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Ultra-Stable Interferometry for Dark Matter and Gravitational Wave Detection

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Ultra-Stable Interferometry for Dark Matter and Gravitational Wave Detection
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Ferguson, Reid
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The stability of an optical bench design is explored. The ALPS collaboration requires a light-tight yet stable central breadboard. However, low-expansion materials (LEMs) commonly used as optical benches in the field of laser optics are not light-tight. The ALPS group at the University of Florida designed a composite breadboard to solve the issue. To test its dimensional stability, an optical resonator (cavity) is constructed by placing two collinear mirrors on optical mounts anchored in an LEM block through a light-tight aluminium layer and is placed in a vacuum chamber. A laser field is aligned and mode-matched to this cavity, and then locked to a resonant frequency via the Pound-Drever-Hall method. The same is done to an ultra-stable LEM reference cavity in another vacuum chamber with another laser. The beat note between the two lasers is analyzed to determine the relative stability of the path length between the anchored optics. The correlation of the beat note frequency with temperature is measured and its periodic and stochastic noisiness is analyzed in a spectral density plot. Both measurements find it to be suitable for ALPS. ( en )
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Awarded Bachelor of Science, summa cum laude, on May 8, 2018. Major: Physics
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College or School: College of Liberal Arts and Sciences
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Advisor: Guido Mueller. Advisor Department or School: Physics

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Copyright Reid Ferguson. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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Ultra-StableLaserInterferometryforDark MatterandGravitationalWaveDetection AThesisPresentedinPartialFulllmentof theHonorsBachelor'sDegree ReidFerguson DepartmentofPhysics UndertheSupervisionofProfessorGuidoMueller UniversityofFlorida April2018

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Abstract Thestabilityofanopticalbenchdesignisexplored.TheALPScollaborationrequiresalight-tightyetstable centralbreadboard.However,low-expansionmaterialsLEMscommonlyusedasopticalbenchesinthe eldoflaseropticsarenotlight-tight.TheALPSgroupattheUniversityofFloridadesignedacomposite breadboardtosolvetheissue.Totestitsdimensionalstability,anopticalresonatorcavityisconstructedby placingtwocollinearmirrorsonopticalmountsanchoredinanLEMblockthroughalight-tightaluminium layerandisplacedinavacuumchamber.Alasereldisalignedandmode-matchedtothiscavity,andthen lockedtoaresonantfrequencyviathePound-Drever-Hallmethod.Thesameisdonetoanultra-stableLEM referencecavityinanothervacuumchamberwithanotherlaser.Thebeatnotebetweenthetwolasersis analyzedtodeterminetherelativestabilityofthepathlengthbetweentheanchoredoptics.Thecorrelation ofthebeatnotefrequencywithtemperatureismeasuredanditsperiodicandstochasticnoisinessisanalyzed inaspectraldensityplot.BothmeasurementsndittobesuitableforALPS.

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Contents Abstract Acknowledgments 1MotivationsforExperiment1 2DesignoftheCompositeBreadboard1 2.1LowExpansionMaterials......................................1 3MeasurementTechniques3 3.1OrientationStabilityMeasurement.................................3 3.2DimensionalStabilityMeasurement.................................3 3.2.1Pound-Drever-Hall......................................3 3.2.2ModeMatchingtoanOpticalCavity............................4 3.2.3TestCavityandReferenceCavity.............................5 3.2.4SourcesofNoise.......................................6 3.2.5Carrier-CarrierBeatNoteGeneration...........................7 4Results 8 4.1OrientationStability.........................................8 4.2DimensionalStabilityMeasurement.................................8 4.2.1RoughMeasurementofTemperatureDependence.....................9 4.2.2PreciseMeasurementofFrequencyNoise..........................9 5Outlook 11 5.1ImprovingtheExperiment......................................11 5.2BuildingUpontheExperiment...................................11 5.2.1LISATelescope........................................11 5.2.2ShakeTestsforSpaceQualication.............................12 AGaussianModePropagation13 A.1ShapeofaGaussianBeam......................................13 BOpticalResonators 15 B.1Mode-MatchingtoanOpticalCavity................................16 References

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ListofFigures 1Al-ULECompositeBreadboard...................................2 2OrientationMeasurementAssembly................................2 3OrientationMeasurementDiagram.................................3 4Pound-Drever-HallDiagram.....................................4 5CompositeBreadboardTestCavity.................................5 6ZerodurReferenceCavity......................................5 7CryoTHOROpticalBench......................................6 8TestChamberOpticalBench....................................6 9DiagramofExperimentalSetup...................................7 10OrientationStabilityMeasurement.................................8 11HeatedOrientationStability.....................................8 12CorrelationBetweenTemperatureandLengthShift.......................9 13DierentialNoiseSpectrum.....................................10 14FutureReferenceCavity.......................................11 15FutureHeterodyneSetup......................................12 16GaussianBeamProle........................................13 17ShapeofPropagatingLaserBeam.................................14 18GaussianBeamIntensityandWidth................................14 19HemisphericalOpticalCavityDiagram...............................15 20RelativeIntensitiesinanOpticalResonator............................16 ListofTables 1TableofLow-ExpansionMaterialExpansionCoecients.....................1 2TableofMode-MatchingParameters................................4 3TableofResonatorProperties....................................5

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Acknowledgments IamextremelygratefultoDr.SimonBarkeforhiscoachingandmentorshipoverthelastyearandahalf ofworkingwithhim,andforhissupportandfeedbackinwritingthisthesis.Iwouldalsoliketothankto MicheleMartinazzooftheUniversityofPaduaforhisassistanceinconstructingthemeasurementscheme. Last,butcertainlynotleast,IamgratefultoProfessorGuidoMuellerforallowingmetheopportunityto workwithandlearnfromhim.

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1MotivationsforExperiment TheLaserInterferometerSpaceArrayLISAisaprojectdesignedtodetectandmeasuregravitational wavesatextremesensitivities.ComprisedofthreespacecraftfollowingEarthinorbitaroundtheSun,itwill detectgravitationalwavesfromsuper-massiveblackholecollisionsandothersourcestoolowinfrequencyfor ground-baseddetectionLIGO.InLISA,thedistancebetweenopticalcomponentsonthebencheswithin eachspacecraftmustbeconstantwellbelowthenanometerlevel.Thisdegreeofstabilityiscommonly achievedbymountingfusedsilicaopticsontoalow-expansionopticalbenchbyhydroxidecatalysisbonding, anon-reversible,costlyandtime-consumingtechniquethatonlyworkswithalimitednumberofbench materials. TheAnyLightParticleSearchALPSisaprojectwhoseultimategoalistodetectnew,weaklyinteractingdarkmattercandidateparticlescalledaxions.Theproposeddesigncontainstwocollinearoptical resonators-aproductioncavityandaregenerationcavity-separatedbyalight-tightbarrier.Theproductioncavitygeneratesaxionsbyexposingthecirculatinglighttoastrongmagneticeld.Theregeneration cavitydoestheopposite,transformingtheproducedaxionsintoameasurable,circulatinglasereld. SimilartoLISA,thestabilityofthepathlengthofthelaserisalimitingfactorinthesensitivityofthe experiment.InALPS,thedistancebetweentheendmirrorsoftheproductionandregenerationcavities muststayconstanttowithin100nmoverthedurationoftheexperimentruntime.Thisisaminimumof twoweeksandideallyaslongasseveralmonthstoayear.Furthermore,theorientationsofthemirrorsin bothexperimentsmustbestabletolessthan10 radtomaintainthealignmentoftheinterferometer.The additionalcomplicationinALPSisthattheopticalbenchbetweenbothcavitiesmustbepartofalighttight barrier. Ireportheretheresultsofvarioustestsofanewopticalmountingstrategywhichweinitiallydeveloped forALPSbutmightalsobeapplicableforgroundtestingofLISAcomponents. 2DesignoftheCompositeBreadboard 2.1LowExpansionMaterials Thelow-expansionmaterialsusedinthismeasurementwereZerodur,madebySchottA.G.,andthe previously-mentionedULE.ZerodurisasiliconglassceramicmaterialwhileCorningULEisatitaniasilicateglass.Bothhaveveryreliable,verylowcoecientsofthermalexpansion,makingthemidealfor opticalexperimentation.ZerodurwasusedtoconstructtheopticalbenchontheLISAPathndermission, whichpasseditssensitivitybenchmarksbyawidemargin. X X X X X X X X X X Grade LEM Zerodur Clearceram-Z ULE LowGrade 0 0.1ppm/K 0 0.1ppm/K 0.0 0.1ppm/K HighGrade 0 0.020ppm/K 0 : 0 0 : 020ppm/K 0.0 0.030ppm/K Table1:Coecientsofthermalexpansionforthelowestandhighestgradescommerciallyavailablefor dierentLEMs,withintherangesof273K-323K.Eventhelowest-gradeLEMisstillextremelystable.Low ToolingGradeULEisusedtomakethecompositebreadboard,andthereferenceLEMisZerodur.A Clearceram-Zreferencecavitywillbeusedinfutureiterationsofthisexperiment.Allthreematerialsare functionallyidenticalforthelowgrades,thoughULEhaslowerstabilityinitshigh-grade. Alow-expansionglass/ceramicmaterialismadebycombiningmaterialswithpositiveandnegativecoefcientsofthermalexpansionininterlockingsubregions".Whenonesubregionexpands,theothercontracts, andthenetchangeinvolumeremainsnearzero.LEMblockscanbecustom-madeforagiventemperature

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rangetohaveevenlowernetexpansion.AllLEMsconsideredinthisexperimentarewithin 0.1ppm/K thermalexpansivity,asshowninTable1. Thecompositeboard,builtfromLowGradeULECorningcallsitToolingGrade"andaluminium plate,isaproposedsolutionforthedesignoftheALPScentralbreadboard.ULEisverystableunder temperatureuctuations,whilethealuminiumislight-tight. Figure1:CompositebreadboardcomprisedofaluminiumandULE.Itislight-tight,andtestingitsstability isthegoalofthisexperiment.Totherightisacross-sectionoftheanchoringpostxingtheopticalmounts totheULE.ImagescourtesyofJoeGleasonandSimonBarke. ThemountingdesignshownexplodedinFigure1wasdesignedbyJoeGleasonandconstructedhereat theUniversityofFlorida.ThepoststhatanchortheopticalmountsintheULEaredesignedtoexpandinto theULE.O-ringsonthefourscrewsfasteningthealuminiumtotheULEandthegreeno-ringsonthemirror mountssucceedinmakingtheboardcompletelylight-tight,butitmustalsobestable.Thealuminiumis allowedtobreathe"undertheo-ringsastemperatureuctuateswithoutaectingthepathlengthbetween opticalcomponents,andthelight-tightnessismaintained.Across-sectionoftheanchoringdesignisalso showninFigure1. Twoassemblieswerecompletedformeasuringthenewmountingstrategy:Oneformeasuringorientation stabilityofthemirrorholders,showninFigure2,andoneformeasuringthedimensionalstabilityofthe mountinganchors. Figure2:Testassemblyformeasuringtheorientationstabilityofmirrorsmountedviathenewmounting strategy. 2

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3MeasurementTechniques Dierenttechniqueswereusedtomeasurethepropertiesofthecompositeboard.Weutilizedacommercial o-the-shelfCOTSautocollimatortotracktheorientationofmountedmirrors.Measuringthedimensional stabilitybetweentwomountedmirrorsrequiredamorecomplexsetup. 3.1OrientationStabilityMeasurement ThemeasurementsoftheorientationstabilitywereconductedbyJoeGleasonandSimonBarkeusingan autocollimator.TheautocollimatorsetupshowninFigure3tracksthecenterofthereectedlightforeach mirror. Figure3:Diagramoftheautocollimatororientationmeasurement,trackingthepositionofeachmirror reection. Thedevicewascapableoftrackingtworeectionsfromtwodierentmirrorsatonce.Thecommonmodejitterbetweentheautocollimatorandtheassemblycancelsinthedierentialmotionofthereected beams,leavingonlythedierentialmotionbetweenthemirrorholders. 3.2DimensionalStabilityMeasurement Themeasurementofthedimensionalstabilitywascarriedoutusinganopticalsetupspanningtwoindependentvacuumchambers.PartoftheCryoTHORexperimentconductedbyJohannesEichholz[2]was appropriatedforuseasthestablereferencematerial.Luckily,workontheexperimentwasstillbeingconducted,sobothreferencecavitiesfromCryoTHORwerestillwell-alignedandmode-matched. Tomeasurethedimensionalstabilityofthenewmountingstrategy,anopticaltestcavitywasassembled onthecompositeboard.ThisisshowninFigure5.Theresonantfrequencyofthecavitydependsonthe lengthofthecavityby L = n 0 = nc 0 ; foranypositiveinteger n .Thechangeinresonantfrequencycanbescaledtochangeinthelengthofthe cavity,i.e.,thedistancebetweenthemirrors,by L L = )]TJ/F11 9.9626 Tf 8.944 6.74 Td [( 0 : Areferencecavityisrequiredtocomparetherelativestability.Trackingthefrequencyofabeatnote betweentheresonantfrequencyofthestablereferencecavityandthatofthetestcavitygivesamoreaccurate measurementofthedimensionalstabilityofthetestcavity.However,thefrequenciesofthelasersusedare notconstant,andneitheristheresonantfrequencyofthecavity.Thelasersarelockedtothechanging resonantfrequencyofthecavityusingthePound-Drever-Hallmethod. 3.2.1Pound-Drever-Hall ThePound-Drever-Hallmethodreliesonthefactthatphase-modulatedlightincidentonanopticalcavity reectsasamplitude-modulatedlightwhennotonresonance,anddoesnotreectatallwhenexactlyon resonance.Thelaserisfedthroughanelectro-opticphasemodulator,andthereectedsignalismeasured onaphotodetector.ThiselectricalsignalisthenmixedwiththeelectricalsignalcontrollingtheEOMand 3

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low-pass-ltered,creatingaphase-sensitiveerrorsignal.Thiserrorsignalisfedbackintoapiezoelectric crystalthatactuatesthefrequencyofthelaser.Thefeedbackcircuitisusedtodrivetheerrorsignalto 0,whichhappensatresonancewhenthereectedlightdisappears[4],Ch.11.Abasicschematicofthe opticalsetupforPDHlockingisshowninFigure4.Theerrorsignalcanbeoptimizedbymaximizingits Figure4:AbasicdiagramofthePound-Drever-Hallmethod.Sidebandsarerejectedbythecavity,andso becomeoutofphasefromthereectedeld.Thisgeneratesabeatnoteonthephotodetector. slope-thisallowsthefeedbackcircuittocorrectmoreecientlyandmoreaccurately.Thecalculationfor thisshowsthatthisisdonebysettingthemodulationdepthforthesidebandgenerationto =1 : 08[1]. Inthisexperiment,theJenoptikberEOMsusedhavea V of6V,butdrivingtheEOMsatover6Vis notnecessary.Astablelockwithbothcavitieswasachievedwith 0 : 105,whichismoreoptimalforthis particularexperimentaldesign. ReferencingaheadtoFigure9,bothcarriersandtheirsidebandsareincidentontheCarrier-Carrier Beatphotodetector.Thismeansthatbeatnotesformbetweeneachcarrierorsidebandfrequencyandevery othercarrierandsidebandfrequency.Loweringthevoltagefedintothemodulatorslowerstheintensityof thesidebandbeatnotes,makingthecarrier-carrierbeatnotemoreprominentandeasierfortheMoku:Lab phasemeterseeSection4tolockonto. 3.2.2ModeMatchingtoanOpticalCavity Theshapeofanopticalcavity,denedbytheradiiofcurvatureofthemirrorsandtheirdistanceapart, determineswhichspatialmodesitsupportsinresonance.Spatialmodesaredenedbythesizeofthebeam waistanditslocation.Modematchinguseslensestocreateanimageofthelaserwaistwhosesizeand locationmatcheswiththeeigenmodeofthecavity[4],Ch.15and19,summarizedinAppendicesAand B.1. Forahemisphericalcavitysuchastheoneusedinthisexperiment,theradiusofcurvaturefortheplanar mirrorisinnite.Thismeansthatthebeamwaistmustoccuratthesurfaceofthismirror.Thelengthof thetestcavityisabout12cm,andtheradiusofcurvatureofthefarmirrorisabout44cm.Thisgivesthe mode-matchingparametersshowninTable2. DistanceFromMirror1toWaist z 1 0 DistanceFromMirror2toWaist z 2 12cm RayleighRange z R 10.23cm BeamWidthatMirror1 w 1 = w 0 0.186mm BeamWidthatMirror2 w 2 0.258mm Table2:Tablecontainingtheparametersthelasermustpossesinordertobematchedtothetestcavity. 4

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3.2.3TestCavityandReferenceCavity Figure5showsthetestcavity.Figure6showsthereferencecavity.Thetestcavityistwomirrorsmounted approximately12cmapart,opposingeachotheronanchorsintheULESeeFigure1.Thereferencecavity isleftoverfromCryoTHOR[2],ismadeofpureZerodur,andis23.1cminlength.Table3liststheparticular relevantdetailsofthecavitiesmoresuccinctly. Figure5:Twomirrorsmountedinthe Al-ULEcompositeboard,forminganopticalresonator.Thedistancebetweenthe mirrorsis 12cm. Figure6:Twomirrorsopticallybonded totheendsofasquareZerodurtubeform anopticalresonatorofveryhighstability. Thelengthofthiscavityis23.1cm. Cavity Test Reference Length 12cm 23.1cm Finesse 600 21,000 RoC 1 1 50cm RoC 2 44cm 100cm Table3:Tableofpropertiesforthetwocavities. Thetwoprimarypartsofthemeasurementsetup-theCryTHORvacuumchamberandthetestvacuum chamber-arepicturedinFigures7and8.AsimplieddiagramoftheexperimentisshowninFigure9. ThemeasurementsetupconsistsoftwolasersthatarefedthroughFaradayisolatorsandahalf-waveplate/polarizingbeamsplitter/beamdumpattenuatornotpicturedbeforebeingindividuallyphasemodulatedatfrequenciessetbyafunctiongeneratorRF1andRF2.ThemodulatorsareJenoptikPM1060 ber-coupledelectro-opticphasemodulators.Thebeamsareoverlappedatapowerbeamsplitterandthe tworesultantbeamsaresentintoopticalbercouplers.Fiber1andFiber2feedthelasersintoseparate vacuumchambers. ThebeampassedintoCyroTHORVacuumChamberisopticallyisolatedwithapolarizingbeamsplitter andaquarter-wave-plateandmode-matchedintotheZerodurreferencecavity.Thesameisdoneforthebeam passedviaFiber2intoTestVacuumChamberforthetestcavity.Thetransmittedmodesaremonitoredin ordertondTEM 00 resonance. 5

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Figure7:TheCryoTHORoptical bench,containingreferencecavitiesalreadyalignedandmode-matched,appropriatedforthisexperiment.Theshort referencecavitySRC",asitisreferred toinEichholz'sthesis[2],wasusedinthis experiment. Figure8:TheTestChamberoptical bench,showingthetestcavityinplace ontheright. Thereectedsignalsfromthecavitiesaredirectedontophotodetectors,labeledPDre1and2.These signalsarefedthroughADCsintoacomputerrunningaLabVIEWprogramwrittenbyJohannesEichholz [2]forCryoTHOR.Thisprogramcontrolsaeld-programmablegatearrayFPGArunningcustomcode thatcontainsthefeedbackcircuitsforthelasers.Additionally,itcontainsfreecontrolsforthetemperature circuitofeitherlaser.Thisallowsustomanuallychangethefrequencyofthelaserstondresonancesin thecavitiesbeforethePIDproportionalintegrationanddierentiationfeedbackcircuitsareturnedon. FromtheCryoTHORVacuumChamber,acarrier-carrierbeatnoteisimmediatelypickedobyanother powerbeamsplitterandsentoutawindowtoaphotodetector. 3.2.4SourcesofNoise Althoughslowly-varyingparametersliketemperaturearelikelytobecommon-modebetweenthetwovacuum chambers,mostnoisesourceswillnotbe.Vibrationsintheopticalbench,forexample,willbedierentfor thetwochambers.Inaddition,whilebothchambersareclosedandair-tight,neitherchamberispumped downtolowpressures,soairdriftisdierentforbothchambers. Becausethisisultimatelyaheterodynemeasurementscheme,theonlyconcernsfornoisearethosethat arenotcommon-modeinthenalsignal.Assumingbothlasersareresonantintheirrespectivecavities, whichisonlypossiblewhenthenoiseslevelsfortheindependentPDHsystemsaretolerable,theonlynoise contributionsthatmatterarethosealongtheopticalpathbetweenthelasersandthecarrier-carrierbeat note.However,everythingafterthepowerbeamsplitterthatoverlapsthetwosignalsiscommon-modenoise andisthereforenegatedinthebeatnote. 6

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Figure9:Simplieddiagramoftheexperimentalsetup,madeinwiththehelpofcomponentlibrarycreated byAlexanderFranzengwoptics.org. 3.2.5Carrier-CarrierBeatNoteGeneration WitheachlaserlockedtooneofthecavitiesbythePound-Drever-Hallmethod,bothlasersthemselves outputtheresonantfrequencyofeachcavity.Onelaserisresonantwithanultra-stableLEMcavityandthe otherisresonantwiththecavityconstructedonthetestbench.Therefore,anyshiftsinthepathlengths ofthecavitiesshowupinthefrequenciesofthelasers.Thebenchmarkforpathlengthstabilityisapure LEMbench,sothefrequenciesofthetwolaserscanbecomparedtondthestabilityofthetestbreadboard relativetopureLEM. Anybeatnotebetweenthecarrierscontainsthedierencefrequencybetweenthem.Pickingoneofrom theopticalpath,thisbeatnoteisfedintoaphotodetector.Thissignalislow-pass-lteredtoisolatethe dierencefrequencyandfedintoadevicecalledaphasemeter,whichmeasurestherelativephasebetween theinputsignalandaninternalreferencebytrackingthefrequencyoftheinputsignal[3].Sincetheinternal frequencyisverystablecomparedtotheexpectedfrequencychangesofthebeatnote,thefrequencyreadout ofthephasemetertellsustheactualfrequencyofthebeatnote. Wearenot,however,tryingtomeasuretheactualdierencefrequencyinthebeatnote,butrather thechangeinthedierencefrequency,asthisistherelativestability.Weareinterestedincomparingthe stabilityofthetestcavitytothatofthepureZerodurcavity,sowetreattheZerodurreferencecavity resonantfrequencyasthe"pointforrelativefrequencychanges,i.e.,weassumethatallchangesinthe dierencefrequencyareduetodriftinthetestcavity.Fromthis,thefrequencychangescanbescaledto thelengthofthe test cavitylinearlywithEquation2,givingitsdimensionalstability. 7

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4Results Inmeasurements,wefoundthecompositeboardundertesttobecompatiblewithallALPSrequirements. Initialresultsindicatethatwithsomefurthereort,compliancewithLISAgroundtestingrequirements mightbepossible. 4.1OrientationStability ThemeasurementsoftheorientationstabilitywereperformedbyJoeGleason.Therstmeasurementwas ofthereectionanglestrackedbytheautocollimatoroverthreedays.TheresultscanbeseeninFigure10. Figure10:Measurementoforientationstabilityoverthreedays.Anglesareconstantwithin1 rad. Theorientationofthemirrorsisstabletowithin1 radforthreedaysofmeasurement.Thismeetsand exceedstheprescribedorientationstability.Thesecondmeasurementwasconductedwiththeassembly heatedslightly.Thereectionanglesweretrackedastheassemblycooled.TheresultsareshowninFigure 11. Figure11:Measurementoftheorientationstabilityasthetemperaturevariesoverseveraldegrees.The orientationismuchlessstableforrapidtemperaturechangesbutstillmeetsthe10 radrequirement. ThehorizontalXdirectionofthereectionangleseemsmorestablewithtemperatureuctuationsthan theverticalYdirection,buttheoverallorientationstabilityismaintainedwithinthe10 radrequirementas thetemperaturedriftsbyseveraldegrees.Thesepromisingresultsmotivatedustoseehowwellitperformed indimensionalstability. 4.2DimensionalStabilityMeasurement Twosetsofmeasurementswereconducted.Forthelong-runroughtemperaturemeasurement,thefrequency ofthebeatnotewasreadoutintermittentlyfromanelectro-opticalspectrumanalyzer.Fortheshortrun noisemeasurements,aMoku:Labwasusedtorecordatimeseriesoffrequencydata.Moku:LabsaremultifunctiondevicesproducedbyLiquidInstrumentscontainingalmostanykindofmeasurementdeviceone wouldneedinelectro-opticexperimentation. 8

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4.2.1RoughMeasurementofTemperatureDependence Aroughmeasurementofthedependenceofthelengthofthecavityontemperaturewasconductedby tapingathermometertothesideoftheTestVacuumChamberandrecordingthetemperatureoveraweek. Duringthistime,thebeatnotefrequencybetweenthetwolockedcarrierswasrecorded.Thereisanobvious correlationbetweenlengthshiftandtemperature,showninFigure12.Fromthiswecanconcludethatthe relativetemperaturestabilitybetweenthetwocavitiesisabout50nm/ C.Assumingthisisentirelythetest cavity,for12cmofpathlengththiscorrespondstoapproximately0.42ppm/K.Thetemperatureonlydrifted byalittlemorethanhalfadegreeinaweek'stimeinopenairinamoderatelycontrolledenvironment,so itishardtoimaginethisbeingaprobleminvacuuminahighlycontrolledenvironment. Figure12:Rough,butvisiblecorrelationbetweenrelativecavitylengthshiftandtemperature.Thebeat notewasmeasuredatdierenttimesandscaledtochangesinthecavitylengthblue,withthetemperature measuredcontinuouslyred.TheordinatewasscaledtolengthbyEquation2. Weexpectatmost0.1ppm/KfromtheexpansionoftheULE.At12cm,thisis12nm/K.Ifweassumethat theremaining38nm/Karefromthemirrormountsalone,thenthisvaluedoesnotscaleintolargerbench sizes.TheactualbenchdesignforALPSis1meterinlength.Scalingtherelativeexpansivityaccordingly meansthat100nm/KcanbeexpectedfromtheULE,butthe38nm/Kfromthemountsdoesnotchange;the netexpansivitycanbeexpectedtofallaround138nm/K.Thiscorrespondstoapproximately0.138ppm/K, whichisfairlyclosetostandardlowgradeLEMcoecients. 4.2.2PreciseMeasurementofFrequencyNoise Analysisofspectraldensityistheprevailingmethodinopticalinstrumentationfordepictingnoise;dierent sourcesofnoisehavecharacteristicbandwidths.Low-frequencycontributionstonoisesuchastemperature driftorairdriftcanbecompensatedforwithtemperaturecontrolsystemsandbypumpingtheenvironment downtovacuum.Veryhigh-frequencynoisesourcesaremorediculttocontrol,andthesearetypically thelimitingmeasurementfactors.Therearealsostochasticnoisesources.Theseincludethermalnoisein substratesBrowniannoiseordiscretenoisefromlasereldsandelectronicsshotandpinknoise.Explicitly periodicnoisesourcesarecharacterizedbysharppeaksinthespectrum. Figure13showsapairofmeasurementswithdierentsamplingfrequencies.Thebluelineisameasurementtakenwithasamplingfrequencyof30Hzoverthecourseofonehour.Fromthisplot,wecansee thattherearesomehigh-amplitudedriftsinlowfrequencieswhichmaybeattributabletoslowtemperature uctuations.Thereisapeakatalittlemorethan0.1Hzthatcouldbeairow.Thereisanother,very localizedpeakat10Hz.Thisisprobablyvibrationfromafanoutsidethelab,over20metersaway. TheredlineinFigure13showsamuchhighersamplingratemeasurementofthenoisespectraldensity. Fromit,wecanseethatthetestcavityiswithin1 picometer ofthepureZerodurreferencecavityforhighfrequencylengthchanges,andiswellwithin1nanometerforalmostallothers.Thechangesofmorethan 9

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Figure13:Dierentialnoisebetweenreferencecavityandtestcavity.Blue:NoiseAmplitudeSpectral Densityat30Hz,takenoveronehour.Red:NoiseAmplitudeSpectralDensityat125kHz,takenover30 seconds.TheordinateisrescaledbyEquation2. 1nmonlyoccurinthemHzrange,whichareeasiertocompensate.Therewasinitialconcernwiththerst measurementthattherelativenoisewouldgoas1 =f astraightlineofnegativeslopeinlog-log,assuch systematicswouldmeanthatthenoiseoorofthemeasurementishigherthanthedata.However,thenoise onlytakesthisshapebelowthepicometerlevel,andeventhenthereisaslightcurvingtrend. Oneconclusioniscommonacrossthemeasurementsconducted:Thepathlengthstabilityofthetest benchiswithin100nmacrossallmeasuredfrequencyranges,makingthismountingstrategysuitablefor ALPS.Additionally,thesub-picometerstabilityshownontheright-handsideofFigure13impliesthat thismountingstrategymayalsobesuitableforgroundtestingLISAopticsoncethelow-frequencynoiseis sucientlymitigated. 10

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5Outlook Whilewehaveshownthatwehavesucceededincreatingastable,light-tightcentralbenchbreadboardfor ALPS,thestabilityrequirementisrelativelylow,comparedtoprojectslikeLIGOandLISA. 5.1ImprovingtheExperiment Eortsarecurrentlyinprogresstoestablishawell-aligned,mode-matchedopticalpathtooneofthereference cavitiesvisibleinFigure8,showninFigure14.Thiswillmaketemperatureandairdriftnoisecommon-mode, andalsoallowforheterodynelocking,whichrequiresnofunctiongeneratorsormodulators,asopposedto thecurrentdouble-PDHscheme.Forheterodynelocking,eachlaserwillberesonantinitsowncavitywhile theotherreplacesthesidebandsasthebeatnoteconstituent.Theerrorsignalsneededforlockingwillbe createdbymixingeachreectedbeatnotewithanotherbeatnotebetweenthecavities-onepickedofrom thepathbeforethelasersareincidentontheresonators. Figure14:24cm-longhigh-gradeClearceram-Zreferencecavityforuseinanimprovedversionofthisexperiment. Inadditiontomakingallthenoisecommon-mode,thechambercanbebroughtdowntovacuumpressures togreatlydecreaseairdrift,temperatureuctuation,andseismicnoise.Thiswouldpotentiallydownthe low-frequencynoiseoorbyordersofmagnitude.AdiagramofthisheterodynesetupisshowninFigure15. Thehigher-precisionmeasurementsconductedbytheimprovedexperimentaldesigncouldshowthatthis opticalbenchdesignisvalidevenforextremelysensitivedetectorssuchasLISA.Itwouldcostmuchless toanchormonolithicopticsinLEMthanitwouldtoopticallybondtheopticstotheLEMaswasdonefor LISAPathnder. 5.2BuildingUpontheExperiment Therearesomesmallmodicationsthatcanbedonetothisexperimenttomeasurethestabilityofother bulkopticalcomponents,aswellasmakeitmorediverse. 5.2.1LISATelescope OneproposeduseforthismeasurementschemeistomodifyittomeasurethestabilityoftheLISAtelescope -thecomponentoneachspacecraftthatwillsend/receivethelaserlightto/fromtheotherspacecrafts.Since theinformationfromagravitationalwaveisencodedinthephaseofthelaserlightforgravitationalwave interferometry[3],itisvitalthatthepathlengthsoftheLISAtelescopesareasstableaspossible. 11

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Figure15:Simpliedexperimentalschematicforthefutureheterodyneversionofthisexperiment. 5.2.2ShakeTestsforSpaceQualication Forspace-basedinterferometrymissionslikeLISA,theopticalpathlengthandalignmentmustremainstable throughoutalaunchintospace.Itwouldbepossibletoouttthevacuumchamberopticalbenchwithspacers thatmakesomethinglikethecompositeboardcavityeasytoinstallandremovewithonlyminoralignment adjustmentsneeded.Shaketestscanbeconductedonthecompositeboardtoseehowwelltheanchoring postsweatherthevibrationwithoutlosingalignmentorlooseningfromtheLEM. 12

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AGaussianModePropagation Optimizingthecouplingofalasereldtotheopticalcomponentsinuseisthecruxoflaseroptics.Most myselfincludedprefertousesimplecomputerprogramslikeJAMmTJustAnotherMode-matchingTool tocalculatewhatlensesand/ormirrorstouseinordertoachieve > 99%modematching,butthemathematics arenottoogymnastic. ElectricFieldIntensity Figure16:ShapeofafundamentalTEM 00 Gaussianlaserbeamataninstantintime. A.1ShapeofaGaussianBeam ThepropagationshapetakenbyaGaussianbeam,showninFigure16,isdependentontwofactors:the wavelength andthebeamwaist w 0 ,whichistheminimumwidthofthebeamalongitspropagationaxis, i.e.,thefocusofthebeam.Thewidthofthebeamisdenedas w z = w 0 r 1+ z R z 2 ; where z isthedistancealongtheopticalaxisfromthefocus. z R istheRayleighrange,thepointwherethe widthofthebeamis p 2 w 0 ,andisgivenby: z R = w 2 0 : Itisalsothepointwherethecurvatureofthewavefrontisgreatest. TheradialshapeofaGaussianbeamis,naturally,Gaussian.Theintensityofthebeamatagivenpoint z alongitspropagationaxisasafunctionofdistancefromtheopticalaxis r isgivenby I r = e )]TJ/F6 4.9813 Tf 11.055 2.677 Td [(2 r 2 w z 2 ; whichismaximizedat r =0. FromEquation5,thebeamwidth w z isalsodenedintermsofthedistancefromtheopticalaxisas p 2 r w where I r w = 1 e 2 I 13

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Figure17:Plotshowingthefocusinganddivergenceofalaserbeamasitpropagates.Thebluelinesmark theRayleighrange,thepointswhere w z = p 2 w 0 .Thisgureisthetheoreticaloptimizedmodeforthe testcavity,with w 0 =0.186mm,and R z =120mm=44cm. Figure18:Thisplotwasgeneratedwithnormalizedintensity, w 0 =2,andtheblueline I = 1 e 2 ,demonstrating howthebeamwidthisdeterminedbytheintensityofthebeam. 14

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TheshapeofthebeamasitpropagatesisshowninFigure17,anditsintensityatagivendistance r fromtheopticalaxisisshowninFigure18.Thesetwoguresarecomponentsoftheequationthatgenerates Figure16,whichistheproductofthetwoandandanadditionaloscillatingcomponentforspacialwave propagation. AsonecanseefromFigure16,asthebeampropagatesthecurvatureofthewaveprolechanges.Atthe waistofthebeam,thecurvatureis0,ortheradiusofcurvatureisinnite-atthispointtheeldproleisa planewave.Asthebeampropagatestothefareld,theeldprolebecomesasphericalwavewhoseradius isthedistancefromthewaist.Astheeldcontinuesto z !1 ,theeldbecomeslocallyplanarforany denedarea,and R !1 .Fromthesearguments,anequationfortheradiusofcurvatureofapropagating beamcanbeconstructed: R z = z + z 2 R z : Dierentiatingthisequationwithrespectto z showsthattheradiusofcurvatureismaximizedatthe Rayleighrange. BOpticalResonators Thissectionwilldiscussthesecondmostcriticalcomponentinthisexperimentapartfromthelaser-the opticalresonator,orcavity.Thisisanyopticalpaththatcreatesaclosedloop.Lighttransmitsintothe resonatorthroughamirrorandcirculates,buildinguppowerintheloop.Inthisexperiment,asimple, two-mirrorhemisphericalcavitywasconstructedonthetestboard. Figure19:Diagramofahemisphericalopticalcavity.Onemirrorisplanar,andtheotherisconcave.The shapeofthebeamcirculatinginthecavityblueisdeterminedbytheradiiofcurvatureofthemirrorsand thedistancebetweenthem.Thisgurewasconstructedusingtheparametersofthetestcavityusedinthis experiment,whicharelabeledexplicitlyinTable3. Thereisanotherpropertyofopticalresonators-nesse-givenbytheratiobetweentheFSRandthe widthofaresonancepeakSeeFigure20. F = FSR : Thelinewidth isdependentonthereectivityofthemirrors;thehigherthereectivity,thesharper thepeaks.Finesseplaysapartindeterminingwhethertwonearbyresonancepeakscanberesolved.The nesseforthecavitiesinvolvedinthisexperiment,however,issucientlyhighthatitneednotbetaken intoconsideration. 15

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Whenthefrequencyofthelightincidentonthecavityisequalto 0 n FSR foranyinteger n ,the beamcirculatesinsidethecavity.Onresonance,thecirculatingintensityisoftenmanytimestheincident intensity.Thenetreectedeldatresonancedropstonear0:Theinitiallyreectedlightisshifted180 in phase,whilethecirculatinglightthattransmitsbackoutthroughtherstmirrorhaspickedupa360 shift fromtwotransmissionsandareectionandisapproximatelyequalinmagnitude.Theresonanttransmitted lightinisapproximatelyequalinmagnitudetotheincidentlight.ThesepropertiescanbeseeninFigure 20.Resonancescanthereforebefoundbyputtingeitherthetransmittedorreectedeldonaphotodetector,feedingthesignalintoanoscilloscope,andscanningthefrequencyofthelaserbackandforth.The oscilloscopewillshowsharpdipsforreectedsignalsandsharppeaksforthetransmittedsignals. Figure20:Intensitiesoftheeldsreectingfrom,circulatingin,andtransmittingthroughanopticalresonatorcomparedtotheintensityoftheincidenteld.Resonanceoccursatintegermultiplesofthefree spectralrange FSR B.1Mode-MatchingtoanOpticalCavity TheimportantinformationforafundamentalTEM 00 Gaussianmodeisneatlypackagedinaquantity calledthe q parameter,whichisafunctionalofradiusofcurvatureofabeam R andwidthofabeam w whichareinturnfunctionsofdistancefromthefocusofthebeam z )]TJ/F11 9.9626 Tf 9.963 0 Td [(z 0 : 1 q = 1 R z )]TJ/F11 9.9626 Tf 9.963 0 Td [(z 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i w 2 z )]TJ/F11 9.9626 Tf 9.963 0 Td [(z 0 : LensesandMirrors:RayTransfer Theshapeofabeamcanbechangedbyplacingalensormirrorwithaknownradiusofcurvatureinthe pathofthebeam.Forcomponentswheretheradiusofcurvatureismuchgreaterthanthethicknessofthe component,thepropagationofthebeamfromamode q 1 toanothermode q 2 isfoundbyleft-multiplying 16

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the q parameterbytheraytransferABCD"matrices: q 2 1 = AB CD q 1 1 ; Forlens/mirrorraytransferwithfocus f = R= 2: 1 q 2 = 1 q 1 )]TJ/F8 9.9626 Tf 11.642 6.74 Td [(1 f ; whichonlyaectstherealpartof q 1 -thetermcontainingtheradiusofcurvature.Thisgeneratestheradius ofcurvatureforthebeampropagatingafterthelens/mirror: R 0 z = Rf f )]TJ/F11 9.9626 Tf 9.963 0 Td [(R 1 q 2 = 1 R 0 z )]TJ/F11 9.9626 Tf 9.963 0 Td [(z 0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i w 2 z )]TJ/F11 9.9626 Tf 9.963 0 Td [(z 0 Forfreespacetheraytransferequationgives q 2 = q 1 + d; where d canbesubsumedintothedistancefromthewaist z )]TJ/F11 9.9626 Tf 9.962 0 Td [(z 0 StableResonance Stableresonanceoccurswhenthecirculatinglasermodeisconstant.Forthistooccur,certainparameters ofthelasereldmustcoincidewiththeparametersoftheresonatoritself.Therearethreefactorswhich determinetheoptimalresonantmodeforatwo-mirrorcavity:Theradiusofcurvatureofeachmirrorand thelengthofthecavity. Awell-alignedcavityisonesuchthatthecentersofbothmirrorsarehitbythelaserwithnormal incidence.Ifitisnotwell-aligned,themostprominentcirculatingmodesareofmuchhigherorderwith potentiallydozensoflobes,andthe q parameternolongerdescribestheresonantortransmittedlight.For amodetoresonateinbetweenmirrors,theradiusofcurvaturesofthebeammustbeequaltothatofeach mirroratthepointofincidence. Foratwo-mirrorresonatoroflength L thereisdenedaresonator g i parametergivenby g i =1 )]TJ/F11 9.9626 Tf 13.208 6.74 Td [(L R i ; where g i istheparameterforeithermirrorand R i isthecorrespondingradiusofcurvatureofthatmirror. Thisparameterappearsinequationsusedtocalculatethemodeneededforoptimalresonance.Thedistance betweeneithermirrorandthewaistisgivenby z i = g j )]TJ/F11 9.9626 Tf 9.962 0 Td [(g i g i + g j )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 g i g j L; wherethe j subscriptssimplyrefertotheopposingmirror.Thewidthofthebeamsatthesepointsisgiven by w 2 i = L r g j g i )]TJ/F11 9.9626 Tf 9.963 0 Td [(g i g j : TheRayleighrangeofthestableresonantmodeisgivenby z 2 R = g i g j )]TJ/F11 9.9626 Tf 9.963 0 Td [(g i g j g i + g j )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 g i g j 2 L 2 ; andthewaistsizeiscalculatedfromthisequationbyusing w 0 = q z R Becausethelengthofacavityisneverpreciselyknownapriori,theparametersthemselvesareonly ballparkgures.Theyonlyneedbecloseenough;thelackofprecisionintheparametersiscompounded againbythefactthatitisnearlyimpossibletoplacetheneededlensesin exactly therightspots.Thisdoes 17

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notmatteratall.Themodeofthebeamdependsalsoonthewavelengthofthelasereld,whichcanbe modulatedbychangingthetemperatureoforthepiezoelectricfeedbacktothelasingcrystal. Anylightthatpropagatesataresonantfrequencywillcirculateinthecavitytosomeextent.An optimized,well-alignedmodeischaracterizedbyastrongTEM 00 Hermitemodetransmittingoutofthe cavity.Scanningthefrequencyofthelaser,onecaneasilyndhigher-ordermodeswhentheirfrequencies arenowtheresonantfrequencyofthecavity.Complicationsmayarisewhenthedistancebetweendierent nm modefrequenciesforalaserisclosetothefreespectralrangeofthecavity. 18

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References [1]EricBlack.NotesonthePound-Drever-Halltechnique.Technicalnote,LIGO,1998. [2]JohannesMichaelEichholz. DigitalHeterodyneLaserFrequencyStabilizationforSpace-BasedGravitationalWaveDetectorsandMeasuringCoatingBrownianNoiseatCryogenicTemperatures .PhDthesis, UniversityofFlorida,2015. [3]DanielShaddocketal.OverviewoftheLISAPhasemeter.Online;accessed9-April-2018. [4]AnthonyE.Siegman. Lasers .UniversityScienceBooks,1986.