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A Low Loss Faraday Isolator for Squeezing Injection in Advanced LIGO, and Radio-Frequency Amplitude Modulation

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Title:
A Low Loss Faraday Isolator for Squeezing Injection in Advanced LIGO, and Radio-Frequency Amplitude Modulation
Creator:
Goetz, Ryan Michael
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (187 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
TANNER,DAVID B
Committee Co-Chair:
MUELLER,GUIDO
Committee Members:
LEE,YOONSEOK
SIKIVIE,PIERRE
CONKLIN,JOHN

Subjects

Subjects / Keywords:
faraday -- isolator -- laser -- ligo -- optics
Physics -- Dissertations, Academic -- UF
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:
Advanced LIGO is currently in the middle of its second science observation run, and operating with strain sensitivity below $10^{-23} \mbox{ Hz}^{-1/2}$ over an observation band of 30 Hz to 1000 Hz. To date, the LIGO Collaboration has published on three separate solar-mass binary black hole mergers, all directly observed by the Advanced LIGO detectors. These detections are the first direct evidence of gravitational radiation as predicted by Einstein's Theory of General Relativity (GR), considered by many to be the final major prediction of GR \citep{ligooverview2009}. Additionally, these three merger events are the first evidence of the existence of solar-mass black holes above $20 M_\odot$, and provide a novel probe into astrophysical processes that cannot be observed with traditional electro-magnetic telescopes. The work described in this dissertation is motivated by the effort to improve the sensitivity of the LIGO telescopes thereby increasing the volume of spacetime that the detectors can see. As a means of surpassing the standard quantum limit, squeezed vacuum states will be injected into the output port of the Advanced LIGO detectors. The effectiveness of squeezing in reducing quantum noise in the interferometer is strongly dependent on the optical losses in the injection path of the squeezed beam, and will require the development of a Low Loss Faraday Isolator (LLFI). The majority of this dissertation is devoted to the design, construction, and performance of the LLFI at the University of Florida. Chapter \ref{chap:RFAM} discusses efforts by the author to trace the impact of radio-frequency modulation of the laser field amplitude in the LIGO Livingston detector (LLO). Chapter \ref{chap:futurework} outlines measures for future work to improve the performance of the LLFI. ( 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, 2017.
Local:
Adviser: TANNER,DAVID B.
Local:
Co-adviser: MUELLER,GUIDO.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2018-06-30
Statement of Responsibility:
by Ryan Michael Goetz.

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UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
6/30/2018
Classification:
LD1780 2017 ( lcc )

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ALOWLOSSFARADAYISOLATORFORSQUEEZINGINJECTIONINADVANCEDLIGO,ANDRADIO-FREQUENCYAMPLITUDEMODULATIONByRYANGOETZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017RyanGoetz

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ForthengersIsacricedtothemagnetgods.

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ACKNOWLEDGMENTSIwouldrstliketoacknowledgethecontributionsofmyadvisorsDavidTannerandGuidoMuellertothisthesiswork.Withouttheirpatience,support,andguidanceIwouldbelostinthebevyofinformationthatistheLIGOexperiment.IowemuchthankstoPaulFuldaforhishelpwithagreatmanythings,includingbutnotlimitedtoteachingmeaboutFinessemodelingsoftwareandtroubleshootingbugsinmycode,introducingmetoPDHservos,andproofreadingpapers.IowethankstoRichardOttensforsharinghisexpertiseincryogenicsandallthingsvacuum.IwillreturnyourGEvarnishsomeday.ToGiacomoCianiforhishelpwithComsol,whichwascentraltotheLLFImagnetdesign,aswellasallofthequestionshewouldaskatpresentationsandgroupmeetings.ToRodicaMartinforbeingmyrstmentoratUFandintroducingmetotheIFIandcleanroomprocedures.Herextensiveknowledgeofouropticsandlabhistorywasinvaluable.ToJayHortonforhishelpoutttingtheLISALabforourpulsetubecryocooler,andallthetoolsI'veborrowedfromhim.IamgratefultoKentaroSomiya,forprovidingmetheopportunitytospendtimeworkingatKAGRA,aswellasYuuKataoka,KazushiroYano,MasayukiNakano,andChrisMuellerfortheirhelpwiththeworkonKAGRA'sIFI.SpecialthanksareduetotheUFMachineShop,especiallyMarcLink,BillMalphurs,andEdStorchfortheirrecommendations,explanations,andpatienceduringmycountlesstripstoshop.Edsacricedagreatdealoftimeandspacetohelpmeassemblemagnetdisks;withouthimIwouldstillbewrestlingwithmagnetwedges.Onapersonallevel,IwouldliketothankPatrickGettinoforgivingmeaplacetolivewhenIwasindirestraits.BobbyBondforswimmingandclimbingwithme,drivingmeeverywhere,andhisenthusiasmfortrivia.MichaelHartmanforhisfrustratinglysuccessfulHalloweencostumesandthemanyhundredsofWikipediasearchestosettlearguments. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1TheGravitationalUniverse .......................... 15 1.2LaserInterferometryforGravitationalWaveDetection ........... 19 1.2.1LIGO ................................... 19 1.2.2GravitationalWaveDetectorNetwork ................. 20 1.3ImprovingGravitational-WaveDetectorCapabilities ............. 20 2SQUEEZEDLIGHTFORADVANCEDDETECTORS .............. 23 2.1NoiseinAdvancedLIGO ............................ 23 2.1.1ClassicalNoise .............................. 23 2.1.2QuantumNoise ............................. 24 2.2SqueezedLight ................................. 28 2.3ImplementationinALIGO ........................... 32 3LOWLOSSFARADAYISOLATORDESIGN ................... 36 3.1LightPropagationinOpticalMaterials .................... 36 3.2TheFaradayEect ............................... 40 3.3OpticalPrinciplesofaFaradayIsolator .................... 46 3.3.1Time-ReversalSymmetry ........................ 46 3.3.2WedgePolarizers ............................ 50 3.4CurrentInputFaradayIsolatorDesign .................... 52 3.5OpticalRedesignforLowLossFaradayIsolator ............... 54 3.6MagnetDesignforLowLossFaradayIsolator ................ 57 3.7LossBudget ................................... 60 3.7.1LossToOpticalElements ........................ 60 3.7.2MisalignmentLosses .......................... 61 3.7.3InhomogeneityLosses .......................... 70 3.7.4EtalonEectsinOpticalElements ................... 71 3.7.5UninvestigatedLosses .......................... 76 5

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4LOWLOSSFARADAYCONSTRUCTIONANDPERFORMANCE ...... 78 4.1OpticalComponentTesting .......................... 78 4.2MagnetAssembly ................................ 79 4.2.1DiskAssembly .............................. 80 4.2.2DiskLiberationandFieldTests .................... 85 4.2.3FullRotatorAssembly ......................... 86 4.3IsolatorAssembly ................................ 98 4.4OpticalLossTests ............................... 100 5LASERAMPLITUDEMODULATIONINLIGO ................. 105 5.1RadioFrequencyAmplitudeModulation ................... 105 5.1.1PhaseModulation ............................ 105 5.1.2AmplitudeModulation ......................... 109 5.2RFAMintheLIGOInterferometer ...................... 111 5.2.1RFAMMonitorInstallationandCalibration ............. 111 5.2.2RFAMMonitoring ............................ 118 5.2.3FINESSESimulations .......................... 120 6FUTUREWORK ................................... 124 6.1ElectromagneticFieldTuningfortheFaradayRotator ........... 124 6.2MonolithicFaradayIsolator .......................... 127 6.2.1IndexMatching ............................. 128 6.2.2CoherentCancellation ......................... 130 7CONCLUSION .................................... 135 7.1LowLossFaradayIsolator ........................... 135 7.2Radio-FrequencyAmplitudeModulation ................... 136 APPENDIX ACRYOGENICINVESTIGATIONSOFBOSEMLEDS .............. 138 A.1CryogenicsforFutureGravitationalWaveDetectors ............. 138 A.2CryogenicLight-EmittingDiodePerformance ................ 138 BJONESANDMUELLERCALCULUSFORBEAMPROPAGATION ...... 143 B.1JonesMatrices ................................. 143 B.2StokesParameters ............................... 144 B.3MuellerMatrices ................................ 145 CMAGNETICFIELDCALCULATIONSFORREALSOLENOIDS ........ 147 C.1AnalyticApproach ............................... 147 C.1.1SingleCoil ................................ 147 6

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C.1.2MultipleConcentricCoils ........................ 148 C.1.3CoaxialCoilDisks ............................ 149 C.2ComputerModeling ............................... 150 DFINESSEMODELOFTHEALIGOINTERFEROMETER ........... 151 EIMCDITHERLOCKINGFORRFAMCOMPENSATION ............ 176 REFERENCES ....................................... 182 BIOGRAPHICALSKETCH ................................ 187 7

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LISTOFTABLES Table page 3-1ExpectedopticallossofeachcomponentintheUFLLFIdesign. ......... 61 4-1RevisedopticallossofeachcomponentinthecurrentUFLLFIdesignaftermeasuringreectancesofeachoptic. .............................. 79 5-1RFsignalsforOctober19,2015observation. ................... 119 8

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LISTOFFIGURES Figure page 1-1Adiagramoftheeectsofapassinggravitationalwaveonaringoftestparticles 19 1-2AdiagramofthemajoropticalcomponentsoftheAdvancedLIGOinterferometers 21 2-1AnnominalstrainsensitivitycurveforAdvancedLIGOoperatingat125Wofinputpower.Quantumuctuationsintheformofshotnoiseandradiationpressurenoiseareexpectedtolimitthedetectorthroughouttheobservationband.ProducedwiththeGravitationalWaveInterferometerNoiseCalculator(GWINC)(1) ..................................... 29 2-2Acartoonofthequadraturerepresentationofanunsqueezed(left)andsqueezed(right)coherentstate ................................. 32 2-3Theoptimalsqueezeangleforadual-recycledFabry-PerotenhancedMichelsoninterferometeroperatingatPSQL .......................... 33 2-4Aplotofthereducedsqueezefactorasafunctionoftheopticalthroughputofthesqueezinginjectionpath ............................. 34 2-5Adiagramofthetentativelayoutforsqueezinginjection ............. 35 3-1Anillustrationoftheeectofahalf-waveplate(HWP)onthepolarizationofanincidenteld .................................... 46 3-2AnillustrationofthebasicprinciplebehindaFaradayrotator .......... 49 3-3Aray-tracingdiagramforawedgegeometry .................... 51 3-4AconceptualdiagramofthefunctionofaFaradayisolator ............ 53 3-5AconceptualdrawingofthecurrentaLIGOInputFaradayIsolator ....... 54 3-6TheKAGRAInputFaradayIsolatorasitisbeingconstructedinthepre-stabilizedlaser(PSL)cleanroom ................................ 55 3-7AconceptualdrawingoftheLLFIopticallayout .................. 56 3-8Theunfortunatefateofacalcitecrystaldroppedintheprocessofinstallation 56 3-9ACOMSOLmodelofamagnetdisk ........................ 58 3-10OutputsofCOMSOLsimulationsforradiallyandaxiallymagnetizeddisks ... 59 3-11Acartoondepictingthepolarizationorientationsofeachmagnetregion ..... 61 3-12Aplotoftheregionofparameterspacethatisallowedbyapowermeterwith1%measurementuncertaintyinthebalancedTGGalignmentscheme ...... 67 9

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3-13Plotsofthealignmentsignalrelativepowerandassociatedlossfortheretro-reectionTGGalignmentscheme ............................... 69 3-14Thedistributionofsingle-passlossduetomisalignmentsofopticsintheLLFI 70 3-15Theaxialeldmagnitudewithinthemagnethousingasafunctionofdistancealongtheopticalaxisforapotentialmagnetconguration ............ 72 3-16Theworst-casescenarioopticallossfrometaloneectsinanoptic,asafunctionofthereectanceofeachsurface ........................... 73 3-17AplotoftherelativepowercontributedtothereectedbeambytheetaloneectinourTGGasafunctionoftheparallelismofthecrystal ............. 75 3-18Aplotofthephasedierencebetweenpromptandsecondaryreectionsasafunctionofthetiltoftheoptic ........................... 77 4-1Apictureofthesetupusedtotakereectancemeasurementsoftheindividualopticalcomponents .................................. 80 4-2MeasuredreectancesfortheTGGandKTPcrystalsusedintheLLFI ..... 80 4-3Theassemblyofamagnetdisk ........................... 84 4-4Theprocessofremovingacompletedmagnetdiskfromthealuminumassemblyblock .......................................... 86 4-5AnassembledmagnetdiskfortheFaradayrotator ................ 87 4-6Measuringthemagneticeldproleofafullyassembledmagnetdisk ...... 87 4-7AcomparisonofthemeasuredaxialeldprolesoftheassembleddiskswiththeinterpolatedprolesfromCOMSOLsimulations ................ 88 4-8Aninitialattemptatjoiningthethreecentral-mostringstocreateamonolithicmagnet ........................................ 89 4-9Anassemblyforsecuringthewedgesofthecompletedcentraldisk ........ 90 4-10Asecond,saferattemptatjoiningthethreecentral-mostringstocreateamonolithicmagnet ........................................ 91 4-11Theprocessbywhichtherstmagnetdiskisplacedinsidethedrum ...... 93 4-12Theprocessbywhichthesecondmagnetdiskisplacedinsidethedrum ..... 94 4-13Theprocessbywhichtheremainingthreemagnetdisksareplacedinsidethedrum(part1) ..................................... 95 4-14Theprocessbywhichtheremainingthreemagnetdisksareplacedinsidethedrum(part2) ..................................... 96 10

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4-15Theprocessbywhichthedrumiscappedandsecured .............. 97 4-16Atestofthemagneticeldproleforthefullyassembledmagnetdrum ..... 98 4-17AplotoftheinducedpolarizationrotationintheTGGasafunctionofthepositionoftheTGGwithinthemagnetdrum ........................ 99 4-18TheassemblyprocedurefortheLLFI ........................ 101 4-19AbasicschematicoftheopticallosstestingfortheLLFI ............. 104 5-1Acartoonspectrumoffrequencymodulatedlight ................. 106 5-2Aphasordiagramforphase-modulatedlight .................... 107 5-3AsimpliedcartoonoftheLIGOlengthcontrolscheme .............. 109 5-4Aphasordiagramillustratingaparticularkindofamplitudemodulation .... 110 5-5AphotographofthefastphotodiodeonIOT2Lwiththebeampathillustrated 112 5-6AcartoonofourRFAMdetectionscheme ..................... 113 5-7TheresponseoftheRFPDtoRFAMinducedbydetuningtheIMC ....... 115 5-8Aowchartdiagramillustratingtheconnectionsbetweenourobservedandmodeledquantities ....................................... 116 5-9ThesimulatedresponsesofidealphotodetectorstothedetuningoftheIMC .. 117 5-10The(very)approximatecurvetstotheresponsesofIMC-T,SQ9,andSQ45totheIMCdetuning ................................. 118 5-11AnamplitudespectraldensityoftheamplitudemodulationindexintheL1detector 120 5-12SimulatedcouplingofRFAMtoDARMasafunctionofthephaseoftheamplitudemodulationrelativetothephasemodulationofthecarrier ............ 123 5-13RFAMcontributiontothestrainchannelofL1aspredictedbyourFINESSEmodeloftheinterferometer ............................. 123 6-1AplotillustratingtherequiredmachiningtoleranceforaTGGholderusingaparticularsolenoiddesign .............................. 125 6-2TheworstcasescenariolossduetotemperaturedriftsoftheTGGcrystal ... 126 6-3TwocompetingmagnetcongurationsfortheLLFI ................ 127 6-4Aconceptualdrawingofapotentialmonolithiclow-lossisolatordesign ..... 129 6-5AplotofthereectanceofaKTP!TGGboundaryasafunctionofthenumberofintermediatelayersbetweenthecrystalsintheevenlyspacedindexscheme 130 11

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6-6Adiagramillustratingthesystemofequationsfortheelectriceldsinak)]TJ /F1 11.9552 Tf 12.61 0 Td[(1layertransitionregion ................................ 131 6-7Aplotofthereectanceofaquarter-waveintermediarylayerasafunctionofthelayer'sindexofrefractionforaHWP!TGGjunction ............. 132 6-8Aplotofthecombinedpandspowerlossfora=2isotropicmonolayerasafunctionofthelayer'sindexofrefraction ...................... 134 6-9Aplotofthecombinedpandspowerlossforatuned-indexbirefringentmonolayerasafunctionofp .................................. 134 A-1Acartoonoftheworkingprinciple(left)andaschematicdrawing(right)ofaBOSEM ........................................ 139 A-2Adiagramoftheexperimentalsetupfortestsofthelow-temperatureperformanceoftheVishayTSTS7100 ............................... 140 A-3TheactivationforwardvoltageoftheVishayTSTS7100asafunctionoftemperature 141 A-4I-VcurvesfortheVishayTSTS7100asafunctionoftemperature ........ 141 A-5AplotoftheestimatedexternalquantumeciencyoftheVishayTSTS7100asafunctionoftemperature .............................. 142 A-6AplotofthemeasuredphotopoweroutoftheVishayTSTS7100asafunctionofforwardcurrentforseveraldierenttemperatures ................ 142 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyALOWLOSSFARADAYISOLATORFORSQUEEZINGINJECTIONINADVANCEDLIGO,ANDRADIO-FREQUENCYAMPLITUDEMODULATIONByRyanGoetzDecember2017Chair:DavidB.TannerMajor:PhysicsAdvancedLIGOiscurrentlyinthemiddleofitssecondscienceobservationrun,andoperatingwithstrainsensitivitybelow10)]TJ /F6 7.9701 Tf 6.586 0 Td[(23Hz)]TJ /F6 7.9701 Tf 6.586 0 Td[(1=2overanobservationbandof30Hzto1000Hz.Todate,theLIGOCollaborationhaspublishedonthreeseparatesolar-massbinaryblackholemergers,alldirectlyobservedbytheAdvancedLIGOdetectors.ThesedetectionsaretherstdirectevidenceofgravitationalradiationaspredictedbyEinstein'sTheoryofGeneralRelativity(GR),consideredbymanytobethenalmajorpredictionofGR( 34 ).Additionally,thesethreemergereventsaretherstevidenceoftheexistenceofsolar-massblackholesabove20M,andprovideanovelprobeintoastrophysicalprocessesthatcannotbeobservedwithtraditionalelectro-magnetictelescopes.TheworkdescribedinthisdissertationismotivatedbytheeorttoimprovethesensitivityoftheLIGOtelescopestherebyincreasingthevolumeofspacetimethatthedetectorscansee.Asameansofsurpassingthestandardquantumlimit,squeezedvacuumstateswillbeinjectedintotheoutputportoftheAdvancedLIGOdetectors.Theeectivenessofsqueezinginreducingquantumnoiseintheinterferometerisstronglydependentontheopticallossesintheinjectionpathofthesqueezedbeam,andwillrequirethedevelopmentofaLowLossFaradayIsolator(LLFI).Themajorityofthisdissertationisdevotedtothedesign,construction,andperformanceoftheLLFIattheUniversityofFlorida.Chapter 5 discusseseortsbytheauthortotracetheimpactofradio-frequencymodulationofthelasereldamplitude 13

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intheLIGOLivingstondetector(LLO).Chapter 6 outlinesmeasuresforfutureworktoimprovetheperformanceoftheLLFI. 14

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CHAPTER1INTRODUCTION 1.1TheGravitationalUniverseOnFebruary11,2016theLIGOScienticCollaborationandVirgoCollaborationannouncedthedetectionofgravitationalwavesfromthemergeroftwoblackholesofmasses36Mand29M( 39 ).Thismergerevent,referredtoasGW150914,markedtherstdirectdetectionofgravitationalradiation,morethan100yearsafterEinsteinrstpresentedtheeldequationsthatwouldpredicttheirexistence( 16 ).Formany,thiswasseenasthenalmajorexperimentaltestofthetheoryofgeneralrelativity;followingthemeasurementoftheprecessionofMercury'sperihelion,thediscoveryofblackholes,andthevericationoftheequivalenceprinciple.Beyonditsimportanceinverifyingthetheoryofgeneralrelativity,GW150914wasalsotherstdirectdetectionofabinaryblackholemerger,andtherstdirectevidencefortheexistenceofsolar-massblackholesbeyond20M.Inthefollowingyear,twomorebinaryblackholemergereventswereannounced,ociallysignalingintheeraofgravitational-waveastronomy( 38 ; 40 ).Generalrelativityisageometrictheoryinwhichgravityisanemergentpropertyoftheuniversebasedontheinteractionofmass-energywithspacetime.Inthistheory,mass-energyfollowsgeodesicsinspacetimewhicharedeterminedbyametricandtheEinsteineldequationswhichread(withoutcosmologicalconstant): G=8GT(1{1)Therightsideof( 1{1 )isknownasthestress-energytensor,itisdeterminedbythedistributionofmass-energy,whereasthetensorontheleftside,knownastheEinsteintensor,ispurelygeometric.Theconceptofpathlengthisgivenbytheinnitesimalspacetimeintervaldenedas: (ds)2=gdxdx(1{2) 15

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wheregisasymmetric44tensoroftenreferredtoasthemetrictensor(orsimplythemetric),whichencodesthecurvatureofaspace.Intheabsenceofmass-energy,spacetimeisdescribedbytheso-callednullmetricgivenby: =0BBBBBBB@)]TJ /F1 11.9552 Tf 9.298 0 Td[(10000100001000011CCCCCCCA(1{3)ThismetricdenesEuclidean(at)spacewiththespacetimeinterval(ds)2=)]TJ /F3 11.9552 Tf 9.299 0 Td[(c2(dt)2+(dx)2+(dy)2+(dz)2.Underthenullmetric,familiargeometricresultsholdtrue;forexampleparallellinesdonotintersectandthesumoftheinterioranglesofatriangleis180degrees.Asithappens,ouruniverseislargelyempty( 25 ),andsoourapproachnowistoconsiderametricfornearlyemptyspace: g=+h(1{4)Weassumehisaperturbativetermwhichisverysmallcomparedtothenullbackgroundmetric,andsowewillgenerallyignorealltermsbeyondlinearorder.Thisscenarioisoftenreferredtoastheweak-eldlimit.Inaccordancewiththeirbehaviorunderrotations,wecandecomposetheperturbationmetricintoscalar,vector,andtensorterms(followingthenotationconventionin( 52 )): h00=)]TJ /F1 11.9552 Tf 9.298 0 Td[(2h0j=wjhij=2sij)]TJ /F1 11.9552 Tf 11.955 0 Td[(2ij(1{5)where: =)]TJ /F1 11.9552 Tf 10.494 8.088 Td[(1 6ijhijsij=1 2hij)]TJ /F1 11.9552 Tf 13.151 8.087 Td[(1 3klhklij(1{6) 16

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HereweareusingthestandardconventionthatGreekindicesrunoverallcoordinates0,1,2,3,whileRomanindicesrunonlyoverthespatialcoordinates1,2,3.IntheweakeldlimitwesetT=0,andifwemakethegaugechoice: @isij=0@iwi=0(1{7)convenientboundaryconditionsyield,wj,andallzero.Thisisknownasthetransverse-tracelessgauge,anditresultsinaverysimpleformfortheperturbingmetric: h=20BBBBBBB@00000s11s12s130s21s22s230s31s32s331CCCCCCCA(1{8)Theonlynontrivialequationin( 1{1 )tosurviveinourweak-eldtransverse-tracelessgaugechoicetakestheform: 2sij=0(1{9)whichwecouldalternativelywriteas2h=0.Thisisawaveequationfortheperturbingmetricwhichadmitsplanewavesolutionsoftheform: h=Aeikx(1{10)foraconstanttensorAandwave-vectork.Themetrictensorisreal,andsowewrite( 1{10 )withtheunderstandingthatweareonlytotaketherealpart.Liketheperturbationmetric,Aisatraceless,symmetrictensorthatisentirelyspatial: A=0BBBBBBB@00000A11A12A130A12A22A230A13A23A331CCCCCCCA;ijAij=0(1{11) 17

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Thewave-vectordeterminesthepropagationdirectionoftheplanewave,andwemaketheconventionalcoordinatechoicesuchthatthisisthex3=zdirection.Werecognizethatkx=kzz)]TJ /F3 11.9552 Tf 12.129 0 Td[(k0t,forwave-numberkzandfrequencyk0,butthisismorecumbersomeandsowesuppressthesimplication.Becausehistransverse,thisforcesallA3jcomponentsofAtozero,andwehave: A=0BBBBBBB@00000h+h00h)]TJ /F3 11.9552 Tf 9.299 0 Td[(h+000001CCCCCCCA(1{12)forstrainamplitudesh+andh.Themotivationbehindthenameandnotationcanbeunderstoodbyevaluating: hdxdx=eikxh+(dx)2)]TJ /F3 11.9552 Tf 11.955 0 Td[(h+(dy)2+2hdxdy(1{13)whichallowsustorewritethespacetimeintervalgivenby( 1{2 )as: (ds)2=)]TJ /F1 11.9552 Tf 9.298 0 Td[((dt)2+1+h+eikx(dx)2+1)]TJ /F3 11.9552 Tf 11.956 0 Td[(h+eikx(dy)2+2heikxdxdy+(dz)2(1{14)Weseefrom( 1{14 )thattheh+termcausesdisplacementsinthexandydirectionstooscillateoutofphasewithoneanother,whilethehtermcausesoscillationsinthexandydirectionswhichareinphasewithoneanother.ThisbehaviorisexplainedgraphicallyinFigure 1-1 .Thetwoindependentterms,h+andh,areamplitudesfortwoorthogonalpolarizations(called\plus"and\cross")ofgravitationalwaves.IfweconsidertwopointsinspaceinitiallyseparatedbylengthL,wecanseefrom( 1{14 )thatanappropriatelypolarizedgravitationalwavewithamplitudehwillresultinamaximaldisplacementofthepointsbyL=hL.Forthisreasonwerefertohasthestrainofagravitationalwave.Theinspiral,merger,andringdownofcoalescingneutronstarorblackholebinarysystems,gammaraybursts,andsupernovaeareallexpectedtoproducegravitationalradiation.Theweakeldlimitisappropriatelynamedforourpurposes,aseventhese 18

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violentastrophysicaleventsinourlocaluniverseareexpectedtoproducestrainsatEarthontheorderof10)]TJ /F6 7.9701 Tf 6.587 0 Td[(2010)]TJ /F6 7.9701 Tf 6.587 0 Td[(22( 14 ; 60 ).ItistheaimofLIGOtodetectandcharacterizethesegravitationalwavesforthepurposeofprobingastrophysicaleventsbeyondthelimitsofelectro-magneticinvestigation. Figure1-1. Adiagramoftheeectsofapassinggravitationalwaveonaringoftestparticlesinfreespaceforthecaseofpluspolarization(red)andcrosspolarization(blue)foronefullperiodofthewave.Inbothcases,thegravitationalwaveispropagatinginto(oroutof)thepage. 1.2LaserInterferometryforGravitationalWaveDetection 1.2.1LIGOTheLaserInterferometerGravitational-WaveObservatory(LIGO)Collaborationoperatesapairofground-based,long-baselineinterferometerslocatedinHanford,WashingtonandLivingston,Louisianatodirectlydetectgravitationalwaves.ThedetectorsoperateonthesamebasicprinciplesasasimpleMichelsoninterferometer;asimplieddiagramisgivenasFigure 1-2 .Alasereldissplitintotwoorthogonalbeams,whicharereectedoofmirrors(moreproperlyreferredtoastestmasses)andthenrecombinedatthebeamsplitter.Anydierenceinpathlengthofthetwobeamsresultsinarelativephaseshift,andwhenthebeamsinterferethispathlengthdierenceisreadoutaslightpowerintheanti-symmetricportofthebeamsplitter.Withthisbasictopology, 19

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relativelengthchangesofthetwoarmsoftheinterferometerduetopassinggravitationalwavesareconvertedtoaphoto-signal.Toreachsucientsensitivityforgravitationalwavedetection,thearmsoftheLIGOinterferometersweremadetobe4kmlong.Tofurtherincreaselightstoragetimeintheinterferometer,thesimplearmsofaMichelsonhavebeenreplacedbyresonantFabry-Perotcavitieswithanesseof200.Recyclingcavitiesinboththesymmetricandantisymmetricbeamsplitterportsfurtherincreasesensitivity( 6 ; 45 ).ThecurrentgenerationofLIGO,knownasAdvancedLIGO,isoperatingwithstrainsensitivityontheorderof10)]TJ /F6 7.9701 Tf 6.587 0 Td[(23overthefrequencybandfrom30Hztoafewkilohertz. 1.2.2GravitationalWaveDetectorNetworkThereareseveralotherinterferometricgravitational-wavedetectorsaroundtheworld.TheVirgodetector,a3kmbaselineobservatorylocatedinCascina,Italy(nearPisa)isaFrench-ItaliancollaborationandhasthegreateststrainsensitivityoutsideoftheAdvancedLIGOdetectors.GEO600,a600mbaselineobservatorylocatednearHanover,GermanyisrunbytheAlbertEinsteinInstituteandservesasatestbedfornewinterferometertechnologies.KAGRA,a3kmbaselineobservatorylocatedintheKamiokaminenearToyama,Japanisanundergroundcryogenicdetectorcurrentlyundergoingcommissioning( 59 ).Atthemoment,athirdLIGOdetectorcalledINDIGOisplannedforconstructioninIndia.Theseobservatoriesworkcloselywithoneanotherandconstituteadetectornetwork.Witheachadditionaldetectoronline,thecondenceincoincidentdetectionsincreasesandsourcelocalizationsignicantlyimproves. 1.3ImprovingGravitational-WaveDetectorCapabilitiesThestrainsensitivitylimitoftheLIGOdetectorsdeterminesamaximaldistanceatwhichaparticulargravitational-wavesourcecanbedetected.Thisdistanceinturndenesanobservablevolumeoftheuniverse(centeredaboutEarth);sourceswithinthisvolumewillbemeasuredbyourdetectors,whilesourcesoutsidecannotbedisentangledfrom 20

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Figure1-2. AdiagramofthemajoropticalcomponentsoftheAdvancedLIGOinterferometers.TheInputModeCleaner(IMC)isresponsibleformodeshapingofthebeamsenttothemaininterferometer.Twoorthogonal4kmFabry-Perotcavitiesserveasthearmsoftheinterferometer,whereinadierentialpathlengthduetogravitationalradiationwillmanifestasarelativephaseshiftofthelightreectedfromthecavities.ThePowerRecylcingMirror(PRM)increasesthelightpowercirculatingintheinterferometer.TheSignalRecyclingMirror(SRM)canbetunedforresonantextractionofsignalsinaparticularfrequencybandofinterest,orcanoperatedetunedforbroadbanddetectorsensitivityimprovement.TheOutputModeCleaner(OMC)isusedtostripthelightofanyhigherordermodecontentduetoscatteringormodemismatch,whichisnotaresultofagravitationalwavesignalandwouldotherwiseaddnoisetoourphoto-signaloutput. detectornoise.(Thisissimplifyingabit,asafullanalysiswouldaccountfordirectionalsensitivityoftheantennanetwork).Becauseweexpectsourcestobeuniformlydistributedabouttheuniverse,thedetectionrateshouldscalelinearlywiththeobservablevolume.Asaconsequence,animprovementtothedetectionrateroughlyscalesasthecubeoftheimprovementtothedetectorstrainsensitivity.Squeezedvacuuminjection,sometimessimplyreferredtoassqueezing,isatechniquetoimprovethesensitivityofinterferometricdetectorsbyreducingquantumnoiseofthe 21

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apparatus.InChapter 2 wewilldiscusssqueezing,andwilldemonstratetheneedforlowopticallossFaradayisolatorsinordertosuccessfullyimplementthetechniqueinAdvancedLIGO.Thelong-termsqueezinggoalsforaLIGOwillrequire<1%opticallossperpassofaFaradayisolator.Thisissignicantlylowerthanboththe<5%lossfromhigh-endo-the-shelfproducts( 17 ),aswellasthe3%lossinthecurrentOutputFaradayIsolator(OFI)usedfortheinitialintroductionofsqueezedinjection( 4 ).Chapter 3 introducesthefundamentalprinciplesbehindFaradayisolatorsandtheInputFaradayIsolator(IFI)designedbytheUniversityofFloridaLIGOGroup.ThechapterthenpresentstheeortsbytheauthortomodifytheIFIdesigntoserveasalow-lossFaradayisolator(LLFI)forsqueezedinjectioninAdvancedLIGO.Chapter 4 describesworkbytheauthortoconstructandcharacterizetheLLFI.Chapter 6 explorespotentialimprovementstotheLLFIthatmaybeimplementedinfutureresearch.Chapter 5 discussesworkbytheauthortoinvestigateamplitudemodulationofthelasereldinAdvancedLIGO.ThisworkwasmainlyperformedduringafourmonthstayattheLIGOLivingstondetector. 22

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CHAPTER2SQUEEZEDLIGHTFORADVANCEDDETECTORS 2.1NoiseinAdvancedLIGO 2.1.1ClassicalNoiseBecausetherelativelengthchangesoftheinterferometerarmsduetoapassinggravitationalwavearesosmall,agreatdealofeortmustbemadetoreducemotionofthetestmassescausedbyenvironmentalexcitations.Groundmotionduetoseismicorhumanactivity,expansionandcontractionoftheinterferometerduetotemperatureuctuationsandoceantides,residualgasinthevacuumchambers,andgravitygradientnoise( 27 )allcontributetopositionnoiseofthetestmasses.Tocompensate,theLIGOcollaborationhasdevelopedmanytechniquesandcontrolschemestoisolatethetestmassesfromtheirsurroundings.Residualgasintheevacuatedinterferometerisasourceofnoiseacrosstheentireobservationband.Straygasmoleculespassingthroughthebeampathinthearmsoftheinterferometerinteractwiththecarriereldandresultinadditionalaccumulatedphaseofthebeam.Becausethisprocessisrandominbotharmcavities,theextraphaseineacharmisnotcommonmodeandisultimatelyphasenoiseatthedarkport.Further,residualgastendstotrapitselfinsmallvolumes,suchasthosebetweenthetestmassesandtheirneighborreactionmasses.Thistrappedgasresultsinuncorrelatedpositionnoiseinthetestmasseswhichmanifestsassignalatthedarkport.Toreducetheimpactofresidualgasontheinterferometeroutput,allcomponentswithinthevacuummustconsistofapprovedmaterialsandberigorouslycleaned(andoftenbaked)beforeinclusion.Aseriesofturbomolecular,ion,andcryo-pumpsareusedtomaintainvacuumatlevelsatorbelow10)]TJ /F6 7.9701 Tf 6.586 0 Td[(9torr.Disturbancestotheobservatory,duetoseismicactivity,strongwindsorrain,temperatureuctuations,passingtrains,orthelikecancausephysicaldierentiallengthchangestotheexperimentalapparatus,whichisexactlywhataninterferometer 23

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isintendedtomeasure.Topreventtheinstrumentfromregisteringdisturbancesthatarenotcosmologicalinnature,agreatdealofeortisputintobothpassiveandactiveisolationelements( 57 ; 58 ).Thetestmassesare40kgfusedsilicaoptics,suspendedfromaquadruplependulumby400mthickfusedsilicabers.Othereectssuchasscatteredlight,laserfrequencyorintensityinstability,andelectronicnoisedonotaectthepositionsofthetestmassesbutcreatenoiseatthereadout.Lightscatteredfromopticsatonepointintheinterferometercaninteractwithothercomponentsofthevacuumenvironment,suchasthewallsofthevacuumtubes,andre-enterthebeampathlateron.Thisprocessbypassestheisolationoftheinterferometeropticsandimprintsmechanicalnoiseofthevacuumenclosureonthecarrier( 50 ).Toreducethisrisk,themajoropticalcomponentsaresuper-polishedbeforecoatingtoreducethelightscatteredfromtheirsurfaces,andbaesareplacedregularlythroughoutthearmcavitiestopreventscatteredlightfromrecombiningwiththemainbeam.Noisethatcannotbecancelledismeasuredbyauxiliaryinstrumentsinaneorttounderstandthecorrelationbetweenenvironmentalmonitorsanddarkportoutput.Thesemonitorstherebyallowscientiststovetodatacollectedduringespeciallynoisytimes( 42 ). 2.1.2QuantumNoiseInadditiontothemanymacroscopicsourcesofnoise,theincrediblesensitivityoftheLIGOinterferometersrequiresthatwealsotakeintoaccountthenoiseassociatedwiththequantumnatureofthelaserlightusedintheinstrument.Theradiationeldforasinglemode(frequency)hasanassociatedHamiltonianoperator( 24 ): H=1 2h(aya+aay)(2{1)whereayandaarethefamiliarcreationandannihilationoperators.TheeigenstatesofHarephotonnumberstatesjniwith: Hjni=hn+1 2jni(2{2) 24

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whichrepresentthequantaofeldexcitations.Acoherentlaserstatecanbeconstructedby: ji=D()j0i(2{3)whereDiscalledthedisplacementoperatorandcanbeexpressedintermsofthecreationandannihilationoperatorsas: D()=eay)]TJ /F4 7.9701 Tf 6.587 0 Td[(a(2{4)Thecompletenessofthephotonnumberstatesallowsustowrite: ji=1Xn=0jnihnji(2{5)where: hnji=n p n!e)]TJ /F13 5.9776 Tf 7.782 3.258 Td[(1 2jj2(2{6)Notethatjiisnotitselfanumberstate;thatis,itdoesnothaveanumberofphotonsassociatedwithit.Instead,thereisaprobabilitydistributionassociatedwithjithatdeterminesthelikelihoodofmeasuringnphotonsinthatstate.Thisprobabilityissimply: jhnjij2=jj2n n!ejj2(2{7)whichisaPoissoniandistributionwithanexpectedvalueofjj2=hni.Becausephotonnumberstatesareenergyeigenstates,weseethatjj2isproportionaltotheexpectedenergyassociatedwiththecoherentstateji.Forthisreason,wemaythinkofasthequantumanalogoftheclassicalcomplexwaveamplitude.Thestandarddeviationofthedistributiondenedby( 2{7 )isthesquarerootofthemean: n=p hni(2{8)thatis,theuncertaintyinphotonnumberforacoherentstatescalesasthesquarerootoftheexpectednumberofphotonsforthatstate.Itfollowsthatthemeasuredopticalpower 25

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ofthestatehasasimilarbehavior: P/p P(2{9)QuantumnoiseintheLIGOinterferometersisresultofthisfundamentaluncertaintyinlaserintensity,anditmanifeststhroughtwodistinctmechanisms.Therstisthroughdirectuctuationsofthepowerintheantisymmetricportofthedetector.Thisnoiseisreferredtoaslasershotnoise.Atverylowfrequencies,thestrainmeasuredbyaninterferometercouplesproportionallyintorelativeintensityattheoutputport,andsotheamplitudespectraldensityofthestrainnoiseduetoshotnoisebehavesas: ~hsh/P LP/1 Lp P(2{10)whereListhelengthofthearmcavities.Athigherfrequenciestheinterferometertransferfunctionformirrordisplacement~xtophotosignal~iisnotaconstant,butinsteadhasafrequencydependency( 41 ): jC()j/1 q 1+(=)2(2{11)forthecavitypolefrequencyassociatedwiththeFabry-Perotarmsoftheinterferometerandisinverselyrelatedtothelightstoragetime.Inthecasewhere>>,thestrainshotnoisegoesas: ~hsh()/1 jC()jLp P Lp P(2{12)andsoweseethatbeyondthecavitypolefrequency,shotnoiseincreaseslinearlywithfrequency.Thesecondmechanismbywhichquantumnoiseentersthegravitational-wavestrainchannelisthroughdierentialradiationpressureonthetestmasses.Fluctuationsoftheeldintensitycirculatinginthecavitiescausespositionnoiseofthetestmasses,whichisconvertedtostrainnoisebytheinterferometer.Theforceoneachtestmassisproportionaltothepoweroftheeldreectingoofit,andsotheamplitudespectral 26

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densityfollows: ~F/P/p P(2{13)Likethefundamentalintensityuctuations,theuctuationsintheforceonthetestmassesisatinfrequency.Intheapproximationwherethetestsmassesarefreelyoating,thisforcecouplesintodisplacementnoiseby( 54 ): ~x()/~F m2/p P m2(2{14)wheremisthemassofatestmass.Asforthecasewithshotnoise,wecanrelatethedisplacementnoisetomeasuredstrainnoiseby: ~hrp()/~x() jC()jL/p P mL2s 1+ 2(2{15)Forlowfrequencies,wecansimplifythisto: ~hrp()p P mL2(2{16)Weseethatthisradiationpressurenoisefallssharplywithfrequencyfor<<.From( 2{12 )and( 2{16 )wecanseethatforagiventestmassweight,theamplitudespectraldensityofthestrainsensitivityduetoradiationpressurenoisescalesasthesquarerootofthecirculatinglaserpower,whilethatduetoshotnoisescalesastheinversesquareroot.Andso,assumingweareunabletoproducearbitrarilylargetestmasses,shotnoiseandradiationpressurenoisewillalwaysbeincompetitionwhenscalingthelaserpower.Thecompetitiondenestheso-calledstandardquantumlimit(SQL);aregionofstrainandfrequencyspacewhichcannotbereachedbyclassicalinterferometryforagiventestmassweightandcavityarmlength.Quantitatively,thestandardquantumlimitforAdvancedLIGOisdescribedby( 11 ): ~hSQL()=r 8~ m2L2(2{17) 27

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Thetotalquantumnoiseinthegravitational-wavestrainisthequadraticsumoftheshotandradiationpressurenoises( 28 ): ~hQ=q ~h2sh+~h2rp(2{18)whichcanbewrittenas( 23 ): ~hQ=~hSQL p 2r K+1 K(2{19)where: K=24 2(2+2)P PSQL(2{20)Here,PSQListheinputlaserpowerrequiredtoreachthestandardquantumlimitoftheinterferometer.AsK>0,wecanconrmfrom( 2{19 )that~hQ~hSQL.AnominalnoisebudgetforALIGOisgiveninFigure 2-1 .IgnoringBrowniannoiseofthetestmasscoatings,thesensitivityoverthedetectionbandislimitedbyquantumnoise,whichisdominatedbyradiationpressurenoiseatmiddlefrequenciesandshotnoiseathighfrequencies.ThismotivatesthesearchfortechniquestoreducequantumnoiseinAdvancedLIGO. 2.2SqueezedLightSqueezingisatechniqueforreducingquantumnoiseinaninterferometerrstproposedin1981thatallowsfordetectorsensitivitiesthatbeattheSQL( 12 ; 22 ; 30 ; 35 ).Tounderstandthephysicalprinciplesbehindthisphenomenon,webeginbyconsideringthewaveequationforelectriceldsinvacuum: r2)]TJ /F1 11.9552 Tf 13.151 8.088 Td[(1 c@2 @t2E=0(2{21)whichadmitssolutionsthatcanbewrittenas: E=X1(r;t)cos!t+X2(r;t)sin!t(2{22) 28

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Figure2-1. AnnominalstrainsensitivitycurveforAdvancedLIGOoperatingat125Wofinputpower.Quantumuctuationsintheformofshotnoiseandradiationpressurenoiseareexpectedtolimitthedetectorthroughouttheobservationband.ProducedwiththeGravitationalWaveInterferometerNoiseCalculator(GWINC)( 1 ) Therealamplitudes,X1,X2,ofthecosineandsinecomponentsarereferredtoasquadraturesofthewave.Thearbitraryphaseofthewaveisgivenby: tan=X2 X1(2{23)Intheconventionwhereweset0,itfollowsthatX2!0,andweseefrom( 2{22 )thatX1correspondstotheamplitudeofthewave.From( 2{23 )wecanalsoseethatX2X1forX2<
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transitiontothisquantumpicturewepromoteX1andX2tooperators,denedby: X1=1 2(a+ay)X2=i 2(ay)]TJ /F7 11.9552 Tf 11.955 0 Td[(a)(2{24)Becausea;ay=1;thatis,thecreationandannihilationoperatorsdonotcommute,itfollowsthatX1andX2donotcommute,andsotheyhaveanuncertaintyrelationgivenby( 21 ): 2X12X21 16(2{25)where2Xi=hXi2i)-301(hXii2.Theuncertaintyrelationservestolimitourabilitytosimultaneouslymeasuretwoquadraturesofthesamelighteld.Torepresentlaserstates,weintroducethedisplacementoperator: D()=eay)]TJ /F4 7.9701 Tf 6.587 0 Td[(a(2{26)andactonthevacuumstatetocreateanewstate: ji=D()j0i(2{27)Thisstatejiisknownasacoherentstate,whichourlasereldswillapproximate.Theconstantisthequantumanalogueofaclassicalwave'scomplexamplitude.Foracoherentstate,theuncertaintiesinbothquadraturesareequalandsatisfytheequalityin( 2{25 ).Inthissense,coherentstatesaretheintrinsicallyleastnoisyexcitationsoftheelectriceld.TheleftgraphinFigure 2-2 illustratesthequadraturespaceuncertaintyregionofsuchacoherentstate.Weseethattheregionissymmetricinanyorthogonalquadraturesthatwecouldchoose.Thissymmetryisnotanecessity,however.Imagineamaterialinwhichtheindexofrefractionwasafunctionoftheelectriceldamplitudeappliedtothematerial(theKerreect).Passingacoherentstatethroughsuchamaterialwillcorrelateamplitudequadratureuctuationswithphasequadratureuctuations,deformingthequadrature 30

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spaceuncertaintyregion(rightgraphinFigure 2-2 ).Suchastateiscalledasqueezedstate,anditistransformedfromapurestatej ibytheunitarysqueezingoperator: j ()i=S()j i=e1 2(a2)]TJ /F4 7.9701 Tf 6.587 0 Td[(ay2)j i(2{28)where=Re2iisacomplexnumbercalledthesqueezingparameter.Itiscommontowritethesqueezingoperatorasanexplicitfunctionoftherealparameters:S(R;).OperatorstransformasA()=S()ASy(),andsoas: h ()jA()j ()i=h jSy()S()ASy()S()j i=h jAj i(2{29)weseethatthesqueezingoperatordoesnotchangetheexpectationvaluesforastate.IfwedenethequadraturesX=X1cos+X2sinandX?=)]TJ /F7 11.9552 Tf 9.298 0 Td[(X1sin+X2cos,wendthat( 21 ): 2X()=e)]TJ /F6 7.9701 Tf 6.586 0 Td[(2R2X2X?()=e2R2X?(2{30)TheeectofthesqueezingoperatorhasbeentoreducethevarianceinthequadraturemakingananglewiththeX1quadraturebyafactorofe)]TJ /F6 7.9701 Tf 6.587 0 Td[(2R.Forthisreason,werefertoasthesqueezeangle,andRasthesqueezeamplitudeor(real)squeezeparameter.WerefertothequadratureXasthesqueezedquadrature.ThereductioninvarianceinXhascomeatacostofincreasedvarianceinX?;notethattheequalityin( 2{25 )ispreserved.atthecostofincreasedvarianceintheother.Indiscussionsofsqueezing,itiscommontocomeacrossthesqueezefactordenedby( 55 ): SF)]TJ /F1 11.9552 Tf 21.918 0 Td[(10log10)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(42X(2{31)Thefactorof4intheargumentofthelogarithmisincludedsothatsqueezefactormeasurestheratioofthevarianceofthesqueezedquadraturetothatofthelowestquadraturevarianceforanunsqueezedcoherentstate,whichisgivenbyforcingequalityin( 2{25 )andtakingthesquareroot.Ofteninliteraturethenormalizedvarianceisquoted,andsothefactorof4canbedropped. 31

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BecausequantumnoiseinLIGOisdominatedbyradiationpressurenoiseatlowfrequencies,andshotnoiseathighfrequencies,squeezingintheamplitudequadratureatlowfrequenciesandthephasequadratureathighfrequenciescanpushthedetectorsensitivitybelowthestandardquantumlimit( 30 ; 35 ).Suchatechniqueiscalledfrequency-dependentsqueezedvacuuminjection. Figure2-2. Acartoonofthequadraturerepresentationofanunsqueezed(left)andsqueezed(right)coherentstate.Whiletheunsqueezedstatehasasymmetricuncertaintyregion,thesqueezedstatehasrelativelylessuncertaintyinthenewXquadraturewhilesimultaneouslygainingmoreuncertaintyinX?. 2.3ImplementationinALIGOTomakeuseofsqueezedlightforsensitivityenhancementinAdvancedLIGO,asqueezedvacuumstatej0(R;)i=S(R;)j0iisinjectedintotheantisymmetricportoftheinterferometer.Itcanbeshown( 30 )thatfor: ()=)]TJ /F1 11.9552 Tf 11.291 0 Td[(cot)]TJ /F6 7.9701 Tf 6.587 0 Td[(1K()(2{32)thequantumnoiseamplitudespectraldensitybecomes: ~hQ=~hSQL p 2e)]TJ /F4 7.9701 Tf 6.587 0 Td[(Rr K+1 K(2{33)Comparingto( 2{19 ),weseethattheeectofsqueezinghasbeentoreducetheamplitudespectraldensityofthequantummechanicalstrainnoisebyafactore)]TJ /F4 7.9701 Tf 6.586 0 Td[(R.TheoptimalsqueezeangleasafunctionoffrequencyisplottedinFigure 2-3 .Atlowfrequencies, 32

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0,andsoX=X1cos+X2sinX1;thatis,theoptimalsqueezequadratureistheamplitudequadratureofthecoherentlaserstate.Converselyathighfrequencies=2,andsotheoptimalsqueezequadratureistheamplitudequadratureXX2.Thisresultisinagreementwiththeheuristicmodelthatradiationpressurenoiseisamplitudenoiseofthecarrierincidentonthetestmasses,whileshotnoiseisthephasenoisefromthebeatingofthecarrierwithvacuumexcitationsattheinterferometeroutput.Thepracticeofproducingstateswithasqueezeanglethatisafunctionoffrequencyisreferredtoasfrequencydependentsqueezing,andisachievedinpracticebyreectingthesqueezedstateoofaltercavity( 18 ; 28 ) Figure2-3. Theoptimalsqueezeangleforadual-recycledFabry-PerotenhancedMichelsoninterferometeroperatingatPSQL.Weseethatbelowtheinterferometerbandwidth(<),itisoptimaltosqueezeinamplitudequadrature,whileaboveit(>)theoptimalsqueezequadratureisthephasequadrature. ThetentativeopticallayoutforsqueezinginjectionisseeninFigure 2-5 .Thesqueezedbeamisgeneratedinthein-vacuumOpticalParametricOscillator(VOPO)andinputtoaltercavitywhichtunesthesqueezingangleasafunctionoffrequency.Fromtheltercavity,thesqueezedbeamissentbacktotheVOPO,whereitisredirectedandinjectedintotheantisymmetricportoftheinterferometerviaaFaradayisolator.Opticallossinthesqueezedpathwillreducethesqueezefactorandpurityoftheinjectedsqueezedstate( 55 ).Ifistheopticalthroughputofthesqueezingpath,also 33

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referredtoasthedetectioneciency,thevarianceofthesqueezednoisetransformsas( 35 ): 2X!2X+(1)]TJ /F3 11.9552 Tf 11.955 0 Td[()(2{34)Hereweareusingthenormalizedvariance.ApreliminarytestofsqueezedvacuuminjectionintheH1interferometerfoundathroughputof=0:44,reducingthemeasurablesqueezingfrom10dBto2.2dB( 36 ). Figure2-4. Aplotofthereducedsqueezefactorasafunctionoftheopticalthroughputofthesqueezinginjectionpath.Curvesfor3dB,10dB,and20dBofinjectedsqueezingareshown. FromFigure 2-5 ,wecanseethatthesqueezedbeampathpassesthroughaFaradayisolatorthreetimes;onceattheVOPOafterreturningfromtheltercavity,andtwicethroughtheOutputFaradayIsolatorlocatedaftertheSignalRecyclingCavity(SRC).Therefore,itisvitaltolimittheopticallossinthefutureoutputFaradayisolator.Toachievethelongtermgoalof10dB( 36 )measuredquantumnoisesuppressionthecongurationrequires<1%lossperpass( 32 ; 37 )oftheisolators.Presently,theInputFaradayIsolator(IFI)installedinAdvancedLIGOhas3)]TJ /F1 11.9552 Tf 12.422 0 Td[(4%singlepassopticalloss( 47 ),andtheoutputFaradayisolator(OFI)hasameasuredlossof3:3%( 4 ),andsoitisnecessarytodesignanewisolatorthatcanmeetthethroughputrequirementsforfrequencydependentsqueezing. 34

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TheworkingdesignattheUniversityofFloridaforthenewLowLossFaradayIsolator(LLFI)isbasedonthecurrentALIGOIFIdesign,allowingustodrawfromyearsofexperiencewiththesetup.Specically,ourapproachistoidentifysourcesoflossinthecurrentIFIandmodifythedesigntobringthetotallossbelowthe1%threshold. Figure2-5. Adiagramofthetentativelayoutforsqueezinginjection.Thesqueezingopticalpath,shownasthedottedredline,requiresthreepassesthroughaFaradayisolatoraftertheltercavity.AdaptedfromLisaBarsotti,LIGODocumentT1400274( 32 ). 35

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CHAPTER3LOWLOSSFARADAYISOLATORDESIGN 3.1LightPropagationinOpticalMaterialsManyopticalmaterialsarecrystallinestructures;solidsthatarehighlyorderedatthemolecularlevelwithregularlyrepeatingstructures.Thismolecularstructure,anditscorrespondingelectronicstructuregiverisetomanyopticalphenomenathatwedon'tobserveinfreespacesuchasfrequencydoubling,dichroism,polarizationrotation,polarizationsplitting.Particularsofatomicstructureguaranteethatallnon-cubiccrystalsareopticallyanisotropic,meaningtheiropticalpropertiesaredependentuponthedirectionofpropagationandtheorientationoftheincidentlight.ParticularlycentraltotheoperationofFaradayisolatorsisbirefringence,thephenomenonwhereindierentpolarizationsoflightexperiencedierentindicesofrefraction.Consideracrystalsubjectedtoanexternalelectriceld.Theelectricdisplacementeldisdenedtobe: D=0E+P(3{1)forEthetotalelectriceldinthematerial,andPthepolarizationofthematerialasaresponsetotheeld,whichisdenedastheelectricdipolemomentperunitvolume.Inlinearmedia,thepolarizationisrelatedtotheeldbytheelectricsusceptibilitytensor:P=0$E.Dening$=$1+$,weareabletorewrite( 3{1 )as: D=0$E(3{2)Thedielectrictensor,,relatestheeldcontributionsfrombothfreeandboundcharges(theelectricdisplacementeld)toanappliedelectriceld.Itisoftencoventionaltoabsorbthe0constantinto,howeverforthefollowinganalysisitissimplertokeepthetwodistinct.Ananalagousauxiliarymagneticeldisalsodened0H=B)]TJ /F3 11.9552 Tf 12.413 0 Td[(0M,whereMisthemagneticdipolemomentperunitvolume,alsocalledthemagnetization.Similarly,for 36

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linearmediathemagnetizationisrelatedtotheauxiliarymagneticeldbythemagneticsusceptibilitytensor:M=$mH.Deningthepermeability$=$1+$m,wecanthenwriteB=0$H.Formostnon-magneticmaterials,$istakentobetheidentityandsothemagneticeldandauxiliarymagneticeldaresimplyrelatedbythepermeabilityoffreespace0.Intheabsenceoffreechargesorfreecurrents,Maxwell'sequationsinmattertaketheform: rD=0rB=0rE=)]TJ /F3 11.9552 Tf 10.494 8.088 Td[(@B @trH=@D @t(3{3)Wecantakethecurlofthethirdequationin( 3{3 ),makinguseofavectorderivativeidentity,andthensubstituteinthefourthequationtoobtain: r2E)-222(r(rE)=00$@2 @t2E(3{4)Sinceweareconcernedwithopticalpropertiesofthemediumunderinvestigation,wewillconsiderplane-wavesolutionsto( 3{4 )oftheform: E=E0ei(kr)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)(3{5)forwhichwehavetheusualrE=0asisthecaseinvacuum.Herekiscalledthewave-vectoranddeterminesthedirectionofpropagationofthewave.Themagnitudeofthewavevectorisrelatedtothewavelengthbyk=2=andthephasevelocityofthewaveisv=!=k.NowsupposethereisabasisfeigwithcorrespondingtransitionmatrixA=(e1je2je3)suchthat$0=A)]TJ /F6 7.9701 Tf 6.587 0 Td[(1$Aisdiagonal.Thatis,feigisthesetofeigenvectorsofthedielectrictensor,calledtheprincipalcoordinatesorprincipalaxes.Ifwewriteourwaveexplicitlyasthelinearcombinationoftheeigeinvectorsofthedielectric 37

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tensor: E=XiEiei(3{6)thenwecanrewrite( 3{4 )as: Xieir2Ei=00Xiiei@2 @t2Ei(3{7)whereiistheeigenvaluesatisfying$ei=iei.Putting( 3{5 )into( 3{7 )yields: k2XiEiei=00!2XiiEiei(3{8)Becausefeigarelinearlyindependent,( 3{8 )canbetreatedasthreelinearlyindependentequations.Wenowdivideouranalysisintothreecases:1)alliareequal,2)exactlytwoiareequal,and3)noiareequal.Therstcaseisthatofanisotropicmaterial.Here,wecancalli=andwrite( 3{8 )as: k2E=00!2E(3{9)Weseefrom( 3{5 )and( 3{9 )thattherearenorestrictionsonE0,andwhilekisperpendiculartoE0byassumption,itisotherwiseonlyrestrictedinmagnitude.Thatis,wavesinthismediumcanhaveanypolarizationandtravelinanydirectionwithspeed: v=! k=1 p 00=c p (3{10)wherecisthespeedoflightinvacuum.Werecognizep astheindexofrefractionofthematerial.Thesecondcaseisthatofauniaxialcrystal.Wecanassumewithoutlossofgeneralitythat1=26=3.Then( 3{8 )yields: 0=(001!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E1=(001!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E2=(003!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E3(3{11)ThiscanonlybetruesolongaseitherE1=E2=0,orE3=0.Evidentlytherearetwodistinctclassesofsolutions:onewithE0lyinginthe1-2plane,travelinginthe3-direction 38

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withwavevelocityv=c=p 1,calledtheordinaryray,andonewithE0inthe3-directionandtravelinginthe1-2planewithwavevelocityv=c=p 3,calledtheextraordinaryray.Inthiscase,werefertothe3-axisastheopticaxisofthemedium.Thethirdcaseissimilartothesecond,withtheexceptionthatnowwehave: 0=(001!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E1=(002!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E2=(003!2)]TJ /F3 11.9552 Tf 11.955 0 Td[(k2)E3(3{12)whichonlyholdsifexactlyoneofE1,E2,E3isnon-zero.Therearethreedistinctclassesofsolutionsinthiscase:eachwaveispolarizedalongoneoftheeigenvectorseiofthedielectrictensorandtravelswithwavevelocityvi=c=p i.Inthisinstance,werefertothecrystalasbeingbiaxial.Anotherobjectofinterestthatcanbeconstructedfromthedielectrictensoristheso-calledindexellipsoid,denedtobethesurfacesatisfying: Xijxixj ij=1(3{13)Theellipsoiddeterminestheindexofrefractionandpolarizationdirectionsforlinearlypolarizedlightwitharbitrarypropagationdirectionbythefollowinggeometricconstruction.Giventhewave-vectork,takeasectionoftheellipsoidthatcontainstheoriginandisorthogonaltok.Thissectionwillbeanellipse,withsemi-majorandsemi-minoraxesrunningparalleltotheallowedpolarizationsandthesemi-majorandsemi-minorvaluescorrespondingtotherespectiveindicesofrefractionforthosepolarizations.Intheprincipalcoordinatesystem( 3{13 )reducesto: u21 1+u22 2+u23 3=1(3{14)Inthissystem,theprincipalaxesarealsosemi-principalaxesoftheellipsoid,andhavelengthsp 1,p 2,andp 3. 39

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Asisillustratedabove,anunderstandingofthepropagationoflightinanopticalmediumrequiresaconstructionofthedielectrictensor.Inthenextsection,wewillattempttoderivethedielectrictensorforaFaradayrotatorcrystalfromrstprinciples. 3.2TheFaradayEectTheFaradayeectisanobservedrotationoflinearlypolarizedlightwhenpassingthroughcertainmagneto-opticmediawhicharesubjectedtoamagneticeld.AswewillseeinSection 3.3 ,thiseectiscentraltotheoperatingprinciplebehindanopticaldiode,otherwiseknownasaFaradayisolator.Theeectcanbedescribedphenomenologicallybytheequation: =VZBds(3{15)whereistheanglethroughwhichthepolarizationvectorisrotated,V,calledtheVerdetconstant,isapropertyofthemedium,andtheintegralistakenalongthepathofthebeaminthedirectionofpropagation.Thehandednessoftherotationisdenedbythepropagationdirection.ForthepurposeofdesigningandconstructingaFaradayrotator( 3{15 )isasucientdescriptionoftheFaradayeect,thoughitisnotilluminating.Toarriveatamodelfor$wewanttoexaminethecaseofaplanewaveopticaleldincidentonacrystalmediumwithabackgroundappliedmagneticeldparalleltothepropagationdirectionoftheopticaleld.Webeginbyconsideringthemicroscopiccase:asingle,classicalelectronwithpositionvectorr,boundtoanucleusattheoriginbyasimpleharmonicpotentialwithrestoringforcegivenby)]TJ /F3 11.9552 Tf 9.299 0 Td[(!20mr.Here,misthemassoftheelectronand!0isthenaturalfrequencyofoscillationsoftheelectron.ThefullclassicalequationofmotioncombinesthisrestoringforcewiththeLorentzforceontheelectron: d2r dt2=)]TJ /F3 11.9552 Tf 9.298 0 Td[(!20r+q mE+dr dtB(3{16)HereqisthechargeoftheelectronandEistheappliedopticaleld.Strictlyspeaking,Biscomposedofboththeappliedstaticmagneticeldaswellasthemagneticeldcomponentoftheincidentlight,howeverthecontributionofthelattertotheequation 40

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ofmotionismuchsmallerthantheformerandcanbeignored.TheappliedopticaleldisoftheformE(r;t)=E0ei(kr)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t),howeverasthewavelengthoflight(10)]TJ /F6 7.9701 Tf 6.586 0 Td[(6m)weareconsideringismuchlargerthanthecharacteristicdimensionofanatom(10)]TJ /F6 7.9701 Tf 6.587 0 Td[(10m)wecanignorethespatialdependenceoftheeldinouranalysis.Theelectronmotionisdrivenbytheopticaleld,andsohasthesametimedependence:r=r0e)]TJ /F4 7.9701 Tf 6.587 0 Td[(i!t.Wecannowrewritethedierentialequationin( 3{16 )asasimpleralgebraicequation: )]TJ /F3 11.9552 Tf 11.956 0 Td[(!2r0=)]TJ /F3 11.9552 Tf 9.299 0 Td[(!20r0+q m(E0)]TJ /F3 11.9552 Tf 11.956 0 Td[(i!r0B)(3{17)Theelectricdipolemomentinducedbytheoscillatingelectronissimplyp=qr.Ifwewishtoreturntoamacroscopicanalysis,wemultiplytheinducedelectricdipolebythenumberdensityofdipoles,Ne,toobtainthepolarizationP.Themacroscopicanalogof( 3{17 )isthus: (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)P0)]TJ /F3 11.9552 Tf 11.956 0 Td[(i!!cP0^B=0!2pE0(3{18)where!c=)]TJ /F3 11.9552 Tf 9.299 0 Td[(qB=misacyclotronfrequencyand!2p=Neq2=m0istheplasmafrequency.(Reallytheelectronmassshouldbereplacedwithaneectivemassintheexpressionforplasmafrequency,butforourpurposesweneednotdierentiatethetwo).Tothispoint,wehavebeenworkingwithcoordinate-freeexpressionstokeepouranalysisasgeneralaspossible,howeverinordertowriteoutanexplicitformforitisnowhelpfultointroduceacoordinatesystem.Callthedirectionofthebackgroundmagneticeldthezdirection,sothatB=B^z.ItimmediatelyfollowsthatP0^Bhasnozcomponent,andsothemagneticeldonlyaectsthexandycomponentsofthepolarization.Makinguseofourexplicitcoordinates,wecanwrite( 3{18 )asasystemofthreeequations: (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)Px)]TJ /F3 11.9552 Tf 11.955 0 Td[(i!!cPy=0!2pEx(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)Py+i!!cPx=0!2pEy(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)Pz=0!2pEz(3{19) 41

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Solvingforthecomponentsoftheelectricpolarizationgives: Px=0!2p [(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2!2c](!20)]TJ /F3 11.9552 Tf 11.956 0 Td[(!2)Ex+i!!cEyPy=0!2p [(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2!2c])]TJ /F3 11.9552 Tf 9.298 0 Td[(i!!cEx+(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)EyPz=0!2p (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)Ez(3{20)whichallowsustoreadothesusceptibilitytensor: $=!2p0BBBB@!20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2 (!20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2)2)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2!2ci!!c (!20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2)2)]TJ /F4 7.9701 Tf 6.587 0 Td[(!2!2c0)]TJ /F4 7.9701 Tf 6.586 0 Td[(i!!c (!20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2)2)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2!2c!20)]TJ /F4 7.9701 Tf 6.587 0 Td[(!2 (!20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!2)2)]TJ /F4 7.9701 Tf 6.587 0 Td[(!2!2c0001 !20)]TJ /F4 7.9701 Tf 6.586 0 Td[(!21CCCCA(3{21)Ifwemakethefollowingsubstitutionsintheinterestofbrevity: =1+!2p(!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2) (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2!2cz=1+!2p (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)Q=)]TJ /F3 11.9552 Tf 9.298 0 Td[(!!c!2p (!20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2)2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2!2c(3{22)thenwecanexpressthedielectrictensor,$=$1+$,inCartesiancoordinatesas: $=0BBBB@)]TJ /F3 11.9552 Tf 9.298 0 Td[(iQ0iQ000z1CCCCA(3{23)Here,Qisknownasthemagneto-opticalVoigtparameter.NoticethatasB!0,wehave!c!0andhenceQ!0aswell;thatistosayQvanishesintheabsenceofanexternalmagneticeld.Thoughourderivationdoesnotshowit(becausewedidnotconsiderdispersiveprocesses),ingeneralQisacomplexnumberthatwewriteasQ=Q0+iQ00. 42

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Considernowtheinvertiblematrix: A=1 p 20BBBB@110)]TJ /F3 11.9552 Tf 9.299 0 Td[(ii000p 21CCCCA(3{24)itisstraightforwardtoshowthatA)]TJ /F6 7.9701 Tf 6.586 0 Td[(1=Ayand: A)]TJ /F6 7.9701 Tf 6.587 0 Td[(1$A=0BBBB@)]TJ /F3 11.9552 Tf 11.955 0 Td[(Q000+Q000z1CCCCA(3{25)thatis,thematrixAdiagonalizes$.Putanotherway,( 3{24 )and( 3{25 )tellusthatthedielectrictensorhaseigenvectors: e1=1 p 20BBBB@1)]TJ /F3 11.9552 Tf 9.298 0 Td[(i01CCCCAe2=1 p 20BBBB@1i01CCCCAe3=0BBBB@0011CCCCA(3{26)witheigenvalues: 1=)]TJ /F3 11.9552 Tf 11.955 0 Td[(Q2=+Q3=z(3{27)Noticethate1ande2areexactlythepolarizationvectorsforrightandleft-circularlypolarizedlightpropagatinginthezdirection,respectively.AsaconsequenceofourdiscussioninSection 3.1 ,weconcludethatanonzeroVoigtparameter,Q,resultsinadierenceinthe(complex)indexofrefractionforrightandleft-handedpolarizations: NR=p )]TJ /F3 11.9552 Tf 11.955 0 Td[(QNL=p +Q(3{28) 43

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Thoughitisinelegant,itishelpfultowriteouttheVoigtparameterasQ=Q0+iQ00andexpandthesquarerootsin( 3{28 )toobtain: NR=1 p 2"r q ()]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0)2+Q002+)]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0+isgn(Q00)r q ()]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0)2+Q002)]TJ /F3 11.9552 Tf 11.955 0 Td[(+Q0#NL=1 p 2"r q (+Q0)2+Q002++Q0+isgn(Q00)r q (+Q0)2+Q002)]TJ /F3 11.9552 Tf 11.955 0 Td[()]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0#(3{29)Inpractice,Q0andQ00areoftentwoorthreeordersofmagnitudelessthat( 9 ),andsowecanapproximate: Re(NR)p )]TJ /F3 11.9552 Tf 11.956 0 Td[(Q01)]TJ /F3 11.9552 Tf 28.902 8.088 Td[(Q00 4()]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0)p )]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0Im(NR)jQ00j p 2()]TJ /F3 11.9552 Tf 11.955 -.001 Td[(Q0)(3{30)andsimilarequationsforNLsuchthat: Re(NR) Re(NL)=Im(NL) Im(NR)=p )]TJ /F3 11.9552 Tf 11.955 0 Td[(Q0 p +Q0(3{31)From( 3{30 ),weseethattherealpartoftheVoigtparameter,Q0,isresponsibleforadierenceintherealindexofrefractionbetweenthetwomodes,aphenomenonreferedtoasmagneticcircularbirefringence.Passingthroughacrystal,rightcircularandleftcircularpolarizationsobserveadierentopticalpathlength,andexitthecrystalwitharelativephaseshift.ThisresultsinarotationofthepolarizationvectortoproducethefamiliarFaradayeect.Q00manifestsadierenceintheimaginarypartoftheindexofrefraction,thatis,itcausestherightandleftcircularlypolarizedlighttoexperiencedierentabsorptioninthemedium.Thisisreferredtoasmagneticcirculardichroism,andservestointroduceellipticitytothebeam,howeveratroomtemperaturethistermisusuallynegligible( 43 ).Forourpurpose,letQbeentirelyreal.Consideranormalizedplanewavetravellinginthepositivezdirection,andpolarizedinthexdirection.Wecanexpressthewaveasa 44

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combinationofeigenvectorsin( 3{26 ): E=ei(kz)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)0B@101CA=1 2ei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)2640B@1)]TJ /F3 11.9552 Tf 9.299 0 Td[(i1CA+0B@1i1CA375(3{32)Ifthiswavepassesthroughamagneto-opticmediumoflength`,thentheright-handandleft-handcomponentswillexperiencedierentindicesofrefraction,andwhencombinedattheoutputofthematerialwillbe: E0=1 2eikR`0B@1)]TJ /F3 11.9552 Tf 9.298 0 Td[(i1CA+1 2eikL`0B@1i1CA(3{33)whereweignorethecommontemporalandspatialdependencies.HerekR=2NR=0for0thewavelengthoflightinvacuum,andkLisdenedsimilarly.Wecandene: k=kR+kL 2k=kR)]TJ /F3 11.9552 Tf 11.955 0 Td[(kL 2(3{34)torewrite( 3{33 )as: E0=1 2ei kl0B@eik`+e)]TJ /F4 7.9701 Tf 6.586 0 Td[(ik`)]TJ /F3 11.9552 Tf 9.299 0 Td[(ieik`+ie)]TJ /F4 7.9701 Tf 6.586 0 Td[(ik`1CA=ei k`0B@cosk`sink`1CA(3{35)Werecognizetheright-handsideof( 3{35 )asavectorthatmakesananglek`withthex-axis.Since,byassumption,jQj<<,wecanapproximate: k= 0(NR)]TJ /F3 11.9552 Tf 11.955 0 Td[(NL)=p 0 r 1)]TJ /F3 11.9552 Tf 13.15 8.088 Td[(Q )]TJ /F8 11.9552 Tf 11.955 19.018 Td[(r 1+Q !)]TJ /F3 11.9552 Tf 28.068 8.088 Td[(Q 0p (3{36)soweseethatthisrotationisproportionaltothe(real)Voigtparameter.Asimilarargumentwillshowthat: k2p 0(3{37)Since kdetermineshowthephaseofthelinearlypolarizedwaveevolvesthroughoutthecrystal(ignoringthefactthatitisrotating),( 3{37 )suggeststhatwecanthinkofp asaneectiveindexofrefractionforlinearlypolarizedlight. 45

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3.3OpticalPrinciplesofaFaradayIsolator 3.3.1Time-ReversalSymmetryAFaradayisolatoristheopticalanalogueofanelectricaldiode:itismeanttoallowlighttopassthroughinonedirectionandrejectcounter-propagatinglight.Suchadeviceisusefulwhereitispreferabletoisolatealasereldsourcefromitsexperimentalapplication.Forthepurposesoffrequencyandintensitystabilization,itiscommonpracticetouseFaradayisolatorstopreventback-reectedeldsfromfeedingbackintothelasermedium. Figure3-1. Anillustrationoftheeectofahalf-waveplate(HWP)onthepolarizationofanincidenteldtoaccompanythediscussionin 3.3.1 .AslightpropagatesforwardthroughtheHWP,itspolarizationismirroredabouttheopticaxisofthecrystal.Uponretroreection,itspolarizationisonceagainipped.Asx0=)]TJ /F3 11.9552 Tf 9.299 0 Td[(x,andy0=y,weseethatthefourthpictureisequivalenttotherst;thisisbecausetheHWPpreservestime-reversalsymmetry. Faradayisolatorsworkonthebasisthatanappliedexternalmagneticeldinducescircularbirefringenceinamagneto-opticcrystal.Thisbirefringencebreakstime-reversalsymmetry,andsotheforwardandbackwardpropagatingbeamscannotbetreatedwiththesamerules.Tounderstandthis,rstconsideranopticaldevicethatdoesobeytime-reversalsymmetry:ahalf-waveplate.Considerahalf-waveplateinthex-yplanewithfastaxisparalleltothex-axis,andsupposewehaveanincomingbeamoflinearlypolarizedlighttravelinginthez-directionwithpolarizationmakingananglewiththex-axis.Theincomingeldcanberepresentedas: E1=(^xcos+^ysin)ei(kz)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)(3{38) 46

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wheretheamplitudehasbeennormalizedandthearbitraryphasetermisdroppedforsimplicity.Theeectofthehalf-waveplateistoaddarelativephaseoftotheeldcomponentorthogonaltothefastaxis,andsotheoutgoingbeam'seldis: E2=)]TJ /F1 11.9552 Tf 6.102 -9.518 Td[(^xcos+^yeisinei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)=[^xcos()]TJ /F3 11.9552 Tf 9.298 0 Td[()+^ysin()]TJ /F3 11.9552 Tf 9.298 0 Td[()]ei(kz)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)(3{39)Weseethatafterpassingthroughthehalf-waveplate,thepolarizationoftheeldnowmakesanangle)]TJ /F3 11.9552 Tf 9.298 0 Td[(withthex-axis,thatistosayithasippedaboutthefastaxis,beingrotated2withrespecttotheincomingpolarization.Nowsupposeweweretoretro-reectthebeamandpassitonceagainthroughthehalf-waveplate,butthistimetravelingintheoppositedirection.Toremainintheframeofthebeam,wemakeacoordinatetransformationx!x0=)]TJ /F3 11.9552 Tf 9.298 0 Td[(x,y!y0=y,andz!z0=)]TJ /F3 11.9552 Tf 9.298 0 Td[(z.Inthesenewcoordinates,theretro-reectedbeambeforeasecondpassthroughthehalf-waveplatecanbewritten: E3=()]TJ /F1 11.9552 Tf 9.921 .166 Td[(^x0cos)]TJ /F1 11.9552 Tf 12.671 .166 Td[(^y0sin)ei(kz0)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)(3{40)Asthexandx0axesareparallel,thefastaxisisthex0axis,andsotheeectofthehalf-waveplateuponasecondpassfromthebeamistoagainaddarelativephasetermtotheeldcomponentinthey=y0direction: E4=)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[()]TJ /F1 11.9552 Tf 9.921 .166 Td[(^x0cos)]TJ /F1 11.9552 Tf 12.671 .166 Td[(^y0eisinei(kz0)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)=(^xcos+^ysin)ei()]TJ /F4 7.9701 Tf 6.586 0 Td[(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)(3{41)Comparing( 3{38 )and( 3{41 ),weseethattheretro-reectedbeamisreturnedtoitsinitialpolarizationstate;theonlydierencebetweeninitialandnaleldsbeingthedirectionofpropagation(andthephasetermthatwehavedropped).Thetime-reversalsymmetryofthehalf-waveplateisnotcharacteristicofalloptics.Consideropticalmaterialsdescribedintheprevioussection,whichwereshowntoexhibitcircularbirefringence.Letusexploretheimplicationsofthisphenomenon.Againwecanconsiderthesameincomingeld,nowincidentonourmagneto-opticcrystal,andwe 47

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explicitlysplittheexpressionintorightandleft-handpolarizations: ~E1=(^xcos+^ysin)ei(kz)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)=ei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t) 2ei(^x)]TJ /F3 11.9552 Tf 11.955 0 Td[(i^y)+e)]TJ /F4 7.9701 Tf 6.587 0 Td[(i(^x+i^y)(3{42)Thetildeisusedtodistinguishfromthepreviouscaseofthehalf-waveplate.Withoutlossofgenerality,wecansaythattheeectofthecrystalistoaddarelativephaseof2Ctotheright-handcircularpolarizationforwavestravelinginthepositivezdirection.Theeldexitingthecrystalisthus: ~E2=ei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t) 2ei+i2C(^x)]TJ /F3 11.9552 Tf 11.955 0 Td[(i^y)+e)]TJ /F4 7.9701 Tf 6.587 0 Td[(i(^x+i^y)=ei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t) 2eiCei(+C)(^x)]TJ /F3 11.9552 Tf 11.955 0 Td[(i^y)+e)]TJ /F4 7.9701 Tf 6.586 0 Td[(i(+C)(^x+i^y)=ei(kz)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)eiC[cos(+C)^x+sin(+C)^y](3{43)wherewecandroptheextraneousphasetermeiC.WeseethatthecrystalrotatedthepolarizationoftheeldbyanangleC.Asinouranalysisofthehalf-waveplate,weretro-reectthebeamandmakethecoordinatetransformationx!x0=)]TJ /F3 11.9552 Tf 9.298 0 Td[(x,y!y0=y,andz!z0=)]TJ /F3 11.9552 Tf 9.299 0 Td[(ztoremainintheframeofthebeam: ~E3=)]TJ /F3 11.9552 Tf 9.299 0 Td[(ei(kz0)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)[cos()]TJ /F3 11.9552 Tf 9.299 0 Td[()]TJ /F3 11.9552 Tf 11.955 0 Td[(C)^x0+sin()]TJ /F3 11.9552 Tf 9.298 0 Td[()]TJ /F3 11.9552 Tf 11.956 0 Td[(C)^y0]=)]TJ /F3 11.9552 Tf 10.494 8.088 Td[(ei(kz0)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t) 2e)]TJ /F4 7.9701 Tf 6.586 0 Td[(i(+C)(^x0)]TJ /F3 11.9552 Tf 11.956 0 Td[(i^y0)+ei(+C)(^x0+i^y0)(3{44)Herewemightbeinclinedtoaddanother2Cofphasetotheright-handcircularpolarization,butthatwouldbeinhaste.Noticefrom( 3{26 )thatp 2e1=)]TJ /F1 11.9552 Tf 9.299 0 Td[((^x0+i^y0)andp 2e2=)]TJ /F1 11.9552 Tf 9.299 0 Td[((^x0)]TJ /F3 11.9552 Tf 12.335 0 Td[(i^y0),thatis,theleft-handcircularpolarizationvectorinourprimedcoordinatesiswhatwehavebeenreferringtoastheright-handcirculareigenvectorofthemagneto-opticmaterial'sdielectrictensorandsimilarlyfortheright-handcircularpolarization.Consequently,forwavespropagatinginthenegativezdirection,weadd2C 48

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ofphasetotheleft-handcircularpolarization: ~E4=)]TJ /F3 11.9552 Tf 10.494 8.088 Td[(ei(kz0)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t) 2e)]TJ /F4 7.9701 Tf 6.587 0 Td[(i(+C)(^x0)]TJ /F3 11.9552 Tf 11.955 0 Td[(i^y0)+ei(+3C)(^x0+i^y0)=)]TJ /F3 11.9552 Tf 9.298 0 Td[(eiCei(k0z)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t) 2e)]TJ /F4 7.9701 Tf 6.587 0 Td[(i(+2c)(^x0)]TJ /F3 11.9552 Tf 11.955 0 Td[(i^y0)+ei(+2C)(^x0+i^y0)=)]TJ /F3 11.9552 Tf 9.298 0 Td[(ei(k0z)]TJ /F4 7.9701 Tf 6.586 0 Td[(!t)eiC[cos()]TJ /F3 11.9552 Tf 9.299 0 Td[()]TJ /F1 11.9552 Tf 11.955 0 Td[(2C)^x0+sin()]TJ /F3 11.9552 Tf 9.298 0 Td[()]TJ /F1 11.9552 Tf 11.955 0 Td[(2C)^y0]=ei()]TJ /F4 7.9701 Tf 6.587 0 Td[(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t)eiC[cos(+2C)^x+sin(+2C)^y](3{45)Insteadofbeingrotatedbacktoitsincomingpolarizationstate,comparing( 3{42 )and( 3{45 )weseethattheretro-reectedlighthasapolarizationthathasbeenrotatedbyanangle2C. Figure3-2. AnillustrationofthebasicprinciplebehindaFaradayrotator.Incomingverticallypolarizedlightisrotated-45degreesbythehalf-waveplate,andthen+45degreesbythemagneto-opticelementresultinginnonetrotationofthepolarization.Returningverticallypolarizedlightisrotated+45degreesbythemagneto-opticelement,andthenanother+45degreesbythehalf-waveplate,resultingin90degreerotation.Apolarizerbeforetherotatorcanthereforebeusedtoseparatetheforwardandreturnbeams. Figure 3-2 providesanillustrationoftheworkingprinciplebehindakeypartofaFaradayisolator.Lightwithapreferredpolarizationentersoneendoftheisolatorandisrotated45degreesbyahalf-waveplate(HWP).Thelightthenpassesthoughamagneto-opticelement(MOE)whichrotatesthepolarizationbackthroughthesame 49

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45degrees,resultinginnonetpolarizationrotation.Lightback-reectedthroughtheisolatorisrotatedbytheMOEagainthrough45degreesinthesamedirectionasitwasintheforwardpropagatingcase.However,theHWProtatestheretro-reectedlight45degreesintheoppositedirectionasitdidtotheforwardtravelingbeam.Thenetresultisa90degreerotationofthepolarizationuponback-reection.Theadditionofpolarizersateitherendofthesetupallowsforanisolatortobeconstructedfromtherotator:thepreferredpolarizationwillpassunperturbedintheforwarddirection,butuponreturnthewillberejected.Terbiumgaliumgarnet(TGG)iscommonlychosenasamagneto-opticelementmaterial,asithasarelativelylargeVerdetconstantof)]TJ /F1 11.9552 Tf 9.298 0 Td[(40rad=Tmfor1064nmlight.From( 3{15 ),wecancalculatethatthiscorrespondstoanintegratedmagneticeldof19.64Tmmoverthebeampathinordertoachieve45degreerotationintheTGG.ForTGGcrystalsontheorderof20mminlengthwerequiremagneticeldsof0.982T1T. 3.3.2WedgePolarizersIn 3.3.1 wediscussedhowtheFaradayeectbreakstimereversalsymmetryandallowsustodistinguishbetweenforwardandbackwardpropagatingbeams.Inordertoconstructanisolatorusingthisprinciple,wemustselectforonepolarizationoranother.Todothis,thecurrentIFIuseswedgepolarizers.Unlikedichroicorthinlmpolarizers,wedgepolarizersoperatebyusingabirefringentcrystaltospatiallyseparateorthogonallinearpolarizations.Thisdevicehastheadvantageofaveryhighisolationratio,theabilitytoavoidlossyabsorptivecoatings,andnearlyequivalenttreatmentofthetwopolarizations. 50

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Figure3-3. Aray-tracingdiagramforawedgegeometry. AverysimplediagramofawedgegeometryisgiveninFigure 3-3 .Usingthenamingconventionsofthegure,basicgeometricargumentsandSnell'slawofrefractiongive: =arcsinsin n=)]TJ /F3 11.9552 Tf 11.955 0 Td[(=arcsin(nsin)(3{46)wherenistheindexofrefractionofthewedgematerial,istheincidentangle,andishalfthewedgeangle.Theanglethattheoutgoingbeammakeswiththeoutgoingfacenormal,,canbeexpressedas: =arcsinnsin)]TJ /F1 11.9552 Tf 11.955 0 Td[(arcsinsin n(3{47)Thisisclearlynotlinearinor,butinthecasewherebothanglesaresmallthiscanbeapproximatedas: n)]TJ /F3 11.9552 Tf 11.956 0 Td[((3{48)Theanglethattheoutgoingbeammakeswithrespecttotheincomingbeamis: w=j)]TJ /F3 11.9552 Tf 11.955 0 Td[()]TJ /F3 11.9552 Tf 11.955 0 Td[(j(3{49) 51

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whichinthesmallanglescenarioapproximatesas: w(n)]TJ /F1 11.9552 Tf 11.955 0 Td[(1)(3{50)Whileonlyapproximatelytrue,( 3{50 )makescleartheexplicitdependenceofthedeectionofthebeamontheindexofrefractionofthewedge.Thiscontrastswithplanaropticalelements,forwhichthedeectionangleofthebeamiszeroregardlessofmaterialindex.Supposethewedgewasmadeofabirefringentmaterial,withindexofrefractionnpforlightpolarizedintheplaneofthepage(ppolarization)andnsforlightpolarizedorthogonaltotheplaneofthepage(spolarization).Inthiscasethedeectionofthebeamoutofthewedgewillbedependentonitsincomingpolarization,andsothewedgeservestospatiallysplitorthogonalpolarizations.Theseparationangleis: w=j(np))]TJ /F3 11.9552 Tf 11.955 0 Td[((ns)j(3{51)While( 3{51 )isgenerallynotlinearinor,forsmallanglesthiscanbeapproximatedas: wjnp)]TJ /F3 11.9552 Tf 11.955 0 Td[(nsj(3{52)whichislinearinthedierenceinrefractiveindexforthesandppolarizations.TheutilityofwedgepolarizersinaFaradayisolatorisstraightforward.WorkingfromFigure 3-2 ,wecanseethatifweweretoplaceapolarizeralignedwiththeincominglightbeforetheHWP,thenuponretro-reectionthebeamwouldbedeectedfromtheincomingbeambythewedge.Thiswillpreventback-reectedbeamscarryinginformationabouttheopticalsetupfrominteractingwiththelasersource. 3.4CurrentInputFaradayIsolatorDesignAcartoonofthecurrentIFIdesignisgiveninFigure 3-5 .Theisolatorconsistsofsevenopticalcomponents:twocalcitewedgepolarizers,ahalf-waveplate,twoterbiumgaliumgarnet(TGG)crystals,aquartzrotator,andadeuteratedpotassiumphosphate 52

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Figure3-4. AconceptualdiagramofthefunctionofaFaradayisolator.Theisolatorcanbethoughtofasadirectionaldevicewithfourports,twooneachsidecorrespondingtothetwoorthogonalpolarizationspands.Inonedirection,theFaradayispolarizationmaintaining,andintheoppositedirectionitipsonepolarizationintoitsorthogonalpolarization.Ourabilitytoselectforapolarizationbecomesanabilitytoselectforaparticulardirection. (DKDP)crystal( 13 ).TheIFIispartoftheInputOpticssubsystemlocatedonHAM2intheAdvancedLIGOdetectors.Atthisplaceintheexperiment,laserpowersareintendedtoreachupto125W.Withanabsorptioncoecientof1:510)]TJ /F6 7.9701 Tf 6.587 0 Td[(3cm)]TJ /F6 7.9701 Tf 6.586 0 Td[(1,thepowerabsorbedby20mmofTGGisover300mW.Thisheatingwillgiverisetoathermalgradientacrossthecrystal,whichinducelinearbirefringencethatwillmanifestasellipticityandultimatelyopticallossandareductioninisolationratio( 29 ).Toreducethisrisk,theTGGisissplitintotwocrystals,withaquartzrotatorplacedbetweenthem.TherstTGGrotatesthepolarizationby22:5,andthenthequartzrotatorrotatesby67:5,sothatthepolarizationincidentonthesecondTGGisat90tothatoftherst.SolongasthetwoTGGcrystalshavethesamelengthandsimilarthermalprole,thismethodwillaverageoutthephasedelaysineitherpolarization,therebycancelinganybirefringenceeects.ThesecondTGGalsorotatesthepolarizationby22:5,bringingthetotalrotationto112:5.Thehalf-waveplatefastaxisisthenchosentoprovide67:5degreesofrotationintheforwarddirection,resultinginanetrotationof180.Heatingfromthehighpowerlaserbeamwillalsoleadtothermallensingoftheopticalcomponents( 44 ),causingmismatchbetweenthespatialmodeoftheIFIoutputandtheacceptedmodeofthecoreopticalcomponentsdownstream( 5 ).Toaccountfor 53

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this,aDKDPcrystalwithnegativethermo-opticcoecientisincludedtoserveasanegativelens.Onthebench,wheretheusercanmanuallyadjustthecomponents,theopticalthroughputofthisconstructionhasbeenmeasuredtobeanywherefrom96%to98%,whichmeetsthe>95%requirementfortheIFIbutfallsshortofourgoalof99%throughput. Figure3-5. AconceptualdrawingofthecurrentaLIGOInputFaradayIsolator.Thequartzrotator(QR)placedbetweenthetwoTGGcrystalscompensatesforthethermallyinducedbirefringenceinTGG.TheDKDPisanegativelensmaterialusedtocompensateforthermallensingoftheTGG. 3.5OpticalRedesignforLowLossFaradayIsolatorThemethodforLLFIdesignwastoimproveuponthedesignoftheIFIuntilasucientlyhighopticalthroughputwasobtained.WeexpectasmuchashalfoftheopticallossintheIFItobefromreectionsonopticalsurfaces,andsoreducingthenumberofelementsintheFaradaypathiscriticaltoreachingourthroughputgoal.Fortunately,theLLFIisintendedtositaftertheSignalRecyclingCavity(SRC)oftheinterferometer,whereitwillcombinethesqueezedbeamfromtheVacuumOpticalParametricOscillator(VOPO)andthegravitational-wavesignalbeam.Becausetheinterferometeroperatesnearadarkfringe,theopticalpoweroutoftheSRCisverylow, 54

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Figure3-6. TheKAGRAInputFaradayIsolatorasitisbeingconstructedinthepre-stabilizedlaser(PSL)cleanroom.TheKAGRAisolatorisbasedontheaLIGOdesignandisnearlyidentical.Theentireassemblyhasitsowndedicatedaluminumbreadboard,andisshownsittingonamarbleblocktominimizetheinteractionbetweentherotatormagnetsandthestainlesssteelopticaltablebelowtheblock.Fromrighttoleft,theelementsare:1)inputcalcitepolarizerinaSiskiyoumount,2)half-waveplateincustomrotationmount,3)magnetassemblywithTGGandquartzrotatorcrystals,4)unpopulatedmountwheretheDKDPwilleventuallybeinstalled,and5)theoutputcalcitepolarizerinaSiskiyoumount.Photocourtesyofauthor. andlaserheatingoftheopticsisnolongeraconcern.ThisallowsustoeliminatethethermalcompensationelementsintheIFIdesign.Byexcludingthequartzrotator,thenegativelensDKDP,andmakingtheTGGmonolithicwereducethenumberofreectivesurfacesfromfourteentoeight.WeareleftwiththefouressentialcomponentsofaFaradayisolatorwhosecharacteristicsshouldbechosentomaximizeopticalthroughput.TheHWPisano-the-shelfitemthatisnotexpectedtobeasignicantsourceofopticalloss.Similarly,TGGiscurrentlytheoptimalmagneto-opticmaterialfortheFaradayasithasrelativelylowabsorptionandalargeVerdetconstant.Thecurrentcalcitewedgepolarizers,however,areo-the-shelfitemswhichcannotbesuperpolishedandhaverelativelyhighdesignreectivities(guaranteeoflessthan2500ppmpersurface).TheUFLLFIreplacesthecalcitewedgeswithpotassiumtitanylphosphate(KTP)wedges.TheadvantageoftheseKTPwedges 55

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isthatthematerialisharder,canbesuperpolished,andconsequentlycanhavebetteranti-reectivecoatingsandlowerscatterlosses. Figure3-7. AconceptualdrawingoftheLLFIopticallayout.ThequartzrotatorandDKDPfromthecurrentIFIhavebeenremoved,theTGGcrystalisnowmonolithic,andthecalcitewedgepolarizershavebeenreplacedwithKTPwedgepolarizers. Figure3-8. Theunfortunatefateofacalcitecrystalwedgepolarizerdroppedintheprocessofinstallation.Thewedgeremainedserviceableuntilareplacementcouldbeprocured.CalciteisabrittlematerialandhasaMohshardnessofabout3(comparedto5forKTP).Photocourtesyofauthor. 56

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3.6MagnetDesignforLowLossFaradayIsolatorAsexplainedinSection 3.3 ,theaveragemagneticeldrequiredtoprovide45degreesofrotationina20mmTGGcrystalisnearly1T.InthecontextoftheaLIGOsetup,thiscanonlybeachievedpracticallythroughtheuseofrare-earthpermanentmagnets.Neodymiummagnetsareboththestrongestintermsoftheirremanenceorresidualmagnetism,whichisthemagnetizationremaininginthematerialoncethemagnetizingeldisremoved,andintermsoftheircoercivity,whichistheexternalmagneticeldmagnituderequiredtoreturnthemagnetizedmaterialtozeromagnetization.Thesepropertiesmakeneodymiummagnetsoptimalforuseincompact,high-eldapplications.ThecurrentIFIusesacylindricalmagnetenclosurecomposedofsevenmagnetdisks.Withtheexceptionofthecentraldisk,eachIFIdiskisfurtherdividedinto16individualmagnetwedges.FortheLLFI,thenumberofmagnetdiskswasreducedtove,andthenumberofwedgesperdiskreducedfrom16to12.Eachdiskhasaninnerradiusof1/2inandanouterradiusof2in,andathicknessofeither1/2or1in.Eachwedgecanbeeithermagnetizedthroughitsthickness(alongthecylindricalaxisofamagnetdisk),orthroughitslength(throughtheradiusofamagnetdisk).ThedisksaremodeledusingCOMSOLsoftware,andtheeldprolesareusedtodesignthearrangementofmagnetdisks.Speciyingtheremanenceandmagnetizationdirection,aswellastheboundaryconditionthattheeldgotozeroveryfarfromthemagnetdisk,COMSOLusesniteelementanalysistocomputethemagneticeldateverypointonagridwithinaspeciedvolume.Becauseofthehighcoercivityofthemagnets,weassumethateachindividualmagnetwedgecanbetreatedasanindependenteldsource;thatis,weassumethepresenceofothermagnetsdoesnotchangethemagnetizationofanyindividualmagnet.ThisassumptionisimplicitintheCOMSOLsimulationsofmagneticeldsfromindividualmagnetdisks.Further,itallowsustomodelthemagneticeldofthefullmagnetassemblyassimplythesumofcontributionsfromeachmagnet.Putexplicitly,fora 57

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Figure3-9. ACOMSOLmodelofamagnetdisk.Ontheleftisthebaredisk,andontherightthediskisshownwiththesimulationmeshoverlaced. Figure3-10. OutputsofCOMSOLsimulationsforradiallyandaxiallymagnetizeddisks.Attoparethe3-dimensionalvectorplotsofthemagneticeld,andbelowareprolesofthemagneticeldmagnitudealongthezaxis. collectionofmmagnetdisks,eachwithcorrespondingeldprolesBi(r),thenthetotalmagneticeldforanarbitraryarrangementofmagnetdisks: B(r)=mXi=1(LTiLRiBi)(r)(3{53)whereLTiandLRiareappropriatetranslationandrotationoperations,respectively.Weseefrom( 3{15 )thatweshouldseektomaximizetheeldalongtheopticalpath. 58

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Thesymmetriesofourmagnetconstructionallowustosimplify( 3{53 )signicantly.Becausewehaveacylindricalrotatormagnet,allmagnetdisksmustbecenteredonaline,whichwedesignateasthezaxis.Thisrestrictsourtranslationstobealongthezaxis.Further,allmagnetdisksmusthavetheiraxesparalleltooneanother,andsotheonlyallowedrotationsareeitherby0(thetrivialrotation)orby180aboutalineinthex-yplane.Asthemagnetdisksareassumedtobesymmetricabouttheiraxis,arotationby180aboutonelineinthex-yplaneisequivalenttoa180rotationaboutanyotherlineinthesameplane.Moreover,allofthemagnetdisksareeithersymmetric(magnetizedthroughthethickness)orantisymmetric(magnetizedthroughradius)aboutthex-yplane,andasaconsequencesoaretheirrespectivemagneticelds.Thus,asymmetricdiskhasaeldthatisinvarianttoallowedrotations,whileanantisymmetricdiskhasaeldwhosesignipsunderallowed(nontrivial)rotations.Further,thesymmetryofeachdiskaboutthezaxisdictatesthatthemagneticeldforpointsalongtheaxisisitselfalongtheaxis.Allofthesesymmetriesallowustoturnthevectorproblemin( 3{53 )toascalarone: B(z)=mXi=1()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)iBi(z)]TJ /F3 11.9552 Tf 11.955 0 Td[(si)(3{54)wheresiisthecoordinateofthecenteroftheithdisk,and: i=8>><>>:1,iftheithdiskisantisymmetricandrotated0,otherwise(3{55)Thesymmetricdiskshavemaximaleldamplitudeattheirorigin,whiletheantisymmetricdiskshavetwoamplitudemaximaoneithersideoftheorigin(seeFigure 3-10 ).Asaconsequence,theoptimalarrangementsplacesasymmetricdiskwiththicknessroughlyequaltothecrystallengthatthecenter,andantisymmetricdisksoneitherside.Figure 3-11 illustratesthebasicconcept.Diskswiththeirnorthpolesontheoutercircumferencearegroupedtogetherandcontactthesouthfaceofthesymmetric 59

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centralmagnet;ontheothersidesitsthegroupofmagnetswithnorthpolesontheinnercircumference.Throughexperimentationwitheldproles,itwasdeterminedthatitwaspossibletoconstructamagnetcapableofachieving45degreesofrotationina14mmlongTGGcrystalwithintheparametersspeciedatthebeginningofthesection.ToreducetheTGGlengthbeyondthiswouldrequireamodicationofthemagnetradiiornewmagneticmaterials.Ournaldesignusesa1inthickaxiallymagnetizedcentraldiskcomposedoftwelveN42gradeneodymiummagnets,andfour1/2inthickradiallymagnetizeddiskseachcomposedoftwelveN52gradeneodymiummagnets.Thereareintendedtobenospacesbetweenthemagnets,andmodelingsuggeststhatitshouldachievemaximalrotationof<60ina22mmTGGcrystal. Figure3-11. Acartoondepictingthepolarizationorientationsofeachmagnetregion.Themagnetsarethegreyregions,andmagneticeldlinesaredrawninorange.ThecentrallylocatedTGG(yellow)experiencestheregionofstrongesteld. 3.7LossBudget 3.7.1LossToOpticalElementsWeanticipatethegreatestsourceoflossintheFaradaytobereectionatthesurfaceofandabsorptionwithinthebulkoftheopticsthemselves.Forthisreason,boththeKTP 60

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Table3-1. ExpectedopticallossofeachcomponentintheUFLLFIdesign. IsolatorelementOpticalloss(ppm) KTPreection(perface/total)500/2000KTPabsorption(percrystal/total)25/50HWPreection(perface/total)300/600HWPabsorption50TGGreection(perface/total)500/1000TGGabsorption(20mmcrystal)3000 Isolatortotal6700 wedgesandTGGcrystalshavebeensuperpolishedbyPhotonLaserOptikandARcoatedbyMLDTechnologies.Thecoatingsaredepositedwithanion-beamsputteringtechnique(IBS),whichprovidesforthebestabsorption,surfaceroughness,andreproducibilityoftheconventionalcoatingmethods.Thedesigngoalis200ppmreectivitypersurface,with<500ppmguaranteed.Table 3-1 givestheexpectedlossesduetoeachopticalcomponent. 3.7.2MisalignmentLossesBeyondlosstoreections,absorption,andscattering,amisalignmentoftheopticscanmanifestasareducedpowerthroughput.WhiletheselossesarenotfundamentaltotheFaradaydesign,inpracticetheyareimpossibletoeliminatecompletelyandthusshouldbecharacterized.Asanexample,supposetheopticaxesofthetwoKTPwedgepolarizersaremisaligned.Inparticular,supposethesecondpolarizerhasanaxisthatisrotatedfromthatoftherstbyasmallangleintheplaneperpendiculartothebeamaxis.Asaconsequence,theacceptedandrejectedpolarizationsdierbetweenpolarizers.IfthebeaminputtotheFaradayisperfectlypolarizedintheacceptedpolarizationoftherstwedge,evenwithidealalignmentoftheHWPandTGGtherelativepowerexitingtheisolatorintherejectedpolarizationis: Prej Pin=1)]TJ /F1 11.9552 Tf 11.955 0 Td[(cos22(3{56)Foran=2,thiscorrespondsto0:1%oftheincomingbeampowerendingupintherejectedpolarization. 61

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Wenowseekarigorousapproachtoestimatinglossesduetomisalignmentofoptics,andindoingsohopetooptimizethealignmentprocedureinordertominimizeloss.Restrictingourconsiderationonlytomisalignentsintwodimensions,thatiswesupposeallappropriatecrystalaxesareperpendiculartothebeamaxis,wecanuseJonescalculustoformalizeourproblem.InJonescalculus,theelectriceldinputtoanopticalsystemisrepresentedbyanormalizedcolumnvectorwiththespatialandtemporaldependencestrippedaway: E(t)=E0ei(kr)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t))166(!a=1 0B@ExEy1CA(3{57)wheretheconventionistoletkbeinthepositivezdirection,ExandEyarethecomplexeldamplitudesinthexandydirectionsrespectively,andischosensothataya=1.Thisnotationalhasanunfortunateoverlapwithconventionsinquantummechanics;itisimportanttonotethattheseJonesvectorsareinnowayrelatedtothecreationandannihilationoperators.OnecanthinkofaJonesvectorasthecomplex-valuedpolarizationvectorofaplanewave.Astheeldpropagatesthroughanopticalsystem,thiscomplexpolarizationmaychange,andthisisformalizedbyrepresentingeveryopticalelementwithasquarematrix,J.Theoutputeldofanopticalelement,a0,isgivenbythismatrixactingontheinputeld: a0=Ja(3{58)andconsequentlytheintensityoftheoutputeldrelativetothatoftheinputeldis: I0 I=(a0)ya0 aa=ayJyJa(3{59)ThespecicformofJforagivenopticalelementisgenerallyinferredfromaknowledgeoftheelement'spropertiesorpulledfromalibraryofknownmatricesforcommonopticalelements(Appendix B hassomeexamples).ToapplythisformalismtotheLLFI,considerthespecicsoftheopticalsetup.Wewilldenethedirectionoftheincomingbeamtobethepositivezdirection.The 62

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LowLossFaradayIsolatorconsistsoffourorderedcomponents:anincomingpolarizer(KTP1),ahalf-waveplate(HWP),aTGGcrystal,andanoutgoingpolarizer(KTP2).Eachpolarizerhastwooutputs,appolarizedbeamandanspolarizedbeamwhicharespatiallyseparatedandshouldbeconsideredindependently.Considerthepoutputofeitherpolarizer:supposingthatthepolarizermakesananglewiththexaxis,itseectistoprojecttheincomingeldalongthevector^p=cos^x+sin^y.Assigningtotheincomingeldthegeneralforma=ax^x+ay^y,thecorrespondingJonesvectorsatises: Ppa=proj^pa=a^p j^pj2^p=(axcos2+aycossin)^x+(axcossin+aysin2)^y(3{60)fromwhichwecanreadothematrixelements: Pp(i)=0B@cos2icosisinicosisinisin2i1CA(3{61)wheretheisubscriptremindsusthattherearetwopolarizers,eachwithauniquei.Asimilaranalysisforthesoutputyields: Ps(i)=0B@sin2i)]TJ /F1 11.9552 Tf 11.291 0 Td[(cosisini)]TJ /F1 11.9552 Tf 11.291 0 Td[(cosisinicos2i1CA(3{62)Alternatively,wecouldhaverecognizedthatsincetheoutgoingpandspolarizationsperfectlyreconstructtheincomingpolarization,itmustbethatPp+Ps=1.TheHWP,withfastaxismakingananglewiththexaxis,servestoiptheincomingeldvectoraboutthevector^f=cos^x+sin^y.Againassumingageneralformfora,thisisequivalentto: Wa=2proj^fa)]TJ /F7 11.9552 Tf 11.955 0 Td[(a=(2axcos2+2aycossin)]TJ /F3 11.9552 Tf 11.955 0 Td[(ax)^x+(2axcossin+2aysin2)]TJ /F3 11.9552 Tf 11.955 0 Td[(ay)^y=(axcos2+aysin2)^x+(axsin2)]TJ /F3 11.9552 Tf 11.955 0 Td[(aycos2)^y(3{63) 63

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andsothecorrespondingJonesmatrixisgivenby: W=0B@cos2sin2sin2)]TJ /F1 11.9552 Tf 11.291 0 Td[(cos21CA(3{64)TheconclusioninSection 3.3.1 thathalf-waveplatessatisfytime-reversalsymmetryisreectedinthefactthatitsJonesmatrixisinvoluntary,thatis:W=W)]TJ /F6 7.9701 Tf 6.587 0 Td[(1.TheJonesmatrixfortheTGGrotatoristhesimplestoftheelements:itrotatesapolarizationby,andsoitcanberepresentedastheusualrotationmatrix: F=0B@cos)]TJ /F1 11.9552 Tf 11.291 0 Td[(sinsincos1CA(3{65)Exceptinthespecialcasewhere=n,forintegern,theJonesmatrixfortheTGGisnotinvoluntaryandinsteadF)]TJ /F6 7.9701 Tf 6.587 0 Td[(1()=F()]TJ /F1 11.9552 Tf 9.299 0 Td[().WealsoseethatFistheonlyJonesmatrixofthefourthatisnotHermitian,asFy()=F()]TJ /F1 11.9552 Tf 9.298 0 Td[()=F)]TJ /F6 7.9701 Tf 6.586 0 Td[(1().Noneofouraboveanalysisinvokedthedirectionofthewaverepresentedbya(excepttochooseaplaneofpolarization).Infact( 3{61 )through( 3{65 )arevalidforboththeincomingandretro-reectedbeams.ThereareafewquantitiesofparticularinterestincharacterizingaFaradayisolator,therstofwhichistheisolationratio.Thisistheratioofincomingpowertothatofunrejectedback-propagatingpower.Forourisolator,itmaybecalculatedusingtheJonesvector: a0=Pp(1)F()W()Pp(2)Pp(2)W()F()Pp(1)a(3{66)Thesecondandthirdquanitiesaretheforwardandbackwardlosses.Thesearerelatedtoratiosofoutgoingpowertoincomingpowerintheforwardandbackwarddirectionsrespectively.Theforwardlossinourisolatoris: 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(f=ayPyp(1)Fy()Wy()Pyp(2)Pp(2)W()F()Pp(1)a(3{67) 64

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andthebackwardslossissimilar: 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(b=ayPys(1)Fy()Wy()Pyp(2)Pp(2)W()F()Pp(1)a(3{68)InthecaseoftheLLFI,boththeforwardandbackwardlossaretreatedwithequalimportance,asthesqueezedpathincludesonepassineachdirection.Wehavenowtheframeworkforaquantitativecharacterizationofmisalignmentlosses.SupposeweareworkingwithaninputalignmentbeamwithpowerP,andourpowermeterhasapowernoiseoor.Chooseacoordinatesystemsuchthattheincomingpolarizationisdescribedbya=(1;0)T.AswillbediscussedinmoredetailinSection 4.3 ,thealignmentoftherstKTPwedgepolarizertoanincomingpolarizedbeamamountstominimizingthepowerintherejectedsdirection.Accordingto( 3{59 ),theupperlimitonmisalignmentisdeterminedbytheequation: P=ayPys(1)Ps(1)a=aT:P2s(1)a(3{69)Inpractice,wemighthavea70mWinputalignmentbeamandourpowermeteranoiseoorof100W(notuncommonforo-the-shelfpowermeters).Inthiscase( 3{69 )hasthesolution1=2:2,correspondingtoalossofabout1500ppm.AsthealignmentoftherstKTPdoesnotsignicantlyreducethepowerinthetransmittedbeam,thealignmentofthesecondKTPwedgeisalsogovernedby( 3{69 ),excepttheangle1intheargumentofthepolarizerJonesmatricesisreplacedwith2)]TJ /F3 11.9552 Tf 12.337 0 Td[(1.Thisyieldstheresult2=21=4:3.Intheworst-casescenario,thesemisalignmentsalonecouldaccountfor2800ppmloss.TheTGGshouldbeplacedintotherotatormagnettoadepththatresultsin45ofrotation.Thiscanbedoneinoneoftwoways:1)measurethepowerinbothpandspolarizationsoutputfromKTP2andadjusttheTGGpositionuntiltheyareequal,whichwewillrefertoasthebalancedscheme,or2)placearetro-reectorimmediatelyaftertheTGG,measurethepowerintheppolarizationheadinginbackwardsoutofKTP1,and 65

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adjusttheTGGpositionuntilthepowerisminimized.Inconsideringtherstoption,ifwecanmeasurebeampowertowithinanaccuracyP=P,thentheequationthatdeterminesthemaximalmisalignmentoftheTGGcanbeexpressedas: P p 2P=(a01;p)ya01;p)]TJ /F1 11.9552 Tf 11.955 0 Td[((a01;s)ya01;s(3{70)subjecttotheconstraintsthatj1j2:2and2)]TJ /F3 11.9552 Tf 11.955 0 Td[(14:3,where: a01;p=Pp(2)F()Pp(1)aa01;s=Ps(2)F()Ps(1)a(3{71)Thesquarerootin( 3{70 )comefromthefactthat(a01;p)ya01;p(a01;s)ya01;s1=2,andsothestandarddeviationassociatedwitheithermeasurementisP=2P.ForanygivenP=P,theallowedmisalignmentsdeneavolumeofthree-dimensionalparameterspace.AnexampleisincludedinFigure 3-12 ;herewehaveassumedameasurementuncertaintyof5%,consistentwiththepowermetersinourlaboratory( 49 ).Aswemightexpect,thesurfaceapproximatesaregionoftheplanedenedby()]TJ /F1 11.9552 Tf 12.445 0 Td[(45)+1)]TJ /F3 11.9552 Tf 12.444 0 Td[(2=0,subjecttotheconstraintsthatj1j2:2j2)]TJ /F3 11.9552 Tf 12.619 0 Td[(1j.Foragiven1and2,thisalignmentmethodisverysensitivetochangesin.Wecanunderstandthisfortworeasons:rst,thatthepolarizationincomingtoKTP2isnearlyat45withrespecttoitsp(ors)axis.Rememberingthatthepoweroutofapolarizerscalesasthesquaredcosineofthisdeviationangle,thenas@cos2j==4=)]TJ /F1 11.9552 Tf 9.299 0 Td[(1,achangeinanglewillhaveaone-to-onecorrespondencetoareductioninpowerinthatpolarization.Further,becauseweareconcernedwiththedierenceinpowersbetweenthepandspolarizations,anysubtractioninpowerfromoneaddsthesamepowertotheother,meaningthatachangeinangle(inradians)hasaone-to-twocorrespondenceto( 3{70 ).ConsiderthesecondalignmentoptionfortheTGG,whereintheobjectiveistotuneapowerreadingtozeroratherthanbalancetworeadings.Thecorrespondingmisalignment 66

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Figure3-12. Aplotoftheregionofparameterspacethatisallowedbyapowermeterwith5%measurementuncertaintyinthebalancedTGGalignmentscheme.Theboundson1and2comefromassuminga100Wnoiseoorona70mWalignmentbeam. oftheTGGisdeterminedby: P=(a02)ya02(3{72)where: a02=Pp(1)F()F()Pp(i)a(3{73)Here,isthefractionofpowerinthebeamincidentbeamthatwouldbeexpectedtomakeittoourpowermeterinthecaseofperfectalignment.Weincludeitbecausetheseparationbetweenpolarizationsisontheorderof1,andsoitisoftenthecaseinpracticethatthetwocannotbedistinguisheduntiltheyhavebacktrackedasignicantwaydowntheopticalpath,interactingwiththeinputopticalcomponentsalongtheway.Aplotoftherelativepowerintheppolarizationasafunctionofand1isincludedastherstplotinFigure 3-13 .BecauseKTP2isnotusedduringthisalignmentscheme,thereisnorelationshipbetweenand2aswefoundwiththebalancedscheme.Unlikeinthebalancedalignmentscheme,theretro-reectionschemehasanaturalwayto 67

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characterizethelossduetoTGGmisalignment,namely: 1)]TJ /F7 11.9552 Tf 11.955 0 Td[(ayPys(1)Fy()F()Pp(1)a(3{74)Thisisthedierenceinpowerbetweenincomingbeamandtheoutgoingspolarizedbeam.Aplotof( 3{74 )fordierent1andisincludedasthesecondplotinFigure 3-13 .Thedashedlinescorrespondtotheboundsplacedonfordierentvaluesof.NoticethatbecauseweareoperatingatpolarizationsverynearlyalignedwiththepandsaxesofKTP1,thealignmentlossintheretro-reectionalignmentschemeisnotverysensitiveto1.Weseethatfor=1,theboundsonarenearly451ComparingthetwoTGGalignmentmethodsisnotnecessarilystraightforward,thoughingeneralwewillfavoraschemethatboundsastightlyto45aspossible.Forthecasewhere0:5,theretro-reectionschemewilloutperformthebalancedschemebythiscriterion.Inthiscase,ourparameterspaceisrestrictedtoj1j2:2j2)]TJ /F3 11.9552 Tf 12.027 0 Td[(1jand=451:52. Figure3-13. Plotsofthealignmentsignalrelativepowerandassociatedlossfortheretro-reectionTGGalignmentscheme. ThenalcomponenttoalignistheHWP.Insimilarfashiontothepolarizers,thisisdonebyminimizingthepoweroftheoutgoingspolarizedbeam.Thegoverningequation 68

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is: P=(a0)ya0(3{75)for: a0=Ps(2)W()F()Pp(i)a(3{76)Alongwiththepreviousconstraintsimposeduponthemisalignmentsfrompreviousstepsinthealignmentprocedure,( 3{76 )denesthefullparameterspaceofmisalignments.Overthisspacewedenethedouble-passJonesvector: a0dp=Ps(1)F()W()P2p(2)W()F()Pp(1)a(3{77)fromwhichwecalculatethedouble-passloss: Ldp=1)]TJ /F1 11.9552 Tf 11.955 0 Td[((1)]TJ /F3 11.9552 Tf 11.955 0 Td[(f)(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(b)=1)]TJ /F1 11.9552 Tf 11.955 0 Td[((a0dp)ya0dp(3{78)Onaverage,thesingle-passlosswillsimplybehalfofthedouble-passloss.Fromapracticalstandpoint,itisdiculttographicallyrepresentafour-dimensionalspace,butwecanevaluate( 3{78 )overtheallowedparameterspacetondthatthemaximumopticallossduetomisalignmentis5,500ppm,or0.6%ofthepowerlostperpass.Initially,thisisalarminglyhigh,butifweassumethatwithinourparameterspacethevaluesof1,2,,andarerandomvariablesdistributeduniformly,wendthatthesinglepasslosshasameanandstandarddeviationgivenby: hLspi=1 2hLdpi=1250ppmq hL2spi)-222(hLspi2=Lsp=950ppm(3{79)wherethestatisticsaretakenfromagridof87,000pointsintheparameterspace.ThedistributionoflossesisillustratedinFigure 3-14 3.7.3InhomogeneityLossesAninhomogenietyinthemagneticeldovertheproleofthebeam(asseeninFigure 3-15 )withintheTGGresultsininhomogeneouspolarizationrotation,andwillthusresultinalossinopticalthroughput.Foragaussianbeamwithwaistwpropagatinginthez 69

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Figure3-14. Thedistributionofsingle-passlossduetomisalignmentsofopticsintheLLFI.Theplotisgeneratedbydiscretizingthefour-dimensionalmisalignmentparameterspaceintoagrid,eliminatingparameterspacepointsthatarerejectedbyouralignmentprocedure,andthencalculatingthelossfortheremainingpoints. directionwecanexpresstheeldintotheTGGas: Ein=A(z;t)e)]TJ /F10 5.9776 Tf 8.702 3.259 Td[(r2 w2^p(3{80)forrtheradiusaboutthecentralbeamaxis,and^pthepolarizationvector.TheFaradayrotatorassemblyisintendedtobesymmetricaboutthebeamaxis,andsowecanapproximatethatthepolarizationrotationinducedbytheTGGis(r;)=(r).Ideally,theoutputpolarizeracceptslightatanangleof=4withrespectto^p,andsotheamplitudeoftheoutputeldofthepolarizerisgivenby: Eout=A(z;t)e)]TJ /F10 5.9776 Tf 8.703 3.258 Td[(r2 w21 p 2cos(r)+1 p 2sin(r)=A(z;t)e)]TJ /F10 5.9776 Tf 8.703 3.258 Td[(r2 w2cos(r)(3{81) 70

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where==4)]TJ /F3 11.9552 Tf 12.288 0 Td[((r)isthedierencefromidealrotation.Theopticalthroughputcanbecalculatedbytakingtheratiooftheintegratedeldintensities: RSE2out RSE2in=4 w2Z10e)]TJ /F13 5.9776 Tf 7.782 3.259 Td[(2r2 w2cos2(r)rdr(3{82)wherethesurfaceofintegrationisaplaneperpendiculartothedirectionofpropagation.FortheparticularmagnetcongurationwithelddistributiongiveninFigure 3-15 wecanexpectapowerlossof1.5ppmforabeamwithw=2mmand0.6ppmforabeamwithw=1mm.WeconcludethatinhomogeneousrotationshouldneverbealimitinglossmechanismintheLLFI. Figure3-15. Theaxialeldmagnitudewithinthemagnethousingasafunctionofdistancealongtheopticalaxisforapotentialmagnetconguration.Thebluecurvecorrespondstor=0mmandtheredcurvetor=5mm.Thoughtheeldsarevisiblydierentatthisscale,theconsequentpowerlosstoinhomogeneouspolarizationrotationisnegligible. 3.7.4EtalonEectsinOpticalElementsInconstructingTable 3-1 ,wetreatedeachopticalsurfaceasanindependentcontributortotheopticallossoftheLLFI.In 6.2.2 wewillderivethereectanceofa 71

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losslessopticalcavity,whichcanbeexpressedas: Rcav=R1+R2)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p R1R2cos' 1)]TJ /F1 11.9552 Tf 11.956 0 Td[(2p R1R2cos'+R1R2(3{83)whereR1andR2arethereectancesattherstandsecondopticalsurfacesrespectively,and'=4nL=istheroundtripphasethattheeldaccumulatesintheopticalmediumoflengthLwithindexofrefractionn.(Whencomparingto( 6{12 ),notethatwehavemadeuseofthefactthatR+T=1intheabsenceofopticallosses,wehaveassumedthatr1andr2haveoppositesign,andwehavedened'=2forsimplicity).ThoughR1andR2correspondtocoatingsthatarenominallyidentical,wewillnotyetmakethesimplicationofsettingthemequal.Thereectancegivenby( 3{83 )diersfromournaivelyestimatedreectancesbyacorrectivefactor: Re=Rcav)]TJ /F3 11.9552 Tf 11.955 0 Td[(R1)]TJ /F3 11.9552 Tf 11.956 0 Td[(R2=2p R1R2cos'(R1+R2)]TJ /F1 11.9552 Tf 11.956 0 Td[(1))]TJ /F3 11.9552 Tf 11.955 0 Td[(R1R2(R1+R2) 1)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p R1R2cos'+R1R2(3{84)Fromexaminationof( 3{84 ),wendthattheetalonreectanceismaximizedwhen'isanoddintegermultipleofandminimizedwhenitisanevenintegermultiple.Thisagreeswithourintuition,asthemaximizing'isthecasewheretheroundtriptravelthroughtheopticisanintegerplusone-halfwavelengths.AplotoftheetalonreectanceforthiscaseisgiveninFigure 3-16 .Ifweareonlyinterestedinaroughestimateofthereectancefrometaloneects,theformforRegivenin( 3{84 )isunecessarilycomplicated.WecangreatlysimplifyitbymakingtheassumptionthatR1=R2R,andbyignoringhigherordertermsinR: Re2R(2R)]TJ /F1 11.9552 Tf 11.955 0 Td[(1)cos')]TJ /F1 11.9552 Tf 11.955 0 Td[(2R3 1)]TJ /F1 11.9552 Tf 11.956 0 Td[(2Rcos'+R2)]TJ /F1 11.9552 Tf 9.299 0 Td[(2Rcos' 1)]TJ /F1 11.9552 Tf 11.955 0 Td[(2Rcos'(3{85)ForARcoatingsupto1000ppmreectance,theapproximationin( 3{85 )isgoodtowithin5ppmof( 3{84 ).Withthissimplication,itisveryeasytoseethattheworst-casescenarioisroughlytwicethenominalreectancefromtheoptic.BecausetheKTPpiecesarewedgedtheyareunaectedbythisresult,butconsideringTable 3-1 weseethatwe 72

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Figure3-16. Theworst-casescenarioopticallossfrometaloneectsinanoptic,asafunctionofthereectanceofeachsurface.Theopticisassumedtobelosslessinthiscalculation. mightestimatetoloseasmuchas1600ppmfromtheetaloneect.Thiswouldpushthetotallossduetotheopticsthemselvesbeyond0.8%forasinglepassoftheLLFI;consideringouranalysisin 3.7.2 thiswouldmake<1%single-passlossuponassemblyverydiculttoachieve.Initially,thiswouldseemanalarmingresult,howeverunlikelyitmaybe.Wemighthopethatthisetaloneectcouldbewashedoutbythermaleects,andsowecanconsider: @(') @T=4 @(nL) @T=4 nL+@n @TL(3{86)whichisthechangeinroundtripaccumulatedphaseperchangeintemperature.Hereisthecoecientofthermalexpansionand@n=@Tisthethermo-opticcoecient.FortheTGGcrystaloflength20mmwith=7:110)]TJ /F6 7.9701 Tf 6.587 0 Td[(6K)]TJ /F6 7.9701 Tf 6.587 0 Td[(1,@n=@T=17:510)]TJ /F6 7.9701 Tf 6.586 0 Td[(6K)]TJ /F6 7.9701 Tf 6.587 0 Td[(1( 20 ),andusing1064nmlaserlight,( 3{86 )evaluatesto7:410)]TJ /F6 7.9701 Tf 6.587 0 Td[(3,meaningthatrandomtemperatureuctuationsinthecrystalhavenomeasureableeectontheetalonreectance. 73

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Thecalculationsabovehaveassumedthattheopticalsurfaceisperfectlyat.Ofcourse,thisisnotthecaseinpractice,thoughsuperpolishedopticscanbeprovidedwithverylowatness(so-called=10inourcase),roughness(<0:5nmRMS),andhighparallelism(<30arcsec)specications.(Thoughopticalcoatingsareprovidedwithnosuchspecications,theirperformancerestslargelyontheiruniformity).Toestimatewhetherdeviationsfromperfectlyat,parallelsurfaceswillmitigatetheetaloneect,wecanconstructasimplemodel.Consideragaussianbeamwithwaistw=1mmpropagatinginthezdirection,normallyincidentontheperfectlyatfrontsurfaceofaTGGcrystal,andsupposetherearsurfaceoftheopticmakesasmallangle withthefrontsurfaceinthex-zplane.Forsimplicity,wewillassumethebeamisretro-reectedattherearsurface,eventhoughitisnotnormallyincident.Therelativepowerinthereectedbeamduetotheetaloneectinthiscasecanbeapproximatedas: Re=RSIr RSI0=)]TJ /F1 11.9552 Tf 17.813 8.087 Td[(2 w2ZS2Rcos' 1)]TJ /F1 11.9552 Tf 11.955 0 Td[(2Rcos'e)]TJ /F13 5.9776 Tf 7.782 4.025 Td[(2(x2+y2) w2dxdy(3{87)whereSisthefrontsurfaceoftheTGG,and'isnowspatiallydependent.(WeareignoringthechangeintheradiusofthebeamthroughouttheroundtripintheTGG).Inparticular,becauseabsolutephaseisinconsequential,wecanchoose: '=4nx +2(3{88)where0
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Figure3-17. AplotoftherelativepowercontributedtothereectedbeambytheetaloneectinourTGGasafunctionoftheparallelismofthecrystal.SeveraldierentvaluesforarechosentodemonstratetherangeofpossiblecontributionswemightobservewiththeLLFIcrystal.Noticethatthecurvescantakenegativevalues,whichcorrespondstothecasewherethepowerinthereectedbeamisdiminishedthroughdestructiveinterference.WehavechosenR=500ppmasthenominalreectanceofeachsurfaceandabeamsizeofw=1mm. example,istoincreasetheopticalpathlengthofthelasereldpassingthroughit,thuscontributingadditionalphasetothebeam.Theetaloneectinhighlyanti-reectiveopticsismainlycausedbytheinterferenceofthebeamthatisimmediatelyreectedattherstopticalsurface,thepromptreection,withthebeamthatisreectedatthesecondopticalsurface,thesecondaryreection,andsothephaserelationshipbetweenthesetwobeamsiscentraltotheeect.Callingitheangleofincidenceofthebeam,andttheangleofthebeamwithrespecttothenormalinsidethemedium,thenthephasedierencebetweenthepromptandsecondaryreectioncanbegeometricallyreasonedtobe: '=4nL 1 cost)]TJ /F1 11.9552 Tf 11.956 0 Td[(tantsini(3{89)AnapplicationofSnell'slawallowsustoevaluate( 3{89 )intermsoftheincidentanglei.Aplotof'fortheLLFITGGisincludedinFigure 3-18 .Weseethatthephase 75

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dierencecanbepushedthroughafull2withatiltof20arcminorless.Thisisaverypromisingresult:intheunluckyeventthatetaloneectslimittheopticalthroughputoftheLLFI,opticscanalwaysbetiltedslightlysothatthereectedbeamslosespatialcoherenceandthusinterferemoreweakly. Figure3-18. Aplotofthephasedierencebetweenpromptandsecondaryreectionsasafunctionofthetiltoftheoptic.Herewehaveassumeda20mmthickperfectlyparallelTGGcrystal. 3.7.5UninvestigatedLossesTherearesomesourcesofloss,oreectivesourcesofloss,thathaveyettoberesearched.ThoughtheyareunlikelytolimittheperformanceoftheLLFI,theyareworthenumerating.Theuseofwedgesforpolarizershastheadvantagethatorthogonalpolarizationscanbesplitwithhighextinctioncoecientsandlowopticalloss.However,awedgedopticwillinduceastigmatisminthebeam( 26 ),whicheectivelypushespoweroutofthecoreopticalcomponentresonantmodeandintohigherordermodes( 10 ; 48 ; 62 ).Tocompensateforastigmatism,theKTPwedgesareippedwithrespecttooneanother.However,norigorousattemptatcharacterizingorquantifyingtheeectoftheresidualastigmatismhasbeenmade. 76

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WhilewehaveacknowledgedpreviouseortstocombatthermallyinducedbirefringenceintheInputFaradayIsolator,wehavenotperformedananalysisonpotentialellipticityoftheTGGoutputbeamfortheLLFI.Wedonotexpectthistobeasignicantfactor,butnonethelessacknowledgethatwehaveundertakennoformalstudy.Likelythemostsusbstantialomissioninouranalysisislossesduetomisalignmentinthreedimensions.InSection 3.7.2 ,welimitedthescopeofourinvestigationtoincludeonlymisalignmentsintheplaneperpendiculartotheopticalaxis.Thisisonlyaveryroughapproximationofwhatactuallyhappensinthelaboratory.Generalmisalignmentsarenotconsideredbecausetheinformationtheyaddtotheworkdoesnotwarrantthecomplexitytheyaddtotheanalysis. 77

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CHAPTER4LOWLOSSFARADAYCONSTRUCTIONANDPERFORMANCE 4.1OpticalComponentTestingFortroubleshootingandfuturedesignpurposes,itisimportanttocharacterizetheopticalcomponentsindependently.OfparticularimportancearecharacterizationsoftheKTPandTGGoptics,astheyarelesscommonitemsandwereprovidedwithcustomspecications.InestimatingthetotalopticallossintheLLFIinChapter 3 ,thenoisebudgetwasdividedintolossesduetotheindividualopticalelementsandlossesduetothecomponentalignments.Themajorityofindividualcomponentlossisexpectedtocomefromreectionsattheopticalsurfaces,andsoitisnecessarytocharacterizetheperformanceoftheanti-reectivecoatingsprovidedbyMLDTechnologies.AsimpletestofreectancewasrunontheLLFIcomponentstoconrmthattheperformanceoftheanti-reectivecoatingsagreedwithdesigngoals.ApictureofthesetupisprovidedasFigure 4-1 .Alaserispassedthroughahalf-waveplateandcalcitewedgepolarizertopolarizeandmodulatethepowerofaprobebeam.Thebeamthenpassesthroughanadditionalhalf-waveplate,whichcanbeusedtorotatetheprobebeampolarization,andisthenincidentatasmallangleonthetestoptic.Thereectedbeamismeasuredbyacalibratedpowermeterandcomparedtoameasurementoftheincominglaserpower.Duetothewedgegeometry,thepromptandsecondaryreectionsfromtheKTPseparateandcanbemeasuredindependentlyandaveraged.Atsmallangles,thereisnosuchseparationforthereectionsfromtheTGG,andsotheymustbemeasuredcoincidentallyandhalfoftheirsumistakentobetheaveragereectance.BoththeTGGandKTPweretestedoverarangeofsmallangles.BecausetheKTPisbirefringentandwillseebothpolarizations,thewedgesweretestedforbothpandspolarizedincominglight.TheresultsofthereectancetestsareshowninFigure 4-2 .ComparingthemeasuredvaluestothosequotedinTable 3-1 ,weseethattheTGG 78

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anti-reectivecoatingunderperformsexpectation,havingameasuredreectanceof650ppmcomparedtothegoalof500ppm.Fortunately,theanti-reectivecoatingsfortheKTPsignicantlyoutperformexpectation,havingameasuredreectanceof20ppmcomparedtothesamegoalof500ppm.BecausetherearefourKTPsurfacesintheLLFIpath,andonlytwoforTGG,thebetter-than-expectedKTPARcoatingscompensatefortheslightlydisappointingTGGARcoatings.TheinitiallexpectedopticallossduetoreectionfromtheTGGandKTPwedgeswas3000ppm,butthereectancemeasurementssuggestthatthiscanbereducedto1400ppminourlossbudget.Thissignicantimprovementisovershadowed,however,bytheunexpectedlypoorperformanceoftheHWP,measuringapproximately1300ppmlosspersurface.UnliketheKTPandTGG,theHWPwasnotcustomordered,polished,andcoated,butwasinsteadaspareLIGOInputOpticspart.Thisdecisionwasmadepartlyintheinterestoftimeandmoney,butlargelyduetoamisplacedcondenceintheoptic.ArevisedversionofTable 3-1 withmeasuredvaluesisprovidedasTable 4-1 .Weseethatthetotalexpectedlossduetotheopticalcomponentshasnotchangedsignicantlyfromtheinitialestimate. Table4-1. RevisedopticallossofeachcomponentinthecurrentUFLLFIdesignaftermeasuringreectancesofeachoptic. IsolatorelementOpticalloss(ppm) KTPreection(perface/total)20/80KTPabsorption(percrystal/total)25/50HWPreection(perface/total)1300/2600HWPabsorption50TGGreection(perface/total)650/1300TGGabsorption(20mmcrystal)3000 Isolatortotal7080 4.2MagnetAssemblyAsdiscussedin 3.6 ,themagnetassemblyoftheFaradayrotatorisnotmonolithic.Thelimitationsofmagnetmanufacturers'abilitytoproducelargehigh-eldpermanent 79

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Figure4-1. Apictureofthesetupusedtotakereectancemeasurementsoftheindividualopticalcomponents.ATGGtestcrystalcanbeseenintheforeground.Photocourtesyofauthor. Figure4-2. MeasuredreectancesfortheTGGandKTPcrystalsusedintheLLFI.TheTGGwasmorereectiveat1064nmthanexpected(designgoalof<500ppmpersurface),howevertheKTPoverperformedthegoalbynearlyanorderofmagnitude. magnetsrequirethatwedividethemagnetblockintoindividualmagnetdisks.ThesedisksarethenpositionedwithinanaluminumdruminordertocreatethemagneticeldprolerequiredfortheFaradayrotator. 4.2.1DiskAssemblyAswerecallfromChapter 3 ,theFaradayrotatordesignrequiresthreeuniquemagnetdisktypes:onetypemagnetizedradiallywiththenorthpolarizationontheouteredgeofthedisk,onetypemagnetizedradiallywiththenorthpolarizationontheinneredgeof 80

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thedisk,andonetypemagnetizedaxially.Eachdiskconsistsoftwelveindividualmagnetwedges.Forbothtypesofradiallymagnetizeddisk,theindividualmagnetwedgesrepeloneanotherradially.Thisisadvantageousoncethedisksareassembled,astheoutwardforceofeachmagnetontotheretainingringservestokeepthewedgesinplace.However,duringassemblythispresentsaproblem,asrepellingforcespreventthewedgesbeingsimplyputintoplacebyhand.Theaxiallymagnetizeddiskisanevengreaterchallenge,asnotonlydothemagnetwedgesstronglyattractoneanotherduringassembly,butthediskcongurationisunstable.Awedgethatbeginstoescapethediskwillbepushedinthatdirectionbytheeldsfromthesurroundingwedges,oftenresultinginacascade.AllofthemagnetwedgesusedintheLLFIwereprovidedcustom-builtbyK&JMagnetics.Theradiallymagnetizedwedgesare1/2inchthick,1/2inchinnerradius,2inchouterradius,gradeN52.Theaxiallymagnetizedwedgesare1inchthick,1/2inchinnerradius,2inchouterradius,gradeN42.Thedisksarelabeledaccordingtotheirnalorderingintherotator:D1,D2,D3,D4,andD5.Thecentraldisk,D3,isaxiallymagnetized,whiletheotherfoursidedisksaremagnetizedalongtheirradii,withD1andD2havingnorthpolesontheouterradius,andD4andD5havingnorthpolesontheirinnerradius.Eachdiskisassembledinanaluminumblockthatsitsinaviceonamillingmachine.Theblockhasaninsetregionwiththesameradiusasadiskintowhichthedisk'sretainingringisinitiallyinserted.Twelveacryllicwedgesareplacedinsidetheringtoserveasplaceholdersforthemagnets.Eachacryllicwedgehasathroughholewhichcanbealignedwithholesdrilledintotheblockevery30.Copperdowelscanbethreadedthroughtheseholestoprovideorpreventrotationofthedisk.Asafetycapwithagaplargeenoughforonewedgetotthroughisscrewedtotheblockthroughitscenter;itsradiusisroughlyhalftheradiusofthering.Atthebottomoftheinsetisawedge-shapedwell,largeenoughforexactlyonewedgetofalldownthroughatatime.Belowthe 81

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wellaretwotrapdoors,onebrassandonealuminum,thatcanbeopenedbyslidingthedoorhorizontallyawayfromtheblock.Ingeneral,thebrasstrapdoorisusedduringtheassemblyofD1,D2,D4,andD5,whilethealuminumtrapdoorisusedduringtheassemblyofD3(thecentraldisk)Inthespindleofthemillingmachineisaholder,withaninsetthattsthedimensionsofanindividualmagnetwedge;themillingmachineisadjustedsothattheholdersitsdirectlyabovethewell.Theholderhasahorizontalguideleverforprovidingorpreventingrotationoftheholder.Atthetopoftheholderaretwoplasticpushdowelssittingverticallyinthroughholeswhichareusedtoreleaseamagnetwedgefromtheholder.Aclampwrapsaroundtheholderandsecuresmagnetwedgesfrombelow.Tobeginassembly,theacryllicwedgesarealignedsothatone,calledthesacricewedge,sitsdirectlyabovethewell.Thesafetycapisrotatedsothatitsgapleavesthesacricewedgecompletelyexposedfromabove.Amagnetwedgeisplacedinsidetheholder,andtheclampisattachedtokeepitsecure.Usingthepilotfeedleverofthemillingmachine,andholdingtighttotheguidelever,theholderwithwedgeisslowlyloweredtotheblockuntiltheclampisnearlytouchingtheacryllicwedges.Theclampiscarefullyremoved,andtheholderisloweredagainuntilthemagnetwedgemakescontactwiththeacryllicwedge,andultimatelyforcesthearcyllicwedgeintothewellandtakesitsplaceinthedisk.Thepushdowelsarepressed,separatingthemagnetandholder,andtheholderisraisedslowlyawayfromtheblock.Thesafetycapisrotatedsothatthemagneticwedgeissecuredfromabove,andthentheentirediskisrotatedsothatthemagnetwedgeisnolongersittingabovethewell.Afterwards,thetrapdoorisopenedandthesacricewedgefallstotheground.Toplacethesecondwedge,thetrapdoorisclosed,andthediskisrotatedintheblocksothatthepreviouslyplacedmagnetwedgesits120awayfromthewellposition.Thesafetycapisrotatedsothatthenewsacricewedgeisexposedfromaboveandtheproceduredescribedinthepreviousparagraphisrepeatedoncemore. 82

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Toplacethethirdwedge,thetrapdoorisclosed,andthediskisrotatedintheblocksothatbothpreviouslyplacedmagnetwedgessit120awayfromthewellposition.Fromthispointforwardthecopperdowelsareusedtosecuretheorientationofthedisk,andtheplacementprocedureisrepeated.Theringcontinuestobepopulatedwithmagnetwedgesinassymmetricalapatternasisallowed.Inordertopreventthewedgesfromjumpingoutoftheholderoncetheclampisremoved,theeldproleofthedisknearthewellshouldbeasweakandsymmetricalaspossible.Theprocessabovecanberepeateduntilelevenmagnetwedgeshavebeenplacedinthering.Atthispointtherearenomoreholesintheblockavailableforthecopperdowelstositin.InthecaseofD1,D2,D4,andD5,itispossible(thoughnotconsistentlyreproducible)forthenalwedgetobeloweredintoplacefollowingthesameprocedurewithoutcopperdowels.InthecaseofD3,becausethewedgesareattractedastheyareloweredtothedisk,thediskwillalwaysattempttorotateinonedirectionortheothersothatthebottomfaceofthenalwedgecollideswiththetopfaceofanotherwedge.Thispositionishardtorecoverfromwithoutcompletelyandsuddenlydisassemblingtheentiredisk.Foralldisks,thenalsacricewedgeisloweredthroughthewellsothatitsupperhalfissittinginsidetheringanditslowerhalfisbelow.Thispreventsthediskfromrotatingwhileholdingspaceopenforthenalmagnetwedge.ForD1,D2,D4,andD5,a1intallacryllicwedgecalledthesandwichisplacedontopofthesacricewedge.ForD3therearetwosandwichwedges:a1inwedgethatsitsdirectlyontopofthesacricewedge,anda1/2inwedgethatsitsontopofthat.Thesandwichesarechosensothatanacryllicwedgeisalwayspartlyabovethewellandpartlybelowthewellwhenthebottomfaceofthenalmagnetwedgeisparallelwiththetopfaceofthedisk.Thisisthemostreliablewaytopreventthediskfromrotating.Thenalmagnetwedgeisloweredintheholderuntilittouchesthesandwichwedge.Theguideleverisusedtoensurethatthenalwedgeandthesandwicharealigned,andtheholderbeginstopressthemagnetwedge 83

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downontothesandwich.Thesandwicheventuallypushesthesacricewedgeallofthewaydownthewell,andthetrapdoorisopenedtoremovethesacrice.Thetrapdoorisimmediatelyclosedandtheholdercontinuestolowerthemagnetandsandwichwedges.Whenthemagnetwedgeispressedfullyintoplace,thesandwichwillbefastenedbetweenthemagnetandthetrapdoor;thispreventsthemagnetwedgesfromescapingoutthebottomoftheblock.Thesafetycapisturnedsothatthegapdoesnotfullyexposeanymagnetwedge. Figure4-3. Theassemblyofaamagnetdisk,inthiscasethecentraldisk(D3).Thestrategyforassemblydiersslightlythanfortheotherdisks,asthewedgesareattractedtooneanotherastheyareloweredinplace,asopposedtorepelled.Photoscourtesyofauthor. 84

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4.2.2DiskLiberationandFieldTestsAssemblingallofthemagnetwedgesintheirrespectiveringsisonlyhalfwaytoobtainingmagnetdisks.Onceassembledintheblock,theymustbesafelyremoved,secured,cleaned,andstoredforlatertestingandeventualrotatorassembly.Whenamagnetringhasbeenfullypopulated,astorageplateisslippedoverthetopofthediskandisscrewedtotheblock.Theinnerradiusoftheplateisslightlygreaterthantheradiusofthesafetycap,andsothesafetycapisremovedvertically.Theblockisremovedfromitsviceandbroughttoanopenworkspacefreeofunecessarymagneticmaterials.Two1=2inthicksteeltrianglesarelayedatontopoftheplate.Thesteelismagneticandwillsitsecurelyinplace.Theplateisunscrewedfromtheblock,andtheentireassemblyisturnedupside-downtobesittingonthesteeltriangles.Theblockisslowlyliftedoandthetopfaceofthemagnetdiskisexpose.Asecondstorageplateisplacedontopofthedisk,andisboltedtotherstplate.Thesteeltrianglesarewrestledoandthediskisnowinasafeandportablecontainer.AnexampleoftheprocessisshowninFigure 4-4 .Inlieuofultrasonicbathing,thedisksweresoakedinbothacetoneandisopropylalcohol,andaftercleaningwerebroughtintoacleanroomenclosure.Beforeassemblingthedisksintoarotatormagnet,eachindividualdiskneededtobetestedtoensureitsmagneticeldprolewasasexpected.Eachdisk,securedinitsstorageplates,wasfastenedtoaluminumrods10inaboveanopticalbenchtopreventanyinteractionswithmagneticmaterials.ASypris6000SeriesaxialHall-eectprobewasaxedtoamountonatrackrunningalongthecentralaxisofthemagnetdisk,andthemagneticeldvalueswerereadoutonthedisplayofaSypris6010Teslameter.Figure 4-6 showsthetypicalsetup.PlotsoftheeldmeasurementsareincludedinFigure 4-7 .Theproleofeachdiskwaswithin90%ofthesimulatedvaluethroughouttheextentoftheprobe.Smallchipsandmissingcornersfromtheassembly 85

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Figure4-4. Theprocessofremovingacompletedmagnetdiskfromthealuminumassemblyblock.Thebrassclampsarespecictothecentraldisk,asthemagnetcongurationislessstablethanforthefoursidedisks.Photoscourtesyofauthor. procedurelikelyresultedinslightlyweakermagnetdisksthananticipated,howevereverydiskwaswellwithinthedesigntolerance. 4.2.3FullRotatorAssemblyOnceallvediskshavebeenindividuallyassembledandcleaned,theymustbejoinedtogetherwithinthedrumtoeectivelycreateasinglemagnet.Therequirementthatassemblybedoneinacleanroomenvironment,therelativelackofaccesstospaceinsidethemagnetdrum,andtheincreasedeldstrengthsfromincorporatingmoremagnetsatoncemakesthisprocesssignicantlymoredicultthandiskassembly.TheassemblyprocedureisadaptedfromtheprocedureusedtoconstructtheIFI( 51 ).Asidefromeldprole,therearetwosignicantdierencesbetweentheIOIFIand 86

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Figure4-5. AnassembledmagnetdiskfortheFaradayrotator.Eachindividualwedgeismagnetizedalongthelength,whichwhencombinedapproximatesasinglemagnetdiskwithonepolarizationontheinneredgeandtheoppositepolarizationontheouteredge.Thebrightblueplatesunderneaththediskaresteeltriangles,whicharemagneticandhelptokeepthewedgessecurewithinthering.Photocourtesyofauthor. Figure4-6. Measuringthemagneticeldproleofafullyassembledmagnetdisk(herethecentraldisk).Photocourtesyofauthor. theLLFImagnetswhichimpacttherotatorassemblyprocedure.TherstisthattheLLFImagnetwedgesareallnickelplated(whereastheIOIFIwedgeswerebare).Whilethismakesforsignicantlyeasiercleaning,theplatingalsoreducesfrictionbetweenthewedges,makingthemmoreslipperyagainstoneanotherandthereforemorelikelytoloose 87

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Figure4-7. AcomparisonofthemeasuredaxialeldprolesoftheassembleddiskswiththeinterpolatedprolesfromCOMSOLsimulations. themselvesfromtheirretainingring.Theseconddierenceisrelatedtotherst:thecentralmagnetisnotmonolithicandisunstablewhenexposed.TherststepintheIOIFIrotatorassemblyprocedureistojointhethreecentermagnets.Analuminumtableprovidesasurfaceabovethemagneticsteelsurfaceofanopticalbenchonwhichtoassemblethemagnets.Asteelrodisboltedtothetablesothatitrestsupright;therodservesasaguideforthemagnetsthroughouttheassembly.Removedfromitsstorageplates,andwithsteeltrianglesaxedtothebottomfacetokeepthemagnetwedgessecure,D2islowereddowntheroduntilitsitsonthetable.Thenextdisk,D2,issecuredwiththreethinsteelkeys,whichserveinasimilarmannerasthesteeltrianglesbuthavetheadvantagethattheyareeasiertoremovewithoutdisruptingthewedges.D2isslowlylowereddowntheroduntiltherepellingforcefromD2preventsitfromfreelydescendinganyfurther.MirroringD2,D4issecuredbysteeltrianglesonitstopfaceandlowereddowntheroduntilittoositssuspended.Atthispoint,procedurecallsforthethreediskstobepressedtogetherfromabove.However,aswasalludedtointhepreviousparagraph,thecentralringD3isnotinagreementwiththisstep,andtendstodynamicallydisassemble.AbriefpictorialexplanationisprovidedinFigure 4-8 .WiththeneedtocontainthewedgesofD3apparent,aspecialassemblywasdesigned.Twoaluminumplateswithsixgroovesrunningradiallysitoneitherfaceofthecentral 88

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Figure4-8. Aninitialattemptatjoiningthethreecentral-mostringstocreateamonolithicmagnet,demonstratingtheneedtoproperlysecurethecentralringwedges.Photoscourtesyofauthor. disk.Brassclampsslideintothegrooves,securingtheplatesandwedgeswithin.IntheinterestinpreservingtheintegrityofD3wedges,someofwhichatthispointhadexperiencedmultipledisassemblies,thedecisionwasmadetokeepthecentralringbetweenitsprotectiveplatespermanently.Thisaddsanunexpected1/8inofseparationbetweenD3andtheotherfourmagnets,butmodelingwiththediskeldprolessuggestedthattherotatoreldwouldstillbestrongenoughtoachieve45rotationintheTGG.Asanothersafetymeasure,athinaluminumcylinderabout1ftlongislowereddowntherodbeforeanyofthedisks.Thiscylinderclosesthegapbetweentheinnerradiusofthedisksandtherod,makingitharderforwedgestoescape.Again,D2isaxedtosteeltrianglesandlowereddowntherodtothetable.D3,nowcompletelycontained,islowerednextandcomestorestafewinchesaboveD2.LastlyD4isloweredwithsteeltrianglessittingonitstopface.Cylindricalaluminumweightsareloweredontotherod,pushing 89

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Figure4-9. Anassemblyforsecuringthewedgesofthecompletedcentraldisk.Brassclampstintogroovesinthetopandbottomaluminumplates,holdingtheminplace.Holeswerelateraddedtotheclampstoaidintheirremoval.Photocourtesyofauthor. downonD4.AsD4movesfurtherdown,D3compensatesandthedisksremainnearlyevenlyspaced.Eventuallytheforceoftheweightsisnotenoughtocompressthethreedisksanymore,andabrassnutisattachedtothetopofthering.Thenutiswounddowntheroduntilitsbottomedgemeetsthetopmostweight,atwhichpointacomedicallyoversizedwrenchisusedtoscrewthenutdown,therebyforcingthediskstocontinuecompressing.Atapoint,thedisksgetcloseenoughandtheforcesofeachoneontheotherquicklyfalltozero.Fromthere,thediskscanbebroughttogetherbyhand,andoncetouchingtheysticktogetherasonemagnet.Unfortunately,becauseofthesafetyplatesonD3,thespacingbetweenthemagnetsistoolargeandtheattractiveforcebetweenthemisveryweak.Thethreediskscanbebrokenaparteasily,atwhichpointtheyareagainahazard.Therotatorassemblyneedstoproceedwithouttheadvantageofasinglecentral3-diskmagnet.TheneedforsafetyplatesonD3,andtheresultthatD2,D3,andD4willnolongersticktogetherwithoutanexternalforcemaketheLLFIrotatorassemblysignicantlymoreinvolvedthanthatoftheIOIFI.Nevertheless,itisimportanttodocumentallofthestepsindetail. 90

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Figure4-10. Asecond,saferattemptatjoiningthethreecentral-mostringstocreateamonolithicmagnet.Unfortunately,thealuminumplatesthatsecuredthecentralring'swedgeswerethickenoughtopreventthethreeringsfromcomingcloseenoughtogethertostickandbecomeonemonolithicmagnet.Photoscourtesyofauthor. Thealuminumtableisagainplacedonanopticalbenchinacleanroom,withathreadedsteelrodstickingverticallyoutofitsbase.Thealuminumguidecylinderisloweredontotherodandtwosteeltrianglesareplacedatitsbasesoastomakeasquare.Thebottomcapoftherotatorisscrewedtightlyintothedrum,andthedrumislowereddowntherodontothetable.Becausethedrumistallerthanthesumoftheheightsofthedisks,aluminumspacersareplacedinsidethedrumatthebottom;thespacershaveaninnerdiameterslightlygreaterthanthedisksandandouterdiameterslightlyless.Aluminumblocksaresetuponeithersideofthedrum,andsupportrodsarerunthroughtheventchannelsofthedrumsothatthetopofthesupportsareevenwiththetopofthechannels.Sandwichedbetweentwospacers,D1isloweredslowlydowntheroduntilitcomesintocontactwiththetopofthealuminumsupports.Acylindricalaluminumweight 91

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isplacedontopofD1,theblocksholdingupthesupportrodsareremoved,andD1isslowlyloweredbyloweringthesupportsuntiltheyhitthebottomofthedrumchannels.ThreesteelmagnetkeysareslidthroughopenchannelsinthedrumandaxedtothebottomofthespacerthatD1sitson.Thesupportsarecarefullyslidoutandthediskislowereduntilthemagnetkeyshitthebottomofthedrumchannels.Themagnetkeysareremovedoneatatimebyslowlypullinghorizontallyawayfromthedrum,andthediskfallsontothealuminumspacersatthebottomofthedrum.Atthispoint,theattractionbetweendiskandthesteeltrianglesbelowthedrumisenoughtosecureD1.TheweightandspacerontopofthediskareslowlyremovedandD1sitsinthedrumwithitstopfaceexposed.Toaddaseconddisktotheassembly,thealuminumblocksandsupportsareputbackinplace.Tokeepthemagnetsascloseaspossibleinsidethedrum,D2cannothaveaspaceronitsbottomface.Instead,aspacerisplacedonthetopfacealongwithsixsteelmagnetkeys,andthediskisslowlylowereddowntheroduntilitsitsonthesupports.Atthispointthekeysatthetopofthediskneedtobetransferredtothebottomofthediskinordertoallowittodescendfurther.AcylindricalaluminumweightisloweredontoD2,andoneatatimeakeyisslidoofthetopofthediskandplacedontothebottomthroughthechannelsinthedrum.Oncethreeofthekeyshavebeentransferred,thesupportsareremovedtofreeupaccesstothebottomofD2.Theremainingkeysaretransferredfromthetopofthedisktothebottom,withafewextraaddedforgoodmeasure.TheweightandspacerontopofD2areremoved,andthedisksitssuspendedaboveD1.Toaddthenalthreediskstotheassembly,thealuminumblocksandsupportsaresetupsothattheypreventD2fromdescending.Clampedinsideitssafetyplates,D3islowereddowntheroduntilitsitssuspended.SixsteelmagnetkeysareaxedtothetopfaceofD4,anditislowereddowntheroduntilitissuspended.Withsixmagnetkeysonitsbottomfaceandtheremainingspacersanddrumcapsittingonitstopface, 92

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Figure4-11. Theprocessbywhichtherstmagnetdiskisplacedinsidethedrum.(Lefttoright,toptobottom).Withbottomcapattached,thedrumisplacedontopofmagnetictriangleswiththeopenendfacingup.Spacersareloweredintothedrum,andsupportsareplacedthroughtheventchannels.Therstdiskislowereddowntheguideroduntilitsitsontopofthesupports.Alargealuminumweightisputontopofthedisk,andthenthesupportsarelowereduntiltheytouchthespacersatthebottomofthedrum.Magneticsteelkeysareplacedunderthedisk,thesupportsareremovedfromthedrum,andthediskislowereduntilitissittingonthespacersatthebottomofthedrum.Thekeys,weight,andtopaluminumplateareremoved,leavingthetopsurfaceofthediskexposed.Photoscourtesyofauthor. D5isaddedtotherodandlowereduntilallthreeofthenalmagnetsaresuspended.Cylindricalaluminumweightsareloweredoneatatimedowntherodtogradually 93

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Figure4-12. Theprocessbywhichthesecondmagnetdiskisplacedinsidethedrum.(Lefttoright,toptobottom).Supportsareputinplaceandthediskislowereddowntothem.Magnetkeysaretransferredfromthetopofthedisktothebottomandthenthesupportsareremoved.Photoscourtesyofauthor. compressthedisks.Oncealloftheweightshavebeenadded,eachdiskislowenoughtobeincontactwiththealuminumguidecylinder.Thealuminumblocksholdingupthesupportsarereplacedwithadjustableplatforms,andthebrassnutisscreweddowntherod.Againwiththegiantwrench,theassemblyisslowlycompressedandthecenterthreediskscomeintocontact.D5isbroughttowithinlessthananinchofD4andthemagnetkeysonbothdisksareremoved.Theadjustableplatformsareloweredslightly,andthenthebrassnutisturnedslightly,repeatinguntil 94

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Figure4-13. Theprocessbywhichtheremainingthreemagnetdisksareplacedinsidethedrum(part1).(Lefttoright,toptobottom).Thedisksareloweredinsuccessionuntiltheysitsuspended.Spacersandthetopcapofthedrumareplacedontopofthenaldisk,andweightsareaddedtobegincompressingthedisks.Photoscourtesyofauthor. theplatformsnearlycannotbeloweredfurther.Aluminumblocksareplacedbetweentheplatformsandthedrum,thesupportsareswitchedoutforlongerrods,andtheadjustableplatformsareslidothealuminumtableandontotheopticalbench.Theadjustableplatformsareraisedtocomeintocontactwiththesupports,andthealuminumblocksareremoved.Againthebrassnutisscreweddownandtheadjustableplatformsarelowered,repeatinguntilthebottomofthesupportscomeintocontactwithD1.Thesupportsare 95

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Figure4-14. Theprocessbywhichtheremainingthreemagnetdisksareplacedinsidethedrum(part2).(Lefttoright,toptobottom).Theassemblyiscompresseduntilthecenterthreedisksaretouching.Unecessaryclampsandkeysareremovedandadjustablesupportsareusedtoguidethedisksdownintothedrumuntilonlythenaldiskisoutside.Photoscourtesyofauthor. removedandtheassemblyiscompresseduntilD2isnearlytouchingD1.ThemagnetkeysonthebottomfaceofD2areremovedandscrewsareputinplaceonthetopcap.Slowlytighteningthebrassnut,thescrewsinthecapareusedtoguideitdowntothedrum.Oncethecaphasmadecontact,thescrewsarefullytightened,thebrassnutthreadedallthewaybackupandotherod,theweightsareremoved,thealuminumguidecylinderisremoved,andtherotatorsitscompleted.Themagnetsarenolongerathreat. 96

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Figure4-15. Theprocessbywhichthedrumiscappedandsecured.(Lefttoright,toptobottom).Theadjustableplatformisswitchedoutforonewithalowerrange,andtheassemblyisfurthercompresseduntilthecapreachesthedrumandisscrewedinplace.Photoscourtesyofauthor. Nowfullyassembled,themagneticeldproeoftherotatorismeasuredinmuchthesamemannerasfortheindividualdisks.Figure 4-16 showstheresultsoftheeldtests,andweseethatthereisgoodagreementbetweenmeasurementandmodel,servingasvericationofourworkinSection 3.6 .Knowingtheeldprole,wecanalsocalculatethepolarizationrotationinducedintheTGGasafunctionofthepositionofthecrystalcenterwithinthemagnetdrum.IfweassumeboththebeamandTGGarecenteredon 97

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Figure4-16. Atestofthemagneticeldproleforthefullyassembledmagnetdrum.Ontheleftisaplotoftheaxialmagneticeldprole.Individualdatapointsareshowninred,whilethebluecurveistheexpectedeldprolederivedfromthemeasuredeldprolesofeachindividualmagnetdisk.Ontheleftisapictureofthesetup.Photocourtesyofauthor. themagnetaxis,then( 3{15 )becomes: (z;L)=VZz+Lz)]TJ /F4 7.9701 Tf 6.586 0 Td[(LB(s)ds(4{1)wherezisthepositionofthecenteroftheTGG,andthelengthoftheTGGcrystalis2L.Aplotof( 4{1 )forseveraldierentlengthsofTGGisincludedinFigure 4-17 .Weseethatwecanachieveasmuchas65ofrotationinourcurrent22mmcrystal,wellbeyondtherequired45.TheeldtestssuggestalowerlimitonTGGlengthof15mmforthecurrentmagnetdesign,whichwouldreduceabsorptionlossesinthecrystalby32%. 4.3IsolatorAssemblyWiththerotatormagnetassembled,theconstructionoftheisolatorcanbegin.AnillustrationoftheprocedureisincludedinFigure 4-18 .AnInnoLightNPROprovidesalaserbeamwhichissentthroughahalf-waveplateandthenacalcitepolarizer.ThepolarizerdeterminesthepreferredpolarizationoftheFaradayisolator,andthehalf-waveplateallowsforcontrolofopticalpowerinthesetup.UsingtheAdvancedLIGOconvention,thelessdeectedbeamoutofthecalciteisselectedastheprobebeam,whichcorrespondstotheppolarization,andthemoredeectedspolarizationisdumped.An 98

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Figure4-17. AplotoftheinducedpolarizationrotationintheTGGasafunctionofthepositionoftheTGGwithinthemagnetdrum.Thedesiredrotationof45degreesisgivenbytheredline.Thebluecurvecorrespondstoa22mmlongTGGcrystal,whichisusedintheinitialLLFIbuild,butweseethatthecurrentmagnetcongurationcansupportsmallercrystals,withalimitaround15mm. OphirPD300-1Wpowermeterisplacedintheprobebeamandthehalf-waveplateisadjustedtomaximizepowerinthebeam.TherstKTPpolarizer,calledKTP1,isplacedinthebeampathandispositionedsothatthebeamincidentangleisnogreaterthanthewedgeangle.TheKTP1holderisthenrotatedandtiltedsothatthebeammaintainsitsheightdownstreamandthepowerinthemostdeectedbeamoutofKTP1isminimized.Bydoingso,wearealigningtheaxesofKTP1withtheincomingpolarization,therebyminimizingopticallosstotheelement.Inordertolocatethes-polarizedoutputformeasurement,itishelpfultoplaceasparehalf-waveplatebetweenthecalcitewedgeandKTP1tomaketherejectedbeamvisible,alignthebeamtoapowermeter,andthenremovethehalf-waveplate.ThesecondKTPpolarizer,KTP2,isplaceddownstreamofKTP1,deningthefootprintoftheLLFI.SimilartoKTP1,KTP2ispositionedsoastonothavetoogreatanangleofincidencewiththebeam,andtominimizethepoweroutputinthemostdeectedbeam,whichforsimplicitywillalsobecalledthespolarization.Whenthespolarization 99

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outofKTP2isminimized,wehaveoptimizedthealignmentofthecrystalaxesofKTP1andKTP2.ThemagnetdrumisplacedbetweenKTP1andKTP2suchthatthebeamentersandexitscenteredontheaperture.TheTGGinitsholderisinsertedintothedrum(itwillbepulledinsomewhatbythemagnets).Threethreadedrods(positioningscrews)screwintotheholderandpushagainstthedrumface.ThreemorethreadedrodsaxedtothedrumfaceserveasguidesforthelocksleevewhichholdstheTGGholdertighttothedrum.ThepositioningscrewsadjusthowfartheTGGsitsinsidethemagnetdrum,thuscontrollingthetotalpolarizationrotationwithintheTGG.UndertheguidanceoftheresultsfromSection 3.7.2 ,aretro-reectingmirrorisplacedbetweentheTGGandKTP2,andtheprobebeamissentbacktowardsthelaser.IftherotationintheTGGisnow45,mostofthepowerpropagatingbackwardsoutoftheFaradayfootprintisinthespolarization,soweputourOphirpowersensorontheppolarizedbeamandadjustthepositionoftheTGGuntilthepowerisminimized.Thiscorrespondsto45degreerotationintheTGG.LastlytheHWPisplacedbetweenKTP1andtheTGG,andispositionedsothattheincidentangleisassmallaspossible.TheHWPisthenrotateduntilthepowerintherejecteds-polarizationisminimized.Thenalratioofpowersintheoutgoingpandspolarizationsisoftenabout2000,anextinctionratioof33dB. 4.4OpticalLossTestsTomeasuretheopticallossintheFaradayisolator,weutilizeabeamsplittertomonitorinputandoutputpowersimultaneously.AsimplieddiagramoftheexperimentalsetupisshowninFigure 4-19 .Lightfromafree-runningInnoLightNPROlaserpassesthroughahalf-waveplateandcalcitewaveplate,whichactstoxthepolarizationandadjustthepowerofthetrasmittedbeam.ThebeamthenpassesthroughaStanfordResearchSystemsModelSR540opticalchopper,whichmodulatesthebeambyasquarewaveat151Hz;thisallowsustobypassnoisefromdarkcurrents.Afterpassingthrough 100

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Figure4-18. TheassemblyprocedurefortheLLFI.TheKTP1wedgeispositionedandrotatedaboutthebeamaxissoastoalignitsaxeswiththoseofthecalcitepolarizer.TheKTP2wedgeisplacedandrotatedaboutthebeamaxistoalignitsaxeswithKTP1.ThemagnetdrumandTGGarepositioned,andtheTGGismovedwithinthemagneticenvironmentuntilthepowertheretro-reectedppolarizationisminimized.Lastly,theHWPispositionedandrotateduntilallofthepoweroutofKTP2isintheppolarization. appropriatemode-matchinglenses,thebeamisincidentonapowerbeamsplitter.ThelighttransmittedthroughthebeamsplitterissenttoaphotodiodecalledPD-POW,whichservesasamonitorofthepowersuppliedbythelaser.ThelightreectedatthebeamsplitterissentthroughtheLLFIandthenretroreectedthroughtheisolator.Someofthisretroreectedbeamistransmittedthroughthebeamsplitter,andisthensenttoaphotodiodecalledPD-RET,whichservesasamonitorofthedouble-passthroughputoftheLLFI.Eachphotodiodesignalissenttoalock-inamplier,usingthebeamchopperasareferencefrequency.TheexperimentisthenrepeatedwithouttheLLFItoestablishbaselinereadings. 101

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IfP0isthepowerfromthelaserincidentonthebeamsplitter,Rp;sarethereectivitiesofthebeamsplitterforpandspolarizations,isthesinglepasspowerlossoftheLLFI,andRretisthereectivityoftheretroreector(forppolarizationinourcase),thenthepowerincidentonPD-POWissimply: PD-POW=P0(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp)(4{2)WhentheLLFIispresent,thepowerincidentonPD-RETisgivenby: PD-RET=P0Rp(1)]TJ /F3 11.9552 Tf 11.956 0 Td[()2Rret(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(Rs)(4{3)andwhentheLLFIisremoved,thepoweris: PD-BASE=P0RpRret(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp)(4{4)Ingeneral,thepowerfromthelaserwillbeafunctionoftime(whereastheotherparametersshouldnotvarysignicantly),andsoadirectcomparisonofthePD-RETtothePD-BASEreadingwillbelimitedbylaserpowernoise.However,wecancomparesimultaneousmeasurementsofPD-RETandPD-POW,andPD-BASEandPD-POW: RAT-LLFIPD-RET PD-POW=Rp(1)]TJ /F3 11.9552 Tf 11.955 0 Td[()2Rret(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rs) 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp(4{5) RAT-BASEPD-BASE PD-POW=RpRret(4{6)Theseratiosaredeterminedbyexperimentalparametersthatareindependentofthetimeofthemeasurement.Takingtheratiooftheratiosremovesanydependenceonthereectanceoftheretroreector: LLFI/BASERAT-LLFI RAT-BASE=(1)]TJ /F3 11.9552 Tf 11.956 0 Td[()21)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rs 1)]TJ /F3 11.9552 Tf 11.956 0 Td[(Rp(4{7)AnexpressionforthesinglepassLLFIlossisthen: =1)]TJ /F8 11.9552 Tf 11.955 22.107 Td[(s LLFI/BASE(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp) 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rs(4{8) 102

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Ifthebeamsplitterisaperfectpowerbeamsplitter,thatisRp=Rs,thenthesinglepasslosscanbemeasuredsimplycomparingthetestsetupwiththeLLFItothatwithout.Ingeneral,however,thisisnotthecase.Toaccountforthedierentialreectionbetweenpandspolarizationatthebeamsplitter,ameasurementliketheonedescribedatthebeginningofthissectionistakenwithaquarter-waveplateinplaceoftheLLFI.Ifisthesinglepassopticallossofthequarter-waveplate,whenthewaveplateisalignedsothattheretroreectedbeamreturnstothebeamsplitterinppolarization,theratioofthepowerincidentonPD-RETtothatonPD-POWis: PD-P=Rp(1)]TJ /F3 11.9552 Tf 11.955 0 Td[()2Rret(4{9)andwhenthequarter-waveplateisalignedsothattheretroreectedbeamreturnstothebeamsplitterinspolarization,thesameratioofpowersis: PD-S=Rp(1)]TJ /F3 11.9552 Tf 11.956 0 Td[()2Rret1)]TJ /F3 11.9552 Tf 11.956 0 Td[(Rs 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp(4{10)Therefore,theratioofthetwomeasurementsin( 4{9 )and( 4{10 )is: S/P=1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rs 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rp(4{11)Wecanputthisresultinto( 4{8 )toexpresstheLLFIsinglepasslossintermsofmeasuredvalues: =1)]TJ /F8 11.9552 Tf 11.955 18.586 Td[(r LLFI/BASE S/P(4{12)Inthelaboratory,LLFI/BASEismeasuredtobe0.349,andS/Pismeasuredtobe0.360,givingasinglepasslossfortheLLFIof1:620:128%.Thisresultfallsabout6000ppmshortofthe<1%statedgoalfornear-termsqueezingimplementation. 103

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Figure4-19. AbasicschematicoftheopticallosstestingfortheLLFI. 104

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CHAPTER5LASERAMPLITUDEMODULATIONINLIGOAmplitudemodulationreferstoaphenomenonwheretheamplitudeofarelativelyhighfrequencysinusoidhasarelativelylowfrequencytimedependence.ItistheprinciplebehindAMradiosandallowsforthetransmissionoflowfrequency(audiofrequencies,forexample)signalsthroughahighfrequency(GHz,forexample)carrierwave.InLIGO,amplitudemodulationofthecarrierlasereldwillinterferewithinterferometercontrolschemes,aswillbeexplainedinthischapter. 5.1RadioFrequencyAmplitudeModulation 5.1.1PhaseModulationTounderstandthesignicanceofamplitudemodulation,itishelpfultoexploreatechniquecalledphasemodulationandexplainitsusefulnesstoLIGO.Mathematically,wecanrepresentaphasemodulatedlighteldwithmodulationfrequencyas: EPM=ei(!0t+mcost)(5{1)Here,miscalledthemodulationindex,ormodulationdepth.Notethatthepowercarriedbytheeld,whichisproportionaltojEPMj2,isunaectedbythispurephasemodulation.Toreplicatethiswithlasersystems,abeamispassedthroughadeviceknownasanelectro-opticmodulator(EOM).TheEOMhastwomajorsubsystems:aresonantcircuitthatamplieselectricalsignalsataparticularfrequency(themodulationfrequency),andanelectro-opticcrystalwhoseindexofrefractionisafunctionoftheelectriceldappliedacrossit.Asignalsentthroughtheresonantcircuitputsahighvoltageoscillatingpotentialontoelectrodessurroundingtheelectro-opticcrystal,causingatimedependenceoftheindexofrefraction.Thebeampassingthroughthecrystalexperiencesatimedependentoscillationoftheopticalpathlengththroughthecrystal,andconsequentlytherelativephaseoftheeldexitingthecrystaloscillatesatthemodulationfrequency. 105

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WecanusetheJacobi-Angeridentitytoexpand( 5{1 )intermsofBesselfunctions: eimcost=1Xn=inJn(m)eint="J0(m)+1Xn=1inJn(m))]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(eint+e)]TJ /F4 7.9701 Tf 6.586 0 Td[(int#(5{2)Forpositiveintegersn: Jn(x)=1Xl=0()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)l l!(l+n)!x 22l+n(5{3)andwecanwrite: J0(m)+1Xn=1inJn(m))]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(eit+e)]TJ /F4 7.9701 Tf 6.587 0 Td[(it=1+im 2)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(eit+e)]TJ /F4 7.9701 Tf 6.586 0 Td[(it+O(m2)(5{4)Solongasthemodulationissmallenough(m<<1),wecanuse( 5{1 )and( 5{4 )toapproximate: EPMei!0t+im 2ei(!0+)t+im 2ei(!0)]TJ /F6 7.9701 Tf 6.586 0 Td[()t(5{5) Figure5-1. Acartoonspectrumoffrequencymodulatedlight.Thetwosidebandsarespacedatthemodulationfrequency,,awayfromthecentralcarrierfrequency,!0.Therelativeheightofthesidebandsisdeterminedbythemodulationindexm. Thersttermontheright-handsideof( 5{5 )iscalledthecarrierwave,andtheremainingtwotermsarereferredtoasthesidebands.Noticethatthefrequencyofthesidebandsdierfromthecarrierfrequency,!0,bythemodulationfrequency.AfrequencyspaceillustrationofthismodulationisgiveninFigure 5-1 .Wecanconsidereachcomponent, 106

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thecarrierandtwosidebands,tobevectorsinthecomplexplane,thesumofwhichrepresentsthetotaleld.ThisisknownasthephasorpictureandisillustratedinFigure 5-2 .Choosingareferenceframeinwhichthecarriervectorisxed,thesidebandvectorsrotateincounter-propagatingdirectionsatthemodulationfrequency. Figure5-2. Aphasordiagramforphase-modulatedlight.Thecarrier,showninred,isxedinourframe.Thesidebands,ingreenandpurple,rotatewithoppositehandednessatthemodulationfrequency.Weseethatforsmallsidebandamplitudes,theneteectistorotatethecarrierinphasespace. Ifwerewrite( 5{5 )as: EPM=ei!0t(1+imcost)(5{6)thenisiteasytoseethat: jEPMj2=jj2(1+imcost)(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(imcost)=jj21+O(m2)(5{7)andso,torstorderinm,ourapproximationofthemodulatedeldleavesthepowerintheeldunchanged.WecanmakethesameargumentqualitativelyfromanexaminationofthephasordiagraminFigure 5-2 .Observingthecarrierandsidebandvectorsoverafullperiod2=,weseethattheprimaryeectofthesidebandsistorotatethesumofthethreevectors(thefulleldvector)throughasmallangle. 107

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Supposingtheopticalsystemhasa(frequencydependent)transmissionfunctiongivenbyT,theoutputeldwillbe: EOUT=T0ei!0t+im 2T+ei(!0+)t+im 2T)]TJ /F3 11.9552 Tf 7.085 1.793 Td[(ei(!0)]TJ /F6 7.9701 Tf 6.586 0 Td[()t(5{8)whereT0andTareshorthandforT(!0)andT(!0)respectively.Theresultantsignalonaphotodetectorwillbeproportionalto: jEOUTj2=jj2jT0j2+mIm(X)cost+mRe(X)sint(5{9)forX=T0T+)-290(T0T)]TJ /F1 11.9552 Tf 7.085 1.793 Td[(,andwherewehaveagaindroppedtheO(m2)terms.TheDCcomponentof( 5{9 )canbelteredoutwithahighpasslter,leavingasignal: Srf=Gmjj2[Im(X)cost+Re(X)sint](5{10)whereGisafactorthatconvertstheincidenteldintensitytoavoltage.Thisresultantphotodetectorsignalcanbemixedwithalocaloscillatorsignal,Slo=Acos(0t+),toget: SrfSlo/Im(X)fcos[cos()]TJ /F1 11.9552 Tf 11.956 0 Td[(0)t+cos(+0)t])]TJ /F1 11.9552 Tf 11.955 0 Td[(sin[sin()]TJ /F1 11.9552 Tf 11.955 0 Td[(0)t)]TJ /F1 11.9552 Tf 11.955 0 Td[(sin(+0)t]g+Re(X)fcos[sin()]TJ /F1 11.9552 Tf 11.955 0 Td[(0)t+sin(+0)t]+sin[cos()]TJ /F1 11.9552 Tf 11.955 0 Td[(0)t)]TJ /F1 11.9552 Tf 11.955 0 Td[(cos(+0)t]g(5{11)withproportionalityconstant1 2Gmjj2A.Now,ifthelocaloscillatorfrequencyismatchedtothemodulationfrequency(=0),thesignalcanbelow-passlteredtoget: Serr=1 2Gmjj2A[Im(X)cos+Re(X)sin](5{12)From( 5{12 )weseethatbyadjustingthephaseofthelocaloscillator,wecanselectforeithertherealorimaginarypartsofX.Tosumup,phasemodulatingacarrierwaveanddemodulatingtheresultingsignalatthemodulationfrequencywillresultinaDCerrorsignalthatcontainsinformationaboutthefrequencydependenttransmissionfunctionof 108

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anexperimentalsetup.Ifthistransmissionfunctionisknownbeforehand,theresultingsignalcanbeusedtodeterminethefrequencyofthecarrierwaveandcanbeusedinfeedbacktocontroltoholditconstantatadesiredvalue.ThisistheoperatingprinciplebehindthePound-Drever-Halllockingtechnique( 15 ),aswellassimilarcontrolschemesinLIGO(seeFigure 5-3 ). Figure5-3. AsimpliedcartoonoftheLIGOlengthcontrolscheme.VariousdemodulatedsignalsgatheredattheREFL,POP,andASportsareresponsibleforcontrollingfourlengthdegreesoffreedomoftheinterferometer.FromKeikoKokeyama,LIGODocumentG1301236. 5.1.2AmplitudeModulationIntheprevioussubsectionwediscussedaprocesscalledphasemodulation,whichcanbeusedtoproduceerrorsignalsforlengthsensingandfeedbackcontrolschemes..Supposewemodulatetheeldgivenby( 5{6 )sothat: AM=(1+cost)(5{13)foramplitudemodulationfrequencyand(complex)amplitudemodulationindex.Noticethatsince: jAMj2=jj2(1+2Re()cost+jj2cos2t)(5{14) 109

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unlikeinthecaseofphasemodulation,amplitudemodulationdoesaectthepowerinthelasereld.Tounderstandhowsuchamodulationmightarise,considerthephasorpicturewithacarrierandfrequencysidebands,butsupposethesidebandshavebeenshiftedinphasesothattheynolongeralignwhentheyareperpendiculartothecarrier(Figure 5-4 ).Weseethattheeectofthesidebandsisnolongersolelytorotatethecarrierinphasespace,butitnowappearsthattheeldamplitudeismodulatedatthesidebandfrequencyaswell.Wecouldimagineasimilareectinthecasewherethefrequencysidebandshaddierentamplitudes.Ingeneral,anydispersiveprocesswillresultinamplitudemodulationofthelighteld( 33 ; 53 ).Thisshouldnotbesurprising,asitwasimplicitlyintroducedin 5.1.1 with( 5{8 ),andisthebasisforphasemodulationcontrolschemes:phasemodulationisconvertedtoamplitudemodulationbytheopticalsystem,whichisthenusedasanerrorsignaltocontrolsomedegreeoffreedomofthesetup.Drivingtheerrorsignaltozerowithfeedbackcontrollockstheopticalsystemtoadesiredworkingpoint. Figure5-4. Aphasordiagramillustratingaparticularkindofamplitudemodulation.Thefrequencysidebandshavebeenphaseshiftedsothattheirsumisnolongerperpendiculartothecarrierphasor.Thesumofthesidebandandcarriereldthushasatimedependentamplitude,representedbythemagnitudeoftheresultantvector. 110

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Returningto( 5{13 ),ifweignorethefrequencydependenceoftheopticaltransmissionfunctionfortheamplitudemodulationterms,theresultantACsignaloutofthephotodiodewillbe: SAMrf=SPMrf+2GRe()cost)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(jj2jT0j2+SPMrf(5{15)whereSPMrfisdenedby( 5{10 ),andwehaveignoredalltermsbeyondrstorderin.If=or=2,thendemodulatingthephotodiodesignalatfrequencywilladdatermproportionaltointheDCerrorsignalgivenby( 5{12 ).Weseethattheeectofamplitudemodulationofourcarrieristoaddosetstoourerrorsignals.Inpractice,anerrorosetinacontrolsignalcansimplybezeroedaway.However,whenhasalowfrequencytimedependence,thiserrorosetdriftswithtimeandwouldrequirere-optimizationofcontrolmatrices.ThispresentsapotentialproblemforAdvancedLIGO,whichaimstomaintainadutycycleof75%perinterferometerduringobservationruns. 5.2RFAMintheLIGOInterferometer 5.2.1RFAMMonitorInstallationandCalibrationInordertomeasureRFAMonthecarrierintheL1detectorinLivingston,Louisiana,anRFphotodiode( 56 )wasplacedonanin-airtable,namedIOT2L,adjacenttothein-vacuuminterferometer.LaserlightexitingtheInputModeCleaner(IMC)ontheHAM2opticalbenchispickedo,thensentthroughaperiscopeoutofvacuumandontoIOT2L.Thisbeamissteeredacrossthetable,beingpickedotwicemoreforothermonitors,andeventuallyterminatedonthephotodiode,hereaftercalledtheRFPD.ApictureoftheRFPDsittingonIOT2LisgiveninFigure 5-5 .TheDCoutputisinputtoaDCphotodiodeboxsittingonIOT2LwhichisthensenttoanADCandrecordedat2048HzunderthechannelnameL1:IMC-IOT1 SPAREIN1.Inprinciple,thisDCchannelcanbeusedasareferenceforthepowerincidentontheRFPD,howeverin 111

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practiceweusetheL1:IMC-IM4 TRANS SUMchannel,asitisalreadycalibratedtodirectlyreadthepoweroutintransmissionoftheIMC,whichwewillrefertoasIMC-T.TheRFPDACoutputissentthroughaZSC-2-1powersplitterandtheoutputsaresenttotheinputchannelsofanI-Qdemodulatorfor9and45MHz.AschematicoftheI-QdemodulationisgiveninFigure 5-6 .Thisdemodulationconvertsthesignalamplitudesat9and45MHztoDCsignalsthatarerecordedat2048Hz.Fromthesedemodulationsignals,thefullACsignalcanbereconstructedifneedbe.ThedemodulatedRFsignalsarestoredunderthechannelnamesL1:LSC-POPAIR A RF9 I,L1:LSC-POPAIR A RF9 Q,L1:LSC-POPAIR A RF45 I,L1:LSC-POPAIR A RF45 Q,forthe9MHzIandQ(RF9-IandRF9-Q),and45MHzIandQ(RF45-IandRF45-Q)datarespectively.AllrawdatachannelsinLIGOaresavedwithunitsof\counts."ItisworthnotingthattheveRFPDdatachannelsoriginallyservedotherpurposesandthenfelloutofuse,hencetheirnamesarenotdirectlyrelatedtotheirfunction.ItiseasiertoswitchsomecablesaroundandleaveanotethanitistochangeaLIGOchannelname. Figure5-5. AphotographofthefastphotodiodeonIOT2Lwiththebeampathillustratedinpink.Photocourtesyofauthor. Asarsttestofthesetup,whilemaintainingcavitylocktheIMCwasdetunedbymanuallyvaryingavoltageosetintheIMCcontrolservocalledL1:IMC-REFL SERVO COMOFS. 112

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Figure5-6. AcartoonofourRFAMdetectionscheme.LightintransmissionoftheIMCispickedoandsenttoourfastphotodiode,thesignalissplitintoalowfrequencycomponentwhichservesastheDCpowermonitor,andahighfrequencycomponentthatispassedthroughademodulationphase.Thehighfrequencysignalissplitfourtimesandmixedwithsineandcosinewavesat9MHzand45MHz,theresultingmixturesarelowpassedtorecovertheso-calledIandQquadraturesoftheRFsignal. Thisosetwasselectedbecauseadjustmentstoallotherosetsintheservoappearedtobecompensatedforbyanotherstage.TheIMCisdesignedsothatwhenitisonresonancewiththecarriereld,boththe9and45MHzsidebandsareaswell.WhentheIMCisslightlydetuned,theupperandlowersidebandsateachfrequencyexperienceadierentopticaltransferfunctionandRFAMisproduced(asinFigure 5-4 ).ThisRFAMwillexistat9and45MHz,andfrom( 5{15 )weexpecttherespectiveACphotosignalsattheRFPDtobeoftheform: SAM9;45=2Gjj2jT0j29;45cos(9;45t)g(t)+SPM9;45(5{16)where9;45are29MHzand245MHzrespectively,andg(t)describesthelowfrequencymodulationoftheIMCoset.NotethatSPM9;45correspondtomodulationtermsthatarisefromsourcesotherthantheIMCdetuning.Henceforthwedropthefrequencysubscriptswiththeunderstandingthatthecalculationsarevalidforbothradiofrequency 113

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bands.Afterdemodulationandlowpassltering,thesignalsintheIandQchannelsare: RFI=Gjj2g(t)cos+(SPMterms)RFQ=Gjj2g(t)sin+(SPMterms)(5{17)whereGhasaccountedforallopticalandelectronicgainsandisthearbitrarydemodulationphase.IfweassumethatthetermsfromtheSPMmodulationremainconstantovertheintervalofourtest,thentheycanbesubtractedoasDCterms.Supposingaswellthatg(t)integratestozerooverourmeasurementperiod,wewillhave: RFI)]TJ ET q .4782 w 109.061 -193.377 m 130.037 -193.377 l S Q BT /F1 11.9552 Tf 109.061 -203.22 Td[(RFI=Gjj2cosRFQ)]TJ ET q .4782 w 293.65 -193.377 m 319.506 -193.377 l S Q BT /F1 11.9552 Tf 293.65 -203.22 Td[(RFQ=Gjj2sin(5{18)fromwhichitisclearthat: q (RFI)]TJ ET q .4782 w 172.76 -265.103 m 193.735 -265.103 l S Q BT /F1 11.9552 Tf 172.76 -274.946 Td[(RFI)2+(RFQ)]TJ ET q .4782 w 262.459 -265.103 m 288.316 -265.103 l S Q BT /F1 11.9552 Tf 262.459 -274.946 Td[(RFQ)2=Gjj2(5{19)Thefactorofjj2isleftinexplicitlyasareminderthatthesignalisdependentonthecarrierpower.Wewillrefertotherighthandsideof( 5{19 )astheSQ9andSQ45channels.Figure 5-7 showstheresultofthedetuningtest.ThetransmittedIMCpower,SQ9,andSQ45channelscanallbeseentorespondatapproximatelytwicethefrequencyoftheservomodulation.ThetransmittedIMCpower,L1:IMC-IM4 TRANS,isnoticeablynoiser.Ourinterestisinmonitoring,andsowerequireanestimateofG.Inpractice,measuringGisverydicult:duringobservationperiodsaccesstotheLargeVacuumEnclosureArea(LVEA)isverylimitedandgenerallyonlyallowedformaintenanceandothernecessarywork.However,wemayusetheIMCdetuningasaroughcalibration.WemakeuseofFINESSE,anopticalsimulationsoftware,toconstructmodelsoftheIMCanditsoutput,andthencollaboratetheresultswithmeasurement.Figure 5-8 givesanillustrationoftheprocess.WecanseethatderivingGjj2isequivalenttotheproblemofconstructingGN)]TJ /F6 7.9701 Tf 6.587 0 Td[(1A.TheonlyunknownintheowdiagramisthemappingG,and 114

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Figure5-7. TheresponseoftheRFPDtoRFAMinducedbydetuningtheIMC.TheIMCwasdetunedbymanuallyvaryingaservooset,shownattheverytop.ThesecondplotisameasureofpowertransmittedthroughtheIMC(notthepowermeasuredattheRFPD).ThenaltwoplotsaretheappropriatesumofsquaresofRFchannels. tocalculateitwewillevaluate: G=SQ (N)]TJ /F6 7.9701 Tf 6.587 0 Td[(1DC)]TJ /F6 7.9701 Tf 6.587 0 Td[(1N)(IMC-T)(5{20)Thoughtheyshareasymbol,thetwonormalizationfunctionsin( 5{20 )dierslightlyintheirmeaning.WhenactingonIMC-T,Nnormalizeswithrespecttothemaximaltransmissionpowerwithinthemodel.Generally,thiscorrespondstothepowerwhenorareheldtobezero.However,whenactingonSQ,Nnormalizeswithrespecttothecoincidentpowertransmitted.ThisisadistinctioninprinciplebecauseRFAMpresentintheSQchannelswillalsobepresentinIMC-T.Inpractice,however,thechangeinIMC-Tduetoamplitudemodulationisminimalcomparedtoitsnaturaloperatingvalue. 115

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Figure5-8. Aowchartdiagramillustratingtheconnectionsbetweenourobservedandmodeledquantities.Mappingsareitalicized,whilemeasurablesarebolded.ThemapsA,B,C,andDareobtainedthroughsimulationsinFINESSE,Nisanormalizationmapping,andGistheunmodeledtransferfunctionbetweentheRFphotodiodeandtheRFdatachannels. Wewillapproach( 5{20 )inpieces.BecauseIMC-Tisalreadycalibrated,thedegreeofdetuningoftheIMC,herecalled,canbefoundby: =(C)]TJ /F6 7.9701 Tf 6.586 0 Td[(1N)(IMC-T)(5{21)whereCisthefunctionthatmapsadetuningphasetoarelativepowertransmittedthroughtheIMC.Figure 5-9 providesaplotofCastherstimagewhichhastheempiricalt: C(x)=)]TJ /F1 11.9552 Tf 11.235 8.088 Td[(34:2 deg2x2+1(5{22)Weseethatinthestrictsense( 5{22 )isnotinvertiblebecauseitisnotone-to-one.However,becauseitisanevenfunctionaboutzero(asisD,whichisthesecondimage),wecanconsiderthemaximalamplitudeofoscillationinIMC-Tand,withoutlossofgenerality,attributeittothemaximaldetuningoftheIMC.InFigure 5-10 roughtsofthedetuningsignalsareshown.ThetofIMC-Tapproximatesthepeak-to-peakoscillationfromdetuningtobe12:610)]TJ /F6 7.9701 Tf 6.587 0 Td[(3watts,whichwhennormalizedtothemaximalpowerreadoutbyIMC-Tbecomes12:210)]TJ /F6 7.9701 Tf 6.587 0 Td[(3.Thisvaluecorrespondstomaximallossin 116

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Figure5-9. ThesimulatedresponsesofidealphotodetectorstothedetuningoftheIMC.OntheleftisthetotalpowertransmittedthroughtheIMCnormalizedtothevalueatzerodetuning.OntherightistherootsumsquaredIandQchannelsforboth9and45MHz,normalizedtotheinstantaneoustransmittedpower.ThesimulationwasperformedusingFINESSEopticalmodelingsoftware. relativepowerduringthedetuning,andsowederiveourmaximal: max=C)]TJ /F6 7.9701 Tf 6.586 0 Td[(1[1)]TJ /F1 11.9552 Tf 11.955 0 Td[((12:210)]TJ /F6 7.9701 Tf 6.586 0 Td[(3)]=r 12:210)]TJ /F6 7.9701 Tf 6.586 0 Td[(3 34:16=1:8910)]TJ /F6 7.9701 Tf 6.586 0 Td[(2deg(5{23)Havingcalculatedthemaximaldetuningphaseduringthetest,weseefromFigure 5-8 thatthemeasuredSQchannelsarerelatedtoby: SQ=(GN)]TJ /F6 7.9701 Tf 6.586 0 Td[(1D)()(5{24) D9(x)=4:1810)]TJ /F6 7.9701 Tf 6.587 0 Td[(6jxjdeg)]TJ /F6 7.9701 Tf 6.586 0 Td[(1D45(x)=2:0910)]TJ /F6 7.9701 Tf 6.586 0 Td[(5jxjdeg)]TJ /F6 7.9701 Tf 6.586 0 Td[(1(5{25)ThetsforSQ9andSQ45inFigure 5-10 approximatethenormalizedpeak-to-peakoscillationsfromdetuningtobe0.946and1.99respectively.From( 5{24 )and( 5{25 )wecanthereforeevaluate: G=SQmax (N)]TJ /F6 7.9701 Tf 6.586 0 Td[(1D)(max))166(!G9=0:946 (4:1810)]TJ /F13 5.9776 Tf 5.756 0 Td[(6)j1:8910)]TJ /F13 5.9776 Tf 5.756 0 Td[(2j=1:19107G45=1:99 (2:0910)]TJ /F13 5.9776 Tf 5.756 0 Td[(5)j1:8910)]TJ /F13 5.9776 Tf 5.756 0 Td[(2j=5:03106(5{26)ThemodelingfrommodulationindextoRFPDsignalisverystraightforwardandcanbeapproximatedtoverygoodprecisionwithpen,paper,andthetrigonometricidentity: 117

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Figure5-10. The(very)approximatecurvetstotheresponsesofIMC-T,SQ9,andSQ45totheIMCdetuningtest.Thepeak-to-peakamplitudeoftheIMC-Ttisusefulforestimatingthemaximaldetuningphaseduringthetest. 2cos2(u)=1+cos(2u).Nevertheless,weuseFINESSEtoproducethemodel: A(x)=0:498x(5{27)Notethatthisresultholdstrueregardlessoftherelativephaseoftheamplitudemodulation.Thecoecientin( 5{27 )isnotexactly1=2becausethesourceofamplitudemodulationisbeforetheIMConthebeampath,andsothe(small)opticallossintheIMCreducesA.Wecannowcombine( 5{19 ),( 5{26 ),and( 5{27 )tocalculatetheamplitudemodulationindex: 9=1:6810)]TJ /F6 7.9701 Tf 6.586 0 Td[(7SQ9 IMC-T45=3:9910)]TJ /F6 7.9701 Tf 6.586 0 Td[(7SQ45 IMC-T(5{28)WeseethatisitconvenienttodenethesignalSQN=SQ=IMC-T. 5.2.2RFAMMonitoringThoughwederived( 5{28 )forthespeciccaseoftheIMCdetuningtest,theresultgeneralizestoallmeasurementsofRFAMontheIMCoutputsolongaswemodifythedenitionofSQslightlytobe: SQ=q (RFI)]TJ /F1 11.9552 Tf 11.955 0 Td[(DI)2+(RFQ)]TJ /F1 11.9552 Tf 11.955 0 Td[(DQ)2(5{29) 118

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whereDIandDQarethedarksignalsfortheIandQchannelsrespectively.Asacheckonthemonitor,theRFAMsignalswereanalyzedforOctober19,2015.Theinterferometerwaslockedandinobservingmodefrom09:00:00to11:00:00UTC.Thedarksignalsareestimatedfromtheinterval02:45:00-04:45:00UTC,duringwhichlaserlightwasnotexitingfromthePSLintothemaininterferometer.TheresultsofthemonitoringperiodareincludedasTable 5-1 .(ThoughthedarksignalismeasuredforIMC-Tandusedtocomputethetransmittedpower,thevaluesareexcludedfromthetableforaestheticreasons). Table5-1. RFsignalsforOctober19,2015observation. SignalChannelNameBrightCountsDarkCountsRF)]TJ /F1 11.9552 Tf 13.2 0 Td[(D RFI9L1:LSC-POPAIR A RF9 I7:550:166)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:490:0739:040:181RFQ9L1:LSC-POPAIR A RF9 Q)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:1960:1622:340:061)]TJ /F1 11.9552 Tf 9.298 0 Td[(2:530:173RFI45L1:LSC-POPAIR A RF45 I)]TJ /F1 11.9552 Tf 9.299 0 Td[(1:090:0910:6180:051)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:710:104RFQ45L1:LSC-POPAIR A RF45 Q0:7620:0816:340:064)]TJ /F1 11.9552 Tf 9.298 0 Td[(5:580:103IMC-TL1:IMC-IM4 TRANS22:50:00722:50:007 Using( 5{28 )and( 5{29 )withthissampledata,wearriveatvaluesfortheeectiveamplitudemodulationindicesontheinterferometer'scarrierbeam: 9=7:0310)]TJ /F6 7.9701 Tf 6.586 0 Td[(81:2410)]TJ /F6 7.9701 Tf 6.587 0 Td[(945=1:0410)]TJ /F6 7.9701 Tf 6.586 0 Td[(71:5610)]TJ /F6 7.9701 Tf 6.587 0 Td[(9(5{30)Ithasbeenshownthatfor=m<10)]TJ /F6 7.9701 Tf 6.586 0 Td[(4amplitudemodulationwillnotsignicantlydisruptlengthsensingandcontrolschemesinLIGO( 31 ).Thevaluesin( 5{30 )correspondto=m<10)]TJ /F6 7.9701 Tf 6.587 0 Td[(6,andareroughlytwoordersofmagnitudelowerthanwerepreviouslymeasured( 46 );thissuggeststheneedforanothermethodofRFPDcalibration.Aresultlike( 5{30 )isonlymeaningfulforstationarymodulationindices;thatis,ifthemodulationindicesremainconstantoverlongperiodsoftime.Inreality,theamplitudemodulationindexisafunctionoftime(t),andsoamoreusefulcalculationmightbetoconstructtheamplitudespectraldensityoftheamplitudemodulationindex.Naively,wemightexpectthat~(!)=G~SQN(!),butwemustalsoaccountforelectronicnoisein 119

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theSQNsignal.AtDC,thiscouldbeachievedbysimplysubtractingthemeanchannelvalueswhennolightwasincidentontheRFPD,andforthespectraldensitywehaveananalagousmethod: ~(!)=G IMC-Tq ~SQ2(!))]TJ /F1 11.9552 Tf 16.832 3.022 Td[(~SQ2d(!)(5{31)whereSQdcorrespondstothechannelsignalwhennolightisincidentontheRFPD.Wehaveassumedthatthecouplingbetweentheamplitudemodulationindexandtheresultantradio-frequencysignalisconstantoverfrequenciesofinterest.Aplotof( 5{31 )isincludedasFigure 5-11 Figure5-11. AnamplitudespectraldensityoftheamplitudemodulationindexintheL1detector.DataistakenfromJune7,201700:00:00UTC.TheauthorstressesthatthecalibrationoftheRFPDiscurrentlyinquestion,andsothescaleofeachcurvemaydierfromthetruevalue. 5.2.3FINESSESimulationsIntheprevioussubsectionwemadeuseofresultsfromFINESSEwithoutexplanation.FINESSE,orFrequencydomainINterferomEterSimulationSoftwarE,isasoftwarepackagethatisusedtoquicklymodelinterferometersandotheropticalsystems( 19 ).Everymodeliswritteninakatle,andconsistsofelementsofzerodimensionlocatedatuniquenodes.Elementscanbelasers,modulators,mirrors,lenses,beamsplitters,photodiodes,etc.Eachnodeisconnectedtoanothernodebyaspace,havingone 120

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dimensionoflength(whichcanbezero).Whenasimulationisrun,theprogramsolvesasetoflinearequationsfortheelectriceldamplitudesateverynode.FINESSEcancomputesolutionsforbothstaticcasesaswellasinthefrequencydomaintocomputeopticaltransferfunctions.Moreimportantthancalibratingphotodiodes,FINESSEcanalsohelptotracetheeectsofRFAMthroughouttheinterferometer.Inordertodoso,wetakeadvantageofthelockingfunctionalityofFINESSE:auserspeciesanerrorsignal,again,andaprecision,andthesystemiterativelysolvestheforasteady-statesolutionthatsetstheerrorsignaltozerowithintheprecisiongiven.Thefullinterferometerkatledeneserrorfunctionsfortheveinterferometerdegreesoffreedomas: PRCL err=POP 9 IMICH err=POP 45 QCARM err=REFL 9 ISRCL err=REFL 45 IDARM err=OMC DC)]TJ /F17 11.9552 Tf 11.955 0 Td[(DARM offset(5{32)Theseerrorsignalsinturndenelockfunctions: PRCL err!PRCL lockMICH err!MICH lockCARM err!CARM lockSRCL err!SRCL lockDARM err!DARM lock(5{33)Thelockfunctionstakeintoaccounttheopenloopopticalgainofthefeedbackroutine.Theselocksignalsbecomeactuationsignalsforopticalcomponentsoftheinterferometer 121

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as: PRCL lock!PRMMICH lock!ITMX)]TJ /F17 11.9552 Tf 9.298 0 Td[(MICH lock!ITMYCARM lock+MICH lock+DARM lock!ETMXCARM lock)]TJ /F17 11.9552 Tf 11.955 0 Td[(MICH lock)]TJ /F17 11.9552 Tf 11.956 0 Td[(DARM lock!ETMYSRCL lock!SRM(5{34)wheretheactuationisinthephase,,ofeachoptic;themicroscopicpositionosets.Thesephasesareeasilyconvertedtophysicaldistancesintheusualfashion:x=0=2.Thus,bysweepingoverdierentAMindicesinoursimulation,wecanndthecorrespondingosettotheerrorsignalforagivendegreeoffreedom,whichitselfcorrespondstoaphysicalactuationwithunitsoflength.AnexamplefortheDARMfeedbackloopisgiveninFigure 5-12 ;herewehaveprovidedtheratioofdisplacementnoiseinDARMtothecorrespondingamplitudemodulationindexthatcausesthisnoise.Weseethattheratiocanvaryoverroughlyanorderofmagnitudedependingonthephaseoftheamplitudemodulationrelativetothatofthephasemodulation.Wecanusethisinformationtoconstructtheexpectedstrainnoisefrommeasuredamplitudemodulation.AnexampleofsuchaconstructionisgiveninFigure 5-13 ,whereweuseL1datafromJune7,201700:00:00UTC,andassumethecouplingistheaverageofthevaluesinFigure 5-12 .Aswewouldexpect,thecalculatedstrainnoiseduetoRFAMisbelowthecurrentstrainsensitivityofAdvancedLIGO.However,ifithappensthattheRFPDcalibrationiscorrect,ourresultssuggestthattheRFAMnoiseoorcouldpossiblybesittinganorderofmagnitudebelowcurrentsensitivity,andwouldbecomealimitingnoisesourceintheeventthatquantumandcoatingbrowniannoisecouldbemitigatedbyafactoroften. 122

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Figure5-12. SimulatedcouplingofRFAMtoDARMasafunctionofthephaseoftheamplitudemodulationrelativetothephasemodulationofthecarrier.AsinFigure 5-11 ,the9MHzdataisshowninblueandthe45MHzingoldenrod. Figure5-13. RFAMcontributiontothestrainchannelofL1aspredictedbyourFINESSEmodeloftheinterferometer.ThemodeldoesnotaccountforfrequencydependenceoftheDARMcontrolloop.TheauthorstressesthatthecalibrationoftheRFPDiscurrentlyinquestion,andsothescaleofeachcurvemaydierfromthetruevalue. 123

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CHAPTER6FUTUREWORK 6.1ElectromagneticFieldTuningfortheFaradayRotatorInthecurrentaLIGOIFIconstructionthepositionsoftheTGGcrystalswithinthemagnethousingaremanuallyadjustedtoachievethedesiredpolarizationrotation.Whilerelativelysimple,thisprocesscanprovetobeverytediousanddiculttoreproduce.Further,itisnotclearwhetherpositioningchangesoftheTGGduringpumpdowncontributetoanobservedreductionintheisolationratiofrominthelabtoinsitumeasurements.TheUFLLFImagnetdesignaimstoeliminatethehumanelementofTGGalignment.LiketheaLIGOIFI,largepermanentmagnetsareusedtocreatethestrongbackgroundeldnecessaryfor45rotationwithintheTGG,however,netuningoftheeldwillbeperformedwithweakersolenoidelectromagnets.Inprinciple,theprecisioninachievedrotationisdeterminedbytheprecisionofthecurrentsourcesupplyingthesolenoidelectromagnets,andisnolongerdependentonmechanicaladjustments.Becausetheneadjustmentiselectricalinnature,thedesignallowsforinvacuumadjustmentoftheFaradayrotation.ElectromagneteldtuningalsoallowsforaLLFIdesignwhichhasaTGGcrystalxedwithrespecttothemagnethousing.Suchaschemewouldbeusefulifitisevershownthatpumpdownmisalignmentsleadtoreducedisolationratio.Figure 6-1 illustratesthattheplacementaccuracyforaTGGcrystalinasolenoiddesigniswellwithinmodernmachiningtolerances.PossiblythegreatestadvantageofhavingsolenoideldtuningistheabilitytocompensateforthermaldriftsoftheTGG.WhilethelasereldsarenotexpectedtosignicantlyheattheopticsintheLLFI,therearestilllowfrequencythermaldriftsassociatedwithdailyenvironmentaltemperaturechanges.TheVerdetconstantofTGG 124

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Figure6-1. Aplotillustratingthetuningrangeofasolenoidelectromagnetaidedrotatormagnet.ThehorizontalaxisisdisplacementofthecenteroftheTGGcrystalfromthecenterofthemagnethousing,whiletheverticalaxisistheintegratedmagneticeldthroughtheTGG,normalizedsothatavalueof1gives45degreesrotation.Thedashedbluelinecorrespondstothecaseofnosolenoidelectromagnet,sothereisonlythepermanentmagnetcongurationtoprovidethemagneticeld.Thedarkestshadedregioncorrespondstothepossibleintegratedeldforoperatingpowerupto1Winthespecicsolenoiddesign,whilethelightershadedregionisforupto2Wofoperatingpower. roughlyobeys: V= T(6{1)where1:17104radK/Tm.From( 3{15 ),weseethat: @ @T=)]TJ /F3 11.9552 Tf 13.343 8.088 Td[( T2ZBds(6{2)Theintegralin( 6{2 )isxedbytheeldandcrystaldimensions,chosensoastogiveavalueof1:9610)]TJ /F6 7.9701 Tf 6.586 0 Td[(2Tm,andsoaroundroomtemperaturewegetabout0:154degreesofmisrotationper1KchangetotheTGGtemperature.Thelossduetomisrotationisnotlinearinmisrotationangle,andingeneralisdependentuponthealignmentofotheropticsintheLLFI.TheworstcasealignmentscenarioisshowninFigure 6-2 ,weseethatdriftsof1Kcancorrespondtoadditionallossofasmuchas300ppm. 125

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Figure6-2. TheworstcasescenariolossduetotemperaturedriftsoftheTGGcrystal.Thisscenarioiscomputedbyallowingopticalcomponentstobemisalignedwithintolerancesuchthatthetotalthroughputlossismaximized. Twopossiblesolenoidcongurationsarepresentlyunderconsideration,bothofwhichareshowninFigure 6-3 .Therstisthesolenoidsleevedesigninwhichasinglecylindricalcoilsurroundsthestrongerpermanentmagnets.Thesecondisthesolenoidpancakedesigninwhichtwosolenoiddisksarespacedbetweenpermanentmagnetdisks.Becausethesleevedesignallowsfortighterpackingofthepermanentmagnets,itpermitsstrongermagneticeldsandismorecompactintheopticalpathdimension.However,becauseoftheproximityofthesolenoidtothemagneto-opticelement,thepancakedesignallowsforgreatereldtuningrangeforagivensolenoidpower.OfsignicantconcernistheheatdumpedbythesolenoidintotheTGG.Initialinvestigationssuggestthatinordertoobtainatuningrangeof0:5werequire5Wforasleeveand1:5Wforapancakesolenoid.MoreworkmustbedoneinvestigatingheatsinkgeometriesinordertopreventsignicantthermalloadingoftheTGG.Ultimately,bothdesignsmustbeexperimentallytestedtoensureagreementwithmodelingofboththeeldsandheattransfer. 126

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Figure6-3. TwocompetingmagnetcongurationsfortheLLFI.Ontheleftisthesolenoidsleevedesign,whichhasalargecoilencasingstrongpermanentmagnets.Ontherightisthesolenoidpancakedesign,whichhassmallersolenoiddisksspacedbetweenstrongpermanentmagnets. 6.2MonolithicFaradayIsolatorForthirdgenerationdetectorsandbeyond,itmaynolongersucetohaveeven1%singlepasslossisolatorsforsqueezinginjection.Currentestimatesarethat15dBofsqueezingwouldrequireasinglepasslossof<3000ppm( 37 ).AsshowninTable 4-1 ,lossrequirementsthislowareatthefrontierofwhatispossibleevenwithsignicantimprovementstotheLLFIopticalelements.ThenextmajorstepinlossreductionwilllikelyrequireanoverhauloftheLLFIdesign.Onepromisingavenueistomaketheisolatormonolithic.Super-polishedcrystalscanbeopticallycontacted:aprocessbywhichintermolecularforcesbindtheopticalelementstogetherwithouttheneedforanyadhesives.Evencoatedopticscanbebondedtogetherthroughaprocessknownaschemicallyactivateddirectbonding(CADB).ThesetupforamonolithicisolatormightlooksomethinglikethatinFigure 6-4 .Aswesee,thisreducesthenumberofreectingsurfaces(from8to5),withtheaddedadvantagethat 127

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italsoreducesthefootprintoftheisolator.Thedisadvantageofamonolithicdesignis,ofcourse,thattheisolatorisnolongermodular.Oncecontacted,crystalaxesalignmentscannotbetunedanddamagedordefectiveelementscannotbereplaced.Assumingallofthesetechnicaldicultiescanbeovercome,amonolithicdesignisworthexploring.Asastartingpoint,weconsiderthenaivemonolith:acasewhereallopticsaresuper-polished,leftuncoated,andopticallycontactedintheseriesthattheyarecurrentlyfoundintheaLIGOIFI.Ignoringabsorptionandinterferenceeects,wecanapproximatethereectanceduetointernalboundariesasbeingthesumofthecontributionsfromeachsurface: Rinr21+(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r21)r22+(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r21)(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(r22)r23(6{3)wherer1isthereectioncoecientfortheKTP!HWPboundary,r2isthatfortheHWP!TGG,andr3isforTGG!KTP.Thisapproximationgives3.3%reectionforthep-polarizationand3.6%forthes-polarization,whichisfartoohighforanextgenerationisolator.Clearly,wemustdesignatransitionlayerforeachinterface,muchthesamewayanti-reectivecoatingsaredesignedforthesurfacesofoptics.Whileconceptdesignofthemonolithicisolatorisinitsearlystages,twopossibledesigntechniquesareexploredinthesectionsthatfollow. 6.2.1IndexMatchingBecausethereectionataboundaryisafunctionofthedierenceofindexofrefractionbetweenthetwointerfacingmaterials,onewaytoreducethereectionwouldbetolayermaterialsofintermediateindexbetweenthetwocrystals,eectivelysmoothingoutthetransitionfromoneindextoanother.Ifweignorelossesinthemediaandinterferenceeects,fork)]TJ /F1 11.9552 Tf 11.955 0 Td[(1intermediarylayersthereectanceis: R=n0)]TJ /F3 11.9552 Tf 11.955 0 Td[(n1 n0+n12+k)]TJ /F6 7.9701 Tf 6.586 0 Td[(1Xj=1 jYi=12p ni)]TJ /F6 7.9701 Tf 6.587 0 Td[(1ni ni)]TJ /F6 7.9701 Tf 6.587 0 Td[(1+ni!2nj)]TJ /F3 11.9552 Tf 11.955 0 Td[(nj+1 nj+nj+12(6{4) 128

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Figure6-4. Aconceptualdrawingofapotentialmonolithiclow-lossisolatordesign.Alloftheopticalelementsareopticallycontactedinseries,furtherreducingthenumberofreectingsurfacesfrom8to5.Thoughitisnotshowninthedrawing,becauseofitssizethesingleopticwouldlikelysitentirelywithinthemagnethousing. wheren0correspondstotheinitialmediumandnktothenalmedium.Oneschemewemightuseistochoosematerialssuchthat: nj=n0+j k(nk)]TJ /F3 11.9552 Tf 11.955 0 Td[(n0)(6{5)thatis,theindexofprogressivelayersincreasesordecreasesmonotonicallybetweenn0andnk;thisschemeprovidesthesmoothesttransitionandisthusagoodcandidateforinitialconsideration.ForaKTP!TGGboundary,thereectanceasafunctionofnumberoflayersisgiveninFigure 6-5 .Modernanti-reectivecoatingscanprovide200ppmreection,andsoinordertobecompetitivethemonolithicdesignmusthave<<400ppmperboundaryforbothpolarizations.Fromtheplot,weseethatthiswouldrequirenofewerthan8layersfortheppolarizationand3forthespolarizationinthecurrentLLFIdesign.Thissuggeststhatitwouldbebenecialtorotatetheordinaryandextraordinaryaxesby90degreeswithrespecttothewedgegeometryinafuturemonolithicdesign.Thisiscurrentlyanimpracticallylargenumberoflayers;therearesimplynotenoughopticallyusefulmaterialswiththeappropriaterefractiveindex.Intheeventofa 129

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breakthroughinmaterialsengineeringthatallowsfornetuningtherefractiveindicesofopticalmaterials,thestepped-indextechniquemaybecomeviable.Untilthen,weareatthemercyofnatureandmusttakeadierentapproach. Figure6-5. AplotofthereectanceofaKTP!TGGboundaryasafunctionofthenumberofintermediatelayersbetweenthecrystalsintheevenlyspacedindexscheme.Thebluecurvecorrespondstothep-polarization,andtheredcurvetos-polarization.Weseethattoreach400ppmreectionfortheppolarizationwerequireatleast8layersofprogressivelyincreasingindexmaterials. 6.2.2CoherentCancellationIn 6.2.1 weignoredinterferenceeectsinourcalculationofthereectance;nowwewouldliketotakeadvantageoftheseeects.Consideragainaseriesofk)]TJ /F1 11.9552 Tf 12.08 0 Td[(1intermediatelayersbetweentwocrystalsasshowninFigure 6-6 .Itistediousbutstraightforwardtoworkoutthatfor1ik)]TJ /F1 11.9552 Tf 11.955 0 Td[(1: Ei;1=)777(!tiEi)]TJ /F6 7.9701 Tf 6.586 0 Td[(1;2+ )]TJ /F3 11.9552 Tf 3.177 -5.147 Td[(riEi;4Ei;2=iEi;1Ei;3=)777(!ri+1Ei;2+ )]TJ /F3 11.9552 Tf 3.864 -7.354 Td[(ti+1Ei+1;4Ei;4=iEi;3(6{6) 130

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whereE0;2Ein,E0;3=Er,Ek;1=Et,andi=eiiforitheaccumulatedphaseofthewavetraversinglayeri.Here,thevaluesriandtiarethereectionandtransmissioncoecientsrespectively,whichforthecaseofnormalincidencearedenedby: )777(!ri=ni)]TJ /F3 11.9552 Tf 11.955 0 Td[(ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1 ni+ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1 )]TJ /F3 11.9552 Tf 1.649 -5.148 Td[(ri=ni)]TJ /F6 7.9701 Tf 6.587 0 Td[(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(ni ni+ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1)777(!ti=2ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1 ni+ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1 )]TJ /F3 11.9552 Tf 2.174 -7.354 Td[(ti=2ni ni+ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1(6{7)Noticethat)777(!ri=)]TJ 9.298 5.148 Td[( )]TJ /F3 11.9552 Tf 1.65 -5.148 Td[(ri,andsowecandropthearrownotationandsetri=)777(!ri.Thetransmissioncoecientsdonothavethesamesimplesymmetry,thoughinmanycasestheyarenotunpairedbutinsteadshowupastheproduct: )777(!ti )]TJ /F3 11.9552 Tf 2.173 -7.353 Td[(ti=4ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1ni (ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1ni)2(6{8)andsoitisusefultodene: ti=2p ni)]TJ /F6 7.9701 Tf 6.586 0 Td[(1ni ni)]TJ /F6 7.9701 Tf 6.587 0 Td[(1+n1(6{9)Ingeneral,therewillbe2k)]TJ /F1 11.9552 Tf 12.379 0 Td[(2degreesoffreedomassociatedwiththetransitionregion:onefortheindexofrefraction,andonefortheaccumulatedphase(length)ofeachindividuallayer.Itishelpfultobeginbyconsideringthesimplestcases. Figure6-6. Adiagramillustratingthesystemofequationsfortheelectriceldsinak)]TJ /F1 11.9552 Tf 11.955 0 Td[(1layertransitionregionbetweenmaterialofindexn0andindexnk. Thissystemof4k)]TJ /F1 11.9552 Tf 12.718 0 Td[(2linearequationsin( 6{6 )canbesolvedforarbitrarilylargek,butcanbecomeunwieldy,andsointhisdissertationwedonotconsiderthecaseofmultiplelayers.Fork=2(asingleintermediarylayer),wendthatthereectedeldis: Er=r1+r2(r21+t21)2 1+r1r22Ein(6{10) 131

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andsothereectanceofthesinglelayeris: R=Er Ein2=jr1j2+2Re[r1r2(r21+t21)2]+jr2j2jr21+t21j2 1+2Re[r1r22]+jr1r2j2(6{11)Thisisthefamiliarcaseofasimplelinearopticalcavity.Intheabsenceofopticallosses,r1,r2,andt1areallrealand( 6{11 )becomes: R=r21+2r1r2(r21+t21)cos(2)+r22(r21+t21)2 1+2r1r2cos(2)+r21r22(6{12)Ourgoal,thenistominimize( 6{12 )withourchoicesfortheindexofrefraction,n1,oftheintermediatelayerandtheaccumulatedphase,.Inagreementwithintuition,ithappensthatthelowestreectanceisachievedwhenn1=(n0+n2)=2and==2,whichcorrespondstoaquarter-wavelengthtransitionlayer.Aplotof( 6{12 )for==2attheHWP!TGGboundaryisgivenbyFigure 6-7 .Weseethatthereectancerisessharplyastheindexoftheintermediarylayerdeviatesfromitsoptimalvalue. Figure6-7. Aplotofthereectanceofaquarter-waveintermediarylayerasafunctionofthelayer'sindexofrefractionforaHWP!TGGjunction. Theuseofbirefringentmaterials,likeKTP,forpolarizerscomplicatesourschemeas:1)theintermediarylayerwillalsoneedtobebirefringenttosatisfythemeanindexcondition,and2)foraxedlayerthicknesstheaccumulatedphaseforaparticular 132

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polarizationwillbedeterminedbytheindexofthelayerforthatpolarization.Together,thesedictatethatwecanneversimultaneouslyoptimizetheindexandtheaccumulatedphaseoftheintermediarylayerforamixedpolarizationstate.LookingbacktothetentativesqueezinginjectionlayoutinFigure 2-5 ,weseethataftertheltercavitythereisoneKTP!ELEMENTboundarywhichwillrequirelowlossforbothpolarizationspassing,whichwecalltheinjectionpoint.WehavethechoiceofwhetherthesecondopticalelementinthisjunctionistheHWPorTGG.ThepoweroutputofthefullisolatorpathforaninputpowerPinwillfollow: Pout/(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(Rp)(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(Rs)Pin(6{13)andsothepowerlossisgivenby: L=Rp+Rs)]TJ /F3 11.9552 Tf 11.955 0 Td[(RpRs(6{14)Wecanuse( 6{14 )asagureofmeritforourlayerdesign;specicallyweaimtominimizeL.Ifweconsideranisotropicintermediarylayer,wecanxalayerthicknesssoastoset==2.Thelossin( 6{14 )isplottedasafunctionofthelayerindexinFigure 6-8 .Weseethatthechoiceofelement,eitherahalf-waveplateorTGG,islargelyinconsequential:bothinterfaceshavealowerlimitofabout300ppmloss.Alternatively,wecouldsupposethatcanproduceabirefringentlayerwithtunedindicessatisying: n1;p(s)=1 2n0;p(s)+n2(6{15)Thethicknessofthelayeris(ofcourse)xed,andsotheaccumulatedphaseinthepandspolarizationsarerelatedby: p s=n1;p n1;s(6{16)AplotofthelossforabirefringentlayerisshowninFigure 6-9 .UnlikeforthecaseofanisotropiclayerwhereboththequartzHWPandTGGperformaboutthesame,withabirefringentintermediarylayertheTGGboundaryissignicantlylesslossythanthatof 133

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Figure6-8. Aplotofthecombinedpandspowerlossfora=2isotropicmonolayerasafunctionofthelayer'sindexofrefraction.ThebluecurvecorrespondstoaKTP!TGGjunction,andredforKTP!HWP. theHWP.ThisisbecausebothindicesofrefractionintheKTParemuchclosertotheindexforTGGthantoquartz. Figure6-9. Aplotofthecombinedpandspowerlossforatuned-indexbirefringentmonolayerasafunctionofp.ThebluecurvecorrespondstoaKTP!TGGjunction,andredforKTP!HWP. 134

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CHAPTER7CONCLUSION 7.1LowLossFaradayIsolatorInthepost-detectionera,gravitationalwaveobservatoriesarepoisedtoincorporatenewerandmoreadvancedtechnologiesinthehopesofpushingtheirdetectorsensitivitieshigher,therebyopeningupagreatersliceoftheobservableuniversetogravitational-waveastronomy.Squeezedinjectionpromisestotakemoderndetectorsbeyondthestandardquantumlimit,allowingforgreaterobservationalrangewithouttheneedforheaviertestmasses,coldertemperatures,orlongerarms.ItisthegoaloftheLIGOcollaborationtoincludesqueezinginthethirdobservationrunoftheAdvancedLIGOdetector.Theimplementationofsqueezedinjectionrequiresamethodforverylow-lossinjectionofthesqueezedbeamintotheinterferometerbeampath.TheFaradayisolatoristheidealtoolforsuchaninjection,solongastheinjectedbeamcanbemadetohaveanorthogonalpolarizationtothecarrierbeam.Becausethesqueezedbeammakesthreepassesthroughanisolatorbeforecombiningwiththegravitational-wavesignalattheoutputportoftheinterferometer,thelimitsontotalopticallossinthesqueezedpathputstrongconstraintsonthelossintheFaradayisolators.ThisdissertationdescribesaFaradayisolatordesignbasedontheInputFaradayIsolatorinAdvancedLIGOthatisintendedtoachievethenear-termlossgoalof<1%opticallosspersinglepass.ThenumberofopticalelementswasreducedfromthesixoftheIFItojustfournecessarycomponentsintheLLFI.CalcitewedgepolarizerswerereplacedwithKTPwedges,whichweresuperpolishedandARcoatedtohaveR<50ppmpersurface.Themagneto-opticelement,TGG,wasmadetobemonolithic,thensuperpolishedandARcoatedtohaveR<700ppmpersurface.Theisolator,includingtherotatormagnet,wasassembledandtestedintheUF-LIGOcleanroom,andfoundtohavemaximalopticalthroughputof98.38%(1.62%loss)persinglepass.Itisunlikelythatthislossisduetomisalignment,asitisconsistentthroughoutseveralconstructions. 135

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ThoughthecurrentLLFIdesignfallsshortofthe<1%opticallossgoal,itsignicantlyimprovesupontheperformanceofthecurrentOutputFaradayIsolator,whichwaspreviouslymeasuredtohave3:3%single-passloss.Futureimprovementstosqueezedinjectionwillrequirefurtherreductionofopticallossintheinjectionpath.Asexplainedin 3.7.2 ,evenwithperfectlylosslessopticalmaterials,themisalignmentofFaradayisolatorelementswilllimittheopticalthroughput.Topushsignicantlylowerthan1%singlepasslosswilllikelyrequireautomatedalignment. 7.2Radio-FrequencyAmplitudeModulationTheLIGOinterferometersmeasurethedierentialdisplacementbetweenfreeoatingtestmassesinorthogonaldirectionsinordertodetectthedisturbanceofthespacetimemetriccausedbypassinggravitationalwaves.Inpractice,however,thetestmassesmustbeactuatedoninorderthattheinterferometerbeoperatinginitsdesignconguration,withallofitsdegreesoffreedommaintainedattheirrespectivesetpoints.Todosoweutilizeatechniquecalledphasemodulation,whichaddsfrequencysidebandstothecarrierwave.Thesesidebandsinteractdierentiallythroughouttheopticalsetupandtheresultantsignalprovidesameansforfeedback.Amplitudemodulationofthecarrierwaveaddsexcesssignalwhichcarriesnoinformationaboutthegeometryoftheexperiment,andsoleadstoosetsinfeedbackerrorsignals.Inordertomonitorforpotentialradio-frequencyamplitudemodulation(RFAM),aphotodiode(RFPD)wasinstalledintransmissionoftheInputModeCleaner(IMC),andthephotosignalisdemodulatedatboththe9and45MHzsidebandfrequencies.AtestwasperformedinwhichtheIMCwasdetunedfromresonanceandaresponsewasmeasuredbythephotodiode.ThestrengthofthesignalswerecomparedtosimulationsusingFINESSEopticalmodelingsoftware,andthephotodiodewascalibratedaccordingly.Withthiscalibration,themonitorsuggeststhatRFAMexistsintheLLOinterferometersatmodulationabout=10)]TJ /F6 7.9701 Tf 6.586 0 Td[(7,whichgivesaratioofamplitude 136

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modulationtophasemodulationofroughly10)]TJ /F6 7.9701 Tf 6.586 0 Td[(6.Thisresultdisagreeswithpreviousmeasurements,andthediscrepancyispossiblyduetoerrorinthesimulationresults.Currently,anumericaldierenceinsimulationresultsbetweensoftwareversionsisbeingexaminedinthehopethataresolutionwillprovidemoreclarityontheissueofRFPDcalibration. 137

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APPENDIXACRYOGENICINVESTIGATIONSOFBOSEMLEDS A.1CryogenicsforFutureGravitationalWaveDetectorsChapters 2 through 4 wereconcernedwithboththefundamentalphysicsbehindaswellasthepracticalneedsforimplementingsqueezinginjectionasameansofreducingquantumnoiseinAdvancedLIGO.Whilesqueezingpresentsanopportunitytoincreasestrainsensitivityofextantgravitational-waveinterferometerswithoutsignicantcongurationchanges,therearealsoeortsunderwaytoimprovethedesigncapabilitiesoffuturethirdgenerationdetectorsandbeyond( 2 ).Oftheselong-termimprovements,oneofthemostpopularistheproposaltooperatetheinterferometersinacryogenicenvironment( 3 ).Particularly,cryogenicopticsareviewedasamethodtomitigatethermalnoiseinthesubstratesandcoatings.KAGRAhasalreadytakenthesteptousecryogenics,anditisexpectedthatfuturedetectorswillfollow. A.2CryogenicLight-EmittingDiodePerformanceAconsequenceofcoolingthetestmassesisthatthecontrolinfrastructurewillalsorequiretooperateincryogenicconditions.OfparticularinterestforthisappendixarethepositionsensorscalledBirminghamOpticalSensorsandElectro-Magneticsacutators(BOSEMs).AdiagramofthebasicworkingprinciplebehindtheBOSEMisincludedinFigure A-1 .Smallmagneticagsareattachedtomassesinthemainandreactionsuspensionchains.Theendofaagsitsbetweenalight-emittingphotodiodeandaphotodetectorsothatitblockssomeofthelightfromtheLED.Translationalmotionoftheagisthenreadoutasachangeinthephotocurrentatthephotodiode.Overthelinearresponseregionofthedetector,theBOSEMissensitiveto10)]TJ /F6 7.9701 Tf 6.586 0 Td[(10mHz)]TJ /F6 7.9701 Tf 6.586 0 Td[(1=2at10Hz( 8 ).TheLEDusedintheBOSEMdesignistheVishayTSTS7100( 61 ),aGaAsnearinfrared(950nm)emitterchosenforitsrelativelylownoise( 8 ).AsameansofdeterminingitsviabilityasanLEDforcryogenicopticalsensors,thedevicewasaxedto 138

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FigureA-1. Acartoonoftheworkingprinciple(left)andaschematicdrawing(right)ofaBOSEM.TheBOSEMisashadowsensorthatcandetectmotionofagaxedtoamassandactuateontheagtocontrolthepositionofthemass.RightschematiccourtesyofStuartAston( 7 ) thecoldngerofapulse-tubecryocoolerandcharacterizedovertemperaturesfrom30Kto300K.Specically,threequnatitiesweremeasured:1)thevoltageacrossthedioderequiredtoproduceacurrent,calledtheactivationforwardvoltage,2)theresponseofthecurrentasafunctionofthevoltageacrossthediode,calledtheI-Vcharacteristiccurve,and3)thenumberofemittedphotonsperelectronpassingacrossthediodejunction,calledexternalquantumeciency.AdiagramoftheexperimentalsetupisprovidedinFigure A-2 .TomeasuretheactivationforwardvoltageandI-Vcurves,avoltagesourcewasusedtoproviderampingvoltagepulsestotheLEDwhileinacryogenicvacuumenvironment.ThecurrentthroughtheLEDandvoltagedropacrosstheLEDwererecordedastimeseries.FromFigure A-3 weseetheactivationforwardvoltageriseswithdecreasingtemperature,asonewouldexpectbecausethebandgapenergyofGaAsisincreasing.Startingat100K,weobserveaverysteepriseintheactivationforwardvoltagewhichisnotexplainedsimplyasariseinbandgapenergy.I-VcurvesforaparticularLEDtemperaturearezeroattheactivationforwardvoltageandthensharplyrisewithincreasingvoltage.WeseeinFigure A-4 thatbelow70K,theI-Vcurveshavepointswheretheslopeisnegative.Ofcourse,thisisnotduetoanegativeresistanceofthediodejunction,butinsteadisduetoself-heatingofthe 139

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FigureA-2. Adiagramoftheexperimentalsetupfortestsofthelow-temperatureperformanceoftheVishayTSTS7100.TheLEDisaxedtothecoldngerofapulse-tubecryocooler(PTC)withinavacuumtankheldbelow10)]TJ /F6 7.9701 Tf 6.586 0 Td[(6torr.Asource,whichcaneitherbeavoltageorcurrentsourcedependingonthetest,feedstheLED,andtheemittedlightpassesthroughavacuumwindowandisincidentonaphotodetector.ThecurrentthroughtheLED,thevoltagedropacrossit,aswellasthephotocurrentthroughthephotodetectorareallmeasuredsimultaneously.Photoscourtesyofauthor. LED.Asthevoltagepulsebeginstoramp,theLEDisatambienttemperatureanddoesnotproduceacurrentuntilthevoltagepassestheactivationforwardvoltagethreshold.Onceitbeginstoemit,somepowerisdissipatedintheLEDasheat,whichraisesthetemperatureandlowersthejunctionresistance.Tomeasuretheexternalquantumeciency,acurrentsourcewasusedtoprovideasteady35mAacrosstheLED,andthelightemittedwaspassedthroughavacuumwindowandontoaphotodiode.Solongastheemissionspatialproleremainsunchangedacrossthetemperaturesofinterest,thequantumeciencyislinearinmeasuredphotopower.FromFigure A-5 weseethattheequantumeciencyoftheVishayTSTS7100ismaximalnear100K. 140

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FigureA-3. TheactivationforwardvoltageoftheVishayTSTS7100asafunctionoftemperature.ThebandgapenergyofGaAsisprovidedastheredcurveasareference. FigureA-4. I-VcurvesfortheVishayTSTS7100asafunctionoftemperature.Ontherightiscollecteddata;weseethatfortemperaturesbelow70K,thecurvesexhibitnegativeslopesatcertainpoints.Thisisduetoselfheating,andisillustratedintheplotontheright.TheI-VcurveforagivenLEDtemperaturewillhaveasharpandpositiveslope(shownasthereddashedlines),astheLEDheatsuptheI-Vbehaviorjumpsfromonexedtemperaturecurvetothenext. Anotherinterestinglow-temperatureeectistheapparentfreeze-outofnonradiativerecombination,asisshowninFigure A-6 141

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FigureA-5. AplotoftheestimatedexternalquantumeciencyoftheVishayTSTS7100asafunctionofambienttemperature.Weassumethattheemissionproleremainsunchangedoverthetemperaturerange. FigureA-6. AplotofthemeasuredphotopoweroutoftheVishayTSTS7100asafunctionofforwardcurrentforseveraldierenttemperatures.Weseethatforlowertemperaturesthelinearrelationshipispreserved,asnonradiativerecombinationprocessesarefrozenout. 142

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APPENDIXBJONESANDMUELLERCALCULUSFORBEAMPROPAGATION B.1JonesMatricesApolarizedplane-wavetravelinginthez-directioncaningeneralberepresentedbyitscomplexeldcomponents: Ex=E0xei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t+x)Ey=E0yei(kz)]TJ /F4 7.9701 Tf 6.587 0 Td[(!t+y)(B{1)ThecorrespondingJonesvector(?)hecht2002)isgivenby: a=0B@E0xE0yei1CA(B{2)where=y)]TJ /F3 11.9552 Tf 11.955 0 Td[(x,anditisconventionaltoxthenormalizationconstantsuchthat: aya=1(B{3)TheJonesvector,then,encodesthenormalizedamplitudeandrelativephaseinformationofthetwoorthogonaleldcomponents.WhentheeldpassesthroughanopticalelementwithcorrespondingJonesmatrixJ,theresultingoutputJonesvectoris: Ja=a0(B{4)Theintensityoftheoutputeldrelativetotheinputeldcanbefoundbyevaluating: I0 I=(a0)ya0=ayJyJa(B{5)Jonescalculusallowsustopredictandrelateeldintensitymeasurementsatdierentpointsinanexperimentalsetup.Often,theformofoutputeldforagivenopticalelementisknown,andsothecorrespondingJonesmatrixcanbefoundfrom( B{4 ).ItishelpfultolistJonesmatricesforcommoncomponents: 143

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Linearpolarizeratfromxaxis:0B@cos2cossincossinsin21CALinearretarder,fastaxisatfromxaxis:0B@cos2+eisin2(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(ei)cossin(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(ei)cossineicos2+sin21CARotationthroughanangle:0B@cos)]TJ /F1 11.9552 Tf 11.291 0 Td[(sinsincos1CA B.2StokesParametersOnefailingoftheJonestechniqueisthatitcannotbeusedtodescribeunpolarizedlight.Instead,todosoweusetheStokesparameters: S0=E20x+E20yS1=E20x)]TJ /F8 11.9552 Tf 11.956 9.684 Td[(E20yS2=h2E0xE0ycos(y)]TJ /F3 11.9552 Tf 11.956 0 Td[(x)iS3=h2E0xE0ysin(y)]TJ /F3 11.9552 Tf 11.955 0 Td[(x)i(B{6)Forperfectlypolarizedlight,wecanreplacetheexpecationvalueoperatorswithabsolutevalues: S0=jE0xj2+jE0yj2S1=jE0xj2)-222(jE0yj2S2=2Re(E0xE0y)S3=)]TJ /F1 11.9552 Tf 9.299 0 Td[(2Im(E0xE0y)(B{7)Theso-calledStokesvectoris: S=0BBBBBBB@S0S1S2S31CCCCCCCA(B{8) 144

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ItiscommonusagetorefertotheStokesparametersbytheconventionS0=I,S1=Q,S2=U,andS3=V. B.3MuellerMatricesTopropagatetheStokesvectoraswedoJonesvectors,weconstructaMuellermatrixMvia: M=A(JJy)A)]TJ /F6 7.9701 Tf 6.587 0 Td[(1(B{9)whereJisthecorrespondingJonesmatrixfortheopticalelementand: A=0BBBBBBB@1001100)]TJ /F1 11.9552 Tf 9.298 0 Td[(101100i)]TJ /F3 11.9552 Tf 9.298 0 Td[(i01CCCCCCCA(B{10)Here,istheKroneckerproduct,whichactsonmatricesas: 0B@abcd1CA0B@efij1CA=0BBBBBBBB@a0B@efij1CAb0B@efij1CAc0B@efij1CAd0B@efij1CA1CCCCCCCCA=0BBBBBBB@aeafbebfaiajbibjcecfdedfcicfdidf1CCCCCCCA(B{11)Notethat: A)]TJ /F6 7.9701 Tf 6.586 0 Td[(1=1 20BBBBBBB@1100001)]TJ /F3 11.9552 Tf 9.298 0 Td[(i001i1)]TJ /F1 11.9552 Tf 9.299 0 Td[(1001CCCCCCCA(B{12) 145

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TheStokesvectorsarepropagatedinanalogousfashiontotheJonesvectors: MS=S0(B{13) 146

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APPENDIXCMAGNETICFIELDCALCULATIONSFORREALSOLENOIDSWeaimtoexpressthemagneticeldfromarealsolenoidasafunctionofpositionforallpointsinsidetheintitecylinderboundedbythesolenoidcoil.Tosimplifyourmodel,weassumeperfectlycylindricallysymmetriccoils.Inthemagnetostaticregime,wecanusetheBiot-SavartLawtondtheeldatanygivenpointinspace: dB=0I 4^rds r2(C{1)fordstheinnitesimalwirevector,Ithecurrentthroughthewire,andrthevectorfromourobservationpointtotheinnitesimalelement. C.1AnalyticApproach C.1.1SingleCoilWebeginouranalysisbyconsideringtheeldcontributionfromasinglecoilofradiusR.Wechooseourcoordinatesystemsothatourobservationpoint,O,liesinthex)]TJ /F3 11.9552 Tf 12.514 0 Td[(yplaneandthecurrentringliesinthey)]TJ /F3 11.9552 Tf 12.419 0 Td[(zplane.Towithinasign,symmetryallowsustospecifyObytwocoordinates(=D=R;=L=R),whereDisthedistancefromtheobservationpointtotheplaneofthecoil,andListhedistancefromtheobservationpointtothecenteraxisofthecoil.(Forourpurposes,willalwaysbepositiveandlessthan1).Apoint,S,onthesolenoidcanbespeciedby,theanglebetweentheyaxisandtherayfromthecenteroftheringtoS.ThevectorfromOtoSisthen: r=0BBBB@D001CCCCA+0BBBB@0RcosRsin1CCCCA)]TJ /F8 11.9552 Tf 11.955 38.377 Td[(0BBBB@0L01CCCCA=R0BBBB@cos)]TJ /F3 11.9552 Tf 11.955 0 Td[(sin1CCCCA(C{2)with: r2=R2)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[(1+2+2)]TJ /F1 11.9552 Tf 11.955 0 Td[(2cos(C{3) 147

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Choosingthecurrentdirectionsuchthattheaxialeldpointsinthepositivexdirection,thecurrentunitvectoratSis: ^s=0BBBB@0)]TJ /F1 11.9552 Tf 11.291 0 Td[(sincos1CCCCA(C{4)Thisallowsustoevaluate: r^s=R0BBBB@1)]TJ /F3 11.9552 Tf 11.955 0 Td[(cos)]TJ /F3 11.9552 Tf 9.298 0 Td[(cos)]TJ /F3 11.9552 Tf 9.298 0 Td[(sin1CCCCA(C{5)WecanintegratetheeldcontributionfromallpointsalongtheringtondthetotaleldatO: B=0IR 4Z20^r^s r2d=0IR 2Z0(^x+^y)^r^s r2d=0I 2R"^xZ01)]TJ /F3 11.9552 Tf 11.956 0 Td[(cos (1+2+2)]TJ /F1 11.9552 Tf 11.956 0 Td[(2cos)3=2d)]TJ /F1 11.9552 Tf 12.672 .166 Td[(^yZ0cos (1+2+2)]TJ /F1 11.9552 Tf 11.955 0 Td[(2cos)3=2d#(C{6)Wecanusenumericalintegrationtoevaluatethetwointegrals.Itishelpfultowritetheeldgivenby( C{6 )asanexplicitfunctionofthetwodimensionlesscoordinates:B=bR(;). C.1.2MultipleConcentricCoilsSupposewenowwanttondthemagneticeldfromadiskofn+1evenly-spacedconcentriccoils,allcarryingthesamecurrentI.Ifweindextheradiusofeachcoilloopwithinthedisk,beginningwithinnermostcoilradiusR0,thenwehave: Rk=R0+k n(Rn)]TJ /F3 11.9552 Tf 11.956 0 Td[(R0)(C{7) 148

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for0kn.Theindexedandcoordinatesforaxedpointinspacesatisfy: k=R0 Rk0k=R0 Rk0(C{8)Calling=Rn=R0,wehave: R0 Rk=n n+k()]TJ /F1 11.9552 Tf 11.956 0 Td[(1)k(C{9)andthetotaleldcanbeexpressedas: B(D;L)=nXi=0bRi(i;i)=nXi=0b)]TJ /F13 5.9776 Tf 5.756 0 Td[(1iR0(i0;i0)(C{10) C.1.3CoaxialCoilDisksConsider2m+1identicalcoaxialcoildisks: k;l=Dl Rkk;l=Ll Rk=L Rk=k(C{11)If0;06=0,then: k;l=R0 RkDl D00;0(C{12) Dl=D0+l m(Dm)]TJ /F3 11.9552 Tf 11.955 0 Td[(D0)(C{13)Calling=Dm=D0,wehave: Dl D0=1+l m()]TJ /F1 11.9552 Tf 11.955 0 Td[(1)l(C{14) k;l=kl0;0(C{15)andtheexpressionfortheeldis: B(D;L)=mXj=)]TJ /F4 7.9701 Tf 6.587 0 Td[(mnXi=0bRi(i;j;i)=mXj=)]TJ /F4 7.9701 Tf 6.587 0 Td[(mnXi=0b)]TJ /F13 5.9776 Tf 5.756 0 Td[(1iR0(ij0;0;i0)(C{16) 149

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C.2ComputerModelingThebasicideabehindthecodeistoexpresstheeldinunitsof0I=2R,whereRistheinnerradiusofthesolenoid.Todoso,weaddtogethercontributionsfromconcentriccurrentringstoformcurrentdisks.Thesediskscanthenbestackedtosimulatearealsolenoidofsomegivencoilnumberandlength.Thecodeinputsshouldbe:1)numberofringsperdisk,2)numberofdisks,3)eectivethicknessofcoilring,4)eectivethicknessofdisk,5)valueofinterest(wherecorrespondstotheinnerradiusofthesolenoid).InpracticewewillevaluateBxandByseparately. TRn)]TJ /F3 11.9552 Tf 11.955 0 Td[(R0)]TJ /F1 11.9552 Tf 11.956 0 Td[(1=T R0(C{17) H2(Dm)]TJ /F3 11.9552 Tf 11.955 0 Td[(D0))]TJ /F1 11.9552 Tf 11.956 0 Td[(1=H 2D0(C{18) Ix(;)=Z01)]TJ /F3 11.9552 Tf 11.955 0 Td[(cos (1+2+2)]TJ /F1 11.9552 Tf 11.955 0 Td[(2cos)3=2d(C{19) Iy(;)=Z0cos (1+2+2)]TJ /F1 11.9552 Tf 11.955 0 Td[(2cos)3=2d(C{20) Bx(D;L)=^xmXj=0nXi=0b)]TJ /F13 5.9776 Tf 5.757 0 Td[(1iR0(ij0;0;i0)=0I 2R0mXj=0nXi=0iIx(ij0;0;i0)(C{21) By(D;L)=^ymXj=)]TJ /F4 7.9701 Tf 6.587 0 Td[(mnXi=0b)]TJ /F13 5.9776 Tf 5.756 0 Td[(1iR0(ij0;0;i0)=0I 2R0mXj=)]TJ /F4 7.9701 Tf 6.587 0 Td[(mnXi=0iIy(ij0;0;i0)(C{22) 150

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APPENDIXDFINESSEMODELOFTHEALIGOINTERFEROMETERBelowisthefullmodeloftheAdvancedLIGOinterferometerusedforFINESSEsimulations. #--------------------------------------------------------------------------#LLO_IFO_maxtem2.kat##FINESSEkatfileforthemode-matchedL1dualrecycledMichelsonwithFParm#cavities.#Doesnotincludealignmentsensingandcontrol##Design2referstotheupdatesfromtheoriginaldesign,suchas#includingmeasuredmirrorparameters.Somelengthsmayhavechanged#fromtheoriginaldesigntoaccountforthenewmirrorparameters.#LengthsaretakenfromE1200274.##NoIMCorPMMT,thinPRMandSRM##Mirrorspecstakenfromhttps://nebula.ligo.caltech.edu/optics/asof#2015/09/08##IMC,HAM2,HAM6,OMCpathandOMCarebasedonthegenericaLIGOdesignfiles#foundhere:#https://dcc.ligo.org/L1300231-v10#withupdatesbasedongalaxypagewhereappropriate.##ThermallensinITMsmodeledasathinlensinfrontoftheARsurfaces 151

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#ITMsincludenon-thermalsubstratelensesascalculatedfrommeasurements#reportedinthepolisher'sreportsfoundonthenebulapage.Theseareturned#offbydefaultherethough.##**Updatednon-thermallensfocallengths(notusedherethough)**#ITMYlensfocallength=-82.4km#ITMXlensfocallength=305km#CalculationisshowninLIGOT1300954##-UpdatedwithGalaxypage(https://galaxy.ligo.caltech.edu/optics/)values#forinstalledSRM-w#-IncludeslocksforDARM(~10pmoffset),PRCL,MICH,CARMandSRCL##CharlotteBond,PaulFulda,DanielBrown,AntonioPerreca,AndreasFreise#2015-09-08#--------------------------------------------------------------------------%%%FTblockPSL#################################################################################laserlL01250n0slmod11n0n1######9MHzEOMmodmod1$f10.181pmn1n2smod1toamsourcef10n2nam_source_f1_in 152

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######9MHzAMonthe1stEOMmodam_source_f1$f101am0nam_source_f1_innam_source_f1_outslmod21nam_source_f1_outn3######45MHzEOMmodmod2$f20.181pmn3n4smod2toamsourcef20n4nam_source_f2_in######45MHzAMonthe2ndEOMmodam_source_f2$f201am0nam_source_f2_innam_source_f2_outslmod31nam_source_f2_outn5######24MHzEOMmodmod3$f30.11pmn5n6smod3toamsourcef30n6nam_source_f3_in######24MHzAMon3rdEOMmodam_source_f3$f301am0nam_source_f3_innam_source_f3_outsmod3toMC11nam_source_f3_outnMC1AR1in###########################################################################%%%FTendPSL%%%FTblockIMC########################################################################### 153

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###MC1IMCF-01bs2MC1AR1343u6.849u044.59nMC1AR1indumpnMC1AR1transdumpssMC1substrate10.0845$nsilicanMC1AR1transnMC1HRinbs1MC1HR6150u0.7u028.9661nMC1HRinnMC1HRreflnMC1HRtransnMC1HRfromMC3HRssMC1substrate20.0845$nsilicanMC1HRreflnMC1AR2inbs2MC1AR2343u6.849u028.9661nMC1AR2indumpnMCREFLdumpssMC1HRtoMC216.2405708nMC1HRtransnMC2in###MC2IMCC-03bs1MC23.5u12.5u00.82nMC2innMC2reflnMC2transdump1ssMC2toMC316.2405708nMC2reflnMC3inattrMC2Rc27.178###MC3IMCF-02bs1MC3HR6130u0.7u044.59nMC3innMC3reflnMC3transnMCreturn_reflssMC3toMC10.465nMC3reflnMC1HRfromMC3HRssMC3substrate0.0845$nsilicanMC3transnMC3ARinbs2MC3AR305u6.85u028.9661nMC3ARindumpnMC3ARtransdumpssMC3ARtoIM10.4282nMC3ARtransnIM1in###########################################################################%%%FTendIMC%%%FTblockHAM2########################################################################### 154

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#IM1a.k.a.SM1bs1IM100053nIM1innIM1reflnIM1HRtransdumpssIM1sub0.02995$nsilicanIM1HRtransnIM1ARinbs2IM1AR00033.4nIM1ARindump5nIM1ARtransdumpsIM1ARtonanoscan3nIM1ARtransnIOT2Lnanoscan#AOE1ssIM1toAOE10.1955nIM1reflnAOE1inlensAOE1infnAOE1innAOE1trans#IM2a.k.a.PMMT1ssAOE1toIM21.0983nAOE1transnIM2inbs1IM20007nIM2innIM2refldumpdumpattrIM2Rc12.8ssIM2toIM31.1704nIM2reflnIM3in#IM3a.k.aPMMT2#ssIM2toIM31.1704nIM2reflnIM3inbs1IM30007.1nIM3innIM3refldumpdumpattrIM3Rc-6.24#AOE2ssIM3toAOE21.041nIM3reflnAOE2inlensAOE2infnAOE2innAOE2trans 155

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#IM4a.k.a.SM2ssAOE2toIM40.134nAOE2transnIM4inbs1IM42400u0045nIM4innIM4reflnIM4transnIM4rettransssIM4toPRMAR0.4135nIM4reflnREFL###########################################################################%%%FTendHAM2%%%FTblockPR############################################################################PRMPRM-02#ARsurfacem2PRMAR26u11.45u$phi_PRMnREFLnPRMARb#SubstratessPRMsub10.0737$nsilicanPRMARbnPRMHRa#HRsurfacem1PRMHR0.0315.9u$phi_PRMnPRMHRanPRMHRbattrPRMHRRc11.009#DistancebetweenPRMandPR2slp116.6107nPRMHRbnPR2a 156

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#PR2PR2-02bs1PR2243u8.6u0-0.79nPR2anPR2bnPOPdumpattrPR2Rc-4.545#DistancefromPR2toPR3slp216.1647nPR2bnPR3a#PR3PR3-03bs1PR35.3u17u00.615nPR3anPR3bdumpdumpattrPR3Rc36.027#DistancefromPR3toBSslp319.5381nPR3bnPRBS###########################################################################%%%FTendPR%%%FTblockBS############################################################################BSbeamsplitter##------------------------------------------------------------##BS##^##toIMY|##|,'-.##|+`. 157

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##nYBS|,':'##nPR3b|+i1+##---------------->,:._i2,'##fromthePRCnPRBS+\`-.+nXBS##,'i3\,'--------------->##+\+toIMX##,'i4.'##`._..##`._,'|nSRBS##-|##|totheSRC##|##v##------------------------------------------------------------#BSBS-02bs1BS0.58.6u$phi_BS45nPRBSnYBSnBSi1nBSi3sBSsub10.0685$nsilicanBSi1nBSi2sBSsub20.0684$nsilicanBSi3nBSi4bs2BSAR130u1.7u$phi_BS-29.1951nBSi2dumpnXBSnPOXbs2BSAR230u1.7u$phi_BS29.1951nBSi4dumpnSRBSdump###########################################################################%%%FTendBS%%%FTblockYarm########################################################################### 158

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sly14.847nYBSnCPYAR1a#YarmcompensationplateCP-08m2CPYar148.9u0.4u0nCPYAR1anCPYAR1bssCPY0.10032$nsilicanCPYAR1bnCPYAR2am2CPYar230.5u0.3u0nCPYAR2anCPYAR2bssCPYtoITMYar0.02nCPYAR2bnITMYTLa#YarminputmirrorITM-08#ThermallenslensITMYTL$TL_fnITMYTLanITMYTLbsITMYTL_null0nITMYTLbnITMYconstLa#ConstantITMYsubstratelenslensITMYconstLinfnITMYconstLanITMYconstLbsITMYTL_null20nITMYconstLbnITMY1m2ITMYAR250u0.6u$phi_ITMYnITMY1nITMYs1slITMY0.19961$nsilicanITMYs1nITMYs2m1ITMYHR0.014814.3u$phi_ITMYnITMYs2nITMY2attrITMYHRRc-1940.7#Y-armsLYarm3994.515nITMY2nETMY1#ETMYETM-09 159

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m1ETMYHR3.5u9.3u$phi_ETMYnETMY1nETMYs1sETMYsub0.2$nsilicanETMYs1nETMYs2m2ETMYAR230u0$phi_ETMYnETMYs2nPTYattrETMYHRRc2242.4attrETMYHRmass40attrITMYHRmass40###########################################################################%%%FTendYarm%%%FTblockXarm###########################################################################slx14.829nXBSnCPXAR1a#XarmcompensationplateCP-06m2CPXar133u0.6u0nCPXAR1anCPXAR1bssCPX0.10031$nsilicanCPXAR1bnCPXAR2am2CPXar28u0.6u0nCPXAR2anCPXAR2bssCPXtoITMXar0.02nCPXAR2bnITMXTLa#XarminputmirrorITM-04#ThermallenslensITMXTL$TL_fnITMXTLanITMXTLbsITMXtl_null0nITMXTLbnITMXconstLa 160

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#Non-thermalITMlenslensITMXconstLinfnITMXconstLanITMXconstLbsITMXTL_null20nITMXconstLbnITMX1m2ITMXAR164u0.5u$phi_ITMXnITMX1nITMXs1sITMXsub0.20027$nsilicanITMXs1nITMXs2m1ITMXHR0.014810.4u$phi_ITMXnITMXs2nITMX2#X-armsLx3994.485nITMX2nETMX1#ETMXETM-07m1ETMXHR3.7u10.9u$phi_ETMXnETMX1nETMXs1sETMXsub0.2$nsilicanETMXs1nETMXs2m2ETMXAR200u0$phi_ETMXnETMXs2nPTXattrITMXHRRc-1937.9attrETMXHRRc2239.7attrITMXHRmass40attrETMXHRmass40###########################################################################%%%FTendXarm%%%FTblockSR 161

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############################################################################DistancefromBStoSR3sls319.3661nSRBSnSR3b#SR3SR3-01bs1SR35u19.1u00.785nSR3bnSR3adumpdumpattrSR3Rc35.97#DistancefromSR3toSR2sls215.4435nSR3anSR2b#SR2SR2-04bs1SR218.3u6.1u00.87nSR2bnSR2adumpdumpattrSR2Rc-6.406#DistancefromSR2toSRMHRsls115.7566nSR2anSRMHRa#SignalrecyclingmirrorSRM-w14m1SRMHR0.36888.3u$phi_SRMnSRMHRanSRMHRbsSRMsub0.0749$nsilicanSRMHRbnSRMARam2SRMAR50u0$phi_SRMnSRMARanSRMARbattrSRMHRRc-5.715###########################################################################%%%FTendSR 162

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%%%FTblockFI############################################################################TheFIisonaplatformdelimitedbytheInput/OutputBufferAssy(I/OBA)#ThephysicaldistanceIBA-->OBA=0.5034(D0901920-V13)#OFIdesignbasedon:D0900464,D1002598#DistancefromSRM(ARsurface)totheinputbuffleassy(IBA)inOFIsusslIBAin0.491516nSRMARbnIBAinm1IBA100nIBAinnIBAout#DistancefromIBAtoinputofOFI(Prisminbetweennotconsidered)slOFIin0.16nIBAoutnOFIin#InputPolarizerIP(Silica)bs1IP1000nOFIindumpnIPtransdumpslIP0.019$nsilicanIPtransnROTin#Rotator(TGG)m1ROTin100nROTinnROTbslROT0.08285$nTGGnROTbnROToutam1ROTout100nROToutanOPa#OutputpolarizerOP(Silica)slOP0.019$nsilicanOPanOPbm1OP100nOPbnOFIout 163

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#WaveplatethicknessslWP0.0127$nCalcitenOFIoutnWPam1WP100nWPanWPb#DistancefromWaveplatetoOBAofOFI(Prisminbetweennotconsidered)slOBA0.2098563nWPbnOBAinm1OBA100nOBAinnOBAout###########################################################################%%%FTendFI%%%FTblockOMCpath############################################################################(LoctionsandanglesbasedonthesolidworkfileD1000342-v14give~5%#mismatch.Thuslom1,lom3omchavebeenadjustedtoget~99.7%overlapattheOMC)#(lom1=2.6334,lom3omc=0.24.8give99%overlapatOMC)#DistanceOBA-->OM1slom12.724nOBAoutnOM1a#OM1#Tissetforhighpower;Lossisaguessbs1OM1800u$Mloss02.251nOM1anOM1bdumpdumpattrOM1Rc4.6#DistanceOM1-->OM2slom21.395nOM1bnOM2a 164

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#OM2#Tisaguessbs1OM210u$Mloss04.399nOM2anOM2bdumpdumpattrOM2Rc1.7058#DistanceOM2-->OM3#TisafromT1200410-v2slom30.631nOM2bnOM3abs1OM30.01$Mloss030.037nOM3anOM3bnOM3transdump#DistanceOM3-->OMCinputcouplerIC(ARside)#Bydesignshouldbe~0.31slom3omc0.196nOM3bnOMC_ARIC_in#DistanceintransmissiontoOM3usedfortesting#slomOM3trans0.1nOM3transnOMC_ARIC2_in###########################################################################%%%FTendOMCpath%%%FTblockOMC############################################################################OMC(asbuiltparameters:D1300507-v1)#InputCouplerIC(flatmirror)bs1OMC_ARIC1004.004nOMC_ARIC_indumpnOMC_ARIC_transdumpssubOMC_IC0.01078$nsilicanOMC_ARIC_transnOMC_HRIC_in 165

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bs1OMC_HRIC0.007610u02.7609nOMC_HRIC_indumpnOMC_HRIC_transnOMC_HRIC_ret#DistancefromICtoOCsOMC_ICOC0.28151nOMC_HRIC_transnOMC_HROC_in#OutputCouplerOC(flatmirror)bs1OMC_HROC0.007510u04.004nOMC_HROC_innOMC_HROC_reflnOMC_HROC_transnOMC_HROC_retssubOMC_OC0.01078$nsilicanOMC_HROC_transnOMC_AROC_inbs1OMC_AROC1002.7609nOMC_AROC_indumpnOMC_AROC_transdump#DistancefromOCtoCM1sOMC_OCCM10.28421nOMC_HROC_reflnOMC_CM1_in#CurvedMirrorCM1bs1OMC_CM136u10u04.004nOMC_CM1_innOMC_CM1_refldumpdump#DistancefromCM1toCM2sOMC_CM1CM20.28151nOMC_CM1_reflnOMC_CM2_inattrOMC_CM1Rc2.57321#CurvedMirrorCM2bs1OMC_CM235.9u10u04.004nOMC_CM2_innOMC_CM2_refldumpdumpattrOMC_CM2Rc2.57369#DistancefromCM2toICsCM2OC0.28421nOMC_CM2_reflnOMC_HRIC_ret###########################################################################%%%FTendOMC 166

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%%%FTblocktunings###########################################################################constphi_SRM89.9989780433498constphi_PRM-0.00104420144479276constphi_ITMX0.000561571624295843constphi_ITMY-0.000561571624295843constphi_ETMX0.00179545319662242constphi_ETMY-0.00184678655646107constphi_BS0###########################################################################%%%FTendtunings%%%FTblockconstants###########################################################################constnsilica1.44963098985906constnTGG1.954constnCalcite1.65846constMloss37.5uconstDARM_DC_offset0.147303constTL_f34.5kconstfdither100###The9MHzmodulationfrequency 167

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constf19099055constmf19099055###The45MHzmodulationfrequencyconstf245495275constmf2-45495275###The24MHzmodulationfrequency(forIMClocking)constf324000000constnf3-24000000###The36MHzbeatfrequencybetween9MHzand45MHzlinesconstfM36397884constnfM-36397884###Thedoubled9MHzfrequencyconstfd118198110constnfd1-18198110###Thedoubled45MHzfrequencyconstfd290990550constnfd2-90990550###Thetripled9MHzfrequencyconstft127297165constnft1-27297165 168

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###Thetripled45MHzfrequencyconstft2136485825constnft2-136485825###########################################################################%%%FTendconstants%%%FTblockerrsigs##############################################################################DCpowertransmittedthroughtheIMCpdIMC_TRANS_DCnIM1in###9MHztransmittedthroughtheIMCpd1IMC_TRANS_9_I$f1-120.9375nIM1inpd1IMC_TRANS_9_Q$f1-210.9375nIM1inpd2dd_IMC_TRANS_9_I$f1-120.9375$fdither0nIM1inpd2dd_IMC_TRANS_9_Q$f1-210.9375$fdither-90nIM1in###45MHztransmittedthroughtheIMCpd1IMC_TRANS_45_I$f2169.453125nIM1inpd1IMC_TRANS_45_Q$f279.453125nIM1inpd2dd_IMC_TRANS_45_I$f2169.453125$fdither0nIM1inpd2dd_IMC_TRANS_45_Q$f279.453125$fdither-90nIM1in###24MHztransmittedthroughtheIMC 169

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pd1IMC_TRANS_24_I$f30nIM1inpd1IMC_TRANS_24_Q$f3-90nIM1inpd2dd_IMC_TRANS_24_I$f30$fdither0nIM1inpd2dd_IMC_TRANS_24_Q$f3-90$fdither-90nIM1in###DCpowerreflectedofftheIMCpdIMC_REFL_DCnMC1HRrefl###9MHzreflectedofftheIMCpd1IMC_REFL_9_I$f10nMC1HRreflpd1IMC_REFL_9_Q$f1-90nMC1HRrefl###45MHzreflectedofftheIMCpd1IMC_REFL_45_I$f20nMC1HRreflpd1IMC_REFL_45_Q$f2-90nMC1HRrefl###24MHzreflectedofftheIMCpd1IMC_REFL_24_I$f3147.56825nMC1HRreflpd1IMC_REFL_24_Q$f357.56825nMC1HRrefl###REFLporterrorsignalspd1REFL_f1_I$f1105nREFLpd1REFL_f1_Q$f115nREFLpd1REFL_f2_I$f230nREFLpd1REFL_f2_Q$f2120nREFL###POPporterrorsignals 170

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pd1POP_f1_I$f1116nPOPpd1POP_f1_Q$f126nPOPpd1POP_f2_I$f228nPOPpd1POP_f2_Q$f2118nPOP###ASporterrorsignalspd1AS_f1_I$f10nSRMARbpd1AS_f1_Q$f190nSRMARbpd1AS_f2_I$f2130nSRMARbpd1AS_f2_Q$f240nSRMARb###########################################################################%%%FTenderrsigs%%%FTblockpowers###########################################################################pdP_DC_ASnSRMARbpdP_DC_OMCnOMC_HROC_transpdPIMCtransnREFL*pdPxnITMX2pdPynITMY2pdPprcnPRMHRbpdPsrcnSRMHRa*adprc00nPRMHRb 171

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adprcf1$f1nPRMHRbadprcf2$f2nPRMHRbadaoc00nOMC_HROC_transadasc00nSRMARbadasf1$f1nSRMARbadasf2$f2nSRMARbadsrc00nSRMHRa*adsrcf1$f1nSRMHRa*adsrcf2$f2nSRMHRa*###########################################################################%%%FTendpowers%%%FTblockHOMs###########################################################################cavcavIMCMC2nMC2inMC2nMC2reflcavcavPRXPRMHRnPRMHRbITMXHRnITMXs2cavcavPRYPRMHRnPRMHRbITMYHRnITMYs2cavcavSRXSRMHRnSRMHRaITMXHRnITMXs2cavcavSRYSRMHRnSRMHRaITMYHRnITMYs2cavcavXARMITMXHRnITMX2ETMXHRnETMX1cavcavYARMITMYHRnITMY2ETMYHRnETMY1cavcavOMCOMC_HROCnOMC_HROC_reflOMC_HROCnOMC_HROC_inmaxtem2 172

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###########################################################################%%%FTendHOMs%%%FTblocklocks###########################################################################setIMC_errIMC_REFL_24_IrelockIMC_lock$IMC_err-0.01475uput*MC2phi$IMC_locksetPRCL_errPOP_f1_IresetMICH_errPOP_f2_QresetCARM_errREFL_f1_IresetSRCL_errREFL_f2_Ire#setAS_f2_I_reAS_f2_IresetOMC_DCP_DC_OMCrefuncDARM_err=$OMC_DC-$DARM_DC_offsetlockPRCL_lock$PRCL_err-5.180683394745u###was10ulockMICH_lock$MICH_err9.618388810335u###was10ulockCARM_lock$CARM_err6.68149947611e-055u###was10ulockDARM_lock$DARM_err-0.004274351264871ulockSRCL_lock$SRCL_err-1.182556184165u###was-1.1825561841610ufuncmMICH_lock=0-$MICH_lock 173

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funcETMX_lock=$CARM_lock+$MICH_lock+$DARM_lockfuncETMY_lock=$CARM_lock-$MICH_lock-$DARM_lockput*PRMHRphi$PRCL_lockput*PRMARphi$PRCL_lockput*ITMXHRphi$MICH_lockput*ITMXARphi$MICH_lockput*ITMYHRphi$mMICH_lockput*ITMYARphi$mMICH_lockput*ETMXHRphi$ETMX_lockput*ETMXARphi$ETMX_lockput*ETMYHRphi$ETMY_lockput*ETMYARphi$ETMY_lockput*SRMHRphi$SRCL_lockput*SRMARphi$SRCL_lock#noplotPRCL_lock#noplotSRCL_lock#noplotMICH_lock#noplotDARM_lock#noplotCARM_lock#noplotmMICH_lock#noplotETMX_lock 174

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#noplotETMY_lock###########################################################################%%%FTendlocks%%%FTblockcommands%%%FTendcommands 175

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APPENDIXEIMCDITHERLOCKINGFORRFAMCOMPENSATIONConsiderouropticalsystem,whichhasaeldtransformfunction,T,suchthatforinputeldEintheoutputeldisT(Ein).Decomposingthelighteldintoitscomponentwaves,wecanwritethetransformas: T(E)=T"Xjajei!jt#=XjT(!j)ajei!t(E{1)wherewecallT(!j)thecomplexeldtransformcoecientforwavewithfrequency!j.WithoutditheringMC2,theelectriceld(forthesinglesidebandfrequencycase)incidentonthephotodiodeisgivenapproximatelyby: E=Tei!0t1+im 2eit+im 2e)]TJ /F4 7.9701 Tf 6.587 0 Td[(it1+ 2eit)]TJ /F4 7.9701 Tf 6.586 0 Td[(i+ 2e)]TJ /F4 7.9701 Tf 6.587 0 Td[(it+iei!0tT0+im 2T+eit+im 2T)]TJ /F3 11.9552 Tf 7.085 1.793 Td[(e)]TJ /F4 7.9701 Tf 6.587 0 Td[(it+ 2T+eit)]TJ /F4 7.9701 Tf 6.587 0 Td[(i+ 2T)]TJ /F3 11.9552 Tf 7.085 1.793 Td[(e)]TJ /F4 7.9701 Tf 6.587 0 Td[(it+i(E{2)whereT0,T+,andT)]TJ /F1 11.9552 Tf 10.987 1.793 Td[(aretheeldtransformcoecientsforthecarrier,upper,andlowersidebandsrespectively,andistherelativephaseoftheamplitudemodulationattheEOMwithrespecttothephasemodulation.(Weignoretermsbeyondlinearorderinthemodulationindices).Theintensityonthephotodiodecanbefoundbyevaluating: jEj2=jT0j2+mIm(X1)cost+mRe(X1)sint+Re(X2)cos(t)]TJ /F3 11.9552 Tf 9.654 0 Td[())]TJ /F3 11.9552 Tf 9.654 0 Td[(Im(X2)sin(t)]TJ /F3 11.9552 Tf 9.654 0 Td[()(E{3)forX1=T0T+)-134(T0T)]TJ /F1 11.9552 Tf 10.987 1.794 Td[(andX2=T0T++T0T)]TJ /F1 11.9552 Tf 7.771 2.956 Td[(.WecanthinkofX1andX2ascharacteristicfunctionsoftheexperimentalsetupforthephaseandamplitudesidebandsrespectively.Tofurthercondensetheexpression,wemakethesubstitutions: Q0=jT0j2Q1=mjX1jQ2=jX2j(E{4) 176

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and: Q1cos 1=mIm(X1)Q2cos 2=Re(X2)Q1sin 1=mRe(X1)Q2sin 2=)]TJ /F3 11.9552 Tf 9.299 0 Td[(Im(X2)(E{5)sothatwecanwritetheintensityas: jEj2=Q0+Q1cos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[( 1)+Q2cos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()]TJ /F3 11.9552 Tf 11.955 0 Td[( 2)(E{6)WecanthinkofQ0astheDCintensityfromthecarrier,Q1astheamplitudemodulationcausedbypassingthefrequencysidebandsthroughadetunedIMC,andQ2astheamplitudemodulationintroducedattheEOMandthenpassedthroughadetunedIMC.Equation( E{6 )isinaconvenientformifwewanttoconsiderthesignaloutputfromademodulationstage: S/jEj2cos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()=Q1 2cos( 1)]TJ /F3 11.9552 Tf 11.955 0 Td[()+Q2 2cos(+ 2)]TJ /F3 11.9552 Tf 11.956 0 Td[()+cos(t)terms+cos(2t)terms(E{7)whichislow-passlteredtoget: S/Q1cos( 1)]TJ /F3 11.9552 Tf 11.955 0 Td[()+Q2cos(+ 2)]TJ /F3 11.9552 Tf 11.956 0 Td[()(E{8)Weneedtobecarefulhowweintroducedithering,whichisitselfamodulation.DitheringMC2isequivalenttovaryingT0,T+,andT)]TJ /F1 11.9552 Tf 7.084 1.794 Td[(,andsoingeneralwewouldexpectQ0,Q1,Q2, 1,and 2toresponddierently.Recalltheeldtransmissioncoecientforasinglemodewithfrequency!throughalinearcavity: tcav=t1t2ei 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r1r2e2i(E{9)wherer1,r2,t1,arereectionandtransmissioncoecientsfortheinputandoutputmirrors,andisthephaseaccumulatedbytheeldoverthelengthofthecavitygivenby: (!)=!L c(E{10) 177

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Forourpurposes,r1r2,t1t2,andsowecanwrite: tcav=t2ei)]TJ /F3 11.9552 Tf 11.955 0 Td[(t2r2e)]TJ /F4 7.9701 Tf 6.586 0 Td[(i 1+2r2cos2+r4=t2 1+2r2cos2+r4(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2)cos+i(1+r2)sin(E{11)wherewehaveexplicitlysplittherealandimaginarycomponents.Now,weexpectthatourcavityisnearlyresonantforcarrierandsidebandfrequencies,andso`pforsomeinteger`pwherep=)]TJ /F3 11.9552 Tf 9.299 0 Td[(;0;+,correspondingtothelowersideband,carrier,anduppersidebandrespectively.Thisallowsustoapproximate: tcav()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)`pt2 1+r21)]TJ /F3 11.9552 Tf 11.956 0 Td[(r2 1+r2+i(E{12)where=)]TJ /F3 11.9552 Tf 12.569 0 Td[(.Soweseethat,torstorder,aditherinMC2willonlyaecttheimaginarypartofthetransmissioncoecient.Alsonoticethatsinceoursidebandsareevenlyspacedinfrequencyfromthecarrier,`+)]TJ /F3 11.9552 Tf 11.963 0 Td[(`)]TJ /F1 11.9552 Tf 10.987 1.794 Td[(mustbeevenandso()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)`)]TJ /F1 11.9552 Tf 10.093 -3.264 Td[(=()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)`+,meaningwedonotneedtodistinguishbetweentheupperandlowersidebandintegers.Callthesingleintegerforthetwosidebands`s.Wewanttorelate( E{12 )backtotheeldtransformfunctionT.Becausethesidebandsareslightlydetuned,therearecorrespondingphaseosets,+and)]TJ /F1 11.9552 Tf 7.085 1.793 Td[(,fortheupperandlowersidebandsrespectively.Wecandecomposethesephaseosetsintoasumanddierenceoftwoosets: =S+A(E{13)ThevalueSisthesymmetriccomponentofthesidebanddetuning,inthesensethatitrepresentsasymmetricosetofthesidebandfrequencyaboutthenominalcarrierresonanceintheIMC,whileAistheantisymmetriccomponentwhichcorrespondstoanosetofthecarrierfromthenominalresonancemode.WewilltreatAasastaticoset.NowsupposeweweretoditherthepositionofMC2: x(t)=xcos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()(E{14) 178

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Ifc=L>>2,forLthelengthofthecavity,theaccumulatedphaseis: ~(!;t)=+!x ccos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()(E{15)wherethetildedenotesthatthefunctionisforthecaseofdithering.IfwedithertheMC2phasewithdithersignalcos(t)]TJ /F3 11.9552 Tf 12.261 0 Td[(),wecanuse( E{12 )toconstructtheeldtransformcoecients: ~T0=()]TJ /F1 11.9552 Tf 9.298 .001 Td[(1)`0t2 1+r21)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2 1+r2+iA+icos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()~T=()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)`st2 1+r21)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2 1+r2iS+iA+icos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()(E{16)From( E{16 )wecancomputethecharacteristicfunctions: ~X1=()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)`2t4 (1+r2)2Scos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[())]TJ /F3 11.9552 Tf 11.956 0 Td[(iA1)]TJ /F3 11.9552 Tf 11.956 0 Td[(r2 1+r2~X2=()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)`2t4(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2) (1+r2)31)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2 1+r2+Acos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()+iS(E{17)where`=`0+`sisevenorodddependingonwhetherthesidebandsareanevenoroddmultipleoftheFSRfromthecarrier.Wecanuse( E{17 )toevaluateanaloguestotheexpressionsin( E{4 )forthecarrier: ~Q0=j~T0j2t4(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(r2)2 (1+r2)4(E{18)forthephasemodulationsidebands: ~Q1=mj~X1j=2mt4 (1+r2)2s (S)22cos2(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()+(A)21)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2 1+r22(E{19)andfortheamplitudesidebands ~Q2=j~X2j=2t4(1)]TJ /F3 11.9552 Tf 11.956 .001 Td[(r2) (1+r2)3s 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2 1+r2+Acos(t)]TJ /F3 11.9552 Tf 11.956 0 Td[()2+(S)22t4(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2)2 (1+r2)41+Acos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()1+r2 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2(E{20) 179

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Atthispoint,wewishtoconsidertworegimes:1.)A<>S.Fortherstregime,wecanthinkofthisasacasewherethecarrieriswelltunedtotheIMCbutthesidebandfrequencyisnotmatchedthetheFSRofthecavity.Equations( E{19 )and( E{20 )become: ~Q1=2mt4 (1+r2)2Scos(t)]TJ /F3 11.9552 Tf 11.955 0 Td[()~Q2=2t4(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(r2)2 (1+r2)4(E{21)andthecorrespondingphasetermsare: ~ 1=8>><>>: 2,for()]TJ /F1 11.9552 Tf 9.299 -.001 Td[(1)`S>0)]TJ /F4 7.9701 Tf 10.494 4.707 Td[( 2,for()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)`S<0~ 2=8>><>>:0,for`even,for`odd(E{22)Notethatthephasemodulationdepth,thesymmetricsidebanddetuning,andthedithersignalareallcontainedin~Q1,whiletheamplitudemodulationdepthiscontainedentirelyin~Q2.Wecanuse( E{22 )toconstructthedierenceinphasebetweenphaseandamplitudesidebandAM: ~ =~ 1)]TJ /F1 11.9552 Tf 14.349 3.155 Td[(~ 2=8>><>>: 2,forS>0)]TJ /F4 7.9701 Tf 10.494 4.707 Td[( 2,forS<0(E{23)Andsoweseefrom( E{8 )and( E{23 )thatsolongastheirphaseisthesameattheEOM(=0),theninthecaseofsymmetricsidebanddetuningtheAMandPMsignalsareinseparatequadraturesatthephotodetector.Forthesecondregime(A>>S),wecanthinkofthisasacasewherethesidebandfrequencyismatchedtotheFSRoftheIMC,butthecarrierosetfromresonanceofthecavity.Equation( E{19 )becomes: ~Q1=2mt4(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(r2) (1+r2)3A(E{24)and( E{20 )isunchanged.Nowthedithersignaliscontainedentirelyin~Q2. 180

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~ 1=8>><>>:0,for()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)`+1A>0,for()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)`+1A<0~ 2=8>><>>:0,for`even,for`odd(E{25) ~ =8>><>>:,forA>00,forA<0(E{26) 181

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BIOGRAPHICALSKETCHRyanGoetzgrewupinthenorthwestsuburbsofChicago,graduatingfromSchaumburgHighSchoolin2007.HeenrolledatIllinoisWesleyanUniversity,wherehespentfouryearsstudyingphyscisandmathematics.Inthesummerof2009,asarisingjunior,heparticipatedintheUniversityofFlorida'sInternationalResearchExperienceforUndergraduates(IREU)program,wherehewasrstintroducedtothegravitationalwaveresearchcommunity.Inthespringof2011hegraduatedfromIllinoisWesleyanUniversitysummacumlaudewithaBachelorofScienceinmathematics.InJune2011hemovedtoGainesville,FloridatobeginworkingintheUFLIGOLabbeforeociallyenrollinginthephysicsgraduateprograminthefall.HegraduatedwithaPh.D.inphysicsinDecember2017. 187