquantizationnoiseandADCtimingjitter. Theonlylimitationintothe750kHz,1.5MHzand3MHzmeasurementsistheatdigitizationnoiseoorjustbelow210)]TJ /F5 7.97 Tf 6.59 0 Td[(8cycles=p HzinagreementwithEq.( 5{39 ),andforthefasterfrequenciesweseeanincreaseproportionaltofbeatinagreementwiththelevelsweexpectfromtheADCtimingjitter.Thedisplayednoiseliesfarbelowtheneededinstrumentsensitivitytoobservefrequencyuctuationsinducedbycoatingnoiseinopticalcavities. 108

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CHAPTER6DIGITALHETERODYNELASERFREQUENCYSTABILIZATIONThepreferredlasersinGWdetectionassociatedexperimentsarediode-pumpedsolidstatelasersthatuseNeodymium-dopedYttrium-Aluminum-Garnet(Nd:YAG)astheactivelasermediuminanon-planarringoscillator(NPRO)geometrycrystalcut[ 109 ].ThetypicalNd:YAGtransitionhasanarrowamplicationprolethatallowslasingat1064nm,andtheNPROissizedtoallowonlyasinglemodetoresonateinthecrystal.Theplantousecontinuous-wave(CW)Nd:YAGNPROsinLISAhaspersistedinthemissiondesignduetoitsuperiorintrinsicnoisecharacteristicsandenergyeciencycomparedtootherlasertypes.WediscussedinSection 3.3.3 theneedforanabsolutefrequencystabilityofthelaserlinksinLISAthatsatisesEq.( 3{10 ),howevertheintrinsicnoiseoftheNd:YAGNPROinthemeasurementbandistoohighbyupto6ordersofmagnitudeandrequiresadditionalstabilizationmeasures.Amongthescenariosforlaserfrequencypre-stabilizationschemesaretheuseofthe532nmIodinetransitionasalinereferenceforthefrequency-doubledlightofa1064nmlaser[ 110 ],andcompactstabilizationsetupshavebeendeveloped[ 111 ]aimedatoperationinspace.Whileatomictransitionsguaranteethelong-termstabilityofthereferenceforthelaserwavelength,stabilizationsetupsthatinsteaduseopticalcavitiesinvolvefarfewerauxiliarycomponentsanddonotrequirefrequency-doubling.Laserslockedtocavitiesmadefromultra-lowcoecientofthermalexpansion(CTE)materialscanrivalthestabilityofatomictransitionsintheLISAband[ 112 ].ThepopularPound-Drever-Hall(PDH)method[ 113 ]foropticalcavitylaserfrequencystabilizationhasbeenunderinvestigationforLISAsinceitsearlydays[ 114 ],butitinvolvesthephasemodulationofthelasercarriers,eitheraddingsidebandstoanalreadycrowdedspectrumoffrequencycomponents,orrequiringentirelyseparatemodulationcomponentsandcavity-space.CoinedtowardstheLISAmissionwedevelopedanddemonstratedHeterodyneStabili-zation(HS),anovellaserfrequencystabilizationmethodthatissimilartoPDH,butreplaces 109

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theneedforsidebandswiththeeldofaninLISAreadilyavailablesecondlaser.Providedthefeedbackcapacitytothelaserfrequency,adigitalHScontrollercanbeimplementedasanadditiontothePMprogramming,whichmakesHShaveaverysmallauxiliarycomponentandopticalbenchfootprint.AfullydigitallaserstabilizationschemeforLISAcanescapemanylow-frequencynoisesources[ 115 ],andspecicallytheintegrationofanHScontrollerintothePMhardwareoersexibledemodulationoptions.WefurtheradoptedthesimultaneousstabilizationoftwolaserstotwocavitiesinadualHSschemefortheCryoTHORexperimentthatisdiscussedinChapter 7 .Alargeamountofthepresentedworkinthischapterhasbeenpreviouslypublishedin[ 116 ].ThischapterexplainstheoperatingprincipleofHS,forwhichlightisrstshedontheinteractiondynamicsbetweenlasereldsandopticalcavities.Weelaborateonthestabilizationmethod,discusstheimplementationofthedigitalcontrollerintotheAX3065PM,andexploretheintricaciesofadualheterodynelock.Finally,wepresentthesetupandresultsofourdemonstrationexperiment,andbuildasciencecasefortheimplementationofHeterodyneStabilizationinLISA. 6.1OpticalCavitiesOpticalcavitiescanactasltersforboththetransverseproleandthewavelengthofanincidentcoherentlighteld.ThespatiallteringismostconvenientlyanalyzedwiththeGaussianbeamformalism,whichisdiscussedinmoredetailinAppendix B .Theabilitytoseparateandcleanlasermodeprolesisanimportantexperimentalaspectofopticalcavities,butitistheirwavelengthselectivitythatenablesthemtoserveasareferenceforlaserfrequencystabilization.Becausetheinterplayofphasescanbecrucialfortheinterferencephenomenainopticalcavities,wersttakeacloserlookatthereectionprocessinthecavitymirrors. 6.1.1MirrorReectionandTransmissionCoecientsHigh-qualityreectorsmostcommonlyachievetheirreectivitieswithmultiplestacksofpairedlayersofdierentdielectricmaterials.Byalternatingbetweenlowandhigh 110

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Figure6-1. Dielectricmirror.Ateachinterfacebetweenlowindex(nL)andhighindex(nH)materialstherespectiveincidenteldispartiallyreected.Thelayerthicknessesarechosentocauseaphasegainof=4forthedesignwavelengthofthecoating,whichcausesconstructiveinterferenceinreectionanddestructiveinterferencefornexttozeroeldamplitudeintransmission. indicesofrefraction(nLandnH,respectively),theinterferenceofthenumerouspartialreectionsfromeachlayerinterfacecanbetunedwiththelayerthicknessestobedestructiveeitherintransmissionorreection,yieldinghighlyreective(HR)oranti-reective(AR)coatingstructures.Depositionprocessessuchaschemicalvapordeposition(CVD)orionbeamsputtering(IBS)producetheamorphousthinlmlayersonthesubstrate.Whilethereectivitiesofcrystallinecoatings[ 117 ]andgratingstructures[ 118 ]arebeingpushedhigherandhigher,theamorphousdistributedBraggreectorshowninFigure 6-1 representsthemostrobustandbestunderstoodsolutionforhigh-quality,low-lossopticstodate.Thereectionofanelectromagneticwaveatasingleinterfacebetweenindicesofre-fractionnLandnHisdescribedbytheFresnelequations,whicharederivedfromMaxwell'sequationsforthebehavioroftheeldsattheboundary,andfornormalincidencewithnolossesstatethatthereectioncoecientrandthetransmissioncoecienttare r=nL)]TJ /F3 11.955 Tf 11.96 0 Td[(nH nL+nHandt=p 4nLnH nL+nH:(6{1)Thereectedfractionoftheincidenteldisthesamefromeithersideoftheinterface,butforreectionsotheopticallythicker(highern)medium,itexperiencesaphasereversal, 111

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whilethetransmittedeldcontinuesinphaseinbothcases.Thecorrespondingreectivityandtransmissivityforthesingleinterfacearegivenbythesquareoftheeldcoecients, RI=nL)]TJ /F3 11.955 Tf 11.96 0 Td[(nH nL+nH2andTI=p 4nLnH nL+nH2:(6{2)andforthepresentedloss-lesscaseitisnecessarilyRI+TI=1.Inaslabofthicknessditheeldhasgainedthepropagationphase2di=whenitreachestheback-interfaceandexperiencesthenextreection.ForHRcoatings,ordinarilythelayerthicknessesarechosentoeachcauseaquarterwavephasegainuponasinglepass,suchthatthereturninglightfromalayer'sbackinterfaceisinphase(2=4+=2=)withthereectedeldatitsfrontinterface,causingthemtointerfereconstructively(inturn,thetransmittedeldexperiencesonelessphasereversalandthereforefacesdestructiveinterference).ForabeamtravelingthroughvacuumthecompositeRNandTNofNquarterwavestackpairsis RN=nL2N)]TJ /F3 11.955 Tf 11.96 0 Td[(nH2N nL2N+nH2N2andTN=4(nL2NnH2N) nL2N+nH2N2:(6{3)whereweneglectedtheinuenceofthenalinterfacewiththesubstrate.ComparingEq.( 6{3 )toEq.( 6{2 ),weseethatbyincreasingthenumberoflayersoneaddstoanef-fectivemismatchofindicesofrefraction,furtherboostingthereectivity.Commonchoicesofdielectricsfor1064nmarefusedsilica(Silicon-Dioxide,SiO2)forthelow-indexmaterialwithnL=1:4703[ 119 ],andTantala(Di-Tantalum-Pentoxide,Ta2O5)forthehighindexwithnH=2:0962[ 120 ].Asanexample,astackofN=5pairsyieldsareectivityofR=94:4%.Thequarterwavedesignisnotanecessitytoachievehighreectivities,andmodelcalculationsandsimulationscanyieldthenumberofstacksandthicknessesfordesiredmirrorproperties,andmayevenbeusedtominimizeotherconstraintssuchasthermo-opticnoisecancellation[ 121 ].ThebiggerthedierencebetweennLandnHis,thelargerthefractionofthereectedeldateachinterfacebecomes,whichmeansthatfewerlayerscanbeusedtoachievethesameamountofeldextinction. 112

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Toarealmirror,whichwillfacecumulativelossesLduetoscatteringandabsorption,wecanattributeanempiricalreectivityRandtransmissivityT,whereconservationofenergynowrequiresthat R+T+L=1:(6{4)CavityopticswillusuallyhaveanARcoatingtosuppressthenitereectivityatthesubstratebacksideinterface,andwhilestate-of-the-artHRcoatingscanachieveT'saslowasseveralpartspermillion(ppm),ARcoatingswillusuallystillhavehundredsofppmofresidualreectivity.ToseparatethisspuriousreectionfromtheHRreectedlightitiscustomarytoadditionallycutthesubstratewithawedgedbackside.Becausetheempiricalreectionandtransmissioncoecientsjrjandjtjfortheeldsaregivenby jrj=p R=p 1)-221(T)-222(L;andjtj=p T;(6{5)theasymmetryinlossesLforreectionsoneithersideoftheHRcoatingresultsinslightlydierentreectioncoecients,whilethetransmissioncoecientsremainnominallythesame.AssumingARcoatinglossesof500ppm,wend jrHRj jrARj)]TJ /F1 11.955 Tf 11.95 0 Td[(1=p R p R)]TJ /F1 11.955 Tf 24.74 0 Td[(L)]TJ /F1 11.955 Tf 11.95 0 Td[(1=1 p 1)]TJ /F1 11.955 Tf 11.96 0 Td[(L=R)]TJ /F1 11.955 Tf 11.96 0 Td[(11 2L R110)]TJ /F5 7.97 Tf 6.59 0 Td[(4;(6{6)thereforethedierenceisentirelynegligibleforsinglereections.Insideanopticalcavityontheotherhandthisseeminglysmalldierencecanhaveahugeimpactandactuallydominatetheroundtriplosses.InthesecomplexreectorstheoriginalxedphaserelationsbetweentheeldsseeninEq.( 6{1 )softenduetolossesandadditionalpropagationphases(thicknessofthesubstrate).Theexactphaseshiftsarehowevercommonlynotofimportance,sincethepositioningofopticalcomponentsisnotknowntothecorrespondinglevelofprecisiontobeginwith.Theconservationofenergyconstrainsonlytherelativephasesofthecomplexreectionand 113

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Figure6-2. Cavitygeometryandeldreference.Afractiont1oftheincidenteldleaksintothecavitywhereitcanbuilduponresonance.Anotherfractiont2ofthecavityeldistransmitted,whilethereectedeldisasuperpositionofthereturningcavityeldandthedirectlyback-reectedlight. transmissioncoecientsofthesecomplexreectorstoobey 1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[('leftr+'rightr)]TJ /F3 11.955 Tf 11.96 0 Td[('t=(2N+1) 2;(6{7)wherelosseshaveagainbeenneglected[ 92 ].ThereectorfromEq.( 6{1 )isatrivialsolutiontothiscondition.Amoreconvenientdistributionofphasesforanalyzingcomplexopticalarrangementsis r=rleft=rright=jrj;andt=ijtj;(6{8)whichweadopthereforthediscussionofthetwo-mirroropticalcavitytransferfunctions.IfweincludelossesinthepicturetheremaybeviolationsofEq.( 6{7 ),butforthehighqualityreectorsusedforcavityopticstheyaregoingtobeextremelysmall. 6.1.2OpticalCavityTransferFunctionsWelookattheeldnomenclatureandgeometryoftheloss-lesscavityinFigure 6-2 ,whichconsistsoftwomirrorswithreectivitiesRiandtransmissivitiesTirespectively,thatareseparatedbythecavitylengthL.Anopticallystablearrangementofmirrorsrequiresatleastoneofthemirrorstobecurved,whichisdiscussedinmoredetailinAppendix C .DuetothedivergenceofGaussianbeamstwoatmirrorscouldnotreectaspatialmodebackintoitself.Theresultingpick-upofGuoyphasebreaksthehigherordermodedegeneracy,butisirrelevantforthederivationoftheresonancebehaviorofanyparticularmode. 114

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TheeldEi(t)isincidentonmirror1,andwithEq.( 6{8 ),weidentifythecavityeldEc(t),assessedrightbehindthefrontmirror,tobeasuperpositionofthefractionit1oftheincidentlightandafractionr1r2ofitself,delayedbytwicethecavitysingle-passtimeT.Wetransformtheaccordingrecursiveexpression Ec(t)=it1Ei(t)+r1r2Ec(t)]TJ /F1 11.955 Tf 11.95 0 Td[(2T);(6{9)totheLaplacedomain,whereatranslationintime)]TJ /F1 11.955 Tf 9.3 0 Td[(2Tresultsinafactore)]TJ /F5 7.97 Tf 6.58 0 Td[(2sT,andobtaintheclosedform eEc(s)=it1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(r1r2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2sTeEi(s)=C(s)eEi(s);(6{10)whichweusetodenethecavitytransferfunctionC(s)fortheeldinsidethecavity.Inasimilarfashion,thereectedeldEr(t)andthetransmittedeldEt(t)inFigure 6-2 areconstructedfromEi(t)andEc(t)as Er(t)=r1Ei(t)+it1r2Ec(t)]TJ /F1 11.955 Tf 11.95 0 Td[(2T);andEt(t)=it2Ec(t)]TJ /F3 11.955 Tf 11.96 0 Td[(T);(6{11)andwendtheirrespectivecavityeldtransferfunctionstobe R(s)=eEr(s) eEi(s)=r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2sT 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2sT(6{12)inreection,and T(s)=eEt(s) eEi(s)=)]TJ /F3 11.955 Tf 23.89 8.09 Td[(t1t2e)]TJ /F8 7.97 Tf 6.59 0 Td[(sT 1)]TJ /F3 11.955 Tf 11.96 0 Td[(r1r2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2sT(6{13)intransmission.Iftheincidenteldoscillatesataxedangularfrequency!==fsg=2withconstantamplitude(=
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Figure6-3. Cavityresonancebehavior.PlottedarethepowergainsandphasegainsofC(),R(),andT(),normalizedtoinputpowerforT1=T2=0:2. alsocalledthefreespectralrangeofthecavity.ThemagnitudeofC()takesitslargestvaluewhenever=FSRisanintegernumber,inwhichcasethelightisresonatinginthecavity.ThepreviouslyneglectedGuoyphaseGwouldappearasanadditionalround-tripphaseinEq.( 6{9 ),andthereforesimplyaddaconstantoset(m+n+1)GtothephaseexponentinEq.( 6{14 ),wheremandnarethehigherordermodeindicesoftheGaussianprole.Thisshiftstheabsolutefrequencylocationsofthecavity'sresonancesandbreaksthehigherordermodedegeneracy,butdoesnotaecttheirrespectiverelativespacing,whichisstillgivenbytheFSR.Thetwoothersteady-statetransferfunctionsare R()=r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r2e)]TJ /F8 7.97 Tf 6.58 0 Td[(i2=FSR 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2e)]TJ /F8 7.97 Tf 6.59 0 Td[(i2=FSR;andT()=)]TJ /F3 11.955 Tf 23.89 8.09 Td[(t1t2e)]TJ /F8 7.97 Tf 6.59 0 Td[(i=FSR 1)]TJ /F3 11.955 Tf 11.96 0 Td[(r1r2e)]TJ /F8 7.97 Tf 6.58 0 Td[(i2=FSR;(6{16) 116

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andnecessarilyshareC()'sresonancebehavior.WeplotallthreecavityfunctionsinFigure 6-3 forT1=T2=:2,whichareexaggeratedtransmissivitiestobetterillustratetheirinterplay.ThepeaksareLorentzianinshape,andtheirfullwidthathalfmaximum(FWHM)iscalledthelinewidthofthecavity.ThenesseFisdenedastheratioofFSRandFWHM,andcanbeapproximatedby F=FSR FWHMp r1r2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2(6{17)ifthemirrorreectivitiesaresucientlyhigh.Withoutdissipativelossesthesumofreectedandtransmittedpowerequalsthepowerintheincidenteld,whichalsofollowsfromtheconservationofenergy.Theintra-cavityelditselfstorespowerthatisafactort12=(1)]TJ /F3 11.955 Tf 10.56 0 Td[(r1r2)higherthantheinputpower,whichcanexceedvaluesof100,000forextremelyhighnessecavities. 6.2StabilizationMethodTheimprintthattheinteractionwiththecavityleavesonthereectedlight(andalsothetransmittedlight)canbeusedtoobtainafrequency-discriminatingerrorsignaltoeitherlockthefrequencyofalasertoaresonance(usuallytoenhancethestabilityofthelaserlight)orforcethecavitytoadjusttochangesinlaserfrequency(mostlywhenusingitasamode-cleanerforthetransmittedlight).Eitherway,thecriticalinformationforthisispresentinthepropagationphasethatthelaserlightexperiencesinthecavity,andsimilartothediscussioninSection 5.1 ,additional,non-degeneratelasereldcomponents(dierentinfrequencyand/orpolarization,suchthattheydonotresonateinthecavity)areneededtoextractit. 6.2.1ErrorSignalExtractionFarofromanyresonancewecanapproximateR()1,andthecavityactsasasimplemirror.Itisnoteworthythattheeectivereectivityofthiscompoundreectorisactuallyhigherthanthatofthefrontmirroralone,duetothedestructiveinterferenceoftheleakedeld,whichistheprincipleofKhalilicavities[ 122 ].Ifontheotherhandis 117

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Figure6-4. Basicheterodynesetup.AsuperpositionoftwolasersgeneratesabeatY(t)onareferencePDandreectsoanopticalcavity.Onlyonelaser(red)isresonant,andthecavitybeatX(t)containsitsinteractionphase,whichcanbeisolatedbydemodulationagainstthereferencebeat. sucientlyclosetoaresonancefrequency(withinit'slinewidth),=res+,thenR()canbeapproximatedby R()r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2)]TJ /F3 11.955 Tf 11.96 0 Td[(i2 FSRr2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(r21) (1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2)2:(6{18)UsingthedenitionofthenessefromEq.( 6{17 ),wewriteitsimaginarypartas =fR()g)]TJ /F1 11.955 Tf 33.02 8.09 Td[(2 F FWHM1 r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1;(6{19)whichisproportionaltothemismatch.R()isaeldtransferfunction,thereforeapowermeasurement(whichsquaresthereectedamplitude)withaPDcannotrecoverthisdiscriminant.Instead,aseparatecomponentisneededinthereectedeldthatdoesnotresonateinthecavity,astheircross-powerthenfeaturesonlyasingleinstanceofR().PDHusesphasemodulationsidebandstoextracttheinteractionsignal,whileHSusesasecondlaserandareferencebeat.Supposewehavepreparedasuperpositionoftwolaserswithangularfrequencies!1and!2,suchastheoneinEq.( 5{3 ),foruseinthesimplesetupshowninFigure 6-4 ,withmatchingspatialprolesandpropagationdirection.Becauseoftheirperfectoverlapweconsideronly 118

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theon-axiseldsforthefollowingcalculations,andalsoneglectthez-dependency,yielding Ei(t)=E1e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!1t+E2e)]TJ /F8 7.97 Tf 6.58 0 Td[(i!2t:(6{20)Sentontoaphotodetector,thiscombinationofeldsgeneratesavoltageY(t)thatispro-portionaltoitsintensity, Y(t)/jEi(t)j2=hE12+E22+2E1E2cos)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(!ti:(6{21)Ifitisreectedoofacavityinstead,eachofthetwoeldsexperiencesR()fromEq.( 6{13 )individuallybasedonitsfrequency.Weassumelaser1tobenear-resonant,andlaser2tobefarofromanyresonance.Itisthensafetoignorelaser2'sinteractionwiththecavityandtakeittobedirectlyback-reectedatthefrontmirror.Laser1'sreectedamplitudeandphasedependonthefrequencymismatch,andthereectedeldexperiencesR()fromEq.( 6{18 ).Thetotalreectedeldreads Er(t)=R()E1e)]TJ /F8 7.97 Tf 6.58 0 Td[(i!1t+E2e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!2t;(6{22)andifthissuperpositionisincidentonaphotodetector,itcreatesthesignalX(t)/jEr(t)j2=hjR()j2E12+E22+2E1E2
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Figure6-5. Heterodynestabilizationerrorsignal.Foragivennesse,shiftingthetransmissivelossesbetweenfrontandbackmirroraectstheslopeoftheerrorsignal,whichismaximalintheover-coupledlimit. wecanisolate=fR()gifweapplyaphaseshiftof=2toY(t)beforemultiplyingthesignals.Thedoubledfrequencytermsthatarecreatedinthemultiplicationprocessaredisposedofwithalow-passlter,leavingonly X(t)Y(t)LP=` 2ArefAcav=fR()g;(6{25)where`isagaintheconversionlossofthemixerthatwasintroducedinEq.( 5{13 ),andArefandAcavaretheamplitudesofthelaserbeats(oresonanceinthecaseofAcav).Weobtaintheerrorsignalforthestabilizationfeedback e(1)=)]TJ /F1 11.955 Tf 11.11 8.09 Td[(1 F1 r1)]TJ /F3 11.955 Tf 11.96 0 Td[(r1`ArefAcav FWHM1:(6{26)Thefactor1 F1 r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1isoforderunityandrangesfrom0(forr1!1)to4(forr1!0)forxedvaluesofr2,andinthecaseofr1=r2itbecomesexactly2.Drivinge(1)tozerowithfeedbacktothelaserfrequencylocksthelasertothecavityresonance. 120

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Figure6-6. Heterodynestabilizationdemonstrationsetup.Thesuperpositionoftwolasersisreectedotheopticalcavity,inwhichLaser1(red)isresonant.Thereferencebeatisusedtooset-phaselockLaser2toLaser1,andalsotodemodulatedthecavitybeatandextracttheerrorsignalfortheHSloop,whichlocksLaser1toafundamentalcavityresonance. 6.2.2DemonstrationPhaseTherstattemptatimplementingHSwasundertakenwiththebasicsetupthatisshowninFigure 6-6 .WithonlyasinglelaseratdisposalontheHSbench,weroutedasecondlaserfromtheUFLISbenchthrougha10msingle-modepolarizationmaintainingberandinterferedthebeamsintheHSsetup.TheNPRO'ssinglesupportedfrequencycanbetunedbyadjustingthecrystaltemperature,whichchangestheopticalpathlengthinthecrystal.ThisthermalactuationworksonlyontimescalesbelowtheHzrange,andforfasterfrequencycontrolitcanbesubjectedtomechanicalpressurewithapiezo-electricelementthatallowsforsmallsignalfrequencymodulationsupto100kHz.ThetwooutputportsofthebeamsplittersupplythetwoinstancesofthesuperpositioninEq.( 6{20 )forthestabilizationscheme.Thereectedcavityeldpassestwicethroughaquarter-waveplate(QWP)thatisrotatedby45suchthattheoriginallylinearlypolarizedlightresonatesinthecavitywithcircularpolarizationandturnsintotherespectiveorthogonallinearpolarizationonitswayback,whereitispickedobyapolarizingbeamsplitter. 121

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Figure6-7. HSdemonstrationcavity.Four1inchcylindricalZerodurpieceswerechemicallybondedtoserveasthecavityspacer.Themirrorswerebondedusingthesametechnique. Figure6-8. HSdemonstrationthermalshield.The1/8inchthickaluminumplatesoftherectangularboxarecoveredbyaluminizedMylartoreducetheiremissivity. ThedemonstrationcavityitselfwasassembledfromfourindividualpiecesofZerodur,anultra-lowcoecientofthermalexpansion(CTE)material[ 123 ]andispicturedinFigure 6-7 .Eachpiecehadalengthof1inch,andtheyweregluedtogetherusinghydroxidecatalysisbonding[ 124 ],achemicaladhesiontechniquethatusesanetchingsolutiontopartiallydissolvethesurfacestructureoftwosmoothlypolishedsurfaces.Givenenoughtime,thesolutionevaporatesandhydrogenbondsformfromsurfacetosurface,holdingthetwopartstightlytogether.HydroxideCatalysisbondingwasunderinvestigationasanassemblytechniqueforLISAtelescopespacerandopticalbenchassemblies[ 125 ].Themirrorswereadheredtothespacerusingthesamebondingtechnique.ThecavitywasplacedinasmallvacuumchamberinsidethethermalshieldthatispicturedinFigure 6-8 .RatherthanpolishingthesolidAluminumplatestoreducetheiremissivity,theywerecoveredwithaluminizedMylar.Apre-existinganalogservofromtheUFLISexperimentservedasthefeedbackcontroller.AsindicatedinFigure 6-6 ,thedemodulationphasewascoarselysetbyadjustingthereferencePD'spositionandcablelengths.Finecorrectionstoachievetheoptimumphase 122

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Figure6-9. Heterodynestabilizationdemonstrationphaseresult.ShownisthelinearspectraldensityofthedierentialfrequencynoisebetweentheosetphaselockedsecondaryUFLISlaserandanother,independentlyPDH-stabilizedUFLISreferencelaser.Theretroactivelyassessedimpactofspacertemperaturenoisehasbeenaddedtoshowthatatlowfrequenciesthesystemwasmostlikelylimitedbyresidualtemperatureuctuations. wereperformedbyadjustingtheosetfrequencyforthephaselockofthelasers.TheresultofthisrstdemonstrationphasecanbeseeninFigure 6-9 ,whichshowsthefrequencynoisebetweenthephaselockedUFLISlaser(whichcarriesthesamefrequencynoiseastheprimaryHSlaserduetothephaselock)andanindependentlyPDH-stabilizedUFLISreferencelaser,whosestabilityhasbeenshownpreviouslytomeettheLISArequirement[ 126 ].ItwasnotpossibletostabilizethetestsystemtoalevelthatsatisedtheLISArequirement,andunfortunatelybeforewecouldlaunchafullinvestigationintowherethelimitationswerecomingfrom,oneofthespacerpiecebondsfailedandthecavitybrokeintotwopieces.Weattemptedtoreassemblethespacer,butbecauseofdamagefrombreakingandresiduefromthepreviousbondthesurfacesweretoocompromisedtoformasolid 123

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enoughbondagain.Intheaftermathwewereabletotracethelowfrequencyperformanceretroactivelytotemperatureuctuationsinthesomewhatsmallvacuumtankandinsucientthermalshieldingbymeasuringthecharacteristictemperatureuctuationswithasensorthatweattachedtooneofthespacerpieces.TemperatureuctuationseT(f)impactthelengthnoiseofthecavityaseL(f)=LeT(f),whereL=4inchesisthelengthofthecavityand=210)]TJ /F1 11.955 Tf 7.09 -4.33 Td[(8K)]TJ /F5 7.97 Tf 6.59 0 Td[(1isthecoecientofthermalexpansionofZerodur.WeaddedthecorrespondingfrequencyuctuationsofalaserthatislockedtoacavitywhichissubjecttoeT(f)toFigure 6-9 .Thisassessmentofthecharacteristictemperaturenoisemakesitseemverylikelythatresidualtemperatureuctuationswerelimitingthelaserstabilityinthisrange.Thedesignofbetterthermalshieldingandaddingmorelayerswillsolvethisissue.Thehigherfrequencyportionofthelasernoisespectrumwedeterminedtobegainlimited.ThefrequencynoiseofafreerunningNd:YAGNPROgenerallyfollowsroughlya1=fslopeandisthereforesuppressedtoamoreorlessconstantlevelbyasingleintegratorgainstage.AUGFofabout100Hzwouldresultintheobservedlevelofthelasernoise. 6.2.3HeterodyneStabilizationControlTheoryInSection 6.2.1 wehavederivedthediscriminantoftheHSerrorsignal,butweneglectedthedynamicbehavioroftheheterodynelockandfrequencydependentnoisesuppression.Tomaximizethebandwidthoftheheterodynelock,whichresultsinmoregainandthereforebetterin-loopnoisesuppression,weallowforphaseuctuationsoftheinputlightandinvestigatetheirimpactonthereectionphase.LetthecarriereldofE1(t)offrequency1beonresonancewiththecavity, E1(t)=E1(t)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!1t=E1(t)e)]TJ /F5 7.97 Tf 6.59 0 Td[(2i1t;(6{27)andanydeviationfromtheresonanceconditionisexpressedintheslowlyvaryingfactor E1(t)=E1ei(t)E1+iE1(t):(6{28) 124

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Weonlyconsiderperturbationsofitsphase,astheyrelatedtofrequencyuctuationsoftheresonatinglaser.TheLaplacetransformofthissignaldisregardstheconstanteldandreads eE1(s)=iE1f(s):(6{29)Onresonancethereectedamplitudehasaminimumandtorstorderdoesnotrespondtophaseorfrequencyuctuations.Wesimilarlyexpresstheeldusingapurephasemodulationterm Er(t)=E1eir(t)Er+iErr(t)(6{30)ofthesamecarriereld,andletr(t)representthecavityinteractionphaseshift.ItsLaplacetransform eEr(s)=iErfr(s)(6{31)equalsthereectivecavitytransferfunctionR(s)timestheinputeldeE1(s),andwewrite eEr(s)=R(s)eE1(s)=iE1R(s)f(s)=iR(0)E1R(s) R(0)f(s):(6{32)SinceweknowfromEq.( 6{18 )thatEr=R(0)E1,weidentifythereectedphaseshift fr(s)=R(s) R(0)f(s);(6{33)Theuctuations(t)alsoappearinthereferencebeat,sinceitisgeneratedfromthesamesuperpositionofelds.Theresultingerrorsignale(t)/2ErE2hcos)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(!t)]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)isin)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(!t)]TJ /F3 11.955 Tf 11.95 0 Td[((t)==R(0)E1E2r(t))]TJ /F3 11.955 Tf 11.96 0 Td[((t) (6{34)isthereforeproportionaltothephasedierencebetweentheincomingandreectedresonanteld.WeobtainitsLaplacetransform ee(s)/R(0)p P1P2R(s) R(0)f(s))]TJ /F9 11.955 Tf 12.26 3.15 Td[(f(s)=p P1P2R(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R(0)f(s);(6{35) 125

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Figure6-10. Heterodynelockopticalgain.Theopticaltransferfunctionforlaserfrequencynoiserelativetothecavityisatatlowerfrequencies(errorsignalisproportionaltofrequencymismatch)andhasapoleathalfthecavitylinewidthFWHM=2.Abovethepoleitrollsoas1=f.Thethreeexamplesaretransferfunctionsfordierentdistributionsoftransmissivelossesinotherwisedissipation-freecavities. andndthattheoveralltransferfunctionfromphaseuctuationstoerrorsignalhastheDCtermR(0)removedfromR(s).Ifwehadchosentoperturbthephaseofthecavityeldbycratherthanthatoftheincidentlaser,resemblingalengthchangeofthecavity,Eq.( 6{33 )wouldhaveinsteadread fr(s)=R(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R(0) R(0)fc(s)(6{36)forthecombinedreturningeldduetothemissingdirectlyreectedcomponentofthecavityeld. 126

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InAppendix A weshowthatthephasenoisespectrumf(s)ofasignalisequivalenttoitsfrequencynoisespectrumf(s)bytherelation f(s)=f!(s) s=2f(s) s:(6{37)Becauseweuseee(s)forfeedbacktothelaserfrequency,wechangethepicturetoaresponsetoinputfrequencynoiseandobtaintheHSopticalgainfunction Gcav(s)=R(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R(0) s;(6{38)whichweplotinFigure 6-10 versusrealfrequenciesnormalizedtotheinputpower.Weincludedthreeexamplesfor(loss-less)over-coupled,impedance-matched,andunder-coupledcavities,whichcauseidenticalphaselossinacontrolloop,butoerdierentlevelsofopticalgain.Themorethe(transmissive)cavitylossesareconcentratedonthefrontmirror,themoredominantthecavityeldisinthereection,andthereforehigherthepurelyopticalgainisinthesystem.Thishasimportanttechnicalimplications,asitcanforexamplehelptoovercomelimitationsbysensingnoise. 6.3ControllerImplementationinPhasemeterHardwareInparallelwiththeinitialHSexperimentwebegantobuildanFPGA-baseddigitalfeedbackcontrollerusingtheAX3065card.HSdoesnotperserequirephasemeterlogic,asallsignaloperationstoobtainandmanipulatetheerrorsignalcanbeperformedbytheFPGAcompletelyindependentlyofallPMfunctionality.TherstcontrollerweprogrammedintotheAX3065wasinfactasimpleemulationoftheanalogPI-controllerweusedinthedemonstrationphase.WhilewewereabletoimplementthebasiccontrollerarchitectureintheAX3065,itsfunctionalitywasatrstseverelylimitedduetoaninsucientlydevelopedinterfacebetweenFPGAandtheDACunitandauser-unfriendlycommand-promptinterface.TheadvancesoftheAX3065PMprogrammingincludedagraphicalinterfacebasedonLabviewthatisabletocallallrelevantdrivercommands,andthesetupofabasiccommunicationsprotocoltoissuemorecomplexcommandstotheFPGAlogic,whichenabled 127

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Figure6-11. AX3065D/Aconversionprotocol.TheFPGAlogicdetectschangesinthesignallinesandtriggersthebueringofdataandchanneladdressbytheAD5547.TheD/Aoutputisthenupdatedfromthebuereddata.Thewriteprocesstakesatleast4clockcyclesperchannel.Ifbothchannelvalueschangeatthesametime,priorityifgiventochannel1. ustochangethecontrollerbehaviorwithouttheneedtorecompileanewPMcore.Inparallel,weimplementedtheD/AconversionprotocolthatisshowninFigure 6-11 .ItretiredtheAX3065'spre-programmedFirst-In-First-Out(FIFO)component,whichseverelylimitedthebandwidthofthedigitalcontrollerduringthedemonstrationphase.TheupdatedPMlogicdetectschangesinthesignallinesandimmediatelyproceedstoupdatetheDACoutputwiththefreshdata.TheAD5547hasaninternalbuerforreceivingdataanditsintendedchanneladdress.Thesebuersmustbewrittentorst,andthenasecondcommandwilltriggertheupdateoftheDACoutputofthecorrespondingchannelwiththebuereddata.Asaresult,ittakesatleast8clockcyclestoupdatetheoutputofbothD/Achannels(bueraddress,updateaddress,buerdata,andupdateoutputforeachchannel). 6.3.1DigitalLaserControllerTheimplementationofdigitalfeedbackcontrolsinLISA-typehardwarefeaturesthedemodulationofthebeatsignalsbytheFPGA.Incontrast,duringthedemonstrationphasethereectedcavityPDsignalwasdirectlymultipliedwiththereferencebeatbyananalogmixer.Itsconversiongain`scaledtheresultingerrorsignalby`ArefAcav=2.InthedigitaldemodulationschemethesignalsarenormalizedtotheADCreferencevoltageVrefbeforetheyaremultiplied,andsincethereisnodigitalconversionloss,theerrorsignalfromEq.( 6{26 ) 128

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becomes e(1)=1 2ArefAcav Vref2Gcav1;(6{39)whereGcavistheopticalcavitygainthatweintroducedinEq.( 6{38 ).TheoutputoftheD/Achannelsrangesfrom-5to+5Vin216steps,resultinginsmallestpossibleincrementsof152:6V.TypicaltuningcoecientsoftheNd:YAGlasersusedintheexperimentrangefrom2to5MHz/V,whichmeansthatlaserfrequencyactuationbytheoutputofthedigitalcontrolleroccursinstepsof300to750Hz,whichrepresentsasourceofquantizationnoiseinthedigitalcontrollerimplementation.UpdatingtheDACoutputindualchannelmodetakesatleast8clockcycles,thereforedrivingthecardwitha64MHzclockresultsinamaximumDACupdaterateof8MHz.TheprecisioncutobytheDACissubjecttoitsdynamicswitchingbehavior,whichiswhyEq.( 5{37 )canonlygivearoughestimateoftheintroducederror.Theresultingatvoltagenoisesitsatalevelofcirca20nV=p Hz,whichiscomparabletothe12nV=p HzinstrumentalvoltagenoisespecicationfortheDACoutput.Thecombinednoiseoorof23nVp Hz,scaledbytypicallasertuningcoecientsyieldsaatfrequencynoiselevelatabout100mHz=p Hzthatisintroducedintothesystematthelaserfrequencyactuationpoint.WhilethisisofnoconcernfortheLISArequirementfromEq.( 3{10 ),naivelyitcouldposeaproblemforthethermalnoiseexperiment(CompareEq.( 5{43 )).Anysuchactuationmisstepshoweverappearasdierencesbetweenlaserfrequencyandcavityresonanceintheerrorsignal,andarethereforesuppressedbytheloopgain.Highenoughgainandbandwidthwillensurethatthisnoise,whichisveryclosetotherequirementtobeginwith,issucientlysuppressedinadierentialcavitymeasurement.WedeterminedthethroughputdelayoftheAX3065forsignalsfromdigitizationintheADCstothesignalre-creationintheDACstobe1:1sinagreementwiththecard'sdocumentation.Thiscorrespondstoaphasedelayof45atafrequencyofabout114kHz,whichturnsintoahardupperbandwidthlimitforthedigitalcontroller.Anyadditionalinternaldelaysforcomputationswilllowerthisthreshold,whichiswhyweplacedthedown-samplinglterfortheDACfeedbehindtheservo,asdepictedinFigure 6-12 ,while 129

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Figure6-12. AX3065digitalPI-controller.Switchingvariablescontrolthegainbehavioranddierentgaincanbeappliedtothefeedbackpath.Thedigitallogicallowsforswiftchangesoftherelativegains,enablingeasygainandbandwidthmaximizationduringlockoperation.ArstorderCIClterdown-samplesthecontrolleroutputfortheDACs.TwosuchlaserfeedbackchannelsareimplementedintheAX3065. theservoitselfrunsatthefull64MHz.WekeptthePI-controllerarchitectureduetoitseaseofimplementationandlittleneedformultiplicativecomputations.Formoreversatilitythecontrollercanswitchbetweenprimaryproportionalfeedback(eitherforosetphase-lockingorlockingtocavitieswithlinewidthssmallerthantheloopbandwidth),andanin-lineintegrator(forcavitylockswithlinewidthslargerthantheloopbandwidth)toachievemaximumbandwidth.Asecondintegratorisintendedtoboostthefeedbackgainatlowfrequenciesineithercase,anditscross-overfrequencycanbeadjustedbywritingagainvaluetoaregisterintheFPGA.Usingtheivariablestoindicateuserchoicesforswitchingthefeedbackbehavior,theservogainisgiveninthezdomainby GS(z)=G1z)]TJ /F5 7.97 Tf 6.59 0 Td[(41)]TJ /F3 11.955 Tf 11.96 0 Td[(1+12)]TJ /F5 7.97 Tf 6.59 0 Td[(6z)]TJ /F5 7.97 Tf 6.59 0 Td[(1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(11+22)]TJ /F5 7.97 Tf 6.59 0 Td[(8G2z)]TJ /F5 7.97 Tf 6.58 0 Td[(1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F5 7.97 Tf 6.59 0 Td[(1;(6{40)whereG1scalesthetotalcontrollergain,andG2determinesonlythegainofthesecondaryintegratorrelativetotheprimaryfeedback,andthereforesettingthecross-overfrequency. 6.3.2FastPhasemeterIndirectDemodulationUsingadirectdemodulationscheme,inwhichthecavitysignalandthereferencebeataredigitizedandmultiplied,justasintheanalogcasethecorrectphasedierenceof=2hastobeachievedwithaphysicaldelaybetweenthesignals(cablelengths,PDplacement,phase 130

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Figure6-13. AX3065indirectdemodulationmethod.APMchannellockstothereferencebeatnote,andthecavitysignalisdemodulatedagainstasecondaryNCOinstancethatisgeneratedwiththephaseformtheintra-PMNCO,correctedbytohavethecorrectdemodulationphase. shifters,ordigitalregisters),whichcausesthecorrectphasesettingtobecomeafunctionofthebeatfrequency.However,thePMchannelshaveaninternalbandwidthmuchhigherthantheactuationrangeaimedforwiththelasercontrols,whichiswhywedesignedafeedbackmodelfortheheterodynecontrollerthatusestheindirectdemodulationschemeshowninFigure 6-13 .Ratherthanmultiplyingthedigitizedbeatnotesdirectly,aPMchannellocksitsNCOtothereferencebeatnote.WecanthenusethephasefeedbackinformationfromwithinitsPLLtocreateasecondaryinstanceoftheNCO{shiftedbyaphaseoset{thatwemultiplywiththecavitysignalinstead.DuetothehighergainandbandwidthofthePMchannelscomparedtothelaserfeedback,thesecondNCOservesasaninsitucopyofthereferencebeatwiththecorrectdemodulationphase.Thisenablestherecordingofthebeatswithidenticaltiming,whichcantheoreticallyeliminatethechangeofthedemodulationphasewiththebeatfrequency.Practically,therewillremainasmallamountofdependency,whichcanbemappedandadjustedforifneeded.ForthetheoryofoperationtheindirectdemodulationresultsintheappearanceoftheclosedloopPMtransferfunctionHPM(s)fromEq.( 5{20 )asacorrectiontotheR(0)term 131

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inGcav,andthecompleteHSopenloopgainbecomes GHS(s)=GPZT(s) 2Acav VrefR(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R(0)HPM(s) sGS(s);(6{41)whereGPZT(s)hasthevalueoftheDCtuningcoecientCPZTatlowfrequenciesbutaccountsforvariationsinthefrequencyresponseofthepiezo-electricelements.ThisequationillustrateswhyhighbandwidthsinthePMareneededfortheindirectphasedemodulation.Actuatingthelaserfrequencieswillalterthebeatfrequency,whichinturnaectsthereferencebeat.ThePMneedstofollowthesechangeswithmuchhighergainthanthelaseractuation,otherwiseitmightcausecomplexcommunicationbetweenmultiple,supposedlyindependentfeedbackloops(twolaseractuationchannelsandthePMreferencechannel).HigherPMgainguaranteesthatHPM(s)canbeapproximatedas1withinthelaseractuationbandwidthwithouttheneedtoconsiderback-reactionsfromthelaserfeedback. 6.3.3DualHeterodyneStabilizationInadirectdemodulationschemetheauxiliarylaserneedstobeoset-phaselockedtotheprimarystabilizedlaser,butwiththeindirectdemodulationitispossibletomatchthemacroscopicpathlengths,whichreducestheimpactofdierentialfrequencydriftsimmensely.Whileforaperfectmatchofopticalpathlengthsfrequencydriftsofthesecondlaserdonotaectthedemodulationphase,acompletelyfree-runningsecondlasercaneasilydrifttoofarfromtheprimarylaserforthePMtomaintainlock,whichiswhyitisstillnecessarytocontrolitsfrequency.Anoset-phaselocktotheprimarylaserwouldagainbethepreferredformoffrequencycontrol,butinlightofthethermalnoiseexperimentofChapter 7 thatrequiresthesimultaneousstabilizationoftwolaserstotwocavities,wedevelopedadualHSschemeinwhicheachlaserisstabilizedtoitsowncavity,whilesimultaneouslyusedastheauxiliarylaserinthedemodulationprocessoftherespectiveotherHSlock.WeimplementedthedualversionoftheoriginalHScontroller(Figure 6-13 )thatisshowninFigure 6-14 .Thesamereferencesignalisusedforthedemodulationofthe 132

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Figure6-14. HSdualservoarchitecture.BothlaserchannelsdemodulatetheirHSerrorsignalfromthesameNCOanddrivetheirrespectivelaser'sfrequency.Sincethephasenoiseoftheotherlasercouplesintotheirsensingsignal,residualmutualmodulationsofthelaserfrequenciesandchannelcross-talkarepossible. twoHSerrorsignals,andtheAX3065'sdualDACoutputenablesthesimultaneousdigitalcontrolofbothlaserstofollowtheirrespectivecavities.ThemeasurementofthedierentialfrequencynoiseofthetwostabilizedlaserscanbeproducedbythereferencePMchannel,butbecauseitalsofeedsitsphaseestimateintobothHSchannels,caremustbetakenthatthisnotstrictlyout-of-loopmeasurementdoesnotmaskcross-talkbetweenthechannelsasdierentialcavitynoise,andalsodoesnotsuppressdierentialcavitynoiseintheoutput.Ideally,thelaserdierencefrequencyL1)]TJ /F3 11.955 Tf 12.45 0 Td[(L2,asreadoutbythereferencePMchannel,wouldbeequaltothedierentialevolutionofthecavityresonancesC1)]TJ /F3 11.955 Tf 11.93 0 Td[(C2.BecausetheHSchannelsindependentlydrivethephaseL1oflaser1andL2oflaser2,butsensethedierentialphasenoisebetweenbothlasersrelativetothereferencePMestimate,residualphasemodulationsmaybepersistentinthelaserbeatthatdonotreectthecavitybehavior.WeanalyzethispossibilitywithFigure 6-14 ,inwhichweindicatetheinputandrespectiveoutputsignalsintheLaplacedomain.Inthe 133

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linearizedmodelthereferencechannelproducesthephasenoiseestimate eRef=HPM(s)heL1)]TJ /F9 11.955 Tf 13.07 3.16 Td[(eL2i;(6{42)whichisusedinthecavitychannelstoobtainthedemodulatederrorsignals eeHS1=1 2R1(0)R1(s) R1(0)eL1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(R1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R1(0) R1(0)eC1)]TJ /F9 11.955 Tf 13.07 3.15 Td[(eL2)]TJ /F3 11.955 Tf 11.96 0 Td[(HPM(s)heL1)]TJ /F9 11.955 Tf 13.07 3.15 Td[(eL2i(6{43)and eeHS2=1 2R2(0)eL1)]TJ /F3 11.955 Tf 13.32 8.09 Td[(R2(s) R2(0)eL2+R2(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R2(0) R2(0)eC2)]TJ /F3 11.955 Tf 11.96 0 Td[(HPM(s)heL1)]TJ /F9 11.955 Tf 13.08 3.15 Td[(eL2i(6{44)respectively,whereweusedEqs.( 6{32 ),( 6{33 ),and( 6{36 ),andassumedthebeatnotestolltheADCinputregisterswithAref=Acav=Vref.ThecontrollerappliesthechannelgainsGi,inwhichD/Aconversionandthefrequencytuningresponsehavebeenincluded,toEqs.( 6{43 )and( 6{44 ),andadjuststhefrequencyoflaseriaccordingto eLi=s 2eLi=GieeHSi:(6{45)Switchingbacktothefrequencynoisepicture,wecansolvefortheactuatedlaserfrequenciesandobtain eL1=2 sG1[R1(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R1(0)] 2 sG1[R1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R1(0)HPM(s)])]TJ /F1 11.955 Tf 11.96 0 Td[(1eC1+2 sG1R1(0)[1)]TJ /F3 11.955 Tf 11.96 0 Td[(HPM(s)] 2 sG1[R1(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R1(0)HPM(s)])]TJ /F1 11.955 Tf 11.96 0 Td[(1eL2;(6{46)and eL2=2 sG2[R2(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R2(0)] 2 sG2[R2(s))]TJ /F3 11.955 Tf 11.95 0 Td[(R2(0)HPM(s)])]TJ /F1 11.955 Tf 11.96 0 Td[(1eC2+2 sG2R2(0)[1)]TJ /F3 11.955 Tf 11.96 0 Td[(HPM(s)] 2 sG2[R2(s))]TJ /F3 11.955 Tf 11.96 0 Td[(R2(0)HPM(s)])]TJ /F1 11.955 Tf 11.96 0 Td[(1eL1:(6{47)ForHPM=1thersttermscausethenormalin-looptrackingbehaviorofCibyLi,andthesecondtermsvanish,whichagaindisplaystheneedforhighbandwidthandgaininthePMchannelsforthepresentedindirectdemodulation.AnotherobservationisthatforimpedancematchedcavitiestheimpactofHPMisdiminishedbecausetheyaresuppressedbytheRi(0)terms.Weillustratethispotentialcross-talkwiththetransferfunctionsin 134

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Figure6-15. Potentialchannelcrosstalkinindirectheterodynelocks.TheleftgraphshowsacomparisonbetweentheinteralPMgainandtypicalHSlockgains.Therightgraphshowsthetransferfunctionsforcavitynoisetolasernoise(black)andsecondarylasernoisetoprimarylasernoise(red). Figure 6-15 .TheleftgraphshowsacomparisonbetweenthePMchannelopenloopgainwiththetypicalcompleteloopgainoftheheterodynelock(includingcavity,digitaldemodulation,D/Aconversionandlasertuning).AcrossthebandthePMhasatleastafactorof100moregainthantheheterodynelock.Therightgraphshowstheclosedlooptransferfunctionfromcavitynoisetolasernoise(black)anditsdeviationfrom1(blue).Theredcurveisthecouplingofnoiseinlaser2tolaser1foracavitywithT1=600ppmandT2=400ppm.Incavitiesthatarenotdominatedbydissipativelosses,andespeciallyinimpedancematchedcavities,thelasernoisecouplingissucientlysuppressedtobeofnoconcern. 6.4DigitalHeterodyneStabilizationforLISATheworkweshowedinSection 6.2.2 wasindicativeofthepotentialofHS.Withagoodunderstandingofthetheoryofoperationandadigitalcontrollerwhosetechnicallimitationsareknown,itwasclearthatthebiggesteorttomeetingtheLISAfrequencyrequirementwastothermallystabilizeandisolatethecavityenvironment.Sincethemoststraightforwardwaytodeterminethefrequencystabilityofalaseristobeatitwithareferencelaserwith 135

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similarorbetterperformance,wewantedthenextphaseoftheHSdemonstrationexperimenttofeaturetwocavitiestowhichwestabilizetwolasersandmeasuretheirdierentialfrequencyuctuationswiththephasemeter.ThecondencewegainedfromtheinsightintothedualdigitalstabilizationschemeassuredthatwecanusethedierentialstabilityoftwonominallyidenticalHSsystemsastheindicatorforitsabsoluteperformance.AtthesametimewehadmadethedecisiontolaunchthethermalnoiseexperimentsTHORandCryoTHOR,whichwediscussindetailinthenextchapter,andduetovastsynergiesbetweentheirexperimentalimplementations,wedesignedtheLISAHStestbedasaprecursorexperimenttotheroomtemperatureTHOR. 6.4.1CavityParametersTheplantomeasurethedierentialfrequencynoisebetweenthetwoHSsystemswiththeLISAphasemeterplacedsomeconstraintsonthelengthsofthecavities.Sinceitisclockedat64MHzitcanonlymeasurelaserbeatsuptoabout30MHz.Thetheoreticalmaximumof32MHzisolimitsforthephasemeterbecausethemixingoftwosignalsatthatfrequencywouldproduceasumfrequencyof64MHz,whichaliasestoDCandthereforeintotheactuationbandwidth,whichneedstobeavoided.ThetwocavitiesfortheLISAHSexperimentthereforerequireanintentionalmacroscopicmismatchinFSR,suchthatalternatebeatfrequenciescanbefoundbymovingthelasersbetweenresonances.LongercavitieshavesmallerFSRs,whichmakesiteasiertondreasonablebeatsbetweenresonances,andbecausethecoecientofthermalexpansionindicatesthefractionallengthchangewithtemperaturevia T=L(T) L=(T) ;(6{48)theoveralllengthofthecavityspacerdoesnotaectthefrequencyresponsetotemperatureuctuations(itdoeshoweverhaveimplicationsforthermalmass,contactpointsandexposuretothermalradiation).THORwilleventuallyincludetestcavitieswithFSRsupto10GHzwithonlyafewresonancesavailabletotheNPROswithintheirtuningrange,andlonger 136

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Table6-1. Ultra-low-expansionspacermaterialproperties. PropertyULEZerodurCCZHS CTE110)]TJ /F5 7.97 Tf 6.58 0 Td[(8K)]TJ /F5 7.97 Tf 6.59 0 Td[(1210)]TJ /F5 7.97 Tf 6.58 0 Td[(8K)]TJ /F5 7.97 Tf 6.59 0 Td[(1210)]TJ /F5 7.97 Tf 6.58 0 Td[(8K)]TJ /F5 7.97 Tf 6.59 0 Td[(1Density2:21103kg=m32:53103kg=m32:55103kg=m3Young'sModulus67:6109N=m290:3109N=m290109N=m2PoissonRatio0.170.250.25MechanicalLoss210)]TJ /F5 7.97 Tf 6.58 0 Td[(5310)]TJ /F5 7.97 Tf 6.58 0 Td[(4210)]TJ /F5 7.97 Tf 6.58 0 Td[(5 cavitieshelptoreducethedierencefrequencyoftheclosestresonances.Consideringthatexcessivecavitylengthhoweverexacerbatesaccelerationnoise,wedeterminedacavitylengthofabout25cmtobeagoodcompromisewiththeavailabilityofresonances.Morepragmatically,thematerialcostandbothavailablevacuumandbreadboardspacewouldnothaveallowedformuchlongercavities.TocreatethemismatchinFSRswevariedthisnominallengthona10%levelandchose24cmand27cmforthelengthsofthespacers,withFSRsof625MHzand556MHz,respectively.Themismatchof69MHzisalittleoutsidetheoriginalcomfortzoneof60MHz,whichwetradedforslightlybettercoverageoftheresonancespectrumofthetestcavities.TherstdemonstrationexperimentwasusingaZerodurcavityspacer,andwasstillshowingsignsoftemperature-drivencavitylengthmodulationsinexcessoftheLISAre-quirement.Itisthereforeimperativethatthenewcavitiesaremadefromamaterialwithsimilarthermalproperties.Ultra-LowExpansionGlass(ULE)[ 127 ],whichismanufacturedbyCorning,andSchott'sZeroduraretheclassicalchoicesforlowCTEcavityspacers,buttobroadenthescopeofcandidatematerialsforlow-expansion-reliantinterferometrywedecidedtoobtainClearCeramZ(CCZ)spacersoftheCCZHSvarietyfromOHARA.WhileZerodurandULEbothhaveashallowzero-crossingsoftheirCTEsatroomtemperature,CCZHSfeaturesalocalmaximumbetweentwozero-crossings[ 128 ].InTable 6-1 wecompiledrelevantpropertiesofthedierentlowexpansionmaterials.Thespacershaveadiameterof7.5cm,a1cmclearanceholealongtheiraxisforthebeampropagation,anda5mmventingholeintheirgeometriccenter,sincetheendsoftheaxialclearancewillbetightlycoveredbymirrors. 137

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Figure6-16. Opticalgainversusmirrortransmissivities.Theleftplotshowsthatforaxedbackmirrorreectivity(here200ppm)weoptainthehighestopticalgainifthefrontmirrormatchesitstransmissivities.ForT1>T2thecavitybecomesover-coupled,butthenesseisreduced,andforT1
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Table6-2. Referencecavityassemblyparameters. PropertyCavity1(long,LC)Cavity2(short,RC) Length27cm24cmFrontMirrorROC11BackMirrorROC1m1mg-factor0.730.76Waist388m380mFSR555.1MHz624.5MHzFWHM27.4kHz30.8kHz thatofanimpedance-matchedcavityofamuchhighernesse.Therightplotillustratesthatinordertomaximizetheopticalgainforthegivenxednesse,transmissivecavitylossesneedtobeconcentratedonthefrontmirror.WeorderedfromCoastlineOpticstwoseparateHRmirrorcoatingbatcheswithtargettransmissivitiesof15ppm(intendedforthebackmirrors)and300ppm(forthefrontmirrors),whichwelledwithmirrorsofvariousradiiofcurvature.AfterthecompletedcoatingrunsCoastlinedeterminedfromwitnesssamplesactualtransmissivitiesof280ppmand10ppm,respectively,withexpecteddissipativelossesineachmirrorontheorderof10ppm.Wedesignedthedemonstrationcavitiesashalf-symmetricwithatfrontmirrorsand1mradiusofcurvature(ROC)backmirrors.Toattachthemirrorstothespacerstheyweresuppliedwithasuper-polishedannulus,andtheendfacesoftheCCZHSspacersweresimilarlypolishedtoasurfacequalityof10-5scratch-digandaroughnessof=10atthecalibrationwavelengthof632.8nm.Thisenabledustoopticallycontactthemirrorstothespacers,atechniquethatworkswithoutthehelpofanyadhesivewhatsoever.ThesheeratnessofthejoinedsurfacesissucientforthesurfacemoleculesoneithersidetogetcloseenoughthatattractivedipolevanderWaalsforcesstartreachingacrosstheinterface,pullingtheobjectstogether[ 129 ].TheparametersofthenalassembledcavitiesarepresentedinTable 6-2 .BothTHORreferencecavitieswereassembledwithplaneT1frontmirrorsand1mROCT2backmirrors.Weperformedtheopticalcontactingourselvesinourin-houseclass100clean-room.WhilethecurvedbackmirrorseasilyattachedtotheCCZHSspacers(whichwasaconcernuptothatpoint),ittookmultipleattemptswithintermittentcleaningtocontact 139

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Figure6-17. DualHSsetupforLISAbandmeasurements.Twolasersareinjectedintoanopticalberthatcarriestheirsuperpositiontothein-vacuumtestbench.Thebeamissplit,reectedothecavities,andsentoutthroughopticalwindows. theatsubstrates.Thecoatingmaskthatwasusedtocontainthereectivecoatingtothecarvedoutportionofthecurvedmirrorsandnotdepositontotheannuliwasnoteectiveincaseoftheatsubstrates.Instead,itreducedthecontactareabetweensubstrateandspacertotheouterregionofthecoatedarea,whichmadecontactingmoredicult,andhadtheunwantedside-eectthatthecontactwasactuallyformedbetweenthecoatingandthespacer. 6.4.2HeterodyneStabilizationSetupforLISATheopticalcavitiesfortheUFLIStestbenchwerelocatedwithin5layersofgold-platedpolishedstainlesssteelthermalshields,achievingexcellentthermalstability[ 130 ].Incontrast,thedemonstrationexperimentusedamuchsimplerthermalshield,consistingofAluminumplateswithaluminizedMylarsheetsattachedtoboththeinsideandoutside.Weadoptedasimilarlysimple(andcheap)thermalshieldsolutionforthesecondgenerationHSexperimentwithacubicalskeletonstructureofAluminumstrutswithaluminizedMylarspannedacrossthecubefaces.Thecavitieswereplacedonabreadboardthatwasattachedtoasizableframestructureforeventuallysuspendingthebreadboard,allofwhichwasinsidethethreeshieldlayerswithnodirectlinktotheoutside,LowthermalconductivityMaycorblockswereusedasspacersbetweentheshieldlayers. 140

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AIn-vacuumbreadboardwithopticalcavities BExternalbreadboardsFigure6-18. HSdemonstrationexperiment.ThetwoCCZHScavitieswereplacedinorthogonalorientationonthebreadboard,whichislocatedwithinthreelayersofthermalshielding.Theexternalassemblypreparedthelasersuperpositionintoasingleopticalberthatcarriedthelasersinsidethevacuum,andthereectedlightleftthroughanopticalwindowintheplaneoftheupperbreadboard. AsketchofthesecondphaseHSdemonstrationsetupisshowninFigure 6-17 .TwodedicatedHSlasersareinterferedonanexternalinjectionbenchandcoupledintothesamepolarizationmaintainingber.Theberisintegratedintoavacuumfeed-throughangeandterminatesonthetestbenchinsidethevacuumtank.Itsoriginalpurposewastodecoupledierentialmotionofbreadboardandinjectionbenchfromthemeasurement,butadditionallyitactsasamode-cleanerforbothlasers,andforcesthemintothesamespatialmode,whichmaximizesthecontrastonallPDs.Onthetestbenchthebeamissplittwiceintothreeseparateinstances,twoofwhichreectothetwocavitiesandthenjointhereferencebeatonthewayoutthroughanopticalwindowtothedetectionbench.SinceaPDisnotmode-selective,thedierentialmotionisnotaconcernanymoreatthispoint.TheAX3065dualfeedbackcontrollerdigitizedthebeatnotesandactuatedthelaserfrequencies,closingtheloopwiththeinjectionbench.Becausethelasersneedtostayonresonancefor 141

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anextendedperiodoftime,duringwhichthelaser'snativecenterfrequencymaywandertoofaroforthepiezo-electricactuator(PEA)tofollow,asecondaryservousedthefastmodulationfeedbackasitserrorsignalandmovedthelasercrystal'stemperaturetokeepthePEAwithinitsdynamicrange.Thetankinsidewiththecavitiesonthein-vacuumbreadboardwithinthethermalshieldsisillustratedinFigure 6-18 ,alongwithapictureoftheexternalinjectionbench.WhiletheheatcapacitanceofthethermalshieldsthemselveswasfarlowerthanthatoftheUFLISshields(duetolackingsubstance),whichaectstheirlow-passlteringability,thethermalmassthatwasexposedtotheresidualtemperatureuctuationsinsidetheshieldswasmuchhigher.Thechangesinheatowthereforehadtocatertotheentirebreadboard,andaonlyareducedamounthitthecavities,positivelyaectingtheirthermalstability. 6.4.3StabilizationResultsTheimplementationofHSforthecavitystabilizationinthissecondphaseworkedasexpected.WewereabletoperformsingleaswellasdualHSlocks,andconrmedthatafreerunningsecondarylaserdoesnotthrowtherstlaseroutofitslock.Moreimportantly,however,wefoundthatHSseemstobeextremelyreliable,andduringtheprimeoftheexperimenttherewereperiodswherethelasersstayedlockedwithoutinterruptionsforweeksatatimewithaperfectdutycycle.Alossoflockwasgenerallyassociatedwithexternaldisturbances,suchaspersonalinterferenceorautonomouscomputerupdates.ThefrequencystabilityofthebeatnoteofthedualHSlockedlasersweachievedwiththisnalHStestsetupisdisplayedinFigure 6-19 .ThedierentialfrequencynoiselevelisbelowtherequirementsbyaboutanorderofmagnitudethroughouttheLISAband.Thenoisecurvedisplaystwopeaksnear100mHz,whichmarkthecross-overfrompiezo-electrictotemperatureactuation.ThetemperaturecontrollersamplesthePEAsignalwitha16bitADCandrunsinternallywithdoubleprecision.TheDACoutputalsohas16bits,andsincethereisnocontinuousfrequencytuningpossiblewithdiscretevoltagesteps,thePEAsignaltothelaserwillusuallybeslightlyosetfromzero.Whenthetemperature 142

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Figure6-19. DualHSfrequencynoiseintheLISAband.ShownarethedierentialnoisebetweenthetwodualHSlasers,andalsothenoisewithrespecttoanindependentlystabilizedUFLISlaser.Atlowfrequenciesweseecommon-modesuppressionofthein-tanktemperatureuctuationsoftheHSsystem. controllerhasaccumulatedthisosetforlongenough,itmovesthelaserfrequencybytheconstantstepsizegivenbytheresolutionofthetemperatureDAC(ontheorderof1MHz),whichwillovershootthenecessarycorrectionofthePEAsignal.Thetemperaturefeedbackvoltageisthensittingrightinbetweentwotruncatednumbers,andwillrepeatedlyswitchbackandforth,causinganoscillationinthePEAatthecharacteristicfrequencyseenintheplot.Oneneedstokeepinmindthatthetwocomparedcavitieswerelocatedinthesamevacuumtank,whichcausescommon-modesuppressionoftemperaturedrivenfrequencynoise 143

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duetotheirsimilardimensions.LISAhoweverneedstoguaranteetheabsolutestabilityofastabilizedlaser.ForanindependentvericationweadditionallyusedopticalberstoguideafractionoftheHSstabilizedlasertoaseparateopticaltable,whereitwasinterferedwiththeUFLISreferencelaser.ThedierentialnoisebetweenanHSlaserandtheUFLISlaserwasaddedtoFigure 6-19 ,whichshowsthatitmeetstherequirement,butisworseatlowfrequenciesthanthedualHSmeasurementindicated.Weexpectacommon-modecancellationfactorofthetemperaturenoisethatisontheorderof 1)]TJ /F1 11.955 Tf 13.15 8.09 Td[(0:24cm 0:27cm=0:11;(6{49)whichmatchestheobservedspacingbetweenthetwonoisecurvesverywell.Below100mHzoursystemisthereforelimitedbytemperatureuctuations,butthestabilizationmethodworksasintended.ThekeypointaboutthiscomparisonisthattheUFLIScavityislocatedinanentirelydierenttankonadierentopticaltable,andthereforethetemperatureenvironmentsforthecavitycansafelybeconsideredascompletelyuncorrelated.ThismeansthattheabsolutestabilityoftheHSsystemissatisfactorywithregardstoLISA. 144

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CHAPTER7CRYOGENICTHERMALNOISETESTBEDThemechanicallossintheHRcoatingsofthetestmasseshasbeenidentiedtolimitthesensitivityoftheadvancedGWdetectorsofthesecondgeneration.InpreviouscongurationstheexistingdetectorsdidnotreachthecoatingBrowniannoise(CBN)limitduetolowerinputlightpowers,andbecausetheirseismicisolationwaslesseective.ThefactthatthedetectorshadtoprogressintotheadvancederatoconsiderCBNisindicativeofhowelusiveanoiseitcanbe.Althoughthecouplingmechanism,whichoriginatesintheFDT,isrelativelywellunderstood,relevantdirectmeasurementsofthebroadbandmechanicallosswithintheLIGOfrequencybandthatwouldscaledirectlyintotheinstrumentsensitivityarerare.ModelsfortheCBNinLIGOuseextrapolationsofthelossanglesassessedinring-downmeasurementsthatexcitespecicmechanicalresonancesinsamplemirrorsandmeasurethetimeconstantofthedecayingamplitude[ 131 ].TheexcitedeigenmodesarecommonlyabovetheLIGObandandalsoonlyprovidealossestimateattheresonancefrequenciesoftheinvestigatedsample.AdierentapproachtoassesstheimpactofCBNistomeasurethebroadbandthermalnoisedirectlyinanopticalexperimentthatcanprobetheaccordingcavitylengthnoise.Whilethisisonlyanindirectmeasurementofthemechanicalloss,itcandirectlydeterminetheopticalmanifestationofCBNthatwillbeseenbythefull-scaleGWdetectors.ThequesttomeasureCBNincryogenicallycooledsamplesisdrivenbytheprospectofusingcryogenicstolowerthethermalnoiseinfutureGWdetectors.Becauselarge-scalecryogenicfacilitiesinvolvemassivenancialcommitments,itisofspecialimportancetoperformfeasibilitystudieswellinadvanceofsuchupgrades.Theimplementationofacryogeniccoolingcapacityhoweveraddsimmenselytothecomplexityoftheexperiment,asitdirectlyclasheswitheortstoseismicallyisolatethetestbenchfromoutsideinuences.ThischapterdescribestheconstructionoftheUFCryogenicThermalNoiseOpticalResonator(CryoTHOR)experiment,whichwasdevelopedasacryogenicspin-oofthe 145

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roomtemperatureTHOR.WeexplainthemeasurementprincipleoftheTHORapproachandaddresstheengineeringchallengesoftheseismicisolationandthecryogenicsubsystem.TheassemblyoftheopticalcavitiesforuseinCryoTHORisdiscussed,andthecompleteopticalsetupispresented.Aftercharacterizingthecryogenicsubsystemandassessingthenoiselimitofthereferencesystem,weobtainedpreliminarynoisemeasurementsfromimprovisedtestcavitiesbothatroomandcryogenictemperatures. 7.1CryoTHOROverviewCavitylengthnoiseinducesvariationsoftheresonancefrequencies,suchthatwecansensethelengthuctuationsofacavitybymonitoringoneofitsresonances.Bylockingalasertoit,weextractcavitylengthchangesfromthefrequencyuctuationsoftheopticalbeatwithasecondlaser.Thisotherlaserneedstoeitherhavebetternoisecharacteristicsandserveasareferencesystem,orbestabilizedtoanominallyidenticalcavity,inwhichcasetheobserveduncorrelatednoiseliesafactorofp 2abovetheindividualcavitynoiselevels.Anequal-cavityexperimenthasbeencompletedrecentlythatconrmedtheexpectedlevelofthermalnoiseinTantala-Silicacoatings[ 132 ].WiththethermalnoisetestbedattheUniversityofFloridawedecidedtotakethereferencesystemrouteinstead.Whilerequiringmoreengineeringworkupfront,itmakesnoisebudgetingeasierinthelongrun,asthereferencesystemcanbeindependentlycharacterized. 7.1.1MeasurementPrincipleWeshrinktheLIGOarmcavitiestoatable-topsizedopticalexperimenttoboosttheimpactofCBN.LookingatEqs.( 3{9 )and( 4{24 ),wecanscalethelengthnoiseforanorderofmagnitudeestimateofthestabilitylevelsthatneedtobeachieved.FromFigure 3-4 weseethatthereferencesystemshouldbeabletoseetheequivalentfrequencynoisefromstrainuctuationsontheorderof2:410)]TJ /F5 7.97 Tf 6.58 0 Td[(24=p Hzat100Hz.Ifwecompressthe4kmcavitiestoalengthof2cm,andnarrowthebeamwaistsfrom5cmto100m,weobtainforthe 146

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Figure7-1. THORprinciplesetup.Twolasersaresuperimposedandsenttothetestbenchviaanopticalber.Onthetestbenchthesuperpositionissplitandreectedothecavities.ThereectedlightleavesthevacuumtankandproducesbeatnotesthatareusedforthelaserfrequencystabilizationusingHS. frequencynoiseattheCBNlevelinahypotheticaltestcavityanaiveestimateof e(f)3108m=s 1064nm5cm 100m4km 2cm2:410)]TJ /F5 7.97 Tf 6.59 0 Td[(24 p Hz68mHz p Hz:(7{1)ArigorousreferencesystemintheTHORexperimentsneedstooerbetterstabilitythanthisrequirement.AchievingthefrequencynoiselevelinEq.( 7{1 )ischallenging,butwithinthereachofbench-topopticalexperiments.AswasdiscussedattheendofChapter 6 ,therststepstowardstheTHORexperimentwereundertakenwiththeHSdemonstrationsetup,intowhichsomeofTHOR'sexperimentalaspectshadalreadybeenintegrated.Firstandforemost,THORneededtobeabletoresolvetheexpectedfrequencyuctuationsfromCBNinalaserbeatnote.SincetheinstrumentalnoiseofthefastLISAPMwasfoundtoliefarbelowtheexpectedlevels,weintegrateditasthemainreadoutmethodinCryoTHOR.Asforthecavitystabilizationmethod,wedecidedtostickwithHSduetothesuccessithadintheLISAdemonstrationphase,butalso 147

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becauseitenabledustomeasurethelaserbeatsinreectionratherthanintransmissionofthecavity,suchthatwecouldimplementover-coupledcavitieswithhigheropticalgains.WeshowthebasicprinciplesetupofoutthermalnoisetestbedinFigure 7-1 .Thetwolasersareinterferedonanexternalinjectionbenchandcoupledintoapolarization-maintai-ningopticalber.Arrivingonthein-vacuumtestbench,thesuperpositionissplitseveraltimesandreectedothecavities.Backoutsidethetank,thelaserbeatsarerecordedonthediagnosticbenchandfedtotheAX3065,whichdemodulatestheerrorsignalsfortheHSlocks.ThiswaywecompareanoisytestcavitythatisdominatedbyCBNtoaquietreferencecavity.TheideabehindthisapproachisthatthealldierentialfrequencynoisecanbeblamedonthetestcavityandtracedtoCBN. 7.1.2CryogenicTankUpgradeTheconstructionofCryoTHORwasdecidedwhenavacuumtankthathadpreviouslybeenusedforlong-termstabilityinvestigationsofamock-upLISAtelescopestructure[ 133 ]wasvacated(tobeseeninFigure 7-2A ).Tocreatearealistictemperaturegradientacrossthetelescopestructure,thetankhadbeenequippedwithareservoirforliquidnitrogenthatwasmountedontothetopofasuspensionframethatwastheprototypeforthespringblade-pendulumcombinationsuspensionthatwecopiedfortheoriginalTHOR.Thetelescopespacerhadbeenmountedonona2:52:5feetsolidaluminumbreadboardwithathicknessof0.75inches.SomeessentialcomponentsforacryogenicversionofTHORwerethereforepre-existing,butbeforewecouldstarttosetuptheopticalinfrastructure,wedecidedtomakesomemajoralterationstothetank.WhilebeingslightlysmallerintermsofvolumethantheTHORtank,itoersmoreverticalclearancefromthebaseup.Allitsportsarepartofthestainlesssteelbelltopthatrestsonthesolid2inchthickaluminumbaseplate,whichwasmountedonsteelI-beamsforstructuralsupport.Adouble-sided(topandbottom)opticalbenchwasattacheddirectlyasanextensionofthebaseplateinfrontofaviewportwith4inchclearaperture.Thefactthatallvacuumportsareinthetoppart,whichisnecessarilyliftedowhenthetankis 148

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ATankandexternalbench BSuspensionsandtestbench CDewarwithllingtubeFigure7-2. CryoTHORinitialinventory.Startingpointfortheexperimentwasalargetankequippedithaframeforsuspensionsandareservoirforliquidnitrogen. opened,collidedwithourplantousevacuumberfeed-throughsandalsomakestheroutingofcablesthroughelectronicfeed-throughsdicult.Therstupgradestepwasthereforetointegratevacuumportsontheunderside,whichwillremainstationaryevenwhenthetankisopened.Forthisweneededtoraisethebaseplate,anddecidedtousetheadditionalverticalspacefortherststageofseismicisolationasshowninFigure 7-3A .Thetestbenchandcubicalframewiththespring-blade-pendulumcombinationsuspension(Figure 7-2B )couldbeusedinCryoTHORwithoutmajormodications.Thein-vacuumdewar,whichismountedonthetopofthesuspensionframeandispicturedinFigure 7-2C wasoriginallydesignedtobetop-loadedthroughabellowedtube,whichwasstickingoutofthetankthroughacustomizedangepiece.Thereweretwoproblemswiththisconguration.Firstly,thecustomangewaspronetocreateleakswhilethedewarwasbeinglledwithliquidN2duetodierentialthermalexpansionandthefactthatthehermeticsealwasaccomplishedwithaVitonring.Secondly,thedewarwasmadefromstainlesssteel,whichisgenerallyabadconductorforheat.Thecoldpointsofthedewaronitsundersidewerethusfarfromreachingactualliquidnitrogentemperatures. 149

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ARaisedtankonisolationfeet BDewarangeandT-tubingFigure7-3. CryoTHORhardwaremodications.Weraisedthetankontoactiveseismicisolationfeetandaddedvacuumportstoitsunderside.Thellingmechanismforthedewarwasalteredfromopen-topaccesstoallingandexhaustlinethroughabottomport. InanengineeringeorttosalvagethedewarforCryoTHORweimplementedthesolutionthatisillustratedinFigure 7-3B .Weturnedthedewarupsidedownanddesignedacustomangepiecemadeentirelyfromhigh-conductivitycopper.WithaT-adapterforstainlesssteeltubingwelaidaliquidN2supplylineandanexhaustpipethatleavesthetankthroughoneoftheangesonthebottomofthetank.Sincethecopperangepieceislocatedatthelowestpointofthedewar,itwillremainincontactwiththeliquidN2untilitisallboiledo.Duetoitssuperiorthermalconductivity,itcandeliverheatswiftlytothethermalbathfrompointsofcontact.Weequippeditwithscrewterminalstoattachheatpipelinerodsthatreachdowntothesamplecavitymount. 7.1.3TestBenchSuspensionsThecubicaluminumframestructure(Figure 7-4A )supportsthedewaratthetopofthetankandalsoprovidesthemountingpointforfourspring-bladesinitscorners.BuildingvibrationshadproventocauseexcessaccelerationnoiseinTHOR'sreferencecavities,andthisinheritedframewasusedasatemplatetoaddapendulum-spring-bladecombinationsuspensionstagetoTHOR.Wethereforeadoptedthesamesuspensiontechnique,which 150

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ATestbenchandframe BSpring-bladewithdampingsheetFigure7-4. CryoTHORtestbenchsuspensions.Thepre-existingframestructurewithitsspring-bladesuspensionswastheinspirationforthesuspensionsysteminTHORandkeptforCryoTHOR. provideshorizontalisolationduetothependulummotionofthesuspendedopticalbenchandverticalisolationfromtheelasticdeformationofthespringblades(Figure 7-4 ).Thesizabletankwithitsplentifulverticalclearanceenabledustouseapendulumwireswithalengthof90cmforthesuspension,whichwerecutfromPhosphor-Bronze-wire.Theresonancefrequencyfhofthetwohorizontalpendulumdegreesoffreedomweestimatewith2fh=p g=Ltobeabout0.5Hz.Theverticalresonancefrequencyfvdependsonthetotalweightofthetestbench,asitsinertiaworksagainsttherestorativeforceofthespringblades.Toavoidaspreadofresonancefrequenciesbetweenthefoursuspensionpoints,theweightdistributionacrossthetestbenchneedstoputequalstrainoneachsuspensionwire,andideallyhaveitscenterofmassatthemid-pointofthetestbench.Withadditionalweightswebalancedbreadboardandequalizedthestrainofthewiresusingthewire'sviolinmodefrequenciesasanindicatoroftheirstrain.Thenalresonancefrequencywas8Hz. 7.1.4ActiveSeismicIsolationWeusedanactiveseismicisolationstagethatwehadacquiredfromLIGOcollaboratorsatMITtogivetheCryoTHORtanktheadditionallyneededunderneathclearancefortheaddedvacuumportsinthebaseplate.TheStacis2000system,manufacturedby 151

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AStacis2000unit BStacisgainadjustcircuitmapFigure7-5. CryoTHORactiveseismicisolationsystem.OurStacis2000systemconsistsofthreepiezo-electricallyactuatedfeetthatobtaintheircontrolsignalsfromintegratedaccelerometersandfeedbackelectronics.Thesupportforthisobsoletesystemwaslimitedtociruitmapsfortheadjustmentofthefeedbackbehavior. BarryControls,consistsofthreelargelyautonomouslyoperatingisolationunitspicturedinFigure 7-5A .Eachfootconsistsofamushroom-shapedtoppartwhosestemsitsonaheavy-dutypiezoelectricactuator(PEA)socketinthelowerpartforverticalisolation.Twomoreorthogonalpiezo-electricelementspressthesidesofthepillaragainstthickrubberdiscs,enablinglateralaccelerationsuppression.AccelerometersandservosinthetoppartprovidefeedbacksignalstothePEAs.TheStacis2000systemisquotedtohaveasuppressioncapabilitydownto1%oftheenvironmentalseismicnoisebetween1and10Hz,andatleast10%athigherfrequencieswhentunedproperly.Theactivebandwidthreachesuptoabout100Hz,andabovethatanysuppressionislargelyduetoapassiveroll-o.InTHORwehadobservedthattwosuspensionstageswereneededtosuppressexcessaccelerationnoiseinthelongreferencecavities.Becausethesuppressionofseismicnoisesuppressiontowardslowfrequenciesisinherentlydicultusingpassiveisolation,theadditionofanactivestagewaspromisingtohelplowertheaccelerationnoiseinthelowerportionoftheCryoTHORmeasurementband.UnfortunatelytheStacis2000systemhadalreadybeen 152

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Table7-1. Stacisfeetgainsettings. Foot1Foot2Foot3 X-AxisGain4:5k5:0k6:0kY-AxisGain4:5k5:0k6:0kZ-AxisGain8:0k8:5k5:0k obsoleteformanyyears,andtheusualprocedurestosetitupwithmanufacturersupportcouldnotbefollowed,andwehadnocomputerizedinterfaceavailabletodebugthesystem.Out-of-theboxtheStacisactuationoftheCryoTHORtankwasfoundtobeunstable.Duetoimproperlysetgains(notoptimizedforoursystem),therewascross-talkbetweenthenormallylargelyindependentlyactingunitsandfrequentrailingofthefeedbacksignals,alongwithresidualoscillationsinmultiplechannels.Dependingonthedistributionofthesupportedweightandtheplacementofthefeet,theservogainsinthe9actuationchannels(x,y,andztimesthree)havetobetunedtostabilizethesystem.Thefeedbackiscontrolledwithpotentiometersthatcanonlybeaccessedbyopeningasidepaneloftheunitstoextractthecircuitboardwhilethesystemiso-line.WiththegainadjustmentmapshowninFigure 7-5B thatwedidmanagetoobtainfromthemanufacturerwetunedthegainsinall9channelssuchthattheabovesymptomsofanimproperlyconguredsystemvanished.ThenalizedresistancevaluesthatdeterminethegainforthedierentdirectionsinthedierentfeetarelistedinTable 7-1 .AcomparisonofboththeverticalandhorizontalaccelerationnoisewithandwithoutactivefeedbackisshowninFigure 7-6 .Althoughthenoiselevelsareneartheresolutionlimitoftheaccelerometerusedforthismeasurement,asuppressionofmorethananorderofmagnitudecanbeseenupto30Hzintheverticalchannel,whichisagoodcomplementoftheverticalspring-bladeisolation.Thehorizontalchanneldoesnotperformquitethatwell,whichislikelyduetothefactthatthex-actuationinfoot2isdeadanddoesnotproducefeedbacksignals.Sincethehorizontalnoiseislowertobeginwith,andbecausethecut-ofrequencyofthependulumsisabout.5Hz,asopposedtoabout8Hzforthespring-blades, 153

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AHorizontalsuppression BVerticalsuppressionFigure7-6. Stacissystemsuppressionofaccelerationnoise.Intheverticaldirectionweachievedareductionofmorethananorderofmagnitudeupto30Hz,whilethehorizontalsuppressionturnsoutlesseective,whichislikelyduetoasingulardeadfeedbackchannelinoneofthefeet. thehorizontalpassiveisolationofthetestbenchismuchmoreeective,andthedeadchannelbecomeslessofaconcern. 7.1.5TestCavityMountingSincetheinvestigatedmirrorsaregoingtobecryogenicallycooledbybeingputinthermalcontactwiththeliquidnitrogendewar,themountingstructureforthetestcavitiesmustkeepthemwellinsulatedfromtherestofthetestbench.Thermalleakswillraisetheobtainabletemperatureandmaycoupletemperatureuctuationstothesamplemirrors.TheexperiencewebroughtinfromthethermalinsulationofthetankinsideintheLISAHSexperimentshelpeddesignaproperthermalinsulation.AconceptualsketchofthetestcavitymountisshowninFigure 7-7A .Itcanacceptmirrorsubstratesupto1inchindiameterand0.25inchesthickandcavityspacerswithadiameterof1.5inchesandalengthbetween0.75and2inches.Thetestcavitiesaresupportedattheundersideoftheirspacer,andthemirrorsthemselvesdonottouchthemount,butareattachedtothespacer.Themountwasmanufacturedfromhighconductivitycopperandconsistsoftwohalf-shellsthatleaveagapbetweenthemdependingonthecavity 154

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ATestcavitymountclose-up BGeneralassemblyFigure7-7. Cryogenictestcavitymount.ThetestcavityissupportedinsidethetwoCopperhalf-shells,whichsupportspacerlengthsbetween0.75and2inches.Themountisplacedcentrallyunderneaththedewarandhasthermalrodsprotrudeupwardsasheatpipeplines. length.Thetoppartpressesagainstthespacerwithaspring-loadedscrewmechanismandholdsthecavitytightlyinplace,whileallowingforsmalldierentialmotionofthepartswhenthesystemundergoesachangeintemperature.Thermcontactisnecessaryfortheecienttransferofheat,butatightclampingmayshattertheglasswhenthepartsstartcontracting/expandingwithtemperature.Thethermalcontactbetweenmountandspaceroccursacrossanannularregionofabout.2incheswidthbothatthetopandthelowershell.Themountsitsinsidetwolayersofthermalshielding,whichjustliketheLISAthermalshieldswerebuiltfromAluminumplateswithaluminizedMylarcoveringallsurfaces.Themountisplaceddirectlyunderneaththedewarwithitsthermalrodsreachingupthroughclearanceholesinthethermalshields,asshowninFigure 7-7B .Duetodierentialthermalexpansionthecavitywillmovewithrespecttothetestbenchwhenthesystemiscooled,whichwillnegativelyaectthealignmentoftheprobelasers. 155

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Theintroductionoftiltisatthispointworsethanasimpledisplacement,becauseoverthelongopticalleversfromthecavitiestothetankoutsidethebeamcanwalkobothinreectionandtransmissionandmaycausealossofsignalwhilethesystemiso-limits.Thiscanturnoutespeciallyproblematicinthecaseofhighnessecavities.Wethereforedesignedthemountandthermalshieldswithathree-pointstandbetweeneachlayerusingsphericalpiecesofPEEK,anultra-highvacuumcompatibleplasticwithverylowCTE,inakinematicmountconguration.Itisdesignedtofavordisplacementovertiltswhendierentialexpansionoccurs,whichwillkeepthetestcavitieslevelduringthecool-down. 7.1.6FlexibleThermalLinksThesampleholderismountedonthetestbench,whilethedewarisusingthesuspensionframeforsupport,becausetheshakingofthedewarfromtheboilingnitrogenmustbekeptfromreachingthecavities.Thethermallinksthatextracttheheatfromthetestcavitythereforeneedtobeexibleinordertonotcompromisethetestbenchsuspension.Furthermore,theymaybesubjecttomaterialstieningduringcryogenicoperation,andstillmustleavethecavitiesundisturbed.Usingcopperbraidsisthemostattractivesolution,astheyspreadtheavailablecross-sectionforthermalconductivityamongmanyindividualstrandsthatcanindividuallyremainexible.Weusecopperbraidswitharope-likestructurethatisshowninFigure 7-8A .Itisspunfrommorethan10,000individualstrandsofGauge48copperwire,withatotaleectivecrosssectionof20mm2.Justlikenormalpiecesofrope,theyaretightlypackedandresiststretching,butareextremelyexibleforallbendingdeformations.Weuseatotalofsixlinkswithalengthof8incheseach,loopedinahalfcircletoallowforsomeslackascanbeseeninFigure 7-8B .ThreesolidCopperrodsreachdownfromthecoldspotofthedewarandmeetthethreesolidCopperrodscomingupfromthesampleholderthroughthethermalshields.TheCopperbraidsareonlyusedtoclosethelastbitofremainingdistance. 156

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ACrudecopperbraid BLinkswithweldedblocksFigure7-8. Flexiblethermallinks.Theleftphotoshowsastringoftherawbraid,andtotherightthenalassemblywithbutt-weldedterminationblockscanbeseenbridgingtheconnectionbetweenupperandlowerthermalrods. 7.2CavityAssemblyWhileonlyasinglereferencecavityisneededforadierentialmeasurementwiththetestcavity,therearetwogoodreasonstobuildCryoTHORwithtworeferencecavities.Firstly,toconrmthattheperformanceofthereferencesystemisbetterthanthelevelseeninthetestcavities,asimilarlystablereferenceisneeded.Secondly,cm-scaletestcavitieswillhaveFSR'scloseto10GHz,andasecondreferencecavitywithadierentFSRincreasestheoddstoformbeatnoteswithreferenceresonancesatlowerfrequencies.Throughoutprecisionopticsandcavitystabilizationexperimentsopticalcontacting[ 129 ]isapreferredmethodformountingmirrorstocavityspacersbecauseitposeslessdangersofcompromisingtheoptics,andalsobecauseitisareversibletechnique.ThetwocavitiesforthereferencesysteminCryoTHOR,whichweretakenoutoftheretiredUFLISexperiment,featuredopticallycontactedmirrors.TheyhadbeenbuiltfromZerodurspacerswithasquarecross-sectionof3.8cmwidthwithfusedsilicasubstratesanddielectric1064nmHR 157

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coatings.TheexactcoatingspecicationswereofnoimportanceinUFLIS,andhadthereforenotbeenrecordedatthetimeofthemirroracquisition.Atlengthsof23.1cmand25.9cmtheyhadsimilardimensionstotheTHORcavities,andwiththethreatoftemperatureuctuationsduetotheactivecoolingandnotestbench-widethermalshielding,thelowthermalexpansionZerodurmadethemanaturalchoiceforCryoTHORoverthepurchaseofnewspacers.Theirnessesatthelasttimeofusehadbeendeterminedas4,600fortheshorter,and11,500forthelongercavity.Becauseofitslowernesse,whichresultsinreducedopticalgain,wedecidedtoattempttoremovethemirrorsfromtheshortercavityandreplacethemwithmirrorsoutoftheTHORcoatingbatch.Usingacleanroom-gradeovenweplacedthecavityinatemperaturecontrolledheatedenvironmentandincreasedthetemperatureinstepsof5Cstartingatroomtemperature.Thisway,thedierenceinthermalexpansionoftheZerodurandtheFusedSilicawasgoingtograduallyloosenthebondandeventuallycausethemirrortodetach,leavingbehindacleansurface.Thereisofcourseariskofcompromisingthesurfaceifthedetachmentistooviolent,inwhichcasethespacerwouldneedtobere-polished.Atatemperatureof85Cthemirrorshadstillnotcomeobythemselves,butwewereabletogentlyprythemowithoutanymacroscopicallyvisibledamagetothespacer. 7.2.1MirrorContactingInSection 6.4.1 wearguedthatfortheHSloopoperationwewerelookingforalinewidthontheorderof30kHzinourcavities.Forcavitiesofabout25cmlengththiscorrespondstoanessevalueofabout20,000,whichinturnisachievedwithtotalopticalround-triplossesof314ppm.Aftergettingintouchwithmirrormanufacturers,weorderedtwodierentcoatingrunsonfusedsilicasubstratesfromCoastlineOpticswithT1=300ppmandT2=15ppmasthetargettransmissivities.Weincludedavarietyofmirrorswithdierentradiiofcurvature(8cm,15.2cm,50cm,1m,andnumerousats)withasuper-polishedannulusforopticalcontacting(exceptfortheatmirrors)ofbothHRvarieties.Allmirror 158

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AAssembledreferencecavityinxedmount BTestcavityFigure7-9. CryoTHORassembledreferencecavity.ThetwoZerodurcavitiesforthereferencesystemweretakenfromUFLIS,andafreshmirrorpairwasgiventotheonewithlowernesse. back-sidesweretreatedwithanARcoatingwitharesidualreectivityof500ppm.Theendproductmirrorswerequotedtohavetransmissivitiesof280ppmand10ppm,asmeasuredbyCoastline,withexpecteddissipativelossesontheorderof10ppm.ManyofthemirrorshadbeenaccountedforinTHOR,suchthatforCryoTHORweonlyhadtheleftovermirrorsfromthetwocoatingbatchesatourdisposal.TheatcavitymirrorshadturnedouttobediculttocontactinTHOR,thereforewechosetousetwocurvedmirrorsforthenewCryoTHORreferencecavity.Withafrontmirrorof1mROCandatransmissivityof280ppmandabackmirrorof0.5mROCwith10ppmtransmissionwere-assembledthecavityinthesameover-coupledcongurationthatwaschosenfortheTHORcavities(thatwerealsousedfortheLISAHSdemonstration).Whilethecontactingofthe0.5mmirrorwentwithoutcomplications,ittookseveralattemptswithmultiplecleaningstepstocontactthe1mbackmirror.Becauseithadahighernessetobeginwith,andoutofworrythatthecontactingmaynotgosmoothlyasecondtime,wedecidedtokeepthelongreferencecavityinitsoriginalstate.Moreimportantly,wewererunningoutofopticstobuildover-coupledcavities.Figure 7-9A showsthere-assembledZerodurcavity.TherewerestillseveralshortROCmirrorsleftthatcouldnotbeusedtobuildstablelongcavities.Instead,wedecidedtoturnthemintotestcavitiesandpurchasedacoupleof 159

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Table7-2. CryoTHORcavitycongurations. PropertyShortcavityLongcavityTestcavity Length23.1cm25.9cm1.9cmFrontROC1m115.2cmBackROC0.5m0.5m8cmg-factor0.4130.4820.655Waist284m291m101mFSR648.6MHz578.6MHz7.582GHzFinesse21,00011,500210,000Linewidth31kHz50kHz36kHz short,super-polishedspacers.Becausethesetestcavitieswilleventuallybecooled,thespacerneedstobemadefromthesamematerialasthesubstrate,orelsethedierentialcontractionmaycausethemirrortopopo.Thespacershavealengthof1.9cmandadiameterof1.5inches,asisrequiredbythemountwedesignedandpresentedinSection 7.1.5 .Becausethespacerissomuchshorter,anesseof20,000asintheothercavitieswouldresultinalinewidthofabout400kHz,andthereforeareductioninopticalgain.Instead,tocompensateweused10ppmmirrorsforboththefrontmirrorwith15.2cmROCandthebackmirrorwith8cmROCtoassemblethetestcavityshowninFigure 7-9B .Table 7-2 liststhegeometricparametersofthecavitiesthatwillbeusedinCryoTHOR.Thedisplayedvaluesfornesseandlinewidthareestimatesusingquotedtransmissivityvalues.InSection 4.3.1 wediscussedthatfusedsilicaisnotactuallysuitableforcryogenicapplications.OurchoicewaspurelymadebecauseoftheavailabilityofhighqualityopticsfortheinitialtestingofCryoTHOR..Forprogressivecryogenicthermalnoisemeasurementssapphirecavityopticsandspacersneedtobeacquired. 7.2.2CavityMountsForeseeingthatthelackofasecondglobalseismicisolationstageforthetestbenchandthefactthatadewarfullofboilingnitrogenwillshaketheframestructurethatitisattachedto,weaddedanadditionalsuspensionstageforthereferencecavities,whicharemoresusceptibletoaccelerationnoisethanthetestcavity.ForalowerverticalresonanceweusequadrupleextensionspringsinthecavitymountthatisshowninFigure 7-10A .Both 160

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AReferencecavityhammock BAssembledtestcavitymountFigure7-10. Cavitymountingsolutions.Wedesignedanextensionspringhammockforthereferencecavities,andequippedthetestcavitymountadditionallywithheatertape. theverticalandhorizontalresonancesofthishammockforthereferencecavitiesareabout4Hz.ThenalproductofthetestcavitymountdesignpresentedinSection 7.1.5 canbeseeninFigure 7-10B .Thethermalrodsterminateontheuppershell,andtheheatowfromthelowerisachievedwithshort,atCopperbraidsthatallowforthecontaction/expansionofthefusedsilicaspacer.Weaddedfourpiecesofshort,self-adheringheatertapearoundthecylindricalCopperpiecesuchthatinthefuturewewillbeabletoholdthetestcavityatspecictargettemperatures.Thelaserbeamhitthecavityfrombelowthroughclearanceholesinthemountanditsthermalshields,andthetransmittedlightescapesthroughthetop. 7.3InterferometricComponentsTheopticsassemblyforCryoTHORcanbebrokendownintothreeparts.TheinjectionbenchpreparesthesuperpositionforHSintothevacuumberfeed-through,thein-vacuum 161

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AInjectionbenchtopview BFiber-coupledphasemodulatorsFigure7-11. CryoTHORinjectionbench.Thebeamsareindividuallyber-coupledintointegratedphasemodulators,andafractionoftheirlightisusedtocreateanunmodulatedlaserbeatontheinjectionbench. testbenchwiththebeam-splitting,mode-matchingandpolarizationcontrolopticsforthecavities,andthediagnosticbench,wherethePDsrecordthebeatnotesinreectionandcamerasobservethetransmittedlight.Usingbothitstopandbottomsurfaces,theexternalbreadboardthatisattachedtotheCryoTHORtankoersenoughspacefortheinjectionanddiagnosticoptics. 7.3.1InjectionBenchWeusetwoCoherent(FormerlyLightwave)NPRO125lasersystemstoseedthehetero-dynelasereldinCryoTHOR.Atopview(oftheundersideoftheexternalbreadboard)oftheinjectionbench,wherethetwolasersarecoupledintobersisshowninFigure 7-11A .Faradayisolators(FI)suppressback-reectionsintothelaser,andmode-shapingoptics(lenses)andmirrorssteerthebeamsintothebers.Smallportionsofeachlaserarepickedoandinterferedforanon-benchbeatsignal.Weincludedaber-coupledphasemodulatorfromJenoptik(picturedinFigure 7-11B )inthepathofeachlaserforadvancedlockingtechniquesthatwediscussinSection 7.5.1 .Theyenablethebroadbandphasemodulationupto2GHzwithouttheneedforhighvoltageampliers.Onathirdpartoftheinjectionbench(notshown)theEOMoutputsarejoinedintoacommonber,whichterminatesonthetestbenchinvacuum. 162

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Table7-3. Comparisonofphotodetectors. PropertyPDA10CFET-3000ADBBPD NEP12pW=p Hz0:03pW=p Hz10pW=p HzGain5kV/A700V/A5kV/ABW125MHz2GHz25MHzArea.2mm2.008mm24mm2 7.3.2DiagnosticBenchThepurposeofthediagnosticbenchistorecordthevariousbeatnotesandmonitorthepowerlevelsandspatialmodesofthetransmittedlightelds.AnynoisethatisintroducedtotheelectronicsignalsbythePDsinreection(RPDs)isgoingtocoupleassensingnoiseintothecavitylockandcannotbesuppressedbytheloop.Electronicnoiseinanalogcomponentsincreasesrapidlyatlowfrequencies(DCuptoseveralkHz),butisverymuchconstantatRFfrequencies.HeterodynetechniquesshiftthislowfrequencynoisetotheRFbandinthemixingprocess,fromwhereitisthenltered,whiletranslatingthevariationofRFsignalstoDC.BecausethePMdigitizesthelaserbeatsdirectly,theRPDnoiseattheheterodynefrequencywillbetheonlyelectronicnoisesource(besidesthepreviouslydiscussedADCnoiseinthePM)thatcausessensingnoise.WehadseveraldierentPDsatourdisposaltouseinTHOR,andnarrowedourchoicetothreedierentmodels,eachofwhichhaddierentadvantagestooer.TheThorlabsPDA10CFisaDC-coupledall-aroundRFPDwithamoderatebandwidth.Incontrast,theET-3000AbyEOTechisaveryfastphotodetectorwithrelativelylowdetectorgain,butalsoalownoise-equivalentpower(NEP).Finally,wecustomizedthedesignoftheRPD3detectorthatisusedintheLIGOdiagnosticbreadboard(DBB)[ 134 ],whichhaslowbandwidth,butaverylargeareaphotodiodeandcanoutputlargervoltages.ThecharacteristicsofthethreemodelsarelistedinTable 7-3 .AnyelectronicsensingnoisethattheexperimentwillbefacedwithcanberevealedinadierentialchannelphasemeasurementwiththeAX3065PM.TosettleforaPDmodelweassembledthesetupinFigure 7-12A .Anosetphase-lockedlasersuperpositionwascoupled 163

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ATestSetup BInvestigatedPhotodetectorsFigure7-12. Photodetectordierentialphasenoisemeasurement.Twolaserswerephase-lockedandcoupledintoacommonber,justasinCryoTHOR.TheberoutputissplitandmeasurewithtwoseparatePMchannels.DierentialphasenoiseinthismeasurementgivesinsightintothePDperformanceinaHSscenario.ThethreePDmodelswecomparedcanbeseentotheright. intoanopticalberformodecleaning,andtheberoutputwassplitwitha50/50powerbeamsplitter.ThethreedierentPDmodels,whicharepicturedinFigure 7-12B ,werepluggedintothesetupinpairstomeasuretheirnoisecharacteristics.Toensurethatinputphasemeternoisemasksthedetectorperformanceaslittleaspossible,weequalizedthelaserpowersinthebeatnoteandtunedthemtoproducebeatnotesnearthemaximumoftheiroutputrangethatweattenuatedtoa.95Vamplitude.TheresultsareplottedinFigure 7-13 forrecordinga14.4MHzbeatwithtwindetectorpairs.TheET-3000AandtheDBBPDdisplaysimilarperformanceonlymarginallyabovethephasemeterlimit,whilethePDA10CFshowedahighernoiseoorthatisinagreementwithitsmanufacturernoise-equivalentpower(NEP).Becauseoftheircustomizabilityintransimpedancegainandlargearea,wedecidedtousetheDBBdetectorsintheexperiment. 7.3.3TestBenchLikemanyothercomponentsofCryoTHOR,theassemblyofthetestbenchprogressedinmultiplestepswithincrementalimprovements.Becausethedewarishoveringcentered 164

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Figure7-13. PhotodetectorNoiseInvestigation.TheRPD3andtheET3000Ashowsimilarperformanceinthetargetfrequencyrange,whilethePDA10CFsurprisinglyshowedanincreasednoiseoor.OurchoicefellontheRPD3. abovethetestbench,andforabettersymmetryinthedistributionofweight,thetestcavitymountisplacedatthecenterofthebreadboard.Thereferencecavitiesarethenplacedparallelwitheachotheroneithersideofthecentralmount.Thebeampropagationandmode-shapingopticswerethendistributedacrossthebreadboardinaneorttoequalizetheweightdistribution.Additionalweightswereusedforne-tuningthestraininthesuspensionbersforequalverticalresonancefrequencies.Becauseabercarriesthelaserstothetestbench,allalignmentopportunitiesareolimitsduringoperation.Smallalignmenterrorsduetoshiftsduringpump-downsorcool-downscansignicantlyaectthevisibilityofthecavities,whichiswhyweintegratedremotelycontrolledpiezo-motordrivenkinematicmounts.AschematicviewofthecompletenallayoutofCryoTHORisshowninFigure 7-14 . 165

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Figure7-14. SchematicviewofcompleteCryoTHORopticslayout.TwoNPROlasersarecoupledintophasemodulatorsandeventuallycombinedintothesameopticalber.Onthetestbenchtheirsuperpositionissplitseveraltimesandreectedoofeachcavity.Thepolarizingbeamsplittersthatseparatetheincidentfromthereectedcavitylightweregroupedinfrontofthemainopticalwindow.Thetransmittedbeams(dashedlines)arereectedupwardsandguidedoutofthetankthroughaseparateopticalwindowonahigherlevel.ThedewarisinthermalcontactwiththesamplemountviatheCopperbraids,andtheentiretestbenchissuspendedinvacuum. 166

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7.4ExperimentCharacterizationThemostpressingconcernforbringingCryoTHORoperationalasaplug-and-playmeasurementapparatusfortestcavitiesistheperformanceofthereferencesystem.UsingEq.( 4{26 )wecanestimatetheCBNlevelinthetestcavitythatweincludedinTHORtobe eTC(f)=0:16Hz p Hzs 100Hz f;(7{2)fromwhichwewouldliketoseeaclearseparationinthedierentialfrequencynoisebetweenthetworeferencecavities.Ourcharacterizationofthecavitiesshowedthattheyhadsigni-cantlyhigherdissipativelossesthananticipated,whichactuallyexceededthetransmissivelosses.TheoriginaltreatmentfromSection 6.1 needstobemodiedtocorrectlyassesstheopticalgainsoflossycavities. 7.4.1OpticalCavitiesRevisitedThecavitybehaviorthatwederivedinChapter 6 ignoreddissipativelossesinthecavity,whichisagoodapproximationaslongastheydonotrivalthetransmissivelossesinmagnitude.Inthehigh-nessecavitiesthatweassembledforTHORwewerefacedwithunexpectedlyhighopticallossesthatweredominatedbydissipation(scatterandabsorption),whichwereinstarkcontrastwiththemanufacturervalues.Individualmirrordefectscancertainlycausesuchanomaliesincavitybehavior,whichiswhyforCryoTHORwetookspecialcaretoinspectthemirrorsbeforeopticalcontacting.Wefoundhowevernothingthatcouldexplaintheagreementoflossesacrossdierentassembledcavities.InthepresenceoflossesEq.( 6{9 )needstobemodied.Ifwetakeanempiricalstandpointanddonotinferreectioncoecients,butuseonlythetotalcavityround-triplossLandthetransmissivityT1ofmirror1toexpresstheeldrelations,weobtain Ec(t)=ip T1Ei(t)+(1)-221(L=2)Ec(t)]TJ /F1 11.955 Tf 11.96 0 Td[(2T):(7{3) 167

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Table7-4. CryoTHORmeasuredcavityproperties. PropertyShortcavityLongcavityTestcavity FSR648.6MHz578.6MHz7.875GHzLinewidth31.6kHz81.9kHz78.5kHzFinesse20,4706,830100,300Res.Re.0.0890.02220.908L307ppm920ppm63ppmT1199ppm391ppm1.5ppmOpticalGain4:4410)]TJ /F5 7.97 Tf 6.58 0 Td[(5rad/Hz2:0810)]TJ /F5 7.97 Tf 6.58 0 Td[(5rad/Hz1:2110)]TJ /F5 7.97 Tf 6.58 0 Td[(6rad/Hz TakingthesamestepsasinthederivationinSection 6.2.1 ,weendupwithaversionofEq.( 6{18 )thatreads R()1)]TJ /F1 11.955 Tf 11.96 0 Td[(2T1 L)]TJ /F3 11.955 Tf 11.96 0 Td[(i4T1 L FWHM:(7{4)HereweseethatthefractionT1=Ldeterminesthecouplingofthecavitybythefractionofthetotallossesthatareinthetransmissivityofthefrontmirror.Ifitmatchesallotherlossesexactly,thecavityisimpedance-matchedandtheresonantreectivitybecomeszero.Thenessemustalsobere-evaluated,andwithri=p 1)-222(Ti)-221(LiandL=T1+L1+T2+L2,wecanturnEq.( 6{17 )into F=FSR FWHM2 L:(7{5)Withoutanyknowledgeofthemirrorswecandeterminethenesseandthustotallossesviaalinewidthmeasurement,andusingEq.( 7{4 )wecanndT1fromthedipinthereectedlightpoweronresonance.TheempiricalopticalgainfollowslikewisefromEq.( 7{4 ).BecauseallsensingnoiseintheHSschemeissuppressedbythisgain,unexpectedlyhighlossescanhaveasevereinuenceonthenoisecharacteristics.Inacarefulmeasurementwesubtractedthenon-participatingHOMcontentinthebeamsanddeterminedthecavitypropertiesthatarecompiledinTable 7-4 .Thepreviousnessemeasurementoftheunalteredreferencecavityisconrmed,butsurprisingly,thededucedmirrorspecicationsareinstarkcontrasttothequotedvalues.Weareunabletosaywhatcompromisedthem,asweobtainagreeingresultsbetweencavitiesthatrequiredmultiplecontactingattemptswithrepeatedmirrorcleaningstepsandmirrorsthatcontacted 168

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out-of-the-box.Afterwereceivedthemtheywerekeptintheirsealedoriginalpackaginginsideaclean-roomcabinet,butitispossiblethattheywerecompromisedduetoextendedstorageuptoseveralyears. 7.4.2ReferenceSystemNoiseFloorObtainingthedualHSlockedstatewithCryoTHORrequiresacarefulpreparationphaseforwhichwedevelopedthefollowingprocedure.Onceapairofresonancesintwodierentcavitieshasbeenidentied,thetwolasersareosetphase-lockedtoabestguessoftheirfrequencydierence.Movingthelasersinunison,weadjusttheosetfrequencyuntiltheycoincidentallybecomeresonantintheirrespectivecavity,whichsetsthenominalheterodynefrequencyforthemeasurement.Thephase-lockstaysengagedandthephasemeterislockedtothereferencebeatnote,whilethelasersaretunedoresonancetosetthedemodulationphasebyzeroingtheHSerrorsignal(whichdenesthedemodulatedquadrature).Thesameisrepeatedwiththerolesofbothlasersreversedfortheotherfeedbackchannel.Whenthecorrectdemodulationphasesareset,thephase-lockisdisengaged,butthePMkeepstrackingthenowfree-runningbeat.MovingbothlasersclosetotheirresonancesissucientforHStocatchlockwhenthefeedbackcontrolsareengaged.Subsequentlythefeedbackgainsareoptimizedwithregardstoloopstabilityandnoisesuppression.Figure 7-15 showsthedierentialfrequencynoisebetweenthetwolasersaftertheyhavebeenlockedtotheirrespectivereferencecavitieswiththeaboveprocedure.Thesensinglimitrepresentsthecombinedphasenoiseduetoshotnoise,PDtechnicalnoise,andADCquantizationnoise,allofwhichhavesimilarmagnitudeanddriveaccordingresiduallaserfrequencyuctuations.Thegainlimitrepresentthelaserfeedback'sinabilitytosuppressthedierentialnoisebetweenlaserandcavityduetoinsucientbandwidth.WecomparetwomeasurementsbetweenwhichtheStacis2000wereactivated.Withoutactivecompensationonlyaverynarrowpartofthenoisecurvedipsbelowtherequirementat1kHz,butwewereabletostretchthewidthofthisbuckettospananorderofmagnitudefromroughly200Hzto2kHz.ThemarginissignicantenoughtoobserveCBNinducedfrequencyuctuationsofthe 169

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Figure7-15. CryoTHORreferencesystemperformance.Above1kHzthesystemisgain-limitedandthelasersarenottrackingthecavityresonancessuciently.Atlowfrequenciesthesuspensionpeaksarevisible,andbetween10Hzand300Hzweencounteranunidentiednoisesourcethatreactstotheactiveisolation.Wesuspectthatitismotion-relatedbutnotadirectcouplingofaccelerationnoiseduetoitsdierentcharacteristicshape. testcavity,butunfortunatelythefrequencynoisefeaturesanintermediateplateauwhoseroll-obecomestheloweredgeofthebucket.Ourinvestigationsofthesystemincludedassessmentsofthedierentialfrequencynoisebetweenbeatsignalsamplingpoints,fromwhichwecanexcludeconventionalsensingnoiseduetopathlengthuctuationsandPDnoise.Oureortstodetermineitsnaturehavebeenfruitlesstothispoint,howeverwefoundthatitissafetoexcludethedirect,conventionalcouplingofaccelerationnoisetospacerlength.Acoincidentmeasurementoftheaccelerationspectrumofthetank,pairedwith 170

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thetransferfunctionsofthesuspensionsplacesanupperboundthatliestoofarbelowtheobservednoise.ThecopperbraidsandelectricconnectionsforthePZTmirrors,temperaturesensors,andheatersmaycauseacomplexalterationofthesuspension'stransferfunctions,butnotsignicantlyamplifythenoiseinabroadbandmanner.However,becausetheplateaulevelandedgefrequencyrespondtotheactivationoftheStacissystemweexpectthemtoberelatedtomotionsoftheopticalbench.Asanexample,rotationaldegreesoffreedomofthetestbenchcouldcouplestraylightintothefeedback,orfrequencyup-conversionmayplayarole.Lastly,theissueofthedeadchannelintheactiveisolationremains,anditsnon-actuationmaycoupleintendedlinearactuationtorotationalmotion.Weestimatethatthefrequencybandinwhichcoatingthermalnoisecanbeobservedcanbeincreasedbyanotherfactorof4towardslowfrequencieswithbetterisolation. 7.4.3CoolingCapacityBecauseofthephysicalseparationbetweendewarandtestcavitymountwiththecopperbraidheatowbottleneckitisimpossibletoreachtheactualtemperatureofliquidnitrogen(77.2Kat1atmpressure)withthetestcavity.Tryingtomodelthesteady-stateheatowfromcavitytodewar,takingintoaccountthermalleaksfromcontactpointsofthethermalshieldingandradiativeheattransfertothetankoutside,weestimatedthatwecanreachatemperatureof90Katbest.Figure 7-16B showsthellingprocessofthein-vacuumreservoir.Weuseanexternaldewarwith50gallonscapacitythatwepressurizeat5psi.Thepressurepushesliquidnitrogenthroughthellinglineintothereservoir.At5psitheinitialllingprocesstakesabout2hoursifthereservoirispreviouslywarm,andwhenrellinganalmostempty(butcold)reservoirthetimeshortenstoabout45minutes.Asingleexternaldewarloadcanllthereservoirabouttwotimes.Whenthedewarislledwiththesystembeinginitsstationarycoldstate,ittakesabout40hoursforallliquidnitrogeninthedewartoevaporate.Afterthereservoirisfull,thellinghoseisdetachedandadefrostingcoil(Figure 7-16C )isattached, 171

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ATanktubing BFillingprocess CBoil-odefrosterFigure7-16. CryoTHORcooldownimpressions.Thetestcavityinsideitsthermalshieldsisconnectedvia6copperbraidstothedewar.Onceinvacuum,anexternalreservoirpumpsliquidnitrogenintothedewar.Afterthellingiscompleteadierentialvalve-terminatedcoppercoilclosesthetubingo. cappingbothlineswithadierentialoverowvalve.Thecoppercoilgivestheexhaustgasenoughtimetoreachroomtemperatureandpreventsthefreezingofthevalve.Initialcooldowntestswereperformedwithadummycavity,asolidAluminumpiecethatwasmanufacturedintotheshapeofthetestcavities.InFigure 7-17 weshowthetimeseriesofacooldownsession.Thebluecurvewasmeasuredwithsaiddummycavity,andtheredcurvewastakenaftertheactualfusedsilicacavityandheatershadbeeninstalled.Thedewarcheckpoint,whichisclosetooneofthethermalrods,quicklyreachesastationarytemperatureof90K,andthesamplemountbreaks100Kafterabout10hoursofcooling.Aftertheinstallationoftheheaterstheobtainabletemperaturerosefrom98Kto104Kduetotheaddedthermalcoupling.Thecomparisonshowsthatthepresenceoftheheatersdoublestheresidualheatow,astheinitialloadofliquidnitrogenlastsonlyhalfaslong. 172

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Figure7-17. CryoTHORcooldowntemperature.Abouttwohoursafterstartingthellingprocessthetemperaturecurveattens,andwithoutadditionalllingsittakesalittleover30hoursuntilallliquidnitrogenisevaporated. Inthenalcongurationthereservoircanrunonallcycleofabout36hours(withidlingheaters). 7.5CryoTHOROperationWhilethereisroomforimprovementforthereferencesystem,itperformsatalevelwhereobservationsofCBNinthetestcavityaretheoreticallypossible,aslongastheyarenotmaskedbyothernoise.Ananticipatedproblemforthelocktothetestcavitywasthatreferenceresonanceswillusuallybefurtherawaythan32MHz,whichwasthemaximumfrequencytheAX3065controllercanprocess.Apossiblesolutionistodemodulatethehigherfrequencybeatnotesagainstacommonfunctiongeneratorsignal,whichtranslatesthesignalstoanintermediateheterodynefrequencyforthePM.Thelow-noiseCryoTHORPDshoweverhaveabandwidthofonly25MHz,sotheyphysicallycannotoutputfasterbeats.Usingtheber-coupledphasemodulatorswedevelopedanosetheterodynelockingschemeusingsidebandmodulationthatsolvedthisproblem. 173

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Figure7-18. Osetheterodynesidebandlocking.Bothlasersaremodulatedwithdierentfrequenciesthatplaceoneoftheirrespectivesidebandsclosetotheotherlaser.TheresultaretwobeatsoneveryPD(mostotherbeatswillbeoutsidetheirbandwidth)thatcanbeusedfortwoindependentHSlocks. 7.5.1HeterodyneSidebandLockingTheoriginalHSneededthesinglesecondaryeldtodemodulatethecavityinteractionphase,andwithregardstoLISAtheeldofasecondlaserwasmostwelcomeforthis.Whilethissecondlaserisnowtoofarawayforadirectbeatnote,wecanphase-modulateitandcreatesidebandsthatarecloseenoughtotheoriginallasertoformabeatnotethattheAX3065canprocess.Inturn,wealsomodulatetherstlasertoprovideaheterodyneeldforthesecondHSlock.ThismethodisillustratedinFigure 7-18 .Themodulationfrequenciesmustbedierent,otherwiseabeatnotedegeneracyisintroducedthatmasksthecavityinteraction.AninterestingfeatureofthismethodisthatthereisnomoreneedforadedicatedreferencePD,asareferencebeatforeachcavitysignalexistsintheotherrespectivecavityPD.Tomeasurethedierentialfrequencyuctuationsbetweenthecavities,thelaserbeatnotewouldneedtobeassessedwithafasterPD,butcanalsobetakendirectlyfromthePMchannelsfortheheterodynelockifthefrequencynoiseofthemodulationsignalsissucientlystable.Totestthephasedelityofthemodulationweperformedano-resonancefrequencynoisemeasurementforwhichweosetphase-lockedacarrier-sidebandbeatandmeasuredtheotherbeatwiththePM.TheresultisplottedinFigure 7-19 ,forwhichthetwolaserswere 174

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Figure7-19. Osetheterodynesidebandlockingperformance.ThedierentialfrequencynoisebetweenthemodulationsignalsastheytravelwiththecarriersislowenoughtoobservetheCBNinducedfrequencyuctuationsinthetestcavity. modulatedwith60MHzand72MHz,respectivelyandthefasterbeatlockedtoa19MHzNCO.Theresultingdierencefrequencyof79MHzresultsinasecondbeatof7MHz,whosefrequencynoiseisshownintheplot.TheobservednoiseoorisaperfectmatchwiththeexpectedsensingnoiseinthePLLduetoPDnoise,ADCquantizationnoise,andADCtimingjitter.Onedisadvantageofthismethodisthatthereislesspoweravailableinthebeat,sincethesidebandborrowspowerfromitscarrierbutismuchweaker. 7.5.2RoomTemperatureMeasurementsWiththeosetHSmethodwewereabletoobtainanoiseestimateforthe1.9cmtestcavity.Theinitiallockacquisitionprovedtobedicult,becauseitsunder-couplednaturediminishestheopticalgain.Evenwithoutvisualfeedbackfortheerrorsignalthe 175

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demodulationphasecanbesetoresonanceasdescribedearlier,butitonlyroseminutelyfromitsrmsbackground.However,therewasenoughgaintomaintainthelockandobtaintheplotinFigure 7-15 ,whichcomparesthefrequencynoiseofthetestcavitywiththepreviouslyassessedreferencesystemlimit.Thegoodnewsisthatwecansafelyattributealloftheobservednoisetothetestcavity,butthatunfortunatelymeansthatitisquitefarfromCBN.Athighfrequenciesthesystemisnowlimitedbysensingnoise,inparticularshotnoisehasbecomeamajorplayerbecausewehadtoincreasethelightpoweronthephotodetectortobringtheweakerbeatbetweenlaserandsidebandtoahigheramplitude.Whilethelaserisnotsettolaseatitsfulloutputpowerpotential,weareencroachingonthedamagethresholdofthebermodulators,suchthatareductioninsensingnoiseisnotpossibleinthissituation.Inadditiontothefrequencynoise,weassessedtheaccelerationnoiseatatestpointontheexternalbench(whichisattachedtothetank)andaddedtheprojectedimpactonthecavitynoiseusingonlythetransferfunctionofthesinglesuspensionstagebetweenthetankandthetestbench,towhichthetestcavityismounted(withnoothersuspensionstagesinbetween).Thematchisnotstriking,howeverthemanyelectricalconnections,andinparticularthecopperbraids,arelikelyresponsibleforareducedsuspensioneciency.Upto200Hzweobservethetypicalpeak-richshapeofaccelerationnoise,andbetween200Hzand2kHzthereisaportionofthenoisethatfollowsapowerlawthatisclosetothef)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2behaviorthatistypicalforthermalnoise.Itsproximitytoshotnoiseandwhatislikelyaccelerationnoisemakesithardtoidentifywhatkindoffeatureitisexactly,butitistoohightobecausedbyCBNbyafactorofabout20.AtthispointitisclearthatobservingCBNinthiscavityisoutofthequestion,butsinceitappearstoseeaccelerationnoisewecanatleastuseittosenseanincreaseinaccelerationcouplingwhenthesystemissubjectedtocryogeniccooling. 176

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Figure7-20. Testcavityfrequencynoise.Thesensingnoisehasincreasedmassivelybecausetheopticalgainislowerbyafactorof20,andthesystemisnowshotnoiselimitedduetothenecessaryincreaseinlightpowerforthecarrier-sidebandbeat.Itappearsthatthesystemisaccelerationnoiselimited,andthatthein-vacuumcablingandcopperbraidsseverelyshortthesuspensions.ObservingCBNwillbeimpossiblewiththiscavity. 7.5.3FrequencyNoiseofaCryogenicallyCooledTestCavityAlthoughtheopticallossesinthetestcavityprovedprohibitiveofobservingCBN,coolingitwithliquidnitrogenstillservedthreemainpurposes.Firstandforemost,therewastheconcernthatthethreepiececavityisgoingtobecooledfromthecentralpieceoutwards.Amorphousfusedsilicaisapoorconductorofheat(=1:38Wm)]TJ /F5 7.97 Tf 6.58 0 Td[(1K)]TJ /F5 7.97 Tf 6.59 0 Td[(1),andwithathermalexpansioncoecientofFS=5:510)]TJ /F5 7.97 Tf 6.58 0 Td[(7K)]TJ /F5 7.97 Tf 6.58 0 Td[(1itisnotconsideredalow-expansionmaterial.Theheatowthroughtheopticallycontactedsurfacesmayoccuratareducedeciencyand 177

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Figure7-21. Cryogenicallycooledtestcavityfrequencynoise.Below100Hzthelevelofthecavitynoisehasnotchangedsignicantly,whichindicatedthatthecoolingofthecopperbraidsdidnotincreasetheirstinesssignicantly.Moreacceleration-likenoisehasappearedbetween100Hzand1kHz,mostlikelyduetotheboilingnitrogeninthedewar.Theperformanceofthereferencesystemwasassessedduringthecryogenicoperation,andwouldbesucienttoseeCBN. causedierentialexpansionbetweenthepieces,whichcouldleadtoabreakingofthebond.Wecanreportthatnosuchthinghappened.Thetestcavityhasbynowenduredmultiplecoolingcycles(andwarmingup,whichhappensmuchfasterasseeninFigure 7-17 )withoutdisassemblingitself.Forfuturematerialswithbetterthermalconductivitysuchassapphireandsiliconthisisevenlessofaconcern.Thecool-downofthemountandthethermalshieldsdidnotcauseasignicantmisalign-mentofthecavity.Acorrectionofthebeamalignmentwiththein-vacuumpiezo-mirrors 178

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producedonlymarginalimprovementsontheorderof5%atmost.Are-assessmentofthelinewidthshowedachangefrom78.5kHzto90kHz,whichmeansthatthenessedecreasedfrom100,300to87,500,whichreducestheopticalgainfurther,butnotbyordersofmagnitude.Wewereabletolocktothecavity,theresultforwhichisshowninFigure 7-21 .Inadditiontothenormalaccelerationnoise,duringcoldoperationthereistheaddedshakingofthesuspensionframeduetotheboilingnitrogen.Thetank'sisolationfeetcannotsuppressthisnoise.Above100Hzazooofpeakshascomeintoexistence,butbelow100Hzthetrendofthespectrumhasnotchangedmuch.Weinterpretitsuchthatthebraidsdidnotstiensignicantlyduringthecool-down,andthattheincreaseinaccelerationnoiseiscomingtothenitrogenboil-o. 179

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CHAPTER8CONCLUSIONSThepresentedworkconcernedspecicinterferometricaspectsofLISAandLIGO,betweenwhichwewereabletobuildabridgeandprotfromtechnologytransferbetweentheprojects.Wesummarizethethreemaintopicsandtheirindividualnalassessmentandconclusionsinthischapter. 8.1FastStreamingLISAPhasemeterforLIGOBandTheLISAphasemeterapproachwassuccessfullyportedtoacompactdesktopFPGAcardwithA/DandD/Acapability.Auser-friendlygraphicalinterfacewasdeployedthatallowsforaccessofdeepphasemeterfunctionality,andadatastreaminginterfacewasimplemented.AdvancedprototypingofthephasemetercoreincreasedtheinternalbandwidthandwasabletoextendthemeasurementbandwellbeyondLISA-relevantfrequencies,beingmainlylimitedbythequantizationnoiseintheA/DprocessandtoasmalldegreebyADCtimingjitter.ThePM'sabilitytoresolvethedierentialphasenoiseofheterodynesignalsonsplitopticalpathsprovedtobeavaluabletool,anddueitsadditionalusabilityasalasercontrolleritisnowimplementedattheheartoftheUFthermalnoisetestbed. 8.1.1VersatileUsageLISAPMprototypinghasadvancedsignicantlyoverthepastfewyears,andcompletephasemetersolutionshavebeenengineered.ThisdoeshowevernotobsoletetheAX3065project,whichwasneverintendedforprogressiveprototypingofLISAPMhardware,butratherasaninterimmeasurementdevicewithlasercontrolcapability.Unfortunately,itisturningofageandsupportforitsFPGAwilleventuallyceasetoexist.Thegoodnewsisthatthecoreprogrammingisportable,andcanbeimplementedintonewerFPGAs.ThePM'sGUI-basedprogramminginterfaceiseasytolearnevenforpeoplethathaveneverworkedwithFPGAsbefore.Whileitcanbeusedforanykindofheterodynephasemeasurement,itsrealstrengthcomesforthwhenatime-varyingsignalneedstobetrackedwithhighfractionalphaseresolution,asforexamplebeatnotemeasurements.Other 180

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experimentsinourlabuseitasanaccuratedigitallock-inampliertoreliablymeasuretheamplitudesofminutebeatnotes.Itwasforexampleusedinback-reectionbeamoverlapmeasurementsandmaybeplayingakeyroleinanupcomingsingle-photonheterodynedetectionschemeinlightparticlesearchexperiments. 8.2HeterodyneLaserFrequencyStabilizationThealternativeandnovellaserfrequencystabilizationschemeHeterodyneStabilizationwasdevelopedanddemonstratedinaproof-of-principleexperiment.AfterthesuccessfulrsttestweimplementedthefeedbackcontrolsintotheAX3065LISAPManddevelopedadual,indirectlydemodulatingHSschemeforthesimultaneousstabilizationoftwolasersystems.Anadvanceddemonstrationsetupimplementedtheultra-lowCTEcandidatematerialClearCeramZHSandshowedthatitwasabletomeettherequirementswithverysimplethermalinsulation.ThePMwasusedforthelaserfrequencycontrolandthebeatmeasurement,emphasizingtheexperiment'sclosenesstoLISA. 8.2.1ApplicabilitytoLISABecauseLISAisadeep-spacemissionalllocksneedtobeacquiredautomaticallyandmanagedautonomously.ThereforetheprimaryfrequencyactuationofalllaserswillbedirectlycontrolledbyFPGAinput.BecauseallbeatnotesaredigitizedandavailabletothePM,implementingHStogivethelaserstheneededfrequencystabilityonlyrequirestheadditionofacavitytotheopticalbench.Nophasemodulationisneededtoextractthecavityinteractionphase,andthefeedbackcontrolscanbeplacedinthePMlogic.InFigure 8-1 weshowaheavilysimplied,butfullyfunctionalsketchoftheLISAopticalbench.IntheoriginalLISAdesigneachspacecraftisequippedwithanopticalbenchandadedicatedlaserforeachofitstwolaserlinks.Themajorityofthepowerofthelocallaserissentdirectlyouttothefarspacecraft,andamuchsmallerfractionispickedoforthelocalinterferometry.Thebi-directionalback-linkbercarriesthelightfromtherespectiveotherlocallasertotheopticalbenchforheterodyneinterferometricreadouts.VariousPDsrecordthebeatbetweenthedierentlasers,suchastheSC-SCIFO,inwhichtheincoming 181

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Figure8-1. PossibleopticalbenchimplementationofHeterodyneStabilization.Mostoftheavailableprimarylaserpower(about2W)isdirectlysenttothefarspacecraft,whileafraction(severalmW)ispickedoforthelocalinterferometry.Severalinterferometerswithintendedcontingenciesontheopticalbenchobtaintheindividualphaseuctuationdatastreams.Apossibilitytoaddanopticalcavityistousethelow-expansionZerodurbenchitselfandbondaminimalamountofadditionalopticstoit. weaklightisinterferedwiththeprimarylocallaser.TheSC-PMIFOyieldsthepositionreadoutoftheproofmass(PM),andtheREFIFOprovidesareferencephasefortheotherbeatnotes.SincetheopticalbenchismadefromZerodur,onepossibilityinthissimplisticlayoutistosimplyusethesecondoutputoftheREFIFObeamsplitterandaddacavitybybondingadditionalopticstotheZerodurbaseplate.Whilecavitiescouldbeaddedtoeachopticalbench(forcontingency)andthelaserscouldbeindividuallystabilizedwithadualHSschemeoneachspacecraft,thefrequencystabilityofasinglemasterlaserthatisfrequencystabilizedcanalsobetransferredtootherlaserswithphaselocks.InthescopeofDopplershiftcorrections,itispreferabletousethephase-lockmethod,becauseitallowstosteerthelaserfrequencieswiththeosetfrequency.Thephasedelityofweak-lightphase-lockshasbeendemonstrated,thereforetheloweredfrequencynoisecanbedistributedacrossallthelasersviatheinter-spacecraftlinksand 182

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back-linkbers.Thismeansasinglepre-stabilizedlasercanseedthefrequencystabilityforalllasersintheinterferometer,andwedemonstratedthatHSisabletoprovidethenecessarystabilitybothintermsofopticaltechniqueandtechnicalimplementationinLISA. 8.3CryogenicThermalNoiseTestBedBuildingontheaccomplishmentsofthePMandfrequencystabilizationworkwebegantheconstructionofCryoTHOR,aimedattheabilitytomeasuretheCBN-inducedlengthuctuationsinopticalcavitiesatroomandcryogenictargettemperatures.Theobservednoisetermswouldenabletheestimateofthecontributingnoisetermstothesensitivityofafull-scaleGWdetector.Theexperimentwouldthushelpsupportandhopefullyadvancethesearchforbettercoatingstobeusedinnextgenerationobservatories.Alargeportionoftheupfrontworkwasthemechanicalandcryogenicupgradeofthehosttank,andthedesignofathermalpipelinethatcoolsthemirrorsbutleavesthecavityundisturbedinitssuspendedenvironment.ThedesignofareferencesystemwhichfeatureslowerintrinsicnoisethanthemodelpredictionsfortheCBNinthetestcavitiesevolvedoutoftheHSdemonstrationsetup,andweassembledthefullCryoTHORexperimentaroundtheduallaserstabilizationprinciple.WedevelopedacopingstrategyforthelargeFSRsofshortcavities,andwereabletocreateamargininthereferencesystemperformancethatallowsfortheobservationofCBN.TheonlypossibilitywehadtoobtainasamplecavitywastouseleftovercavityopticsoftheoriginalTHORexperiment,whicharenotsuitableforcryogenicsbecausethemechanicallossoffusedsilicasignicantlyrises.Ratherthanobservingnoisecausedbymechanicallosses,thecavitiesturnedouttobeexceptionallyopticallylossyandsusceptibletoaccelera-tionnoise,inwhichanyCBNinuencedrowned.Wetestedthecryogeniccapabilityofthesystemandfoundthatwecanreachtemperaturesnear100Kwiththetestcavities. 8.3.1NecessaryImprovementsFromthecurrentstandpointweidentifytwoseparateimportantstepstobeabletoobserveCBNinatestcavity.Firstly,theexcessaccelerationnoisemustbebroughtunder 183

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control.ThisisacommonthemeinLIGOandalotofknow-howexistsonhowtoconstructappropriatesuspensions,fromwhichtheexperimentwillprot.Wesuggesttore-evaluatetheStacisisolationsystemandpotentiallyndbettersolutionsforthemountingofthetank.Heavy-dutycompressionspringscouldbethebudgetsolution,butonecansimilarlyimagineanequivalentdesigntooatingfeetforlaboratoryopticaltables.Allopticspathsandcomponentswouldthenbeproperlyisolatedandnodubiousfeedbackchannelscouldcauseacouplingofdegreesoffreedomincomplexways.Theinternalsuspensionwithits90cmwirescouldhaveanintermediatestageaddedthatcausesanevensteeperroll-o.Atthesametime,theseismicnoiseduetotheboilingnitrogeninthedewarneedstobelteredbeforeitreachesthetestcavity.Addinganintermediatesuspensionstagewouldprovideamountingpointforthebraidsandimpedetheirtuggingeortswithinertia.Secondly,higherqualitycavityopticsarerequiredforsucientopticalgaintobeatthesensingnoiseintheexperiment.Themirrorsneedtobemadeofsapphire,ratherthanfusedsilica,forwhichtheaccordingspacersandpossiblycoatingrunsneedtobeordered.TheconnectedcostwasprohibitivetoCryoTHORinitsearlystage,whichiswhyweusedfusedsilicaforthersttests.Lastly,itneedstobeguaranteedthatofalltheknownnoisesourcestoaectacavity,CBNwillbethedominatingterm,whichcanforexamplebedonebyvaryingthebeamsizeintheproducedcavities. 184

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APPENDIXAESSENTIALSOFSIGNALPROCESSINGANDNOISESPECTRAAGWdetectorgeneratesadatastreamX(t),inwhichaGWsignalfGW(t)maybepresent,inadditiontotheinherentnoiseitproduces.Thisnoiseisaninterplayofrandomprocessesinvariousstagesandaspectsofthedetector,bothfundamentalandtechnicalinnature,thatwecanallintegrateinacombinedn(t).Wethuswrite X(t)=fGW(t)+n(t);(A{1)andpointoutthat,sincewelackabinitioknowledgeofn(t)duetoitsrandomnature,wedonotknowifaGWsignalispresent,andifthereis,wecannotdirectlydetermineit.Whatwedoknowhowever,isthatintheabsenceofGWsignalsthedetectorproducesonlynoise,suchthatwecancharacterizethebehaviorofn(t)anddetermineboundsfortheexpecteddetectoroutputintheabsenceofGWsignals.IfthereisanunanticipateddetectorresponseinX(t),whichdistinguishesitselfsuf-cientlyfromtheexpectedvariationduetonoise,wetreatitasapotentialGWimprintandattempttoreconstructthetransientsignal.Forlikelycandidatewaveformsf(t)wecandetermineasignal-to-noiseratio(SNR),whichisdenedastheratiobetweenthepowerinf(t)andnf(t)=X(t))]TJ /F3 11.955 Tf 11.95 0 Td[(f(t), SNR=t2Rt1[f(t)]2dt t2Rt1[X(t))]TJ /F3 11.955 Tf 11.96 0 Td[(f(t)]2dt:(A{2)IftheSNRislargeenough,andnoauxiliarydatachannelsvetotheuseofthedatabecauseofareportedunusualdetectorbehavior,weconsidertheeventadetection. A.1LaplaceandFourierTransformationsAcharacteristicexpectationvalueforthenoisepowerisonlyaverycollapsedfromofinformationaboutthedetectorsensitivity.Sincen(t)isacombinationofvariousnoise 185

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mechanismsthatarepresentinthedetector,itisinsightfultostudyitsspectralshape,whichwecandobymeansofLaplaceandFouriertransformation,whicharecloselyrelated.WecanchangetherepresentationofasignalF(t)tothefrequencydomain,whereitischaracterizedbyafrequencydependentcoecientfunctioneF(f).TheFouriertransform(FT)ofF(t)isdenedas eF(f)=1ZF(t)e)]TJ /F5 7.97 Tf 6.59 0 Td[(2iftdt;(A{3)whichcanbeunderstoodastheinnerproductofF(t)withtheelementsofthecontinuousbasisei2ft.Theinversetransformation,whichreconstructsthetimeseriesF(t)fromthepotentiallycomplexspectralinformationeF(f),isthen F(t)=1ZeF(f)e2iftdf:(A{4)Fortheanalysisoflinearsystemsincontroltheorythemoregeneral(bilateral)Laplacetransform(LT)iscommonlyused,whichisdenedas eF(f)=1ZF(t)e)]TJ /F8 7.97 Tf 6.59 0 Td[(stdt;(A{5)whichusesthecomplexvariables=+i!.andthereforereducestotheFTforthechoices=0and!=2f.InastrictmathematicalsensetheFTofasignalonlyexistiftheintegralinEq.( A{3 )converges,whichisnotthecasefornoisetimeseries.However,thereareothermethodstoextractthespectralcharacteristicsofnoise,whichavoidapplyingadirectFT. A.2Wiener-KhinchinTheoremandSpectralDensitiesIfthenoiseisproducedinwhatiscalledawide-sensestationaryprocess,wecanapplytheWiener-Khintchinetheoremandobtainanestimateforthenoisespectrum[ 135 ].ThetheoremstatesthatinthiscasetheautocorrelationfunctionRF()ofF(t),whichisdened 186

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astheconvolutionofthesignalwithitself,parametrizedinatimeoset, RF()=1ZF(t+)F(t)dt;(A{6)istheFTofthenoisespectrumSF(f)ofthefunctionF(t).SF(f)isthepowerspectraldensity(PSD)ofF(t),whichprovidesameasureforhowmuchpoweriscontainedF(t)perunitfrequencyatfrequencyf,andisthusfoundbyevaluating SF(f)=+1ZRF()e)]TJ /F5 7.97 Tf 6.59 0 Td[(2ifd=+1Z1ZF(t+)F(t)dte)]TJ /F5 7.97 Tf 6.59 0 Td[(2ifd:(A{7)TheunitsofthePSDSF(f)arethoseofF(t)squaredperunitbandwidth.Asanexample,thePSDofafrequencyserieswouldbeinunitsofHz2=Hz.Thelinearspectraldensity(LSD)LF(f),denedby LF(f)=p SF(f);(A{8)whichissometimescalledamplitudespectraldensity(ASD),isthesquarerootofthePSD,andthereforecomesinunitsofF(t)perp Hz.ThekeytotheWiener-KhintchinetheoremisthatLF(f)convergestowardseF(f)asmoreandmoredataisbeingtaken,andwegenerallyusetheFTsymbolinsteadoftheLSDthroughoutourdiscussion.Fouriertransformshaveanimportantpropertythatcanbederivedfromthesymmetryofthesineandcosinefunctions.ForthiswereadEuler'sformula ei2ft=cos(2ft)+isin(2ft)(A{9)asthedecompositionofei2ftintoitssymmetric(cosine)andandanti-symmetric(sine)componentsunderthetransformationf!)]TJ /F3 11.955 Tf 24.58 0 Td[(f.Anintegraloverananti-symmetricfunctionvanisheswithinsymmetricintegrationboundaries,andtheproductofasymmetricfunctionwithananti-symmetriconereturnsyetanotheranti-symmetricfunction.ItfollowsthatasymmetricspectralfunctionwitheF(f)=eF()]TJ /F3 11.955 Tf 9.29 0 Td[(f)producesarealinverseFTF(t).Thisargumentisreversible,andsincephysicalobservablesaregenerallyreal-valued,theirFTwill 187

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producesymmetriceF(f)'s.Sincethereisnophysicalmeaningtonegativefrequencies,itiscustomarytocollapsethefullspectrumofftothesinglesidedspectrumthatassumesftorunfrom0to+1bysetting eF+(f>0)=eF(f)+eF()]TJ /F3 11.955 Tf 9.3 0 Td[(f)=2eF(f):(A{10)WeextendthistreatmenttoPSDsandLSDsandwillexlusivelygraphsingle-sidedspectraldensitiestodisplaythecharacteristicspectralshapeofsignals. A.3LinearSystemsTheFourierformalismisusefulforthedescriptionofsignalevolutioninlineartime-in-variantsystems.Oneofitsadvantagesisthatsomenon-trivial,thoughlinearoperationsonfunctionsinthetimedomaindrasticallysimplifyinthefrequencydomain.Therearethreeexamplesofspecialimportanceforthisdiscussion.First,wetakenotethattimederivativesoffunctionsbecomemultiplicationswithanumber, d dtF(t)=1Zi2feF(f)ei2ftdf;(A{11)sincethetimederivativeactsexclusivelyontheexponentialfunction.Similarly,anintegrationresultsinthemultiplicationwiththeinversefactor, tZF()d=1Z1 i2feF(f)ei2ftdf:(A{12)Finally,thetransferfunctionofalinearsystemproducesanoutputY(t)inthetimedomain,whichisalinearsuperpositionofallpastinputsX(t), Y(t)=+1Z(t)]TJ /F3 11.955 Tf 11.96 0 Td[()X()d:(A{13)AnyphysicalsystemcannothaveY(t)dependonfuturevaluesofX(t),andthereforetheresponsefunction()mustreturnzerofor<0.Eq.( A{13 )ismathematicallyknownasaconvolutionoperation,andinFourierspaceitbecomesamultiplicationoftherespective 188

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individualFTs, eY(f)=e(f)eX(f):(A{14)Thisfactisextremelyimportantfortheanalysisanddescriptionoflinearsystems. A.4FrequencyandPhaseNoiseThroughoutthisdissertationthenoiseinfrequencyandphasetimeseriesisdiscussed,andsometimestheyareusedinterchangeably.Thisisbecausethereisnolossofinformationifwechoosetokeeptrackofthesignalfrequency(t)ratherthanitsphase(t),saveforaninitialphasethathasnoeectonsubsequentphasevariations.Bydenitionitis (t)=1 2d dt(t);(A{15)andwiththeinformationweobtainedinSection A.3 ,weknowthatthedierentiationbecomesafactor2if.Forthereal-valuedLSDsofbothphaseandfrequencynoisethisturnsinto e(f)=fe(f);(A{16)whichmeansthatthenoisespectrumobtainedfrom(t)isequivalenttothatobtainedfrom(t),shapedbyafactorf. 189

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APPENDIXBGAUSSIANBEAMSThederivationofthewaveequationforthepropagationofelectromagneticwavesthroughvacuumisatypicalapplicationofMaxwell'sequationsinundergraduatephysicscourses.Fortheelectriceld~E(t;x;y;z)ittakestheform )]TJ /F1 11.955 Tf 12.46 8.09 Td[(1 c2@2 @t2+@2 @x2+@2 @y2+@2 @z2~E(t;x;y;z)=0:(B{1)Planewavesei(!t)]TJ /F8 7.97 Tf 6.17 2.1 Td[(~k_~x)withj~kj=!=crepresentacompletebasisforsolutionsofEq.( B{1 ),suchthatageneralcoherentsolutioncanbeconstructedasthelinearcombination ~E(t;x;y;z)=XiZ~eiAi(~k)ei(!t)]TJ /F8 7.97 Tf 6.16 2.11 Td[(~k~x)d3k(B{2)withcoecientfunctionsAi(~k)forthetwopolarizations~ei.ArbitrarywavepacketsandspatialamplitudedistributionscanbedescribedwithEq.( B{2 ),howevermostinterferometricapplications{GWinterferometryincluded{utilizecontinuouswave(cw)lasers,whichprovidecoherent,narrowlybeamedmonochromaticlight. B.1HelmholtzEquationandParaxialApproximationTostudythephaseandamplitudedynamicsofacwlasereld,weassumeastationaryplanewavesolutionofangularfrequency!withwavevector~k,andgiveitatime-independentamplitude-phaseproleA(x;y;z), E(t;x;y;z)=A(x;y;z)expn)]TJ /F3 11.955 Tf 9.29 0 Td[(i~k~x)]TJ /F3 11.955 Tf 11.96 0 Td[(!to:(B{3)ForthistesteldEq.( B{1 )turnsintotheHelmholtzequation )]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2+k2hA(x;y;z)expn)]TJ /F3 11.955 Tf 9.3 0 Td[(i~k~xoi=0:(B{4)Wechoosezasthepropagationdirection,andfurtherassumethatA(x;y;z)variesonlyslowlywithzcomparedtoexpf)]TJ /F3 11.955 Tf 15.27 0 Td[(ikzg.Thisparaxialapproximationneglectsthe@2zAterm 190

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intheresultingpartialdierentialequation,whichbecomes @2xA+@2yA)]TJ /F1 11.955 Tf 11.95 0 Td[(2ik@zA=0;(B{5)andshowsthatAseparatesmultiplicativelyintoitsx,y,andzdependencies.BecauseofitssymmetryinxandyitiscustomarytoassumerotationalsymmetryandparametrizeAinthedistancer=p x2+y2fromthezaxis.Thetrialfunction A(r;z)=A0exp)]TJ /F3 11.955 Tf 9.3 0 Td[(ip(z))]TJ /F3 11.955 Tf 11.96 0 Td[(ikr2 2q(z)(B{6)resultsinasystemoftwodierentialequationsinzforq(z)andp(z) d dzq(z)=1andd dzp(z)=)]TJ /F3 11.955 Tf 18.85 8.09 Td[(i q(z):(B{7)Thesolutionsaregivenby q(z)=z+izRandp(z)=)]TJ /F3 11.955 Tf 9.3 0 Td[(iln1)]TJ /F3 11.955 Tf 11.96 0 Td[(iz zR=)]TJ /F3 11.955 Tf 9.3 0 Td[(iln)]TJ /F3 11.955 Tf 9.3 0 Td[(iq(z) zR;(B{8)wheretheRayleighrangezRemergesasanintegrationconstantanddenesacharacteristiclengthscaleforA.Since expf)]TJ /F3 11.955 Tf 15.27 0 Td[(ip(z)g=exp)]TJ /F1 11.955 Tf 11.29 0 Td[(ln)]TJ /F3 11.955 Tf 9.3 0 Td[(iq(z) zR=izR q(z);(B{9)allpropagationcharacteristicsarecontainedinthecomplexbeamparameterq(z).Substitu-tionintoEq.( B{6 )yields A(r;z)=A0zRi q(z)exp)]TJ /F3 11.955 Tf 18.84 8.09 Td[(i q(z)kr2 2:(B{10)Severalquantitieswithsignicanceforbeamcharacteristicscanbederivedfromq(z),forwhichwewritei q(z)=i z+izR=zR z2+zR2+iz z2+zR2=1 p z2+zR2expitan)]TJ /F5 7.97 Tf 6.58 0 Td[(1(z=zR); (B{11) 191

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FigureB-1. Gaussianbeamparameters.Thebeamradiusw(z)isnarrowestatthewaistwithavalueofw0.TheROCR(z)ofthewavefrontdistinguishestheGaussianbeamfromaplanewave,andtheGuoyphase(z)indicatestheirrelativephasedeviation.WithintheRayleighrangezRtotheleftandrightofthewaistthisphasepickupoftheGaussianbeamequals=2. andusethattowritethefullresultforE(x;y;z)fromEq.( B{3 )as E=A0 p z2+zR2exp)]TJ /F3 11.955 Tf 10.5 8.08 Td[(kzR 2r2 z2+zR2)]TJ /F3 11.955 Tf 11.95 0 Td[(ikz 2r2 z2+zR2)]TJ /F1 11.955 Tf 11.95 0 Td[(tan)]TJ /F5 7.97 Tf 6.59 0 Td[(1z zR)]TJ /F3 11.955 Tf 11.96 0 Td[(kz)]TJ /F3 11.955 Tf 11.96 0 Td[(!t:(B{12)Basedonthisequationwecandenethreesecondarybeamparameterswithintuitivephysicalinterpretations. B.2GaussianBeamParametersIntroducingthequantitiesw0,w(z),R(z),and(z),whosegeometricinterpretationisillustratedinFigure B-1 ,Eq.( B{12 )isrewrittenas E=E0w0 w(z)exp)]TJ /F3 11.955 Tf 19.58 8.09 Td[(r2 w(z)2)]TJ /F3 11.955 Tf 11.95 0 Td[(ir2 R(z)2)]TJ /F3 11.955 Tf 11.96 0 Td[((z))]TJ /F3 11.955 Tf 11.96 0 Td[(kz)]TJ /F3 11.955 Tf 11.95 0 Td[(!t:(B{13)Thetransverseeldmagnitudedistributionisdeterminedonlybythersttermintheexponentialfunction,whichgivesitaradiallysymmetricGaussianproleatanypointz. 192

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Thebeamradius w(z)=w0s 1+z2 z2R;(B{14)isthe1=etransverseexpanseoftheeldamplitudeatlocationz,andcorrespondinglythe1=e2characteristicradiusoftheintensitydistribution.Forz=0itbecomesminimal,withavaluegivenbythewaist w0=r 2zR k:(B{15)Theradiusofcurvature(ROC)ofthewavefrontispositiondependentanddenedas R(z)=z1+z2R z2:(B{16)Whenevaluatingtheeldo-axis,forexampleforspatialoverlapcalculationsbetweendierentmodes,R(z)isthephaseretardationwithrespecttoatwavefronts.Lastly,theGuoyphase (z)=arctanz zR:(B{17)isanon-axiscorrectivetermtothepropagationphasekzoftheplanewave.WithinthedistancezRonbothsidesofthewaistthisadditionalphasepick-upis=2,halfofthephasedierenceofbetweenz!andz!+1.TheGuoyphaseisresponsibleforbreakingthehigherordermodedegeneracyinopticalcavities. B.3Hermite-GaussModesThetrialA(x;y;z)thatledtoEq.( B{12 )isnotauniquesolutiontoEq.( B{5 ).NotonlydodierentvaluesforzRrepresentdierentsolutions,evenforagivenzRthereexistsawholefamilyofsolutions.Forexample,wecantakeastepbacktoEq.( B{4 )andbreaktheradialsymmetrybyfurthermodulatingA(x;y;z)withafunctionHasin E(x;y;z)=E0w0 w(z)H p 2x w(z)!exp)]TJ /F3 11.955 Tf 9.3 0 Td[(ik(x2+y2) 2q(z)expn)]TJ /F3 11.955 Tf 9.3 0 Td[(ikz)]TJ /F1 11.955 Tf 11.96 0 Td[((1+=2)(z)o;(B{18) 193

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FigureB-2. HermiteGaussmodes.ThenomenclaturefortheorthogonalmodesusesthenumberofnodesofthespatialdistributionintheTransverse-Electro-Magnetic(TEM)modes.Toprow,lefttoright:TEM-00,TEM-10,TEM-01.Bottomrow,lefttoright:TEM-20,TEM-11,TEM-02. whereweaddedaparameterwhichappearsintheresultingdierentialequationforH @2uH(u))]TJ /F1 11.955 Tf 11.95 0 Td[(2u@uH+H=0:(B{19)ThisequationisknownasHermite'sdierentialequation,andhasbeenextensivelystudiedintheliterature.SolutionsHnexistif=2nfornon-negativeintegersnandareoftheform Hn(x)=()]TJ /F1 11.955 Tf 9.3 0 Td[(1)neu2dn dune)]TJ /F8 7.97 Tf 6.58 0 Td[(u2:(B{20) 194

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WecanrepeatthistreatmentforydependencyandlearnthatE(x;y;z)=E0w0 w(z)Hm p 2x w(z)!Hn p 2y w(z)!::::::exp)]TJ /F3 11.955 Tf 9.29 0 Td[(ik(x2+y2) 2q(z)expn)]TJ /F3 11.955 Tf 9.3 0 Td[(ikz)]TJ /F1 11.955 Tf 11.96 0 Td[((1+m+n)(z)o: (B{21)equallysolvesEq.( B{5 )foranycombinationofm,n.ThesearecalledtheHermite-Gaussmodesofthebeam,ofwhichthe(0,0)-modeiscalledthefundamentalGaussmodeandcoincideswiththeoriginalsolutionfromEq.( B{12 ).ItisimportanttopointoutthathigherordermodesaccumulatemoreGuoyphasethanthefundamentalmodebyafactor1+m+n. B.4MatrixFormalismforGaussianBeamsForagivenq(z)wecanimmediatelycalculatew(z),R(z),and(z).ThismeansthatwheninvestigatinghowdispersiveopticalelementsactonGaussianbeams,weonlyneedtondinwhichwaytheymanipulateq(z).Asanexample,thefreespacepropagationoveradistancedwilltakeaninitialq1to q2=q1+d:(B{22)IncomparisonthepropagationmatrixintheABCDformalismofgeometricopticsforatranslationbydis Td=0B@ABCD1CA=0B@1d011CA:(B{23)Ifweutilizeitscoecientsanddeneq2intermsofq1as q2=q1(ABCD)=Aq1+B Cq1+D;(B{24)theninsertingTdcorrectlyyieldsEq.( B{22 ).ThesameistrueforotherABCDmatrices,forexamplethatofthethinlenswithfocallengthf Lf=0B@ABCD1CA=0B@10)]TJ /F5 7.97 Tf 10.85 4.71 Td[(1 f11CA;(B{25) 195

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whichalsodescribesamirrorwithradiusofcurvatureR=2f.TheeectofasequenceofABCD-operationscanbeevaluatedbymatrixmultiplicationsofthecorrespondingABCD-ma-trices,whichsignicantlysimpliestheanalysisofcomplexopticalsystems 196

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APPENDIXCOPTICALCAVITIESWhenaclosedpathexistsconnectinganarrangementofmirrorsthatreectaspatiallasermodeintoitself,theyformanopticalcavity.Lightcanenterthecavitythroughamirrorwithnitetransmissivity,andonceinside,willcontinuetotravelthroughthecavity,graduallylosingpowertoeveryreectiononmirrors,andabsorptionanddiractioneects.WewillseethatopticalcavitiesactasspatialltersforGaussianbeams,bothfortransverseandlongitudinalmodes.Werestrictourdiscussiontothesimplestscenario,acavityformedbytwomirrorsthatfaceeachotherasshowninFig. C-1 .Themathematicaltreatmentformorecomplicatedgeometriesproceedscompletelyanalogicallytothepresentedcase.Werstanalyzethestabilityofsuchamirrorarrangement,andthenformulatetheopticaltransferfunctionofthegenerallinearcavity. C.1StabilityCriterionWeapplytheABCDformalismtondtheround-tripeectonthebeamparameterq.FollowingtheillustrationinFig. C-1 ,wedenotethemirrorseparationwithLandattributeradiiofcurvatureR1andR2tothetwomirrors.UsingEqs.( B{23 )and( B{25 )wendthattheround-tripABCD-matrixM1forq1atthefrontmirrorisgivenby M1=0B@ABCD1CA=0B@1)]TJ /F1 11.955 Tf 11.96 0 Td[(2L R22L1)]TJ /F8 7.97 Tf 15.53 4.7 Td[(L R2)]TJ /F5 7.97 Tf 13.64 4.71 Td[(2 R1)]TJ /F5 7.97 Tf 16.3 4.71 Td[(2 R21)]TJ /F1 11.955 Tf 11.95 0 Td[(2L R1)]TJ /F1 11.955 Tf 9.29 0 Td[(2L R1+1)]TJ /F1 11.955 Tf 11.96 0 Td[(2L R11)]TJ /F1 11.955 Tf 11.96 0 Td[(2L R21CA:(C{1)TherequirementthatMisstablerequiresthatthereareq1whichareprojectedintothemselvesbyMinEq.( B{24 ),sowewrite q1=q(M)=Aq1+B Cq1+D:(C{2)Wecanrewritethisequationas B1 q21+(A)]TJ /F3 11.955 Tf 11.95 0 Td[(D)1 q1)]TJ /F3 11.955 Tf 11.96 0 Td[(C=0;(C{3) 197

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FigureC-1. Twomirrorresonator.Forthecavitytobeopticallystable,itmustbepossibletoreectalasermodeintoitselfuponaround-tripthroughthecavity. whichisaquadraticformin1=q1withsolutions 1 q1=)]TJ /F1 11.955 Tf 9.3 0 Td[((A)]TJ /F3 11.955 Tf 11.96 0 Td[(D)p (A)]TJ /F3 11.955 Tf 11.96 0 Td[(D)2+4BC 2B;(C{4)fromwhichallotherbeamparameterscanthenbederived.Beforewesolvethisequation,welookbackatEqs.( B{14 )and( B{15 ),andseethatanitevalueforwrequireszR6=0.InEqs.( B{8 )and( B{11 )itisapparentthatthisinturnrequires=f1=qg6=0,andthereforethattheterminthesquarerootisnegative, (A)]TJ /F3 11.955 Tf 11.96 0 Td[(D)2 4+BC<0:(C{5)ForstablesolutionsqmaynotgrowinniteorvanishforarbitrarymanyiterationsofM,consequentlyitmustbedet(M)=AD)]TJ /F3 11.955 Tf 11.95 0 Td[(BC=1.ThissimpliesEq.( C{5 )to (A+D)2<4;(C{6)whichisequivalentto 0
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andwecaninsertAandDfromEq.( C{1 )tondthestabilitycriterion 0<1)]TJ /F3 11.955 Tf 16.6 8.09 Td[(L R11)]TJ /F3 11.955 Tf 16.6 8.09 Td[(L R2<1:(C{8)Wedenetheindividualstabilityg-factorspermirroras gi=1)]TJ /F3 11.955 Tf 15.92 8.08 Td[(L Ri:(C{9)Theoverallcavityg-factoristheproductofallindividualgi,andthestabilitycriterionbecomes 0
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inwhichwechoosethenegativebranchforapositivezR,andsolveforq1, q1=L1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1+ip g(1)]TJ /F3 11.955 Tf 11.95 0 Td[(g)=g2 (1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1)2+g(1)]TJ /F3 11.955 Tf 11.95 0 Td[(g)=g22:(C{14)Weidentify z1=L1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1 (1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1)2+g(1)]TJ /F3 11.955 Tf 11.95 0 Td[(g)=g22andzR=p g(1)]TJ /F3 11.955 Tf 11.96 0 Td[(g)=g2 (1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1)2+g(1)]TJ /F3 11.955 Tf 11.95 0 Td[(g)=g22(C{15)asthedistanceofthewaistfromthefrontmirrorandtheRayleighrangeofthesupportedGaussmode,respectively.WeseethatfromR1!1followsthatz1!0,whichisconsistentwithourpreviousthoughts.Theroundtripphasegainistheback-and-forthpropagationphase2kL,withk=2=c,plusthespatialmode-dependentgainofGuoyphase m;n=[m+n+1]2G=[m+n+1]2[(z2))]TJ /F3 11.955 Tf 11.95 0 Td[((z1)]:(C{16)Neglectingdiractionlossesfromthenitegeometryofthecavity,theround-tripcoecientrtfortheintra-cavityeldisthus rt=()]TJ /F3 11.955 Tf 9.3 0 Td[(r1)()]TJ /F3 11.955 Tf 9.3 0 Td[(r2)expfi2kL+im;ng=r1r2expfi2kL+im;ng:(C{17)Combiningtheaboveinformation,theintra-cavityeldEcavatmirror1,whereweassumeEintobeincident,isasuperpositionofinnitelymanyelds En=nrt(it1Ein);(C{18)whichresultsfromtheinterferenceofallcontributionsafterncavityroundtripsoftheleakedeldit1Ein.Weobtain Ecav=1Xn=0En=it11Xn=0nrtEin=it1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2expfi2kL+im;ngEina(C{19)andseethatwithexpfi2kL+iGgithasatermthatisperiodicin. 200

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wewritethecavitycouplingtransferfunctionFc()as Fc()=it1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2exp2i FSR+im;n:(C{20)ThereectivetransferfunctionFr()thatgivesthereectedeldEr=Fr()Einsuperimposesthedirectlyback-reectedeldwiththeeldthatleaksoutofthecavityatmirror1,andreads Fr()=r1)]TJ /F3 11.955 Tf 11.95 0 Td[(r2exp2i FSR+im;n 1)]TJ /F3 11.955 Tf 11.95 0 Td[(r1r2exp2i FSR+im;n:(C{21)ThetransmissivetransferfunctionEt=Ft()Einforthetransmittedeldisgivenby Ft()=)]TJ /F3 11.955 Tf 18.85 8.85 Td[(t1t2expi FSR+im;n=2 1)]TJ /F3 11.955 Tf 11.96 0 Td[(r1r2exp2i FSR+im;n:(C{22)ThefunctionsFc()andFt()exhibitmaximaif FSR+[m+n+1]G 2=N(C{23)forsomepositiveintegerN,whileFr()isataminimum.Wecallm;nforwhichEq.( C{23 )holdsresonancefrequenciesofthecavity. 201

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BIOGRAPHICALSKETCHJohanneswasborninBerlin-MarzahnintheformerGermanDemocraticRepublicandmovedtoLangenhagen,Germanywhenhewasfour.AftertheeighthgradehetransferredfromhishometownhighschoolGymnasiumLangenhagentotheJugenddorf-Christophorus-schuleboardingschoolinBraunschweig,Germany,fromwherehegraduatedin2003withtheGermanAbitur.HespentayearofsocialserviceworkingattheParacelsusklinikamSilberseehospitalandenrolledattheLeibnizUniversityofHannoverinageneralphysicsDiploma(Master's)programin2004.Afterhisrsttwoyearshestartedworkingasastudentteachingassistantforundergraduatecourses,andin2008hejoinedtheLaserInterferometerSpaceAntenna(LISA)projectgroupattheMax-PlanckInstituteforGravitationalPhysics(Albert-Einstein-Institute)inHannoverforhisDiplomathesisresearch,whichfocusedontheimplementationofweak-lightopticalranginginatable-topexperiment.HegraduatedfromtheLeibnizUniversityofHannoverwithdistinctionandjoinedthegroupofDr.GuidoMuellerattheUniversityofFloridaasaPhDstudent,whereherstfocusedfurtheronLISAresearch,buteventuallytransitionedintoaLIGOposition. 212