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Sensitivity Enhancement in Future Interferometric Gravitational Wave Detectors


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ForemostIwouldliketothankmyfamilyfortheirunconditionalsupport.IwouldliketothankDr.DavidTannerandDr.DavidReitzeforkindnessandnumeroushelpfuldiscussions.ThanksgotoDr.GuidoMueller,forimaginativeideas,helpfulexplanations,andforregularandsoundgurativeposterior-kicking,aswellasego-deating.ThanksalsogotoDr.VolkerQuetschke,forcheerfullyansweringprobablythousandsofquestions,andtoDr.BernardWhiting,forverythoroughreviewingofmycalculationsandrepeatedlyshowingmetheerrorofmymathematicalways(andsomeveryexcellentcontradancing).Forfriendshipthatwaswortheverybitofoccasionaldistractionfromtheacademictaskathand,ImustthankJoeGleason,AmrutaDeshpande,andMalikRakhmanov.SomedistractionsI'vehadevenlonger-RachelHaimowitz,AhmedRashed,andJames\Tweety"Pencehelpedmegethere.IcouldalwayscountonndingDea'sHalf-FastJamfolksor50MilesofElbowRoomreadytoplaysoulfultunesonddleandbanjoonaweekdaynight,fulllingamusicalneed. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... x CHAPTER 1INTRODUCTIONTOGRAVITATIONALWAVEEXPERIMENTS .. 1 1.1GeneralRelativity ........................... 1 1.2DirectObservationofGravitationalWaves ............. 3 1.2.1ResonantBarDetectors .................... 4 1.2.2InterferometricDetectors ................... 5 1.3InterferometerNoise ......................... 10 1.3.1SeismicNoise ......................... 10 1.3.2ThermalNoise ......................... 10 1.3.3QuantumNoise ........................ 11 1.4Third-GenerationDetectorsandBeyond .............. 12 1.5InThisWork ............................. 14 2PARALLELPHASEMODULATIONFORADVANCEDLIGO ..... 15 2.1InterferometerControl ........................ 15 2.1.1PhaseModulation ....................... 16 2.1.2Pound-Drever-HallCavityLocking .............. 17 2.1.3AdvancedLIGO ........................ 19 2.1.4Electro-opticModulators ................... 21 2.2ParallelPhaseModulationNoise ................... 24 2.2.1MirrorMotions ........................ 25 2.2.2FluctuationsatNon-zeroFrequencies ............ 33 2.3Mach-ZehnderParallelPhaseModulationPrototype ........ 34 3NEAR-FIELDEFFECTHEATTRANSFERENHANCEMENT ..... 42 3.1Near-eldTheory ........................... 42 3.2FluctuationalElectrodynamicsTheory ............... 44 3.2.1Green'sFunctionMethod ................... 45 3.2.2FluctuationDissipationTheorem ............... 47 v

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........... 50 3.3.1EectofDielectricFunction ................. 51 3.3.2EectofLayeredMedia .................... 52 3.4NoiseCouplingDuetoFluctationalElectrodynamics ....... 53 3.4.1VanderWaalsandCasimirForces .............. 53 3.5ImplementationinanInterferometer ................ 54 3.5.1Congurations ......................... 54 3.5.2StabilityRequirementsforColdMass ............ 55 3.6ExperimentalProof .......................... 56 4WHITE-LIGHTINTERFEROMETRY ................... 57 4.1TheSensitivityTheoremQuandary ................. 57 4.2TheLinewidth-EnhancedCavity ................... 58 4.2.1TheNatureoftheGain-bandwidthDilemma ........ 58 4.2.2\White-light"Cavities .................... 59 4.3GravitationalWaveResponseintheTimeDomain ......... 61 4.4RevisedTheory ............................ 68 4.5Plane-waveTreatmentofaGratingCompressor .......... 69 4.6TestsofCorrectedTheory ...................... 72 4.6.1ExperimentalSetup ...................... 72 4.6.2ExperimentalResults ..................... 74 4.6.3AOMs ............................. 77 4.7AlternativeSolutions ......................... 78 4.8IfAlternativesSucceed ........................ 78 5CONCLUSION ................................ 80 APPENDIX AOPTICALTRANSFERFUNCTIONS ................... 81 A.1Fabry-PerotCavity .......................... 81 A.2Three-MirrorCoupledCavity .................... 81 BHEATTRANSFERTHROUGHFIBERS ................. 84 CCIRCUITDIAGRAM ............................ 85 REFERENCES ................................... 87 BIOGRAPHICALSKETCH ............................ 91 vi

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Table page 2{1AdvancedLIGOdegreeoffreedomintermsoflengthinFigure 2{3 .. 20 2{2Controlmatrixfor40mprototypewithoutsecond-ordersidebands[ 20 ] 21 2{3Controlmatrixfor40mprototypewithsecond-ordersidebands[ 20 ] .. 22 C{1ValuesofelementsusedintheMZfeedbackcircuit ........... 85 vii

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Figure page 1{1Theeectofaplus-polarizationgravitationalwavepropagatingintopageonastringoftestmasses .................... 4 1{2Theeectofacross-polarizationgravitationalwavepropagatingintopageonastringoftestmasses .................... 4 1{3Thepower-recycled,cavity-enhancedopticalconguration ...... 7 1{4Thedual-recycledopticalconguration ................. 8 1{5NoisebudgetforAdvancedLIGO .................... 13 2{1SchematicforPound-Drever-HallFabry-Perotcavitylocking ..... 17 2{2Fabry-PerotintracavityintensityandPound-Drever-Hallerrorsignal 19 2{3LengthsinAdvancedLIGOthatrequirefeedbackcontrol ....... 20 2{4SerialphasemodulationofIFOinputlight ............... 23 2{5ParallelphasemodulationofIFOinputlight .............. 23 2{6PossiblenoisemotionintheMach-ZehnderopticsandresultantnoiseonAdvancedLIGOinputlight.C:carrier,SB:sideband. ...... 26 2{7PhasordiagramofMach-Zehndercommon-modeuctuation'seectonrecombinedcarrierandonepairofsidebands. .......... 26 2{8PhasordiagramofMach-Zehnderdierential-modeuctuation'seectonrecombinedcarrierandonepairofsidebands. .......... 28 2{9RatiooffrequencynoisetorelativephasenoisetransferfunctionstoFabry-Perotcommonmodeerrorsignal ................ 30 2{10PermissibleresidualdierentialdisplacementofMach-Zehnderarmlengths .................................. 31 2{11LayoutoftheprototypeMZphasemodulationscheme ......... 35 2{12Spectrumoflighttransmittedbyparallel-phasemodulatingMach-Zehnderinterferometer .............................. 36 viii

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.... 37 2{14ComparisonofMZerror-pointnoisewithrequiredstabilityforfree-runningandclosed-loopoperation ....................... 40 2{15Out-of-loopmeasurementofresidualMZdierentialdisplacementnoise 40 2{16AttenuationtransferfunctionofMZfeedbackelectronics ....... 41 3{1Comparisonofquantum-mechanicalsensitivitylimittotestmassinternalthermalnoiselimitforAdvancedLIGO[ 22 ] ............. 43 3{2Geometryofdielecticmediafornear-eldeectcalculation-twosemi-inniteslabsseparatedbyavacuumgapwithwidthd. ........... 45 3{3Poweruxperunitareaasafunctionofvacuumgapseparatingtwosemi-innitechromiummassesat40and10K ............ 51 3{4Poweruxperunitareaasafunctionofvacuumgapseparatingtwosemi-innitedopedsiliconmassesat40and10Kforvariousdopantconcentrations ............................. 52 3{5Poweruxperunitareaasafunctionofdopedsiliconmetalicityata1.4micronseparation .......................... 53 3{6Possiblecongurationsforevanescentcoolingofamirror ....... 54 4{1Grating-enhancedcavity ......................... 60 4{2Comparisonofround-tripphasesforgrating-enhancedandstandardFabry-Perotcavities .......................... 61 4{3Linewidthsofgrating-enhancedandstandardcavities ......... 62 4{4White-lightcavitygravitationalwaveresponse ............. 68 4{5Apairofidentical,parallel,face-to-facediractiongratingsandtwomirrors(M1andM2)formaresonantcavity. ............ 69 4{6Experimentaldesigntotestgratingphase ................ 72 4{7Measuredandtheoreticaldataformotionparalleltothegratingface 75 4{8Mislignmentangleasdterminedfromtsofmeasureddata ...... 76 A{1ElectriceldsinaFabry-Perotcavity .................. 82 A{2Electriceldsina3-mirrorcoupledFabry-Perotcavity ........ 82 C{1DiagramofanalogciruitusedforMZfeedbackcontrol ......... 86 ix

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1.1 GeneralRelativityJustwhentheworldwasbeginningtothinknothingelseremainedforphysicistsbuttomeasurethephysicalconstantsoftheuniversetothenextdecimalplaceofprecision,theearly20thcenturybroughtthewondersofquantummechanicsandrelativity.In1916AlbertEinsteinrevolutionizedourunderstandingoftheuniversewithhisgeneraltheoryofrelativity(GR)[ 1 ].Initiallyseekingaframe-independentdescriptionofthelawsofmotion,hecametothestartlingconclusionthatmatterandenergywarpspaceandtime.Non-Euclideangeometriesariseinthisfour-dimensionalspace-time\curvature"which,inturn,directsthemotionofmass/energy.Inthisway,weseegravitationasapropertyofspace-timeitself.Whenvelocitiesaresmallcomparedtothespeedoflight,generalrelativity'sconsequencesarenotmuchdierentfromIssacNewton'sdescriptionofgravity.Inregionsofstrongcurvatureorhighvelocity,however,generalrelativityismarkedlyandmeasurablydierent.TheveryrsttestofthenewhypothesiswasthebendingoflightbyourSun'sgravitationaleld.Today,GRispartofeverydaylife,usedforexampleinthetimingoftheGlobalPositioningSystemsatellites.AlthoughmostpredictionsofEinstein'stheoryhavenowbeensatisfactorilyveriedbyexperiment,directconrmationofoneimportantpredictionremainsoutstanding{thatofgravitationalwaves.Oneoftheprimarymotivationsforpursuingtheoreticalalternativestothepre-20thcenturytheorywasthatNewton'sgravityallowedforinstantaneouscommunication.Ifamassmovedtoanewlocationinspace,theeectonatestmasswouldbeimmediate,regardlessofthedistanceseparatingthesemasses. 1

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InformationencodedinthemotionofanobjectononesideoftheMilkyWayGalaxycouldtheoreticallyberead(assumingasensitiveenoughinstrumentexists,anissuewe'llgettolater)ontheothersidewithnotimedelay.Einsteininsistsinthespecialtheoryofrelativitythatnoinformationcantravelfasterthanthespeedoflight[ 1 ].Likewavesspreadingoutwardfromahandsweepingacrossthesurfaceofapond,changesinspace-timecurvaturefromamovingmasspropagateoutfromtheirsourceatthenite,althougheet,speedoflight,about3108meterspersecond.Justasacceleratingchargesproduceelectromagneticwaves,acceleratingmassisneededforgravitationalwaves(GW).Itisamuchweakerforcethanelectromagnetism.Forexample,thegravitationalforce'sattractionbetweentwoelectronsis1042timesweakerthantheelectromagneticrepulsion.Moreover,whereasthedominantcomponentofelectromagneticradiationisthedipoleterm,theabsenceofanegativemasscorrespondingtonegativechargemeansthelowestorderanddominanttermingravitationalwavesisthequadrupole.Wewillhavewaveswheneverthesecondtimederivativeofamassdistribution'squadrupolemomentQisnonzero.Anoft-citedexample[ 2 ]ofgravitationalwavegenerationinalaboratorydemonstratesthedicultyindetecting(orcreating)thesesmallripples.Insomeenormousdiabolicalresearchfacility,scientistscouldcreatetheirownwavesbyrotatingaM=500,000kgironrodaboutanaxisperdendiculartoitscylindricalsymmetryaxisatthemaximumpossibleangularvelocity.This!max=28rad/sisdeterminedbythetensilestrengthofiron.ThebarisL=20mlong.ThegravitationalwavepowerradiatedisP=2 45M2L4!6G c5=2:35W(whereGisthegravitationalconstantandcthespeedoflight),whichisasmallfractionoftherotationalenergyML2!2=3=5:231010J.Gravitational

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perturbationofspace-timegeometrycanbeinterpretedasarelativelengthchange,orstrainh,ofspace.Inthenear-Newtonianlimit(i.e.,gravitationalwavesareasmallperturbationtootherwiseatspace-time),thestrainisanestimatedhGQ c4r;whererthedistancefromsourcetoobserver.Thisstrainisverysmall.Fortwosolar-massneutronstarsintheVirgoClusterthatarenearlyinthemergerphaseoftheircoalescence,hisoforder1021.Einsteinhimselfwasveryskepticalaboutwhetherhumanscouldeverdetectthesewavesdirectly.Thebrightestsourcesweexpecttoseeintheuniversearecompactandmassiveobjectssuchascoalescingblackholesandcoalescingneutronstars,andmaybemovingatrelativisticvelocities,asinthecaseofsupernovaexplosions.Althoughthewavescancarryenormousamountsofenergy,theireectonmatterisveryweakbecausespace-timeisstiandnotveryresponsive.Wedohavesomeindirectevidenceforgravitationalwaves.The1993NobelPrizeforPhysicswenttoHulseandTaylorwhocarefullytimedradiosignalsfromapulsarandfoundthestarwaslosingenergythatmatchedthepredictedenergyradiatedawayingravitationalwaves[ 3 ]. 1.2 DirectObservationofGravitationalWavesThewaytoseegravitationalwavesdirectlyistomeasuretheresultanttidalforcesusingtestmassesfreetorespondtoperturbationofspace-timecurvature.Farawayfromeventhemostferocioussource,thewavewillproduceonlyasmalldeviationinthegeometryofspace-time.Thegeometryisdescribedbythe\metric"gsuchthattheintervaldsbetweentwopointsinfour-dimensionalspace-timeisds2=gdxdx:

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Theeectofaplus-polarizationgravitationalwavepropagatingintopageonastringoftestmasses Theeectofacross-polarizationgravitationalwavepropagatingintopageonastringoftestmasses LetusassumethewaveispropagatingthroughanotherwiseatMinkowskispace-timegeometry.Inthetransverseandtracelessgaugetheperturbedmetricduetoawavetravellinginthez-directionisg=266666664100001+h+h00h1h+00001377777775:Gravitationalwaveshavetransverseforceelds.Thesubscriptsonthestraincomponentsh+andhdenotethetwopossiblepolarizationsofthewave.Ifithastheh+\plus"polarization,thewave'seectonaringoftestmasseswouldbeasinFigure 1{1 .Thelinearlyindependent\cross"polarizationhhastheeectshowninFigure 1{2 ,withthedisplacementsrotated45ofromtheotherpolarizationstate. 1.2.1 ResonantBarDetectorsJosephWeberbuilttherstgravitationalwavedetectorin1960[ 4 ].Hisinstrumentwasalarge(1ton)aluminumcylinderwithpiezoelectrictransducers

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toreadoutrelativeaccelerationoftheendsurfaces.Gravitationalwavesexciteoscillatorymodesinthisresonantbardetectorwhentheycontaintheproperfrequencies,muchlikeahammerstrikingamassive,pure-tonedbell.Gravitationalsignalsmakeverysmallhammers,however,soresonantbarexperimentstodayuseclevermulti-stagetransducerstoregistersmallbarmovementsandcryogeniccoolingtoreducenoise.AlthoughWeberclaimedaGWdetection,subsequentexperimentscastdoubtonitsauthenticity,andwearestillwaitingtodayfortherstdirectobservationofgravitationalwaves.ModernresonantmassGWdetectorsincludeALLEGROatLSU,USA,EXPLORERatCERN,theItalianAURIGAinLegnaroandNAUTILUSinFrascati,andNIOBEinPerth,Australia[ 5 ].Therearealsospheres,whichworkinthesamewaybutcanwithlimitedprecisionlocatethesourceinthesky.TheseincludeBrazil'sMarioSchenberg[ 6 ]projectandMini-GRAILintheNetherlands[ 7 ]. 1.2.2 InterferometricDetectorsWeber'sinstrumentcouldhaveseengravitationally-inducedstrainsassmallas1016=p

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bars,asagravitationalwavecompressesonearmwhilestretchingtheorthogonalarm,doublingtheeect.Anequal-arm-lengthMichelsoninterferometeralsohasthesamesensitivitytosignalsofallfrequencies 8 ],aswelltheFranco-ItalianVIRGO[ 9 ],JapaneseTAMA[ 10 ],andBritish-GermanGEO600[ 11 ]haveallalteredtheirfrequencyresponsefromthatofthesimpleMichelsoninterferometer.TherstthreeincludeFabry-PerotopticalcavitiesintheMichelsonarms.Thecavitieshavetheeectofincreasingtheinteractiontime,andthusthestrength,ofgravitationalsignalswithinthecavitylinewidth,whiledegradingperformanceoutsidethebandwidth.TocountertheminisculenatureofGWstrainsinthispartoftheuniverse,weshouldmakethelengthofthearmsasgreataspossible.Earth'scurvaturelimitshowfarwecantakethis{amirrortoofarawaywillbebelowtheopticalhorizon

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Thepower-recycled,cavity-enhancedopticalconguration wetendtokeeptheseinterferometers'outputs\dark"intherestposition.Thatis,microscopic(laser-wavelength-order)lengthdierencesmakethelightexitingtheantisymmetricportinterferedestructively.Thereforemostallthelightwillbereectedbacktowardthelaser.Towastenot,andtowantnot,LIGO,VIRGO,GEO,andTAMAaddapartiallyreectivemirrorbetweenthelaserandMichelsonbeamsplittertothecongurationtoreectthislightbackintothedetector.Itispositionedsothatlaserlightreectedbacktowardtheinstrumentwillbeinphasewithfreshlaserlightpassingthrough.Wecallthispower-recycling.GEO600sitsoutsideHannover,Germany.ItdoesnothaveFabry-Perotcavities,butitimprovessignalintegrationtimebyaddingamirrortotheinterferometeroutputthatpreferentiallyreectsacertainbandofsignalfrequenciesbackintotheinstrumentforanotherroundwiththegravitationalperturbation.Thisschemeisknownassignalrecycling.Basicpower-recycled-cavity-enhancedandsignal-recycledMichelsoninterferometercongurationsareshowninFigures 1{3 and 1{4 .ThisworkwillfocusonLIGO,presentandfuture.

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Thedual-recycledopticalconguration TheLIGOprojectcurrentlyconsistsofthreeinterferometricdetectorsattwosites.Apower-recycledMichelsonwith4kmFabry-PerotarmcavitiessitsintheswampypinewoodsofLivingston,LA.About2000miles(or10milli-light-seconds)awaythereisanother4kmandaco-located2kmdeviceofthesamedesignattheHanfordReservationineasternWashington.Atthetimeofthiswriting,LIGOisinthemiddleofitsfth\ScienceRun,"S5,aperiodofcoincidentdata-takinginhopesofdetectingtherstgravitationalwaveor,atleast,settinganastrophysicallyrelevantupperlimitoneventratesforsomesources.Onthebestdaysitisreachingitsdesignsensitivitybutforanexcessof1.5to3timesthepredictednoiseinthe40to150Hzband,acriticalregion.S5willlastaboutoneandonehalfyears,withagoalofa70%triplecoincidencedutycycleandtheabilitytosee1.5solarmassneutronstarinspiralsoutto10Mpcforthe4000meterinstrumentsand5Mpcforthe2k.LIGOisscheduledtoaquire1yearofdataintriplecoincidencetobesearchedforgravitationalsignalsatthedesignsensitivitybeforeweupgradeittothenextincarnation.ResearchforthisAdvancedLIGOhasbeenconcurrentwithLIGOcommisioningandscience.

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ThebasicthrustofAdvancedLIGOdesignis,unsurprisingly,greatersensitivity[ 12 ],soastomakeaGWdetectionpracticallyguaranteed.TheeventrateforsignalsthatwouldbevisibletoinitialLIGOisunfortunatelyquitelow.Forexample,somereasonableandpopularcosmologicalmodelspredictbinaryneutronstarcoalescenceswithstrainsvisibletoinitialLIGOtooccuratarateofupto1every1.5years[ 13 ].AdvancedLIGOwillhavetentimesthesensitivity,andthusaviewofathousandtimesthevolumeofouterspace.TheupgradewillbeabletopeerintotherichVirgocluster.ThisbringstheestimatedeventratesforAdvancedLIGOto20to1000peryear[ 13 ].AccordingtoJunMizuno'sinterferometersensitivitytheorem[ 14 ],thetotalintegratedsensitivityofadetectoroverallsignalfrequenciesisafunctiononlyofthestoredlightpower.Toimprovechancesofadetection,onecanincreasethelaserpowerorcavitystoragetimes,ortailorthefrequencyresponsetofavorregionswhereeithercosmologicalsignalstrengthsaregreatestornoiseinterferenceisleast,whilethetotalsensitivityremainsunchanged.AdvancedLIGOgetsalittleofbothimprovements.Werstincreasetheinputlaserpower20times,from6to120Watts.Itisdiculttoproduceacleansingle-spatialmodelaserwithgreaterthanabout200Watpresent,andbeyondthispowerthermallensingeectsinouropticsmakeincreasesmoreproblematicthanadvantageous.Theupgradewillalsouseasignal-recyclingmirrortomovethepeaksignalgainfrequencyofthearmcavitiesfrom0Hz(DC)toabout300Hz.Althoughanticipatedsignalsbecomegreaterasfrequencydecreases,aroom-temperatureground-basedinstrumentissubjecttothermalandseismicnoisedisturbancesthatdestroylow-frequencysensitivityanyway.Thereforeweplaceourgreatesthopesfordetectionascloseaspossibletothisnoise\wall".

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1.3 InterferometerNoiseThephysicsofagravitationalwave-interferometerinteractionisrelativelysimple.Themostchallengingtaskistocreateadevicequietenoughtomeasuretheseminisculeeects.SomehavecomparedgravitationalwavesearchestotryingtodetectaboatlaunchedfromthecoastofAfricabylookingatthewavesonFlorida'sshore. 1.3.1 SeismicNoiseRecallourestimatedneutronstarinspiralstrainestimateofh=1021.IninitialLIGOthiswouldcausea1018mapparenttestmassdisplacement{lessthanthewidthofanatomicnucleus.Throughoutmostofthespectruminwhichwearetryingtomakemeasurements,theearthitselfismovingfarmorethanaGWstrainproduces.Todecreasesensitivitytogroundmotionnoise,andbecausewemustinsomewaykeepthetestmassesfromfallingtoearth,themirrorshangaspendulafromseismicisolationstacks.Althoughconstrainedintheverticaldirection,theyarefreetorespondtolocalspace-timecurvatureinthelongitudinaldegreeoffreedom.Abovetheirresonancefrequencies!r,thetestmasses'responsetoperturbationsofthesuspensionpointsdecreasesas(!r=!)2.Withncascadedharmonicoscillators,weget(!r=!)2nnoiseattenuation.Evenso,ourbesteortspushtheresonancefrequencyofthependulum/isolationstacktoabout1Hz,soseismicnoisewillsetthelow-frequencysensitivitylimit. 1.3.2 ThermalNoiseIfthemasseswhosepositionswemeasuresocarefullyareatanitetemperature,theirconstituentmoleculeswilldgetinthermalexcitation.Thismotioncanexcite

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resonancesinthemassorthesuspensionsthatmovethemirrorsurfacealongtheopticalaxis,mimickingthemotionduetoagravitationalwave.Inthependulumsuspensionwires,weattempttomaketheseresonancesasnarrowinfrequencyaspossible,toconcentrateallthenoisemotionintoabandthatwewillhavetosacrice.Themassesarealsomadeofmaterialwithhigh-qualitymechanicalresonance.Additionally,wecantunetheexcludedfrequenciesoutoftheGWmeasurementbandbychangingthetestmassaspectratio.Atpresent,noneoftheinterferometricdetectorsiscryogenicallycooled.Anotherthermalnoisesourceoccurswhenlowthermalconductivityanddierentlaserintensityondierentpartsofthetestmasscauselocalizedrefractiveindexandradiusofcurvaturechangesinthecoating.Themicroscopicallybubblingtestmasssurfaceisexhibitingthermorefractivenoise.Thermoelasticnoise[ 15 ]resultsfromlocaltemperatureuctuationscausinglocalizedthermalexpansion.Samplingalargepartofthesurfacewithawidelaserbeammakestheeectaverageoutsomewhat.Onecanalsoghtthermoelasticandthermorefractivenoisewithnovellaserintensityproles,perhapsusingabeamwithaatintensityproleinsteadofGaussianshape[ 16 ]. 1.3.3 QuantumNoiseIninterferometrytherewillbenoiseinameasurementsimplyduetothefactthatlight,despiteitswave-likebehavior,isalsocomposedofdiscreteparticles[ 17 ].LightofagivenintensityIisassociatedwithanaveragenumberNofphotons,withanuncertainyinthatnumberduetothefactthatlightparticlesobeyPoissonianstatistics:N=p N:Clearly,thegreaterthelightintensity,thesmallertheuncertaintyinitsmeasurement.Toseegravitationalwavesontheinterferometeroutputwemuststoresucientpowerinthearmsthatphoton-countingstatisticalerrorsaresmallerthanthe

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changeduetoawave.BecausewitharmcavitiesLIGOhasfrequency-dependentlightintensity,theshotnoisealsohasfrequency-dependentbehavior.AsEinsteinnoted,photonscarrymomentum.WhenNNphotonspersecondstrikeLIGO'ssuspendedmirrors,theycreateauctuatingforce,andthusdisplacement,knownasradiationpressurenoise.Astheradiationpressurenoiseinthearmsisuncorrelated,itwillappearintheinterferometergravitationalwavechanneloutput.Radiationpressurenoiseincreaseswithlaserpower.Wecanghttheeectbyusingheavytestmasses{initialLIGOhas10kgfusedsilicaarmcavitymirrors,andtheupgradewilluse40kgmirrors.Loweringthelaserpowerisnotthepreferredsolutionbecauseshotnoisescalesinverselywithintensity.InamachinesuchasLIGOthequadraturesumofradiationpressureandshotnoisedenesa\standardquantumlimit"(SQL)ofsensitivityforagivenlaserpower.Withtheadditionofsignal-recyclingithasbeennotedthatthesequantumnoisetermscanbecomecorrelated,leadingtobetterthanSQLsensitivityatsomefrequencies[ 18 ].AllothernoisesourcesmustcontributelessspuriousGW-signalthanthesemainthree.Acombinationofinterferometerimperfections(inevitable)andnoisyinputlightcancauseadditionalnoisetoappearonthedetectoroutput.Thisoccurswhensomecommonmodesignal\leaks"totheantisymmetricportorwhencontrolsignalsusedtostabilizetheinstrumentarenoisy.Atypical\noisebudget"plotforaLIGO-likeinterferometerisshowninFigure 1{5 [ 22 ]. 1.4 Third-GenerationDetectorsandBeyondAfterAdvancedLIGO,innovationsandprogresswillcontinuetoreachfartheroutintotheuniverse.LISAistheLaserInterferometerSpaceAntenna,agigameter-longinterferometerthatwillyinformationalongtheearth'ssolarorbit.Itwillfeaturethreelaser-linkedtestmassessurroundedbyspacecraftthatshieldthetestmassfromsolarwindandothersourcesofdrag.Withoutterrestrial

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NoisebudgetforAdvancedLIGO seismicconstraints,LISA'smostsensitivebandwidthwillbeinthemilliHertzrange.Thisprojectpresentsformidableengineeringchallenges,butresearchonthemyriadsubsystemsthatwillmakeaspaceantennaiswellunderway.LISAiscurrentlyscheduledtolaunchin2012.Onearth,weexpectfutureinterferometricdetectorswilladdressseismic,thermalandshotnoiseinimaginativeways.Itwillbediculttomakeatestmassbothcryogenicallycooledandseismicallyisolated,butifwehopetoseefartherthanAdvancedLIGOwemustndawaytolessentestmassinternalthermalnoise.Anotherideaforthefutureistobeatthermallensingontheopticswithanall-reectivedesign.Wewouldthenbeabletoincreasetheinputlaserpower(reducingshotnoise)many-fold.Somehavesuggestedusingdiractiongratingsasbeamsplittersandcavityinput-couplers.Toalleviatetheeectofseismicnoisewecouldputdetectorsunderground,andsomewhatescapeseismicharassmentfromhumanactivitysuchasplanes,trains,andloggingoperations,aswellasnaturalphenomenalikewindandearthquakes.Becausemostofaseismicwavetravelsalongearth'ssurface,gravitygradientuctuationsaremuchlessinsubterraneanspaces.TAMA'supgrades,aswellasthepan-EuropeanEUROproject,willbeunderground.Lookingevenfartherahead(andtomoredistantGWsources),detectordesignbecomesevenmorefantastic.

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Canwendsometrickorinnovationthatdefeatstherulethatbandwidthmustbesacricedforsensitivity,andviceversa? 1.5 InThisWorkEachofthesestrategiesimprovesadetectorbyamplifyingthesizeofthesignalrelativetonoisesourcespresent{thesignal-to-noiseratio,orSNR.ThisdissertationpresentsthreetechniquesaimedatimprovingtheSNRofinterferometricdetectorsbeyondinitialLIGOanditssiblings.TherstisanimprovementforAdvancedLIGO'scontrolsystem.Aparallelphasemodulationschemeisneededtoenjoyadiagonalcontrolmatrix.Thesecondtechniquealsoreducesnoise,thermalnoise,bycoolingtestmasseswithnear-eldelectromagneticcouplingofhotmirrortocold.ThelastproposaldiscussedinthisworksuggeststhatsignalstrengthcanbeimprovedwithoutbandwidthlossbyputtingdiractiveelementsinFabry-Perotcavities.Istheresomewaytodefeatthesensitivitytheorem[ 14 ]?Theultimateandthusfarratherdistantgoalofallworkongravitationalwavedetectorsisgravitationalwaveastrophysics.Allsignalswecurrentlyusetomapoutandunderstandourcosmicsurroundingsareelectromagneticvariants,fromthenaked-eyeastronomyofancient(andcurrent)mantomicrowaveandgamma-rayantennae.Aseachnewpartoftheelectromagneticspectrumopenedforus,wediscoveredwondersintheuniversewebeforehadneverevenimagined.Whatawaitsusinanewgravitationalwaveparadigm?

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2.1 InterferometerControlInterferometricdetectoropticsmustbestabilizedagainstseismicnoise,thermalexpansion,andearth-moontidalforcestokeeptheseuctuationsfromaectingthegravitationalwavesignaloutputandtokeeptheappropriateresonanceconditionswithintheinterferometer.Macroscopicdistancesdeterminewhatlaserspatialmodecouplestothespace,butmicroscopiclengthsdecidewhetherlightcanbuildupinthecavity.Onemustmeasurewithgreatprecisionchangesintherelativeseparationofmirrorsandfeedacorrectivesignalback 15

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topositioningactuators. 2.1.1 PhaseModulationInitialandAdvancedLIGOuseelectro-opticmodulators(EOMs)toconvertsomeoftheinputlaserlightintosidebands.EOMsconsistofabirefringentcrystalacrosswhichanelectricalpotentialisapplied.Theeldreorientsdipolesinthematerialandchangesitsindexofrefractionproportionally.Ifoneappliesasinusoidalelectricalsignal,lightpassingthroughthecrystalperpediculartotheeldwillhaveasinusoidalphasemodulation.Forasmallrelativeindexchangem,thelasergainsfrequencysidebands.Putasignalwithtimevariationsin(t)ontheEOMscrystal,andanoriginallymonochromaticlighteldE0ei!tbecomesE0ei!t+imsint'E0ei!tJ0(m)+iJ1(m)eit+iJ1(m)eit;

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SchematicforPound-Drever-HallFabry-Perotcavitylocking andlowerrst-ordersidebands,separatedfromthecarrierfrequencyby.Theremainingtwotermsaresecond-ordersidebandsat2. 2.1.2 Pound-Drever-HallCavityLockingThebasicheterodynetechniqueisknownasPound-Drever-Hall(PDH)locking,andourcontrolschemeisavariationonthesame.Forasimpleopticalresonator,PDHworksasfollows.AFabry-PerotcavityshowninFigure 2{1 ismadefromtwopartiallytransmissivemirrors,M1andM2,separatedbyalengthL.Alaserintroduceslightintothecavityfromtheleft.Therstandsecondmirrorsthelightencountershaveamplitudereectivitiesandtransmissivitiesr1,t1,andr2,t2,respectively.Inthesteady-state,oncetransienteectfromturningonthelaserhavedissipated,theintracavityeldEcavandreectedeldErofthesystemintermsoftheinputE0are(derivedinAppendixA):

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TheratiosTcavandTrarethecavitytransferfunctions.Giveninputlightwithcarrierandapairofsidebands,withamplitudeEin=E0ei!t+im 2.1 validforverysmallm.ThereectedtotalamplitudefromthecavitywillbeEtot=E0TCei!t+im

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Fabry-Perotintracavityintensity(lightline)andPound-Drever-Hallerrorsignal(heavyblackline)asfunctionsoflengthdetuningfromresonance 2{2 .Herethesidebandsserveas\localoscillators"thatdon'tdependuponthequantitywearetryingtomeasure,butallowustobringtherelevantsignalsfromMHztonear-DClevelsthroughthebeat-notephenomenon.Wedemodulatetheamplitude\beat"signalatthedierencefrequencytoremovetheACportionofthesignal.Whatremainswillbeproportionaltotherelativephaseofthetwolightfrequencies.Dierentinformationisobtainedbydemodulatingwithasinusoidalsignalatthedierencefrequencythatisinphasewiththelightphasemodulation,resultinginanappropriatelynamed\in-phase"errorsignal,orwithasinusoidthatis90ooutofphase{the\quadrature"signal[ 19 ]. 2.1.3 AdvancedLIGOWiththenewsignal-recycling(SR)mirrorattheinterferometeroutput,AdvancedLIGOwillhavevelengths,longitudinaldegreesoffreedom(DOF),

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LengthsinAdvancedLIGOthatrequirefeedbackcontrol thatmustbecontrolledtokeeptheinstrumentatitsoperatingpoint.TheseareillustratedinFigure 2{3 .TheveDOFsarethecommonanddierentialFabry-Perotarmcavitylengths,L+andL,respectively,thedierentiallengthoftheshortMichelsonarms,l,thelengthofthepower-recyclingcavitycomposedofthecommonshortMichelsonarmsandbeam-splitter-to-power-recyclingmirrorlengths,l+,andthedistancefromthebeamsplittertothesignal-recyclingmirror,ls.IntermsofthelengthsA-Fshowninthegure,theDOFsaredenedasinTable 2{1 :Inordertomonitorthevariouslengths,weuseaheterodynetechnique Table2{1. AdvancedLIGOdegreeoffreedomintermsoflengthinFigure 2{3 DOFdenitionL+(C+E)=2L(CE)=2l+A+(B+D)=2l(BD)=2lsF+(B+D)=2 similartoPound-Drever-HalltomeasurechangesinthephaseoflightthathasbouncedoofdierentpartsoftheIFO.

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Adual-recycledinterferometerrequirestwopairoffrequencysidebands.Thearmcavitycommonanddierentialerrorsignalscomefrominterferencebetweenasideband(SB)andthecarrier.ThecontrolschemeweexpecttouseinAdvancedLIGOusesbeatsbetweensidebandsforthethreeremainingdegreesoffreedom.The40meterAdvancedLIGOprototypeatCaltechtestsacontrolschemewiththisdivisionoflaborbetweenlightfrequencies.Ideally,onewouldndanerrorsignalforeachDOFthatwasindependentofallothers,but,realistically,somecouplingalwaysexistsbetweenDOFs.ThiscanbeminimizedmakingvariousfrequenciesresonantindierentpartsoftheIFO,concentratingtheirpowerintheseareas,andbyappropriatechoicesofdemodulationphases.Donewell,thiswillproduceapleasinglydiagonalcontrolmatrix.Table 2{2 isanexamplefromthe40mprototypeofanormalizedmatrixshowingthedependenceofeacherrorsignaloneachdegreeoffreedom.Suchacontrolschemewouldberelativelyeasytoimplement.ScientistsatCaltechnoticed,however,thatwhentheyinjectedlightintotheirinstrumentthatwasalreadyatthedierencefrequenciestheywishedtomeasure,theirmatrixincludedsignicanto-diagonalelements,asshowninTable 2{3 [ 20 ].Forthisreason,AdvancedLIGOwillneedanalternatemethodofaddingphase-modulationsidebandstotheinputlaser. Table2{2. Controlmatrixfor40mprototypewithoutsecond-ordersidebands[ 20 ] Dem.!Dem.L+Ll+lls 2.1.4 Electro-opticModulatorsIninitialLIGO,and,theinitial40mprototype,theIFOinputopticsincludeaseriesofelectro-opticphasemodulators(EOMs)throughwhichthelaserpasses.

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Table2{3. Controlmatrixfor40mprototypewithsecond-ordersidebands[ 20 ] Dem.!Dem.L+Ll+lls110o111081103110661062271o1107111051103710612184o73104131021101125o61043271011710212161o32210141021 IfthelaserisfedthroughtwosequentialEOMs,drivenwithfrequencies1and2,theresultisE0J0(m1)J0(m2)ei!t+iJ0(m2)J1(m1)ei(!+1)t+iJ0(m2)J1(m1)ei(!1)t+iJ0(m1)J1(m2)ei(!+2)t+iJ0(m1)J1(m2)ei(!2)tei!tJ0(m1)J2(m2)e2i2t+J0(m1)J2(m2)e2i2tei!tJ0(m2)J2(m1)e2i1t+J0(m2)J2(m1)e2i1tei!tJ1(m1)J1(m2)ei(1+2)t+J1(m1)J1(m2)ei(12)t 2{4

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SerialphasemodulationofIFOinputlight ParallelphasemodulationofIFOinputlight

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we'lluseanarrangementthatmakesmixedsidebandsimpossible.Splitthelaserwithabeamsplitter,asshowninFigure 2{5 .SendthetransmittedlightthroughoneEOMandthereectedlightthroughanother.Recombinewithanotherbeamsplitter.ThiscongurationiscommonlycalledaMach-Zehnder(MZ)interferometer.Whatcouldbesimpler?Naturally,however,thereareafewcaveatsforthescienticemptortoconsider.BecauseeachEOMinteractswithonlyhalftheinputlaserintensity,andbecausetheresultantsidebandsarehalvedagainbythesecondbeamsplitter,withthesameradio-frequencypowerattheEOM,ourparallel-creationsidebandswillhaveonlyaquarterofthepowerintheirseriescounterparts.WithAdvancedLIGO'slargelaserpowerandveryhigh-nesseopticalcavitiesthisshouldpresentnoproblemtondingadequatelysensitiveerrorsignalswiththesameSNR.Mirrormotionsorindexofrefractionchangesinthetwoarms,particularlyifdierential,cancreatenoisethatinterfereswithgravitationalwavedetection,sowewilllikelyneedtoactivelystabilizetheMZinsomeway.Parallelphasemodulationamountstohavingonemoredegreeoffreedomtocontrolthantheseriescase. 2.2 ParallelPhaseModulationNoiseThebasicnoiserequirementforcomponentsofLIGOcanbesummarizedasfollows:notechnicalnoisesourceshouldproduceadetectoroutputthatismorethan10%ofthatmadebytheweakestgravitationalwavetheinstrumentisdesignedtoregister.Theamplitude,frequencyandpointing(transversespatialmode)cleanlinessoftheMZ'soutputmustberegulated.Additionally,becausethecombinedcarrierandsidebandsnolongertravelacommonpath,noiseintheirrelativephases,amplitudes,andpointingcanoccur.Agravitationalwavesignalcanbereadfromthedetectoroutputwitheitheraheterodyneorhomodynetechnique,andthenoiserequirementswilldependuponwhichisused.Withheterodyneor\RF"read-out,radio-frequencysidebandlight

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iseverpresentattheantisymmetricIFOportandactsasalocaloscillator.TheGWsignalisinthebeatbetweenthesideband,whichdoesnotresonateintheFabry-Perotarmcavities,andanycarrierlightthatleaksoutwhenaGWstrainchangesthearmsdierentially.Inthe\DC"read-outstrategy,wemeasuretheeectofaGWstrainasachangeinintensityofcarrierlightitselfattheoutputport.Asonewouldexpect,thereareadvantagesanddisadvantagesassociatedwitheachmethod.Overall,theDCread-outtechniquehasmorelenientcarrierandsidebandnoiserequirements,andhasbeenselectedforthebaselinedesign[ 21 ]. 2.2.1 MirrorMotionsMirrorsintheMZinterferometermayuctuateineithertheirlongitudinalorangulardegreesoffreedom.Bothlongitudinalandangularnoisemotionscanbedividedintothosethatarecommontobotharms,andthosewhicharedierential.Inthecaseofangularnoiseafurtherdivisionoccursbetweencommonanddierentialmovementofopticsinasinglearmandthatbetweenopticsindierentarms.Figure 2{6 isatreediagramthatexhauststhepossiblemirrormotionsandconsequentnoiseonthedetectorinputlight.Theeectsofthetwomainbranchescanbemixedif,forexample,abeamisnotcenteredonanopticorhitsthemirrorwithnon-normalincidentangle.Letusdealwitheacheectinturn.Whenwendthattheparallelphasemodulationcongurationaddsnoiseinawaysimilartoserialmodulation,wewillbesatisedthatweneedonlyrecreatethestabilityofthatoriginalscheme. LongitudinalCommonMotion Longitudinalcommon-modeuctations,whenthemirrorsmovealongtheopticalaxisoftheMZ,willcausethesamephase(andthusfrequency)noisetoappearonthecarrierandsidebands.ThisisillustratedinthephasordiagraminFigure 2{7 .LettheeldtravellingaclockwisepatharoundtheMach-ZehnderIFO

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PossiblenoisemotionintheMach-ZehnderopticsandresultantnoiseonAdvancedLIGOinputlight.C:carrier,SB:sideband. PhasordiagramofMach-Zehndercommon-modeuctuation'seectonrecombinedcarrierandonepairofsidebands.

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intermsoftheoriginalinputeldamplitudeE0beEc=1=p LeikLJ0(m1)+iJ1(m1)ei1t+iJ1(m1)ei1t LongitudinalDierentialMotions Whenopticsmovedierentiallyalongtheopticalaxis,twonoiseeectoccur.First,therecombinedcarrierlightsuersamplitudeuctuations.AmplitudeandintensitystabilityofthecarrieriscriticalwithDCreadout[ 21 ],but,asmentioned,noisewillbesuppressedbyfeedbackloops.

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PhasordiagramofMach-Zehnderdierential-modeuctuation'seectonrecombinedcarrierandonepairofsidebands. Thesecondnoiseeectistherelativephasebetweencarrierandsidebands.WithlongitudinaldierentialmotionstheMach-Zehndercongurationallowsfoructuationsincarrierandsidebandrelativephase,whichmaybeinterpretedasnoiseinthesidebandfrequency.ThishasthesameeectasEOMphasenoiseintheserialmodulationcase(whichwillalsooccurinparallelmodulation).Weexpectthatrelativephasenoisebetweenthecarrierandcontrolsidebandswillpresentthelargestproblemofallnoisetermsdiscussedhereasitappearsasfrequencynoiseonthelaserviathefrequencystabilizationloop,ascurrentlyconceived,andnofeedbackloopsupressesit.Themodecleanerthatservesasalterforlaserfrequency,pointing,andspatialmodenoiseislockedtotheFabry-Perotarmcavities'common-mode,andinturnservesasafrequencyreferenceforthelaser.Thefeedbackloopispoorlyequippedtodealwiththiskindofnoise,andcannotdistinguishsidebandfrequencychangesfromlaserfrequencychanges.Withrelativephasenoise,thecontrolloopimposesamisguidedcorrectiontothelaserfrequency,andtherebyrelativephasenoisebecomeslaserfrequencynoise,which,asmentionedabove,caninterferewithGWdetection.AsymmetriesinthearmsoftheMichelsoninterferometer,ofboththeintentionalandtheinadvertentbutinevitablevarieties,allowsuchcommon-modenoiseasfrequencyuctuationstoappearinthedarkportsignal,wheretheyarenotdistinguishablefromagravitationalwavesignal.TomakeMZ-assistedparallel

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phasemodulationworkforAdvancedLIGO,onemustcalculatetheallowablemagnitudeofcarrier-sidebandrelativephasenoisefromAdvancedLIGO'sfrequencynoiserequirement.UsingDC-readoutforthegravitationalwavesignal,AdvancedLIGOhasa1%mismatchbetweenitstworesonatorcavities.Giventhisoperatingpointfortheinterferometer,thereisafrequencynoiserequirement[ 22 ]fortheinputlightmadebycomparingdetectoroutputduetocarrierfrequencynoisetooutputresultingfromthewavesweseek.Wewishtotranslatethisintoarequirementonrelativephasenoiseamplitude.TheprocedureistocomparetheeectoflaserfrequencynoiseontheFabry-Perotcommon-modesignaltotheeectofrelativephasenoise[ 23 ].TheL+errorsignalisinthebeatbetweencarrierand2sidebandlightthatisreectedfromtheIFObacktothelaser.EnlistingtheFINESSEmodellingtooldevelopedbyAndreasFriese 2{9 .Itisrelativelyatovertheregionofinterestforourdetectors.Giventheallowablefrequencynoisef(f)discussedforlongitudinalcommonmotion,wecanndacorrespondingrelativephasenoiselimitrel(f): rel(f)=drel dS=drelf(f);(2.6)wheredS=dfanddS=darethetransferfunctionsoffrequencyandrelativephasenoise,respectively.AlimitonrelativephasenoisetranslatesintoalimitondierentialdisplacementsLMZoftheMZarmsbyLMZ=

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RatiooffrequencynoisetorelativephasenoisetransferfunctionstoFabry-Perotcommonmodeerrorsignal PermissibleLMZ(f)isshowninFigure 2{10 .Itshouldbenotedthatthisassumesthefrequencystabilizationloop'sgainishighenoughthatitimposespracticallyalloftherelativephasenoiseonthelaserfrequency.ThiswillbethemoststringentstabilityrequirementfortheMach-Zehnderinterferometer.AsmirrorsonanopticaltablesuchastheinputopticsinAdvancedLIGOwillrestupongenerallyhavemuchlargeructuations,theMZwillneedactivecontrol. AngularMotions Whenmirrorshaveangulaructuations,adisparityarisesbetweentheopticalaxesoftheMach-ZehnderIFOandAdvancedLIGO'scoreoptics.Gaussianopticsdescribetheamplitudeofalightbeamintheplaneperpendiculartoitspropagationintheparaxialapproximation(thatis,whenitstransversevariationismuchlessthanthelongitudinalvariation.)TheGaussianmodesconstituteanorthonormalbasissetintowhichanyspatiallightdistributionmaybedecomposed.Aninterferometerwillgenerallyuselightbeamsmadealmostentirelyfromthelowest-ordertermofthebasisset,TEM00.ThisisthetermthathasthenointensityzeroesandaGaussianshape.Whentherearesmalldisplacements

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PermissibleresidualdierentialdisplacementofMach-Zehnderarmlengths ortiltsinthebeamrelativetotherstopticalaxis,thenewspatialprole,whenexpandedintheoriginalbasisset,willcontainsmallcontributionsfromhigher-orderGaussianmodes[ 24 ].Foralasertoberesonantinagivenopticalcavity,itmusthavenotonlytheproperwavelength(thecavitylengthdividedbyaninteger),butalsoaparticularspatialmodedenedbythelengthandmirrorcurvatures.Spatialmodenoiseisaproblembecauseitcausesapparentlaseramplitudenoisesincenon-TEM00modescouplelessecientlytotheresonatorcavities.Noisealsoresultsfromthereconversionofnoisyhigher-ordermodestothefundamentalmodeviamisalignedcoreoptics[ 25 ].IntheHermite-Gaussianeigenbasis,inwhichatransversespatialamplitudeistheproductoftwoone-dimensionalfunctions,therstthreemodesare: 2q,thefundamentalmode weikx2 2q,the\tilt"mode 8!p 2q,thebull's-eyemodewithwavenumberk,beamsizew(z),Guoyphasor(z),andcomplexradiusq(z).

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Letandbebeamtiltsrelativetothex-andy-axes,respectively.Inaddition,lettherebesmalldisplacementsandofthebeamfromthecoreopticsaxis.ThetotaleldatthesymmetricportoftheMach-ZehnderIFOwillbeEtot/"2 Lei x=qeik xeik y=qeik y"eikLeieikx=qeikxeiky=qeiky w+eikLeieikx=qeikxeiky=qeiky w#Whereaquantitylabeled w)tolinearorderinsmallperturbations.TheeldisE/r Lei x=qeik xeik y=qeik y2coskL++kx q+kx+ky q+ky wsinkL++kx q+kx+ky q+ky(2.8)

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Termssuchaseik refertoMZcommon-modetiltsanddisplacements,andthesearethesamephasesfoundintheelectriceldsofserialmodulationschemes.WhenthenewpointingoftheMZisexpressedintermsoftheIFOopticalaxis,therearesmallcontributionsfromhigher-ordertiltmodes.Inthex-direction,E(x)/U0(x)2+2k x q+2k x+2kx q+2kx+2k2x qL2iw wkx q2iw wkxwhichis2U0(x)+ q+k +k q+k+k2 qLiw wk qiw wkU1(x) 2.9 denesatiltmodeamplitudeatilt.Weseefromtheexpression 2.8 thatdierentialpointingeectsareallofsecondorder,andthereforemostlikelynoproblematall. 2.2.2 FluctuationsatNon-zeroFrequenciesNowassumethattheMZmirrorpointingnoiseoccursnotatDC,butatafrequencyfsuchthat

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maximumpermissibleamplitudeofatilt 2:5103 2.9 tocalculatepermissiblecommon-modetiltsinMach-Zehnderopticssuchasmirrorsandbeamsplitters.Theinputmodecleanersupresseshigher-ordermodesbyanadditionalfactorof2000.Weassumeabeamsizeontheopticinquestionof5104mandthathalftheallowableuctuationsareallocatedtotilts,andhalftodisplacements.Asanexample,themaximumcommontilt w=8:5109rad 2.3 Mach-ZehnderParallelPhaseModulationPrototypeHavingfoundthestabilityrequirementforparallelphasemodulation,atable-topprototypewillexploretheattainabilityoftheselimitsondierentialopticalpathchanges.ForourpurposestheMach-Zehnderinterferometershouldbecompact,inexpensive,andeasilycontrolledwithafeedbackloop.Inthisdesigntheopticsareallmountedtoasinglealuminiumplate.Thesearesmall,o-the-shelfmirrorsandmounts.Onemirrorisgluedtoastackofpiezoelectricceramics(PZT),whichgrantsuslengthcontrolaccesswithonlysmalldrivingvoltages(2Vcancoverafulllaserwavelength).Toavoidaddingpointingnoisewithour

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LayoutoftheprototypeMZphasemodulationscheme actuatorweuseacombinationofpolarizingbeamsplittersand=4-waveplatesinthecornersoftheMZ,asshowninFigure 2{11 .Inthisway,thePZTpushesandpullsthemirroralongthelightpropagationaxis.Asabonus,thewaveplatesalsoallowonetomaketheimpedancematchingofthearmsnearlyperfect,sothatdarkinterferencefringesattheoutputcanbetrulyblack.TheEOMsemployedareo-shelfLiNbO3resonantphasemodulators.Theirfrequenciesintheclockwiseandcounterclockwisearms(asviewedinthegure)are31.5and12.0MHz,respectively.Itiswiththe12MHzsidebandwelocktheMZdierentialmode.Thisrequiresahigh-speedphotodetector-oursisa1GHzSiphotodiode.WemixtheACportionofthephotodiodeouputsignalwiththeRF-signalthatdrivesthe12MHzEOM.Thelow-pass-lterederrorsignalisampliedandfrequency-lteredbyananalogcircuitbeforeitisfedbacktotheMZlengthcontrolPZT.AlthoughoneshouldeasilybeabletoguessthespectralcontentoftheMZinterferometeroutput,experimentalsciencecanbefullofsurprises(seeChapter4),soweputtheMach-Zehnderoutputlightthroughahigh-nesseopticalspectrum

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Spectrumoflighttransmittedbyparallel-phasemodulatingMach-Zehnderinterferometer analyzertoseewhatfrequencieswerecreated.Theinterferometer'sbehaviorwasasexpected,however,asseeninFigures 2{12 and 2{13 .Therearetwopairoffrequencysidebands.Thejaggedpeakstotherightofthecarrierresonancearehigher-orderGaussianmodes.Welookcloselyatthemixturefrequencies.Smallsecondharmonicsidebandsarevisibleforbothfrequencies,soweknowwewouldbeabletodiscernwhetherSBsofSBswerepresent.Nopeakispresentat19.5MHz,butafewsmallpeaks(amixedsidebandshouldhaveapproximatelythesameamplitudeasasecond-ordersideband)appearat43.5MHz.Uponcloserinspection,lookingatthelightcorrespondingtothesepeaksonaCCDcamera,wefoundthatthesewerehigher-orderspatialmodesresonatingintheanalyzercavity,andnotmixedsidebands.TogetasucientlyquietMZinterferometerwewillneedaquieterrorsignalwithstrongdependanceonthequantityitintendstoregulate.Ourtechniqueisheterodynelengthsensinginthebeatsignalbetweenthe=12MHzsidebandandtherecombinedcarrier.TheeldatthesymmetricportoftheMach-ZehnderIFO

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ExpandedviewofMZoutputspectrum.Mixedsidebandsareabsent{the43.5MHzpeaksareduetohigher-orderspatialmodes. (withtheeectofthe31.5MHzEOMomittedforclarity)is 2.1 is

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dierencebetweenbright-anddark-fringelockingisinversionofthesignal.UnliketheFabry-PerotcavityanalyzedabovetheMach-Zehnderheterodyneerrorsignalhasaninnitelockingrange.Thatis,nomatterhowfarremovedthedierentiallengthbecomesfromtheproperinterferencecondition,thePZTwillalwayspush(orpull)thelengthclosertoperfection,seekingthatzerointheerrorsignal.ThedisadvantagerelativetoFabry-PerotPound-Drever-Halllockingisthatwedonothavesharpresonancesthatheightenourawarenessoflengthchangesthroughsteeperrorsignalslopes.Thestrengthofacorrectivesignalisinthesizeofitsderivativewithrespecttotheparameteritcontrols.AswehaveafeeblesinusoidalvariationinourerrorsignalasafunctionofL,wemustaddslopeelectronically.Todeterminehowmustamplicationtheerrorsignalneedswemeasurethenoiseinthefree-running(nofeedback)Mach-Zehnderoutputandcomparetothestabilityrequirement.Ananalogproportional-integratingampliercircuitwithadjustablegainmakesupthedierence.Wewouldbecontenttosetthegaintothehighestlevelnecessaryandfeedthisbackatallnoisefrequencies,butwemustbemorecareful.Mechanicalresonancesinthesystem,particularlyintheactuator,willbeexcitedifthegainisgreaterthan1atthesefrequencies.At10kHz,thePZT'sinternalresonanceisthelowest-frequencyandmostproblematic.Fortunately,measurementsofthefree-runningMZ'serror-pointnoiseindicatethatitisalreadysucientlyquietat10kHz,sothefeedbackloopneedsnogainatthisorhigherfrequencies.InFigure 2{14 oneseestheperformanceofthestabilityfeedbackloopforvariousnoisefrequenciescomparedtotherequirement.Theopen-loopmeasurementindicatestheinterferometerneedsactivefeedbacktosupressexcessnoisebelowabout500Hz(where,unfortunately,thereisasmallresonanceduetothePZTmount).Withthefeedbackloopclosed,theresidualdierentialmotionnoisefallswithinanacceptablerangeofthecalculatedrequirement.Noteworthy

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exceptionsaretheaggravatinglyever-present60Hzlinenoisespikes,aswellasthe500Hzresonance.Wealsomadeanindependentmeasurementoftheresidualdierentialmotionbydemodulatingthephotodiodeoutputwiththeradio-frequencysignalthatdrovethesecondEOM.ThemeasurementpointislabelledSoutinFigure 2{11 .Figure 2{15 depictstheout-of-loopnoisemeasurementaloneversusthetarget.Althoughasmallamountofexcessnoisepersistsbelowabout100Hz,wefeelthisresultdemonstratestherelativeeasewithwhichonecanachievethenecessarystabilityforparallelphasemodulation.IfweimplementedthisschemeinAdvancedLIGO,wecouldimprovethenoiseperformancebyraisingthefrequencyofthePZTresonancebygluingalightermirrortoit,whichwouldallowthefeedbackloopahigherunitygainfrequencytosuppressnoisebelow100Hz.ThenoiseattenuationtransferfunctionfortheelectronicsusedinthisexperimentisfoundinFigure 2{16 (AppendixCcontainsthecircuitdiagram).Groundloopsandlinenoisecouldbereducedwiththevariousblackmagictechniqueselectricalengineersknow.TheentireMZinterferometercouldalsobeenclosedinvacuumforbetterpassiveisolation.Intheend,parallelphasemodulationisvyingwithseveralothertechniquesforinclusioninAdvancedLIGO'slength-controlscheme.ComplexmodulationofasingleEOMhastantalizinglyfewerdegreesoffreedomtocontrol.Itsstrictestrequirementiscoordinatingthephasemodulationwithamplitudemodulation,thatis,lockingtwooscillators.Itmaybethatdouble-demodulationfortheinnerAdvancedLIGODOFsisabandonedentirelyforadierentmixofbeatsignals.AlongwiththeCaltechprototypegroupwehaveinvestigatedtherequirementforandperformanceofapossiblesolution,andawaitacoalitiondecisionasthetimetobeginconstructionthetheupgradenears.

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ComparisonofMZerror-pointnoisewithrequiredstabilityforfree-runningandclosed-loopoperation Out-of-loopmeasurementofresidualMZdierentialdisplacementnoise

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AttenuationtransferfunctionofMZfeedbackelectronics

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3.1 Near-eldTheoryInterferometricdetectorswilldisclosethehitherforeimperceptiblepassageofagravitationalwavethroughcarefultimingofphotonstravellingbetweenthemirrorsthatserveastestmassesfortheeld.Unfortunately,ifthemirrorsthemselvesmovewithrespecttoanotherpartoftheinstrument,theeectimitatesagravitationalwave.InthecurrentandadvancedversionsofLIGO,wehaveseveralstrategiesforquietingthesenoisemotions.Thetestmasseshangaspendulafromseismicisolationstacks.InadditiontoactivefeedbackoutsidetheGWmeasurementband,initialLIGOhassinglestagesuspensions,butAdvancedLIGOwillhavefourmasseshanging,onefromtheother,endingwithaveryquietnaltestmassatthebottom 26 ]calculatedthenoiseintheGWchanneloftheIFOforaroom-temperatureAdvancedLIGOtestmass.TestmassinternalthermalnoiseisshowningreeninFigure 3{1 .Accordingtotheclassicalequipartitiontheorem,thereis1 2kBTofenergyineachmirror 9 ] 42

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Comparisonofquantum-mechanicalsensitivitylimittotestmassinternalthermalnoiselimitforAdvancedLIGO[ 22 ] degreeoffreedom,wherekBisBoltzman'sconstantandTthetemperatureindegreesKelvin.Levinfoundinternalthermalnoiseisproportionaltothemass'stemperature.InAdvancedLIGOroomtemperaturetestmassespreventourreachingthequantumlimitofdetectorsensitivity,alimitduetothecorpuscularnatureoflight,insomefrequencyregions.Themirrorsneedtobecooledto10Ktohavethermalnoisecomfortably

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thesuspensionbers.TheeectofthelatterinsuspensionslikethoseinAdv.LIGOwillbefarinsucient. m2K4T4;whereistheemissivityofthematerial.TheT4factorintheStefan-Boltzmannlawmeansthatradiativeheatlossdiminishesrapidlywithtemperature.Afusedsilicamirror(=0:9)with1m2area,at10K,therefore,wouldradiate510W.WithAdvancedLIGO'sintracavitycirculatingpowersof10kWandtheanticipatedmirrorabsorptivities,however,theinputtestmasswillbecollecting2Wfromthelaser.AdvancedLIGOwouldclearlyrequireactivecoolingtobeatthecurrentthermallimit.IntheLIGOupgradethemirrorswillremainatroomtemperature,butwheninevenmorefuturisticincarnationsofinterferometricdetectorsthestandardquantumlimitofsensitivityismuchlower,wewillcertainlyneedtoactivelycoolthetestmasses.Howcanwebothenhanceheatlossandmaintainseismicisolation?Near-eldelectromagneticeectsprovidetheanswer.Wecannottouchthetestmasseswithacoldnger,butwecanbringacoldobjectcloseenoughthatphotonscantunnelthroughthevacuumgapfromhottocoldmass[ 27 ].Classically,theeectmaybedescribedaspenetratingtheregionofsignicantevanescenteldssothattheseeldscantransferenergy,aphenomenonalsoknownasfrustratedtotalinternalreection(FTIR). 3.2 FluctuationalElectrodynamicsTheoryToquantifytheheattransferenhancement,weusetheuctuationalelectrodynamicsmethoddevelopedbyS.Rytov[ 28 ].Thermalexcitationswithinamaterialcause

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Geometryofdielecticmediafornear-eldeectcalculation-twosemi-inniteslabsseparatedbyavacuumgapwithwidthd. oscillatingcurrentsJthatserveaselectromagneticeldsources,describedbyMaxwell'sequations: 3.2.1 Green'sFunctionMethodWithGreen'stheoremsonecanndtheelectricandmagneticeldsduetothethermally-inducedcurrentsusingGreen'sfunctions.

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isotropicandnonmagneticmatterseparatedbyavacuumgapofwidthd.OrienttheCartesianaxessuchthatthez-axisisperpendiculartotheinterfaces,andtheplanez=0coincideswiththesurfaceofonesemi-innitechunkasseeninFigure 3{2 .Thethreeregions1,2,and3aredenedasshown.InsteadofacompleteFourierdecompositionofGreen'sfunctionintoplanewavesoftheformEei(kr!t),whicharenotingeneralguaranteedtosatisfythefree-spaceMaxwellequations,wewilluseapartialspatialtransformEei(Kr!t)+ikzz,whereKisavectorinthex-yplane.Thez-componentkzissubjecttotheconstraintK2+k2z=k20;withk0thephotonwavenumber.ThustheGreen'sfunctionencompassesbothpropagatingwaves,whenK=jKj
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3.2.2 FluctuationDissipationTheoremOurheroonceagain,AlbertEinsteinrstidentiedthefundamentalprinciplebehindtheuctuationdissipationtheoreminhisworkonBrownianmotion[ 30 ].Acoupledecadeslater,in1928,Nyquistfoundaconnectionbetweenvoltageuctuationsinelectroniccircuitsandelectricalresistancethatreliesonthesameunderlyingpostulate[ 31 ].Thegeneralizedtheorem,ofwhichBrownianmotionandJohnsonnoiseareexamples,waspublishedbyCallenandWeltonin1951[ 32 ].TheFDTstatesthatinthermalequilibriumasystemwithdissipationD(!)generatesauctuatingforceFF2=2

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Withtherstofthesetwoclues,Eq. 3.3 becomesE(r;!)=o! u1Jzkz1 3.2 .Transformingtheoperatorrtoiu3,themagneticeldinregion3isH(r;!)=1 82ZVdV0Zd2K1 u1Jzkz1 82ZVdV0Zd2Ku3 u1Jzkz1 3.6 ).TheheatuxperpendiculartotheinterfacesishSz(z;!;T)i=o!20im1

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2=(kz1):Athirdsimplifyingequationisthat=(1)!2 3.11 maynallybewrittenhSz(z;!;T)i=4 3.12 isthesameiftherolesofemitterandabsorberarereversed.TocalculatethenetheatuxPbetweentwodielectricsweintegratethedierencebetweenright-movingandleft-movinguxandintegrateoverallpositivefrequencies:

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Atsmallvacuumgapwidths,wellintotheevanescenteldregion,theheatuxscaleswiththeinversesquareofd. 3.3 NumericalSimulationsofEvanescentCouplingWemayevaluateEq. 3.13 numericallyoncethedielectricfunctionsandvacuumgapbetweenthemediaaredened.Caremustbetakenatthejunctureofpropagatingandevanescentwavesastheequationsarepronetoinnitieshereifnothandledwell.Ourrsttestofournew-foundcalculationalabilitywasapairofmetallicmasseswithvariousmicroscopicseparation.Onemasswasat10Kandtheotherat40K.WeuseaDrudemodelforthedielectricfunction(!)ofchromium:(!)=b(!)+iNq2ef0 3{3 .Anear-eldenhancementofseveralordersofmagnitudeisevidentwhenthevacuumgapbecomesmuchlessthanthedominantthermalwavelengthth,givenbyWien'slaw:th=2:9103mK

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Poweruxperunitareaasafunctionofvacuumgapseparatingtwosemi-innitechromiummassesat40and10K 3.3.1 EectofDielectricFunctionTotestwhatkindofmaterialshavethestrongestevanescentcouplingeect,weevaluatedtheheatuxbetweendopedsiliconmasses.Siliconisofparticularinterestasitisacandidatematerialforall-reectiveinterferometerdesigns.Thedielectricfunctionusedisevaluatedfrom(!)=014!2p 3{4 .TheStefan-Boltzmanradiationforaperfectpairofemittersisdepictedbythehorizontallineat0.15W/m2.Themoremetallicsamplesareclearlypooreremittersinthefareld,butintheneareldthebehaviorismuchmorecomplex.

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Poweruxperunitareaasafunctionofvacuumgapseparatingtwosemi-innitedopedsiliconmassesat40and10Kforvariousdopantconcentrations Thereisnomonatonicprogresswithdopantconcentration.Infact,iftheplotisslicedalonganaxisextendingintothepage,plottingheattransferversusdopantconcentrationforasinglevacuumgapwidth(Figure 3{5 ),weseerathercomplexbehaviorasthematerial'sdielectricfunctionchanges.Thereisanenhancementinthermalcouplingwhenresonancesintheabsorbercorrespondtothethermalwavelengthsoftheemmitter.ThismakesclearthatifwewanttouseevanescentcoolinginLIGO,wewillhavetocarefullyselectmaterialsappropriatetothetemperatures,geometries,andgapwidthswewishtouse.Overall,however,theeectofdielectricfunctionspecicsonheattransferismuchlessthantheeectofsimplyclosingthegapwidth. 3.3.2 EectofLayeredMediaWealsoinvestigatedwhetherlayereddielectriconthematerialsurfacedcouldenhanceheattransferbyactingasanti-reectioncoatings.TodothisonechangestheGreen'sfunctionsinthecalculationslightlytoincludereectionsatmultipleinterfaces.Thischangesthets;ps.Wefoundthatthishadalmostnoeecton

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Poweruxperunitareaasafunctionofdopedsiliconmetalicityata1.4micronseparation thethermalcoupling.Thereasonisthatsuchcoatingsaectthepropagatingmodesprimarily,whichatthesesmallvacuumgapwidthscontributelittletoheattransfer. 3.4 NoiseCouplingDuetoFluctationalElectrodynamicsAlthoughwecanextractheatfromtheinterferometertestmasswithouttouchusingthenear-eldeect,itisnottruethatthecoldmassexertsnoforceonthehotmass.Infact,theverysamedipoleinteractionthatgivesusheat-transfercouplingalsoleadstotheVanderWaalsinteraction[ 33 ]. 3.4.1 VanderWaalsandCasimirForcesThesimilaritytoproximity-enhancedheattransferisevidentintheequationfortheattractiveforcebetweentwoatsemi-innitedielectricsseparatedbyd:FVdW(d;T)=h (kz1K)(kz3K)e2iK!d=c11+(1kz1+K)(3kz3+K) (1kz1K)(3kz3K)e2iK!d=c11#(3.14)

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Possiblecongurationsforevanescentcoolingofamirror Ifdislargerelativetoth,theaboveequationbecomes 3.15 validis 3.5 ImplementationinanInterferometer 3.5.1 CongurationsThereareseveralwaysafutureinterferometricdetectormighttakeadvantageofthenear-eldheattransferenhancement.Thesmallseparationdistancecouldbe

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alongthelaseropticalaxis,asshownintheleft-handsideofFigure 3{6 .Itcouldbearingwithanapertureforthelightbeamifthemirrorisatransmissiveoptic.Inanall-reectivedesignwecouldaccessmorearea,andthustransfermoreheat.Alternatively,onemaychoosetoplacethecoolingsurfacesalongthesidesoftheoptics,asseenintheright-handsideofFigure 3{6 .Thisway,noiseforcesmakethemirrorjitteraboutinadirectiontransversetothecavitylengthswearemeasuring,greatlyreducinghowstrictstabilityrequirementswillbe. 3.5.2 StabilityRequirementsforColdMassDisplacementsoftheITMarerelatedtotheforceexertedbythenearcoldobjectbyd~x(f)=d~F(f) 4mf2:TheFouriertransformofdF(t)isd~F(f)=1 2Z1dF 2Z1ei(!+)tdt:TheAdvancedLIGOdisplacementrequirementat10Hzis1019m

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WithdFspecied,wecancalculatethemaximumallowabledisplacementofthecoldmassatthenoisefrequency(10Hz,inthisexample):dF=5:71015N 3.6 ExperimentalProofInthenearfuture,theUniversityofFlorida'sLIGOgroupwillconductexperimentstomeasuretheenhancedheattransfereectbetweenlarge,atobjects.TheywillevaluatethepertinenceofthesetechniquestofutureversionsofLIGO.

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14 ]insiststhatthesearetheonlyoptionsforcatchingmoregravitationalwaves.Whendetectorbandwidthisbroad,withbroadercavityresonances,thesensitivityhitsashotnoiseceilingratherquickly.Whenthecavitieshavehighnessewestubourtoesagainsttheedgesofacrampedbandwidth.Wewantspaciousaccomodationsinbothdimensions.Whatwewouldreallylikeistodefeatthesensitivitytheorem.Warning:suchanambitiouschaptercannotendwell. 4.1 TheSensitivityTheoremQuandaryAdvancedLIGO[ 12 ]willhaveapeaksensitivityatabout300Hz,witha1=fdeclineinresponsivenessabovethepeak.Theanticipatedsourcesintheseveral100Hzrangearepulsars,thefundamentalfrequencyoftheintermediatephaseofaneutronstar-neutronstarmerger,andthenalfundamentalfrequencyofsmallblackhole-blackholecoalescences.Highfrequencypulsars,theharmonicsofneutronstarandblackholemergers,stellarcorecollapses,andneutronstaroscillations,however,areallexpectedtoemitgravitationalwaveswithinreachofAdvancedLIGO'ssensitivity,butaboveitsbandwidth.Aninstrumentwithatleast20kHzofbandwidthisneededforthesesources.ThebandwidthofLIGO-liketerrestrialinterferometricgravitationalwavedetectorsissetbythepoleoftheFabry-PerotcavitieswithinthearmsoftheMichelsoninterferometer.Thisconstraintarisesbecausethegainofgravitationalwave-inducedsignal 57

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sidebandsislimitedtofrequencieswithinthelinewidthofthecavities.ThenatureofstandardFabry-Perotcavitiesissuchthatonecannotindependentlyadjustforincreasedgainwithoutsueringalossofbandwidth.Ifthesequantitiescouldbedecoupled,theresultingimprovementinbandwidthmayleadtoviablehighfrequencydetectors.Weguessedthatadiractiveelementplacedwithinanopticalresonatorcouldincreasethecavitybandwidthwithoutlossofpeakintensity.Aswediscovered,thisexpectationwasbasedonerroneous(butnotuncommon,evenamongopticalscientists)understandingofdiractiongratingfunction. 4.2 TheLinewidth-EnhancedCavity 4.2.1 TheNatureoftheGain-bandwidthDilemmaConsiderasimpleFabry-PerotcavitywitharmlengthLandinputandoutputmirroramplitudereectivitiesandtransmissivitiesr1,t1andr2,t2,respectively.Thenormalizedintensityoflightofaparticularfrequency!=2c=withinthecavityis

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cavities,withhighlyreectivemirrors.Onendsthat t21=)FWHM/t21(4.3)ItisclearthatlinewidthandpeaklightintensityareinextricablyintertwinedinastandardFabry-Perotcavity.Asaresult,gravitationalwavescientists,whowouldliketomaximizebothquantities,mustcompromiseinchoosingthecavity'sparameters. 4.2.2 \White-light"CavitiesThelogicofthediraction-enhancedcavitywasasfollows.Theround-tripphaseshift'svariationwithfrequencyhasbeenidentiedasthesourceofthegain-bandwidthdilemma,immediatelysuggestingitasthefocusofdesignalterations.Perhapscouldbemadeinvariantwithfrequencyifthetheopticalpathlengthinsidethecavitywerealsofrequency-dependent: (!)=2!L(!) 4{1 wouldworkasfollows: 1. Monochromaticlightisinjectedintothecavity. 2. Agravitationalwavemodulatesthelight-transittimebetweentheinputmirrorandthedistantrstdiractiongratingsuchthatfrequencysidebandsaregeneratedontheoriginallaserlight.

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Figure4{1. Grating-enhancedcavity 3. Theparallelgratingsallocateadierentpathlengthtolightofdierentfrequencies. 4. Asseeninthegure,theredderlowersideband(dashedline)travelsalongerdistancethanthebluercarrier(solidline)oruppersideband(dot-dashedline)light.Thisadditionalpathcancancelthevariationoftheroundtripphaseshiftrelativetothecarrierlight.Figure 4{2 showstheround-tripphaseresultingfromthefrequency-dependentopticalpathlengthandcomparesthistothefrequency-invariant(!)ofastandardcavitywiththesamenominallength.Ifchanginglengthwithfrequencyweretheonlyeectgratingshadonthelightphase,thegrating-enhancedcavitywouldhaveasuperiorlinewidthwhereveritsslopeislessthanthatofthestandardcavity.Figure 4{3 showstheintensitybuild-upinaLIGO-scaleenhancedcavityasonevariesthelaserfrequency,usingthetheoreticalcalulationsabove.Thiscurveisthetheoreticalperformanceofagrating-enhancedcavitywith4134mtotalone-waylength,aspacingbetweenthegratingplanesof71m,agratingconstantof1633lines/mm,agratingincidentangleof54o,andalaserwavelengthof1064nm.Theplotassumestheslightlyidealizedcaseoflosslessgratings.Alsoshownistheresonancewidthofastandardcavity.Thegrating-enhancedlinewidthhasincreasedbyaboutamilliontimes.Wenoteatthispoint,however,thatthisstaticresponsetochangingfrequencyisnotequivalenttogravitationalwaveresponse,asdiscussedbelow.Realgratingswithlosseswillrequiregreaterlaserpowertoreachthesamemaximumsignalbuild-upinsidethecavitiesasthestandardcavitycase.Thereisnotheoreticallimittodiractiongratingeciency.Gratingdesigns

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Figure4{2. Comparisonofround-tripphasesforgrating-enhanced(solidline)andstandard(dashed)Fabry-Perotcavities.Frequenciesonabscissa=fflaser 34 ].Actualgratingfabricationisrapidlycatchinguptotheory.Finally,then,ifoneputsmirrorsattheplacesshowninthegure,andadjuststhelengthsofthecommonpathsandtheinter-gratingspacingDcorrectly,itshouldbepossibletoarrangeforeachcolorthattheratiooffree-spacepathtothewavelengthisthesameintegervalue.Ifthiswereso,andifthegratingshadnoothereectonthephaseofthelightwaves,thenthedeviceshownwouldbeacavityresonantforallwavelengths,a\white-light"cavity.Detailedcalculations[ 35 ]showthatthebandwidthofthiscavitywouldinfactbenite(becauseofthenon-lineardispersionofthegratings)butwouldbemanyordersofmagnitudelargerthanthebandwidthofthetypicalFabry-Perotcavity,suchastheonesinthearmsoftheLIGOdetector. 4.3 GravitationalWaveResponseintheTimeDomainIfanti-paralleldiractiongratingsindeedmakelight'sround-tripphaseshiftwithinacavityfrequency-invariant,itwouldthenbehelpfultocomputethegravitationalwaveinteractionofsuchacavityinthetimedomain.Allpreviouscalculationsconcernedonlyaneectequivalenttovaryingtheinputlaser's

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Figure4{3. Linewidthsofgrating-enhanced(solidline)andstandard(dashedline)cavitites.Thestandardcavitycurveiscutoat75kHzsoasnottoobscuretheenhancedcavityplot.Frequencies=fflaser

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Here,thezeroofthex-coordinateisthecoatedsurfaceoftheITMmirror.TheETMresidesatpositionx=L

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c)Theuppersideband:it1r2Eoi!ho cLo c)1+i!ho

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TheuppersidebandreturnstotheITM:it1r2Eoi!ho cLo c)iit1r2Eoi!ho cLo c)withanuppersignalsideband:t1r2Eo!ho ce3iL ceiL ci

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andthelower:t1r2Eo!ho ce3iL ceiL ciThedesignofthewhite-lightcavity,initsidealizedform,ensuresthecancellationofthevariationinphasewithwavelengthfortheintra-cavityeld.Thewhite-lightconditionis:d d2Lwl() @!d!=Z!+!Ltot

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forlightthathasmadeoneround-tripofthecavity.Carrierlightthathasmadenround-tripsofthegrating-enhancedcavitywillhaveuppersidebands:t1Eo!ho cei(!+)tThepositionsoftheopticalcomponentsarexedwhilethegravitationalwaveisinterpretedasaphasemodulationofthecavity'sinternallighteld.Thediractiongratings(assumedperfectlyecient)will\see"phasemodulatedlight,andwilldiracteachofthethreefrequencycomponentsintoadierentpath.Thelowersideband,carrier,anduppersidebandarenowalsoseparatedintime,emergingsequentiallyfromthecompoundmirror,thoughallwiththesamephase.Thegravitationalwave'smanipulationofspace-timefashionsfurtherfrequencysidebandsfromthecarrierlightasitreturnstothecavityinputmirror,whereitinterfereswiththeincominglaserlightbeforeitpropagatesagainthroughthecavity.Summingoverallsidebandscreatedinthecarrierfromtheinnitepastyieldsanamplitudetransferfunctionfortheuppersignalsidebandoftheform cr1r2e2i!L c ci2ei(!+)t(4.6)Letuscomparethisequationtothatofastandardcavity,whoseuppersidebandtransferfunctionhastheform: cr1r2e2i(!+)L c cih1r1r2e2i(!+)L ciei(!+)t(4.7)Becausethecavity'slengthislockedtothelaser,theterme2i!L=c=1,hence,thee2iL=cterminthedenominatorof 4.7 isthecauseofthestandardcavity'slimited

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Figure4{4. White-lightcavity'sgravitationalwaveresponse.Thesolidlineisthenormalizedintracavityintensityforawhite-lightcavity;thedashedisastandardcavity.BothcurveshaveminimaatmultiplesoftheFSR. bandwidth.Figure 4{4 comparesthesidebandsintensitieswithinstandardandperfectwhite-lightcavitieswithidenticallosslessmirrorreectivities(r21=0:995,r2=1)andnominalcavitylengths(4134m).Thegravitationalwavestrainhasmaximumamplitudeh0=1023.Thezerosofthewhite-lightcurve,duetothesincfunctioninthenumerator,correspondtofrequenciesforwhichthecavityroundtriptimecausesthesignaltobeintegratedoverafullcycle.ThiseectisnotevidentinLIGO'ssensitivitycurves,wherethearmcavitypolesdiminishthesensitivitywellbeforethesefrequencies.Weseethenthataeectoftheallegedwhite-lightcavityontheLIGObandwidthwouldbeathreeorder-of-magnitudeincrease. 4.4 RevisedTheoryWenowdescribemeasurementsofthephaseshiftoflightbysuchaparallelgratingsetwhichshowthatthegrating-compressorwhite-lightcavityconceptisalmostcompletelywrong.Instead,thepairofgratingsprovidesawavelength-dependentphaseshiftnearlycancellingthephasefromtheadditionalfree-spacepathlengthshowninFigure 4{5 [ 36 ],[ 37 ].Webegantosuspecttheaboveanalysiswhen

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Figure4{5. Apairofidentical,parallel,face-to-facediractiongratingsandtwomirrors(M1andM2)formaresonantcavity. repeatedmeasurementsofgrating-enhancedFabry-Perotcavities,MichelsonandMach-Zehnderinterferometersallhasvirtuallyidenticalfrequencybehaviortotheirgratinglesscounterparts.Forexample,measurmentsoftheresonancebandwidthofFabry-Perotcavitiescontaininghigh-eciencygratingsandconguredtobe\white-light"cavitieswereunchangedifweremovedthegratingsandrestoredthesamenominalopticalpathlength.Wefoundnoenhancementofbandwidth.Hereweshowwhytheexpectedenhancementdoesnotoccur.Moreover,wealsoshowthatthephasedependsnotonlyontheinter-gratingspacing,butalsoontheexactrelationshipbetweenthegratingfeaturesasseenbythelight[ 38 ].Thisdependenceleadstothenon-intuitiveresultthatthephaseismodulatedstronglyifoneofthegratingsistranslatedparalleltoitsface,eventhoughtheopticalpathsinFigure 4{5 arewhollyunaected. 4.5 Plane-waveTreatmentofaGratingCompressorTheerrorlieswithaninappropriatemixofgeometricalandphysicaloptics.Considertheeectoftheparallelgratingpaironinniteplanewaves.InFigure 4{5 ,parallelreectivegratingsarelocatedinthey=0andy=Dplanes.Wewillcalculatetheelectriceldatthetwogratingsandinaplanenormaltotheoutgoinglight(wheretheright-handmirrorinFigure 4{5 islocated).

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Thelighteldimpingingfromtheleftontherstgratingis 4{5 ,theelectriceldwillbe

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WemustnowanalyzehowthephaseoftheeldatM2changeswithlightfrequency.Thephase(!;x;y)is =! c[xsin(yD)cos+Dcos]gx0(4.13) cL(!)g(Dtan+x0);(4.14)whereL(!)isthetotal,wavelength-dependent,geometricpathfromtherstgrating(attheorigin)totheendmirror[ 35 ].Wecomputethedispersion@=@!,usingthegratingequationtoeliminate@=@!,andnd cos+tansin=L(!) 4.15 makesitclearthatthevariationofphasewithfrequencycannotbesettozero.TheearliercalculationsbasedonthegratingequationaloneincludedonlythegeometricpathlengthcontributioninEq. 4.15 ,leadingtothe(incorrect)predictionthat@=@!couldbecomezero,thusallowingforthepossibilityofawhitelightcavity[ 35 ].Missingfromreference[ 35 ]wasthesecondtermintheright-handsideofEq. 4.14 ,presentduetotheposition-dependentphaseshiftlightreceivesuponreectionfrom(ortransmissionthrough)thegrating.TheresultofEq. 4.15 isfamiliartoshort-pulselaserphysicistsasthegroupdelay[ 39 ].Theexactformoftheadditionalphaseshiftisrarelyaconcern,asthepulsecompressordoesnotdependuponabsolutephases.Toourknowledge,directexperimentalvericationofthephaseshift'sformhasneverbeenpublished.ThephaseshiftofEq. 4.13 maybeexpressedinaparticularlyilluminatingwayas (!)=! c[Lo+D(cos+cos)]gx0(4.16)whereL0istheperpendiculardistancefromtheorigintotheplaneofM2,denedbyL0=xsinycos.Analyzingthegratingcompressorwithplanewavesrevealstheoriginandsignicanceoftheposition-dependentphaseshiftonreectionfrom

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Figure4{6. ThesecondofapairofparallelgratingswithinonearmofaMach-Zehnderinterferometerisplacedonanx-ytranslationstage.MichelsoninterferometerswithHe-Nelasersmonitorthemotionofthestage. thegratings.Wemaynowresumecalculationswiththegeometricalopticalpath,solongaswedonotneglecttheadditionalphaseassociatedwiththegratings.Thistheorymakesaveryspecicprediction,whichmaybeexperimentallyconrmed,abouthowtheone-wayphaseshiftdependsonthedistanceDbetweengratingsandthespatialosetx0betweengratingproles. 4.6 TestsofCorrectedTheory 4.6.1 ExperimentalSetupTotestthatthephaseshiftdoeshavethespecicformofEq. 4.16 ,weincorporatedapairofgratingsintoonearmofaMach-Zehnderinterferometer,asshowninFigure 4{6 .Weusedreectivegratingswith1500grooves/mmandadesigninputangle=42(sothat=68)Wealsousedaninputangleof=50(=57:3)insomeofourtrials.Thegratingshadahigheciency,with94{96%oftheincidentlightdiractedintotherst-orderbyeachofthetwo.Weplacedthesecondgratingofourgratingpaironatwo-axistranslationstage.Thisstageallowedustovarytheparametersx0andDofthegratingpair.Wealigned

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thegratingfacewithoneofthetwoorthogonalaxesofthestage,whichwewillcallx0andy0,aswellaswaspossiblewithanakedhumaneye.Asmallmisalignmentangleinevitablyremainsbetweenthegratingaxes(x,y)andthestageaxes(x0,y0).Theaxesarerelatedbybx0=bxcos+bysin.Assumingtheconstructionofthetranslationstageisbetterthanourabilitytoplacethegrating,wealsohaveby0=bxsin+bycos.Asthestagemovesalongthex0direction,itwillproduceacombinationoftheeectsonphaseduetothephase'sx0andDdependence;however,becauseissmall,theinuenceofx0,withaperiodequaltothegratingperiod,willdominate.Theconverseistruewhenonemovesthestagealongy0.Tocalibratethedisplacementofthetranslationstage,weattachedtoittheendmirrorofasimpleMichelsoninterferometerilluminatedbyahelium-neonlaser.Infact,therearetwomirrors(andtwointerferometers)setperpendiculartothetwomotionsofthestage.Toensurethatperpendicularity,wemovethestageintheorthogonaldirection,sothatthemirrorslewscrabwiseacrosstheHe-Nebeam,andadjustitsanglerelativetothestageuntilwereducethenumberofoutputintensityfringestoaminimum.Again,thistechniquereliesupongoodinherentperpendicularityinthestage'scrossedaxes.Wheneverthestagemoves,wemonitortheoutputintensityofbothinterferometers.WequantifythestagemotionbycountingtheHe-Nefringes.TheinputtotheMach-Zehnderinterferometerisa1064nm-wavelengthgrating-stabilizeddiodelaser.Forgoodcontrast,thephysicallengthsofthetwoarmsarenearlyequal.Wemovethesecondgratingalongthex0andy0axesandobservetheintensityfringesoftheinfraredinterferometerandcomparewiththeory.

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4.6.2 ExperimentalResultsWhilethelightinputangleandthegratingperioddareknown,remainsasattingparameter.TheoutputintensityoftheMach-Zehnderistto (x0)=2 (cos+cos)x0sin+2 dx0cos;(4.18)forthex0motion,orto (y0)=2 (cos+cos)y0cos2 dy0sin;(4.20)fortheorthogonaldirection.ThequantitiesA,B,andCareratherunimportantttingparameters;theperiodoftheoutputfringesdeterminedbyiskey.Wemakealeast-squaretofthetheorytoourdatabyadjustingA,B,C,and.Figure 4{7 showsexamplesoftypicalresultsforatrialwith=50.Figure 4{7 (top)showstheinterferenceseenformovementalongthex0-direction,i.e.,whenthegratingmovesparalleltoitsface.Thismotiongivesstrongfringes;themeasuredfringecontrastisinthe92{94%range.Now,motionparalleltothegratingfacehasnoeectonthegeometricpathlengthsinsidetheinterferometer.Thus,ourinitialexpectation(basedonthegeometricpathlength)wasthatthelightphasewouldbeunaectedbythismotion.Incontrasttothisexpectation,thephaseofthelightgoesthroughafullcycleasthegratingistranslatedbyanamountd.InFigure 4{7 (bottom),weshowtheinterferencesignalobservedformotionalongy0.Wealsoplot,inadditiontothepredictedoutputfromthetheoryabove,theoutputintensitythatwouldbeobservedifonlythegeometricpathlengthwere

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Figure4{7. (Top)Measured(crosses)andtheoretical(solidline)dataformotionparalleltothegratingface.Thelightisincidentat50.(Bottom)Thecrossesshowthedataformotionperpendiculartothegratingface.Ofthetwocalculations,theresultsfavortheplane-wavetreatment,withtheadditionalphaseshiftonreection(solidline)overonebasedongeometricpathlengthalone(dottedline).

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Figure4{8. Misalignmentangleasdeterminedfromtsofmeasuredinterferencedatatotheory.Themeaningofthesymbolsisasfollows.Crosses:motionalongx0with=50;Circles:motionalongy0with=50;Asterisks:motionalongx0with=42;x's:motionalongy0with=42.ThedatainFigure 4{7 isfromtrials2and8. changedbythegratingmotion.Useofgeometricpathalonepredictsaperiodfortheinterferencepatternthatisdierentfromthemeasuredone,whereasthetheorythatincorporatestheposition-dependentphaseshiftpredictstheperiodthatwemeasured.Theoutcomemakesclearthatthesetupcannotbeunderstoodwithonlythediractionangleandthegeometricpathlength.Theadditionalposition-dependentphaseshiftisreal[ 40 ].Figure 4{8 showsanindicationoftheagreementbetweenexperimentandthetheorypresentedhere.Init,weplotthemisalignmentangleofthegratingforanumberoftrials.Elevenofthetwenty-threemeasurementsarederivedfromgratingmotionparalleltoitsface,andtwelvefromperpendicularmotion.Ineverycase,weobtainedhigh-contrastfringeswithagreementwiththeorycomparabletowhatisshowninFigure 4{7 .Theerrorbarsoneachdatumreectuncertaintyin,d,,,andthemotionofthestage.Clustersofspecicvaluesforindicatethesystematicerrorinthealignmentofthegratingonthestage,buttheoverallerrorsareverysmall.ThequalityofthettothemeasuredinterferencepatternisevidentinFigure 4{7 andinthesmallvaluesforthemisalignmentanglesin

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Figure 4{8 ,averaging=0:030:12,areasonablevalueforalignmentbyhumaneye.Thephaseoflightreectedbyortransmittedthroughadiractiongratingcannotbededucedfromthegratingequationalone.Thatequationomitsthecuriousresult,derivedabove,thattheabsolutephaseisproportionaltothedistancealongthegratingfaceatwhichthelightstrikes.Indeed,theatgratingsbehaveasmirrorstiltedatanglestilt=sin1(m=2d)relativetothex-axisshowninFigure 4{5 .Weconrmedthistheorybytestingthedependenceoflightphaseonthepositionofthegrating.Foragrating-compressorsetup,wefoundgoodagreementbetweenthistheoryandthechangeoflightphaseasthemirrormovedbothparallelandperpendiculartoitsface.Ourresultshowsthatwhite-lightcavitiescannotbebuiltfromgratingpairs.Infact,onemighthaveconjecturedthatcausalityshouldpreventwhite-lightcavitiesfrombeingbuiltinamuchwiderclassofnon-dissipativesystems|notjustgratingpairs.Indeed,wehavefoundthatapairofprismshasasimilareecttothegratingsonthephaseoflightpassingthroughthem.Finally,wenotethatthephaseeectdiscussedhereisnotuniquetothegratingpairandwouldariseinanexperimentutilizingasinglegrating.Inourarrangement,therstgratingisxedandservestopreservethebeamwidthandtokeeptheangleofthelightleavingthesecondgratingconstantaswavelengthisadjusted.Otherwise,itisequivalenttoamirror.Exceptforalossofcontrast,weexpectthatthedataofFigure 4{7 wouldbeidenticaliftherstgratingwerereplacedbyanappropriatelyorientedmirror. 4.6.3 AOMsInterestinglyenough,itseemsacluetothemissingphasemysterywasunderournoses,indeed,inouropticalcabinets,allalong.Ifonewishestomakefurthermeasurementsoftheeectofamovinggratingondiractedlightphase,anacousto-opticmodulator(AOM)isagoodcandidate.InanAOMoneapplies

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anacousticwaveformtoacrystal,sendingpressurewavesmovingthroughthematerialatspeedsontheorderofthousandsofmeterspersecond.Lightdiractedothismovinggratinginsidethecrystalshouldexhibitthesamephaseshiftbehaviorfoundinourslow-movinggratingexperiments.Howwouldoneseethephaseshifteect?Itwouldfrequency-modulatethediractedlight.InfactthisDoppler-shifteddiractionfrequencyshiftisawell-knownphenomenon,butisrarelyifeverconnectedinliteratureordiscussiontogratingcompressors! 4.7 AlternativeSolutionsTheextraphaseshiftduetotransversepositionalongthegratingatrstappearstocancelthelinewidth-broadeningperfectlybycoincidence.However,similarexperimentswithprismpairs,wherenosuchhiddenphasesexist,yieldedthesameresult.Webelievethisisamanifestationofnature'sjealousguardianshipofcausalityandspeedlimits.Ourinitialanalysisconfusedgroupdelaywithphasedelay.Wespeculatethattheonlywayawhiteorevenbroadenedcavityispossibleiswhenthereisawaytodelaythelightsignicantlywithresonantabsorptionoragainmedium.Thereareseveralwaysinwhichthisdelaymodicationmaybeaccomplished.Therequiredanomalousdispersionisfoundinthecenterofatomicabsorptionlines,buttheabsorptionwouldgenerallysuppressanygravitationalwave-inducedsignals.Wichtetal.[ 41 ]studiedopticallypumpedatomicresonancesystemsandprovedthatonecouldhaveanappropriateanomalousdispersionforawhite-lightcavityatapointofvanishingabsorptionandopticalgain. 4.8 IfAlternativesSucceedThenaldesignforafullwhite-lightinterferometerwillincludetwowhite-lightcavitiesinitsarms.Becausewenolongerneedworryaboutlimitingourbandwidthbyincreasingthearmcavitymirrorreectivities,thesemaybematchedtothelossesinthebandwidth-enhancingblackboxforoptimalcoupling.Thuswemayforegobothpower-andsignal-recyclingmirrorsatthesymmetric

PAGE 90

andanti-symmetricports,respectively.Thishastheaddedadvantageofremovingpowerfromthethicksubstratesofthebeamsplitterandinputmirrors,reducingthermalloadingproblems.

PAGE 91

42 ].Asforwhite-lightcavities,itseemstherewillbenosuchthingasafreelunch.Despitea(too)promisingtheoreticalcalculation,line-widthbroadeningremainedelusive,andwerealizedwehadrepeatedasurprisinglycommonmistakeinoptics.Thesensitivitytheoremisnottobetakenlightly.Ifwecouldyetachievethenecessarydispersionwithatomicresonances,westillhaveafarmorecomplicatedsystemthanthefaileddiraction-gratingsetup.Nonetheless,therewardsofabroad-bandFabry-Perotcavitywouldbeatremendousboontogravitationalwavedetection.Weretainaniotaofhopeforthistechnique. 80

PAGE 92

A.1 Fabry-PerotCavityUsingtheconventionthateldspassingthroughapartiallytransmissiveopticreceivea90ophaseshiftrelativetoreectedelds,theamplitudesimmediatelytotherightoftherstopticisEa=it1Ein+r1EbEb=r2e2ikLEa;wherekisthewavenumberofthelight.Combiningtheseequations,onendsthetotalsteady-stateintracavityeld: A.2 Three-MirrorCoupledCavityUnderstandingofthethree-mirrorcoupledcavityaidsinasimplifyingperceptionofanymorecomplicatedopticalsystem,astheseusuallybehaveinasimilarmanner.Athree-mirrornestedcavityisdeciptedinFigure A.2 ,with 81

PAGE 93

FigureA{1. ElectriceldsinaFabry-Perotcavity FigureA{2. Electriceldsina3-mirrorcoupledFabry-Perotcavity

PAGE 94

mirrorparametersasindicated.Havingfoundthetransferfunctionofeldsreectedfromasinglecavityabove,wecantreatthemiddleandrightmostopticasasinglemirrorwithacomplexreectivitygivenbyEq. A.3 .Forexample,theintracavityeldbetweentheleftandmiddlemirroristhen

PAGE 95

mKwithD=0:1mm,Lfiber=0:2m,T1=40K,andT2=10K,gives[ 43 ]P=1:65W:

PAGE 96

TableC{1. ValuesofelementsusedintheMZfeedbackcircuit elementvalueR11kR21kR31kR41kR5100kR61kR71kR81kR91kR102kR111kR121kR13220kC10.1F 85

PAGE 97

DiagramofanalogciruitusedforMZfeedbackcontrol

PAGE 98

[1] A.Einstein,\Relativity:TheSpecialandtheGeneralTheory,"R.Lawson,trans.,(CrownTradePaperbacks,NewYork),1961. 1.1 [2] C.W.Misner,K.S.Thorne,andJ.A.Wheeler,Gravitation(W.H.FreemanandCo.,SanFrancisco),1979. 1.1 [3] J.H.Taylor,L.A.Fowler,andJ.M.Weisberg,\MeasurementsofGeneralRelativisticEectsinBinaryPulsarPSR1913+16,"Nature,vol.277,pp.437,1979. 1.1 [4] J.Weber,\Detectionandgenerationofgravitationalwaves,"Phys.Rev.,vol.117,p.306,1960. 1.2.1 [5] L.Ju,D.G.Blair,andC.Zhau,\DetectionofGravitationalWaves,"Rep.Prog.Phys.,vol.63,pp.1317{1427,2000. 1.2.1 [6] O.D.Aguiar,L.A.Andrade,L.CamargoFilho,C.A.Costa,J.C.N.deArajo,E.C.deReyNeto,S.T.deSouza,A.CFauth,C.Frajuca,G.Frossati,S.R.Furtado,V.G.S.Furtado,N.S.Magalhes,R.M.MarinhoJr.,E.S.Matos,M.T.Meliani,J.L.Melo,O.D.Miranda,N.F.OliveiraJr.,K.L.Ribeiro,K.B.M.Salles,C.Stellati,andW.F.VellosoJr.,\ThestatusoftheBrazilliansphericaldetector,"Class.QuantumGrav.,vol.19,pp.1949{1953,2002. 1.2.1 [7] A.deWaard,L.Gottardi,M.Bassan,E.Coccia,V.Fafone,J.Flokstra,A.Karbalai-Sadegh,Y.Minenkov,A.Moleti,G.V.Pallotino,M.Podt,B.J.Pors,W.Reincke,A.Rocchi,A.Shumack,S.Srinivas,M.Visco,andG.Frossati,\CoolingdownMiniGRAILtomilli-Kelvintemperatures,"Class.QuantumGrav.,vol.21,pp.S465{S471,2004. 1.2.1 [8] A.Abramovici,W.Althouse,R.Drever,Y.Gursel,S.Kawamura,F.Raab,D.Shoemaker,L.Sievers,R.Spero,K.Thorne,R.Vogt,R.Weiss,S.Whitcomb,andM.Zucker,\LIGO:TheLaserInterferomterGravitational-waveObservatory',"Science,vol.256,pp.325{333,1992. 1.2.2 [9] C.Bradaschia,R.DelFabbro,A.DiVirgilio,A.Giazotto,H.Kautzky,V.Montelatici,D.Passuello,A.Brillet,O.Cregut,P.Hello,C.N.Man,P.T.Manh,A.Marraud,D.Shoemaker,J.Y.Vinet,F.Barone,L.DiFiore,L.Milano,G.Russo,J.M.Aguirregabiria,H.bel,J.P.Duruisseau,G.LeDenmat,P.Tourrenc,M.Capozzi,M.Longo,M.Lops,I.Pinto,G.Rotoli,T.Damour,S.Bonazzola,J.A.Marck,Y.Gourghoulon,L.E.Holloway, 87

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F.Fuligni,V.Iafolla,andG.Natale,\TheVirgoproject:awidebandantennaforgravitationalwavedetection,"Nucl.InstrumMethodsPhys.Res.A,vol.289,pp.518{525,1990. 1.2.2 1 [10] K.Tsubono,\300-mlaserinterferometergravitationalwavedetector(TAMA300)inJapan,"inFirstEduardoAmaldiConferenceonGravitationalWaveExperiments,E.Coccia,G.Pizzella,andF.Ronga,eds.(WorldScientic,Singapore),pp.112{114,1995. 1.2.2 [11] K.Danzmann,H.Lueck,A.Ruediger,R.Shilling,M.Schrempel,W.Winkler,J.Hough,G.P.Newton,N.A.Robertson,H.Ward,A.M.Campbell,J.E.Logan,D.I.Robertson,K.A.Strain,J.R.J.Bennett,V.Kose,M.Kuehne,B.F.Schutz,D.Nicholson,J.Shuttleworth,H.WellingP.Aufmuth,R.Rinkele,A.Tuennermann,andB.Wilke,\GEO600-a600-mlaserinterferometricgravitationalwaveantenna,"inFirstEd-uardoAmaldiConferenceonGravitationalWaveExperiemnts,E.Coccia,G.Pizzella,andF.Ronga,eds.,(WorldScientic,Singapore),pp.100{111,1995. 1.2.2 [12] J.Mizuno,K.A.Strain,P.G.Nelson,J.M.Chen,R.Schilling,A.Ruediger,W.Winkler,andK.Danzmann,\ResonantSidebandExtraction:Anewcongurationforinterferometricgravitationalwavedetectors,"Phys.Lett.A,vol.175,pp.273{276,1993. 1.2.2 4.1 [13] V.Kalogera,C.Kim,D.R.Lorimer,M.Burgay,N.D'Amico,A.Possenti,R.N.Manchester,A.G.Lyne,B.C.Joshi,M.A.McLaughlin,M.Kramer,J.M.Sarkissian,andF.Camilo,\TheCosmicCoalescenceRatesforDoubleNeutronStarBinaries,"Astrophys.J.,vol.601,pp.L179{L182,2004. 1.2.2 [14] J.Mizuno,\ComparisonofOpticalCongurationsforLaser-InterferometricGravitationalWaveDetectors,"PhDthesis,MaxPlanckInstituteforQuantumOptics,1995. 1.2.2 1.5 4 [15] V.B.Braginsky,M.L.Gorodetsky,andS.P.Vyatchanin,\Thermodynamicaluctuationsandphoto-thermalshotnoiseingravitationalwaveantennae,"Phys.Lett.A,vol.264,pp.1{10,1999. 1.3.2 [16] E.D'Ambrosio,\Nonsphericalmirrorstoreducethermoelasticnoiseinadvancedgravitationalwaveinterferometers,"LIGO-P030015-00-D,2003. 1.3.2 [17] C.Caves,\Quantum-mechanicalnoiseinaninterferometer,"Phys.Rev.D,vol.23,no.8,pp.1693{1708,Apr.1981. 1.3.3 [18] A.BuonannoandY.Chen,\OpticalnoisecorrelationsandbeatingthestandardquantumlimitinLIGOII,"Phys.Rev.Lett.,vol.,no.,pp.{,July2000. 1.3.3

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[35] S.Wise,G.Mueller,D.Reitze,D.B.Tanner,andB.F.Whiting,\Linewidth-broadenedFabry-Perotcavitieswithinfuturegravitationalwavedetectors,"Class.QuantumGrav.,vol.21,pp.S1031{S1036,2004. 4.2.2 4.5 [36] S.G.Rautian,\OntheTheoryofInterferometerswithDiractionGratings,"Opt.andSpectroscopy,vol.93,no.6,pp.934{940,2002. 4.4 [37] E.B.Treacy,\OpticalPulseCompressionwithDiractionGratings,"IEEEJ.ofQuantumElect.,vol.QE-5,no.9,pp.454{458,Sep.1969. 4.4 [38] Y.Chen,\TreatmentofGratingPairsUsingPlane-waveApproximationNo.LIGO-G040194-00-Z,"LSCConference,Mar,2004. 4.4 [39] S.DBrorsonandH.A.Haus,\Diractiongratingsandgeometricaloptics,"J.Opt.Soc.Am.B,vol.5,no.2,pp.247{248,Feb.1988. 4.5 [40] S.Wise,V.Quetschke,A.J.Deshpande,G.Mueller,D.H.Reitze,D.B.Tanner,B.F.Whiting,Y.Chen,A.Tuennermann,E.Kley,andT.Clausnitzer,\PhaseEectsintheDirationofLight:BeyondtheGratingEquation,"Phys.Rev.Let.,vol.95,no.1,pp.13901{13904,July2005. 4.6.2 [41] A.Wicht,K.Danzmann,M.Fleischhauer,M.Scully,G.Mueller,R.-H.Rinkle,\White-lightcavities,atomicphasecoherence,andgravitationalwavedetectors,"OpticsComm.,vol.134,pp.431{439,Jan.1997. 4.7 [42] T.Uchiyama,K.Kuroda,M.Ohashi,S.Miyoki,H.Ishitsuka,K.Yamamoto,H.Hayakawa,K.Kasahara,M-K.Fujimoto,S.Kawamura,R.Takahashi,T.Yamazaki,K.Arai,D.Tatsumi,A.Ueda,M.Fukushima,S.Sato,Y.Tsunesada,Zong-HongZhu,T.Shintomi,A.Yamamoto,T.Suzuki,Y.Saito,T.Haruyama,N.Sato,Y.Higashi,T.Tomaru,K.Tsubono,M.Ando,K.Numata,Y.Aso,K-I.Ueda,H.Yoneda,K.Nakagawa,M.Musha,N.Mio,S.Moriwaki,K.Somiya,A.Araya,A.Takamori,N.Kanda,S.Telada,H.Tagoshi,T.Nakamura,M.Sasaki,T.Tanaka,K-I.Ohara,H.Takahashi,S.Nagano,O.Miyakawa,andM.E.Tobar,\Presentstatusoflarge-scalecryogenicgravitationalwavetelescope,"Class.QuantumGrav.,vol.21,pp.S1161{S1172,2004. 5 [43] V.Quetschke,S.Wise,G.Mueller,D.Reitze,andD.B.Tanner,\Onthepossibilityofcoolingtestmassesthroughevanescentcoupling,"LIGO-G040406-00-Z,Aug.2004. B

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StacyWisewasborninMaryland,USAin1977toachemistandagradeschoolteacher,whonurturedhercuriosityabouttheworld.Inadolescenceshebeganhangingoutinthesciencesectionofthepubliclibrary,ndingbooksonthequantum-mechanicaldescriptionofmatterandthenatureoftime.SheattendedtheUniversityofMarylandBaltimoreCounty,throughwhichsheinternedatnearbyNASAGoddardSpaceFlightCenter.AtNASA,shestudiedatmosphericphysicsandopticswiththeRamanLIDARtelescopegroupheadedbyDr.DavidWhiteman.AftergraduatingmagnacumlaudefromUMBCin1999withaB.S.inPhysics,shetookayeartotravelandexploremanuallabor,workingasalandscaperandinalocalciderfactory.Shereturnedtoacademiclifeinthefallof2000whenshematriculatedtotheUniversityofFlorida'sgraduateprograminPhysics.Shewasattractedtotheschoolnotonlyforthe6monthsofhotandstickyweather,butalsoforUF'sparticipationinthesearchforgravitationalwaveswiththeLIGOproject.Dr.DavidTannerbecameheradvisor,andshesoonedlearnedthetrialsaswellasjoysofexperimentalresearch.ShewasdeployedtoGlasgow,UKfor6monthsassistingLIGOcolleagesinGEO600in2002,andwasalsoprivilegedtoparticipateinnumerousconferencesanddetectoroperations.Withgraduationinsight,Stacyhopestocontinuelearningaboutthenatureoftheuniverse.Inparticular,shehopetogaininsightintothemind-matterenigma:howdoesamonisticuniversecontainingonlymatterandenergywhichobeythelawsofphysicscreateaphenomenonlikesubjectiveconsciousness?Shefeelsthathumanbrainsarethemostmysteriousthingstheuniversehasproduced. 91


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SENSITIVITY ENHANCEMENT IN FUTURE INTERFEROMETRIC
GRAVITATIONAL WAVE DETECTORS















By

STACY M. WISE


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

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Stacy M. Wise


































I dedicate this work to my family.















ACKNOWLEDGMENTS

Foremost I would like to thank my family for their unconditional support.

I would like to thank Dr. David Tanner and Dr. David Reitze for kindness and

numerous helpful discussions. Thanks go to Dr. Guido Mueller, for imaginative

ideas, helpful explanations, and for regular and sound figurative posterior-kicking,

as well as ego-deflating. Thanks also go to Dr. Volker Quetschke, for cheerfully

answering probably thousands of questions, and to Dr. Bernard Whiting, for very

thorough reviewing of my calculations and repeatedly showing me the error of

my mathematical v--i,- (and some very excellent contra dancing). For friendship

that was worth every bit of occasional distraction from the academic task at

hand, I must thank Joe Gleason, Amruta Deshpande, and Malik Rakhmanov.

Some distractions I've had even longer- Rachel Haimowitz, Ahmed Rashed, and

James "Tv.. Ii- Pence helped me get here. I could .,li.v ,i-l count on finding Dea's

Half-Fast Jam folks or 50 Miles of Elbow Room ready to pl i, soulful tunes on

fiddle and ,-,0io on a we!.di. night, fulfilling a musical need.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

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

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

A B ST R A CT . . . . . . . . . x

CHAPTER

1 INTRODUCTION TO GRAVITATIONAL WAVE EXPERIMENTS 1

1.1 General Relativity ........................... 1
1.2 Direct Observation of Gravitational Waves ............. 3
1.2.1 Resonant Bar Detectors .......... .......... 4
1.2.2 Interferometric Detectors ................... 5
1.3 Interferometer Noise .......... ............... 10
1.3.1 Seismic Noise .......... ....... ....... 10
1.3.2 Thermal Noise ........... ............. 10
1.3.3 Quantum Noise ............... .. .. .. 11
1.4 Third-Generation Detectors and Beyond . . ..... 12
1.5 In This Work ............... .......... .. 14

2 PARALLEL PHASE MODULATION FOR ADVANCED LIGO ..... 15

2.1 Interferometer Control .................. .... .. 15
2.1.1 Phase Modulation .................. .. 16
2.1.2 Pound-Drever-Hall Cavity Locking . . ..... 17
2.1.3 Advanced LIGO .................. ... .. 19
2.1.4 Electro-optic Modulators .. ................. .. 21
2.2 Parallel Phase Modulation Noise .................. .. 24
2.2.1 Mirror Motions .................. .. .. 25
2.2.2 Fluctuations at Non-zero Frequencies . . 33
2.3 Mach-Zehnder Parallel Phase Modulation Prototype . ... 34

3 NEAR-FIELD EFFECT HEAT TRANSFER ENHANCEMENT ..... 42

3.1 Near-field Theory .......... . . ... 42
3.2 Fluctuational Electrodynamics Theory . . ..... 44
3.2.1 Green's Function Method .... . . 45
3.2.2 Fluctuation Dissipation Theorem. . . ...... 47









3.3 Numerical Simulations of Evanescent Coupling .. .........
3.3.1 Effect of Dielectric Function .. ..............
3.3.2 Effect of L ,i ,. I M edia .. .................
3.4 Noise Coupling Due to Fluctational Electrodynamics ....
3.4.1 Van der Waals and Casimir Forces .. ............
3.5 Implementation in an Interferometer .. .............
3.5.1 Configurations .. .. .. ... .. .. .. ... .. ..
3.5.2 Stability Requirements for Cold Mass .. ..........
3.6 Experimental Proof .. .....................

4 WHITE-LIGHT INTERFEROMETRY .. ................

4.1 The Sensitivity Theorem Quandary .. ..............
4.2 The Linewidth-Enhanced Cavity .. ................
4.2.1 The Nature of the Gain-bandwidth Dilemma........


4.2.2 "White-l;III Cavities


4.3 Gravitational Wave Response in the
4.4 Revised Theory .. .........
4.5 Plane-wave Treatment of a Grating
4.6 Tests of Corrected Theory .....
4.6.1 Experimental Setup .....
4.6.2 Experimental Results .
4.6.3 AOM s .. .........
4.7 Alternative Solutions .. ......
4.8 If Alternatives Succeed .......


Time Domain.

Compressor .


5 CONCLU SION . . . . . . . .

APPENDIX

A OPTICAL TRANSFER FUNCTIONS .. ...............

A. 1 Fabry-Perot Cavity .. .....................
A.2 Three-Mirror Coupled Cavity .. ................

B HEAT TRANSFER THROUGH FIBERS .. ..............

C CIRCUIT DIAGRAM .. .......................

REFEREN CES . . . . . . . . .

BIOGRAPHICAL SKETCH .......... ...............


III















LIST OF TABLES
Table page

2-1 Advanced LIGO degree of freedom in terms of length in Figure 2-3 .20

2-2 Control matrix for 40m prototype without second-order sidebands [20] 21

2-3 Control matrix for 40m prototype with second-order sidebands [20] 22

C-1 Values of elements used in the MZ feedback circuit . .... 85















LIST OF FIGURES
Figure page

1-1 The effect of a plus-polarization gravitational wave propagating into
page on a string of test masses .................. 4

1-2 The effect of a cross-polarization gravitational wave propagating into
page on a string of test masses .................. 4

1-3 The power-recycled, cavity-enhanced optical configuration . 7

1-4 The dual-recycled optical configuration ............... 8

1-5 Noise budget for Advanced LIGO ................ 13

2-1 Schematic for Pound-Drever-Hall Fabry-Perot cavity locking ..... .17

2-2 Fabry-Perot intracavity intensity and Pound-Drever-Hall error signal. 19

2-3 Lengths in Advanced LIGO that require feedback control ...... ..20

2-4 Serial phase modulation of IFO input light .... . 23

2-5 Parallel phase modulation of IFO input light ..... . 23

2-6 Possible noise motion in the Mach-Zehnder optics and resultant noise
on Advanced LIGO input light. C: carrier, SB: sideband. . 26

2-7 Phasor diagram of Mach-Zehnder common-mode fluctuation's effect
on recombined carrier and one pair of sidebands. . .... 26

2-8 Phasor diagram of Mach-Zehnder differential-mode fluctuation's effect
on recombined carrier and one pair of sidebands. . .... 28

2-9 Ratio of frequency noise to relative phase noise transfer functions to
Fabry-Perot common mode error signal . . ....... 30

2-10 Permissible residual differential displacement of Mach-Zehnder arm
lengths. .................. .............. .. 31

2-11 Layout of the prototype MZ phase modulation scheme . ... 35

2-12 Spectrum of light transmitted by parallel-phase modulating Mach-Zehnder
interferometer .................. ........... .. 36









2-13 Expanded view of MZ output spectrum. Mixed sidebands are absent-
-the 43.5 MHz peaks are due to higher-order spatial modes. . 37

2-14 Comparison of MZ error-point noise with required stability for free-running
and closed-loop operation .................. .. 40

2-15 Out-of-loop measurement of residual MZ differential displacement noise 40

2-16 Attenuation transfer function of MZ feedback electronics . 41

3-1 Comparison of quantum-mechanical sensitivity limit to test mass internal
thermal noise limit for Advanced LIGO [22] . 43

3-2 Geometry of dielectic media for near-field effect calculation- two semi-infinite
slabs separated by a vacuum gap with width d. ......... ..45

3-3 Power flux per unit area as a function of vacuum gap separating two
semi-infinite chromium masses at 40 and 10 K . . ... 51

3-4 Power flux per unit area as a function of vacuum gap separating two
semi-infinite doped silicon masses at 40 and 10 K for various dopant
concentrations .................. .......... .. 52

3-5 Power flux per unit area as a function of doped silicon metalicity at a
1.4 micron separation. .................. ..... 53

3-6 Possible configurations for evanescent cooling of a mirror ...... ..54

4-1 Grating-enhanced cavity .................. ..... .. 60

4-2 Comparison of round-trip phases for grating-enhanced and standard
Fabry-Perot cavities .................. ....... .. 61

4-3 Linewidths of grating-enhanced and standard cavities . ... 62

4-4 White-light cavity gravitational wave response . ..... 68

4-5 A pair of identical, parallel, face-to-face diffraction gratings and two
mirrors ( ll and M2) form a resonant cavity. .......... ..69

4-6 Experimental design to test grating phase . . ....... 72

4-7 Measured and theoretical data for motion parallel to the grating face 75

4-8 Mislignment angle as determined from fits of measured data . 76

A-1 Electric fields in a Fabry-Perot cavity ................. .. 82

A-2 Electric fields in a 3-mirror coupled Fabry-Perot cavity . ... 82

C-1 Diagram of analog ciruit used for MZ feedback control . ... 86















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

SENSITIVITY ENHANCEMENT IN FUTURE INTERFEROMETRIC
GRAVITATIONAL WAVE DETECTORS

By

Stacy M. Wise

li ,v 2006

C'! In': David B. Tanner
Major Department: Physics

Mankind is poised to directly detect gravitational waves for the first time. To

improve event rates of sources these detectors will be able to measure, we will need

to increase the signal-to-noise ratio of the current first generation instruments in

upgraded versions. Three techniques for improving interferometer sensitivity to

gravitational-wave strains are explored in this work. The first is a parallel phase

modulation technique for adding two radio-frequency sidebands to the laser input

of an interferometric detector like LIGO without sideband cross-products. This

allows for a diagonal control matrix in the interferometer topology planned for

Advanced LIGO. The stability requirements for a Mach-Zehnder interferometer

with an electro-optic phase modulator in each arm is derived. We construct a

prototype parallel phase modulator to test the viability of these requirements,

with positive result. The second method is cryogenic cooling of optics within an

interferometer by evanescent field coupling. We quantify the enhanced heat-transfer

with fluctuational electrodynamics theory for a variety of dielectric materials and

evaluate the effect of stratified media on thermal coupling. Noise force couplings

and the implications for an advanced interferometer are discussed. In the last









chapter we investigate whether a diffractive element placed in a Fabry-Perot cavity

can increase the cavity bandwidth without loss of peak signal build-up. We study

the interaction of such an optical resonator with a gravitational wave. A diffraction

grating technique is tested and ultimately proven unable to enhance the bandwidth

of gravitational wave detectors.















CHAPTER 1
INTRODUCTION TO GRAVITATIONAL WAVE EXPERIMENTS

1.1 General Relativity

Just when the world was beginning to think nothing else remained for

physicists but to measure the physical constants of the universe to the next

decimal place of precision, the early 20th century brought the wonders of quantum

mechanics and relativity. In 1916 Albert Einstein revolutionized our understanding

of the universe with his general theory of relativity (GR) [1]. Initially seeking

a frame-independent description of the laws of motion, he came to the startling

conclusion that matter and energy warp space and time. Non-Euclidean geometries

arise in this four-dimensional space-time "< ,i n II i, which, in turn, directs the

motion of mass/energy. In this way, we see gravitation as a property of space-time

itself. When velocities are small compared to the speed of light, general relativity's

consequences are not much different from Issac Newton's description of gravity.

In regions of strong curvature or high velocity, however, general relativity is

markedly and measurably different. The very first test of the new hypothesis was

the bending of light by our Sun's gravitational field. Tod i-, GR is part of everydi

life, used for example in the timing of the Global Positioning System satellites.

Although most predictions of Einstein's theory have now been satisfactorily

verified by experiment, direct confirmation of one important prediction remains

outstanding-that of gravitational waves.

One of the primary motivations for pursuing theoretical alternatives to the

pre-20th century theory was that Newton's gravity allowed for instantaneous

communication. If a mass moved to a new location in space, the effect on a test

mass would be immediate, regardless of the distance separating these masses.









Information encoded in the motion of an object on one side of the Milky Way

Galaxy could theoretically be read (assuming a sensitive enough instrument exists,

an issue we'll get to later) on the other side with no time delay. Einstein insists

in the special theory of relativity that no information can travel faster than the

speed of light [1]. Like waves spreading outward from a hand sweeping across the

surface of a pond, changes in space-time curvature from a moving mass propagate

out from their source at the finite, although fleet, speed of light, about 3 x 108

meters per second. Just as accelerating charges produce electromagnetic waves,

accelerating mass is needed for gravitational waves (GW). It is a much weaker force

than electromagnetism. For example, the gravitational force's attraction between

two electrons is 1042 times weaker than the electromagnetic repulsion. Moreover,

whereas the dominant component of electromagnetic radiation is the dipole term,

the absence of a negative mass corresponding to negative charge means the lowest

order and dominant term in gravitational waves is the quadrupole. We will have

waves whenever the second time derivative of a mass distribution's quadrupole

moment Q is nonzero.

An oft-cited example [2] of gravitational wave generation in a laboratory

demonstrates the difficulty in detecting (or creating) these small ripples. In some

enormous diabolical research facility, scientists could create their own waves by

rotating a M 500,000 kg iron rod about an axis perdendicular to its cylindrical

symmetry axis at the maximum possible angular velocity. This max = 28 rad/s

is determined by the tensile strength of iron. The bar is L = 20 m long. The

gravitational wave power radiated is

2 M2L4G6C
P 2.35 W
45 c"

(where G is the gravitational constant and c the speed of light), which is a small

fraction of the rotational energy ML22 /3 = 5.23 x 1010 J. Gravitational









perturbation of space-time geometry can be interpreted as a relative length change,

or strain h, of space. In the near-Newtonian limit (i.e., gravitational waves are a

small perturbation to otherwise flat space-time), the strain is an estimated

GQ
h-
C4r

where r the distance from source to observer. This strain is very small. For two

solar-mass neutron stars in the Virgo Cluster that are nearly in the merger phase

of their coalescence, h is of order 10-21. Einstein himself was very skeptical about

whether humans could ever detect these waves directly. The brightest sources we

expect to see in the universe are compact and massive objects such as coalescing

black holes and coalescing neutron stars, and may be moving at relativistic

velocities, as in the case of supernova explosions. Although the waves can carry

enormous amounts of energy, their effect on matter is very weak because space-time

is stiff and not very responsive.

We do have some indirect evidence for gravitational waves. The 1993 Nobel

Prize for Physics went to Hulse and Taylor who carefully timed radio signals from

a pulsar and found the star was losing energy that matched the predicted energy

radiated away in gravitational waves [3].

1.2 Direct Observation of Gravitational Waves

The way to see gravitational waves directly is to measure the resultant tidal

forces using test masses free to respond to perturbation of space-time curvature.

Far away from even the most ferocious source, the wave will produce only a

small deviation in the geometry of space-time. The geometry is described by the

"metric" g,, such that the interval ds between two points in four-dimensional

space-time is


d)2 g,,,dx dxV.










SOCOO
o Ir/4Q ir/2Q 3ir/4Q ir/Q time
Figure 1 1. The effect of a plus-polarization gravitational wave propagating into
page on a string of test masses



00000
I I I I I i
0 rc/4Q 7c/2l 371/4Q 7C/l time
Figure 1-2. The effect of a cross-polarization gravitational wave propagating into
page on a string of test masses

Let us assume the wave is propagating through an otherwise flat Minkowski
space-time geometry. In the transverse and traceless gauge the perturbed metric
due to a wave travelling in the z-direction is

-1 0 0 0
0 1 + h+ hx 0
gp =
0 h 1 h+ 0
0 0 0 1
Gravitational waves have transverse force fields. The subscripts on the strain
components h+ and hx denote the two possible polarizations of the wave. If it has
the h+ pIIl-" polarization, the wave's effect on a ring of test masses would be as in
Figure 1-1. The linearly independent "( ..-- polarization hx has the effect shown
in Figure 1-2, with the displacements rotated 450 from the other polarization state.


1.2.1 Resonant Bar Detectors
Joseph Weber built the first gravitational wave detector in 1960 [4]. His
instrument was a large (1 ton) aluminum cylinder with piezoelectric transducers









to read out relative acceleration of the end surfaces. Gravitational waves excite

oscillatory modes in this resonant bar detector when they contain the proper

frequencies, much like a hammer striking a massive, pure-toned bell. Gravitational

signals make very small hammers, however, so resonant bar experiments todi

use clever multi-stage transducers to register small bar movements and cryogenic

cooling to reduce noise. Although Weber claimed a GW detection, subsequent

experiments cast doubt on its authenticity, and we are still waiting tod i' for the

first direct observation of gravitational waves. Modern resonant mass GW detectors

include ALLEGRO at LSU, USA, EXPLORER at CERN, the Italian AURIGA in

Legnaro and NAUTILUS in Frascati, and NIOBE in Perth, Australia [5]. There

are also spheres, which work in the same way but can with limited precision locate

the source in the sky. These include Brazil's Mario Schenberg [6] project and

Mini-GRAIL in the Netherlands [7].

1.2.2 Interferometric Detectors

Weber's instrument could have seen gravitationally-induced strains as

small as 10-16/v-Hz; current bars can reach 10-21/v/Hz, but the bandwidth is

very limited for any such experiment. There is alv--, a compromise between

sensitivity and bandwidth because a high quality factor Q,,u bar resonance will

be excited by a smaller signal, but will have less bandwidth, since bandwidth

Af w fo/Qul. We want to be able to register a broader span of frequencies,

both for sensitivity to a large range of sources, and high fidelity for a given signal.

Interferometric detectors can have much greater bandwidth than bars. In them

we use light to measure the relative acceleration of geodesic-tracking free masses.

The quadrupole nature of gravitational waves makes a Michelson interferometer

a natural choice for a detector. The mirrors at the end of the interferometer

arms serve as test masses, and we measure their position fluctuations with a

laser. Such a device automatically gains a factor of two in responsivity over the









bars, as a gravitational wave compresses one arm while stretching the orthogonal

arm, doubling the effect. An equal-arm-length Michelson interferometer also has

the same sensitivity to signals of all frequencies1 We gain greater advantage,

however, if we sacrifice sensitivity in some frequency regions in favor of others.

For this reason, the United States' interferometric effort, the Laser Interferometer

Gravitational-wave Observatory or LIGO [8], as well the Franco-Italian VIRGO

[9], Japanese TAMA [10], and British-German GEO600 [11] have all altered their
frequency response from that of the simple Michelson interferometer. The first

three include Fabry-Perot optical cavities in the Michelson arms. The cavities have

the effect of increasing the interaction time, and thus the strength, of gravitational

signals within the cavity linewidth, while degrading performance outside the

bandwidth. To counter the miniscule nature of GW strains in this part of the

universe, we should make the length of the arms as great as possible. Earth's

curvature limits how far we can take this-a mirror too far away will be below

the optical horizon2 and the suspended mirrors pointing toward the center of

the planet will not hang parallel enough to allow us to build an optical cavity.

This and budgetary concerns for building platforms and such set a practical limit

of several kilometers. LIGO has 4 km-long arms, VIRGO near Pisa has 3 km,

GEO near Hannover is 600 m long, and TAMA, in Tokyo, has 300 m cavities.

Light is stored in the arm cavities to integrate the signal, but if we integrate for

longer than half the GW period, the effect will begin to average out, so there is a

limit to this approach-a limit to the reflectivity of the cavitiy mirrors for a given

length. To reduce the sensitivity to noise in the laser's frequency and amplitude



1 for strains that alter the arm lengths much less than the laser wavelength,
which is true for all known sources of gravitational waves
2 even with light bending in the gravitational field of the earth










SETM





S--I ITM



BS
PR



output

Figure 1-3. The power-recycled, cavity-enhanced optical configuration


we tend to keep these interferometers' outputs "d1 il: in the rest position. That

is, microscopic (laser-wavelength-order) length differences make the light exiting

the antisymmetric port interfere destructively. Therefore most all the light will be

reflected back toward the laser. To waste not, and to want not, LIGO, VIRGO,

GEO, and TAMA add a partially reflective mirror between the laser and Michelson

beamsplitter to the configuration to reflect this light back into the detector. It is

positioned so that laser light reflected back toward the instrument will be in phase

with fresh laser light passing through. We call this power-recycling.

GEO600 sits outside Hannover, Germany. It does not have Fabry-Perot

cavities, but it improves signal integration time by adding a mirror to the

interferometer output that preferentially reflects a certain band of signal frequencies

back into the instrument for another round with the gravitational perturbation.

This scheme is known as signal recycling. Basic power-recycled-cavity-enhanced

and signal-recycled Michelson interferometer configurations are shown in Figures 1-3

and 1-4. This work will focus on LIGO, present and future.









TM









Laser7-^----------
LBS
PR

SSR

output

Figure 1-4. The dual-re' vi 1, optical configuration


The LIGO project currently consists of three interferometric detectors at two

sites. A power-recycled Michelson with 4 km Fabry-Perot arm cavities sits in the

swampy pinewoods of Livingston, LA. About 2000 miles (or 10 milli-light-seconds)

away there is another 4 km and a co-located 2 km device of the same design at

the Hanford Reservation in eastern Washington. At the time of this writing,

LIGO is in the middle of its fifth "Science Run," S5, a period of coincident

data-taking in hopes of detecting the first gravitational wave or, at least, setting

an c-1 ii -i- 11 y relevant upper limit on event rates for some sources. On the

best d i it is reaching its design sensitivity but for an excess of 1.5 to 3 times

the predicted noise in the 40 to 150 Hz band, a critical region. S5 will last about

one and one half years, with a goal of a 711' triple coincidence duty cycle and the

ability to see 1.5 solar mass neutron star inspirals out to 10 Mpc for the 4000 meter

instruments and 5 Mpc for the 2k. LIGO is scheduled to acquire 1 year of data in

triple coincidence to be searched for gravitational signals at the design sensitivity

before we upgrade it to the next incarnation. Research for this Advanced LIGO

has been concurrent with LIGO commissioning and science.









The basic thrust of Advanced LIGO design is, unsurprisingly, greater

sensitivity [12], so as to make a GW detection practically guaranteed. The event

rate for signals that would be visible to initial LIGO is unfortunately quite low.

For example, some reasonable and popular cosmological models predict binary

neutron star coalescences with strains visible to initial LIGO to occur at a rate of

up to 1 every 1.5 years [13]. Advanced LIGO will have ten times the sensitivity,

and thus a view of a thousand times the volume of outer space. The upgrade will

be able to peer into the rich Virgo cluster. This brings the estimated event rates

for Advanced LIGO to 20 to 1000 per year [13].

According to Jun Mizuno's interferometer sensitivity theorem [14], the total

integrated sensitivity of a detector over all signal frequencies is a function only of

the stored light power. To improve chances of a detection, one can increase the

laser power or cavity storage times, or tailor the frequency response to favor regions

where either cosmological signal strengths are greatest or noise interference is least,

while the total sensitivity remains unchanged. Advanced LIGO gets a little of

both improvements. We first increase the input laser power 20 times, from 6 to

120 Watts. It is difficult to produce a clean single-spatial mode laser with greater

than about 200 W at present, and beyond this power thermal lensing effects in our

optics make increases more problematic than advantageous. The upgrade will also

use a signal-recycling mirror to move the peak signal gain frequency of the arm

cavities from 0 Hz (DC) to about 300 Hz. Although anticipated signals become

greater as frequency decreases, a room-temperature ground-based instrument

is subject to thermal and seismic noise disturbances that destroy low-frequency

sensitivity anyway. Therefore we place our greatest hopes for detection as close as

possible to this noise i. 1".









1.3 Interferometer Noise

The physics of a gravitational wave-interferometer interaction is relatively

simple. The most challenging task is to create a device quiet enough to measure

these miniscule effects. Some have compared gravitational wave searches to trying

to detect a boat launched from the coast of Africa by looking at the waves on

Florida's shore.3 A daunting task, indeed. There is so much extraneous din on the

earth. The sensitivity-limiting noise will be of three basic types: seismic, thermal,

and quantum noise.

1.3.1 Seismic Noise

Recall our estimated neutron star inspiral strain estimate of h = 10-21. In

initial LIGO this would cause a 10-s8 m apparent test mass displacement-less

than the width of an atomic nucleus. Throughout most of the spectrum in which

we are trying to make measurements, the earth itself is moving far more than a

GW strain produces. To decrease sensitivity to ground motion noise, and because

we must in some way keep the test masses from falling to earth, the mirrors hang

as pendula from seismic isolation stacks. Although constrained in the vertical

direction, they are free to respond to local space-time curvature in the longitudinal

degree of freedom. Above their resonance frequencies wr, the test masses' response

to perturbations of the suspension points decreases as (w,/w)2. With n cascaded

harmonic oscillators, we get (w/Uy )2" noise attenuation. Even so, our best efforts

push the resonance frequency of the pendulum/isolation stack to about 1 Hz, so

seismic noise will set the low-frequency sensitivity limit.

1.3.2 Thermal Noise

If the masses whose positions we measure so carefully are at a finite temperature,

their constituent molecules will fidget in thermal excitation. This motion can excite


3 Thanks, Malik Rakhmanov, for the imagery.









resonances in the mass or the suspensions that move the mirror surface along the

optical axis, mimicking the motion due to a gravitational wave. In the pendulum

suspension wires, we attempt to make these resonances as narrow in frequency

as possible, to concentrate all the noise motion into a band that we will have

to sacrifice. The masses are also made of material with high-quality mechanical

resonance. Additionally, we can tune the excluded frequencies out of the GW

measurement band by changing the test mass aspect ratio. At present, none of the

interferometric detectors is cryogenically cooled.

Another thermal noise source occurs when low thermal conductivity and

different laser intensity on different parts of the test mass cause localized refractive

index and radius of curvature changes in the coating. The microscopically bubbling

test mass surface is exhibiting thermorefractive noise. Thermoelastic noise [15]

results from local temperature fluctuations causing localized thermal expansion.

Sampling a large part of the surface with a wide laser beam makes the effect

average out somewhat. One can also fight thermoelastic and thermorefractive noise

with novel laser intensity profiles, perhaps using a beam with a flat intensity profile

instead of Gaussian shape [16].

1.3.3 Quantum Noise

In interferometry there will be noise in a measurement simply due to the fact

that light, despite its wave-like behavior, is also composed of discrete particles

[17]. Light of a given intensity I is associated with an average number N of

photons, with an uncer' li-:v in that number due to the fact that light particles

obey Poissonian statistics:

AN -
N

Clearly, the greater the light intensity, the smaller the uncertainty in its measurement.

To see gravitational waves on the interferometer output we must store sufficient

power in the arms that photon-counting statistical errors are smaller than the









change due to a wave. Because with arm cavities LIGO has frc qu'-,i-, -dependent

light intensity, the shot noise also has frcE iiqu,- -dependent behavior.

As Einstein noted, photons carry momentum. When N AN photons per

second strike LIGO's suspended mirrors, they create a fluctuating force, and thus

displacement, known as radiation pressure noise. As the radiation pressure noise

in the arms is uncorrelated, it will appear in the interferometer gravitational wave

channel output. Radiation pressure noise increases with laser power. We can

fight the effect by using heavy test masses-initial LIGO has 10 kg fused silica arm

cavity mirrors, and the upgrade will use 40 kg mirrors. Lowering the laser power

is not the preferred solution because shot noise scales inversely with intensity.

In a machine such as LIGO the quadrature sum of radiation pressure and shot

noise defines a -I ,ir, .ird quantum limit" (SQL) of sensitivity for a given laser

power. With the addition of signal-recycling it has been noted that these quantum

noise terms can become correlated, leading to better than SQL sensitivity at some

frequencies [18].

All other noise sources must contribute less spurious GW-signal than these

main three. A combination of interferometer imperfections (inevitable) and noisy

input light can cause additional noise to appear on the detector output. This

occurs when some common mode signal "leaks" to the antisymmetric port or when

control signals used to stabilize the instrument are noisy. A typical ii.- b-l, I

plot for a LIGO-like interferometer is shown in Figure 1-5 [22].

1.4 Third-Generation Detectors and Beyond

After Advanced LIGO, innovations and progress will continue to reach

farther out into the universe. LISA is the Laser Interferometer Space Antenna,

a gigameter-long interferometer that will fly in formation along the earth's solar

orbit. It will feature three laser-linked test masses surrounded by spacecraft that

shield the test mass from solar wind and other sources of drag. Without terrestrial











10
yquant
Int thermal
Susp thermal
10 Residual Gas
Total nolse

1021

1022






.............. ---- ---
102
10


10 10 10 10
f/Hz


Figure 1-5. Noise budget for Advanced LIGO


seismic constraints, LISA's most sensitive bandwidth will be in the milliHertz

range. This project presents formidable engineering challenges, but research on

the myriad subsystems that will make a space antenna is well underway. LISA is

currently scheduled to launch in 2012. On earth, we expect future interferometric

detectors will address seismic, thermal and shot noise in imaginative v- -iv. It

will be difficult to make a test mass both cryogenically cooled and seismically

isolated, but if we hope to see farther than Advanced LIGO we must find a way

to lessen test mass internal thermal noise. Another idea for the future is to beat

thermal lensing on the optics with an all-reflective design. We would then be able

to increase the input laser power (reducing shot noise) many-fold. Some have

si -.-.- -I. 1 using diffraction gratings as beamsplitters and cavity input-couplers.

To alleviate the effect of seismic noise we could put detectors underground, and

somewhat escape seismic harassment from human activity such as planes, trains,

and logging operations, as well as natural phenomena like wind and earthquakes.

Because most of a seismic wave travels along earth's surface, gravity gradient

fluctuations are much less in subterranean spaces. TAMA's upgrades, as well as

the pan-European EURO project, will be underground. Looking even farther ahead

(and to more distant GW sources), detector design becomes even more fantastic.









Can we find some trick or innovation that defeats the rule that bandwidth must be

sacrificed for sensitivity, and vice versa?

1.5 In This Work

Each of these strategies improves a detector by amplifying the size of

the signal relative to noise sources present-the signal-to-noise ratio, or SNR.

This dissertation presents three techniques aimed at improving the SNR of

interferometric detectors beyond initial LIGO and its siblings. The first is an

improvement for Advanced LIGO's control system. A parallel phase modulation

scheme is needed to enjoy a diagonal control matrix. The second technique also

reduces noise, thermal noise, by cooling test masses with near-field electromagnetic

coupling of hot mirror to cold. The last proposal discussed in this work sir-l-- -1 -

that signal strength can be improved without bandwidth loss by putting diffractive

elements in Fabry-Perot cavities. Is there some way to defeat the sensitivity

theorem [14]? The ultimate and thus far rather distant goal of all work on

gravitational wave detectors is gravitational wave ,i-1 'i11r, -i. All signals

we currently use to map out and understand our cosmic surroundings are

electromagnetic variants, from the naked-eye astronomy of ancient (and current)

man to microwave and gamma-ray antennae. As each new part of the electromagnetic

spectrum opened for us, we discovered wonders in the universe we before had never

even imagined. What awaits us in a new gravitational wave paradigm?















CHAPTER 2
PARALLEL PHASE MODULATION FOR ADVANCED LIGO

Any interferometric detector requires control mechanisms to hold the mirrors

in the proper locations. The baseline control scheme for Advanced LIGO requires

phase modulation of the interferometer input light at two frequencies Q1 and f2.

Beat signals between these additional "sideband" frequencies provide error signals

for some interferometer degrees of freedom. To diagonalize the control matrix one

must avoid injecting light that is phase modulated at the beat frequencies ~1 Q2

into the interferometer. For this reason we must change the input optics' optical

layout from serial modulation, as in initial LIGO, to a parallel configuration.

In this chapter we analyze how noise generated by a Mach-Zehnder (ilZ)

interferometer with a modulator in each arm could appear in the gravitational

wave channel of Advanced LIGO to determine the MZ IFO's stability requirements.

We then construct a prototype that meets these requirements and demonstrates the

desired phase-modulating characteristics.

2.1 Interferometer Control

Interferometric detector optics must be stabilized against seismic noise,

thermal expansion, and earth-moon tidal forces to keep these fluctuations

from affecting the gravitational wave signal output and to keep the appropriate

resonance conditions within the interferometer. Macroscopic distances determine

what laser spatial mode couples to the space, but microscopic lengths decide

whether light can build up in the cavity. One must measure with great precision

changes in the relative separation of mirrors and feed a corrective signal back









to positioning actuators.1 At present, the gravitational wave signal is read

in the corrective signal. Because light makes such a superb ruler, we use an

optical/electrical heterodyne technique for this length-sensing and control. The

light injected into the interferometer will contain several different frequencies,

usually separated from the fundamental teraHertz laser frequency by tens of

MegaHertz. In the desired optical configuration each of the radio-frequency

"sidebands" (SB) or original unaltered laser light (the "( ii,, i frequency in radio

nomenclature) resonates in different combinations of cavities that make up the

interferometer. By measuring the beat note between two frequencies we obtain

information about relative length changes of the cavities each component favors.

2.1.1 Phase Modulation

Initial and Advanced LIGO use electro-optic modulators (EOMs) to convert

some of the input laser light into sidebands. EOMs consist of a birefringent

(i -rI I1 across which an electrical potential is applied. The field reorients dipoles

in the material and changes its index of refraction proportionally. If one applies

a sinusoidal electrical signal, light passing through the crystal perpedicular to the

field will have a sinusoidal phase modulation. For a small relative index change m,

the laser gains frequency sidebands. Put a signal with time variation sin (Qt) on

the EOMs crystal, and an originally monochromatic light field Eoe-i"t becomes

Eoe-iwt+imsin Qt Eoe-Wt (Jo (m) + J, (in) eit + iJ (m) e-it)


iwt (J ()) 2it J2 (n) e-2iQt) (2.1)

where Jo, J1, and J2 are the zeroth- and first- and second-order Bessel functions.

The first term in parentheses is the carrier, and the second and third are the upper


1 Coil-magnet pairs attached to the test masses or other parts of the suspension







17

beamsplitter
LZ I/


laser
EOM










RF

osci lator

Figure 2-1. Schematic for


Ml


Sphotodetector




--- -- PDH error signal

mixer

Pound-Drever-Hall Fabry-Perot cavity locking


and lower first-order sidebands, separated from the carrier frequency by Q. The

remaining two terms are second-order sidebands at 2Q.

2.1.2 Pound-Drever-Hall Cavity Locking

The basic heterodyne technique is known as Pound-Drever-Hall (PDH) locking,

and our control scheme is a variation on the same. For a simple optical resonator,

PDH works as follows. A Fabry-Perot cavity shown in Figure 2-1 is made from

two partially transmissive mirrors, M1 and M2, separated by a length L. A laser

introduces light into the cavity from the left. The first and second mirrors the

light encounters have amplitude reflectivities and transmissivities rl, tl, and r2,

t2, respectively. In the steady-state, once transient effect from turning on the laser

have dissipated, the intracavity field Eca, and reflected field Er of the system in

terms of the input Eo are (derived in Appendix A):


Ecav itleikL
Eo 1- r ir_. ,


r1 (72 + 2) t262ikL
1- i = T (k).
Eo 1 r1r_. -


(2.2)


I


I /









The ratios Tc, and T, are the cavity transfer functions. Given input light with

carrier and a pair of sidebands, with amplitude

E, = Eo (et + i Me(r+()t + ie-)t
2 2

which is an approximation of Eq. 2.1 valid for very small m. The reflected total

amplitude from the cavity will be

Eto = Eo (Tc T+ + T i( + i Tei(w- )t

where T = T, (ko), with ko the wavenumber of the carrier light. The quantities

T are similarly defined for the sidebands. A photodetector upon which this signal

falls will register an intensity


Itot ID + m Elo 2 (iT (Tet + T-_e- t)) E- 2 R (T Te2iQt)

where R denotes the real part of an expression.

The next step is to demodulate the photodetector output current with the

frequency equal to the part of signal we wish to view. For the Fabry-Perot cavity

the second term, with sinusoidal Qt variation, makes a good monitor of the cavity

length. Demodulate the signal by multiplying it by sin (Qt + Ad) and integrating

over a period. This is accomplished electronically with an AC-coupled mixer whose

output we then subject to a low-pass filter. The result is an "error" signal for the

cavity's length, which can be fed back to a mirror positioner to keep resonance

conditions satisfied. If the length is nearly an integer number of carrier wavelengths

and non-resonant for the sidebands, Tc alone will depend upon a small length

change 6L from the resonant length. Under these circumstances Tc = Tc (6L), and

the Pound-Drever-Hall error signal is


S(6L, d) = 2m |Eo 2 [R (TA (6L) (T+ + T_)) cos(d)










04
0- Error sig
Trans Int
03-

0O

n 01


0)

-01 -


-02]
-0 23

-3 -2 -1 0 1 2 3
degrees from resonance

Figure 2-2. Fabry-Perot intracavity intensity (light line) and Pound-Drever-Hall
error signal (heavy black line) as functions of length detuning from
resonance


(T (L) (T+ + T))sin(d)] (2.3)


and is shown for d = 0, along with the cavity transmitted light intensity, for

various cavity length tunings in Figure 2-2. Here the sidebands serve as "local

oscillators" that don't depend upon the quantity we are trying to measure, but

allow us to bring the relevant signals from MHz to near-DC levels through the

beat-note phenomenon. We demodulate the amplitude "b, ii signal at the

difference frequency to remove the AC portion of the signal. What remains

will be proportional to the relative phase of the two light frequencies. Different

information is obtained by demodulating with a sinusoidal signal at the difference

frequency that is in phase with the light phase modulation, resulting in an

appropriately named "in-lph i- error signal, or with a sinusoid that is 90 out

of phase-the "quadrature" signal [19].

2.1.3 Advanced LIGO

With the new signal-recycling (SR) mirror at the interferometer output,

Advanced LIGO will have five lengths, longitudinal degrees of freedom (DOF),







20



C



SB ITM ETM
SD E
BS
PR
F

--- SR



Figure 2-3. Lengths in Advanced LIGO that require feedback control

that must be controlled to keep the instrument at its operating point. These

are illustrated in Figure 2-3. The five DOFs are the common and differential

Fabry-Perot arm cavity lengths, L+ and L_, respectively, the differential length of

the short Michelson arms, 1_, the length of the power-recycling cavity composed

of the common short Michelson arms and beam-splitter-to-power-recycling mirror

lengths, 1+, and the distance from the beamsplitter to the signal-recycling mirror,

1,. In terms of the lengths A-F shown in the figure, the DOFs are defined as in

Table 2-1: In order to monitor the various lengths, we use a heterodyne technique

Table 2-1. Advanced LIGO degree of freedom in terms of length in Figure 2-3

DOF definition
L+ (C + E)/2
L_ (C- E)/2
1+ A+(B+D)/2
l_ (B D) /2
1, F+(B+D) /2


similar to Pound-Drever-Hall to measure changes in the phase of light that has

bounced off of different parts of the IFO.









A dual-recycled interferometer requires two pair of frequency sidebands. The

arm cavity common and differential error signals come from interference between a

sideband (SB) and the carrier. The control scheme we expect to use in Advanced

LIGO uses beats between sidebands for the three remaining degrees of freedom.

The 40 meter Advanced LIGO prototype at Caltech tests a control scheme with

this division of labor between light frequencies. Ideally, one would find an error

signal for each DOF that was independent of all others, but, realistically, some

coupling alv--x exists between DOFs. This can be minimized making various

frequencies resonant in different parts of the IFO, concentrating their power in

these areas, and by appropriate choices of demodulation phases. Done well, this

will produce a pleasingly diagonal control matrix. Table 2-2 is an example from

the 40 m prototype of a normalized matrix showing the dependence of each error

signal on each degree of freedom. Such a control scheme would be relatively easy

to implement. Scientists at Caltech noticed, however, that when they injected light

into their instrument that was already at the difference frequencies they wished

to measure, their matrix included significant off-diagonal elements, as shown in

Table 2-3 [20]. For this reason, Advanced LIGO will need an alternate method of

adding phase-modulation sidebands to the input laser.

Table 2-2. Control matrix for 40m prototype without second-order sidebands [20]

Dem. wj Dem. Q L+ L_ 1+ l_ s
21 100 1 -4 x 10-9 -1 x 10-3 -1 x 10-6 -2 x 10-6
22 271 -5 x 10-9 1 -1 x 10- -1 x 10-3 -2 x 10-s
Q1 x 22 189 -2 x 10-3 -3 x 10-4 1 -3 x 10-2 -1 x 10-1
Q1 x 22 50 -6 x 10-4 2 x 10-3 8 x 10-3 1 7 x 10-2
Q1 x Q2 164 4 x 10-3 3 x 10-3 5 x 10-1 -2 x 10-2 1


2.1.4 Electro-optic Modulators

In initial LIGO, and, the initial 40 m prototype, the IFO input optics include

a series of electro-optic phase modulators (EOMs) through which the laser passes.











I
f
f
f
f
f


22

Table 2-3. Control matrix for 40m prototype with second-order sidebands [20]

)em. wo Dem. Q L+ L_ 1+ l_ 1
t1 100 1 -1 x 10-8 -1 x 10-3 -1 x 10-6 -6 x 10
2 271 1 x 10-7 1 1 x 10-5 1 x 10-3 7 x 10-6
21 x Q2 1840 7 -3 x 10-4 1 -3 x 10-2 -1 x 10
21 x 22 50 -6 x 10-4 32 7 x 10-1 1 7 x 10-2
t1 x 22 1610 3 2 2 x 10-1 -4 x 10-2 1


6

1


If the laser is fed through two sequential EOMs, driven with frequencies Q1 and Q2,

the result is


Eo [Jo (umi) Jo (n2) Ce + zJo (n2) J1 (n) e ) + Jo (n2) J1 (in) e )t


+iJo (mi) Ji (m2) e(+"2)t + iJo () Ji (n2) Ce( -"2)t

-eiwt (Jo (mn) J2 (in2) e2iQ2t + Jo (2in) J2 (nn2) e-2il2t)

-eiwt (Jo (im2) J2 (2in) e2,01t + J (n2) 2 (1in) e-2i01t)

-eit (Ji (mi) J1 (m2) 6e 1+"2)t + J1 (n1) (n2) eC(Q"-Q2)t)

eCt (+J1 (min) Ji (172) Cei(Q+Q2)t + J1 (min) J1 (m12) e- i(Q-2)t) .


In the last two lines of the above equation, we find we've made sidebands of

sidebands at the sum and difference of the EOM driving frequencies. As seen

above, this mixes the error signals that feedback mirror positioning control in an

undesireable way.

There are two immediately obvious solutions. It is possible to drive a single

EOM with a complex signal and produce multiple modulation frequencies without

SBs of SBs2 Another, perhaps easier solution is to add sidebands in parallel

rather than in series. We chose to investigate the latter in greater detail. Instead

of sending the laser through a sequence of modulators, as shown in Figure 2-4,


2 Qi-Tse Shu, unpublished. Some current research is also underway.


(2.4)



























SUopLcs to IFO


Figure 2-4. Serial phase modulation of IFO input light


BS1 Q1I


BS2


optics


Figure 2-5. Parallel phase modulation of IFO input light


to IFO









we'll use an arrangement that makes mixed sidebands impossible. Split the

laser with a beamsplitter, as shown in Figure 2-5. Send the transmitted light

through one EOM and the reflected light through another. Recombine with

another beamsplitter. This configuration is commonly called a Mach-Zehnder

( \!Z) interferometer. What could be simpler? Naturally, however, there are a

few caveats for the scientific emptor to consider. Because each EOM interacts

with only half the input laser intensity, and because the resultant sidebands are

halved again by the second beamsplitter, with the same radio-frequency power at

the EOM, our parallel-creation sidebands will have only a quarter of the power

in their series counterparts. With Advanced LIGO's large laser power and very

high-finesse optical cavities this should present no problem to finding adequately

sensitive error signals with the same SNR. Mirror motions or index of refraction

changes in the two arms, particularly if differential, can create noise that interferes

with gravitational wave detection, so we will likely need to actively stabilize the

MZ in some way. Parallel phase modulation amounts to having one more degree of

freedom to control than the series case.

2.2 Parallel Phase Modulation Noise

The basic noise requirement for components of LIGO can be summarized

as follows: no technical noise source should produce a detector output that is

more than 10' of that made by the weakest gravitational wave the instrument is

designed to register. The amplitude, frequency and pointing (transverse spatial

mode) cleanliness of the MZ's output must be regulated. Additionally, because

the combined carrier and sidebands no longer travel a common path, noise in their

relative phases, amplitudes, and pointing can occur.

A gravitational wave signal can be read from the detector output with either

a heterodyne or homodyne technique, and the noise requirements will depend upon

which is used. With heterodyne or "RF" read-out, radio-frequency sideband light









is ever present at the antisymmetric IFO port and acts as a local oscillator. The

GW signal is in the beat between the sideband, which does not resonate in the

Fabry-Perot arm cavities, and any carrier light that leaks out when a GW strain

changes the arms differentially. In the "DC" read-out strategy, we measure the

effect of a GW strain as a change in intensity of carrier light itself at the output

port. As one would expect, there are advantages and disadvantages associated with

each method. Overall, the DC read-out technique has more lenient carrier and

sideband noise requirements, and has been selected for the baseline design [21].

2.2.1 Mirror Motions

Mirrors in the MZ interferometer may fluctuate in either their longitudinal

or angular degrees of freedom. Both longitudinal and angular noise motions

can be divided into those that are common to both arms, and those which are

differential. In the case of angular noise a further division occurs between common

and differential movement of optics in a single arm and that between optics in

different arms. Figure 2-6 is a tree diagram that exhausts the possible mirror

motions and consequent noise on the detector input light. The effects of the two

main branches can be mixed if, for example, a beam is not centered on an optic

or hits the mirror with non-normal incident angle. Let us deal with each effect in

turn. When we find that the parallel phase modulation configuration adds noise in

a way similar to serial modulation, we will be satisfied that we need only recreate

the stability of that original scheme.

Longitudinal Common Motion

Longitudinal common-mode fluctuations, when the mirrors move along the

optical axis of the MZ, will cause the same phase (and thus frequency) noise to

appear on the carrier and sidebands. This is illustrated in the phasor diagram in

Figure 2-7. Let the field travelling a clockwise path around the Mach-Zehnder IFO














Mirror Motions

Longitudinal

Common Differential
/
C: overall phase C: amplitude
SB: overall phase SB: phase rel. to C


A
Single arm

common C. and
SB pointing


Common

Both arms

common C. an
SB displaceme


Angular
Differential
Single arm Both arms
V V


Id
nt


relative SB
displacement


relative SB
pointing


Figure 2-6. Possible noise motion in the Mach-Zehnder optics and resultant noise
on Advanced LIGO input light. C: carrier, SB: sideband.


+


Figure 2-7. Phasor diagram of Mach-Zehnder common-mode fluctuation's effect on
recombined carrier and one pair of sidebands.


I:









in terms of the original input field amplitude Eo be

Ec 1/ /Zoe ikL1 6im si'n(Qt) E, 1/ /2EoeikL2 eim2 sin(Q2t),

where Ec and Ec, are the clockwise and counter-clockwise fields, respectively,

Li and L2 the respective arm lengths, and m and m2 the modulation depths.

Defining an average arm length L = (L1 + L2) /2 and differential length AL =

(L1 + L2) /2, the recombined field at the second beamsplitter is


Ett = E k [iCkAL (Jo(7rn) +iJ1 ( l) lt + iJi, (Mi)e-ilt)

+e-ik (Jo(In2) + z2 *t + (Jl (M2) -iQ2t)] (2.5)

As the noise in L shows on both carrier and sidebands equally, this case is

fundamentally no different than that found in serial modulation. Its effect on

the GW signal is a function of mismatch between the Michelson arms, which allows

light noise to leak to the dark port. Arm .mi-mmetry may be due to different losses

or reflectivities of test mass mirrors, or to length disparity. The common carrier

and sideband frequency noise, and the carrier amplitude noise resulting from MZ

common arm length changes are both suppressed by independent feedback loops in

Advanced LIGO. A low-noise high-power photodiode monitors the laser intensity

behind the triangular mode-cleaner cavity in the input optics, and a correction

signal fed back to the laser power. Another feedback loop locks the laser frequency

to the common mode of the Fabry-Perot arm cavities.

Longitudinal Differential Motions

When optics move differentially along the optical axis, two noise effect occur.

First, the recombined carrier light suffers amplitude fluctuations. Amplitude and

intensity stability of the carrier is critical with DC readout [21], but, as mentioned,

noise will be suppressed by feedback loops.













+ t



Figure 2-8. Phasor diagram of Mach-Zehnder differential-mode fluctuation's effect
on recombined carrier and one pair of sidebands.

The second noise effect is the relative phase between carrier and sidebands.

With longitudinal differential motions the Mach-Zehnder configuration allows for

fluctuations in carrier and sideband relative phase, which may be interpreted as

noise in the sideband frequency. This has the same effect as EOM phase noise

in the serial modulation case (which will also occur in parallel modulation). We

expect that relative phase noise between the carrier and control sidebands will

present the largest problem of all noise terms discussed here as it appears as

frequency noise on the laser via the frequency stabilization loop, as currently

conceived, and no feedback loop supresses it. The mode cleaner that serves as

a filter for laser frequency, pointing, and spatial mode noise is locked to the

Fabry-Perot arm cavities' common-mode, and in turn serves as a frequency

reference for the laser. The feedback loop is poorly equipped to deal with this

kind of noise, and cannot distinguish sideband frequency changes from laser

frequency changes. With relative phase noise, the control loop imposes a misguided

correction to the laser frequency, and thereby relative phase noise becomes laser

frequency noise, which, as mentioned above, can interfere with GW detection.

Asymmetries in the arms of the Michelson interferometer, of both the intentional

and the inadvertent but inevitable varieties, allow such common-mode noise

as frequency fluctuations to appear in the dark port signal, where they are not

distinguishable from a gravitational wave signal. To make MZ-assisted parallel









phase modulation work for Advanced LIGO, one must calculate the allowable

magnitude of carrier-sideband relative phase noise from Advanced LIGO's

frequency noise requirement. Using DC-readout for the gravitational wave signal,

Advanced LIGO has a 1 mismatch between its two resonator cavities. Given this

operating point for the interferometer, there is a frequency noise requirement [22]

for the input light made by comparing detector output due to carrier frequency

noise to output resulting from the waves we seek. We wish to translate this into a

requirement on relative phase noise amplitude.

The procedure is to compare the effect of laser frequency noise on the

Fabry-Perot common-mode signal to the effect of relative phase noise [23]. The

L+ error signal is in the beat between carrier and f2 sideband light that is reflected

from the IFO back to the laser. Enlisting the FINESSE modelling tool developed

by Andreas Friese3 we calculated the ratio of laser frequency noise's effect on the

common-mode error signal S (f) to the effect of relative phase noise. The ratio of

these transfer functions is shown in Figure 2-9. It is relatively flat over the region

of interest for our detectors. Given the allowable frequency noise Af (f) discussed

for longitudinal common motion, we can find a corresponding relative phase noise

limit A4Q,, (f):


AKrif (f) f () A-f (f) (2.6)
df dSldAf,,

where dS/df and dS/dK are the transfer functions of frequency and relative

phase noise, respectively. A limit on relative phase noise translates into a limit on

differential displacements ALMZ of the MZ arms by


ALMZ A-- rel
27


3 available at www.rzg.mpg.de/ adf








30

Ratio of Transfer Functions to CM Signal
28


2 75 -


27o


g 2 65 -


26 -




25 1 : 0I 10 I I
102 10 100 10 102 103
noise frequency (Hz)

Figure 2-9. Ratio of frequency noise to relative phase noise transfer functions to
Fabry-Perot common mode error signal


Permissible ALMZ (f) is shown in Figure 2-10. It should be noted that this

assumes the frequency stabilization loop's gain is high enough that it imposes

practically all of the relative phase noise on the laser frequency. This will be the

most stringent stability requirement for the Mach-Zehnder interferometer. As

mirrors on an optical table such as the input optics in Advanced LIGO will rest

upon generally have much larger fluctuations, the MZ will need active control.

Angular Motions

When mirrors have angular fluctuations, a disparity arises between the optical

axes of the Mach-Zehnder IFO and Advanced LIGO's core optics. Gaussian

optics describe the amplitude of a light beam in the plane perpendicular to its

propagation in the paraxial approximation (that is, when its transverse variation

is much less than the longitudinal variation.) The Gaussian modes constitute an

orthonormal basis set into which any spatial light distribution may be decomposed.

An interferometer will generally use light beams made almost entirely from the

lowest-order term of the basis set, TEMoo. This is the term that has the no

intensity zeroes and a Gaussian shape. When there are small displacements 6










Stability Requirement for MZ



10


10 i
1 10 11



10 12



10 102 10
noise frequency (Hz)

Figure 2-10. Permissible residual differential displacement of Mach-Zehnder arm
lengths


or tilts 0 in the beam relative to the first optical axis, the new spatial profile,

when expanded in the original basis set, will contain small contributions from

higher-order Gaussian modes [24]. For a laser to be resonant in a given optical

cavity, it must have not only the proper wavelength (the cavity length divided by

an integer), but also a particular spatial mode defined by the length and mirror

curvatures. Spatial mode noise is a problem because it causes apparent laser

amplitude noise since non-TEM ,,, modes couple less efficiently to the resonator

cavities. Noise also results from the reconversion of noisy higher-order modes to the

fundamental mode via misaligned core optics [25].

In the Hermite-Gaussian eigenbasis, in which a transverse spatial amplitude is

the product of two one-dimensional functions, the first three modes are:
----ik.-
Uo (x, z) \114 / e q ,the fundamental mode

U (x, z) = ( / e2 ,the "tilt" mode

U2 (x, z) (14 V T 2) e ,the bull's-eye mode

with wavenumber k, beamsize w(z), Guoy phasor '(z), and complex radius q(z).









Let 0 and Q be beam tilts relative to the x- and y- axes, respectively. In

addition, let there be small displacements 6 and T] of the beam from the core optics

axis. The total field at the symmetric port of the Mach-Zehnder IFO will be


Etot oc
1/4 ((L1) -61)2 ik0x

+ ) w(L)J
( ( \1/4 (-1)2 Cik ikL1

S((2 ) 1/4 i (L2 (_-,2)26ikO Li




S(() 1/4 ( L (Y72)2ik 2Y) ikL2 (2.7)


Note the assumption of equal losses in the arms. This may also be written

E oc 2eik(2+y2 )/2q ikL ei&ikSx/q ikOx ik y
TW-

CikzALC 'T CikASx/q 6ik9AOx ikAny/qyikA y
(1 + U)
C-ikAL C-iAl C-ikA,6xl/q --ikAOx-ikAy/q -ikAy
+ ( 1- w)

Where a quantity labeled a is defined as the average 2- = a1 + a2, and

corresponding differential quantity is 2Aa = a2 a2. We now expand the expression

(1 + Aw/w) to linear order in small perturbations. The field is

E o 2 1 ik(x2+y2)/2q ikL ei&6ik6x/q ikOx '- Ciky
W-,

2 cos x +

Aw kAq x kAqy
-2i Wsin kAL+ AT + + + kAOx + + (2.8)
w q q









Terms such as eikO refer to MZ common-mode tilts and displacements, and these

are the same phases found in the electric fields of serial modulation schemes. When

the new pointing of the MZ is expressed in terms of the IFO optical axis, there are

small contributions from higher-order tilt modes. In the x-direction,


E (x) oc




2k2A6x Aw kAsx Aw
+ AL 2i-- 2i kAOx
q w qw
which is

2Uo (x) + we-2 ( + kO + A +kAOAT

k2A6 Aw kA Aw
+-AL --i i kAO UI (x)
q q W w

2Uo (x) + atitUi (x) (2.9)

Eq. 2.9 defines a tilt mode amplitude atilt. We see from the expression 2.8 that

differential pointing effects are all of second order, and therefore most likely no

problem at all.

2.2.2 Fluctuations at Non-zero Frequencies

Now assume that the MZ mirror pointing noise occurs not at DC, but at a

frequency f such that

6i sin(27ft),

for example. Any Uo1 or Uoi light input into the main Advanced LIGO interferometer

(IFO) can be transformed into Uoo light therein if core optics are misaligned. The







34

maximum permissible amplitude of a tilt4 mode injected into the IFO is given by

atilt /2.5 x 10-3 210-10rad 1
af2(f) = \- + (5 x 10-8) A T .
Sf AOlTM VH

Note that this assumes a 10-10 rad differential tilt of the input test masses (IT\ i).

It also includes a factor of 10 safety margin, keeping technical noise at least a

decade below the interferometer's intrinsic quantum noise.

As an exercise, we use 2.9 to calculate permissible common-mode tilts in

Mach-Zehnder optics such as mirrors and beamsplitters. The input mode cleaner

supresses higher-order modes by an additional factor of 2000. We assume a

beamsize on the optic in question of 5 x 10-4 m and that half the allowable

fluctuations are allocated to tilts, and half to displacements. As an example, the

maximum common tilt 0 is found from


S< t 8.5 x 10-9rad
2kw H/z

at 1 kHz.

2.3 Mach-Zehnder Parallel Phase Modulation Prototype

Having found the stability requirement for parallel phase modulation, a

table-top prototype will explore the attainability of these limits on differential

optical path changes. For our purposes the Mach-Zehnder interferometer should be

compact, inexpensive, and easily controlled with a feedback loop. In this design the

optics are all mounted to a single aluminium plate. These are small, off-the-shelf

mirrors and mounts. One mirror is glued to a stack of piezoelectric ceramics

(PZT), which grants us length control access with only small driving voltages

(2V can cover a full laser wavelength). To avoid adding pointing noise with our



4 "Tilt" here refers to either a U1o or U01 spatial mode, as distinguished from a
tilt 0 in a single MZ beam









rf signal 2



SBS polarizer

rf signal 1 I















Figure 2 11. Layout of the prototype MZ phase modulation scheme
-- EO12


D ph( todiode



gain &
frequency
filter mixe *



Figure 2-11. Layout of the prototype MZ phase modulation scheme


actuator we use a combination of polarizing beamsplitters and A/4-waveplates in

the corners of the MZ, as shown in Figure 2-11. In this way, the PZT pushes and

pulls the mirror along the light propagation axis. As a bonus, the waveplates also

allow one to make the impedance matching of the arms nearly perfect, so that

dark interference fringes at the output can be truly black. The EOMs emplovi l1

are off-shelf LiNbOs resonant phase modulators. Their frequencies in the clockwise

and counterclockwise arms (as viewed in the figure) are 31.5 and 12.0 MHz,

respectively. It is with the 12 MHz sideband we lock the MZ differential mode.

This requires a high-speed photodetector- ours is a 1 GHz Si photodiode. We mix

the AC portion of the photodiode ouput signal with the RF-signal that drives the

12 MHz EOM. The low-pass-filtered error signal is amplified and frcqu ii- ,-i,-filtered

by an analog circuit before it is fed back to the MZ length control PZT.

Although one should easily be able to guess the spectral content of the MZ

interferometer output, experimental science can be full of surprises (see ('!i Ilter 4),

so we put the Mach-Zehnder output light through a high-finesse optical spectrum











Spectrum of MZ Output Light
018

016

014

012

01-

0 08-

0 06

004

002

0-,

-0602 3 -----15 I I ------
-80 -63 -31 5 -24 -12 0 12 24 315 40
frequency (MHz)

Figure 2-12. Spectrum of light transmitted by parallel-phase modulating
Mach-Zehnder interferometer



analyzer to see what frequencies were created. The interferometer's behavior was

as expected, however, as seen in Figures 2-12 and 2-13. There are two pair of

frequency sidebands. The j I.--, d peaks to the right of the carrier resonance are

higher-order Gaussian modes. We look closely at the mixture frequencies. Small

second harmonic sidebands are visible for both frequencies, so we know we would

be able to discern whether SBs of SBs were present. No peak is present at 19.5

MHz, but a few small peaks (a mixed sideband should have approximately the

same amplitude as a second-order sideband) appear at 43.5 MHz. Upon closer

inspection, looking at the light corresponding to these peaks on a CCD camera, we

found that these were higher-order spatial modes resonating in the analyzer cavity,

and not mixed sidebands.

To get a sufficiently quiet MZ interferometer we will need a quiet error signal

with strong dependance on the quantity it intends to regulate. Our technique is

heterodyne length sensing in the beat signal between the f =12 MHz sideband and

the recombined carrier. The field at the symmetric port of the Mach-Zehnder IFO








37

Detail of MZ Output Spectrum
0 01

0 009

0 008

0 007

S0 006
-2
0 005

0 004

0 003

0 002

0001

-60 -50 -435 -315 -24 -195 -12 0
frequency (MHz)

Figure 2-13. Expanded view of MZ output spectrum. Mixed sidebands are absent
the 43.5 MHz peaks are due to higher-order spatial modes.


(with the effect of the 31.5 MHz EOM omitted for clarity) is


Esym = EleikLi C isin(Qt) + E26ikL2, (2.10)


and the intensity is


Isym, = IEi 2 + |E2 2 + 2E1E2 Cos (kAL + msin (Qt)). (2.11)


If m is small, and in our experiments it was a very small 0.001 for the 12 MHz

EOM, the slowly time-dependent part of Eq. 2.1 is


Isym,AC 2E1E2 [cos (kAL) msin (kAL) sin (Qt)] (2.12)


Demodulate the photodiode voltage with a sin (Qt) wave from the EOM's

voltage source to get the Mach-Zehnder error signal. The signal is zero when

the Mach-Zehnder's interference is at an extremum, and becomes either positive

or negative if the differential length deviates in one direction or another from this

value. Thus, if fed back to a bi-directional positioning device such as our PZT, the

error signal "locks" the interferometer at a bright or dark fringe condition. The









difference between bright- and dark-fringe locking is inversion of the signal. Unlike

the Fabry-Perot cavity analyzed above the Mach-Zehnder heterodyne error signal

has an infinite locking range. That is, no matter how far removed the differential

length becomes from the proper interference condition, the PZT will alv--i push

(or pull) the length closer to perfection, seeking that zero in the error signal. The

disadvantage relative to Fabry-Perot Pound-Drever-Hall locking is that we do not

have sharp resonances that heighten our awareness of length changes through steep

error signal slopes. The strength of a corrective signal is in the size of its derivative

with respect to the parameter it controls. As we have a feeble sinusoidal variation

in our error signal as a function of AL, we must add slope electronically.

To determine how must amplification the error signal needs we measure the

noise in the free-running (no feedback) Mach-Zehnder output and compare to the

stability requirement. An analog proportional-integrating amplifier circuit with

adjustable gain makes up the difference. We would be content to set the gain

to the highest level necessary and feed this back at all noise frequencies, but we

must be more careful. Mechanical resonances in the system, particularly in the

actuator, will be excited if the gain is greater than 1 at these frequencies. At 10

kHz, the PZT's internal resonance is the lowest-frequency and most problematic.

Fortunately, measurements of the free-running MZ's error-point noise indicate that

it is already sufficiently quiet at 10 kHz, so the feedback loop needs no gain at this

or higher frequencies.

In Figure 2-14 one sees the performance of the stability feedback loop

for various noise frequencies compared to the requirement. The open-loop

measurement indicates the interferometer needs active feedback to supress excess

noise below about 500 Hz (where, unfortunately, there is a small resonance due to

the PZT mount). With the feedback loop closed, the residual differential motion

noise falls within an acceptable range of the calculated requirement. Noteworthy









exceptions are the ..-'.-i ivatingly ever-present 60 Hz line noise spikes, as well

as the 500 Hz resonance. We also made an independent measurement of the

residual differential motion by demodulating the photodiode output with the

radio-frequency signal that drove the second EOM. The measurement point is

labelled Sot in Figure 2-11. Figure 2-15 depicts the out-of-loop noise measurement

alone versus the target. Although a small amount of excess noise persists below

about 100 Hz, we feel this result demonstrates the relative ease with which one can

achieve the necessary stability for parallel phase modulation. If we implemented

this scheme in Advanced LIGO, we could improve the noise performance by raising

the frequency of the PZT resonance by gluing a lighter mirror to it, which would

allow the feedback loop a higher unity gain frequency to suppress noise below

100 Hz. The noise attenuation transfer function for the electronics used in this

experiment is found in Figure 2-16 (Appendix C contains the circuit diagram).

Ground loops and line noise could be reduced with the various black magic

techniques electrical engineers know. The entire MZ interferometer could also be

enclosed in vacuum for better passive isolation.

In the end, parallel phase modulation is vying with several other techniques

for inclusion in Advanced LIGO's length-control scheme. Complex modulation of

a single EOM has tantalizingly fewer degrees of freedom to control. Its strictest

requirement is coordinating the phase modulation with amplitude modulation,

that is, locking two oscillators. It may be that double-demodulation for the inner

Advanced LIGO DOFs is abandoned entirely for a different mix of beat signals.

Along with the Caltech prototype group we have investigated the requirement for

and performance of a possible solution, and await a coalition decision as the time

to begin construction the the upgrade nears.










40






MZ Differential Mode Stability


- requirement
out-loop means
in-loop meas
open loop meas


II ,

I i


200 400 600 800 1000
noise frequency (Hz)


1200 1400 1600


Figure 2-14. Comparison of MZ error-point noise with required stability for

free-running and closed-loop operation


MZ Differential Mode Stability


1013
200 400 600 800 1000
noise frequency (Hz)


1200 1400 1600


Figure 2-15. Out-of-loop measurement of residual MZ differential displacement

noise


10



I
10





10-


I, I

f I


I '



































Feedback Attenuation Transfer Function


noise frequency (Hz)


Figure 2-16. Attenuation transfer function of MZ feedback electronics















CHAPTER 3
NEAR-FIELD EFFECT HEAT TRANSFER ENHANCEMENT

3.1 Near-field Theory

Interferometric detectors will disclose the hitherfore imperceptible passage

of a gravitational wave through careful timing of photons travelling between

the mirrors that serve as test masses for the field. Unfortunately, if the mirrors

themselves move with respect to another part of the instrument, the effect imitates

a gravitational wave. In the current and advanced versions of LIGO, we have

several strategies for quieting these noise motions. The test masses hang as

pendula from seismic isolation stacks. In addition to active feedback outside the

GW measurement band, initial LIGO has single stage suspensions, but Advanced

LIGO will have four masses hanging, one from the other, ending with a very quiet

final test mass at the bottom

If all this works perfectly to hold the mirror, as a macroscopic object, at

position steady to within 10-18 to 10-21m/v H (for LIGO and Advanced LIGO,

respectively), one still has to contend with the fact that the mirror is composed of

molecules with thermally excited motions all their own. These thermal fluctuations

excite resonant modes of the test mass in a process described by the fluctuation

dissipation theorem (FDT), discussed below. Levin et al. [26] calculated the

noise in the GW channel of the IFO for a room-temperature Advanced LIGO test

mass. Test mass internal thermal noise is shown in green in Figure 3-1. According

to the classical equipartition theorem, there is {kBT of energy in each mirror



1 VIRGO, currently under construction, has seven-stage pendula.[9]








43

10-19
yquant
Int thermal
S. Susp thermal
10 20 Residual Gas
STotal noise

10-21


TN 10-22


10
10-23


10-24


10-25 i .
100 101 102 103
f/Hz

Figure 3-1. Comparison of quantum-mechanical sensitivity limit to test mass
internal thermal noise limit for Advanced LIGO [22]


degree of freedom, where kB is Boltzman's constant and T the temperature in

degrees Kelvin. Levin found internal thermal noise is proportional to the mass's

temperature. In Advanced LIGO room temperature test masses prevent our

reaching the quantum limit of detector sensitivity, a limit due to the corpuscular

nature of light, in some frequency regions. The mirrors need to be cooled to 10K

to have thermal noise comfortably2 below the target detector sensitivity, but we

have yet to develop an economical active cooling method that meets our exacting

engineering requirements. Chief among these is maintaining seismic isolation. One

will commonly cool an object by attaching a cryogenic "cold f:i,; i that extends

from a very cold reservoir. Naturally, we are not about to short-circuit the isolation

system by touching test masses so carefully separated from the environment with

any kind of finger, cold or otherwise. The only routes through which heat can

escape the masses suspended in vacuum are radiation and conduction through


2 a factor of ten









the suspension fibers. The effect of the latter in suspensions like those in Adv.

LIGO will be far insufficient.3 By the Stefan-Boltzmann law, the electromagnetic

radiation from an object at finite temperature T in the far-field is


Q =e(l5.67x m2K ) T4,

where c is the emissivity of the material. The T4 factor in the Stefan-Boltzmann

law means that radiative heat loss diminishes rapidly with temperature. A fused

silica mirror (c = 0.9) with 1 m2 area, at 10 K, therefore, would radiate 510

pW. With Advanced LIGO's intracavity circulating powers of 10 kW and the

anticipated mirror absorptivities, however, the input test mass will be collecting

2 W from the laser. Advanced LIGO would clearly require active cooling to beat

the current thermal limit. In the LIGO upgrade the mirrors will remain at room

temperature, but when in even more futuristic incarnations of interferometric

detectors the standard quantum limit of sensitivity is much lower, we will certainly

need to actively cool the test masses. How can we both enhance heat loss and

maintain seismic isolation?

Near-field electromagnetic effects provide the answer. We cannot touch

the test masses with a cold finger, but we can bring a cold object close enough

that photons can tunnel through the vacuum gap from hot to cold mass [27].

Classically, the effect may be described as penetrating the region of significant

evanescent fields so that these fields can transfer energy, a phenomenon also known

as frustrated total internal reflection (FTIR).

3.2 Fluctuational Electrodynamics Theory

To quantify the heat transfer enhancement, we use the fluctuational electrodynamics

method developed by S. Rytov [28]. Thermal excitations within a material cause


3 See Appendix B for quanification of heat transfer through the fibers









1 2 3











Figure 3-2. Geometry of dielectic media for near-field effect calculation- two
semi-infinite slabs separated by a vacuum gap with width d.


oscillating currents J that serve as electromagnetic field sources, described by

Maxwell's equations:

V x H = 47opJ + i cE (3.1)

V x E -iwB (3.2)

assuming the resultant fields have sinusoidal time dependence and the material

is nonmagnetic. Our calculation will be assisted by two physical/mathematical

theorems- that of Green, and the fluctuation dissipation theorem (FDT).

3.2.1 Green's Function Method

With Green's theorems one can find the electric and magnetic fields due to the

thermally-induced currents using Green's functions.


E (r, w) ipow GEE (r, r', u) J (rl, ru) dVI (3.3)


H (r, w)= x GHE (r, r/, a)) J (r/, w) dVI (3.4)

where GXE (r, r/, w) is the dyadic Green's function relating the field at r to the

source at r/.

Because the Green's functions reflect the geometry of the problem, we must

specify this to proceed. Let us consider two semi-infinite half-spaces filled with









isotropic and nonmagnetic matter separated by a vacuum gap of width d. Orient

the Cartesian axes such that the z-axis is perpendicular to the interfaces, and

the plane z = 0 coincides with the surface of one semi-infinite chunk as seen

in Figure 3-2. The three regions 1, 2, and 3 are defined as shown. Instead of a

complete Fourier decomposition of Green's function into plane waves of the form

Eei(k-r-wt), which are not in general guaranteed to satisfy the free-space Maxwell

equations, we will use a partial spatial transform Eei(K'r- t)+ikzz, where K is a

vector in the x-y plane. The z-component kz is subject to the constraint


K2 + k = -k 2,


with ko the photon wavenumber. Thus the Green's function encompasses both

propagating waves, when K = KI < ko:

k = k0 -_ K2
k, y k-


as well as evanescent v -', when K > ko and


kZ i K2 kO.


Let us introduce further simplifying notation. We define the wave vector u

K k'z. The unit vectors is defined as s K x 2, andp = kK + K+ z, so that

u,s, and p constitute one orthonormal set, and K,s, and z another. Our R = x + y.

The Green's function tying fields in region 3 to sources in region 1 is


GE (r, r/, o) d2Kl (s ti+ 3t 1) eik 3z ik zi iK-(R-R'). (3.5)


where the frame decoration G denotes the dyadic form [29], and the Fresnel

coefficients are
2ki
13 kl, + k3

2nin3ki
1cki + 63kA3









3.2.2 Fluctuation Dissipation Theorem

Our hero once again, Albert Einstein first identified the fundamental principle

behind the fluctuation dissipation theorem in his work on Brownian motion [30].

A couple decades later, in 1928, Nyquist found a connection between voltage

fluctuations in electronic circuits and electrical resistance that relies on the same

underlying postulate [31]. The generalized theorem, of which Brownian motion and

Johnson noise are examples, was published by Callen and Welton in 1951 [32]. The

FDT states that in thermal equilibrium a system with dissipation D (w) generates a

fluctuating force F
2 "
(F2) J D (w) (w, T) dw,
7 Jo
where O (a, T) = h with the Boltzman constant k = 1.38 x 10-23 J/K, is

the frequency distribution of an oscillator at temperature T, known as the Planck

function. We need to find the correlation of forces which are due to fluctuating

dipoles. The material has polarization density P. By the FDT

(Pa,) j ( (c) (r, r', w)) (w, T) dw.


The corresponding function for current density J = -iwP is



(J ) (, T) 2 (c) (r, w) 6 (w w') 6 (r r') 6ab du (3.6)
Jo 27r
Here, S (c) (r, w) is the imaginary part of the emitting body's dielectric function.

Note that this form of the FDT presumes a null correlation length, an assumption

which could prove problematic for materials in which the mean free path of charged

particles is long relative to wavelengths given by S (w, T). The other assumption

is that energy levels in the system are densely spaced relative to driving energies,

even at these low temperatures. This enables use of the classical version of the

FDT result.









With the first of these two clues, Eq. 3.3 becomes

E (r, Lw) -P dV d2K-t t eik3ze i'iK(R-R')

f dV d2K-t t J( JK) eikz3zikzlz'iK(R-R') (3.7)
872 J J k U 1 U1
and H is found from Eq. 3.2. Transforming the operator Vx to iu3x, the
magnetic field in region 3 is

H(r, w)= IdV Id2K u3 1x p3 1 ( kJK))
8z v kl Ul U /

.s1 ikz3z iklz' iK(R-R')
31JC C

dv/ d2 l tK-' tl(- 3) 1 +tP ( z- K1J) K
87 V k,I U1 U\
ikz3Z ikzlz'6iK(R-R') 3.

Once we know E and H, we find the heat flux from one object to another by means
of the ensemble-averaged Poynting vector S.

(S)= -R (E (r, w) x H* (r, w)) (3.9)

The cross-product required for the power flux calculation is E (r, w) x H* (r, w)

p dV' f dV" f d2K' d2K" 3 ik'3 ik z6 :6iK(R-R')6iK'(R-R")
(8 2)2 J J J (k' ')

S+ z1 I 1, (3.10)
33 (l Ut1 1 1 I

where J = Ja (r', w). When one takes the ensemble average of the Poynting vector,
terms with the form (JJb*") arise repeatedly. This is the spatial correlation of the
fluctuating currents. We evaluate this quantity with the help of the fluctuation
dissipation theorem (Eq. 3.6). The heat flux perpendicular to the interfaces is

(Sz2T) }= (u;,OT) dl f K fKf *dz' 12C
(S (z, ) (16 73)2 () o K i(k'3-k' ~z i(k'zl-k )z'
( |W--k----l 2 2 < z'







49


1 tU 12 2 \ 2 a/ (3.1 1)

Because the problem has rotational symmetry about the z-axis, we can substitute

KdKdO, with 0 the angle K makes with the x-axis, for d2K. Another simplification

is to evaluate the integral
0 ei(k, dz-k --
-o 2' (kzi)

A third simplifying equation is that =( 2) (kji) (k i). Remembering that

the Fresnel coefficients are
2ki
13 ki + k3

2nin3ki
ciki + C3k3
the expression in Eq. 3.11 may finally be written


(S (z, T)) =

4 J KdK d 2 1 k2 2 (ki) (k
S- r1r3"2iz2d 2 | (k + ki) (k+2 + k,3) 2

I1C2 2 1k2 12 2 (c(k*3) R (Ck3) ]
l t- rp, rp 2id2 z z3~j 12 (3.12)
21 23 113 i l 2 Ikz,2 + 2ki) (C3kz2 + 62kz3) 2
The integral is divided into two regimes. When IKI < Iko k ~ is real, and the

Poynting vector describes propagating electromagnetic waves. The quantity k,3

is imaginary when IKI > Iko and a factor of e2i kz3 makes these terms die away

exponentially with distance from the emitter. These are evanescent fields. Note

that Eq. 3.12 is the same if the roles of emitter and absorber are reversed.

To calculate the net heat flux P between two dielectrics we integrate the
difference between right-moving and left-moving flux and integrate over all positive

frequencies:

P (d, T1, T2) [(Sz (d, w, T))- (S, (0, w, T2)) d (3.13)
JOO









At small vacuum gap widths, well into the evanescent field region, the heat flux

scales with the inverse square of d.

3.3 Numerical Simulations of Evanescent Coupling

We may evaluate Eq. 3.13 numerically once the dielectric functions and

vacuum gap between the media are defined. Care must be taken at the juncture of

propagating and evanescent waves as the equations are prone to infinities here if

not handled well.

Our first test of our new-found calculational ability was a pair of metallic

masses with various microscopic separation. One mass was at 10 K and the other

at 40 K. We use a Drude model for the dielectric function c (w) of chromium:

Nqw fo
C [LI) =- Q (L)) + i-
mU (Y7 iu)

where eb is the dipole contribution, N is the number of molecules per unit volume,

qe the charge of an electron, fo the fraction of free electrons to total electrons per

molecule, m the electronic mass, and 7o a phenomenological damping force. The

calculated heat transfer in W/m2 is shown in Figure 3-3. A near-field enhancement

of several orders of magnitude is evident when the vacuum gap becomes much less

than the dominant thermal wavelength Ath, given by Wien's law:

2.9 x 10-3m K
Ath T


For the hot (that is, 40 K) mass, this number is about 72 microns. For large

vacuum gaps propagating modes dominate, and the heat transfer per unit area

settles to a constant value coinciding with the Stefan-Boltzman law. At small

distances the propagating waves contribute very little, as the density of resonant

modes in the gap diminishes. Evanescent modes more than compensate, however,

as these increase exponentially with proximity.








51

103


102


10


10,





10


10I
10 10 10' 10 10 10'
vacuum gap size (m)

Figure 33. Power flux per unit area as a function of vacuum gap separating two
semi-infinite chromium masses at 40 and 10 K


3.3.1 Effect of Dielectric Function

To test what kind of materials have the strongest evanescent coupling effect,

we evaluated the heat flux between doped silicon masses. Silicon is of particular

interest as it is a candidate material for all-reflective interferometer designs. The

dielectric function used is evaluated from


( 47r 2 i4z'
( w)) -o Qco + w w2+w-))
\2 + LL 2 (aL2 + )

where the plasma frequency up is given by

2 Ne2
2 2
(LLp n *


with N the dopant concentration, e the electron charge, and m* the reduced

electron mass.

Heat transfer enhancement for various dopant concentrations is shown in

Figure 3-4. The Stefan-Boltzman radiation for a perfect pair of emitters is depicted

by the horizontal line at 0.15 W/m2. The more metallic samples are clearly poorer

emitters in the far field, but in the near field the behavior is much more complex.











Doped Si

1o17
10 10
1019
1020




1~* 10
10 1


102
100






103
10-






108 10 10 10 10 10
vacuum gap (m)

Figure 3-4. Power flux per unit area as a function of vacuum gap separating two
semi-infinite doped silicon masses at 40 and 10 K for various dopant
concentrations


There is no monatonic progress with dopant concentration. In fact, if the plot is

sliced along an axis extending into the page, plotting heat transfer versus dopant

concentration for a single vacuum gap width (Figure 3-5), we see rather complex

behavior as the material's dielectric function changes. There is an enhancement

in thermal coupling when resonances in the absorber correspond to the thermal

wavelengths of the emmitter. This makes clear that if we want to use evanescent

cooling in LIGO, we will have to carefully select materials appropriate to the

temperatures, geometries, and gap widths we wish to use. Overall, however, the

effect of dielectric function specifics on heat transfer is much less than the effect of

simply closing the gap width.

3.3.2 Effect of L vli. 1t Media

We also investigated whether 1 ., -1 dielectric on the material surfaced could

enhance heat transfer by acting as anti-reflection coatings. To do this one changes

the Green's functions in the calculation slightly to include reflections at multiple

interfaces. This changes the t',Ps. We found that this had almost no effect on








53

3----------------------. ..... i .... ...
S d=1 39 microns

25-


2-


15-


1-


05-


1016 10" 1018 101 1020 1021 1022
Dopant Concentration

Figure 3-5. Power flux per unit area as a function of doped silicon metalicity at a
1.4 micron separation


the thermal coupling. The reason is that such coatings affect the propagating

modes primarily, which at these small vacuum gap widths contribute little to heat

transfer.

3.4 Noise Coupling Due to Fluctational Electrodynamics

Although we can extract heat from the interferometer test mass without touch

using the near-field effect, it is not true that the cold mass exerts no force on the

hot mass. In fact, the very same dipole interaction that gives us heat-transfer

coupling also leads to the Van der Waals interaction [33].

3.4.1 Van der Waals and Casimir Forces

The similarity to proximity-enhanced heat transfer is evident in the equation

for the attractive force between two flat semi-infinite dielectrics separated by d:


FVdW (d, T) 2- Z j dK duK2U20 (, T) x
27 r2c Jo Jo


(kzj + K) (kz3 + K) .2iKwdc -1 lk, + K) (c3z3 + K) 2iKwd/c -1
[(k K) (kz3 K) (e)kzi K) (3kz3 K)
(3.14)














or hot



cold
hot cold

Figure 3-6. Possible configurations for evanescent cooling of a mirror


If d is large relative to Ath, the above equation becomes

Aa
Fvdw (d, T) 3 (3.15)


where a is the area and A is a material-specific number known as the Hamaker

constant. Metallic masses will also experience an attraction due to vacuum energy

fluctuations in the space between. The Casimir force at separations comparable to

the conditions that make Eq. 3.15 valid is

Cher2
Fcas (d, T) (3.16)
240d4

The dependance of heat transfer on 1/d2 for small distances, and the dependence

of noise forces on 1/d3 and 1/d4 means that there will be a best distance from

the hot test masses at which to place a cold mass. If too far, evanescent modes

will not couple the two strongly. If too close, noise couplings will require us to

suspend and isolate the cold mass as well as the hot, and the problem of heat

transfer simultaneous with isolation is one step removed, but still present.

3.5 Implementation in an Interferometer

3.5.1 Configurations

There are several v--,v a future interferometric detector might take advantage

of the near-field heat transfer enhancement. The small separation distance could be









along the laser optical axis, as shown in the left-hand side of Figure 3-6. It could

be a ring with an aperture for the light beam if the mirror is a transmissive optic.

In an all-reflective design we could access more area, and thus transfer more heat.

Alternatively, one may choose to place the cooling surfaces along the sides of the

optics, as seen in the right-hand side of Figure 3-6. This way, noise forces make the

mirror jitter about in a direction transverse to the cavity lengths we are nii, I-ii:

greatly reducing how strict stability requirements will be.

3.5.2 Stability Requirements for Cold Mass

Displacements of the IT\ i are related to the force exerted by the near cold

object by

d(f dF (f)
4rmf2'
The Fourier transform of dF (t) is

dF (f)=1 + dF
dF (f) (ei(w+n)t + i(w -)t) dt


2 dF [6 (w + ) + 6 ( Q)] ,

this from the fact that


6 (w + ) =( tdt.

The Advanced LIGO displacement requirement at 10 Hz is 10-19 m Therefore

N -
dF = (0-19 ) (47) (40 kg) (10 Hz)2 5.0 x 10-15 N


What requirement does this place on the distance between the two planes?

dF = VdF 5.7 x 10-15
2 2 H/z









With dF specified, we can calculate the maximum allowable displacement of the

cold mass at the noise frequency (10 Hz, in this example):

N Aa x dz
dF = 5.7 x 10-15
VHZH 27z4

With A = 6.6 x 10-20J (for SiO2), a = x (0.2m)2, and zo = 10pm, dz may be no

greater than 4.3 x 10-14 m/vHz. This number can be achieved without suspending

the cold mass.

This calculation assumes the worst-case scenario in which we must put the

distance d along the optical axis.

3.6 Experimental Proof

In the near future, the University of Florida's LIGO group will conduct

experiments to measure the enhanced heat transfer effect between large, flat

objects. They will evaluate the pertinence of these techniques to future versions of

LIGO.















CHAPTER 4
WHITE-LIGHT INTERFEROMETRY

In this chapter we are peering into the misty distant future of gravitational

wave interferometry. Every foreseeable innovation is either a way of reducing noise

or of moving sensitivity .1-liv from unavoidably noisy frequency regions. Without

simply adding more stored light power to the detector, the sensitivity theorem [14]

insists that these are the only options for catching more gravitational waves. When

detector bandwidth is broad, with broader cavity resonances, the sensitivity hits a

shot noise ceiling rather quickly. When the cavities have high finesse we stub our

toes against the edges of a cramped bandwidth. We want spacious accommodations

in both dimensions. What we would really like is to defeat the sensitivity theorem.

Warning: such an ambitious chapter cannot end well.

4.1 The Sensitivity Theorem Quandary

Advanced LIGO [12] will have a peak sensitivity at about 300 Hz, with a 1/f

decline in responsiveness above the peak. The anticipated sources in the several

100 Hz range are pulsars, the fundamental frequency of the intermediate phase

of a neutron star-neutron star merger, and the final fundamental frequency of

small black hole-black hole coalescences. High frequency pulsars, the harmonics

of neutron star and black hole mergers, stellar core collapses, and neutron star

oscillations, however, are all expected to emit gravitational waves within reach

of Advanced LIGO's sensitivity, but above its bandwidth. An instrument with

at least 20 kHz of bandwidth is needed for these sources. The bandwidth of

LIGO-like terrestrial interferometric gravitational wave detectors is set by the

pole of the Fabry-Perot cavities within the arms of the Michelson interferometer.

This constraint arises because the gain of gravitational wave-induced signal

57









sidebands is limited to frequencies within the linewidth of the cavities. The nature

of standard Fabry-Perot cavities is such that one cannot independently adjust

for increased gain without suffering a loss of bandwidth. If these quantities could

be decoupled, the resulting improvement in bandwidth may lead to viable high

frequency detectors. We guessed that a diffractive element placed within an optical

resonator could increase the cavity bandwidth without loss of peak intensity. As

we discovered, this expectation was based on erroneous (but not uncommon, even

among optical scientists) understanding of diffraction grating function.

4.2 The Linewidth-Enhanced Cavity

4.2.1 The Nature of the Gain-bandwidth Dilemma

Consider a simple Fabry-Perot cavity with arm length L and input and output

mirror amplitude reflectivities and transmissivities rl, tl and r2, 2, respectively.

The normalized intensity of light of a particular frequency w = 27rc/A within the

cavity is
t2
I (o) 1 2 (4.1)
21 + 2 + rj-2 Or2coS (I (O))

with ) (w) = 2wL/c, a frequ'-'i- i -dependent round-trip phase shift. When, as in

initial and Advanced LIGO, r2 1, and the mirror transmissivities and losses are

very small, the maximum of this function can be approximated as

i (Wo) 4 (4.2)

The intensity is maximum for one frequency of light (and periodically for every free

spectral range thereafter) for which < (A) = 27r, with n an integer. The intracavity

intensity decreases for larger or smaller frequencies at a rate determined by ri

and r2. The full width at half maximum of the Airy peaks in the intensity that

correspond to cavity resonances is FWHM = FSR/F, with the free spectral range

FSR = c/2L and the finesse F = x/r-r2/ (1 rir2). Again consider the LIGO arm









cavities, with highly reflective mirrors. One finds that


2wr
F 27 FWHMcN t2
t2


(4.3)


It is clear that linewidth and peak light intensity are inextricably intertwined in

a standard Fabry-Perot cavity. As a result, gravitational wave scientists, who

would like to maximize both quantities, must compromise in choosing the cavity's

parameters.

4.2.2 "White-l5!lII Cavities

The logic of the diffraction-enhanced cavity was as follows. The round-trip

phase shift's variation with frequency has been identified as the source of the

gain-bandwidth dilemma, immediately -i-i- -1ii-; it as the focus of design

alterations. Perhaps K could be made invariant with frequency if the the optical

path length inside the cavity were also freC qu-i' ,i-dependent:


2wL(w)
S(U) 2- constant
c


(4.4)


Making the constant an integer multiple of 27 would presumably ensure that

light of any frequency resonates inside the altered cavity. We deemed this the

S.-lite-hl;h cavity.

We posited that a pair of anti-parallel diffraction gratings placed inside the

cavity create a frec -iiu' ,- -dependent cavity length that cancels the free-space

dispersion of the resonator. The arrangement shown in Figure 4-1 would work as

follows:

1. Monochromatic light is injected into the cavity.

2. A gravitational wave modulates the light-transit time between the input

mirror and the distant first diffraction grating such that frequency sidebands

are generated on the original laser light.















Figure 4-1. Grating-enhanced cavity


3. The parallel gratings allocate a different path length to light of different

frequencies.

4. As seen in the figure, the redder lower sideband (dashed line) travels a longer

distance than the bluer carrier (solid line) or upper sideband (dot-dashed

line) light. This additional path can cancel the variation of the round trip

phase shift relative to the carrier light.

Figure 4-2 shows the round-trip phase resulting from the fre uii'- -in'-dependent

optical path length and compares this to the fre-u'ii-_',-invariant + (w) of a

standard cavity with the same nominal length. If changing length with frequency

were the only effect gratings had on the light phase, the grating-enhanced cavity

would have a superior linewidth wherever its slope is less than that of the standard

cavity. Figure 4-3 shows the intensity build-up in a LIGO-scale enhanced cavity

as one varies the laser frequency, using the theoretical calulations above. This

curve is the theoretical performance of a grating-enhanced cavity with 4134 m total

one-way length, a spacing between the grating planes of 71 m, a grating constant

of 1633 lines/mm, a grating incident angle of 54, and a laser wavelength of 1064

nm. The plot assumes the slightly idealized case of lossless gratings. Also shown

is the resonance width of a standard cavity. The grating-enhanced linewidth has

increased by about a million times. We note at this point, however, that this static

response to changing frequency is not equivalent to gravitational wave response,

as discussed below. Real gratings with losses will require greater laser power to

reach the same maximum signal build-up inside the cavities as the standard cavity

case. There is no theoretical limit to diffraction grating efficiency. Grating designs











2 6e+10



phase (rad)





5e+12 4e+12 3e+12 2e+12 le+12 0 le+12 2e+12 3e12 4e12 5b 12
frequency (Hz)

Figure 4-2. Comparison of round-trip phases for grating-enhanced (solid line)
and standard (dashed) Fabry-Perot cavities. Frequencies on abscissa
S f aser


that use methods beyond the scalar approximation routinely produce efficiencies of

essentially 10 r'. [34]. Actual grating fabrication is rapidly catching up to theory.

Finally, then, if one puts mirrors at the places shown in the figure, and adjusts

the lengths of the common paths and the inter-grating spacing D correctly, it

should be possible to arrange for each color that the ratio of free-space path to

the wavelength is the same integer value. If this were so, and if the gratings had

no other effect on the phase of the light waves, then the device shown would be a

cavity resonant for all wavelengths, a v .ite-lh,!i cavity. Detailed calculations

[35] show that the bandwidth of this cavity would in fact be finite (because of

the non-linear dispersion of the gratings) but would be many orders of magnitude

larger than the bandwidth of the typical Fabry-Perot cavity, such as the ones in the

arms of the LIGO detector.

4.3 Gravitational Wave Response in the Time Domain

If anti-parallel diffraction gratings indeed make light's round-trip phase shift

within a cavity frc uii'- i,'-invariant, it would then be helpful to compute the

gravitational wave interaction of such a cavity in the time domain. All previous

calculations concerned only an effect equivalent to varying the input laser's










10,

10

\ 10
N \
\


10 \



10 ,
103

100 10' 102 103 104 105 106 107 10
frequency (Hz)

Figure 4-3. Linewidths of grating-enhanced (solid line) and standard (dashed line)
cavitites. The standard cavity curve is cut off at 75 kHz so as not to
obscure the enhanced cavity plot. Frequencies = f flaser


frequency, which is not the same. We begin by transmitting an input field Eoe(iwto)

through the left-hand mirror, ITM, with amplitude reflectivity and transmissivity

rl and tl. The parameters for the end mirror ETM are similarly defined. After

transmitting through ITM, the field at time t is E(t) = itlEoe(it). Consider

the traceless, transverse gauge interpretation of the gravitational-wave effect.

The positions of the IT\ i and ET\ i remain fixed, but the space-time metric does

not, resulting in a change in the time in which light travels from one mirror to

the other. Two assumptions are used in this calculation. First, assume that the

GW-strain is sinusoidal: h (t) = ho sin (Qt). Second, if the strain h is small, we may

take for granted that the difference between a null ray path and a geodesic will be

of order h2, and thus negligible. A null ray is described by


(ds)2 (dt)2 + (1 + h+)d2 + (1 h+)dy2 + dz2 = 0


Thus the time in which light travels from the ITA\ [to ET\ I a spatial coordinate

distance L/2 away is described by

p14) t i L
/ dt +- / + h, cos (Qt)dx
Jto C 0








Here, the zero of the x-coordinate is the coated surface of the IT\ i mirror. The
ETA F resides at position x = A. The above equation may also be written:

L t(1 ) dt h-
(t o cos(Ot))dt
2c to + ho cos (t) io 2


St(L) to h[ sin (t (L))- sin (Oto) (4.5)
2/ 20/ / 2
Keeping terms of linear order in ho, that is, substituting to = t () within
the argument of the sine function, we approximate to:

to (L L h, (L sin ( ((L\ L\\

(L\ L ho L L L
=-sinm cos (0 t (L
2 2c 02 4c 2 4c}
We now substitute the new interpretation of to into the original input laser
field.
it Eoeiwto

it1Eoeiw(t()-_ ) l 2 sin e)(t()-L) + e-C(t(-)- )

where we have expanded the quantity e sin( ) cos(2(t(4) )) to first order
in ho and expanded the cosine term. One can now interpret the effect of the
gravitational wave on the input laser light as phase modulation creating two
frequency sidebands with frequencies w = W 2. The grating compressor
within a white-light cavity will also see phase-modulated light, and will donate an
appropriate phase shift in reflection to each frequency component.
Let the compressor length for the fundamental carrier frequency be ALo. The
sidebands travel lengths ALL = ALo T 6L, where 6L assumes a linear dependence
of the compressor length on the gravitational wave frequency 0, a valid assumption
as the GW frequencies are much smaller than the optical frequencies. With this
approximation we define:












The quantity t (f) is to be interpreted as the time the sideband fields must
reach the first grating of the compressor in order to emerge simultaneously (in the
local frame) with the carrier light. We now replace t (L) ofor the three frequency
components.
The carrier:
it1r2Eoe~w(t(f+ALo)-L _-o)

The upper sideband:

-i ho QlL\ L+n)L(t(+ALo)-L) i- ia
itlr2Eo sin 2 ce 2c e 4c
2Q 4c 9

The lower sideband:

-iwh0o QL i(> -n)(t( +AL0o) At iL imL
2 s 4c
itlr2E 2Q ( ( 2e g e

Iterating the process above, the time at which the light returns to the ITM
may be approximated:

t + ALo) t (L + ALo) sin ) cos t (L + ALo)
2 2c Q 4c 4c

In propagating the light back to the ITM, we assume that phase modulations
of phase modulated light may be neglected, as these are of order h2, and the
sidebands merely return with the appropriate phase. Modulations of returning
carrier light, however, are significant.
The carrier light becomes:

iw(t(LAL)- ) i + sin i(t(L+AL)- .c.
2Q ( 4c)I


and we note that further frequency sidebands are born.










The upper sideband returns to the ITM:

itrE -iwho. QL i t(L+ALo)- -i
itlEo sm e c e 2c e 4c
2Q 4c 4

Likewise the lower sideband:


who L\ )i(w-n)(t(L+ALo)-_
itlr2Eo sm e
t~r2 0 2Q 4c,2


Thus the total returning field is:


it1r2Eo [e(t(L+Lo)- L AL)]


-iT2lrEoi h sin (IL ei(w+Q)t(L+ALo)

i2h0 ( L4c

itr2E ho sin OL Q)t(L+ALo) C
-Ztl2Eo- sin e e
20 4c s



itr iEho \ i OLh C i(w+Q)t(L+nA C (L
20 sin (4c


AL- -'iL ML
c / 2c e 4c


iw(L+ALo)
c e


-i(L+ALo) MfL
c e 4c



AL+) -( +aL+)
c P c


']1 .
-itlr2Eo- sin
202Q


(OL ei(_n))t(L+ALo) e
4c


i(L ) i(3L +AL+)
e


Thus the total field may be expressed in an illuminating manner as follows.

First the carrier component:


itr2Eoe w(t(L+ Lo)- Ao)


with an upper signal sideband:


tlr2Eowho
sin
20


i(L i(w+Q)t(L+zALo) -(L+ALo) ML i+ L 3iQL -iALo iQ6L
S4ce ) C 4c c C 4c c
4c-









and the lower:

tir2Ewho (L nei( -)t(L+ALo)e -L+Lo) e L i6L 3iL MiALo -iQL
,nsm e e 4c + c C 4c C c C c
2 4c

The design of the white-light cavity, in its idealized form, ensures the

cancellation of the variation in phase with wavelength for the intra-cavity field.

The white-light condition is:

d d (27rLw (A)


where Li (A) is the total half-trip cavity length for a particular wavelength. The

above equation leads to:

dLi (A) L.i (A) dLwl (wi) -Lwi (Lwi)
dA A dwji i UwI

Converting this equation to the language of our previous calculations, in which

dL (uw ) = ':L, duwi = +, L (uw ) = L + ALo T 6L, and uwi = a Q, we find, for

the upper sideband:



TL I[ L ( \1 i
dL -= -dw dwu = -Ltot In +
JAL j Lou ) 0


S(L + ALo 6L) -=> u6L = QL + QALo 6L

When this white-light condition is applied to the above equation for the upper

sideband, its amplitude and phase becomes:


tl2Ewh, sin L i(w+Q)t(L+ALo) i-(L+ALo) [-iQL L]
2 2 \- 4c,)


tEouho I i(w+n)tL -(L+ALo).1 L
(rr2,) C C c. sin 1-)
2Q 2c









for light that has made one round-trip of the cavity. Carrier light that has made n

round-trips of the grating-enhanced cavity will have upper sidebands:


t Eowho (L\ cL+ALo)* ei(w+)t
tE sin I n (rir2) e ot
20 ( 2c
The positions of the optical components are fixed while the gravitational

wave is interpreted as a phase modulation of the cavity's internal light field. The

diffraction gratings (assumed perfectly efficient) will "see" phase modulated light,

and will diffract each of the three frequency components into a different path.

The lower sideband, carrier, and upper sideband are now also separated in time,

emerging sequentially from the compound mirror, though all with the same phase.

The gravitational wave's manipulation of space-time fashions further frequency

sidebands from the carrier light as it returns to the cavity input mirror, where

it interferes with the incoming laser light before it propagates again through the

cavity. Summing over all sidebands created in the carrier from the infinite past

yields an amplitude transfer function for the upper signal sideband of the form1


twLsin ) 2icL
TWLC () Lsin 2 (4.6)
2c 1 rir2e cI

Let us compare this equation to that of a standard cavity, whose upper sideband

transfer function has the form:
2i(w+Q)L
S) t1Lsinc ( r-L) rir2e2i(
TcP Li)=c i-i(Lw+)t (4.7)
2c 1 rIr2e c 1 r1r2e

Because the cavity's length is locked to the laser, the term e-2iwL/c = 1, hence, the
e-2iL/c term in the denominator of 4.7 is the cause of the standard cavity's limited


1 B. F. Whiting, S. Wise, G. Mueller, 2003, unpublished.











10,

10-2



10

10
: 1\






10'
E\









Figure 44. White-light cavity's gravitational wave response. The solid line is the
normalized intracavity intensity for a white-light cavity; the dashed is a
standard cavity. Both curves have minima at multiples of the FSR.


bandwidth. Figure 4 4 compares the sidebands intensities within standard and

perfect white-light cavities with identical lossless mirror reflectivities (r = 0.995,

r2 = 1) and nominal cavity lengths (4134 m). The gravitational wave strain has
10'

10' \\

10' --
10 10\ 10, 10, 104 10,








maximum amplitude h0 = 10-23. The zeo of the white-light curve, due to the s in

unction in the numerator, correspond to frequencies for which the cavityhe dashed is a








time causes the signal to be integrated over a full cycle. This effect is not evident
in LIGO's sensitivityandard cavity. Both curves, where the arm cave minima at multiples ofdiminish the sensitivity


well before these fFigurequencies. We see then that a effect of the alleged white-light
perfect white-light cavities with identical lossless mirror reflectivities (r2 0.995,








2 cavity non the LIGO bandwidth The gravitational wave stincrease.








4.4 Revised Theory
maximumWe now describe measurementhe zeros of the phase shift of light curve, due to the sin

grating set which show that the grating-o frequencies fsor white-light cavity concept is

time causes the signal to be instead, the pair of gratings provides a wavelength-dependent

phase shift nearlnsitivy cancelling the phase from the additional free-spach the path lengthivity

shown in Figurre these 5 [36, [frequencies. We see then that a effsuspect the above analyssleged white-light
cavity on the LIGO bandwidth would be a three order-of-magnitude increase.

4.4 Revised Theory

We now describe measurements of the phase shift of light by such a parallel

grating set which show that the grating-compressor white-light cavity concept is

almost ,,- ,ml. /. h/ wrong. Instead, the pair of gratings provides a wavelength-dependent

phase shift nearly cancelling the phase from the additional free-space path length

shown in Figure 4-5 [36], [37]. We began to suspect the above analysis when















SM2

0

Figure 4-5. A pair of identical, parallel, face-to-face diffraction gratings and two
mirrors (\ll and M2) form a resonant cavity.


repeated measurements of grating-enhanced Fabry-Perot cavities, Michelson and

Mach-Zehnder interferometers all has virtually identical frequency behavior to their

gratingless counterparts. For example, measurements of the resonance bandwidth

of Fabry-Perot cavities containing high-efficiency gratings and configured to be

v lite-h;ht cavities were unchanged if we removed the gratings and restored

the same nominal optical path length. We found no enhancement of bandwidth.

Here we show why the expected enhancement does not occur. Moreover, we also

show that the phase depends not only on the inter-grating -I' ii.- but also on

the exact relationship between the grating features as seen by the light [38]. This

dependence leads to the non-intuitive result that the phase is modulated strongly if

one of the gratings is translated parallel to its face, even though the optical paths in

Figure 4-5 are wholly unaffected.

4.5 Plane-wave Treatment of a Grating Compressor

The error lies with an inappropriate mix of geometrical and physical

optics. Consider the effect of the parallel grating pair on infinite plane waves.

In Figure 4-5, parallel reflective gratings are located in the y = 0 and y = D

planes. We will calculate the electric field at the two gratings and in a plane

normal to the outgoing light (where the right-hand mirror in Figure 4-5 is located).









The light field impinging from the left on the first grating is


Ein= Eoeik(xsin a-ycosa)(48)


The grating at y = 0 bestows a spatial phase modulation on the incoming plane

wave. The phase factor is e6kG x-0o); the periodic function G (x x') represents the

grating profile, with origin at x'. This phase factor may be expanded in a Fourier

series:

C eik0(r <) meimP(r-<9 (4.9)

where g = 27/d.

Each term of the series is a diffraction order. In the following we consider only

the m = -1 order, set C_1 = and chose x' = 0 for the first grating. The light

field leaving that grating is

EF,out = Eoei[(ksin a-g)x+kycos/] (4.10)


where we have used the grating equation to substitute k sin a + mg for -k sin / < 0.

When the light reaches the second grating, on the y = D line, it again receives a

spatial phase modulation ei9g(o-a). The quantity xo is the x-offset of the second

grating's periodic modulation with respect to that of the first grating. Note that

the second grating is reversed relative to the first, so that its local coordinate runs

in the -x direction. We again use m = -1, making light of all wavelengths leave

the second grating parallel to the incident light. The outgoing electric field at the

second grating is

E2,out Eoi[k( in a+D cosp)-gxo (4.11)

When the light finally arrives at a point (x, y) on the right-hand mirror of

Figure 4-5, the electric field will be


em = Eoei[k{xsin a-(y-D)cosca+Dcos/3}-gxo]


(4.12)









We must now analyze how the phase of the field at M2 changes with light

frequency. The phase K (w, x, y) is


S= [x sin a (y D) cos a + Dcos ] gxo (4.13)


L (w)- g (D tan 0 + o), (4.14)
c
where L (w) is the total, wavelength-dependent, geometric path from the first

grating (at the origin) to the end mirror [35]. We compute the dispersion 0/0uw,

using the grating equation to eliminate 3/j0w, and find


S xsina (y D)cosa + D + tan K sin a (4.15)
OUj c cos )\ c

Eq. 4.15 makes it clear that the variation of phase with frequency cannot be set to

zero. The earlier calculations based on the grating equation alone included only the

geometric pathlength contribution in Eq. 4.15, leading to the (incorrect) prediction

that a0)/aw could become zero, thus allowing for the possibility of a white light

cavity [35]. Missing from reference [35] was the second term in the right-hand side

of Eq. 4.14, present due to the position-dependent phase shift light receives upon

reflection from (or transmission through) the grating.

The result of Eq. 4.15 is familiar to short-pulse laser physicists as the group

d. 1 v [39]. The exact form of the additional phase shift is rarely a concern, as the

pulse compressor does not depend upon absolute phases. To our knowledge, direct

experimental verification of the phase shift's form has never been published. The

phase shift of Eq. 4.13 may be expressed in a particularly illuminating way as


+ (w) = [Lo + D (cos a + cos3)]- gxo (4.16)
c

where Lo is the perpendicular distance from the origin to the plane of M2, defined

by L = x sin a-y cos a. Analyzing the grating compressor with plane waves reveals

the origin and significance of the position-dependent phase shift on reflection from










S/ to photodiode




c ^ Y actuator





stage 'Cact5 V

laser put to leHe N r lterferometer


Figure 4-6. The second of a pair of parallel gratings within one arm of a
Mach-Zehnder interferometer is placed on an x-y translation stage.
Michelson interferometers with He-Ne lasers monitor the motion of the
stage.


the gratings. We may now resume calculations with the geometrical optical path,

so long as we do not neglect the additional phase associated with the gratings.

This theory makes a very specific prediction, which may be experimentally

confirmed, about how the one-way phase shift depends on the distance D between

gratings and the spatial offset xo between grating profiles.

4.6 Tests of Corrected Theory

4.6.1 Experimental Setup

To test that the phase shift does have the specific form of Eq. 4.16, we

incorporated a pair of gratings into one arm of a Mach-Zehnder interferometer,

as shown in Figure 4-6. We used reflective gratings with 1500 grooves/mm and

a design input angle a = 420 (so that 3 = 680) We also used an input angle of

a = 500 ( = 57.30) in some of our trials. The gratings had a high efficiency, with

!I! 'I.'. of the incident light diffracted into the first-order by each of the two. We

placed the second grating of our grating pair on a two-axis translation stage. This

stage allowed us to vary the parameters xo and D of the grating pair. We aligned









the grating face with one of the two orthogonal axes of the stage, which we will call

x' and y', as well as was possible with a naked human c- A small misalignment

angle 0 inevitably remains between the grating axes (x, y) and the stage axes (x',

y'). The axes are related by x' = cos 0 + y sin 0. Assuming the construction of

the translation stage is better than our ability to place the grating, we also have

y' = sin 0 + y cos 0. As the stage moves along the x' direction, it will produce

a combination of the effects on phase due to the phase's xo and D dependence;

however, because 0 is small, the influence of xo, with a period equal to the grating

period, will dominate. The converse is true when one moves the stage along y'.

To calibrate the displacement of the translation stage, we attached to it the

end mirror of a simple Michelson interferometer illuminated by a helium-neon

laser. In fact, there are two mirrors (and two interferometers) set perpendicular

to the two motions of the stage. To ensure that perpendicularity, we move the

stage in the orthogonal direction, so that the mirror slews crabwise across the

He-Ne beam, and adjust its angle relative to the stage until we reduce the number

of output intensity fringes to a minimum. Again, this technique relies upon good

inherent perpendicularity in the stage's crossed axes. Whenever the stage moves,

we monitor the output intensity of both interferometers. We quantify the stage

motion by counting the He-Ne fringes.

The input to the Mach-Zehnder interferometer is a 1064 nm-wavelength

grating-stabilized diode laser. For good contrast, the physical lengths of the two

arms are nearly equal. We move the second grating along the x' and y' axes and

observe the intensity fringes of the infrared interferometer and compare with

theory.









4.6.2 Experimental Results

While the light input angle a and the grating period d are known, 0 remains as

a fitting parameter. The output intensity of the Mach-Zehnder is fit to


I (x') = A + B cos [K (x') + C], (4.17)


where
2xr 2w
KP (x') = (cosa + cos ) x' sin0 + x' cos 0, (4.18)
A d

for the x' motion, or to


I (y') A + B cos [ (y') + C] (4.19)


with
2w 2w
S(y') (cos a + cos /) y' cos 0 y' sin 0, (4.20)
A d

for the orthogonal direction. The quantities A, B, and C are rather unimportant

fitting parameters; the period of the output fringes determined by + is key. We

make a least-square fit of the theory to our data by adjusting A, B, C, and 0.

Figure 4-7 shows examples of typical results for a trial with a = 50.

Figure 4-7 (top) shows the interference seen for movement along the

x'-direction, i.e., when the grating moves parallel to its face. This motion gives

strong fringes; the measured fringe contrast is in the 92 !' range. Now, motion

parallel to the grating face has no effect on the geometric path lengths inside

the interferometer. Thus, our initial expectation (based on the geometric path

length) was that the light phase would be unaffected by this motion. In contrast to

this expectation, the phase of the light goes through a full cycle as the grating is

translated by an amount d.

In Figure 4-7 (bottom), we show the interference signal observed for motion

along y'. We also plot, in addition to the predicted output from the theory above,

the output intensity that would be observed if only the geometric path length were






























0.6 -

S0.4 -

0.2

012345678910






0 1 2 3 4 5 6 7 8 9 10
Displacement along x'(pm)



0.8

S0.6 -

0.4 -

0 .2 -'

0 1 2 3 4 5 6 7 8 9 10
Displacement along y'(pm)

Figure 4-7. (Top) Measured (crosses) and theoretical (solid line) data for motion
parallel to the grating face. The light is incident at 500. (Bottom) The
crosses show the data for motion perpendicular to the grating face. Of
the two calculations, the results favor the plane-wave treatment, with
the additional phase shift on reflection (solid line) over one based on
geometric path length alone (dotted line).










05
04-
03





-0 2-
-02-
-0 3-
-0 4-
-0 5
0 5 10 15 20 25
trial

Figure 4-8. Misalignment angle 0 as determined from fits of measured interference
data to theory. The meaning of the symbols is as follows. Crosses:
motion along x' with a = 500; Circles: motion along y' with a = 500;
Asterisks: motion along x' with a = 420; x's: motion along y' with
a = 420. The data in Figure 4-7 is from trials 2 and 8.


changed by the grating motion. Use of geometric path alone predicts a period

for the interference pattern that is different from the measured one, whereas the

theory that incorporates the position-dependent phase shift predicts the period

that we measured. The outcome makes clear that the setup cannot be understood

with only the diffraction angle and the geometric path length. The additional

position-dependent phase shift is real [40].

Figure 4-8 shows an indication of the agreement between experiment and

the theory presented here. In it, we plot the misalignment angle 0 of the grating

for a number of trials. Eleven of the twenty-three measurements are derived from

grating motion parallel to its face, and twelve from perpendicular motion. In every

case, we obtained high-contrast fringes with agreement with theory comparable to

what is shown in Figure 4-7. The error bars on each datum reflect uncertainty in

a, d, 0, A, and the motion of the stage. Clusters of specific values for 0 indicate

the systematic error in the alignment of the grating on the stage, but the overall

errors are very small. The quality of the fit to the measured interference pattern

is evident in Figure 4-7 and in the small values for the misalignment angles in









Figure 4-8, averaging 0 = 0.030 0.120, a reasonable value for alignment by human



The phase of light reflected by or transmitted through a diffraction grating

cannot be deduced from the grating equation alone. That equation omits the

curious result, derived above, that the absolute phase is proportional to the

distance along the grating face at which the light strikes. Indeed, the flat gratings

behave as mirrors tilted at angles Otilt = sin-'(mA/2d) relative to the x-axis shown

in Figure 4-5. We confirmed this theory by testing the dependence of light phase

on the position of the grating. For a grating-compressor setup, we found good

agreement between this theory and the change of light phase as the mirror moved

both parallel and perpendicular to its face. Our result shows that white-light

cavities cannot be built from grating pairs. In fact, one might have conjectured

that causality should prevent white-light cavities from being built in a much wider

class of non-dissipative systems-not just grating pairs. Indeed, we have found that

a pair of prisms has a similar effect to the gratings on the phase of light passing

through them. Finally, we note that the phase effect discussed here is not unique

to the grating pair and would arise in an experiment utilizing a single grating. In

our arrangement, the first grating is fixed and serves to preserve the beam width

and to keep the angle of the light leaving the second grating constant as wavelength

is adjusted. Otherwise, it is equivalent to a mirror. Except for a loss of contrast,

we expect that the data of Figure 4-7 would be identical if the first grating were

replaced by an appropriately oriented mirror.

4.6.3 AOMs

Interestingly enough, it seems a clue to the missing phase ili-. i,- was under

our noses, indeed, in our optical cabinets, all along. If one wishes to make further

measurements of the effect of a moving grating on diffracted light phase, an

acousto-optic modulator (AOM) is a good candidate. In an AOM one applies









an acoustic waveform to a crystal, sending pressure waves moving through the

material at speeds on the order of thousands of meters per second. Light diffracted

off this moving grating inside the crystal should exhibit the same phase shift

behavior found in our slow-moving grating experiments. How would one see the

phase shift effect? It would fre lpii- -modulate the diffracted light. In fact this

Doppler-shifted diffraction frequency shift is a well-known phenomenon, but is

rarely if ever connected in literature or discussion to grating compressors!

4.7 Alternative Solutions

The extra phase shift due to transverse position along the grating at first

appears to cancel the linewidth-broadening perfectly by coincidence. However,

similar experiments with prism pairs, where no such hidden phases exist, yielded

the same result. We believe this is a manifestation of nature's jealous guardianship

of causality and speed limits. Our initial analysis confused group delay with phase

d 1 i We speculate that the only way a white or even broadened cavity is possible

is when there is a way to delay the light significantly with resonant absorption or

a gain medium. There are several v--- in which this delay modification may be

accomplished. The required anomalous dispersion is found in the center of atomic

absorption lines, but the absorption would generally suppress any gravitational

wave-induced signals. Wicht et al. [41] studied optically pumped atomic resonance

systems and proved that one could have an appropriate anomalous dispersion for a

white-light cavity at a point of vanishing absorption and optical gain.

4.8 If Alternatives Succeed

The final design for a full white-light interferometer will include two

white-light cavities in its arms. Because we no longer need worry about limiting

our bandwidth by increasing the arm cavity mirror reflectivities, these may be

matched to the losses in the bandwidth-enhancing black box for optimal coupling.

Thus we may forego both power- and signal-recycling mirrors at the symmetric







79

and anti-symmetric ports, respectively. This has the added advantage of removing

power from the thick substrates of the beamsplitter and input mirrors, reducing

thermal loading problems.















CHAPTER 5
CONCLUSION

To move from first detection of a gravitational wave to gravitational

astronomy with future detectors will require innovative noise suppression and

signal enhancement. The parallel phase modulation technique for length-control is

straightforward in theory and execution. We derived a stability requirement for the

extra degree of freedom in the parallel versus serial modulation scheme that was

achievable and reasonable. This technique could be applied to the next generation

of interferometric detectors with little problem, we believe.

The near-field cooling effect is robust in principle, but we still lack definitive

experimental proof that Rytov's heat-transfer calculation method is correct for

planar geometries. Forthcoming proof-of-principle experiments at this institution

will be difficult. We will be challenged to keep two objects separated by a tiny

distance, fighting Van der Waals and Casimir forces. We will compare the outcome

of such experiments with alternative cooling methods such as those used in the

cryogenic Japanese LCGT [42].

As for white-light cavities, it seems there will be no such thing as a free lunch.

Despite a (too) promising theoretical calculation, line-width broadening remained

elusive, and we realized we had repeated a surprisingly common mistake in optics.

The sensitivity theorem is not to be taken lightly. If we could yet achieve the

necessary dispersion with atomic resonances, we still have a far more complicated

system than the failed diffraction-grating setup. Nonetheless, the rewards of a

broad-band Fabry-Perot cavity would be a tremendous boon to gravitational wave

detection. We retain an iota of hope for this technique.















APPENDIX A
OPTICAL TRANSFER FUNCTIONS

A.1 Fabry-Perot Cavity

Using the convention that fields passing through a partially transmissive optic

receive a 90 phase shift relative to reflected fields, the amplitudes immediately to

the right of the first optic is

Ea = itlEiE + rlEb

Eb r22ikLEa,

where k is the wavenumber of the light. Combining these equations, one finds the

total steady-state intracavity field:

itlEj,
L itl (A.1)
Ea 1 rir_ -

from which the cavity transmitted field can be found by transmission through the

left-hand optic:

-tlt2eikfLE,
Et = it2eikLEa 2 Tt (k, L) Ej,. (A.2)
1 r~1ri-2AL

The reflected amplitude is the sum of the light promptly reflected from and the

amplified light leaking through the input mirror:

-r + (r2 + t2 2ikL
Er = rEi, + itlEb 1 l i I T, (k, L) E (A.3)


A.2 Three-Mirror Coupled Cavity

Understanding of the three-mirror coupled cavity aids in a simplifying

perception of any more complicated optical system, as these usually behave in

a similar manner. A three-mirror nested cavity is decipted in Figure A.2, with























Ea



Eb

r1, tl


Figure A-1. Electric fields in a Fabry-Perot cavity


En

Er


Et


r2 t2 r3 t3


Figure A-2. Electric fields in a 3-mirror coupled Fabry-Perot cavity


Ein



Er


Et







83

mirror parameters as indicated. Having found the transfer function of fields

reflected from a single cavity above, we can treat the middle and rightmost optic

as a single mirror with a complex reflectivity given by Eq. A.3. For example, the

intracavity field between the left and middle mirror is then


3 it E (A.4)
1 rircave2ikL1

where
-r 2 (r + t2) 2ikL2
-cav 2r i'2
1 r2r 2















APPENDIX B
HEAT TRANSFER THROUGH FIBERS

A suspension fiber of diameter D and length Lfiber, connecting two heat

reservoirs at temperatures T1 and T2 conducts a power P

p kBD2 T1 T2
4 Lfiber

which for silica fibers with a conductivity K = 1.4K with D 0.1 mm
mK
Lfiber = 0.2 m, TI = 40 K, and T2 = 10 K, gives [43]


P 1.65pW.















APPENDIX C
CIRCUIT DIAGRAM

All operational amplifiers are OP27. The circuit elements are as follows:

Table C 1. Values of elements used in the MZ feedback circuit

element value
R1 1 kQ
R2 1 kQ
R3 1 kQ
R4 1 kQ
R5 100 kQ
R6 1 kQ
R7 1 kQ
R8 1 kQ
R9 1 kQ
R10 2 kQ
Rll 1 kQ
R12 1 kQ
R13 220 kQ
C1 0.1/pF




































Vout


+15Vo
R7 R1 R12

R10


-15V -

Figure C 1. Diagram of analog ciruit used for MZ feedback control















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