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Whispering gallery mode resonators for frequency metrology applications
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Whispering gallery mode resonators for frequency metrology applications
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WHISPERINGGALLERYMODERESONATORSFORFREQUENCY METROLOGYAPPLICATIONS by LukasBaumgartel ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (PHYSICS) December2013 Copyright 2013 LukasBaumgartel TableofContents Preface xi Acknowledgements xiv Abstract xvi Glossaryofacronyms xviii Chapter1 Whisperinggallerymoderesonators 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Modestructureandfields . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 WGMsinaperfectsphere . . . . . . . . . . . . . . . . . 5 1.2.2 Analyticapproximationforoblatespheroids . . . . . . . . 9 1.2.3 FiniteelementmethodforWGMresonators . . . . . . . . 11 1.3 WGMresonatorexperimentalsetup . . . . . . . . . . . . . . . . 14 1.3.1 Opticalsetup . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Evanescentcoupling . . . . . . . . . . . . . . . . . . . . 16 ii 1.3.3 TemperatureControl . . . . . . . . . . . . . . . . . . . . 17 1.3.4 Vacuumchamber . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Quantificationsandfiguresofmerit . . . . . . . . . . . . . . . . . 21 1.4.1 QualityFactor . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4.2 ModeVolume . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.3 ExperimentaldeterminationofQ,FSR,andmodeindex . 25 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter2 Frequencystabilization&metrologyinWGMcavities 33 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Pound-Drever-Hall(PDH)locking . . . . . . . . . . . . . . . . . 34 2.2.1 PDHconcept . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Implementingthelock . . . . . . . . . . . . . . . . . . . 39 2.2.3 Closedlooptransferfunction . . . . . . . . . . . . . . . . 40 2.3 Quantifyinglaserfrequencystability . . . . . . . . . . . . . . . . 43 2.3.1 Comparisontoanindependentreferencelaser . . . . . . . 45 2.4 Trouble-shootingthelaserlock . . . . . . . . . . . . . . . . . . . 46 2.4.1 Error-signalresiduals . . . . . . . . . . . . . . . . . . . . 47 2.4.2 Errorsignalfrequencynoisedensity . . . . . . . . . . . . 49 2.4.3 Lockingtwolaserstotwodifferentcavitymodes . . . . . 50 2.4.4 Differentialtemperaturedependenceofparallelmodes . . 55 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 iii Chapter3 TemperatureCompensation 59 3.1 Dual-modestabilization . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.1 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.1.2 ExperimentalSetup . . . . . . . . . . . . . . . . . . . . . 63 3.1.3 Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.1.5 Frequencystabilityofadual-moderesonator . . . . . . . 79 3.1.6 Outlookfordual-modestabilization . . . . . . . . . . . . 82 3.2 Hetero-compoundresonator . . . . . . . . . . . . . . . . . . . . . 84 3.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.2 Calculationofazero-crossing . . . . . . . . . . . . . . . 86 3.2.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Chapter4 NoiseSources 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Fundamentalthermalnoise . . . . . . . . . . . . . . . . . . . . . 100 4.2.1 Typesofthermalnoise . . . . . . . . . . . . . . . . . . . 101 4.2.2 Thermoconductivenoisemechanisms . . . . . . . . . . . 103 4.2.3 Thermomechanicalnoise . . . . . . . . . . . . . . . . . . 110 4.2.4 Measuringthermalnoise . . . . . . . . . . . . . . . . . . 116 4.3 Technicalnoisesources . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.1 Laseramplitudenoise . . . . . . . . . . . . . . . . . . . . 118 iv 4.3.2 Vibrationallyinducednoise . . . . . . . . . . . . . . . . . 123 4.4 OtherDestabilizingMechanisms . . . . . . . . . . . . . . . . . . 129 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Chapter5 WGM-basedFrequencyCombs 133 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1.1 Micro-combgeneration . . . . . . . . . . . . . . . . . . . 134 5.1.2 Microcombsforfrequencymetrology . . . . . . . . . . . 137 5.1.3 Stabilizingthecomb . . . . . . . . . . . . . . . . . . . . 141 5.1.4 Coherenceandnoise . . . . . . . . . . . . . . . . . . . . 142 5.2 Engineeredcavityfornativecomb . . . . . . . . . . . . . . . . . 143 5.2.1 Fabricationandtesting . . . . . . . . . . . . . . . . . . . 144 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3 MicrocombwithPDHlockedpumplaser . . . . . . . . . . . . . . 148 5.3.1 PDHlockingofthepump . . . . . . . . . . . . . . . . . . 149 5.3.2 Stabilityofopticaldown-conversion . . . . . . . . . . . . 153 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Bibliography 158 AppendixA FabricationofWGMresonators 169 AppendixB Detailsandschematicsoflaserlockelectronics 172 v AppendixC Short-termstabilityoftheMenlofrequencycomb 180 vi ListofTables 4.1 MechanicalproperteisofMgF 2 . . . . . . . . . . . . . . . . . . . 108 vii ListofFigures 1.1 SurfaceplotofWGMfieldmagnitudeinasphere . . . . . . . . . 7 1.2 FEMsimulationofaWGMfielddistribution . . . . . . . . . . . . 12 1.3 EffectsofambientpressurechangeonWGMfrequency . . . . . . 13 1.4 SchematicofatypicalWGMresonatorsetup . . . . . . . . . . . 16 1.5 PhotographsoftheWGMvacuumchambersetup . . . . . . . . . 20 1.6 MeasurementofQ: modulationandfit . . . . . . . . . . . . . . . 26 1.7 MeasuringcavityFSRwithanEOM . . . . . . . . . . . . . . . . 28 1.8 ChangeinFSRwithradialmodeorder . . . . . . . . . . . . . . . 29 1.9 TemperaturedependenceofFSRinanMgF 2 resonator . . . . . . 30 2.1 Schematic: PDHlockingtoaWGMcavity . . . . . . . . . . . . . 36 2.2 Schematic: lock-loopsystemdetails . . . . . . . . . . . . . . . . 40 2.3 Schematic: calculatinglock-looptransferfunction . . . . . . . . . 41 2.4 Bodeplotsofclosedlooptransferfunctions . . . . . . . . . . . . 43 2.5 LaserLockResidualsSpectra . . . . . . . . . . . . . . . . . . . . 48 2.6 Schematic: twolaserslockedtoonecavity . . . . . . . . . . . . . 51 2.7 Measurement: twolaserslockedtothesamecavity . . . . . . . . 53 2.8 Measurement: Differentialdriftsbetweenparallelmodes . . . . . 56 viii 3.1 Schematic: thedual-modetemperaturestabilizedWGMcavity . . 65 3.2 Thermalactuationandlowpassfiltering . . . . . . . . . . . . . . 67 3.3 CleaningtheTEM-mountedresonator . . . . . . . . . . . . . . . 69 3.4 In-loopAllandeviationofWGMRtemperatureactuators . . . . . 70 3.5 FEMsimulationofadual-modestabilizedresonator . . . . . . . . 75 3.6 Dual-modeWGMReffectivetemperaturecoefficient . . . . . . . 77 3.7 Allandeviation: absolutedual-modestability . . . . . . . . . . . 81 3.8 Additionaldual-modestabilitymeasurements . . . . . . . . . . . 83 3.9 Temperature dependence of thermal expansion and refraction for CaF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.10 FEMmodelofthecompoundWGMresonator . . . . . . . . . . . 90 3.11 Thermalcoefficientzero-crossingsofacompoundresonator . . . 91 3.12 MgF 2 compoundresonatorwithnegativeCTEmaterial . . . . . . 92 3.13 Photographsofthecompoundresonatorandtestenclosure . . . . 93 3.14 Compoundresonatorlinewidthandtemperaturecoefficientmeasure- ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.15 Measurement: Reducedtempcoofcompoundresonator . . . . . . 95 4.1 ExpectedthermorefractivenoiseinaMgF 2 resonator . . . . . . . 107 4.2 TheoreticalpredictionsofBrowniannoisespectraldensity . . . . 114 4.3 Schematic: measuringmodevolumetemperatureresponsetoampli- tudenoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 ix 4.4 Measurement: temperature response of mode volume to amplitude fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.5 Frequencynoisespectraldensityfromlaseramplitudenoise . . . . 122 4.6 Couplingsensitivityexperimentschematic . . . . . . . . . . . . . 125 4.7 Couplinggapsensitivitydata . . . . . . . . . . . . . . . . . . . . 126 4.8 Couplinggapsensitivityvs. contrast . . . . . . . . . . . . . . . . 127 5.1 Microcombgeneration: FWM,FSR,anddispersion . . . . . . . . 136 5.2 Microcombforfrequencymetrology . . . . . . . . . . . . . . . . 139 5.3 CavitydispersionparameterD 2 vsradius . . . . . . . . . . . . . 144 5.4 Fabricationofamicrocombwithdiamondturning . . . . . . . . . 145 5.5 Engineeredmicroresonatorfornativecombgeneration . . . . . . 147 5.6 Engineeredmicroresonatorfornativecombgeneration . . . . . . 148 5.7 Schematic: CombgenerationwithaPDHlockedpumplaser . . . 150 5.8 Spectrumofamicro-combwithPDHlockedpumplaser . . . . . 151 5.9 Threeregimesofmicrocombbeatnote . . . . . . . . . . . . . . . 152 5.10 Measurement: Opticalandmicrowaveexcursionsofacomb . . . 154 5.11 Down-conversionstabilityofthePDH-lockedmicrocomb . . . . . 156 A.1 Photos: basicfabricationstepsofWGMresonators . . . . . . . . 170 B.1 Photographsofthelockelectronics . . . . . . . . . . . . . . . . . 173 C.1 ComparisonofFP-stabilizedlaserandfsfrequencycomb . . . . . 181 C.2 ModifiedAllandeviationofcombshowingWvsFphasenoise . . 182 x Preface As I begin wrapping up this dissertation, a recently published Science paper describes the new Yb lattice clock at NIST Boulder [58]. This frequency standard reaches a fractionalinstabilityofacouplepartsin10 18 afteronly7hrsofaveraging;thisexquisite precision can “resolve spatial and temporal fluctuations equivalent to 1 cm of elevation in Earth’s gravitational field.” Such devices clearly enable fundamental experiments suchasmeasurementsofrelativityandtimeevolutionoffundamentalconstants,aswell asmyriadpracticalapplicationslikedatatransfer,navigation,andgeodesy. However, it is difficult to exaggerate the size and complexity of these clocks: mul- tiple large optical tables supporting a handful of different lasers, vacuum chambers, temperature isolation enclosures, and electronics – plus the human resource of the sev- eral grad-students and post-docs it takes to actually run the clock. Many of the excit- ingscience,navigationandcommunicationapplicationsarethusprecludedbecauseit’s impossible to imagine transporting such a clock on Earth, let alone sending one into space. This background is the motivation for the work described here: progress on devel- opinganewgenerationofcompact,highstabilityopticaldevices,withtheultimategoal xi being a compact optical atomic clock. Whispering gallery mode (WGM) resonators wererecognizedearlyintheirdevelopmentashavinggreatpotentialtominiaturizetwo crucialcomponentsofanopticalclock: thefrequencycombandthereferencecavityfor laserstabilization. Opticalatomicclocksuseanultrastable,narrowlinewidthlaserastheirlocaloscilla- tor. Thisoscillatorperiodicallyinterrogatesanelectronictransitionofoneormoreatoms (or ions) that have been carefully prepared, and then its frequency is slightly adjusted to keep it on resonance. The laser oscillator must therefore have narrow linewidth – to efficiently drive the highQ atomic transition – and excellent stability on time-scales exceedingtheclockinterrogationcycle. Currentstate-of-theartclocklasersdemonstrate fractional instabilities at the level of 10 −15 – 10 −16 at several seconds. At their heart is aFabryP´ erotresonatorusuallyabout10–20cmlong. Bycontrast,high-Qwhispering gallerymoderesonatorscanbemillimetersin diameterandjustafewhundredmicrons thick. And although their demonstrated performance to date is a few orders of magni- tudeabovetheinstabilitiesjustgiven,itisonlythroughacomprehensiveunderstanding of the various destabilizing mechanisms, technical and fundamental noises that their truepotentialcanbequantifiedandrealized. Onereasonopticalclocksconferbetterstabilitythantheirmicrowavecounterpartsis theysplitthesecondupintoabouttenthousandtimesmorepieces. Buttheiroscillation frequencyisnotdirectlyaccessiblebycurrentelectronicstechnologies,thusaphotonic technology is required – namely, a self-referenced frequency comb. WGM cavities can also be used for generating combs, with similar form-factor gains. A recent flurry of xii effort from WGM path-finding groups as well as frequency metrology power-houses has led to much better understanding of microcomb dynamics and properties. Yet it remains to be demonstrated that they can provide the frequency down-conversion sta- bility necessary for optical clocks. Thus, as in the case of frequency reference cavities, anon-goingpushtounderstandtheultimateperformancelimitationsofthesedevicesis necessary. xiii Acknowledgements Firstofall,IwouldlikethankmyadvisoratUSC–DrE.S.Kim–andmysupervisorat JetPropulsionLab–DrNanYu. Withoutyoursupportandguidance,noneofthiswould have been possible. Working with each of you, in turn, has taught me myriad technical skills,approachestowardsscienceandengineering,andgivenmetheconfidencetofind mywaythrougharesearchproblem. Next, to the Quantum Sciences Group at JPL, whose members are welcoming, talented, and always willing to stop what they are doing for a few minutes to help. And in particular: Rob Thompson, for technical guidance and encouragement; Dmitry Strekalov, for experimental techniques and ideas; Ivan Grudinin, for sharing freely his exceptionallydeepandbroadunderstandingofWGMresonators,andfordemonstrating how to approach research always with an eye towards productivity, novelty, and accu- racy. And finally, David Aveline, who became in many ways a de-facto mentor, always taking time to patiently describe a technical concept, suggest a course of action, talk throughaproblem–andmakesureIhadagoodoscilloscopetouse. Thank you to the USC MEMS group members who were senior to me – and in particular Shih-Jui Chen – for sharing your knowledge on the details and subtlety of xiv microfabrication,withoutwhichIcouldneverhaveaccomplishedanythingintheclean- room. And last but definitely not least, thanks to my family, for always encouraging intel- lectualcuriosityandacademicachievement,andforgivingmesomuchsupportoverthe years. And to my wife, who inspires me in innumerable ways, and who has taught me morethanathingortwoaboutwhatitmeanstoworkhard. IamgratefultotheUSCCollegeDoctoralFellowshipforpartialfunding. ThisworkwascarriedoutatJetPropulsionLab,CaliforniaInstituteofTechnology, underacontractwithNASA. xv Abstract This dissertation describes an investigation into the use of whispering gallery mode (WGM) resonators for applications towards frequency reference and metrology. Laser stabilizationandthemeasurementofopticalfrequencieshaveenabledmyriadtechnolo- giesofbothacademicandcommercialinterest. Atechnologywhichseemstospanboth motivations is optical atomic clocks. These devices are virtually unimaginable without the ultra stable lasers plus frequency measurement and down-conversion afforded by FabryP´ erot(FP)cavitiesandmodel-lockedlaser combs, respectively. However, WGM resonators can potentially perform both of these tasks while having the distinct advan- tagesofcompactnessandsimplicity. Thisworkrepresentsprogresstowardsunderstand- ingandmitigatingtheperformancelimitationsofWGMcavitiesforsuchapplications. AsystemforlaserfrequencystabilizationtoathecavityviathePound-Drever-Hall (PDH) method is described. While the laser lock itself is found to perform at the level of several parts in 10 15 , a variety of fundamental and technical mechanisms destabilize theWGMfrequencyitself. Owingtotherelativelylargethermalexpansioncoefficientsinopticalcrystals,envi- ronmentaltemperaturedriftssetthestabilitylimitattimescalesgreaterthanthethermal xvi relaxationtimeofthecrystal. Uncompensated,thesedriftspullWGMfrequenciesabout 3 orders of magnitude more than they would in an FP cavity. Thus, two temperature compensation schemes are developed. An active scheme measures and stabilizes the modevolumetemperaturetothelevelofseveralnK,reducingtheeffectivetemperature coefficient of the resonator to 1.7×10 −7 K −1 ; simulations suggest that the value could eventually be as low as 3.5×10 −8 K −1 , on par with the aforementioned FP cavities. A second, passive scheme is also described, which employs a heterogeneous resonator structure that capitalizes on the thermo-mechanical properties of one material and the optical properties of another. Calculations show that a temperature coefficient zero- crossingcanbeachieved,andencouraginginitialexperimentalresultsarepresented. At shorter time scales, fundamental thermal and technical noise sources define sta- bility limits. The relative strengths of thermorefractive, thermoelastic, and Brownian motion are outlined, along with the level at which they can expect to be observed and some approaches to minimize them. It is shown that variations in the coupling gap pull the frequency at about 10 Hz/nm. A method for calculating frequency noise density causedbylaseramplitudefluctuationsispresented. Frequency comb generation in WGM resonators is also discussed. It is shown that cavity dispersion can be engineered through geometric parameters, yielding a micro- comb with initial sidebands at 1 FSR from the pump. Such combs are thought to be coherent. Also described is a microcomb generated by a PDH locked pump laser. The resultingmicrowavebeatnotecanbechangedfromnoisytoquietbychangingtheoffset ofthislock. Aninvestigationofopticaltomicrowavedown-conversionisconducted. xvii Glossaryofacronyms AOM: .................................................Acousto-opticmodulator CTE: ...........................................Coefficientofthermalexpansion DAQ: .........................................................Dataacquisition EOM: ..................................................Electro-opticmodulator FDT: ............................................Fluctuationdissipationtheorem FEM: ................................................... Finiteelementmethod FFT: .................................................... FastFouriertransform FP: ........................................................FabryP´ erot(cavity) FSR: .......................................................Freespectralrange IF: ..................................................... Intermediatefrequency LO: ...........................................................LocalOscillator MLL: .......................................................Mode-lockedlaser OSA: ................................................Opticalspectrumanalyzer PDH: .......................................................Pound-Drever-Hall PI(D): ........................................ Proportionalintegral(differential) RF(SA): ....................................Radiofrequency(spectrumanalyzer) TEM: ..................................................Thermo-electricmodule xviii TR: ...........................................................thermorefractive VCO: .............................................. voltagecontrolledoscillator WGM(R): ...................................whisperinggallerymode(resonator) xix Chapter1 Whisperinggallerymoderesonators 1.1 Introduction The term “whispering gallery” mode originates from a phenomenon observed in the dome of St. Peter’s Cathedral in London. Sounds made within the dome and near the wall, from e.g., a person whispering, will travel around the inside of the dome as the acoustic wave is repeatedly reflected off the smooth stone wall. The sound eventually returns to the location of its origin and it sounds like the original whispering is coming backfrombehind. Although the term whispering gallery mode (WGM) was not used, they were first proposed as electromagnetic excitations in a dielectric material by Richtmyer in 1939 [90]. In this work the mode structure was solved for spheres and it was shown that energy must be radiated away from the resonator. Cryogenic sapphire oscillators operating at microwave frequencies [34,76] represent perhaps the most mature realiza- tion of electromagnetic whispering gallery mode resonators, with demonstrated field installations[43]. Optical whispering gallery modes were observed indirectly in several early laser experiments, but much of the initial work on purpose-built WGM cavities took place withfusedsilicaspheresatMoscowUniversityinthelate80’sandearly90’s(e.g.,[23]). 1 SubsequentworkwascarriedoutbysomeofthesameresearchersatJetPropulsionLab, wherebothfusedsilicaandcrystallineresonatorswerestudied(averycompletereview ofthelattercanbefoundin[78]). OtherpioneeringworktookplaceatCaliforniaInsti- tuteofTechnologyintheKimblegroup[117,118]andVahalagroup[12,69]–wherethe iconic fused-silica micro-toroids were developed. Tobias Kippenberg, one of Vahala’s students, went on to do a number of advanced experiments with WGM resonators at MaxPlankInstituteforQuantumOpticsandEPFLSwitzerland,mostnotablyforappli- cations to frequency combs and optomechanical coupling [60,65,121]. And finally, there has been a wide range of work on microfabricated WGM devices that is not cited here,withtheexceptionoftheworkbyGaetaandFosteratCornellbecauseitishighly related to our WGM comb effort [84,92]. (The preceding review is in absolutely no way exhaustive, but rather a brief survey of the references which have been consulted themostduringthiswork.) A whispering gallery mode resonator can be understood fundamentally as follows. Lighttravelingaroundtheperimeterofadielectricstructurehavingaxialsymmetrywill be repeatedly totally internally reflected. Assuming the loss per round trip is relatively small,strongresonantmodesareexcited. Themodesoccuratfrequencieswhichcanbe thought of as a standing wave condition, or as solutions to Maxwell’s equations within the air-dielectric boundary condition. Either way, the resonances can be very narrow in frequencyandcorrespondtoalargebuild-upofelectromagneticenergy. This buildup leads to many interesting phenomena which have been the subject of study for over 20 years. The very tight localization of whispering gallery modes 2 means extremely high optical intensities – the resulting nonlinear effects have led to phonon lasers [51], myriad harmonic generation experiments [35,42,106], and para- metric oscillators demonstrated in the optical [66] and microwave [102] regime. Cav- ity quantum electrodynamics with WGM cavities facilitates probing the interaction of light with matter and fundamental quantum physics [117,118]. More recently, the cou- pling between optical and mechanical excitations has been getting increasing attention, examples being mechanical WGMs [113], and optomechanical cooling to the quantum groundstate[122]. Theverysharpfrequencyresponseofresonantmodes(i.e. narrowlinewidths)make WGM resonators appealing for sensing applications, and their superb sensitivity has beendemonstratedinseveraldifferentexperiments[77,105]. Dopingwithionsorquan- tum dots facilitates compact lasers which can have narrow linewidths and/or low pump thresholds[68,74]. Finally, narrow linewidths make these devices attractive for frequency reference applications, and strong nonlinear interactions for combs. More extensive backgrounds andsurveysofrelevantliteraturewillbegivenintherespectivechaptersonthesetopics. The purpose of this chapter is to introduce crystalline whispering gallery mode res- onators and outline the properties and characteristics which are most important within thecontextofopticalatomicclocks. Weareprimarilyconcernedwithfrequencystabil- ityandthestructureofmodes–bothspatiallyandinthefrequencydomain. The chapter is organized as follows. The eigenfrequencies and distribution of WGMs is described along with several complimentary ways of understand and calcu- 3 lating them. Next, the basic experimental setup is presented, followed by examples of characterizing the relevant parameters and figures of merit. Interspersed along the way are some small calculations and experiments that were performed in the background, the inclusion of which will hopefully give a flavor of the subtlety and myriad physical mechanismswhichcomeintoplay. 1.2 Modestructureandfields Conducting research with the mesoscopic resonator structures covered in this work requires a well-rounded understanding of whispering gallery mode structure. For such devices, an exact analytic or numerical solution for the eigenfrequencies (and other relevant quantities) simply does not exist. Closed-form solutions exist only for small spheres; when the diameter starts to exceed about 1 mm numerical solvers do not con- vergebecauseoftheveryhigh-order(> 1000)Besselfunctionswhichareneeded. Var- iousasymptoticexpressionshavebeenderivedforspheroids, buttheyareapproximate. Eigenfrequencies found using the finite element method can be quite precise, but they assumeperfectaxialsymmetry–farfromthecaseforhand-madedevices. Compound- ing all this is the fact that other slight imperfections or asymmetries, such as a crack propagating into the mode region or a piece of dust which alights on the surface, may causesuppressionoforscatteringbetweencertainmodes. TheneteffectisthatsuccessfulinvestigationsofWGMresonatorsiscarriedoutonly through a combined understanding that uses a different one of several quantitative (or 4 conceptual)physicalmodelsdependingonthesituation. Theanalysistechniqueswhich weremostbeneficialtothisworkaredescribedbelow. 1.2.1 WGMsinaperfectsphere Solutions to Maxwell’s equations inside a dielectric sphere serve as a starting point to WGM analysis. A convenient method uses the so called Debye potential. From this potential function, it is possible to take various derivatives to explicitly find the vector components of the electric and/or magnetic field. As is given below, many authors follow a method similar to that presented in the review by Oraevsky [85]. For the sake of brevity, many of the intermediate steps are omitted, with an emphasis placed on the resultingequationsmostrelevanttothiswork. The starting point is Maxwell’s equations in a dielectric medium with electric and magneticsusceptibilityϵandµ,respectively, ∇×E =ikH ∇×H =−ikE k = ( ω c ) √ ϵµ. (1.1) Generally speaking, the resonator supports two different “mode families.” Mathemati- cally they correspond to solutions where one of the fields vanishes along the rotational symmetryaxis(usuallytakentobethez,or“3”axis. TheTEmodeisaccomplishedby setting H 3 = 0 while the TM mode has E 3 = 0. Henceforth only the former will be treated. 5 After writing equations (1.1) in spherical coordinates, it can be shown that a scalar functionU exists, whose various derivatives yield the components ofE andH that will satisfy the coupling given in Maxwell’s equations. This Debye potential satisfies the Helmholtzequationinsphericalcoordinates,yieldingaseparatedvariablesolutionof U lm (r,θ,ϕ) =CP l m (cosθ) √ krZ ν (kr)e ±ilϕ . (1.2) HereZ ν (kr)representsBesselfunctionsofthefirstkind,J ν (ka),withinthesphere,and Hankel functions, H (1) ν (ka) outside. P l m (cosθ) are Associated Legendre polynomials. Thewavenumberkisalsoleftgeneral,andisequaltoω/coutsideand √ ϵµω/c =nω/c inside (the material is taken to be non-magnetic). The constant C is determined from matching boundary conditions. Derivatives of U with respect to the different spatial coordinatesgivethefinalvectorcomponentsofE(r,θ,ϕ)[85] E r =Cm(m+1)P l m (cosθ) Z ν (kr) (kr) 3/2 e ±ilϕ (1.3a) E θ =C d dθ [P l m (cosθ)] 1 kr d d(kr) [ √ krZ ν (kr)]e ±ilϕ (1.3b) E ϕ =±iC l sinθ P l m (cosθ) 1 kr d d(kr) [ √ krZ ν (kr)]e ±ilϕ . (1.3c) Very often the most relevant quantity is the eigenfrequency, which can be found throughasolutiontothefollowingcharacteristicequation. Forasphereofradiusa,itis derivedthroughsatisfactionofthecontinuityofthetangentialelectricfieldatthesphere 6 boundary, [ √ kaJ ν (ka)] ′ √ kaJ ν (ka) =n [ √ nkaH (1) ν (nka)] ′ √ nkaH (1) ν (nka) . (1.4) Theprime denotes thederivativewith respectto the Bessel functionargument. Solving therootsof(1.4)withrespecttowavenumberyieldstheeigenfrequenciesofthesphere. Notethatsofarthematerialisisotropic,lossless,andnon-dispersive. Figure1.1showsasurfaceplotof|E r (r,ϕ)|alongtheequatorialplaneofaresonator with a=1000 µm, n=1.3, l=100, and θ = π/2. For these modes with l = m, P l m ∝ sin 2m θ, meaning that the mode is confined very tightly to the equator. Thus, for the fundamental azimuthal modes the spherical treatment just presented is often a useful approximationevenfordisc-shapedresonators. FIGURE 1.1 Surface plot of|E(r,θ = π 2 ,ϕ)| (from equation (1.3)) for a sphere with a =1 mm. Careful inspection reveals the field behavior at the edgeofthesphere: asmallleakagefieldthatrapidlydecays. 7 Modeindexnumbers Whispering gallery modes are generally characterized with three mode numbers, l, m, and q. The angular number l, loosely speaking, represents the number of wavelengths presentinthestanding-wavepatternthatcirculatestheresonator. Itcanbeapproximated simplybyl =c/(2πnR)foradiscorspheroidofradiusR. Thepolarindexmrepresents the mode distribution in the direction of axial symmetry, and the quantity|l−m| gives the number of nodes near the equator (in the direction of increasing θ. The value of the final index q is equal to the number of roots of (1.4) present inside the sphere, or equivalentlythenumberofnodesalongtheradialdirection. Indices l and m appear explicitly in Eqs. (1.3) and (1.4) above, as well as in the asymptoticapproximationgivenbelow. WhenusingFEM,thenumberofeigenfrequen- ciestobecomputedcanbespecifiedtothesolver,correspondingtothenumberof|l−m| values – or nodes along the θ direction (see Fig. 1.2). Radial index q is more difficult to pin down. Experimentally, it may be possible to determine from precise measure- ments of free spectral range (FSR), but this may require an assumption about l and m. The finite element method simulations performed during this work – using the codes from[53,86,87]–didnotfacilitatesolutionswithq > 1. A “fundamental” WGM has|l−m| = 0 and q = 1. Such modes have the small- est mode volume and therefore are superior for nonlinear processes. However, the millimeter-scale, handmade cavities used in this work have several tens of modes per FSR.Unravelingthespectrumisthereforenontrivial. Furthermore,thehighestQ,high- est contrast modes are often not fundamental modes – as determined by FSR measure- 8 ments,describedbelow. 1.2.2 Analyticapproximationforoblatespheroids Therearetwoimportantlimitationsofcalculatingtheeigenfrequenciesbyfindingroots of Eq. (1.4): (1) they are correct solely for a sphere, which can be fabricated only throughfusionthusprecludinguseofcrystallinematerials,and(2)forresonatorslarger than a few hundred micrometers in diameter, numerical solvers fail because they must manipulate Bessel functions of order in the tens-of-thousands. Therefore, asymptotic approximationshavebeendevelopedwhichfacilitatedealingwith“large”spheroids. ThefollowingexpressionsaretakenfromthepaperbyGorodetskyandFomin[45]. AnapproximationfortheWGMeigenfrequenciesofanoblatespheroidwithmajoraxis aandminoraxisbisgivenby nka =l−α q ( l 2 ) 1/3 + 2p(a−b)+a 2b − nχ √ n 2 −1 − 3α 2 q 20 ( l 2 ) −1/3 − α q 12 ( 2p(a 3 −b 3 )+a 3 b 3 + 2nχ(2χ 2 −3n 2 ) (n 2 −1) 3/2 )( l 2 ) −2/3 +0(l −1 ), (1.5) where α q is the negative q th zero of the Airy function, p = |l− m|, and χ =1 for TE modes and χ = 1/n 2 for TM modes. Assuming the major and minor radii of the spheroid are known (from a micrograph, e.g.,), Eq. (1.5) can be used for finding WGM frequencies. Often it is desirable to deduce the mode numbers from an actual WGM device. In 9 this situation, a very useful set of equations gives the change in eigenfrequency with respect to an increment of mode indices. The difference resulting from a change of ±1 in l (with m and q constant) would be the typical definition of FSR: successive longitudinalmodes. Modespacingsforthethreedifferentindicesaregivenby 1 ω ∂ω ∂l ≈l −1 [ 1+ α q 6 ( l 2 ) −1/2 ] (1.6a) 1 ω ∂ω ∂m ≈ (l −1 ) b−a b [ 1+ α q 12 (b−a)(a+2b) b 2 ( l 2 ) −2/3 ] (1.6b) 1 ω ∂ω ∂q ≈ π 2 √ −α ( l 2 ) −2/3 [ 1+ α q 20 ( l 2 ) −2/3 ] . (1.6c) As pointed out in [45], there are certain cases where successive values of a different index are equal. For example, when a = 2b, the spacing between modes of successive m (with l and q constant) is the same as an FSR. This is in analogy to a confocal FP cavity,wherehigherlateralordermodesaredegenerate. Obviously,whenattemptingto experimentally measure FSR, it would be very difficult to determine if successive l or mvalueswerebeingobservedforaresonatorwitha≈ 2b. Another consequence of Eqs. (1.6) is that, since the (negative) Airy function zeros increase with order, the spacing between modes of successive index will be larger for higherradialordermodes. Thisistrueacrossallthreeindices,anditcaneasilybeunder- stoodphysicallyfromthestandpointthathigherordermodesaredistributedclosertothe resonatoraxisandthereforehavelowereffectiveradius(withafirstorderapproximation ofFSR=c/(2πna)). 10 1.2.3 FiniteelementmethodforWGMresonators The finite element method (FEM) is a powerful tool for studying WGM cavities. In additiontocalculatingeigenfrequencies,modespacings,modevolumes,anddispersion, FEMcansimulatearangeofotherrelevantphysicssuchasheattransferandmechanical strain. FEM modeling of the later type was more significant for this work; however a brief discussion of finding eigenfrequencies and field distributions with FEM is given below. FEMeigenfrequencycalculationsforarbitrarygeometries Solutions to the vectorial Helmholtz equation within a domain consisting of the dielec- tric and surrounding air yield WGM eigenfrequencies. An axially symmetric geometry is created with refractive indexn inside the resonator andn air outside, usually taken to be≡ 1. Also input to the model is the radial mode order l which is usually adjusted manuallyorthrougharoutinetoyieldthecorrectvalueoffree-spacewavelength. Then, an FSR can be approximated by incrementing l, or the frequency difference betweendifferentvaluesof|l−m|canbefound. Theseapproximationstendtoberough, since there is a large amount of error in the solutions both from the simulation itself (meshing, etc) and from the idealizing of a non-ideal geometry. Therefore, this type of modelingtendstobemoremeaningfulforrelativeratherthanabsolutecalculations. For example finding the difference in WGM frequency between a resonator surrounded by air (n air = 1.0003) and vacuum (n air ≡ 1). Plots of field distribution are also useful visualaids. 11 FIGURE 1.2 FEM simulations of a WGM excited in a geometry similar to that of resonators used in this work. Left: calculation domain, resonator geometry, and fundamental mode; vertical ticks separated by 22µm. Right: detailofthenextthreehigherordermodes. AtmosphericpressureeffectsuponWGMfrequency Several other experimental parameters can be modeled with FEM, giving insight or an order-of-magnitude approximation if done roughly, or very precise results if adequate care is taken. For example§3.1 contains details of temperature coefficient calculations usingheattransferandthermalstrainFEMmodels. Anothersuchexampleisacalculationofatmosphericeffectsoneigenfrequencies. A varying ambient pressure changes the resonator’s radius through mechanical deforma- tion, a simplistic yet non-negligible contribution to frequency instability. We find that among many benefits of operating a WGM cavity under vacuum is the isolation from changesinatmosphericpressure. Fig.1.3(a)showstheresultsofsimulatingamagnesiumfluorideresonatorundergo- ingcompressionfromanincreaseinambientpressure. Theresultingstrainchangesthe 12 radius and pulls the frequency through ∆f/f =−∆r/r. As a guide for pressure vari- ations, it’s interesting to start with the record setting extremes for pressure at sea level tobe108.3kPa(maximum)and87.0kPa(minimum)[67]. Theseextremesareapprox- imately ±10% of standard pressure. Thus, we simulate a 10% variation from base pressure,forbasepressuresrangingfrom0.1Patofullatmosphericpressure(10 5 Pa). 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 Base Pressure [Pa] Δ r/r for 10% Variation (b) FIGURE1.3 Modelingtheeffectsofchangingambientpressure. (a)Sym- metricallyreducedgeometryrepresentingonequarterofatypical1mmthick MgF 2 resonator with 3.5 mm radius is subjected to 10% variations in pres- sure. (b) Fractional changes in r directly correspond to expected fractional frequency changes, suggesting that vacuum packaging is necessary for high stability. Under a vacuum of 0.1 Pa (corresponding to 0.75 millitorr), a 10% variation would result ina fractional radial strain of a fewparts in 10 14 . Howeverat atmosphere, a 10% pressure variation causes a huge fractional shift of a few parts in 10 8 (see Fig. 1.3(b)). Weagainreiteratethatsuchachangecorrespondstothelargesteverrecorded,sounless a hurricane passes over the resonator, an effect this large would never be seen. More- over, a change in atmospheric pressure – or the pressure in a vacuum chamber – would likelyhappenmuchmoreslowlythanotherdestabilizingmechanisms. Nonetheless,the 13 somewhat surprising results shown here of pressure variations noticeably compressing theresonatorcrystalshouldbekeptinmind. 1.3 WGMresonatorexperimentalsetup Most of the experiments described here were done with a setup like the one shown schematically in Fig. 1.4. The simplicity of using an angle-polished fiber coupler is apparent; only one lens and aperture is needed. In some cases, the angle polished fiber is held by an external 3D piezo stage; in other cases it is glued to a flexure arm on the temperaturecontrolledbase. Forplacinginavacuumchamber,thelatterapproachmust be taken. The setup for comb generation is similar, except often there is an additional output coupler oriented the opposite direction with respect to the mode propagation direction. Thisisusedtopickoffasmallamountoflightforinputtingtoafiber-coupled opticalspectrumanalyzer(OSA). 1.3.1 Opticalsetup For most of the discussion we assume that the laser light travels through an AOM and EOM before the polarization controller shown in Fig. 1.4. The EOM proves valuable forcharacterizationofqualityfactorandFSR,andisrequiredforlaserfrequencystabi- lization. Similarly, the AOM is primarily for the laser frequency lock, but the ability to modulate amplitude quickly and predictably is also useful for troubleshooting, system characterization,orimplementinglaserintensitystabilization. 14 Polarization control and stability is a challenge with all fiber-coupled optics, par- ticularly when some of the fiber pigtails are not polarization maintaining fiber – as is the case here. Initially, we used fiber paddle polarization controllers; these were found to be too unstable with respect to temperature fluctuations on the scale of tens of min- utes. Wefoundthatfiberbenches–wherethelightislaunchedtofree-space,propagates throughwaveplates,thenislaunchedbackintothefiber–hadmuchgreaterstabilityand precisioncontrol. As a first pass, detection is accomplished by simply placing a lens to collect and focus onto a detector the diverging beam exiting the coupler. However, since this con- figuration produces a transmission signal, eliminating stray light will decrease the DC background and increase contrast. An aperture placed close to the lens helps greatly with this. However, being able to adjust the detector’s two degrees of freedom in the direction perpendicular to the beam also proves critical for maximizing contrast. This can be explained as follows. For the angle polished fiber (and particularly for prism- coupledsystems),muchofthelightpropagatingthroughthecouplerissimplyinternally reflected and directed towards the lens; it never enters the WGM evanescent overlap region. Thislightformsa“halo”aroundthelightthatreflectsofftheinteractionregion, but because the coupler system is asymmetric it’s not necessarily in the middle of the beam. Thus, use of an aperture and x-y stage on the detector allows for directing the maximumpossibleamountofWGM-coupledlightontothedetector. 15 FIGURE 1.4 Schematic of a typical WGM resonator setup. Light is sent down a fiber and couples evanescently to the cavity with an angle polished fiber (inset detail). Both promptly reflected and cavity leakage field exit the flat,cleavedendofthefiber,thenpropagatethroughanapertureandfocusing lensbeforehittingthedetector. 1.3.2 Evanescentcoupling The evanescent field from light totally internally reflected off the inner surface of an optical element will couple into a cavity’s WGMs. In order for this process to happen efficiently,aphasematchingcondition φ = cos −1 ( n r n c ) (1.7) mustbemet,wheren r andn c aretherefractiveindicesofthediscandcouplermaterial, respectively. In this case φ is measured from a line tangent to the disc (and parallel to the coupler surface) to the propagation direction of incoming light (see inset detail of Fig. 1.4). The immediate consequence of (1.7) is that the coupler must have a higher 16 index than the resonator – not a challenge for CaF 2 and MgF 2 , but difficult for many other materials of interest like lithium niobate. For such materials, high-index prisms madeofdiamondorrutilemustbeused. Themechanismbywhichlightiscoupledintothecavityisacrucialpartoftheexper- imental setup. Since it is a near-field interaction, coupling efficiency depends strongly upon the resonator-coupler gap and requires nanometer positioning stability. For basic testsorinitialcharacterizationoftheresonator,a3-Dstagecanbeusedtoholdthefiber. The temperature coefficient and drift of piezo actuators, however, means that the fiber mustbeheldmorerigidlyiflong-termstabilityisneeded. Monolithictemperaturecon- trolledmountswithaflexurearmtowhichthefiberisgluedweremade. Theflexurearm allows for slight adjustments of the coupling, which was subsequently found to stable forweeksorevenmonthsinsidethevacuumchamber. 1.3.3 TemperatureControl The amount of effort which goes towards controlling the temperature of a WGM res- onator cannot be overstated; indeed a great deal of this work was spent on sophisti- cated methods of temperature control and compensation. But even for the most basic measurement of quality factor, the device should have some type of active temperature stabilization and basic isolation from air currents caused by air conditioners or people moving around in the room. As we shall see later in great detail, this is because of the high temperature coefficient of an un-compensated WGM resonator. For example, if a device withQ∼ 10 9 is to be measured, a laser sweep of say 10 MHz may be desirable 17 to resolve the 200 kHz linewidth. If the mode is to stay within that sweep, the cavity temperaturecannotchangebymorethan5mK. Thebasictemperaturecontrolimplementationisaresistiveheatergluedintoablock of metal, along with a thermistor. The WGM resonator is then either glued directly on top of the metal block or else mounted on a post which can be clamped to the block. In this way, either a commercial digital temperature controller or a homemade analog circuit can be used to control the heater current. We have found benefits of each, some ofwhichareexpandeduponin§3.1.5. The primary limitation of the metal block, resistive heater combination is the rela- tively long time constant and resulting low gain in the control loop. Another drawback istheasymmetricgainprofile: whileheatingoccursactivelybyrunningcurrentthrough the heater, cooling occurs passively through convection and conduction, thus the loop gainisdifferentdependingonwhethertheactualtemperatureisaboveorbelowtheset- point. Use of a thermoelectric module (TEM) and low thermal-capacity mount would besuperiorforthesereasons. 1.3.4 Vacuumchamber Placing the resonator-coupler assembly in a vacuum chamber increases the achievable frequencystabilitybyseveralordersofmagnitudeoveroperationinatmosphere. There are several reasons for this. As described above, changes in atmospheric pressure can actually destabilize the WGM frequency, though we expect these changes to be slower thanmostotherrelevantsystematics. Thevacuumchamberalsoprovidesisolationfrom 18 changes in atmospheric humidity and gas content, which may play a more important rolebecausetherearemanyatomicandrovibrationallinesinwaterandCO 2 inthenear andmidinfraredwhichcouldinteractwiththeevanescentWGMfield. Vacuumpackagingcontributestostabilityprimarilybecauseoftemperatureeffects. Havingtheresonatorassemblyundervacuumvirtuallyeliminatesconvectiveheattrans- fer, greatly increasing isolation from the environment. Moreover, if care is taken to suspend the resonator structure with thermally insulating materials, conductive transfer canalsobesuppressed. A final, collateral benefit of placing the resonator in a vacuum chamber is that the sensitiveanddelicateopticalsetupisprotectedfromenvironmentalfactorssuchasdust ormechanicaldisturbances. Therefore, a brass heater block (sometimes called an “oven” since it has a top and encloses the resonator completely) with integrated flexure arm for coupling adjustment wasplacedinavacuumchamber. AslightmodificationtoFig.1.4isneededintheform ofanadditionallensplacedrightafterthefibercoupler. Thislenscollimatestheoutput beam, allowing it to be sent out of the chamber through a window, where it can then beapertured,focusedonadetector,ordecomposedonapolarizingbeamcube. Optical inputisaccomplishedviaafiberfeed-through: barefiberrunsthroughaholeinasmall conical teflon piece which is compressed in standard vacuum swage-lock style fitting. Commercially available electronic vacuum feed-throughs provide connections for the heater and thermistor in the heater block, as well as for the TEM in the dual-mode experiment. 19 FIGURE 1.5 The WGM resonator assembly vacuum setup. (a) Detail of brassheaterblockshowingresonatorandfibercoupler(cominginfromleft). (b) The assembly mounted on the chamber base using a fiberglass block for insulation. Scattered visible laser light shows fiber path. (c) The sealed 8- inch chamber with heater assembly and coupling lens visible. Other optics anddetectorsareoutsidethechamber(topofimage). For simplicity, concerns about vibration, and because it was deemed unnecessary, a pump was not run continuously on the chamber. Instead, a pump station was used to periodically pump it down, eventually reaching pressures as low as ∼ 10 −6 torr. After valving off, the pressure would slowly rise; the rate of this rise decreased with subsequent pumping cycles, after 3-4 of which, the pressure stayed in the tens of milli- torrforaboutamonth. We note here that although conductive and convective heat transfer may be effec- tivelysuppressed,radiativethermalcouplingbetweentheresonatoranditsenvironment 20 remains a performance limitation. Described in detail in§3.1.4, we see that the rela- tively high emissivity of optical crystals (in the neighborhood of 0.8) results in strong coupling to environmental temperature changes. Future implementations should there- fore haveone or more low-emissivityheat shields. Coating the crystal with gold would also be a very easy way to decrease the effect of environmental coupling by almost an orderofmagnitude. 1.4 Quantificationsandfiguresofmerit Here we outline the crucial characteristics of a WGM cavity when used for frequency referenceorcombgenerationapplications,andhowtheymaybecalculatedand/ormea- sured. Quality factor – inversely related to the linewidth – is very important to both applications. For frequency stabilization of a laser, increasing the Q directly improves the noise floor limit of the lock (see§2.4.2). Combs are more efficiently created in a highQ resonator because nonlinear generation goes asQ 2 [8]. Moreover, as discussed in§5.2, the ratio of linewidth to cavity dispersion determines whether or not the comb generatedisalow-noise“coherent”comb. Mode volume is another important parameter in both applications because it deter- minesopticalintensity,whichinturnaffectsbothnonlineareffectsandheatdistribution inthecavity. Andfinally,thefreespectralrange,andnumberofmodesperFSRplayan important role in comb generation dynamics. We also see that by measuring FSR, we mayfindthemodeswithsmallestvolume. 21 1.4.1 QualityFactor ProbablythemostuniversallyimportantfigureofmeritforWGMresonators,thequality factor Q is essentially a measure of loss. It can be an indication of how well the res- onatorwasfabricated,howmuchopticalabsorptionistakingplaceinthehostmedium, or the extent to which the evanescent field is being lost to the environment. In sensing applications, changes in Q alone can be sufficient for measuring the presence of parti- cles or gases which surround the resonator and can interact and absorb or scatter away electromagneticenergy. The dominant physical mechanisms which limit Q are: surface scattering, material absorption,andradiativeloss. Theoverallqualityfactorcanthusbefoundthrough 1 Q tot = 1 Q scat + 1 Q abs + 1 Q rad . (1.8) This expression is valid only in the linear regime. Additional losses must be accounted for when the linear Q becomes very high and power buildup in the cavity is suffi- cient to exceed the threshold for processes such as stimulated Raman scattering or non-negligible Kerr induced frequency shifts [50]. Somewhere in the neighborhood of Q ∼ 10 10 , Raman scattering sets the limitation to increasing quality factor and has been investigated in detail [52]. Achieving maximum quality factor was not, however, the goal of this work, and Q ≈ 4.6×10 9 was the highest observed. Moreover, as dis- cussed in§2.4.2, we determined experimentally that Q in the high 10 8 s was sufficient forachievinglocklooperrorsignalnoisedensitiesontheorderof1Hz/ √ HZ. 22 It is worth discussing the terms in Eq. (1.8) in more detail. The first term, Q scat results from scattering losses on or near the surface of the resonator. This loss mecha- nism obviously depends heavily upon the fabrication process. For microfabricated res- onators, surface quality – the presence and number of scattering sites – may be depen- dent upon details of the lithography and etching, while for microspheres the conditions which allow surface tension to form a smooth surface are critical. In the case of crys- talline resonators formed with polishing, the resonator can be successively re-polished, cleaned, and tested. Atomic force microscopy (AFM) measurements of surface rough- ness after a sequence of diamond polishing steps as used for resonator fabrication have shown RMS roughness values on the order of a few hundred picometers, sufficient that other loss mechanisms will begin to dominate [50]. Still, scattering effects are difficult to eliminate completely, if for no other reason than environmental cleanliness becomes crucial, particularly if a resonator is meant to perform for long periods. Just a single specofdustcandestroytheQifitlandsonthewrongpartoftheresonator. Andfinally, since Rayleigh scattering goes as 1/λ 4 , we note that for shorter wavelengths sources of scatteringlossbecomeincreasinglyimportant. Material absorption, represented by 1/Q abs , sets the fundamental limit in the linear regime. If optical propagation loss in the bulk material is well known, the absorption limited quality factor can be approximated as Q abs = 2πn/λα where α is the material absorptioncoefficientinneperspermeter. Radiative loss is generally considered negligible for mm-scale resonators. As seen in Fig. 1.1, the evanescent tail extending “outside” the resonator is quite small. This 23 evanescentfieldradiatesenergyawayfromtheresonatoratasignificantrateonlywhen the radius is less than a few hundred micrometers. However, more subtle forms of radiative loss exist. For example, in making “waveguide” style cavities for frequency comb generation (§5.2.1), quality factor appeared to be limited by the geometry itself. Thissuggeststhateitherelectromagneticenergyisbeinglosttoairbecauseofthesmall lateraldimensionofthewaveguide(andrelativelylargeevanescentfield),orlightinthe waveguidemodeisleakingintothecentral,bulkpartofthecavitywhereitislosttothe macroscopiccylindricalcrystal. 1.4.2 ModeVolume If an analytic solution for the electric field distribution is known, mode volume can be calculatedbysimplyintegratingthefieldstrengthovertheresonatorvolume,normalized toitsmaximumvalue. Acommonlyusedexpressionisgivenin[23]: V m = (∫ |E(r,θ,ϕ)| 2 d 3 r ) 2 ∫ |E(r,θ,ϕ)| 4 d 3 r , (1.9) where, for resonators with diameter more than a couple hundred microns, the integral need only be taken over the resonator geometry (negligible field outside). For a TM modeinasphere,(1.9)canbeapproximatedas3.4π 3/2 (λ/2πn) 3 l 11/6 √ l−m+1where l andmaremodenumbersandnthehostmaterial’srefractiveindex. Thus, finding V m is typically straightforward for spherical cavities. But since such cavitiesarearelativelysmallsubsetofWGMdevices,FEMbecomesausefultechnique 24 forcomputingmodevolume. TheCOMSOLcodedescribedbyOxborrow[86,87]gives a method for mode volume calculation. But for this work, mode volume was estimated fromthe 1/e 2 dimensionsoftheelectricfieldbasedontheGrudinincode[53]. Fig.1.2 shows the mode field distribution used to approximateV m for the fundamental mode in the heat dissipation simulation of§3.1.3. Higher order modes could be approximated similarly, and Eq. (1.9) can also be implemented explicitly with FEM. However, unless a reliable way of accurately determining all mode numbers is developed, this would be awastedeffort;clearlythemodevolumechangessignificantlyfordifferent|l−m|. 1.4.3 ExperimentaldeterminationofQ,FSR,andmodeindex Qualityfactor Determination of quality factor is typically accomplished through time-domain mea- surements of cavity ringdown [101], or frequency domain measurements of linewidth. For this work only the former was used, but it’s worth pointing out that ringdown mea- surementsarelesssusceptibletosomeoftheerrorsourcessuchasthermalbistabilityor inaccuratecalibration,discussedbelow. Typicallinewidthmeasurementswereperformedasfollows(andshowninFig.1.6). Laser light is passed first through an EOM before being coupled into a cavity. Side- bands are generated by applying an RF signal of known frequency to the EOM. These sidebands then serve as a frequency reference for calibrating the horizontal axis of an oscilloscope. Upon calibration, a Lorentzian fit to the mode (usually with sidebands turned off) provides the quality factor through Q = γ/f 0 , where γ is the FWHM line- 25 width, and f 0 is the optical frequency of the mode (usually taken to be the test laser frequency). FortheexampleshowninFig.1.6,thelinewidthof167.2±2.8kHzyields Q = 1.169±.020×10 9 atf 0 =192.16THz=c/1560.01nm. 0.012 0.0125 0.013 0.0135 0.014 0 0.5 1.0 1.5 2.0 2.5 3.0 Time [s] Detector Signal [V] (a) 1 MHz sidebands −2.0 −1.0 0.0 1.0 2.0 0.5 1.0 1.5 2.0 2.5 3.0 Detuning [MHz] Detector Signal [V] Data Lorentzian Fit (b) FIGURE1.6 Measurementofthecavitylinewidth. Asalaserissweptover the resonance, modulation sidebands allow calibration of the oscilloscope trace time axis into frequency (a). Now without modulation, a trace can be plotted in the frequency domain (b). A Lorentzian fit gives the linewidth, in thiscaseγ = 167.2±2.8kHz. The uncertainty given in the above value of Q comes from the residual for γ in the Lorentzianfit. However,therearemanypotentialpitfallstothistypeofQmeasurement. Thermal bistability [25] will make a linewidth broader in one direction of laser sweep and narrower in the other if too much optical power is used; the response of a laser to 26 its ramp signal is often nonlinear – thus the calibration from time to frequency of the scope trace changes across the ramp; with a noisy laser or very low contrast, it can be difficult to determine the locations of modulation sidebands with high precision; the coupler itself introduces loss, and over-coupling the cavity will widen the resonance; andresonatorswithveryhighQmaydemonstratenonlinearprocesseswhichconvolute thelineshape. Mostoftheerrorscanbeminimizedbyusinglowopticalpowerlevels(on theorderofmicrowatts),undercouplingthecavity,andperformingthemeasurementsat the same position along the laser sweep. However, cavity ringdown is superior if very precisevaluesofQmustbeobtained(especiallywhentheQishigh). FSR AlsoperformedwithanEOM,measurementsoffreespectralrangecanbeveryprecise in a high Q cavity. This facilitates determination of the disc radius or refractive index, depending which quantity is better known. The measurement is performed as follows and typical scope traces are shown in Fig. 1.7. A modulation frequency f m close to the FSR is applied to the EOM. The resulting sidebands on the laser interact with the same mode at±FSR, causing sidebands to appear next to the original mode at δ = |f m −FSR|. (At high enough modulation depth, second (or higher) order sidebands arealsovisible.) Whenδ≈ 0,allpeakslineupandthecontrastofthemainpeakdrops dramatically (see detail in Fig. 1.7). With good initial contrast, the frequency of this peak alignment can be determined to about 1/10 of the cavity linewidth, about 20 kHz inthecaseshownhere. Thus,fora7mmdiametercavitywithQ = 2×10 9 ,theFSRcan 27 bedeterminedtoabout10kHz,or1ppm. Therelativeprecisiongetsbetterwithsmaller cavities,thoughwenotethatlaserlinewidthandjitterbegintosetthemeasurementlimit around10kHz. Detail of the peak −5 −4 −3 −2 −1 0 1 2 3 4 5 1 1.2 1.4 1.6 1.8 Detuning [MHz] Detector Signal [V] δ = 2.0 MHz δ = 1.5 MHz δ = 0.5 MHz δ = 0 MHz FIGURE 1.7 Measuring cavity FSR with an EOM. When the modulation frequencyf m approachestheFSR,sidebandscomeinfromeitherside. Here, fourvaluesofmodulationareshownforδ =|f m −FSR|,andsecond-order sidebands can also bee seen. Right: zoom shows that when f m = FSR, a clearincreaseincontrastisobserved. FSR measurements, along with Eq. (1.6a) can be useful when trying to identify the order of modes in a resonator that supports more than 30 well coupled, high Q resonances per FSR. In Fig. 1.8, a scatter plot is shown of measured FSRs for an MgF 2 resonator with 3.45 mm radius. The 33 modes were measured in sequence through one FSR;“mode”onthex-axisissimplytheorderinwhichtheyappeared. Notethat,while thefrequencyspacingbetweenthemodeswasalsomeasured,itisarbitrary. Thisfollows fromthefactthateachmodehasadifferentFSR,sothesequencewillchangedepending upon the absolute optical frequency of the measurement. The vertical position of each point is determined by how much greater the FSR is for that mode than for the lowest 28 measured valued of 10.08797 GHz. Ticks on the y-axis of Fig. 1.8 (and grid lines) correspond to the values computed according to Eq. (1.6a) for the first five radial order modeswithl = 19,065≈c/(2πnR)(ordinarypolarization). 0 5 10 15 20 25 30 35 0.00 6.53 11.89 16.62 20.94 24.97 Mode (arbitrary) FSR−10.0088 GHz [MHz] FIGURE 1.8 Dependence of FSR on radial mode order q for a 3.45 mm radius MgF 2 WGM resonator. For aQ of 10 9 , the FSR can be measured to 10 kHz precision allowing for approximate identification of the mode num- bers. Lines along vertical axis correspond to the first five radial order mode spacingsaccordingto(1.6a). Thedistributionofmodeswithineachradialorder“bin”resultsfromdifferentvalues of|l−m|. The non-degeneracy is not accounted for by Eq. 1.6, but a direct solution of (1.5) reveals that successive|l−m| have mode spacing on the order of a MHz. It is thought that the significant variation in FSR with mode index m results from the high contrast(a/b>> 1)geometryoftheresonator–comparedtomoresphericalcavitiesin which the modes of same radial number have very similar values of FSR [75]. It may not be possible to accurately identify all indices of each mode. Nevertheless, it is clear thatthemodewiththesmallestFSRmustbetheoneoflowestorder. Thismodemayor may not have the highest Q (based on empirical evidence), but out of those coupled to 29 itissuretohavethesmallestmodevolume. A“map”ofFSRsliketheoneinFig.1.8canbeveryusefulifonewishestoidentify and study the same mode day-after-day. The temperature of the laser and/or cavity are likelytochangeslightlyoverdaysandweeks,sorelyingonthosesetpointsmaynotbe effective in getting back to the same mode. But if the FSR of three or four successive modesismeasuredandcomparedtothechart,themodescanusuallybeidentifiedwith respecttoeachother. Whenquantifyingthermalnoiseasproposedin§4.2,itmaybenecessarytomeasure thefrequencynoiseofaWGMasafunctionoftemperature. Sincethethermorefractive noise floors go as 1/V m , it is crucial to measure the same mode at each temperature. The above technique is useful for finding the measurement mode after the resonator temperaturehasbeenchangedbyseveraldegrees(equivalenttomanyGHz). 45 50 55 60 65 70 75 80 85 −3 −2 −1 0 Resonator Temperature [ ° C] FSR−9.9037 GHz [MHz] FSR Data Liner fit FIGURE 1.9 Thesamemodecanbeidentifiedacrossawiderangeofres- onator temperatures by using an FSR “map” as described. Here, the tem- perature dependence of FSR is shown, along with a linear fit, that facilitates calculatingthetemperaturecoefficientoffrequency. Ifsuchatemperaturescanoftheresonatorisperformed,FSR’sdependenceontem- 30 perature can be found, as shown in Fig. 1.9. A linear fit to these data has a slope of -89.2kHz/K,thusthefractionalchangeinFSRyieldsthethermalexpansioncoefficient α l through ∆FSR FSR = ∆r r =−α l ∆T, (1.10) where r is the radius of the resonator. The resulting value of α l = 9.01×10 −6 K −1 is in good agreement with the datasheet value for Corning MgF 2 of 9.25×10 −6 K −1 , especiallyconsideringthatthislinearapproximationneglectsthedependenceofα l itself upontemperature. (Thermorefractionisneglectedherebecauseitisabout100×smaller thanα l inMgF 2 atthistemperature.) Finally,notethatbycontrasttothemethodsoftem- perature coefficient measurement described later in this work, this technique does not requireanyexternalopticalreference;thetemperaturedependenceofopticalpathlength canbedeterminedquiteaccuratelywithasingleresonator,laser,EOM,andmicrowave source. 1.5 Summary Whisperinggallerymoderesonatorsandatheirbasicexperimentalrealizationhavebeen described. Several techniques for calculating eigenfrequencies, FSRs, mode volumes, and spectra have been given. We show that vacuum packaging is necessary to reduce the effects of changing atmospheric pressure and humidity, and that it greatly increases temperaturestabilityandenvironmentalisolation. Wefindthatforalarge,highcontrast resonator, modes of radial order up to 5 can be excited, and that the non-degeneracy of 31 modes with different|l−m| means that tens of modes with different frequencies can be observed per FSR, a map of which facilitates finding the same mode over different temperatures or times. We show that by measuring FSR as a function of temperature, wecanmeasuretheopticalpathlengthtemperaturecoefficientofthematerialwithouta stableexternalopticalreference. 32 Chapter2 Frequencystabilization&metrologyin WGMcavities 2.1 Introduction Increasing the frequency stability of a laser by locking it to an optical reference cav- ity is a rich and varied topic. The technique was developed in the late 70’s and early 80’s, and has been an important aspect of laser physics ever since. Indeed, the perfor- mance increase afforded by such techniques has been a linchpin in many modern laser- based experiments such as precision spectroscopy [10], detection of gravity waves [6] or fields [104], cavity QED [18], and tests of Lorentz invariance [39], to name a few. Andthespecificapplicationwhichmotivatesthiswork–opticalatomicclocks–which are virtually unimaginable without an ultra-narrow-linewidth, cavity stabilized laser as thelocaloscillator[63]. ThischapterdescribeslaserfrequencystabilizationtoaWGMcavityandthedetails of our implementation and characterization thereof. It begins with an explanation of Pound Drever Hall laser locking in the context of WGM resonators. Next, relevant detailsoftheexperimentalsetuparegiven,followedbyadiscussionofcharacterization techniquesandtheresultsachieved. Intheprocessofsuchcharacterization,wefindthat 33 weareinfactmeasuringwithextremelyhighprecisionthedetailsofthemodestructure withintheresonatoranditsdependenceontemperature. Appendix B is meant to act as a supplement to this chapter and contains additional technical details such as schematics of the lockboxes constructed and diagrams of the experimentalsetup. 2.2 Pound-Drever-Hall(PDH)locking This locking technique has become ubiquitous within the field of AMO physics. At its foundationisthegenerationofanerrorsignalwhichissensitivetothephasedifference between the interrogation field and the cavity build-up field. Thus, a high bandwidth lock loop can significantly improve both the phase and frequency stability of a noisy laser. Itwasfirstdescribedcohesivelyintheseminal1983paperbyR.Drever,J.Hall,and others (with affiliations at Caltech, University of Glasgow, Scotland, and JILA/NIST Boulder) [36]. The “Pound” in “PDH” is a nod to R.V. Pound, who described the anal- ogous technique with microwaves in a 1946 paper [89]. The method of obtaining a dispersive signal via frequency modulation had also been previously shown for atomic vaporsamplesin1980byGaryBjorklundatIBM[19]. ManygoodworkshavebeenwrittenonPDHlocking. Forsittingdownandactually putting together a functioning PDH locked laser, the following two American Journal of Physics articles are very useful: the 2001 paper by Eric Black [20] for a conceptual 34 and theoretical background, and the 1996 paper by Boyd et al. [22] for a details on experimentalimplementation,includingthePIDschematicwhichservedasthestarting pointforthelockboxcircuitsdevelopedduringthiswork. 2.2.1 PDHconcept AttheheartofaPDHlockisanasymmetricerrorsignalthatcontainsinformationabout the detuning between laser and cavity resonance, as well as the relative phase between theincidentlightandthecavityfield. Theasymmetryisaprerequisiteforanyfeedback stabilization loop, but it can be generated in other ways for example locking to the side of a peak in cavity transmission. The PDH technique provides an asymmetric, phase coherent error signal that is also relatively insensitive to changes in the laser output amplitude. A schematic of PDH locking is shown in Fig. 2.1. Light from a noisy laser source passes through an electro-optic phase modulator before being directed toward the cav- ity. At the coupler, some of the modulated light field enters the cavity, while some is reflecteddirectlyoffthecoupler–knownasthe“promptlyreflected”field. Alsopresent at the coupler is light which has built up in the cavity and is now being coupled back out – the so called “cavityleakage” field. The leakage and promptly reflected fields are spatially mode matched and may thus interfere efficiently at the coupler surface, after whichthecombinedfieldisdirectedtoafastphotodetector(“fast”beingf bw &ω m /2π, wheref bw is the bandwidth of the detector). A single local oscillator (LO) operating at ω m drives both the EOM and an RF mixer, the other port of which is connected to the 35 detector. Out of the mixer’s IF port comes the error signal, which is put finally through afilteringamplifier(PIDservo)beforebeingfedbacktothelaser,completingtheloop. FIGURE 2.1 PDH laser stabilization to a WGM reference cavity. After being frequency modulated in the EOM, promptly reflected light interferes with cavity leakage field at the coupler. Demodulating the photodetector signal yields an asymmetric error signal because of the cavity’s frequency- dependentphase. Residualsaremeasuredat“A.” Themechanismbywhichthecavityactsasareferencecanbeunderstoodasfollows. Supposethecenterfrequencyofthelaserfieldison-resonancewiththecavity;thelaser “fills”thecavitywithanelectricfield. Nowsupposethelasermakesafast–greaterthan the inverse cavity lifetime, τ – frequency excursion. Because of its finite lifetime, the cavity will not respond instantly to this frequency excursion and the leakage field will stillhavetheoriginal,on-resonancefrequencyandphase. Thedetectorthushasincident upon it a field with two components, the phase difference of which can be extracted at themixerandfedbacktothelasertokeepinonresonance. 36 At longer time scales (greater than the cavity lifetime), it is the optical pathlength of the cavity which serves as the reference. In other words, the value of the resonance frequency does not change with time. The cavity therefore acts as a reference in two regimes: for time-scales< τ, the optical build-up in the cavity serves as a phase refer- ence,andfortime-scales>τ,thelengthofthecavityservesasafrequencyreference. Stabilization of the laser’s phase and frequency to the cavity requires an error sig- nal that is asymmetric about the cavity resonance, which is accomplished using phase modulation. A light field which is being modulated at angular frequency ω m can be representedby E(t) =E 0 e i(ω 0 t+βcosωmt) (2.1) where ω 0 is the original (carrier) field oscillation frequency, and β is the modulation depth. TheJacobi-Angerexpansion[7] e izcosθ = ∞ ∑ m=−∞ i m J m (z)e imθ (2.2) can be used to write the exponential in terms of Bessel functions. Taking only the first ordertermsgives E(t)≈E 0 [J 0 (β)e iω 0 t +J 1 (β)e i(ω 0 +ωm)t −J 1 (β)e i(ω 0 −ωm)t ]. (2.3) In the frequency domain, this represents a carrier with two sidebands spaced at±ω m . Butthemostimportantpropertyofthefieldrepresentedby (2.3)isthatthesesidebands 37 are out of phase with each other by π. If the modulated beam is shone directly on a fast photodetector, no beatnote signal will be generated. The carrier interferes with the sideband at ω 0 + ω m , creating a beatnote at ω m ; however it also interferes with the sideband at ω 0 −ω m , creating a second beatnote at the same frequency, but exactly π outofphase. Thesetwobeatnotescompletelydestructivelyinterfere. The generation of an asymmetric error signal comes from the interaction of this modulatedlightfieldwiththecavity. Thephaseoftheleakagefieldisalsoasymmetric, changing from +π to −π on either side of cavity resonance. If the incident modu- latedlaserlightisslightlyoffresonance,theinterfering,phaseshiftedleakagefieldwill breakthesymmetrycausingdestructiveinterferencebetweencarrierandsidebands,and a beatnote will appear on the photodetector signal. For an equal detuning on the other side of resonance, the leakage field phase shift has the opposite sign, and the result- ing beatnote will have the same amplitude but opposite phase. Simply put, the cavity changes phase modulation into amplitude modulation – with symmetric amplitude but asymmetric phase – in the vicinity of cavity resonance. (For a sufficiently quiet laser, thisamplitudemodulationcanbeobserveddirectlyonthedetectorsignalwithanoscil- loscope.) Note that if amplitude modulation were applied at the EOM, the technique wouldnotworkbecausethephaseshiftedleakagefieldcontributesequallytobothside- bandsandthebeatnotedoesnotchangephaseacrossresonance. BecausetheRFmixerisaphasesensitivedevice,itcanresolvethesignofdetuning, and outputs a “DC” signal proportional to the detuning (for detunings less than the linewidth). TheexactphaserelationshipbetweenopticalbeatnotephaseandRFphaseis 38 not necessarily known, so a phase-shift element must be placed either before the EOM or between the detector and mixer. This is adjusted empirically to obtain the largest possibleerrorsignalslopeand/orcorrectsignofslope. A final comment is warranted about the details of interfering the leakage field with the incident field. For a Fabry P´ erot cavity, the interference takes place at the outer surface of the input mirror. The promptly reflected and leakage field must be separated fromthebeampathwithaquarterwaveplate. WithaWGM,thesituationissimplersince theinputbeamisalreadytravelingalongadifferentpaththanthepromptly-reflectedand leakagebeams. 2.2.2 Implementingthelock A detailed schematic of the laser lock system is shown in Fig. 2.2. The splitter, bias- T, and bandpass filter are discreet components from Minicircuits. The VCO drives the AOM at its 100 MHz resonance plus a small correctional frequency shift f s which is the fast-branch actuation. A bias-T splits the detector signal so that the DC-coupled portioncanbeusedforviewingthemodeabsorptionsignal(duringasweep),whilethe AC-coupledportcontainsthemodulationsignalforerrorsignalgeneration. Foradjust- ing the phase, homemade opamp-based phase-shift circuits were initially used. Later, we discovered that better results and a simpler system could be achieved by placing a bandpass filter centered at the modulation frequency f m between detector and mixer. For a filter with several poles, there is at least 2π of phase shift as f m is tuned around in the passband. This passive approach eliminates adding frequency components other 39 than f m like an active circuit might do, and which would appear as noise after demod- ulation. Not shown is an attenuator between splitter and mixer. Typically the EOM mightneed+20dBmofinputsignal(themaximumerrorsignalslopeisachievedwhen the sidebands contain half the optical power [20]), but the mixer operates optimally at +7dBm. FIGURE 2.2 The opto-electronic lock system used for laser stabilization toaWGMcavity. SchematicsforthelockboxaregiveninAppendixB.Not explicitly shown is a ramp generator for sweeping the laser. Measurements of noise density are made at “A,” after an input/filtering stage but before the PIservos. 2.2.3 Closedlooptransferfunction Atsomepointbeforeandduringassemblyofthelocksetup,itisnecessarytocalculate thetransferfunctionoftheentirefeedbackloop. AschematicisshowninFig.2.3. Each component of the system has an associated transfer function. Often, optimizing loop performance is an iterative process of introducing a new piece of equipment, calculat- ing the overall transfer function, making a measurement, and repeating. Complication 40 comesfromthefactthatthereisagreatdealofsubtletyinthefrequencyresponsesofthe various components. Furthermore, developing electronics from scratch demands opti- mizing the servo/lockbox transfer functions, with its own set of op-amp circuit transfer functionsandstabilityissues. FIGURE 2.3 Schematic showing the topology of the lock-loop employed in this work. Each component of the loop has its own frequency response which must be known or determined. In this configuration, the correction signal to the AOM is used as the input for a servo driving the laser piezo actuator. OncetheindividualtransferfunctionsshowninFig.2.3areknown,theoverallTFis simplycalculatedbymultiplyingthemtogether. Ofparticularimportanceisdetermining the unity-gain frequency and the total phase lag at that frequency. This phase margin determines loop stability and must be carefully controlled if high loop bandwidth is needed. Inthecaseofnarrow-linewidthfiberlasersasusedhere,highloopbandwidthis notneeded–20kHzwassufficientforthiswork. However,notethatforbroadlinewidth diode lasers, great effort must be given to increasing loop bandwidth if the laser line is to be significantly narrowed. What follows below is a brief description of the relevant characteristicsofthecomponentsusedinthiswork. 41 Laser–ThepiezotuningoftheKoheraslaserispreciseandhasrelativelylittlehys- teresis. However, there is a large resonance in the response around 20 kHz. Therefore, H 2 (s) must be such that the response rolls off about a decade sooner for stable opera- tion. In calculating loop bandwidth, we manually measured the transfer functionT L (s) [Hz/V](phaseandmagnitude)ofthelaserbydrivingthepiezosinusoidallyanddemod- ulating it against the cavity resonance. This was necessary because the two Koheras usedhaveverydifferentpiezoresponses–andevensigns. AOM – The acousto-optic modulator is a fiber pigtailed device from Brimrose cor- poration. It is driven by a voltage controlled oscillator (VCO) from Isomet. In princi- ple, each of these components has its own frequency response. For these calculations, the VCO’s nominal tuning coefficient of T A (s)= 4 MHz/V was used; any roll-off was neglected. However, there is also a delay line property from the AOM – the time it takes the propagating acoustic wave to reach the optical interaction region [82]. Likely the combination of VCO tuning roll-off and this delay set the bandwidth limit for our system. Discriminant–Theslopeoftheerrorsignal,D(s)[V/Hz],dependsonmanyaspects of the optoelectronic setup, including resonator Q, coupling condition, contrast, optical power level, detector gain and bandwidth, and RF mixer levels. These effect the “DC” value, and a gain knob in the servo can easily correct for day-to-day changes. The primary frequency response results from the cavity lifetime, and a phase lead of π/2 mustbeintroducedatthehalf-linewidthiftheloopbandwidthistoexceedthisvalue. Servos – Several versions of lock-box were developed in the course of this work 42 withtransferfunctionsoptimizedtogivemaximumbandwidthwithoutlooposcillation. (H 1 and H 2 are shown separately to illustrate the topology; they are implemented in a single lockbox.) As mentioned, the goal for loop bandwidth in this work was relatively modest. More effort was spent mitigating other electronic issues such as ground loops and ambient noise pickup. Additional details and schematics of the lockboxes can be foundinAppendixB. FIGURE 2.4 Bode plots showing TFs from various parts of the lock loop. “Int 1” and “2” represent cascaded integrators in the servo (H 2 ); when mul- tiplied by T L H 1 D the “Total Piezo Gain” is determined. Similarly, “AOM LPF” represents a filter in the servo, and H 1 T A D yields the “Total AOM LoopGain.” 2.3 Quantifyinglaserfrequencystability Upon completion of the feedback stabilization loop, it becomes necessary to character- izetheoscillator’sperformance. Thiscanbeachievedthroughoneofseveraltechniques, some of which are described in the following sections and chapters. In discussing this 43 work, it is helpful to emphasize that the overall (absolute) frequency stability of the lockedlasercanbebrokendownintotwocategories: performanceofthelock,i.e.,how “tightly” is the laser locked to the cavity resonance, and the stability of the resonance itself. The remaining part of this chapter addresses the former, while Chapters 3 and 4 explorewayswhichwecanimprovethelatter. Measuring the stability of an optical oscillator requires special techniques because the frequency of electric field oscillations is far to great for any electronic system to handle directly. The most common technique employed is to compare the laser being measured with a second laser of very similar wavelength. By carefully overlapping light from the two lasers into the same spatial mode and directing it onto a photodiode, a beatnote corresponding to the difference frequency between the two lasers can be generated. When reference and test lasers with frequencies f ref and f test , respectively, are incident upon a photodiode, the output current contains frequency components at f ref ± f test . Thisresultsfromthenonlinearprocessoccurringatthephotodiode,where theoutputcurrentisproportionaltothesquareoftheelectricfield. Thesumfrequencyis evenfurtheroutofthereachofelectronics,butthedifferencefrequencycanbebrought arbitrarily close to zero (assuming a reference laser with the exact same wavelength is available). Thus, therelativestability of the twolasershas been convertedinto the RF domain, where a measurement can easily be made with an RF spectrum analyzer, frequency counter, DAQ, oscilloscope, etc. The major pitfall is that, for a comparison of just two 44 lasers, it is impossible to tell (without some a priori knowledge) which laser is less stable and therefore setting the measurement limit. The ambiguity is compounded by thefactthatthelasersmaywellhavedifferentrelativestabilitiesacrossthemeasurement bandwidth. Consequently, a measurement between two lasers only sets an upper limit ontheirinstabilities. 2.3.1 Comparisontoanindependentreferencelaser Acomparisonmeasurementofoscillatorstabilitycanbemadeoneoftwoways: (1)Two external references are compared to each other. If their relative stability is better than the expected performance of the test laser at time scales of interest, then the relative stability between either reference and the test laser will be limited by the latter and a good quantification of performance can be obtained. (2) If all oscillators have similar performance, comparisons between them can yield the instability of each one via the “three-corneredhat”method[47]. The Quantum Science and Technologies group at JPL has a commercially available (MENLOSystems)fsfiberlaserbasedfrequencycombwhichisstabilizedtoahydrogen maser. Thus, we had an optical reference with the long-term stability of the hydrogen maser, which is well-known and has an approximate value of 1×10 −13 τ −1/2 from 1 s to 10,000 s. This comb constituted the external reference laser for absolute stability measurementsreportedhere(andin[17]). However, fiber lasers have relatively high phase noise, leading to large “instanta- neous” linewidths – the precise value of which was not initially well known. Since we 45 expectthefiberlasersusedintheWGMcavitystabilizationexperimentstohavenarrow instantaneous linewidths (on the order of 10 kHz), we suspected that the measurement resultsforτ < 1smightbelimitedbythecomb’sphasenoise. Wethereforeassembled ahigh-finesseFPcavity(towardstheendofthiswork)thatwasusedasathirddatapoint tocharacterizedtheshort-termcombstability. The details of this characterization are shown in Appendix C where we show that the comb reference laser has a stability of ≈4×10 −13 τ −1 for 10 ms < τ < 4 s and ≈2×10 −13 τ −1/2 for 4 s < τ < 10,000 s. We note that 2×10 −13 τ −1/2 is not adequate foranopticalclock;thus,asWGMstabilityisimprovedabetterreferencelaserwillbe neededtomeasureit. 2.4 Trouble-shootingthelaserlock In conjunction with such independent measurements, other investigations of systemat- ics revealthe performance limitations. Severalsuch investigationsare described below, includingerrorsignalfrequencynoisedensity,“in-loop”errorsignalresiduals,andlock- ingtwolaserstothesamecavity. Goodresultsfromthesemeasurementsprovideconfi- dencethatthelaserlockitselfisnotthelimitingfactorforoursystem;rather,instability ofthecavity’sresonancefrequencyissettingtheperformancefloor. 46 2.4.1 Error-signalresiduals This characterization involves recording the error signal – after the mixer but before the servo (point “A” in Fig. 2.1) – while the laser is locked. Division by the slope of the discriminant when swept over resonance (units of V/Hz) yields the so called frequency residuals. It cannot be over-emphasized that these residuals represent the best-case stability of the laser; any extension of these measurements to conclusions aboutabsolutestabilityshouldbemadewithextremecaution(ifatall). There are primarily two reasons why the stability inferred from residuals can be very different from actual, absolute oscillator stability: (1) the lock itself may be per- fect, but the cavity is unstable, and (2) electronic noise at frequencies within the lock bandwidth will be directly imprinted upon the laser by the feedback loop, yet be invis- ible to residuals measurements. The former is self-explanatory; indeed, most of this thesis is dedicated to making the WGM cavity itself more stable. A classic example of thelatterisphotonshot-noiseatthedetectorbeingimprintedonthelaserstability. With these words of caution, residuals can be a very useful first step towards char- acterizing and optimizing the servo loop. In principle, residuals are the deviation of theactuallaserfrequencyfromthecavityresonancefrequency(i.e., thesetpoint); inan idealfeedbacklooptheywillbezeroforallfrequencieswithintheunity-gainbandwidth. Thus,non-zeroresidualsindicateinsufficientgainatthatfrequency. Furthermore,afre- quency spectrum of the residuals can give a quick-and-dirty value of loop bandwidth; whileobservingthespectrum,onesimplyturnsthegainupuntiltheloopstartstooscil- late. Thefrequencyofthepeakassociatedwiththeoscillationyieldsapproximatelythe 47 loopbandwidth. Residuals of the WGM cavity lock system were measured early on in this work using a fast DAQ module. Error signals were digitized after the “monitor” port (see schematic), then processed with a digital FFT algorithm. As described previously in §2.2.2, the frequency feedback loop used in this work consisted of a slow actuation on thelaser’spiezotuning,andafastactuationviaanAOM.TheefficacyofthisAOMfast branch – and the utility of measuring residuals – can be seen in Fig. 2.5, where the fre- quencyexcursionsaresignificantlysuppressedasthegain(andbandwidth)areincreased withtheadditionalfeedbackelement. Bodeplotscorrespondingtothesespectracanbe seeninFig.2.4. FIGURE 2.5 In-loop residuals measurements for the laser frequency lock toaWGMR.Withonlythefiberlaser’spiezousedforfeedback(a),theloop gainandbandwidthareconsiderablyless,resultinginmuchlargerfrequency excursionsthanin(b),wheretheAOMfastbranchisengaged. 48 2.4.2 Errorsignalfrequencynoisedensity Performanceofthelockisultimatelylimitedtothenoisedensityintheerrorsignalused for feedback stabilization. If the cavity is perfectly stable and the lock loop optimized, then the servo action will imprint upon the laser whatever noise is present on the error signal. Thisiswell-known,forexampleinLIGO,whereopticalshotnoiseisimprinted ontothelaserfrequencyandsetsthelimitatcertaintimescales[6]. Thus, measuring the noise density in the error signal provides a lower-bound for laser frequency instability. The measurement is easy to make. At point “A” in Fig. 2.2, the error signal is split off and sent through a low-pass filter before being measured with an analog AC voltmeter. Equivalently it could be recorded with a DAQ and inte- grated across a certain bandwidth. This must be done while the laser is far from cavity resonanceandideallyfree-running. Next, the laser is swept across the resonance to be used for locking. The slope of the error signal must be precisely recorded, which can be achieved with a digital oscilloscope or DAQ. Finally, the spectral density of the noise is computed through v rms /D/ √ bw, where v rms is the RMS error signal voltage noise, bw is the filter band- width(Hz),andasbeforeD istheerrorsignalslope(unitsofV/Hz). The resulting frequency noise spectral density is a very useful systematic quantity. However,whenusingthisvaluetointerpretothermeasurements,it’simportanttounder- standthefollowingassumptions. Firstandforemostistheassumptionthatthenoisehas whitespectraldensity. Typically,spectrawerecheckedwithanFFTspectrumanalyzer, and for many types of noise – such as shot noise – this is a good approximation. The 49 low-pass filter in conjunction with the RMS voltmeter effectively integrates over the bandwidth giving equal weight to all frequencies. However, there may be some noises (such as 60 Hz line noise) with strong Fourier components at discrete frequencies; this type of measurement with such noises present will greatly over-estimate spectral den- sity. The second major assumption is that the filter bandwidth encompasses the most importantnoisefrequencies. Thepitfallsofthisassumptioncanbemitigatedbychoos- ing a filter with approximately the same bandwidth as the unity-gain frequency of the lock loop. In this case, a 10.4 kHz filter was used (compared to a unity-gain frequency ofabout20kHz). Typicalmeasuredvaluesrangedfrom0.5to6.0Hz/ √ Hz. Severalexperimentalfac- torscanbeadjustedtoimprovethisquantity. Forexample,it’seasytoseehowincreas- ing the cavity Q can directly improve system performance: the discriminant slope D increases linearly with quality factor. Also very important is the mode contrast. In the case of fiber-coupled WGM resonators, this can often be improved through careful alignmentoftheoutputoptics–notablythecollectionlens,detector,andaniris(placed nearthelens). Anotherimportant(butoftenoverlooked)quantityistheLOsignallevel into the mixer. For solid-state mixers, this must be large enough to “open” the transis- tors;toolowanLOsignalwillresultinaweakerrorsignalandhighnoisedensity. 2.4.3 Lockingtwolaserstotwodifferentcavitymodes This well-known measurement (indeed, it’s described in the original PDH paper [36]) yields more valuable information than error signal residuals can. While more compli- 50 cated and requiring more equipment than the latter, it essentially eliminates the second majorproblemwithresidualsasnotedabove: withtwodifferentoptoelectronicsystems, in-loopnoise(includingquasi-DCdriftsinlock-point)willberevealedthroughcompar- isontothesecondsystem. Ifthepropercareistaken,alloptoelectronics,fromdetectors to laser feedback, are independent across the two systems – with only the cavity reso- nancefrequencybeingcommon-mode–andanyproblemsthereinwillberevealed. FIGURE 2.6 Schematic of the setup for locking two lasers to the same cavity. Light from laser+AOM systems A and B passes through separate EOMs,theniscombinedandsentintoonecavityviaanangle-polishedfiber coupler. Fortwomodesofthesamepolarization,onephoto-detector(PD)is used (switch position B); for two orthogonal modes, different detectors are used (switch position A). Relative stability between lasers is measured on a fastPD;absolutefrequencyistrackedsimultaneouslybycomparisonstothe comb. Not shown are fiber polarization controllers before EOMs and before fibercoupler. Fig.2.6showsthesetupusedtoquantifythelocksystem,whichbasicallyconsistsof 51 two copies of the stabilized lasers described above. Each system has a separate EOM, with the light being combined just before it’s sent into the WGM cavity. Modulation frequencies are 400 kHz for one system and 13 MHz for the other; a narrow RF band- pass filter on each respective system provides > 50 dB of rejection between the two. The different laser locks are thus completely independent for all portions of the loops except the fiber coupler and cavity. When the lasers are stabilized to modes of parallel polarization,thedetectoriscommonmode;whenorthogonalpolarizationsareused,the detectorstooareindependent. Multimode WGM resonators display many high-Q, high-contrast resonances that can be used for locking. The best relative stability between the two lasers is achieved when they are locked to the same mode one FSR apart, as this represents the highest level of common-mode rejection. Locking the lasers to other combinations of modes provides additional information about the system. From the perspective of quantifying the lock performance, using orthogonally polarized modes facilitates testing the two detectorsindependently. Usingtwomodesofthesamepolarizationbutdifferentspatial order reveals a differential temperature coefficient – resulting from geometric factors – whichisonthesameorderasthatresultingfrombirefringence. Resultsofthreemeasurementscomparinglaserslockedtodifferentcombinationsof modes are shown in Fig. 2.7. When locked to the same mode, we see the best achieved relativefractionalinstabilitybetweentwolasersof3.7×10 −15 at200ms. Theturn-over in the instability around that time scale could be caused by one of two mechanisms: fractionalchangesintheFSRfromtemperaturedriftsintheresonator,and/orDCerrors 52 10 −2 10 −1 10 0 10 1 10 2 10 −15 10 −14 10 −13 10 −12 10 −11 10 −10 τ [s] σ (τ) Orthogonal Polarizations Same spatial mode, 1 FSR apart Same polarization, different mode SNR limit: 5.4×10 −15 τ −1/2 FIGURE 2.7 Allan deviation plots of the beatnote between two lasers locked to different combinations of modes in the same WGM cavity. The best relative stability result is achieved when locked to the same mode, 1 FSR apart. In all cases, the short-term stability is very close to the noise densitylimit,suggestingahigh-qualitylock. intheanaloglockelectronics. TheFSRchangesfractionallywithtemperaturethrough−∆FSR/FSR = ∆r/r = α l ∆T. These data were taken with the same resonator/vacuum chamber setup as describedforthedual-modetemperaturestabilization,§3.1;theanalogtemperaturecon- trol loop oscillates by about 2-3 mK with≈2000 s period. The mode has an FSR of 10.6718 GHz, thus we could expect a drift of 200-300 Hz at these time scales. The observedinstabilityof10 −12 at1000secondsisconsistentwiththebluedatainFig.2.7 (blue triangles). However, drifts in lock-point from DC errors in the analog electronics are also on this order (as described below). Therefore, it’s difficult to definitively say whatmechanismcausestheturn-overofAllandeviationat200-300ms. Theturn-overin 53 instabilityoftheothertwomeasurementshappenssooner,theoverallstabilityisworse, and the cause is clearly temperature related. The three comparisons have a very similar instability value at the first point, τ =10 ms – which was limited by our counter that hasaminimumgatetimeof10ms. Notethattheturnoverfororthogonalpolarizations is consistent with a differential (birefringence induced) mode drift of 67 MHz/K, or an instabilityof≈10 −9 at1000s. Also shown in Fig. 2.7 is the system noise limit. The error signal noise density was measuredasdescribedaboveyielding0.50and1.40Hz/ √ Hzforthetwodifferentlaser lock systems. Assuming the systems are independent, the noises add in quadrature and can be converted to an Allan deviation by dividing by √ 2 [91]. All three curves agree very well with this noise limit for short time scales. The fact that two of the curves dip slightly below this SNR limit indicates that one of the assumptions in measuring the noise density was not entirely justified: either the noise is slightly blue (higher power at higher frequencies), or there were some discreet components which caused an over- estimateofthewhite-noisedensity. The results of these measurements demonstrate that the laser lock employed in this work is capable of achieving the SNR limit of sub Hz/ √ Hz. Also significant is that, for the results using orthogonal polarizations, two different detectors are used. The photodiodeandassociatedpre-amplificationelectronicsarecrucialtotheoverallperfor- manceofthePDHlock,andthesimilarvaluesofinstabilityfororthogonalandparallel modesat10mssuggeststhatnoisefromthedetectorisnotalimitingfactorinoursetup. (We used a homemade detector following the “bootstrapped cascode” transimpedance 54 amplifierdesignof[59],thedetailsofwhichareshowninAppendixB.) 2.4.4 Differentialtemperaturedependenceofparallelmodes Comparing results from different pairs of modes offers insight into the subtle thermal dynamics and gradient-induced strains within the crystal. A naive treatment suggests that modes of the same polarization will respond to temperature induced changes in radius similarly to the way an FSR does, i.e., −∆f space /f space = ∆r/r where f space is the frequency spacing between the modes. The implication would thus be that by lockingtomodesFSR/10apart,arelativeinstability10timesbetterthanthebluecurve in Fig. 2.7 would be achieved. This is not the case – the red curve in Fig. 2.7 is from modesseparatedbyabout3GHz. To verify that temperature drifts are causing the differential frequency changes between two modes of the same polarization, the following test was performed. As shown in Fig. 2.6, one of the laser systems can be simultaneously compared to the H- maserreferencedfrequencycomb. Thus,boththedifferential(betweentwomodes)and absolutefrequencyofoneofthemodescanbedetermined(thelatteratalevelof≈2Hz at 100 s). The results of two such measurements is shown in Fig. 2.8. Clearly, when lockedtodifferentmodes,thespacingdependsstronglyupontemperaturedrifts–which we can infer from the absolute frequency excursions. When locked to the same mode 1 FSR apart, drifts in lockpoint may dominate, although they are on the same order as theexpectedtemperature-inducedchangeinFSR. The absolute drifts in Fig. 2.8(a) are scaled down by a factor of 400, reflecting that 55 −8 −4 0 4 Absolute Excursions ÷ 400 [kHz] −8 −4 0 4 Differential Excursions [kHz] (a) 0 500 1000 1500 2000 2500 3000 −2 −1 0 1 Absolute Excursions [MHz] Time [s] −2 0 2 Differential Excursions [kHz] (b) FIGURE 2.8 Data from simultaneously monitoring absolute and differen- tial drifts of parallel modes. (a) For the case of two different modes, the absolute excursions are divided by 400, allowing extraction of an approx- imate differential temperature coefficient of 4.28 MHz/K. When locked to the same mode 1 FSR apart, the differential instability seems unrelated to temperature,perhapscausedbyadriftinlockpoint. thedifferentialtemperaturecoefficientisabout4.25MHz/K–basedonanoveralltem- perature coefficient of 1.71 GHz/K. Owing to the rich mode structure in mm-scale res- onators, it is likely that the modes in Figs. 2.8(a) and 2.7 (red curve) are of different, high order, i.e.,|l−m| > 0 andq > 1. As either of these indices increase, the mode’s effective(oraverage)radiusbecomessmaller. Moreover,as|l−m|increases,themode is spread across a larger vertical extent. Thus, slight asymmetries in the resonator may cause the two modes to strain differently under thermal expansion. Taking for example 56 the first 1000 s in Fig. 2.8(a), we have a differential excursion of ∆f diff ≈4 kHz. For effectivemoderadiir 1 andr 2 ,thismeans ∆r 1 −∆r 2 = ∆f diff f r≈ 57fm (2.4) for our resonator having r =3.25 mm. It is not hard to imagine that for different, high ordermodesthatmaybeseparatedbyafeworeventensofmicrons,thattherecouldbe 57fmofdifferentialstrainbetweenthem. Forexample,thisexperimentwasdoneusing a WGM cavity that is glued to the ceramic portion of a TEM. The thermal-expansion mismatch between the MgF 2 and ceramic could easily strain the lower portion of the discwithrespecttotheupperportion. The results of this investigation are noteworthy for two reasons. First, it demon- stratesthepowerofsuchopticalfrequencymeasurements. Wecouldeasilyresolvedif- ferentialdriftsanorderofmagnitudelessthanthoseobserved,meaningwecanmeasure strainsintheresonatorcrystalontheorderofonefemto-meter. Second,thisdifferential tempco could be used for temperature measurement and stabilization – as described in §3.1. Thisexperimentdemonstratesthatabirefringentcrystalneednotbeused,opening the possibility for dual-mode temperature stabilization using isotropic (and even amor- phous)materials. 57 2.5 Summary Laser frequency stabilization can be achieved with a WGM resonator in direct anal- ogytoaFabryP´ erotcavity;interferencebetweenpromptlyreflectedandcavityleakage field takes place at the inner surface of the coupling prism (or fiber), rather than on the front surface of the input mirror as for an FP cavity. Our experimental setup con- sisted of a fiber DFB laser (Koheras Adjustik) operating at 1560 nm. All subsequent optics were fiber coupled, and frequency actuation was provided by an acousto-optic frequency shifter and the piezo tuning of the laser. Using a homemade electronic servo with2-3cascadedintegrators,theoverallloopunitygainbandwidthwasabout20kHz, andwecouldachievelockingatthenoisedensitylimitoflessthan1Hz/ √ Hz. Charac- terizing the systematics by locking two lasers to one cavity demonstrated that a differ- ential tempco exists between parallel modes, suggesting that a dual-mode temperature stabilizationtechniquecouldbeusedinresonatorsmadefromisotropicmaterial. 58 Chapter3 TemperatureCompensation A primary challenge in the realization of a stable frequency reference is minimizing the effect of environmental temperature changes. This is typically achieved through a combination of isolationfrom the environmentand decreasing the reference’stempera- ture coefficient, i.e. the change of oscillation frequency for a given change of temper- ature (often expressed fractionally as parts-per-kelvin). Isolation from the environment amounts to making the system’s thermal time constant as long as possible, so that tem- peraturechangesinthelab–fromdoorsopeningandclosing,toairconditionercycling, to daily changes – are effectively low-pass filtered from the primary reference element. A decreased temperature coefficient results from careful oscillator design, which typi- callyinvolvestheuseofspecialmaterialsandaphysicallayoutengineeredtomakethe oscillationsastemperature-independentaspossible. Hugeeffortsonthesefrontshaveyieldedfrequencyreferencecavitieswithfractional instability at the level of a few parts in 10 15 or even 10 16 ; a brief look at these systems demonstrates how heroic the efforts must be. For example, Alnis et al. [9] describe a large, thermally stabilized and insulated box around a vacuum chamber containing several nested heat shields around a Fabry P´ erot (FP) resonator. By using ultra-low expansion(ULE)glassfortheFPspacerandoperatingitnearthethermalturningpoint withthermo-electricmodulesinsidethechamber,theyachieveafrequencydriftof0.63 59 Hz/s, limited by material aging in the spacer itself. NIST’s most recent ultra-stable cavity resides within two nested, custom made vacuum chambers, and the laser locked to this resonator has the amazing performance of 10 −16 fractional stability at 1 s [63]. Another example is the recent PTB-NIST-JILA collaboration. They describe a spacer machined from a single piece of monocrystalline silicon, with silicon mirrors. The material’s low mechanical loss minimizes thermal noise, while the entire cavity is held at77Ktoexploititsthermalexpansionzero-crossingforanunmatchedstability[64]. In the case of whispering gallery mode resonators, thermal isolation from the envi- ronment is in principle easier because of the resonator’s small size compared to an FP cavity. However, the fact that light travels through solid medium makes reducing the temperature coefficient much more difficult; adding the requirement of very high opti- cal quality precludes specially developed low coefficient of thermal expansion (CTE) glasses. Such glasses used for spacers in modern high-performance FP cavities has a nom- inal, linear CTE of ±30×10 −9 K −1 . This value is specified over a range near room temperature. Somewhere in that range the CTE crosses zero, at which temperature the linear CTE vanishes and only a quadratic term is important. Cavities operated at the zero-crossingtemperature(asisthecasefor[9,63],amongothers)havethebeststabil- ity. By contrast, optical crystals such as magnesium fluoride and calcium fluoride have expansioncoefficientsα l ontheorderof10 −5 K −1 nearroomtemperature. SinceWGM and FP cavities have the same fractional frequency dependence upon temperature, we 60 canexpectthelattertobealmost500timesmorestableinthesamethermalenvironment (basedonthelinear CTE,sayingnothingofthezero-crossing). Reducing the temperature coefficient of a WGM cavity is therefore a primary goal. This chapter describes our efforts on two techniques for such a reduction. The first method – dubbed “dual-mode” stabilization – exploits a differential temperature coef- ficient between orthogonally polarized modes in a birefringent resonator to sense and stabilize the temperature [17,109]. The second approach is a “compound resonator,” wherebyanopticalmaterial’shighexpansioncoefficientismechanicallysuppressedby bondingittoalow-expansionglass. Bywayofclarification,thischapterdealswithinstabilitiescausedby“slow”thermal processes. Roughlyspeaking,thismeansslowerthanthethermalrelaxationtimeofthe resonator,orabout1s. Fasterthermal“noise”isaddressedinthenextchapter. 3.1 Dual-modestabilization Whispering gallery mode resonators made from birefringent materials display a dif- ferential temperature coefficient between the TE and TM modes. Thus, the spacing between orthogonal modes provides a measurement of the mode-volume temperature andcanbeusedastheerrorsignalinafeedbackloopfortemperaturestabilization. We havedevelopedasystemin whicha laseris lockedto aWGM referencecavity, whilea modulation sideband simultaneously provides such an error signal. We experimentally demonstrate that the technique reduces the overall temperature coefficient of the res- 61 onator by more that 50 times, from 8.8×10 −6 K −1 to 1.7×10 −7 K −1 ; our simulations suggestthatwithbetterimplementation,thevaluecouldbeaslowas3.5×10 −8 K −1 ,on parwithultra-lowCTEglasses. 3.1.1 Technique Asdescribedin[109],thefrequencyspacing∆f betweenorthogonallypolarizedmodes dependsuponchangesintemperaturevia d(∆f) dT =−f 0 (α (o) n −α (e) n ), (3.1) where α (o) n and α (e) n are the thermorefractive coefficients of the ordinary and extraor- dinary polarizations, respectively, and f 0 is the optical frequency. These coefficients depend upon both temperature and wavelength, and for magnesium fluoride (MgF 2 ) at theexperimentaltemperatureof40 ◦ Candf 0 =c/1560nm,Eq.(3.1)yieldsadifferen- tialtemperaturecoefficientof67MHz/K. We can find the resolution of this differential temperature measurement by convert- ing the error signal’s electronic noise density into temperature units. This is done fol- lowing the method described in§2.4.2, with the additional step that the discriminant is convertedintotemperaturewiththeabovedifferentialcoefficient. The resulting noise-density is determined by many factors, but most important are the resonator Q, mode coupling contrast, and noise in the electronics. For a typical Q of 3×10 9 and contrast of around 40%, we measure frequency noise densities of around 62 3 Hz/ √ Hz, corresponding to a temperature noise density of 45 nK/ √ Hz. Thus, the noise-limitedresolutionis4.5nKafter100secondsofaveraging. Thereisanobviouspotentialforhightemperaturestabilityifthisprecisionmeasure- mentisusedasaninputtoafeedbackloop. Suchatemperature-stabilizationschemefor WGM resonators was first proposed by Savchenkov et al. [99] in 2007, and has direct analogy to a technique that has been used for decades in quartz acoustic resonator sys- tems [119]. Experimentally, it was first demonstrated in 2011 at JPL by Strekalov et al. [109], where a swept laser and digital feedback to a resistive heater stabilized the mode-volume temperature to the nK level after 10,000 seconds of averaging. Subse- quentworkatMPQ[40],aswellasthecontinuedeffortatJPL[17](theworkdescribed below) implemented a single locked laser and some type of modulation sideband for temperaturemeasurement. 3.1.2 ExperimentalSetup Our implementation of the dual-mode stabilization scheme is shown in Fig. 3.1 and described as follows. A narrow linewidth fiber laser (Koheras Adjustik) is coupled via an angle-polished fiber [61] into a 3.25 mm diameter, z-cut resonator made from excimer-gradeMgF 2 . Modelinewidthsareintherangeof40-80kHz,correspondingto qualityfactorsof2.4–4.8×10 9 ,dependingonthemodeorder,polarization,andloading conditions. Laser light passes first through an acousto-optic modulator (AOM), then an electro-opticphasemodulator(EOM),andfinallyapolarizationcontrollerbeforebeing coupled into the cavity. The input polarization state is linear, with an angle such that 63 bothmodesareexcited. However,therearetimeswhenitisadvantageoustosendmore light into one mode or the other. For example, when laser intensity is used as a fast branch of the temperature stabilization loop (as described below), the mode to which the laser is locked must have a large fraction of the power in order to achieve efficient thermalactuation. TwomodulationfrequenciesareappliedattheEOM:a1MHzsignalforgeneration of Pound-Drever-Hall error signals, and a signal at the frequency spacing ∆f between the chosen pair of orthogonal modes. These two signals are combined in a linear adder – either a summing op-amp circuit or discreet RF power combiner. Thus, the laser is frequency locked to the ordinary mode, while the modulation sideband coincides with the extraordinary mode, providing an error signal for temperature stabilization. In its final form, thermal feedback with a unity gain bandwidth∼1 Hz was provided to the resonator by a thermoelectric module (TEM) glued directly to the disc. An optional fast thermal loop actuates on the in-coupled laser intensity via the AOM, which modu- lates the mode-volume temperature via optical absorption heating. The resonator-TEM assembly is mounted in a small brass “oven,” whose temperature is measured with a thermistorandstabilizedwitheitherahomemadeanalog,orcommercialdigitaltemper- aturecontrolleractuatingaresistiveheater. Theentiresetupishousedinasmallvacuum chamberpumpedtomilli-torrlevels,providingisolationfromatmosphericpressureand humidity fluctuations, as well as decoupling from environmental temperature changes byeliminatingconvectiveheattransfer. Withamulti-moderesonatorsuchastheonesdescribedhere,thereareseveraltensof 64 highQ, high contrast modes per polarization, per FSR. Thus, it is a relatively easy task to find a pair of modes with spacing in the tens of MHz range, allowing this technique tobeexecutedwitha“slow”andreadilyavailablefunctiongeneratororoscillatorchip. FIGURE 3.1 The experimental setup. (a) Schematic of the optical system and stabilization loops. Each proportional-integrator (PI) loop has both a slow and fast actuation branch. Polarization controller (PC) allows for con- trol of excitation ratio between the two modes. We experimented with two thermalactuators,aTEM(b),andresistiveheatersembeddedinasmallbrass ovenenclosingtheresonator(c). Electronics Theexperimentisoperatedviaahomemadelockbox. Bycontrasttothesetupdescribed in§2.2.2,discreetcomponentsarenotusedfortheRFportionoftheloop. Rather,acus- tommadePCBperformsthemodulationsignalgeneration,phase-adjustment,demodu- lation, and filtering for both the laser frequency and resonator temperature stabilization loops. After this board, the two error signals pass through PI servos. Because the system was typically under development, the characteristic frequencies for the temper- aturelockservowereconstantlybeingchangedasloopbandwidthincreased(described 65 below). More details onthe two-channelmodulation-demodulation board can befound inAppendixB. ThermalactuationofaWGMresonator Optimal performance of the dual-mode stabilization technique is achieved when the temperaturefeedbackloophasveryhighgain,enablingthetightestpossiblelocktothe high-resolutionerrorsignal. Basiccontroltheory[108]dictatesthatthisisequivalentto maximizingthefeedbackloopbandwidth. Forthermalstabilization,thisturnsouttobe anon-trivialtask. A body whose temperature is being stabilized is inevitably coupled to the environ- ment through a chain of physical objects. For each object, this coupling has associated withitsomekindofdissipationandheatcapacity. Whentakentogether,theseconstitute aneffectiveRCtimeconstantinalow-passfilter. Andsincetherearemanysuchobjects inseriesbetweenthebodyofinterestanditsenvironment,weenduphavingalowpass filter with as many poles as there are RC time constants within the relevant time-scale of the experiment. A simple schematic is shown in Fig. 3.2. The WGM resonator itself represents one RC time constant; its mass provides some heat capacity, while heat loss through radiation comprises a thermal resistance. The interface between resonator and heatsource–madefromthermalgreaseorepoxy–representsanotherpoleinthefilter, whiletheheatsourceanditscouplingtotheenvironmentprovideathirdpole. The phase-lag associated with the three-pole filter presents a severe limitation to loop bandwidth and therefore overall performance. To optimize the loop, we measured 66 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 −150 −125 −100 −75 −50 −25 0 Frequency [Hz] Normalized Gain [dB] Heater TF fit TEM TF fit TEM measurement Heater measurement τ 1 =700 τ 2 =1 τ 3 =0.9 τ 1 =9 τ 2 =0.1 τ 3 =0.035 (b) FIGURE 3.2 Low-pass behavior of the thermal actuation mechanism. (a) SchematicshowingtherelevantRCtimeconstantsarisingfromthedisc,heat source, and interface between the two. (b) Measurement results and trans- fer function (TF) fits for the two different actuation mechanisms: resistive heaterandTEM.Thefitsrepresentathird-orderlow-passTFwithpolescor- respondingtotimeconstantswiththeindicatedvalues(inseconds). the transfer functions of two potential thermal actuators, shown in Fig. 3.1(b) and (c). The mode-volume temperature was monitored by locking the laser to one mode, then sweeping the sideband (∆f in Fig. 3.1(a)) through a range of a few MHz, centered on the other mode. A step function or sinusoidal excitation could then be applied to the thermal actuator, and the response monitored via a computer program that tracked the movementofthesecondmode. Thestepresponserevealedthedominant(longest)time constant, while the sinusoidal excitation facilitated extraction of both magnitude and phase,yieldingthethreedifferenttimeconstantsthroughaBodeplotfit. TwoofthelattertypemeasurementsareshowninFig.3.2,comparingtheactuation bandwidth of the brass oven and the TEM. The initial dual-mode experiment used the entire brass oven and its resistive heater as the thermal actuation mechanism, which has a very slow dominant pole with τ ≈ 700 s. Perhaps this long time constant is not 67 surprisingforaone-inchcubeofbrass,andtoincreasetheloopbandwidthwechoseto usethebrassovenasthesinkononesideofasmallTEM.Thus,thelargethermalmass of the oven is replaced by the relatively small thermal mass of the top TEM alumina plate,andthenewdominanttimeconstantisreducedtoτ ≈9s. Thisincreaseofnearly two decades in loop bandwidth is directly reflected in loop performance as described belowandshowninFig.3.4. Fabrication In order to use a TEM as the thermal actuator, we modified the brass oven by milling a shallowrecessintoitstopface. Thisallowedformountingofasmallcopperplatewhich heldtheTEM-resonatorassembly,which,inturn,wasfabricatedbygluingtheresonator directly to the TEM. Because any amount of thermal resistance directly decreases loop bandwidth, high-density silver epoxy was used for this glue joint, while the TEM was soldereddirectlyontothecopperplate. Fig.3.1(b)showsadetailoftheassembly. An important consideration in any WGM resonator-based experiment is disc clean- liness. Quality factor can be degraded orders of magnitude by the smallest amount of oil or a piece of dust which alights on the disc edge. Thus, we made a special brass post which holds the resonator-TEM assembly with the disc centered for cleaning on the air-bearing lathe. Shown in Fig. 3.3 is the TEM assembly mounted for cleaning, as wellasitsfinalpositioninthebrassoven. 68 FIGURE 3.3 The TEM-resonator assembly. (a) Mounted on a custom spindle for cleaning, and (b) the assembly mounted in the brass oven. Fiber couplerisvisiblecominginfromtheleft. Characterization Toverifythatthedual-modestabilizationloopisoptimized,weperformedin-loopmea- surements of the temperature residuals. These are made by recording the error signal voltagewithafastdataacquisition(DAQ)module,thenconvertingthesevaluestotem- peratureexcursionsviadivisionbytheerrorsignalslopeandtemperaturecoefficientof 67MHz/K,asdeterminedby(3.1). As a general rule, in-loop measurements represent a best-case scenario for system stability. Since the feedback process will try to compensate for any change in the error signal, any noise inside the loop bandwidth corrupting the error signal will be directly mappedintosysteminstability,yetindiscernibleasanerrorsignalexcursion. Thatsaid, these measurements are extremely useful for diagnostic purposes because if they are bad,truesystemstabilitywillbeevenworse. Theutilityofin-loopmeasurementsisillustratedinFig.3.4. Thefigureshowsresults 69 for both thermal actuators: brass oven (heater) and TEM. As compared to the heater- stabilizedresonator,weseeahugeincreaseinperformancewiththeTEM,toverynear the noise-limited level of 45 nK/ √ τ. Thus, we can conclude that the loop is optimized and performing as well as can be expected. Furthermore, assuming that any significant noisesourcesareatdiscreetfrequencies(i.e.,justappearingasspikesinanAllandevia- tionplotbutnotchangingtheslope),thein-loopmeasurementsyieldanaccurateoverall valueformode-volumetemperaturestability,i.e.,atthenKlevelforafewhundredsec- ondsofaveraging. We also performed out-of-loop measurements by comparing the WGM cavity sta- bilized laser to an independent optical reference. A small amount of power is split off from the dual mode experiment and beat on a photodiode with one tooth of a self ref- erenced femtosecond frequency comb that is disciplined to an H-maser. The results of thesemeasurementsareshownanddiscussedbelow,in§3.1.5. 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 1 10 100 1000 10000 τ [s] Temperature Instability [nK] Heater Stabilized TEM Stabilized FIGURE 3.4 In-loop measurements comparing the two different temper- ature actuation mechanisms: heater and TEM. The large improvement in stability is a direct result of the higher loop bandwidth as indicated by Fig.3.2(b). 70 Systemperformancewascharacterizedbymanyothermeasurementswhosedetailed descriptionsareomittedhereforbrevity. However,itisworthnotingafewoftheresults. As described earlier, there is an optional fast branch in the thermal loop which actuates uponmodevolumetemperatureviamodulationoflaserintensity. Thisadditionalbranch wasveryeffectiveinincreasingtheloopbandwidth,whichhastheneteffectofallowing for higher gains on the TEM branch without forcing the loop into oscillation. Without this fast branch, the in-loop Allan deviation plot had a peak around 1 second (corre- sponding to the oscillation frequency of the TEM loop), and was slightly higher than the red curve in Fig. 3.4 for shorter averaging times. However, for τ &1 s, the plots aresimilarbecausethat’swellwithintheTEMbandwidthandthefastloopmakeslittle difference. Wealsomeasuredthethermalactuationbandwidthoflightintensitymodulation,the results of which are shown and discussed in§4.3.1. Similar to the characterization of heaterandTEM,thelaserwaslockedtoonepolarization,whilethemodulationsideband provided a measure of mode separation (and thus change in temperature). Light inten- sity was then sinusoidally varied with the AOM and the temperature response recorded asafunctionoffrequency. Becausethelaserdriftingfromitslockpointisindiscernible from a temperature-induced change in mode spacing, such a measurement is limited to the laser lock bandwidth (about 20 kHz). However, temperature actuation was seen for intensity modulation frequencies exceeding 10 kHz. This is significant because it sug- gests that active feedback could be used to suppress the fast, local thermal fluctuations –thesourceofcertaintypesoffundamentalthermalnoise. 71 3.1.3 Simulation Initialout-of-loopstabilitymeasurementsofalaserlockedtothedual-modeWGMcav- itysuggestedthattherealizedfrequencystabilitywasnotashighasthatsuggestedbythe mode-volume temperature stability. Thus, to understand the various factors influencing mode volume temperature, as well as the overall heat distributions in the system, finite element method (FEM) modeling was used. We performed two-dimensional, axially symmetric modeling of the resonator-TEM assembly using a commercial FEM pack- age (COMSOL). Material properties were taken from the manufacturer datasheets or materialpropertydatabases;modelgeometrieswerebasedascloselyaspossibleonthe physicalsystem. ScreenshotsoftheFEMmodelareshowninFig.3.5. TheobjectivewastodeterminethefrequencystabilityofaWGMcavitywhosemode volume temperature is stabilized to the nK level via the dual-mode technique, and how this stability is affected by changes in experimental parameters. Our model accounted for the following heat transfer mechanisms: radiative coupling between resonator and oven,opticalheatingofthemodevolume,heatpumpedinto/outoftheresonatorbythe thermoelectric effect, and thermal conduction through the TEM. (We assumed convec- tive transfer is negligible in the vacuum chamber.) Here, we were primarily concerned with dynamics within the TEM feedback bandwidth, i.e., those with times-scales>1 s. Simulation output was the thermally-induced strains on the resonator’s radiusr, which pullthefrequencythrough ∆f/f 0 =−∆r/r. Themodeldidnotaccountforthermore- fraction because the mode volume temperature is taken to be stable, as indicated by the residual error signal of the temperature stabilization loop. Moreover, the fractional 72 thermorefractive coefficient is approximately two orders of magnitude smaller than the thermalexpansioncoefficient. Radiative heat transfer between one body completely enveloped by another can be estimatedas[103] q rad = σ 1/ϵ 1 +A 1 /A 2 (1/ϵ 2 −1) ( T 4 1 −T 4 2 ) , (3.2) whereϵistheemissivity,Athearea,T thetemperature,thesubscriptsindicateinner(1) andouter(2)bodies,andσistheStefan-Boltzmannconstant. Forvariousvaluesoftem- perature difference between resonator and enclosure, q rad is calculated using Eq. (3.2) andthenappliedasaboundaryconditionontheresonatorsurface. Laserlightcirculatinginthemodeisconvertedtoheatthroughmaterialabsorption. For a laser locked on resonance (zero detuning), the heat generated by this absorption canbeestimatedthrough[25] q laser =Iη Q Q abs (3.3) where I is the input laser power, η is the coupling efficiency, and Q (Q abs ) are the observed (absorption limited) quality factors. This heat is included in the model as a volume heat source (units of J/m 3 ) applied to the mode volume (region shown in pink inFig.3.5(a)). Themodevolumecross-sectionalshapeisanellipsewithmajor(minor) axes of 20 (10) µm, which approximates the distribution of the fundamental mode as calculatedbyaseparate,fullvectorialFEMsimulationasdescribedin[53]and§1.2.3. Heat generated in the TEM (q TEM ) is modeled as a volume source distributed over 73 the top alumina plate. The bottom of the TEM is held at the enclosure temperature. Thus, we can model how the three heat sourcesq rad , q laser , andq TEM contribute to the modevolumetemperature. Aninitialsteady-statesolutionisfoundassumingreasonable values of in-coupled laser power, withq TEM = q rad = 0, yielding values of disc radius andmodevolumetemperature. Thedualmodeconditionissimulatedbydemandingthat subsequentchangesinq rad orq laser arecompensatedforbyachangeinq TEM ,maintain- ing the mode volume temperature at its initial value. The resulting strains (changes in radius)areconvertedtofrequency,allowingustomodelhowachangeinenclosuretem- perature or laser intensity that occurs while the dual-mode loops are closed will affect thefrequencyoftheresonator. 3.1.4 Results The dual-mode scheme measures and stabilizes the temperature in the region occupied by the optical modes. The loop guarantees that any uncontrolled change in thermal transport parameters will result in a compensating change in the level of thermal feed- back actuation. As the relative strengths of these thermal parameters changes, the sys- tem moves along lines of constant mode volume temperature. These isotherms do not, however,representlinesofconstantresonancefrequencybecausethecentralportionof the resonator changes temperature and expands/contracts. The isotherm slope, i.e., the ratio of change in optical frequency to the change in a system parameter, represents an effectivecoefficientoffrequency. 74 Environmentaltemperaturedependence Wesimulatedthedual-moderesonator’seffectivetemperaturecoefficientbysolvingthe FEM model for various temperature differences between resonator and enclosure (T 1 andT 2 inEq. (3.2)). TheconditionT 1 =T 2 meanstheaverageresonatortemperatureis inequilibriumwiththeovenandrepresentstheinstantatwhichthedual-modeloopsare closed. Thisq rad = 0 situation is solved first. Next, a small drift inT 2 is imposed,q rad iscalculated,appliedasaboundaryconditionontheresonator,andthenecessaryvalue of q TEM needed to return the mode-volume temperature to its initial value is found – thisrepresentsthefeedbackactionofthedual-modeloop. −2 −1 0 T−T Vm [mK] 0 1 2 3 0 .5 1 1.5 2 r [mm] Δ r [A] 6 pm (c) (d) o FIGURE 3.5 FEMsimulation. (a)Modelsetupshowingthemesh,materi- als,adetailofthemodevolume(pinkregion),andthelinealongwhichtem- perature and strain are plotted (dashed orange). BiTe: bismuth telluride. (b) Temperature map results showing how different gradients exist for different relative strengths of heat sources. Plots of temperature (c) and radial defor- mation(d)alongthediscradiusforT 2 =T 1 (solidred)andT 2 =T 1 +50mK (dashedblue),showinghowthelatterdistributioniscoolerinthecenter,con- tractingthediscradiusby≈6pm. TheresultisillustratedinFig.3.5(b),wherethedifferentthermaldistributionresult- 75 ing from a change in enclosure temperature is clearly visible. AtT 2 = T 1 , no radiative heating of the disc occurs; the gradient forms as optically-generated heat is conducted outthroughtheTEM.Iftheoventemperaturedriftsupby50mK,radiationadditionally heatsthetopandouterportionofthedisc. Maintainingthemodevolumetemperature– outtothenKdigitinthesesimulations–requiresthattheTEMcoolthecentral-bottom portion of the resonator, contracting it and increasing the common-mode optical fre- quency. The plots in Fig. 3.5 show the calculated distribution of temperature (c) and deformation(d)alongthedashedorangeradiallinein(a). Thetemperaturedistribution is normalized to the mode volume temperature (T Vm ), and the difference in strains at the endpoint (R = 3.25 mm) yields the frequency shift. The calculated shift was found tobelinearacrossfivevaluesofoventemperaturedriftspanninghalfadegree. Alinear fit to these simulated data yields an effective temperature coefficient of 6.9 kHz/mK, corresponding to a fractional coefficient of 3.6×10 −8 K −1 . We note that these values arecomparabletothelinearthermalexpansioncoefficientsofcommonULEglasses. The effective temperature coefficient was also measured experimentally. While monitoring its absolute optical frequency via comparison to the comb, the laser was locked to the dual-mode stabilized WGM cavity. We then increased the setpoint on the oven’s temperature controller in 12 mK steps every 360 seconds (eventually returning it to its initial value). The TEM current (converted to heat via values on the TEM’s datasheet), locked laser’s optical frequency, and in-loop temperature error signal were simultaneouslyrecordedduringtherun,andarepresentedinFig.3.6. Inagreementwith the simulation, increased radiative heating of the disc results in compensatory cooling 76 fromtheTEM,leavingthemodevolumetemperaturethesamebuttheopticalfrequency higher. Alinearfitoftheopticalfrequencyasafunctionofchangeinenclosuretemper- aturegivesaneffectivecoefficient,withtheexperimentalvaluebeing25.5kHz/mK.The magnitude of this value is 67 times less than the thermal-expansion determined coeffi- cientof-1.71MHz/mK,anditcorrespondstoafractionalcoefficientof1.33×10 −7 K −1 . 0 500 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 2.5 Time [s] Δ f opt [MHz] −600 −400 −200 0 q TEM [μW] 0 1500 3000 −100 0 100 200 300 T Vm [μK] (a) FIGURE 3.6 Measurement of effective temperature coefficient: Time series showing shift in optical frequency (∆f opt ) and heat delivered to the resonator by the TEM as enclosure temperature changes in 12 mK steps. Inset shows mode volume temperature from the in-loop error signal; aside from spikes at the temperature steps, it remains extremely stable. Taking the average frequency at each step and dividing by ∆T yields the effective coefficientof25.5kHz/mK. Circulatingpowerdependence Despite MgF 2 ’s extremely high transmission at 1560 nm, absorbed optical power is constantly converted to heat in the mode volume. Thus, changes in circulating optical powerwilldestabilizetheopticalfrequencyatanytimescale. Weinvestigatedtheeffect of slow (within the TEM loop bandwidth) circulating power changes. These slower 77 changesmaybehardertocontrolsincetheyresultfromdriftsinpolarizationorcoupling strength, while faster fluctuations are in principle corrected for by the intensity branch of the dual-mode loop. Similar to the case of environmental temperature, we found a multi-parameter dynamic whereby a change in circulating power results in a change in TEMcorrectionsignal,thermalgradient,andresonatorfrequency. Using the same FEM model described above, we calculated the frequency shift as a function of change in circulating optical power. For these simulations q rad was kept at zero. Heat was allowed to leave the resonator radiatively through its surface (to a boundary at infinity) and via conduction through the TEM. For a given change in cir- culating power, the resulting change in q TEM necessary to maintain the mode volume, change in thermal gradient, and shift in frequency was calculated. Using Eq. (3.3) and reasonable values of I = 100 µW at the coupler, η = 0.07, and that the disc’s quality factor is absorption limited, the simulations suggest that a 5% reduction in laser power willyieldafrequencyshiftof-103kHz. Anexperimentalmeasurementofthisisothermslopewasalsomade. Withthelaser intensity branch of the thermal loop open, we changed the optical power sent to the resonator by varying the amount of light transmitted through the AOM while simulta- neouslymonitoringtheTEMcurrentandabsoluteopticalfrequencyofthelockedlaser. Inagreementwiththesimulations,adecreaseincirculatingpowercausesanincreasein TEM heating to maintain the mode volume temperature. The new thermal gradient is warmerinthedisccenter,increasingr anddecreasingtheopticalfrequency. Theexper- imentallymeasuredfrequencypullingwas-15kHzfora5%reductioninintensity,cor- 78 respondingtoaneffectivecoefficientof43kHz/µWofcirculatingopticalpower. Based on this result, a stability of∼47 pW would be needed to reach a benchmark fractional frequency instability of 10 −14 . This requirement on laser intensity stabilization is a readilyachieved7ppm,howeverwestressthattherequirementisoncirculatingpower, so coupling strength and polarization angle (which changes the power ratio between modes)alsorequireppm-levelstability. The discrepancy between simulated and experimentally determined temperature coefficients results from the heavy dependence on setup details such as boundary con- ditions at the domain interfaces and values of material properties; assumptions made in using Eqs. (1) and (2) could also lead to errors, as well as the approximations used for the mode volume and the fact that the experiment may have been performed with a higher order mode. Experimentally, it is difficult to isolate a single variable as can be done with simulation, so various other destabilizing factors can play a role. However, the simulations agree qualitatively with the experiments, provide meaningful insight, andwithrefinementcouldagreequantitativelyaswell. 3.1.5 Frequencystabilityofadual-moderesonator After characterizing the temperature coefficient of frequency, we measured overall sta- bilityofalaserlockedtothedual-modecavity. Theopticalbeatnotewiththecombwas monitoredwithaprecisioncounter(MenloFXM50)using10msgatetimesduringthe course of many 2000 s runs. The results of several such measurements are shown in Figs.3.7and3.8. Wewereabletoquantifythegainsachievedwiththedual-modetech- 79 niquebycomparingthe“free-running”resonator–i.e.,withthedual-modetemperature stabilization loop open – and “dual-mode” stabilized resonator. Furthermore, we made comparisons between using the analog and digital controller to stabilize the resonator’s enclosuretemperature. The Allan deviation plots agree well with the independently measured dual-mode temperature coefficient; despite a decrease in this coefficient of more than an order of magnitude, the overall performance remains limited by environmental temperature stability. This is evidenced by the two stability measurement plots and described as follows. In Fig. 3.7, a time series (inset) and Allan deviation plot is shown comparing dual- mode and free-running resonator when the analog controller is used to stabilize the resonator’s environmental temperature. The analog controller’s imperfect loop design resulted in an oscillation having a peak-to-peak amplitude of≈2-3 mK and a period of ≈2000 s, as determined by measuring the temperature controller’s error signal. These oscillations are clearly visible in the time-series inset and agree well with the free running and dual-mode temperature coefficients of −1.71 GHz/K and 25.5 MHz/K, respectively. However, the analog controller is quieter at time scales of 1-10 seconds, and Fig. 3.7 represents the best stability achieved with the dual-mode experiments: 1.29×10 −12 at700ms. Stabilities below 100 ms are thought to be limited by phase noise in the frequency combreference. Asmentionedin§2.3.1andshownexplicitlyinAppendixC,weexpect the stability of the comb (when disciplined to the maser) to be ≈4×10 −13 τ −1 for 80 10 ms < τ < 4 s and ≈2×10 −13 τ −1/2 for 4 s < τ < 10,000 s. The former value ofinstabilityintersectswiththecurvesinFig.3.7forτ <100ms. Moreover,thebroad hump starting at 10 ms and the peak at 80 ms are distinctive features which are often present in our comb comparisons, including with an FP stabilized laser as shown in Fig.C.1. 10 −2 10 −1 10 0 10 1 10 2 10 3 10 −12 10 −11 10 −10 10 −9 10 −8 Averaging time τ [s] Fractional Instability −10 0 Δ f opt [MHz] 10 0 2500 5000 −100 0 100 Time [s] Δ f opt [kHz] FIGURE 3.7 OverlappingAllandeviationofalaserlockedtotheWGMR cavity, as compared to the frequency comb, under nominally identical con- ditionsexceptthatthedual-modeloopisclosed(bluetriangles)oropen(red circles). Significant improvement is observed for time scales > 100 ms. Inset: Time sequences of the data run showing frequency excursions for thedual-mode(blue,leftscale)andfree-running(red,rightscale)resonator, showing periodic oscillation from the enclosure’s analog temperature con- troller. Fig. 3.8 provides further support for the conclusion that the dual-mode resonator’s frequency stability is limited by environmental temperature changes, and also demon- stratesthetrade-offassociatedwithusingtheanalogordigitaltemperaturecontrolloop. As compared to the analog loop, the digital one uses a pulsed correction signal which has strong Fourier frequency components with 1/f in the 1-10 second range. Thus, 81 the enclosure temperature is less stable at those averaging times. However, the Allan deviation plots cross over and are better for τ >30 s owing to the digital controller’s superior DC stability, as it lacks the bias-drift errors associated with op-amp circuits. The free running and dual-mode stabilities have similar dependence on averaging time (forτ >1 s) but are shifted by the ratio of free-running to dual-mode temperature coef- ficients for both the analog and digital controllers. The consistency of this behavior for two very different environmental temperature stability characteristics suggests that environmentalfactorsareindeedthelimitingones–notsomeunknownanomalyofthe dual-moderesonatorsystem. Note that the data shown in Fig. 3.8 was taken by a slightly different type of comb comparison – one where the comb rep-rate is phase locked to the cw laser and then countedonafastphotodiode. Thecomb’srep-ratephasenoisethuslimitsthemeasure- mentasindicatedbythedashedblackline. 3.1.6 Outlookfordual-modestabilization Dual-mode stabilization of a whispering gallery mode cavity is a powerful technique for effectively reducing the resonator’s temperature coefficient. In frequency reference applications, this is of particular importance because the un-compensated coefficient of optical crystals is very large relative to that of low-expansion glasses. Our simulation results show that the dual-mode technique can, in fact, close the performance gap to suchglassesbyyieldinganeffectivecoefficientof3.6×10 −8 K −1 . Ourexperimentally measuredresultof1.7×10 −7 K −1 couldlikelybeimprovedthroughthoughtfulmodifi- 82 10 −2 10 −1 10 0 10 1 10 2 10 3 10 −12 10 −11 10 −10 10 −9 10 −8 Averaging time τ [s] Fractional Instability Dual−mode, Analog TC Free−running, Analog TC Dual−mode, Digital TC Free−running, Digital TC FIGURE 3.8 Allan deviation plots showing several additional dual-mode resonator stability measurements. The similar shape yet shifted magnitude ofcurvesforfreerunninganddual-moderesonatorsunderdifferentenviron- mental conditions suggest that the latter is the limiting factor. The comb’s rep-rate phase noise (black dashed line) represents the measurement noise floor. cationssuchas: use ofa TEMthatextendstothe discedge, aconductiveplateatopthe resonatortoreducegradients,andheatshieldingand/orgoldcoatingoftheresonatorto reduceradiativeheattransfer. In a WGM resonator where laser light is continuously converted to heat in the opti- cal crystal, however, these limiting gradients are inevitable and cannot be eliminated completely. Our results also demonstrate a principle of thermal feedback control: that temperature can only be stabilized at a single location – that of the sensor (in our case themodevolume). Nonetheless, our results suggest that with more substantial efforts to stabilize its enclosure, a WGM resonator could serve as a respectable frequency reference cavity. 83 Let’s discuss the conditions under which a fractional frequency instability of 1×10 −13 atτ=10scouldbeachieved. Theenvironmentaltemperaturewouldneedtodriftslightly less than 0.6 µK over that 10 second interval. While this is indeed an extremely small temperature change, thermal instabilities bounded at 30 µK for many hours have been realized [116], suggesting that sub-microkelvin drifts at 10 seconds could be achieved. Assuming this 60 nK/s drift, a frequency noise spectral density of 63 Hz/ √ Hz would yield an Allan deviation of 1×10 −13 at ten seconds before turning around from the lineardrift. Ourcurrentexperimenthasanerrorsignalnoisedensitymorethananorder of magnitude less than this, suggesting that the result should be possible. However, as discussed in Chapter 4, we may be limited by fundamental thermal or other technical noiseatthesetimescales. 3.2 Hetero-compoundresonator Inthissectionwepresentasecondapproachtoaddressingthefundamentaldifficultyof highthermalexpansioncoefficientinopticalcrystals. Wediscusssimulationsandinitial experimental results of a passive device in which thermal expansion is mechanically suppressedinacompoundresonatorstructure. 3.2.1 Concept Conventional low-expansion glasses are not of sufficient optical quality for high Q WGM resonators. The materials themselves (such as Zerodur from Schott and ULE 84 from Corning) have a heterogeneous structure containing a mixture of crystalline and amorphous phase silica compounds; adjusting the ratio of these two phases facilitates tuning the coefficient of thermal expansion. These internal boundaries act as scattering sites, making the material optically lossy. Therefore, we wish to devise a structure that hastheoverallCTEofsuchspecialtyglasses,buthighopticaltransmissionintheregion oftheWGM. We have thus investigated a compound resonator structure; one portion of the WGMR consists of a material with high optical quality while the other portion is made fromamaterialselectedforitsthermalproperties. Specifically,adiscoffluorideoptical crystal is sandwiched between two discs of ULE glass. If the bond between materi- alsattheinterfacesissufficientlystrong,thermo-mechanicalpropertiescanbepartially transferredfromonematerialtoanother. We have shown through experiment that the temperature coefficient of a fluoride resonator can indeed be suppressed; our simulations suggest that with the proper com- bination of materials and geometry, a zero-crossing in the overall, linear temperature coefficient can be achieved. This result is significant because it implies that a resonator withvolume<1cm 3 coulddeliveropticalfrequencystabilitycomparabletoFPcavities orders of magnitude larger. The discrepancy between experiment and simulation stems fromthenon-idealboundarybetweenlayers,andreducingitscomplianceremainsakey challengeinthisongoingwork. Thispassiveapproachtothermalcompensationissignificantlysimplerthanthedual- modestabilizationtechnique. Nolongerneededistheadditionalmodulationfrequency 85 source, demodulation electronics, control loop, and thermal actuator on the resonator itself. Successful implementation of a compound resonator promises to unlock the true compactness potential of a WGM frequency reference cavity. In addition to the direct volume savings realized by using a WGM over an FP cavity, the much smaller size meansthatthetechnicalchallengesofoperatingthecavitynearitsthermalzero-crossing temperature and isolating it from environmental fluctuations will be greatly simplified. This benefit is a direct result of the mechanism described in §3.1.2: the smaller size of a chamber needed to enclose the WGM cavity means it would have a smaller ther- mal capacity, enabling a thermal control system that is faster, has higher gain, and can maintainbetterstability. 3.2.2 Calculationofazero-crossing Here we describe calculations which seek to determine if a temperature exists where the overall temperature coefficient of frequency vanishes (crosses zero). Two similar approaches to the compound resonator are investigated: in the first, negative thermore- fractionisemployedtopartiallycompensatethepositiveCTEofCaF 2 ,butthisleavesa large TR coefficient which could raise the thermal noise floor. In the second approach amagnesiumfluorideopticalcrystalisusedandcompensationispurelymechanical;in this scenario glass with negative CTE would need to be obtained, but the thermorefrac- tivenoisefloorcouldbeverylow. Inbothcases,wewishtoshowthatazero-crossingexists. Nearsuchatemperature, the thermal coefficient is expected to be small (at least 2 orders of magnitude smaller 86 than for the optical crystal alone) and small but inevitable changes in environmental temperature would pull the frequency less. Furthermore, operation exactly at the zero- crossing, where only a second-order temperature coefficient is important, should yield furtherstabilitybenefitsforthecompoundWGMresonatorsasithasforFPcavities[9]. Accounting for both thermal expansion and thermorefraction, the overall fractional frequencydependenceofaWGMcanbeexpressedas 1 f 0 df dT =−(α l +α n )∆T (3.4) where f 0 is the optical frequency, f the WGM frequency, and α l = (1/l)(dl/dT) and α n = (1/n)(d n /dT) are the linear material CTE and thermorefractive coefficients, respectively. Both coefficients α l and α n depend themselves upon temperature, thus wecanwriteacombinedtemperaturecoefficient α c (T) =α l (T)+α n (T). (3.5) A quick comment about Eq. 3.5: it is only valid on timescales longer than the thermal time constant of the whole resonator structure, i.e., the time it takes for temperature changestoequilibrateacrossthecrystal. Thisdistinctionbecomesimportantwhenopti- cal power is being converted into heat and thus frequency fluctuations. Such local (to themodevolume)changescantakeplaceontimescalesmuchfasterthanthoseatwhich thermalexpansionchangesthefrequencythroughchangesintheradius(see§4.3.1). 87 Calciumfluoridecompoundresonator Preliminary demonstration of the compound resonator was performed with CaF 2 because of its relatively large, negative value of α n . This means, for example, that at 300 K, the combined coefficient is reduced by about a factor of 2 to∼1×10 −5 K −1 andlessmechanicalsuppressionisneeded. CalciumfluoridealsohasasmallerYoung’s modulus than MgF 2 , so suppression of its expansion is easier to achieve. Similarly, ZerodurischosenoverCorning’sULEbecauseofitslargerYoung’smodulus. Fig. 3.9 shows a plot of the thermal expansion coefficient along with the opposite ofthethermorefractivecoefficient. TheformeristakenfromtheCorningCaF 2 product informationsheet[1],whilevaluesofthelatterwerecalculatedfromthedatain[72]. 0 100 200 300 400 0 5 10 15 20 25 Temperature [K] Fractional Coefficient [ ×10 −6 K −1 ] α l −α n FIGURE 3.9 Temperature dependence of the thermal expansion and refractioncoefficientsofcalciumfluoride. Thelarge,negativethermorefrac- tivecoefficientpartiallycompensatesforthermalexpansion;α c from(3.5)is thedifferencebetweenthecurves. To determine if a zero-crossing exists, we perform the following axially symmetric FEMsimulationofthethermallyinducedstraindistribution. Thestructureconsistsofa 88 fluorideopticalsegmentsandwichedbytwopiecesofZerodur(Fig.3.10(a)). Accounted forinthemodelistheYoung’smodulusandtemperaturedependenceofα l forbothCaF 2 and Zerodur, the latter being taken from a Schott material datasheet [5]. A mechanical expansioncoefficientisthusfoundforthecompoundstructure,denotedα (comp) l . Finally, wecancalculatetheeffectivefrequencycoefficientfortheentireresonatorbyincluding thermorefractionthrough α eff (T) =α (comp) l (T)+α (CaF 2 ) n (T). (3.6) Fig. 3.11 shows typical results from two such simulations. Zero-crossings are expectedattemperaturesT =T zero suchthatα eff (T zero ) = 0. (Fortemperaturesabove 300 K, thermorefraction is extrapolated from the data in [72] via a linear fit to the four pointsat150through300K.)Ascanbeexpected,thevalueofT zero ishighlydependent upon resonator geometry; dimensions for these results are: R=3.5 mm, B = 100 µm (CaF 2 portion),andA=50or60µm. Notethatforthestructurewith60µmthickZero- dur, α eff is less than 2×10 −7 K −1 (an improvement of∼100×) for a 50 degree span aboutT zero . Also extremely important to the results are the details of the boundary condition betweenmaterials. AtechniquecalledopticalcontactingemploysVanderWaalsforces to form a molecular-level bond between two dissimilar materials. For the models here we assume no slippage at that boundary, an assumption which has been validated by FEM analysis of Zerodur rings contacted to fused silica mirror substrates on FP cavi- 89 FIGURE 3.10 Screenshots of the FEM model used to calculate tempera- ture coefficient zero crossings. The setup (a) shows fluoride crystal (either MgF 2 orCaF 2 )sandwichedbetweenZerodur. AsolutionforMgF 2 (b)shows how Zerodur with a negative CTE can suppress the radial strain (indicated bycolor-scale)atthemodelocation(blackarrow)tozero. ties[70]. However,includingthedetailsofthatboundary–eitheroffinitethicknessand adjustable compliance, or no thickness but with some slippage allowed – is an area of ongoingwork. Magnesiumfluoridecompoundresonator The Young’s modulus of MgF 2 is considerably higher than that of CaF 2 – 138.5 versus 75.8 GPa. Moreover, the thermorefractive coefficients of the former are about an order of magnitude smaller. Therefore, the thermo-mechanical suppression achieved by the compound structure must be greater if a zero-crossing is to be achieved; we find that not only must a low-expansion glass be used, it must have a negative CTE. This mate- rialpropertyisingeneralrare,howeverbecausethepropertiesofZerodurdependupon details of fabrication and annealing, some runs indeed have α l < 0 at room tempera- ture. The technical information sheet detailing the CTE of Zerodur shows that there is a distribution around zero from run-to-run, with the largest negative values being 90 150 200 250 300 350 400 −1.5 −1 −0.5 0 0.5 1 Temperature [K] α eff [×10 −6 K −1 ] 60 μm Zerodur 50 μm Zerodur FIGURE 3.11 FEM simulation results of the compound resonator show- ing that zero crossings exist. The curves result from the temperature depen- dencesofmechanicalandopticalpropertiesanddependstronglyupongeom- etry. Foracompensatorylayer10µmthinner,T zero increasesby≈75K. α l ≈−1×10 −7 K −1 [4]. Fig. 3.12 demonstrates that for the correct combination of negative CTE and geom- etry, a zero expansion structure can be achieved. The plot gives radial strain at the mode location for ∆T = 1 K, as a function of the compensation material’s CTE for the indicated radii. The other dimensions (as defined in Fig. 3.10) areA = 350µm and B =100µm. Materialpropertiesinthemodelarebasedontheirvaluesat298K.Also, forthe7mmstructure,theobtainablevalueforCTEofα l =−1×10 −7 K −1 mentioned aboveresultsinaresonatorhavinganeffectivetemperaturecoefficientoffrequencyless than 10 −7 K −1 . Finally, we mention that the benefits in terms of thermorefractive noise of using MgF 2 as the optical material could outweigh the challenges of obtaining glass withalarge,negativeCTE. 91 −10 −7.5 −5 −2.5 0 2.5 5 −3 −2 −1 0 1 2 3 4 α l [×10 −6 K −1 ] Δr/r [×10 −7 ] r = 1.50 mm r = 3.25 mm r = 7.00 mm FIGURE 3.12 Simulationresultsshowingradialstrainasafunctionofthe compensatingmaterial’sCTE(α l )forthreedifferentradiiofdisc. Thelarger thedisc,thelowermagnitudeofnegativeCTEisneeded. 3.2.3 Experiment Initial fabrication and temperature coefficient measurements have been performed on a calcium fluoride compound resonator. A zero crossing is not expected in this proof of concept demonstration for the following reasons: “class 2” Zerodur was used, which has a higher value of CTE than that used in the FEM models, and the structure was assembled with glue rather than optical contacting, thus the decreased strain-transfer at the boundary is expected to adversely affect compensation. Nonetheless, we find that the temperature coefficient of the compound device is indeed reduced by a factor of about3.3,toα eff = 3.38×10 −6 K −1 . The compound resonator is composed of a 70 µm thick CaF 2 disc sandwiched by two 400 µm thick pieces of Zerodur; the overall diameter of the completed structure is 4.45 mm. Successive layers of the disc structure are glued together using a UV- cure adhesive, after which time this compound “blank” is mounted on a brass post, 92 FIGURE 3.13 Photograph of a CaF 2 -Zerodur compound resonator before (a) and after (b) polishing. The diameter is 4.45 mm and the thickness ∼0.5 mm. The resonator/post assemblies are clamped in a temperature- controlled Al enclose for testing (c). Fiber coupler comes in from right, and acapincreasestemperatureuniformity(d). made concentric, then polished. For experimental comparison, a resonator of similar dimensions – but consisting of pure CaF 2 – is also fabricated. For thermal testing, both resonators are mounted on brass posts using high-density silver epoxy for good heat conduction. They are then placed in a small aluminum “oven” similar to the brass one described above; the temperature is adjusted via a thermistor, resistive heater, and commercial, digital temperature controller. Photographs of the completed compound resonatorandthetestingsetupareshowninFig.3.13. Due to challenges of simultaneously polishing two materials with very different mechanical properties, the compound resonator demonstrates a modest Q of 1.7×10 7 . Although the resulting PDH error signal noise density is thus expected to be on the order of hundreds of Hz/ √ Hz, such Q’s are more than adequate for temperature coef- ficient measurements where thermal changes in frequency are on the order of tens-of- megahertz. Similartotheprocessdescribedintheprevioussection,alaser–withsome 93 of the power picked off and compared to the maser-referenced comb – is then locked to the resonator. Next, the enclosure is stepped through a range of temperatures, while recording the beatnote frequency with a counter. A scope trace showing a WGM in the compound resonator, and experimental data from such a temperature coefficient mea- surementareshowninFig.3.14. −100−80 −60 −40 −20 0 20 40 60 80 100 2.8 2.85 2.9 2.95 3 3.05 Detuning [MHz] Detector Signal [V] FWHM=11.4 MHz Q=1.7×10 7 (a) 0 120 240 360 480 600 720 840 960 0 10 20 30 40 Time [s] Beatnote Freq. [MHz] (b) FIGURE 3.14 (a) Mode trace from the CaF 2 + Zerodur compound res- onator showing 11.4 MHz linewidth, limited by difficulty of simultaneously polishing different materials. (b) Raw measurement data: enclosure temper- ature is changed every minute while the frequency of a laser locked to the cavityismeasuredbybeatingitwiththefrequencycomb. The experimental data is processed by taking the average beatnote value for each temperature step; the standard deviation in these data give error bars and a linear fit yields the temperature coefficient. Since there is data for both heating and cooling, any overall, systematic linear drift during the run can be accounted for by averaging the heating and cooling temperature coefficient values. Fig. 3.15 shows results for both the compound and “plain” CaF 2 cavities during heating. Averaging these fits from heat- ing with the fits from cooling (not shown) yields values for temperature coefficients of α (plain) c =−1.12(0.06)×10 −5 K −1 andα (comp) eff =−3.38(0.01)×10 −6 K −1 fortheplain 94 andcompoundresonators,respectively. 0 20 40 60 80 −6 −5 −4 −3 −2 −1 0 1 ΔT (T−317.262) [mK] Fractional Change [×10 −7 ] Plain CaF 2 Compound −1.16×10 −5 K −1 −3.38×10 −6 K −1 FIGURE 3.15 Temperature coefficient measurement results comparing compound resonator to plain CaF 2 . Linear fits yield fractional temperature coefficients;errorbarsarestandarddeviationoffluctuationsateachtemper- aturestep. Based on the data shown in Fig. 3.9 (and extrapolation for thermorefraction above 300 K) and Eq. (3.5), the calculated temperature coefficient for the plain resonator is α (plain) c = -1.03×10 −5 K −1 at the experimental temperature of 317 K. This calculated valueiswithin10%ofthemeasuredone,andthediscrepancymayresultfromincorrect literature values of α n and/or α l , or the slightly higher CTE of brass, to which both resonators are glued. In any case, efforts were taken to make the resonator mounts identical, thus the suppression of observed temperature coefficient by a factor of 3.3 is expectedtobeaccurate. 95 3.2.4 Summary Fabry P´ erot optical cavities take advantage of specially developed ultra-low expansion glasses to decrease frequency coupling to environmental temperature fluctuations, but the poor optical quality of these materials precludes their use for WGM resonators. Therefore, we have proposed, calculated, and demonstrated that the high CTE of opti- cal crystals can be effectively suppressed by incorporating them with ULE glasses in a hetero-compound resonator structure. With FEM modeling we have shown that – given the proper strain-transfer boundary conditions – a resonator can be made that demonstrates a zero-crossing in linear CTE. A similar resonator but with much weaker boundary condition than what was assumed for the simulation has been constructed. Despite this non-ideal fabrication, the device’s thermal coefficient has been suppressed byafactorof3.3,from−1.12×10 −5 K −1 to−3.38×10 −6 K −1 . Accuratemodelingofa physicallyachievableboundarycondition,concurrentlywithimprovingthemechanical strengthofthelatter,isasubjectofongoingwork. Finally,wenotethatheterogeneoustemperaturecompensationschemesusingathin coating of material with strong negative thermorefraction have been reported [55,62, 111]. And despite the fabrication difficulties [15], the technique could yet prove useful forWGMreferencecavities. 96 Chapter4 NoiseSources 4.1 Introduction This chapter deals with the various sources of noise which can destabilize a WGM frequency. Thefocusisprimarilyuponlaserstabilizationapplications,althoughsomeof thediscussionisrelevanttocombgenerationaswell. Thesenoisesourcesmustbewell understood if they are to be suppressed or eliminated, and the purpose of this chapter istoenumeratethevariousrelevantmechanisms,discusstheirphysicalorigin,andgive some idea of their magnitude and/or frequency spectrum. In the case of fundamental thermal noise, these estimates come from applying a survey of theoretical work to our experimental setup; despite our efforts (which will be explained) we did not directly measure thermal noise during this work. Several “technical” noise sources, however, havebeenquantifiedexperimentallyandwillbediscussed. To begin, it is worth clarifying what is meant by the broadly and loosely used term “noise.” In physics and engineering, “noise” typically refers to the degradation of a signalbysomestochasticprocess. Thisdegradationisobviouslyunwanted: itincreases the uncertainty in a measurement, makes data transmission more error prone, or makes an audio-visual experience less appealing. Thus, the ratio of signal to noise becomes a veryimportantparameterinmanysystems. 97 Inthiscontextweareprimarilyconcernedwithfrequencynoise–fluctuationsinthe resonantfrequenciesofanopticalcavity. Becausethefrequencyitselfisjustavalueand doesn’t have a strength, the notion of signal-to-noise ratio becomes more complicated. Usefulextensionsoftheconceptusuallyinvolvetheexperimentalprecisionwithwhich a resonant frequency can be determined, divided by some noise density – discussed in more detail in§2.4.2. When discussing the frequency noise of a WGM without saying anythingaboutthemeasurementitself,itusuallymakesthemostsensetoquantifyitas aspectraldensity,i.e.,whatisthemagnitudeoffrequencyfluctuationatagivenFourier frequency. This is a powerful approach because many types of noise have spectra with a simple mathematical relationship between magnitude and Fourier frequency. More- over,commontypesofstabilityquantificationmethods(e.g.,Allandeviation)havevery straightforward conversions between slopes on a noise spectral density plot and slopes onastabilityplot. An often ambiguous aspect of the term is: at what frequency is a fluctuation noise vs a drift? Probably the best way to distinguish would be based on relevant time scales oftheexperimentorsystem. Ifavariablechangesmonotonicallyduringameasurement cycle, it would be considered a drift; if it fluctuates about a (more-or-less) constant value during the measurement, it is noise. Another way to put this is that noise can be averaged out, while drifts must be subtracted. And the distinction is probably only important insofar and knowing how to deal with a given fluctuation. The frequency range of interest for clock lasers is approximately 100 millihertz to tens or hundreds of kilohertz–maybemegahertzdependingonhowmuchlockbandwidthisneeded. Thus, 98 fluctuations which happen less than once every 10 seconds are considered drifts here. Attheendofthechapterthereissomediscussionofprocessesattheedgeofthisrange –mostlyrelatedtopolarizationrotationsinthefiber. Althoughpotentiallyslowenough tobeconsideredadrift,theydonotstemfromtemperaturechangesinthecavityanddo notbelonginthepreviouschapter. This chapter deals with two types of noise: “fundamental” noise and “technical.” Treatment of the former focuses on thermal noise: spontaneous fluctuations in the tem- perature or volume inside a material, a consequence of the material being at finite tem- perature. It is impossible to eliminate this noise completely, hence the term fundamen- tal. There are certainly other types of fundamental noise, but they are not dealt with herebecausewebelievethethermalnoiseflooristhehighest. Forexample,photonshot noisemayeventuallybealimithereasithasbeeninlaserstabilizationtoFPcavities. Technical noise sources are those which can, in principle, be eliminated completely by ideal experimental conditions. Here we will examine the effects of laser amplitude noise – which can be actively suppressed – and the effects of mechanical vibrations in the mounting setup – which can be minimized via active or passive techniques. The list of potential technical noise sources is virtually endless and we deal only with the aforementioned two because, again, we believe them to be the most likely cause of a limitingnoisefloorinaWGMreferencecavity. Thestartingpointforourdiscussionofnoiseistherelationshipbetweenchangesof 99 aWGMfrequencyandchangesinitsopticalpathlength: ∆f f 0 =− ( ∆n n + ∆r r ) . (4.1) This simple expression can be further broken down to address the specific aspects of a system which may change the resonator’s radius, r, or the index experienced by the mode, n. In the first part of the chapter we shall see how spontaneous thermal fluctua- tions cause changes inn andr, and thus the WGM frequency. In the second part of the chapter,thefocusisonchangesinncausedbyexperimentalimperfections. 4.2 Fundamentalthermalnoise Thermal noise manifests itself as fluctuations in the eigenfrequencies and is considered fundamental because there are not experimental techniques which make it vanish. If suchfluctuationshappenwithinthelaserlockbandwidth,theywillbedirectlytransfered to the laser frequency. This section deals with thermally induced changes in the optical pathlength around the crystal, regardless of whether or not any laser light is circulating in the cavity. The fluctuations become important if there is a spatial overlap with the mode channel, but it is distinct from the thermal or metrological noise sources caused bythelaseritselfthroughintensitynoise–eitherclassical(cw)orquantummechanical (shot-noiserelated). Theprimarypurposeistolayoutthesefundamentalthermalnoisesourcesandgive anumericalestimatesothatwemaydesignanexperimenttomeasurethemordetermine 100 thenoisefloorofagivensystem. 4.2.1 Typesofthermalnoise In addition to the theory directly relating to WGM resonators, the large body of work relating to fiber-based sensors is very relevant here. There has also been extensive the- oretical and experimental work done in the context of thermal noise limits in Fabry P´ erot cavities. In some ways this work is more directly related because the application is locking a laser to the cavities, and the measurement techniques are more applicable. However, the fiber-based sensors are more similar in the sense that light travels com- pletelywithinasolidmedium,ratherthanmostlyinvacuumwithonlyarelativelyshort interactionlengthinthemirrorcoatingsateitherendofthecavity. Becausethereissomeinconsistencyacrosstheliterature,ithelpstofirstclarifywhat ismeantbydifferenttypes(categories)ofthermalnoise. Theterminologylaidoutinthe paper by Bartolo [14] (relating to fiber-based cavities) is the most elucidating because it explicitly relates the noise sources to their fundamental origin: the fluctuation dissi- pation theorem (FDT) of Callen and Green [24]. This theorem spells out a relationship between the spectral density of spontaneous fluctuations in a material and the lossiness of that material. Therefore, the details of how energy can be dissipated from a system becomeimportant. Fluctuations in an optical medium which cause noise in the phase of light traveling through that medium are caused fundamentally by dissipation, and Bartolo uses the terms “thermoconductive” and “thermomechanical” for the two dissipative channels. 101 The former relates to dissipation of thermal energy through a lossy interface, i.e., at the crystal boundary, while the latter describes isothermal loss within the crystal itself (throughfriction)asthermalenergyisexchangedbetweendifferentmechanicalmodes. Relating these terms to those of other authors, we see that “thermorefractive” in [28,46,81] and “thermoelastic” [81] fall into the thermoconductive category, while “Brownian” [81] and “Brownian boundary” [28] would be considered thermomechan- ical. Another (perhaps more intuitive) way to look at this is that thermoconductive noise relates to the fluctuations in the temperature of a crystal given by (from Landau- Lifshitz): ⟨(∆T) 2 ⟩ = k B T 2 C P Vρ (4.2) whereasthermomechanicalnoisecomesfromfluctuationsinthevolumeitselfaccording to ⟨(∆V) 2 ⟩ V 2 =k B T β T V . (4.3) In (4.2) and (4.3), k B is the Boltzmann constant, C P is the specific heat and constant pressure,V therelevantvolume,ρthedensity,β T theisothermalcompressibility,andT thetemperatureofthecrystal. Now, the ways which these fluctuations cause noise in the frequency of a whisper- ing gallery mode can be laid out. Temperature fluctuations within the mode volume cause changes in the optical path length through thermorefraction and thermal expan- sion(“thermoelasticnoise”in[81]). Thermomechanicalnoisecausesfluctuationsinthe optical path length through changes in the resonator radius and the elasto-optic effect 102 (“Brownianboundary”and“elasto-optic”noise,respectively,in[28]). 4.2.2 Thermoconductivenoisemechanisms Thermorefractive noise is commonly believed to be dominant, and is the only WGMRnoisethathasbeenconvincinglydemonstratedexperimentallybyGrudininand Gorodetsky [46]. However, in that work silica microspheres were used, which have two notable properties: a thermorefractive coefficient (α n ) that is more than an order ofmagnitudegreaterthanthethermalexpansioncoefficient(α l ),anddiametersofhun- dredsofmicrometers,meaningverysmallmodevolumes(V m ). Weshallseebelowthat the magnitude of thermorefractive noise scales inversely with mode volume. By con- trast,twoofthemostcommoncrystallinematerialsusedforWGMRs(MgF 2 andCaF 2 ) haveα l >α n (bymorethananorderofmagnitudefortheformer). Magnesiumfluoride has been used for most work on laser stabilization to a WGM resonator [11,17,40], andmoreovertheseresonatorshaddiametersontheorderofmillimetersorcentimeters, meaning vastly larger mode volumes. Thus, it remains to be seen what the dominant fundamentalnoisesourceisinmillimetersizecrystallineresonators. ThermorefractiveNoise Temperature changes in the crystal that overlap spatially with a mode will change the opticalpathlengthviathethermorefractive(TR)effect,andthereforetheeigenfrequency of that mode. The frequency fluctuation spectral density of this noise was calculated andmeasuredinmicrospheresbyGorodetskyandGrudinin[46],andwascalculatedfor 103 oblatespheroidalcrystallineresonatorsbyMatskoetal.[81]. The Matsko paper proceeds along the path laid out in [46] and is summarized as follows. Axiallysymmetricsolutionstothedrivenheatequationcanbeusedtofindthe spectral density and magnitude of thermorefractive noise. Solutions u(r,t) are found suchthat,foraforcingfunctionF(r,t), ∂u ∂t −D∆u =F(r,t) (4.4) over the resonator crystal. Here, u is the temperature at location r and the thermal diffusivityisgivenbyD =κ/ρC (seeTable4.1foralistofmaterialproperties). These fluctuationsarethenaveragedoverthemodevolumeviatheoverlapintegral ¯ u(t) = ∫ u(r,t)|Ψ(r)| 2 dr. (4.5) In this expression, a mode field distribution (represented by Ψ(r)) based on the closed- form solution for WGMs in a spherical resonator is assumed. The spectral density is thenfoundfromtheFouriertransformoftheauto-correlationfunctionby S ¯ u (Ω) = ∫ ∞ −∞ ⟨¯ u ∗ (t)¯ u(t+τ)⟩e −iΩτ dτ. (4.6) Thus, completing the calculation boils down to finding the spatial and temporal solu- tions to Eq. (4.4). This is done for two geometries which have closed form solutions or good approximate solutions: if the resonator is a protrusion on an infinite rod, or if the 104 resonator is a thin disc with radius much larger than the thickness. The authors eventu- ally arrive at the following solution for the spectral density of the fractional frequency fluctuationstobe S ∆ω/ω (Ω) = α 2 n k B T 2 R 2 12ρCV m D [ 1+ ( R 2 D |Ω| 9 √ 3 ) 3/2 + 1 6 ( R 2 D Ω 8ν 1/3 ) 2 ] −1 . (4.7) Finally, the expression is verified by checking that the total temperature fluctuations – found by integrating their spectral density across all frequencies – is consistent with the average of squared temperature fluctuations of the mode volume (T m ) according to (4.2): ⟨(∆T m ) 2 ⟩ =⟨¯ u(0) 2 ⟩ = ∫ ∞ −∞ S ¯ u (Ω) dΩ 2π . (4.8) In Eq. 4.7, k B is the Boltzmann constant, R is the radius of the resonator, V m is the mode volume, and ν is the index for the radial eigenfunction of the electric field distribution(Besselfunction),takentobe1forthefundamentalradialmode. Itisusefultoplotthetemperaturedependenceofthenoisespectraldensityforavari- etyofrealisticexperimentalsituations,predictingthelevelatwhichonecouldexpectto observeTRnoise(Fig.4.1). Thebirefringentmaterialmagnesiumfluorideisinteresting becauseithasturningpointsinthethermorefractivecoefficients(forbothpolarizations) at easily accessible temperatures. Observing a dip in the noise floor near one of these turningpointtemperatureswouldbestrongevidencethatthelimitingnoiseisthermore- fractive. Ontheotherhand,spectraldensitymeasurementsatoffsetfrequencieslessthan severalkHzcanbedifficultbecauseacoustic,electronic,andothertechnicalnoiseinthe 105 measurement system tend to dominate in that region. Fig. 4.1 shows the temperature dependenceofS 1/2 df (T)forbothpolarizationsatthreedifferentoffsetfrequencies. Material property values (for MgF 2 ) used in plotting (4.7) are as follows. Temper- ature independent mechanical properties are those shown in Table 4.1. The thermore- fractivecoefficientsarecalculatedviatheexpressionsgivenin[99],namely α (o) n = 1 n o (0.09797−5.57293×10 −4 T)[10 −5 C −1 ] α (e) n = 1 n e (0.04183−5.63233×10 −5 T)[10 −5 C −1 ] (4.9) withn o = 1.37191 andn e = 1.38341 (ordinary and extraordinary polarization, respec- tively). The specific heat is also a function of temperature, and is calculated through a fitofthedatagivenintheCorningMgF 2 datasheet[2]accordingto C =−1.2×10 −5 T 2 +0.008T −0.33[Jg −1 K −1 ] (4.10) whereT isinkelvin. TheradiusistakentobeR =3.25mm. Fig. 4.1 suggests that TR noise is difficult to measure directly, despite its very dis- tinctive temperature dependence in MgF 2 . The measurement technique must either be able to resolve extremely small frequency noise densities (i.e., millihertz) at moderate offsets, or more modest noise densities but very close in to the carrier (1 Hz offset). Thus, based on the prediction by Matsko et al., TR noise will be difficult to observe in largeradiusmagnesiumfluorideresonators. A very important feature of (4.7) is that for any value of offset or temperature, the 106 40 60 80 100 120 140 160 180 200 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 S df 1/2 (Double−Sided) [Hz/√Hz] Resonator Temperature [ ° C] Aqua to Blue: Ordinary Polarization Red to Brown: Extraordinary Polarization 1 Hz offset 100 Hz offset 10 kHz offset FIGURE 4.1 Expected thermorefractive noise based on Eq. (4.7). Fre- quencynoisespectraldensityisplottedasafunctionoftemperatureforboth polarizations and three different values of offset, as labeled. The vanishing valueofα n meansTRnoiseapproacheszeroat≈176and≈74 ◦ C,forordi- naryandextraordinarypolarizations,respectively. noiselevelgoesasV −1/2 m . ThecurvesinFig.4.1arebaseduponamodevolumeapprox- imationforthefundamentalmodefromtheFEMsimulationsdescribedin§1.2. Clearly, the noise floor decreases as|l−m| and q increase. This fact has two consequences: a measurement of thermorefractive noise can be verified by measuring the level for dif- ferent mode families, as was done in [46]; and, a thermorefractive noise floor can be reducedsimplybycouplingtoahigherordermode. The section on thermorefractive noise is concluded by a brief discussion of two other papers which address it. Chijioke et al. [28] perform an analysis for calcium fluoride using an expression from [46]. However, it is not immediately clear how they 107 TABLE 4.1 Mechanical material property values for MgF 2 atT = 25 ◦ C. Only properties having negligible temperature dependence are shown. The massdensity(ρ),thermalconductivity(κ),thermalexpansioncoefficient(α l ) and Young’s modulus E are from the Corning data sheet [2]. Isothermal compressibility(β T )takenfrom[99](andreferencestherein). D =κ/(ρC) isthermaldiffusivity. ρ κ α l D β T E [kgm −3 ] [Wm −1 K −1 ] [10 −6 K −1 ] [10 −6 m 2 s −1 ] [10 −11 m 2 N −1 ] [GPa] 3176.6 11.6 8.48 3.65 1.0 138.5 derive their value of Γ, and they give no justification of why this expression, evaluated for a 1 mm sphere, should be valid for a high-contrast resonator with 1 cm radius. Nonetheless,theystatethatTRnoisewillbethedominantoneatroomtemperature(for CaF 2 ),thoughonlybyafactorof5orso. Thisisnotabigmarginconsideringthemany approximations made when estimating this type of noise, not to mention the difference inTRnoisefordifferentmodevolumes. And finally, experimental work by Alnis et al. [11] claims to have reached the TR noise floor in an MgF 2 resonator by locking a laser very tightly to a WGM. This claim seems somewhat unsubstantiated, since their experimental Allan deviation plot just touches the theoretical prediction, rather than following it down for a decade (or evenoctave)ofaveragingtimeswhichwouldbeveryconvincingevidencethatTRnoise was indeed the limit. Furthermore, it’s a shame they didn’t do the simple experiment of locking the laser to the other polarization, since this would have provided instant confirmation that they were limited by TR noise if the floor shifted by the ratio of ther- morefractivecoefficients. 108 ThermoelasticNoise Temperature fluctuations in the resonator are transfered into WGM frequency noise through thermal expansion. Generally, this effect is considered to be much smaller than thermorefractive noise because the volume of interest is the entire resonator – not just the mode volume. However, the small values predicted for TR noise are on the same order of magnitude. This can be seen by multiplying (4.2) by the linear thermal expansioncoefficienttoobtaintheexpectedfractionalfrequencyfluctuations: ⟨(∆f TE ) 2 ⟩ f 2 =α 2 l k B T 2 C P Vρ . (4.11) Here, f TE is the frequency excursions due to the thermoelastic effect. The frequency independentfractionalexcursions(∆f TE /f)foranMgF 2 resonatorwith3.5mmradius and 400 µm thickness is≈8×10 −14 at room temperature (increasing linearly with T). This value is not negligible when compared to the TR noise predictions above or the measurementsdonebyAlnisetal. Inthefollow-onpapertoMatsko’s[81],Savchenkovetal.[99]calculatethespectral density of such thermoelastic noise. By assuming that the dominant relevant time scale istheslowestthermaldiffusionratealongtheradiusoftheresonator,theycalculatethat TE noise is approximately white (with a value given by (4.11)) out to a characteristic frequencyofD/r 2 ,afterwhichitdecreasesby10dB/decade. Fortheresonatorspecified above,thischaracteristicfrequencyis1.0Hz. Thus,fromthestandpointofexperimental measurement, the situation is very similar to that of the TR noise: for challengingly 109 smalloffsetfrequencies,thenoisedensitycouldpotentiallybemeasured,butatrealistic offsetsthemeasurementrequiresmHz(orless)resolution. ThisisagoodtimetoclarifythedifferentterminologiesusedintheMatsko[81]and Savchenkov [99] papers. In the former, fundamental temperature fluctuations are con- vertedtofrequencynoiseviathermorefractionandthermalexpansion,called“thermore- fractive”and“thermoelastic,”respectively. Savchenkovusestheterm“ThermalExpan- sion” noise equivalently to Matskos “thermoelastic,” while Savchenkov uses “thermoe- lastic”todescribemechanicalfluctuations,whicharecalledBrownianbyMatsko. This confusionillustratestheutilityofplacingnoisesourcesintothecategories“thermocon- ductive”and“thermomechanical.” 4.2.3 Thermomechanicalnoise In addition to the temperature fluctuations within a rigid body, there are also volume fluctuations, which are familiarly known as Brownian motion. These result from the lossy exchange of thermal energy between different mechanical modes. By contrast to the previous section, these volumetric fluctuations do not depend upon the boundary conditions of the body and would take place whether or not thermal energy crossed the boundary into the surrounding thermal bath. This type of thermomechanical noise has been addressed for WGM cavities in the papers by Matsko and Chijioke et al., as well as for both passive and active (with gain) fibers [14,37,41]. Below we briefly examine these analyses to make an approximation about the level at which the noise could be observedexperimentally. 110 Browniannoise We begin again with the analysis by Matsko [81]. The WGM frequency is perturbed by volume fluctuations in the crystal because the resonator radius is directly affected. ThefollowingexpressionfromLandauandLifshitzgivesthefractionalaveragevolume fluctuationsas ⟨(∆V) 2 ⟩ V 2 =k B T β T V , (4.12) whichisvalidifthevolumeV (takentobethatoftheresonator)ispartofamuchlarger cylinder. A factor of 3 accounts for the differential relationship between volume and radius,yieldingafrequencyindependentBrowniannoiselevelof ⟨(∆f B ) 2 ⟩ f 2 =k B T β T 9V . (4.13) Fromtheseexpressions,thespectraldensityoftheBrownianfrequencynoisedensityis derivedtobe S df/f (f) =k B T β T 9V Γ m (f−f m ) 2 +Γ 2 m (4.14) wheref m isthemechanicalresonancefrequencyandΓ m =f m /(2Q m )isthedecaytime oftheacousticmodewithmechanicalqualityfactorQ m . The paper by Chijioke et al. [28] has an extensive analysis of the Brownian noise in WGM resonators. It includes an FEM-based analysis which explicitly solves for the mechanicaladmittanceforagivendrivingforcetofindthemagnitudeofthespontaneous fluctuations–ashasbeendoneseveraltimesformirrorsubstratesand/orcoatingsinFP 111 cavities. TheyalsoderiveasimpleexpressionforthespectraldensityofBrowniannoise whichtheyfindyieldssimilarresultstothemoredetailedFEManalysis. ThisexpressionisderiveddirectlyfromtheFDT,startingwith S x (f) = k B T π 2 f 2 |Re[Y(f)]|, (4.15) where Y(f) is the susceptibility of a generalized coordinate x under action by a force F(f) conjugate to it. Then, following the approach of Levin [71], the real part of sus- ceptibilitycanbewrittenas |Re[Y(f)]| = 2W diss (f) F 2 = 4π F 2 fU(f)ϕ(f,T). (4.16) Here,W diss is the cycle-averaged dissipation when an oscillating force of magnitudeF is applied to the coordinate (in this case displacement) at frequencyf. The mechanical loss-angle ϕ is assumed to be frequency independent (“structural damping”), and U is themaximumvalueofstrainenergyoveraperiodofoscillation. Both an analytical and numerical solution exist, with the latter being calculated via FEM,andtheformergivenas S df/f (f) = 1 3π 2 k B Tϕ(T) Er 3 f . (4.17) Because this expression approximates the resonator as a sphere, additionally approxi- mating moduli of the (anisotropic) crystal as the Young’s modulus E is claimed to be 112 acceptable. ComparisonofEqs.(4.14)and(4.17)immediatelyrevealsafundamentaldifference: thelatterhas1/f densitywhiletheformerisasecond-ordersystem–relativelyflatden- sity up to a peak at resonance, then decreasing as 1/f 2 . However, both power densities dependlinearlyupontemperature,andthemagnitudeofnoiseisdeterminedbymechan- ical material properties and loss. Experimental work by Hofer et al. [60] suggests that quality factors of mechanical modes in a fluoride resonator vary greatly. The value can beaslowas40forsomeradialbreathingmodeswhentheresonatoriswaxedtoabrass post. But Q m > 10 5 was observed for different types of mechanical modes and when theresonatorwascarefullyclampedbytungstenwiresandheldatlowpressure. To illustrate how these variations might effect spectral density, we plot predicted S 1/2 df/f (f) from Eqs. (4.14) and (4.17) for the aforementioned values of Q m . In plotting (4.14), mechanical resonance frequencies are needed and we use 1 MHz and 10 kHz as illustrative examples. The plots of (4.17) are made assuming that the loss angle is temperature independent, and that ϕ ≈ tan(ϕ) = Q −1 m . All four curves are made for a 400µmthickMgF 2 resonatorwithR =3.25mmat300K. The Chijioke et al. derived curves in Fig. 4.2 can be easily converted to a τ- independent noise floor on an Allan deviation plot by multiplying the value at 1 Hz by √ 2ln(2) [91], yielding 3.2×10 −14 and 6.1×10 −16 for the Q m = 40 and 100,000, respectively. Theheavydependenceofcavityopticalfrequencystabilityuponmechani- callossessuggeststhatmoreinvestigationalongtheselinesshouldbedone(as,again,it has been for FP cavities). The detailed temperature dependence of loss angle, Young’s 113 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 −20 10 −19 10 −18 10 −17 10 −16 10 −15 10 −14 10 −13 10 −12 Frequency [Hz] S df/f 1/2 (f) [Hz/√Hz] Matsko: Q m =10 5 , f m =10 6 Matsko: Q m =40, f m =10 4 Chijioke: Q m =10 5 Chijioke: Q m =40 FIGURE 4.2 Expected Brownian frequency noise density based on Eqs.(4.14)and(4.17). BothanalysespredictthattheBrowniannoiseofopti- cal frequency depend heavily upon mechanical properties of the resonator andsetup. modulus,andmechanicalresonancefrequencymayalsobeimportant. Morediscussiononthe1/f natureofBrowniannoise ItappearsthatthecolorofBrownianfrequencynoiseinaWGMcavitywillplayacru- cial role in determining the overall noise floor. It is easy to see from the above approx- imations that if the noise power density indeed goes as 1/f, the magnitude becomes comparabletothethermoconductivenoisesasdescribedabove. Therefore,itiswortha bitmorediscussion. AnalyticalandexperimentalinvestigationsonthecolorofBrowniannoisehavebeen of interest to the gravity wave and ultra-stable laser community for over two decades. A 1990 paper by Peter Saulson [94] derives – from simple damped spring-mass sys- 114 tems and the FDT – the spectral densities of thermal noise depending on what kind of damping model is used. In particular, if the damping is described by a complex spring constant,thenoisespectraldensitydecreaseswithfrequencyonepoweroff fasterthan if a velocity dependent damping is used. This finding applies to systems of any order, and it’sinteresting to note that for a second order system, the consequence in assuming viscousdampingisthatoneoverestimatesthenoiseaboveresonancebutunderestimates it below resonance. In other words, the solid curves in Fig. 4.2 get “tilted” to the right by one power of f and go as 1/f (in power, f −1/2 on this plot) below resonance if a complexspringconstantisused. Thisisobviouslyespeciallyrelevantifthesystemhas ahighresonantfrequency–thenoiseonanysignalbelowresonanceincreasesas √ f m . Subsequent analysis of interferometers and FP cavities substantiated this prediction of 1/f noise. Two examples are the important paper of Levin [71] – in which a gen- eral form of the FDT, applicable in this context, is derived – and that of Numata et al. [83] – where an experimental measurement of 1/f thermal noise in an FP cavity is first reported. A 1995 paper also demonstrated 1/f thermal noise in a torsional res- onator[44]. More recently and perhaps more applicably since light is traveling within a solid medium, investigations of thermal phase noise in fibers has further supported the 1/f thermal noise model. In particular, the paper by Bartolo et al. [14] shows several mea- surements of phase noise in lengths of optical fibers which increase with decreasing frequency. Toaddressthis,apaperpublishedshortlyafterbyDuan[37]derives–again from the FDT – that while thermorefractive noise is constant with frequency up to a 115 cutoff after which it decreases, thermomechanical noise indeed goes as 1/f below the mechanicalresonancefrequenciesoflengthfluctuationsinthefiber. Thus,thereisgood reasontoquestionwhetherthermorefractivenoiseisthedominantthermalnoisesource inWGMcavitiesasiscommonlybelieved;onlycarefulexperimentalmeasurementwill resolvetheissue. 4.2.4 Measuringthermalnoise Theprecedingsectionsessentiallyconstituteareviewofliteraturewhichestimatesther- mal noise in WGM cavities. A good experimental quantification of thermal noise is naturallyhighlydesirable,notonlytoresolvethediscrepanciesbetweentheories,butto determinethevalidityofthemanyapproximationsmadetherein. The level of thermal noise for any of the above mechanisms will depend upon the temperature of the resonator. An obvious experiment is thus to look for a noise floor which changes with cavity temperature. Based on the above analysis and neglecting temperaturedependenceofphysicalproperties,weseethatthespectralfrequencynoise densitiesshouldgoasfollows(Hz/ √ Hz,atagivenoffset). Thermorefractivenoise(4.7) ∝T 1 ;thermoelasticnoise(4.11)∝T 1 ;andBrowniannoise(4.17)∝T 1/2 . We also saw that magnesium fluoride is a uniquely good candidate for unraveling thermalnoisebecauseofitsturningpointsinα n : ameasurementdonenearoneofthese turningpointswillhaveaneasilyrecognizedtemperaturedependenceifTRnoiseisthe dominant one. Moreover, of those discussed above, TR noise is the only one which dependsuponmodevolume. 116 Thus, we performed an experiment consisting of two MgF 2 resonators in the same small oven. To each of these resonators was locked a nominally identical setup of fiber laserandAOM,asdescribedin§2.2.2. Becauseoftheirverysimilarradius,theFSRsof theresonatorswassimilarenoughthatthetemperatureoftheovencouldbechangedby anamountcorrespondingto1FSRforboth. Inthisway,itwaseasytoidentifythesame modeateachtemperature(viathe“modemapping”describedin§1.4.3),lockthelasers to the two cavities, and compare them to one another. (Measuring the same modes is important because otherwise the mode volume would be unknown.) Using several different methods for analyzing the beatnote between the lasers, we tried to observe a temperature dependent noise floor. We were unable to do so, despite changing the resonators’ temperature over a 35 K range around the extraordinary thermorefractive turningpoint(thepolarizationtowhichthelaserswerelocked). The level of relative noise between the two lasers was above what we would expect to observe based on the preceding estimates. Yet we know the quality of the laser lock is better than 1 Hz/ √ Hz. This combination of results suggest a technical noise source whichiscommonmodetothecoupler-resonatorsystemandistemperatureindependent. Investigationofafewsuchpotentialnoisesourcesisthesubjectofthenextsection. 4.3 Technicalnoisesources In the previous section we investigated fundamental thermal noise in a WGM cavity – spontaneous fluctuations which can never be eliminated, but only minimized through 117 appropriate choice of materials and experimental conditions. By contrast, technical noisereferstosourceswhichcan,inprinciple,bemadearbitrarilysmallwithappropri- ate experimental techniques. The purpose of this section is to investigate and quantify thesetechnicalnoisesourcessothatonemaydetermineaheadoftimetheexperimental conditionswhichmustberealizedtoachieveagivenfrequencystability. For the discussion of technical noise we start once again with Eq. (4.1). Setting ∆r =0yieldsthefractionaldependenceonchangeofindex: ∆f/f 0 =−∆n/n. Thus, for a fractional frequency instability of 10 −14 , the change in refractive index must also be on this order. It is necessary to carefully consider any effect which can change the refractive index sampled by the WGM. In particular, laser amplitude noise is converted tochangesinfrequencythroughthermorefraction. Alsoimportantisthemode’sevanes- cent sampling of the medium surrounding the cavity. In principle this includes air and the coupler. But since much of the work described here involves a WGM in vacuum, onlychangesintheevanescentsamplingofthecouplerareconsideredbelow. 4.3.1 Laseramplitudenoise It is a fact of life that there is some amplitude noise on the output of a laser, often referred to as relative intensity noise, or RIN. When light circulates in a WGM cavity, itisconvertedthroughmaterialabsorptionintoheat. Asdiscussedin§3.1.4,thiscauses an unavoidable, steady-state temperature gradient within the crystal. However, if the laserpowerfluctuatesasmallamountaroundafixedvalue,wecanneglectthisgradient and consider only temperature changes which are much faster than the thermal relax- 118 ation time of the disc. These temperature fluctuations will change the host material’s indexthroughthermorefractionas∆n =n 0 α n ∆T m ,wherethemsubscriptemphasizes that we are only concerned with the mode volume temperature. Thus, the frequency excursionsoftheWGMwillbe∆f =−f 0 α n ∆T m . To calculate the frequency noise spectral density that will result from a given laser amplitude noise spectrum P(f), it is necessary to know a transfer function A(f) such that ∆T m (f) = A(f)∆P(f). We expectA(f) to reflect some kind of low-pass behav- ior because for very fast power fluctuations, the amount of heat delivered and change in temperature will be very small. In principle, this transfer function could be calcu- lated based on the mass and specific heat of the mode volume plus an estimate of the heat delivered by the circulating laser light. However, because of the high precision, highbandwidthmeasurementaffordedbythedual-modeexperiment(§3.1),wechoseto directly measure the mode volume’s temperature response to fluctuations of in-coupled laserpower[16]. ThemeasurementisshownschematicallyinFig.4.3andconductedasfollows. The dual-mode temperature stabilization loops are implemented as described in§3.1.2. The fast-branchofthetemperatureactuationloopisnotclosed–ratherasinusoidalmodula- tionisappliedtotheVCOdrivingtheAOM,causingcontrollableamplitudemodulation ofthelaser. Ifthemodulationfrequencyisgreaterthantheunitygainbandwidthofthe slow-branch temperature actuation, changes in T m will not be corrected for. For these out-of-bandmodulations,theerrorsignalofthetemperaturestabilizationloopisrelated tochangesinT m proportionallythroughtheerrorsignalslope. 119 10 −6 10 −4 10 −2 10 0 10 2 −20 0 20 40 60 80 100 Frequency [Hz] Temp. Loop Gain [dB] Modulate FIGURE 4.3 Measuring the temperature response of the mode volume to amplitude modulation: The fast actuation from§3.1.2 is opened and inten- tionally modulated. If modulation is outside of the slow temperature loop bandwidth,theerrorsignalgivesameasureofT m . Amplitude modulation was applied at a range of frequencies from 1 Hz to 10 kHz andtheerrorsignalresponserecordedwithaDAQ.Subsequently,asinusoidalfittothe recorded data allowed an extraction of the response amplitude that was less susceptible to noise. For each modulation frequency, this amplitude is converted to a temperature response by dividing out the temperature discriminant [V/K]. Also carefully accounted for is the low-pass filtering characteristic of the VCO, i.e., as the input modulation fre- quency increases, the depth of light-level modulation decreases. This decrease was measured and divided out. Finally, an estimate must be made of the light coupled into the cavity during the experiment. This was done by measuring the power at the fiber couplerwithapowermeterandmultiplyingitbythecouplingcontrastη. TheresultisshowninFig.4.4. TheverticalaxishasunitsofµK/mWasneededfor A(f) above. Indeed, we see the expected low-pass filtering characteristic. The upper limitofthemeasurementfrequencyrangeissetbythelaserlockbandwidthbecause,for 120 veryfastchangesinT m ,thelaserlockcannotkeepupwithchangesinWGMfrequency and the differential measurement is no longer accurate. This is likely the reason for the lastpointinFig.4.4beinghigh. 10 0 10 1 10 2 10 3 10 4 10 100 500 Modulation Frequency [Hz] Temperature response [μK/mW] Data Power−law fit FIGURE 4.4 Measurement data showing how fluctuations in laser ampli- tudechangethemodevolumetemperatureT m (bluetriangles). Apowerlaw fit (red line) gives an approximate transfer function that allows calculating thefrequencynoiseresultingfromlaserRIN. A power-law fit to this data is a good empirical approximation of the desired trans- fer function A(f). The fit shown is given by A(f) = Cf x with x = −0.170 and C = 214.8 [µK/mW]. Now, we may proceed to calculate the expected frequency noise densitythatwillresultfromameasuredRINspectrum. First, an FFT spectrum analyzer is used to measure the voltage noise density of the detectorsignalintheWGMcavitylaserlocksetup(withthelaserfreerunningandoff- resonance). Then, this voltage spectrum is converted into an intensity noise spectrum by dividing out the response of the detector. We take this to be the power present at the coupler, P 0 (f). The fractional frequency noise spectrum is then calculated simply 121 through S 1/2 δf/f (f) =α n A(f)ηP 0 (f) (4.18) whereη isthecouplingefficiency. TheresultingnoisedensityforafibercoupledMgF 2 resonatorisshowninFig.4.5. HerewetaketheDCvalueoflaserpowertobeP 0 (0) = 10 µW (a typical value for these experiments) and α n = 2×10 −7 , the value for the extraordinary polarization in MgF 2 near room temperature. Also shown are the results with an additional amplitude stabilization loop engaged. This loop uses an auxiliary detectortomeasureopticalpoweraftertheAOM,thenactuatesontheAOMtosuppress amplitudefluctuations. 10 2 10 3 10 4 10 −19 10 −18 10 −17 10 −16 10 −15 10 −14 10 −13 Frequency [Hz] S δ f/f 1/2 [Hz/√Hz] w/o Ampl. Stab w/ Ampl. Stab FIGURE 4.5 The expected RIN-induced frequency noise spectral density inaWGMcavity. Shownareresultswithandwithoutanauxiliaryamplitude stabilizationloopengaged. A few comments about the resulting frequency noise spectrum: Over the course of thiswork,comparisonsofeithertwolaserslockedtotwodifferentWGMreferencecavi- ties,oroneWGM-stabilizedlasercomparedtothefrequencycombneverdemonstrated 122 stability below a few parts in 10 12 . The frequency noise values below ∼100 Hz are approachingthatlevel. Forthismeasurement,P 0 (0) = 10µWwasassumed;thisresult suggests, however, that a factor of ten higher (also a reasonable experimental value) could have caused RIN to be our limiting factor. On the other hand, locking experi- mentswereoftencarriedoutwithjust1µWofpoweratthecoupler(andevenaslowas 100nW),wheresuchnoisewouldcertainlybenegligible. Anothercomplicationisthatthespectrabelow150Hzaresimilarenoughtosuggest it may be an artifact – for example flicker noise in the electronics used to measure laser power fluctuations. In any case, we did not notice a difference in laser stability performancewhentheamplitudeloopwasopenorclosed. Thissuggeststhateither: (1) laserRINisthelimitingfactor,buttheamplitudestabilizationloophasinsufficientgain at low frequencies (i.e., low frequency hump is real); or (2) RIN is not a limiting factor (andwecannotmakeaconclusionaboutthehump). Therepeatabilityofthe10 −12 noiseflooracrossmanydifferentexperimentssuggests (2),butthissectionisconcludedbynotingthatagain,MgF 2 isaverypromisingmaterial forWGMreferencecavities. IfCaF 2 wereused,withitsthermorefractivecoefficientthat is almost 100 times larger, the curves in Fig. 4.5 would increase commensurately and RINinducedfrequencynoisewouldverylikelybecomeaperformancelimitation. 4.3.2 Vibrationallyinducednoise As described in Ch. 1, coupling to a whispering gallery mode is typically achieved via a near-field interaction. Thus, effective coupling demands that the coupler material be 123 positioned well within the WGM’s evanescent tail that extends outside the resonator. Changes in the coupling gap therefore directly affect the refractive index experienced bythemode. Thepitfallofthistechniqueasthatitplacesstringentrequirementsonthe mechanicalsystemmaintainingthecouplinggap,whichmaybepronetoseismicnoise, vibrations, or even electronic noise in the case of piezo stages. Here, a measurement of couplinggapsensitivityisdescribed. Frequency pulling from changes in coupling gap were measured using the two- resonator, two-laser scheme shown in Fig. 4.6. The setup allows for modulating the coupling gap of one resonator and measuring the resulting frequency excursions by comparisons to a second laser locked to the other cavity. Both cavities were coupled with angle polished fibers. The “test resonator” was mounted in a small, temperature stabilized aluminum enclosure, with the coupler being held by a Thorlabs Nano-max stage. The “reference resonator” was the same device used in the dual-mode tempera- ture stabilization scheme; it was enclosed in a vacuum chamber with the fiber coupler gluedtoaflexurearmonthecavitymount. Because the test resonator has a (comparatively) unstable thermal environment, the two cavities are thermally locked to one another by feeding back to the TEM on which the reference resonator is mounted. This is achieved by mixing down the beatnote between the two locked lasers to about 40 MHz. The RF signal is then compared to a second 40 MHz signal (from a stable digital synthesizer) in a digital frequency com- parator circuit. The output of this comparator is integrated, and then fed back to the TEM. The bandwidth of this “tracking lock” loop is approximately 1 Hz; thus, the two 124 FIGURE 4.6 Experimental setup for measuring coupling gap frequency sensitivity. Two lasers are PDH locked to two resonators (full loops not shown). Some light from each laser is picked off and combined on a fast photodiode, yielding a beatnote at f beat ≈ 6 GHz. This beatnote is mixed down to 40 MHz for counting and thermal “tracking lock” of reference res- onator. stabilized laser systems are effectively independent for noises with Fourier frequency >1Hz. To measure the coupling gap sensitivity, a sinusoidal modulation was applied to the piezo actuator on the stage, along the axis which controls the fiber-resonator distance. The modulation frequency was 40 Hz, chosen to be well outside of the tracking lock bandwidth,butwellwithinthelaserlockbandwidth. Therefore,thelasertightlyfollows changesintheWGMfrequency,whichcanbereadoutasexcursionsintheopticalbeat- notefrequency. PerforminganFFTonthecountedopticalbeatnoteallowsextractionof 125 theamplitudeoffrequencyexcursionsat40Hz. Fig.4.7showstheresultsofthreesuch measurements for three different values of coupling contrast (though the measurement wastakenforfivevaluesofcontrast). 0 10 20 30 40 50 10 −2 10 0 10 2 S df 1/2 [Hz/Hz 1/2 ] Fourier Frequency [Hz] 31% contrast 68% contrast 100 % (critical) 39 40 41 10 −2 10 0 10 2 10 4 FIGURE 4.7 FFTs of beatnote frequency time series, as measured by a counter. The obvious peak at 40 Hz is from modulation of the coupling gap. Theamplitudeofthispeakyieldsthefrequencypullingforagivengap modulation, which in turn is a function of coupling contrast (as can be seen moreclearlyinthepeakdetail–dashedmagentabox). Frequency excursions were converted into a coupling gap sensitivity (units of Hz/nm) as follows. The distance traveled by the fiber coupler during modulation was calculatedbasedonvaluesintheNanomaxdatasheet: thetotaltravelrangewasdivided by the input voltage range to get a displacement coefficient, which was then multiplied bythemodulationvoltageto yieldadisplacementof 13.4nm(rms). Then, thevalueof frequencyexcursionatthe40Hzpeakwasdividedbythedisplacementmodulation. Afewnoteworthyassumptionsaremadeinthiscalculation. First,itisassumedthat theDCvaluesforpiezoactuatorresponselistedinthedatasheetarelinear,havenegligi- bleroll-offat40Hz,andthatthereisnohysteresis. Second,itisassumedthatthereare 126 no mechanical resonances in the stage, clamp, or fiber coupler near 40 Hz. While they are reasonable, these assumptions are the primary source of error. A better experiment couldbeperformedbymeasuringtheactualfiber-couplerdisplacementamplitudewith, forexample,interferometrictechniques. 30 40 50 60 70 80 90 100 0 20 40 60 Contrast [%] Sensitivity [Hz/nm] Measurement data Exponential fit FIGURE 4.8 Couplinggapsensitivityasafunctionofcontrast. Theexpo- nential dependence suggests evanescent sampling of the coupler’s refractive index by the WGM field. Contrast is expressed normalized to critical cou- pling. Coupling gap sensitivity results are shown in Fig. 4.8. The exponential dependence oncontrast(orequivalentlycoupler-resonatordistance)isclearevidencethattheWGM frequency pulling is caused by evanescent sampling of the coupler material. Similar resultshavebeenobtainedinprismcoupledresonatorsystems[73]. To further support this conclusion, an amplitude modulation measurement is made. Thisquantifiestheextenttowhichchangesinthecirculatingopticalpower–alsocaused by coupling fluctuations – pull the frequency. With the gap distance held fixed, an amplitudemodulationat40HzwasappliedviatheAOM.Thedepthofthismodulation was chosen such that the sinusoidal amplitude (at 40 Hz) of the PDH detector signal as 127 seen on the oscilloscope was equivalent to the coupling gap measurements. With the laserslocked,beatnoteexcursionsweremeasuredandfoundtoaccountforlessthan3% oftheobservedcouplinggapinducedfrequencypulling. These results suggest that coupling gap fluctuations must be carefully considered for applications demanding utmost WGM frequency stability. Extrapolating from this plot,weseethatat82%ofcriticalcoupling,fluctuationinthegapof1nmRMSwould limit overall frequency instability at the 10 −13 level. As described in [26], a neutral mountingschemeexistswhichcansuppressWGMfrequencypullingfromaccelerations which induce strains in the resonator itself. However, a mechanical mount which can sustain sub-nanometer gap stability over the relevant frequency range (say, 10 mHz to 1 kHz) may be substantially harder to realize. Indeed, a “neutral mounting” scheme which maintains the coupling gap may be more important than one which minimizes deformations of the cavity crystal itself. Alternatively, a recent free-space coupling technique[13]shouldprovemorerobusttovibrationsbecausenoexternalopticsinteract withtheevanescentfield. Since PDH error signal noise-density depends heavily upon contrast, coupling gap sensitivityalsoactsinoppositiontofrequencylockperformance. Thus,foranoptimized WGMfrequencyreferencesystem,abalancebetweencouplinggapsensitivityanderror signalnoisedensitywillhavetobefound. 128 4.4 OtherDestabilizingMechanisms In addition to the thermal noise, laser RIN, and coupling gap fluctuations, two final destabilizingmechanismsshouldbebrieflymentioned. Theseare: polarizationrotations in the fiber and residual amplitude modulation (RAM) at the EOM. As mentioned in the introduction, these changes take place more slowly than the other types of noise describedabove. Thelockingsystemsusedinthisworkconsistedentirelyoffiber-basedopticalcom- ponents; the only free-space optical path is after the WGM coupling region, where the lightisfocusedontoadetectorwithalens. Suchfiber-basedsystemsareconvenientfor many reasons, including no alignment necessary, potential for compactness/portability, and–ifworkingattelecomwavelengths–readyavailabilityofmanyhigh-qualityopti- cal devices such as modulators. But there is a major drawback to fiber-based optical systems: theyarehighlypronetoproblemscausedbypolarizationrotations. Strain in the fiber’s fused silica causes slight variation in refractive index through the elasto-optic effect and can change the polarization state of propagating light. This can be mostly mitigated by careful implementation of polarization maintaining (PM) fiber, but many of the modulators (and even lasers) used here were pigtailed with non- PM fiber. Thus, changes in room temperature, air currents from air conditioners, and even pressure changes from, e.g., somebody opening a door down the hall, can cause fluctuationsintheoutputpolarizationstateatvaryingtimescales. These fluctuations destabilize the laser lock by changing the overall light level, or the excitation ratio between TE and TM modes. The latter was a particular problem in 129 the dual-mode experiment, while the former was an issue across all experiments. We havealreadyseenhowthechangingamountofheatdeliveredbyavaryingopticalpower destabilizesaWGMonlongtimescales(§3.1.4)andshorttimescales(previoussection, §4.3.1). Here we point out that the problem can also be an optoelectronic one: if the mixer is not well-balanced, changes in light amplitude will change the lock-point – i.e., the DC bias of the error signal. This problem is again directly reduced by using a higher Q cavity, but can still play a very important role especially at time scales > 100 s. A changing light level causes not only a DC offset, butalso a change in error signal slope (amplitude). The resulting change in closed-loop gain can, in extreme cases cause the locktooscillate(increasinglightlevels)orlooseacquisition(decreasinglightlevels). Rotationsofpolarizationarealsorelatedtothefinaldifficultydiscussedhere: RAM. Anelectro-opticphasemodulatorwillinevitablyimpartsomeamplitudemodulationon the light. If the level of this amplitude modulation changes, we get a varying lock- point. The fact that RAM destabilizes an FM spectroscopy experiment was recognized early on, and a 1985 paper by Wong and Hall [123] describes the physical mechanisms causing the problem and an active suppression technique. Because phase modulation dependsheavilyupontherelativealignmentofopticalpolarizationandcrystalaxis,any rotations–eitherinthelightfieldorstrainsinthecrystal–causeshiftsinRAM. TwodifferentEOMswereusedinthiswork: onewhichhasanintegratedanalyzerat theinputandanominally“polarizationmaintaining”waveguidecrystal;andonewhich has no input analyzer. A huge improvement in lock-point robustness with respect to 130 polarization and temperature changes was noticed between the two. An auxiliary loop to actively maintain alignment of crystal axis and polarization via the modulator’s DC biashelpedsomewhat,butincreasessystemcomplexity. 4.5 Summary Some of the noise sources which destabilize a WGM frequency have been discussed. Further experimental work is necessary to fully unravel the relative strengths of these noises. Fundamental thermal noise is one possible limitation. We see that the following experimentally measurable temperature dependencies could be used to identify the dif- ferent types of noise: Thermorefractive noise ∝ T 1 ; thermoelastic noise ∝ T 1 ; and Browniannoise∝T 1/2 . WealsoseethatBrowniannoisemaygoas1/f belowmechan- icalresonance,suggestingitmaybeaveryimportantlimitationinWGMreferencecav- ities. WeseethatthermorefractivenoiseinMgF 2 hasauniquetemperaturedependence signature, but that TR noise is at a relatively low level compared to other fundamental andtechnicalnoises. A technique which estimates the frequency noise density caused by laser RIN is presented; it relies on empirical extraction of a transfer function between optical power fluctuations and mode volume temperature which is facilitated via dual-mode sensing. LaserRINisexpectedtoplayanimportantroleonceinstabilitiesarebetterthanapartin 10 13 . Couplinggapfluctuationsarealsomeasured,andit’sfoundthatsub-nmmechani- 131 cal stability is needed for optical-clock applications. We note that a recently developed free-space coupling scheme based on optical gratings engraved on the resonator sur- face[13]maygreatlyrelaxthisengineeringchallenge. Othertechnicalissuesrelatedtopolarizationrotationsintheopticalfiberarebriefly discussed. 132 Chapter5 WGM-basedFrequencyCombs 5.1 Introduction In2005,halfoftheNobelPrizeinphysicswasawardedtoJohnHallandTedH¨ anschfor “theircontributionstothedevelopmentoflaser-basedprecisionspectroscopy,including the optical frequency comb technique.” [3]. As early as the 1970’s, picosecond pulses wererecognizedashavingapplicationsinspectroscopy[38],andbythelate1990’s,the largerbandwidthaffordedbyTi:sapphiremodelockedlasers(MLL)wasbeingexploited forfrequencymetrologyapplications[115]. Sincethen,frequencycombs(basedaswell uponfiberMLLs)havefoundmanyapplications,withjustafewexamplesbeingmolec- ular “fingerprinting” [112], astronomical spectrograph calibration [107], and extension into the ultra-violet for experiments in fundamental physics [29]. And stabilized fre- quency combs were revolutionary for optical atomic clocks, providing a path to their practical use and comparison to existing microwave standards via frequency down- conversion. AroundthetimeofthisNobelPrize,observationsofparametricoscillation–closely related to comb generation – in WGM resonators were reported [66,102]. Shortly after, the first demonstrations of microresonator-based frequency combs were pub- lished[33,54,98]. It’seasytoseefromtheprecedinglistofapplicationswhytherewas 133 immediateexcitementattheideaofminiaturizingfrequencycombsbygeneratingthem in whispering gallery mode resonators – henceforth dubbed “microcombs.” Therefore, thepast6-7yearshasseenaflurryoftheoreticalandexperimentalresearch,withmajor contributionsbeingmadebytheQuantumSciencesgroupatJPL[27,48,49,110],Tobias Kippenberg’s group [57,65] and in collaboration with Ted H¨ ansh [33,120]; OE-Waves in Pasadena [95,100]; and the Gaeta-Lipson collaboration at Cornell [84,93]. More recently, Scott Diddams’s group at NIST Boulder has initiated a comb research effort and made contributions, particularly towards tuning and stabilization of WGM-based combs[88]. This chapter discusses some of our recent work on microcombs. In particular, we have made progress on producing a low noise, broad comb using relatively low pump power. We have also performed measurements on the relative stability of a comb’s microwave beatnote and the absolute optical frequency of the pump laser which is sta- bilized to the cavity mode via PDH locking. We also find that the comb’s RF beatnote can be changed from a noisy to a quiet regime by adjusting the input-offset to the PDH lock. 5.1.1 Micro-combgeneration Microcombs are generated via the well-known nonlinear optical process of four-wave- mixing(FWM)(seee.g.,[21]). Thenonlinearsusceptibilityχ (3) enablesFWM,whichis whymicrocombsareoftenreferredtoas“Kerrcombs.” Valuesofχ (3) inmostmaterials are relatively small, but since intensities in a WGM cavity can easily reach levels of a 134 GW/cm 2 (andhigher),efficientthird-ordernonlineargenerationcanreadilyberealized. FWMreferstotheprocesswherebythreeelectromagneticfieldcomponentsinteract inanonlinearcrystaltoproduceafourth. Thefrequencyrelationshipbetweenthesefour fieldscanbeexpressedby ω 4 =ω 1 +ω 2 −ω 3 . (5.1) Theprocessisenhancedinanopticalcavityatitsresonantfrequencieswherehighfield energies preferentially build up. Thus, in a WGM microresonator, two of the initiating photons are from the same (pump) mode, and the third is from another excited mode, typically taken to be one cavity FSR away in frequency; by conservation of energy, the fourth field (generated photons) are also separated from the initial mode by one FSR, butwiththeoppositesignoffrequencyspacing. A conceptual representation of FWM comb generation in a WGM cavity is shown inFig.5.1andcanbedescribedby ω 0 +ω 0 →ω + +ω − (5.2) where the initiating mode has frequencyω 0 , the third photon is from an adjacent mode separated by a free spectral range, i.e., ω 3 = ω 0 ± ω FSR = ω ± . Eq. (5.2) describes “degenerate” four wave mixing. Non-degenerate FWM also enables comb growth as it allows a photon from the pump mode to mix with photons from two other non-pump modes,therebyextendingthecombasymmetricallyaboutω 0 . Acombinationofdegen- erate and non-degenerate FWM cascades across the resonator modes, forming a fre- 135 quencycomb [27,78]. FIGURE 5.1 (a) Comb generation in a microcavity via degenerate FWM. TwophotonsfromapumpmodemixwithathirdfromanFSRaway,forming a fourth with the opposite energy spacing (b). Cavity dispersion plays a critical role in comb width, causing the FSR to change with wavelength, adverselyaffectingFWMefficiency(c). It’simportanttopointoutthatthefieldsinvolvedwithcascadedFWMincombgen- eration are not constrained in any way to be separated by just one FSR. In fact we see that combs often initially form on modes separated from the pump by±N FSR where N is an integer on the order of ten. With harder pumping, sub-combs are subsequently generated around the initial lines – with spacings less than N and eventually down to one FSR. Thus, with sufficient pumping, microcombs can extend across a large band- width; however they actually are composed of many different, somewhat independent subcombs, which plays a very important role in short term phase noise and is also inti- mately related to the time-domain notion of mode-locking in microcombs. Below we discusssomeofourworktowardsengineeringaWGMcavityinwhichcombswithini- tial spacingN = 1 preferentially form [49], which has been linked to low phase noise, coherentcombs[57]. As it is for MLL-based combs, cavity dispersion plays a critical role. In the fre- quency domain, dispersion can be conceptualized as a wavelength dependence of FSR 136 acrossthecombspan,shownschematicallyinFig.5.1(c). Dispersionisaccumulatedas the comb span grows, resulting in an eventual mis-match between energy conservation in FWM and the location of modes. This phenomenon, at least in part, limits comb breadth. In the time domain, frequency combs are pulses. Dispersion has the effect of spreading the pulses out, thus the goal is to engineer a cavity with compensated disper- sionsothatveryshortpulsescanbesustained–correspondingtolargebandwidthinthe frequencydomain. As a first-pass, microcombs are in practice formed simply by pumping larger amounts of optical power (∼10’s to 100’s of mW) than might be used for other WGM experiments. In a multimode resonator, it’s crucial to select a fundamental mode – the overall quality factor is also critical – since the nonlinear process at the heart of comb formation goes asQ 2 /V m . Typically, passive thermal locking [25] maintains the WGM frequencyatthepumplaserfrequency. However,belowwedescribeacombpumpedby a PDH locked laser. Thus, microcombs are pumped with a cw laser, and the output can be taken either from the through-port of the input coupler, i.e., as in observing a trans- mission spectrum, or by using a second evanescent coupler to pick off a small amount ofcirculatingcomblight. 5.1.2 Microcombsforfrequencymetrology Combshaverevolutionizedthefieldofopticalmetrologyandfrequencyreference. The fundamentals of comb use in these applications is briefly described here. The physical mechanismsandtechniquesthatwouldbeusedwithamicrocombaresimilaroridenti- 137 caltotheirusewithMLL-basedsynthesizers(comprehensivereviews,e.g.[30,31],can beseenformoredetails). Fig.5.2servesasaschematicreferenceforthediscussion. Combs primarily provide the function of optical frequency stability transfer and down-conversion. The latter is necessary to exploit the stability conferred by long- lifetimeatomictransitionswhichoccuratopticalfrequencies. Inorderforaclockbased on such frequencies to be useful, a microwave signal with equal stability to the optical interrogationfrequencymustbeextracted;combscanprovideaphase-coherent“clock- work” for this conversion – necessary because the optical frequency cannot be directly measured or used electronically. Enabling easy transfer of stability between two arbi- trary optical frequencies has been another consequence of practical importance. For example,ifalabhasasingleultra-stablelaser–referencedtoalarge,delicate,temper- aturestabilizedcavity–itsstabilitycanbetransferredtoanyotherlaserinthelabviaa combandtworelativelystraightforwardopticalphaselockedloops. Priortotheadvent of the frequency comb, either of these applications required incredibly long, complex, andresourceintensivefrequencyconversionchains[31]. For a comb to provide effective down-conversion or stability transfer duties, its two degrees of freedom must be stabilized. These are the repetition rate (ω rep /2π = f rep = FSR) and the carrier envelop offset (ω ceo /2π = f ceo ). Stabilization means that in the frequency domain, the spacing between successive lines (f rep ) is fixed, as well as the overall offset of the comb’s teeth from an integer multiple of the repetition rate f rep . A crucial insight in the development of combs was that the time domain slippage of the pulse-envelope phase with respect to the carrier phase is directly related to this 138 FIGURE 5.2 Combs consist of periodic lines separated byf rep and offset fromDCbyf ceo (Mod(mf rep )),whichisfixedviaanf−2f interferometer. Comparison to a cw laser yields a measurement of – or stabilization to – its opticalfrequency. overall offset of all the teeth (see for example the reviews by Udem, Holzwarth, and H¨ ansch [114] or Cundiff, Ye, and Hall [31]). Thus, a comb can be fully stabilized by stabilizing its two degrees of freedom: f rep and the carrier-envelope offset, f ceo . Moreover, whenf rep andf ceo are known, then the absolute frequency of all comb lines can be found through f = mf rep +f ceo , where m is the comb mode order, an integer approximatelyequaltotheopticalfrequencydividedbytherepetitionrate. The rep-rate (at least in principle) easy to stabilize. Some of the comb light is directed onto a fast photodetector, f rep is measured, and feedback to the appropriate physicalmechanism(e.g. cavitylength)isappliedtostabilizetherep-rateataset-point. MeasuringtheCEOturnsouttobemoredifficult. Neededisaphasecoherentcompari- sonoftwocombteethwhoseseparationspanssufficientcavitydispersion. Thisismost easily accomplished via an f − 2f interferometer. Shown in Fig. 5.2, this is achieved 139 byfrequencydoublinglightfromalongwavelengthtoothandbeatingitagainstatooth with very close to half the wavelength. The technique is called self-referencing – the beatnotefrequencyisdirectlyrelatedtotheoveralloffsetofthecombteeth,andifitcan bestabilizedalongwithf rep ,theentirecombisrigidlyfixedinfrequency. Readilyapparentisthatthisfixedcombhasthepowerfulabilitytoactasan“optical ruler.” Acwlasercanbecombinedwithcomblightyieldingabeatnoteatf beat , i.e. the differencebetweenitsfrequencyandtheclosestcombtooth(aswellashigherharmon- ics). If the cw laser has unknown frequency f u , it can be calculated precisely through f u =mf rep +f ceo ±f beat (theplusorminussigncaneasilybedeterminedbyadjusting the rep-rate). In real applications, the actual values of rep-rate and CEO are fixed via a phase-locked-loop to a local RF reference. This reference may be disciplined in turn to a local RF frequency standard such as a hydrogen maser or Cs clock, having stability and accuracy better than a part in 10 13 . Thus, if we can measuref beat to a similar level (easy with some averaging), we have determined an unknown optical frequency at the leveloftensofHz. The previous is an example of up-conversion; the optical clock enabling down- conversion happens similarly. But now it is the cw laser whose frequency is highly stable and accurate. The CEO is self referenced and locked to the local RF frequency standard. Then, f beat is fixed tightly via a phase locked loop, meaning that f rep is now determinedbytheopticalfrequency. Itsstabilityhasbeentransferredtotheentirecomb, andabeatnoteofanytwoteethhasaccuracyandstabilitypropertiesoftheopticaloscil- lator,frequencydown-convertedbymandoffsetbyf ceo . TheresultingRFbeatnotecan 140 thenbeamplifiedanddistributed,makinganopticalclockuseful. Thus, we see that the first requirement for comb use in frequency metrology is that physical mechanisms exist to adjust both the rep-rate and the CEO, and that the for- mer can be measured. Ability to measure the CEO motivates the need for an “octave- spanning” comb – so that an f − 2f interferometer can be operated. Other types of interferometer schemes exist, e.g. 2f-3f, however an outstanding challenge in either case is obtaining sufficient optical power from a comb line to double in a nonlinear crystal. Nonetheless, increasing the bandwidth of the comb has been the subject of considerableeffort. 5.1.3 Stabilizingthecomb In order for a microcomb to serve as a direct stand-in for MLL-based combs, both the repetitionrateandcarrierenvelopeoffsetmustbeindependentlyadjustable(tosaynoth- ing of measuring the CEO). Since the rep-rate is intimately linked to the inverse cavity round-trip time, it can be adjusted through any mechanism which changes the optical pathlength. Two such examples have been demonstrated: thermal tuning via thermore- fraction and expansion [32], and direct mechanical deformation with a piezoelectric actuator[88]. Bothoftheworksreportagoodresult,withf rep stabilitiesattheparts-per 10 14 level. Bothpapersalsoclaimindependentcontrolofrepetitionrateandthefrequencyofa givenmode(thepumpmode). Theformerisadjustedviapumppower[32]andmechan- ical actuation [88], while the latter is adjusted by changing the pump frequency. How- 141 ever, it’s difficult to see how the two parameters could be independent at times scales longer than the thermal relaxation time of the resonator since the absolute WGM fre- quencyandFSRshoulddependequivalentlyuponopticalpathlength. Ref.[88]touches uponthisbriefly,butonlytosaythatthetwoactuationmethodshavedifferentfrequency responsebandwidths. As mentioned, extending the breadth of microcombs to allow for self-referencing has attracted considerable effort. And while octave-spanning combs have been demon- strated,atthetimeofwriting,wearenotawareofacaseinwhichamicrocomb’scarrier envelopeoffsethasbeenmeasuredorstabilized. 5.1.4 Coherenceandnoise In addition to the requirement that a comb can be self-referenced, there must also be a fixed phase and frequency relationship between each comb tooth. Another way to say this is that f rep measured at any point of the comb span is exactly the same. This was assumed in the above discussion of stabilizing the rep-rate, but in microcombs it turnsouttobenon-trivialandhasinfactbeenthefocusofconsiderablerecentresearch efforts[49,57,79]. “Coherent comb” is the name given to combs with this fixed phase relationship between all teeth. The characteristic can also be understood in terms of the output pulses: if the tooth spacing is slightly different for different portions of the comb, the output pulse train will contain components with slightly different values of f rep – the pulses will not all line up with each other. If the rep-rate from such a comb is viewed 142 on an RF spectrum analyzer (RFSA), the beatnote will be broad and/or noisy. This type of incoherent comb is in fact several smaller subcombs which form during the cascaded four wave mixing process. Cavity dispersion causes the FSR to be different across the comb span, thus there is a slight relative offset between subcombs. It has beenshown[57]thattherearetwodifferentregimesofcombs,withthecoherentvariety havingthefirstsidebandappear1FSRfromthecarrierandtheincoherentcombshaving thefirstsidebandappearatsomemultipleN > 1oftheFSR. 5.2 Engineeredcavityfornativecomb We have demonstrated experimentally that it is possible to engineer the ratio of cavity dispersion and linewidth in order to control whether a microcomb forms with the first sidebands appearing at N = 1 FSR from the pump, or N > 1 [49]. From the theory in [57], we expect the comb with N = 1 – dubbed a “native comb” – to be a coherent one, and that the two regimes N = 1 or N > 1 are determined by the relationship between cavity linewidth,δf, and dispersion,D 2 , with a native comb arising under the condition √ δf/D 2 ≈ 1. Here,cavitydispersionisexpressedasthedifferencebetween successiveFSRsviaD 2 = (f l+1 −f l )−(f l −f l−1 ),wheref l isthefrequencyofaWGM withangularmodenumberl. Thus,wecancomputetheconditionsunderwhichwewouldexpectanativemicro- comb. The expression from [45], along with a Sellmeier equation calculation for mate- rialdispersion,wasusedforcalculatingD 2 inanoblatespheroid. Accountingforakind 143 of geometric dispersion, this calculation yields a constraint on the quality factor that must be achieved for a given size of resonator to meet the condition √ δf/D 2 ≈ 1. As we see in Fig. 5.3, for a microresonator with a 2 mm radius the mode linewidth would have to be about 5 kHz corresponding to Q = 4× 10 10 : about an order of magnitude higher than any reported for MgF 2 . On the other hand, for a smaller resonator with 200 µm radius, the required Q is a much more easily attainable 2×10 8 . Therefore, we fabricatedtworesonators,onewithalargerdiameterbutnotashighaQassuggestedby the plot, and a small cavity which falls very close to the curve; these two are indicated bypointsinFig.5.3. FIGURE 5.3 Microcomb dispersion parameter D 2 vs cavity radius. The curve gives requirements of quality factor in order to achieve √ δf/D 2 ≈ 1 foranativecomb. 5.2.1 Fabricationandtesting The microcombs were made from z-cut MgF 2 , with the larger cavity being fabricated “by hand,” and the smaller one shaped initially with a diamond turning process, then polished by hand to achieve lower surface roughness. The large cavity had a major radius of 1.9 mm and a minor radius – defined by a sharp edge shaped onto the disc 144 – of about 60 µm. The small resonator consisted of a waveguide-like protrusion on a MgF 2 cylinder, designed along the lines of [97] in order to only support one high-Q, wellcoupled modeper FSR perpolarization. Initially, the waveguidewasatrapezoidal shape from the computer-controlled diamond turning process. After subsequent hand polishing with diamond slurries, it took on a nearly Gaussian profile, with approximate dimensions of 20 µm across and 6 µm high. Photographs taken during the diamond turning process are shown in Fig. 5.4. The completed waveguide cavity is seen in the insettoFig.5.6. FIGURE 5.4 The smaller microcomb cavity is initially formed via a computer-controlled diamond turning technique. The diamond tool and cylinderpreformfromthetop(a)andend-on(b). Forcharacterizingandtestingthemicrocombs,afiberlaser(KoherasAdjustik)oper- ating at 1560.3 nm was used, along with an erbium doped fiber amplifier (EDFA) to increase the available pump power. Light was coupled into the cavity with an angle- polished fiber, and an additional fiber coupler was positioned in the opposite rotational orientation in order to couple the generated comb light out, for observation on an opti- cal spectrum analyzer. As designed, the smaller cavity had a greatly limited spectrum, supporting only a few modes-per-FSR. It was intentionally designed to not be com- 145 pletely single mode, however, since we found a trade-off between higher-order mode suppression and overall loss to the bulk material cylinder. Its unloaded quality factor was 1.92×10 8 . The larger resonator had many modes, and the spectrum shown below wasgeneratedbypumpingonewithQ = 1.9×10 9 . 5.2.2 Results Based on the combs generated from these two cavities, the relationship between cavity linewidth and dispersion indeed plays an important role in determining whether or not thecombwillbeanativeone. AsshowninFig.5.5,combformationinthelargecavity begins first with N = 19. As the pump power is increased, secondary combs, again with N > 1 begin to form around the initial lines. Eventually, cascaded FWM fills in all cavity modes and the amplitude of all the lines increases, forming a broad comb with lines spaced by 1 FSR. But the formation dynamics reveal the existence of many overlappingsub-combsassociatedwithanoisy,incoherentcomb. In the small cavity, the first comb lines to appear are separated from the pump by N = 1FSR(Fig.5.6(a)). Aspumppowerincreases,thecombgrowssmoothlythrough the third comb tooth, then suddenly jumps to the much broader spectrum shown in Fig.5.6(b). Themeasuredportionofthisfinalcombisover200nmwide,butthecomb may in fact be broader, since the measurement is limited at longer wavelengths by the rangeandnoise-flooroftheOSA.Thepumppoweratthispointisabout50mW,making this one of the broadest combs for such a low pump power to date. The device has an FSRof172GHz. 146 FIGURE 5.5 Thelargercavitydoesnotmeettheultrahigh-Qrequirement indicated in Fig. 5.3. The first sidebands appear atN = 19, then secondary combs form, filling in the spectrum (A-D). The presence of many overlap- pingsub-combsisassociatedwithanoisyRFbeatnote. 5.2.3 Conclusion We have demonstrated microcombs from two cavities with different ratios of linewidth to dispersion. The smaller of these two meets a requirement for generation of a native comb, √ δf/D 2 ≈ 1, and we find that indeed the first comb lines to be excitedare sep- arated from the pump by 1 FSR. This finding supports the approach of engineering the shapeofacrystallinecavitysothatacombinationofgeometricandmaterialdispersion yield a microcomb with the desired properties. In this case, a native comb is expected to be coherent and therefore generate a very quiet microwave beatnote, though we note that, with a 172 GHz FSR, we did not have a fast enough detector to directly confirm thislatterclaim. 147 FIGURE 5.6 The smaller resonator with engineered spectrum supports a native comb. The first lines appear at a spacing of N = 1 FSR from the pump, grow smoothly through the third sideband (a), then the comb jumps to the broad spectrum shown in (b). Inset (b): completed cavity showing waveguidestructure. 5.3 MicrocombwithPDHlockedpumplaser We performed additional microcomb investigations by pumping the large, multimode cavityusedforthedual-modetemperaturestabilizationexperiment(§3.1). Althoughnot specifically designed for this application, the setup facilitated observation of a micro- comb when the pump laser is actively locked to the cavity. In particular, we were interested to know: will the laser remain locked once a strong nonlinear process like combgenerationsetsin;whatstabilitycanbeachievedatoptical(fromthelockedlaser) and microwave (from the rep-rate beatnote) frequencies; and what if any utility could a 148 microcombinthisconfigurationhaveforfrequencydown-conversion. The setup for observation of the microcomb is shown in Fig. 5.7. The dual-mode experiment was modified by adding a 30/70 non-polarizing beamsplitter to the extraor- dinary port. In this way, 30% of the optical power is split off and launched into a single-mode fiber, while the rest is directed to the detector for mode observation and PDH locking. The fiber-launched portion can then directed to a fast photodetector for observing the RF beatnote, or into an OSA for measuring the comb spectrum. Out- put pump power from the laser is between 40 and 50 mW; after the EOM, splitter, and polarization control optics, power at input to the coupling fiber is roughly 5 to 10 mW. Mostofthecavity’s30+modesperFSRarehighorderandthereforecannotbeusedfor efficient comb generation owing to the large mode volume. A mode with appreciable thermal nonlinearity was found by looking for a mode shape that is asymmetric with respect to laser scan direction. Note that this mode – in which a comb could easily be generated – had coupling contrast about 10× less than some other modes which could not be pumped to a comb, suggesting that the increase inV m for higher order modes is verysignificant(i.e.,morethananorderofmagnitude). 5.3.1 PDHlockingofthepump The Pound-Drever-Hall lock is implemented identically to other experiments in this work,excepttheloopgainmustbedecreased. Additionally,wefindthattheinput-offset bias (a small DC voltage that is added to the error signal to zero it) must be adjusted considerably. This is thought to be because if the pump is stabilized to the center of 149 FIGURE 5.7 Microcomb with a PDH locked pump laser. Modified from the dual-mode experiment, the WGM cavity is under vacuum and temper- ature stabilized. After launching the free-space comb into fiber, it can be viewedonan(OSA)orfastphotodetector(PD). the mode, the resulting strong thermal bi-stability will “fight” the PDH lock, forcing it to oscillate wildly and come unlocked. In other words, the pump is locked at some frequency detuning such that parametric FWM can occur but thermal nonlinearities are not so great as to destabilized the lock. A spectrum of the resulting microcomb can be seeninFig.5.8. TheFSRofthecavityis10.5807GHz. We find that the PDH input-offset bias has a strong influence not only on the exis- tence of a comb and stable locking, but also upon the noise and qualitative proper- ties of the RF rep-rate beatnote. By adjusting said offset, three distinct regimes can be observed: (1) the microwave beatnote consists of several different frequencies and shows up on the RFSA as 3-5 discreet (but noisy) peaks; (2) the beatnote consists pri- marilyofonefrequency,butitisverynoisy;and(3)thebeatnotecomprisesasinglefre- quency and demonstrates a very quiet, resolution-bandwidth-limited linewidth of about 150 1 kHz. Conditions (2) and (3) can be accessed continuously (without breaking lock) by adjusting the input-offset. Condition (1) happens more rarely and was discontinu- ous from the other two, perhaps because the sign of detuning is opposite (i.e. pump laser locked to the other side of the mode). Fig 5.9 shows photographs of the spectrum analyzertraceforthesethreecases. 1558 1558.5 1559 1559.5 1560 1560.5 1561 1561.5 1562 −80 −60 −40 −20 λ [nm] Amplitude [dBm] 61.6 GHz/div FIGURE 5.8 Optical spectrum analyzer trace of the comb pumped in a 3.25 mm MgF 2 resonator. Although the comb is not broad, the RF beatnote canbemadeveryquietbyadjustingtheinput-offsettothepumpsPDHlock Observation and intentional selection of the three different regimes was repeatable. Moreover, regime (3) was very stable, and continuous pumping of the comb and obser- vation of the quiet beatnote could take place over periods of several hours without any intervention. This result is very interesting because it suggests that the coherence of the comb could be actively controlled by locking the pump to a given offset. This is consistent with other recent work [56], and suggests that the physical mechanism can be thought of in terms of mode locking: for certain pump detuning parameters, soli- tons within the WGM cavity become “mode-locked” and pulses are emitted at the FSR frequency. Pulses may also be emitted more frequently and randomly, corresponding 151 to a noisy, incoherent comb. There may also be some regime where pulses are emitted at integer multiples of the rep-rate, analogous to harmonic mode locking in traditional MLLs. At the time of this comb experiment, we did not have an autocorrelator avail- able with sufficient sensitivity to measure the low-power pulses from this comb, nor any technique to make time-domain measurements such as FROG (frequency resolved opticalgating). FIGURE 5.9 ScreencapturesofanRFSAshowingthreedifferentregimes of microwave beatnote. (1) Several different, discrete beatnotes exist, rep- resenting sub-combs with unequal f rep . (2) Most of the RF power is con- centrated at one frequency, and this noisy beatnote can be continuously tuned (2 => 3) from noisy to quiet (3) as the input offset is adjusted. (1): 32kHz/div,logverticalscale;(2)-(3): 20kHz/div;linearverticalscale. 152 5.3.2 Stabilityofopticaldown-conversion Because of the long periods over which it would stay locked, a measurement of the frequency down-conversion stability of the PDH-locked pump microcomb system was appealing. In the case of “traditional” MLL-based combs, f rep will be stabilized to an opticalfrequencyreferenceviaatightphase-lock. Thus,thefrequencydown-conversion is phase coherent; the resulting microwave frequency will demonstrate the same phase and frequency stability as the optical reference (f opt ) divided by the tooth number m ∼ f opt /f rep . Although we cannot expect such a phase-coherent conversion, we are interestedintheextenttowhichtheopticalfrequencyofthelockedlaserandthemicro- comb’sRFbeatnotewillvarytogetherwithafixedratio. Thismeasurementandresults aredescribedbelow. To first order, FSR and WGM mode frequency should depend equivalently upon changes in resonator temperature (via thermorefraction or thermal expansion) or other strain (for example from physical compression of the resonator). Thus, following logic similar to that laid out in [80], the absolute optical frequency should be related rigidly totherep-ratevia f opt =lf rep +∆f (5.3) wherel is the angular mode number and ∆f is a small offset resulting from dispersion andotherdestabilizingeffects. Forexample,thepulsecirculatingintheresonatorcould experiencesomeKerrnonlinearitywhichaffectsitsround-triptime(f rep )incommensu- ratelywith theaverageoptical pathlength aroundthe resonator –which determinesthe 153 WGM frequency and thus that of the locked laser. From this notion we expect ∆f to remainsmall(orfluctuatearoundsomefixedvalue,i.e.,averagedown). AsshowninFig.5.7,bothf rep andf opt canbesimultaneouslymonitoredbyasingle frequency counter (this counter shares a reference with the fs frequency comb so any long-term instabilities should be common-mode). We may thus determine the relative offset between the two, ∆f – and more importantly the excursions of this offset over time. The raw, time domain excursions of the two frequencies are shown in Fig. 5.10. Eachdatasethasbeennormalizedbysubtractingoffthemeanvalueofthefirstfewdata points. 0 200 400 600 800 1000 −100 −50 0 50 100 150 200 Time [s] Excursions in f rep [Hz] −2 −1 0 1 2 3 4 Excursions in f opt [MHz] FIGURE 5.10 By simultaneously monitoring the microwavebeatnote fre- quency and absolute optical frequency (as determined from comparison to the fs comb), the down-conversion stability can be determined. Here, raw measurement data. Note that the comb is operating in the “quiet” regime as describedintext. Clearly, drifts in both frequencies are highly correlated. But to characterize the down-conversionstabilityinafamiliarway,wemakeanAllandeviationplotasfollows. Thevalueofl iscalculatedandthecurvesarenormalizedtoeachotheratthebeginning ofthedatasetbyassuming∆f = 0andtakingl = Floor(f opt /f rep ). Usingthereading 154 from the OSA (Fig. 5.8) to calculate f opt = c/1559.9 nm and the reading from the RFSA forf rep = 10.581 GHz yieldsl = 18,158. Obviously there is some error in this calculation, however we find the resulting stability analysis is relatively insensitive to valuesofl inarange±100. Wethenfindatimeseriesofexcursionsin∆f through ∆(∆f) = ∆f opt −l∆f rep (5.4) where ∆f opt and ∆f rep are the excursions shown in Fig. 5.10. Note that here we are calculating the instability in ∆f at optical frequency, i.e., to get the Allan deviation (showninFig.5.11)theresultof(5.4)isdividedby192.13THz. Howeveranequivalent fractionalinstabilityisreachedbyfirstdividingtheopticalfrequenciesin(5.4)byl,then thatresultbyf rep . Resultsofthisdown-conversionstabilitymeasurementareshowninFig.5.11,along withfractionalinstabilitiesfortherep-rateandopticalfrequencies. Forτ < 10s,thesta- bilityof∆f islimitedbythestand-alonestabilityoftheRFbeatnote. Thisisconsistent with the assertion that the down-conversion is not phase-coherent, however a slightly lower than desired SNR on the microwave beatnote could also contribute to this mea- surementlimit. Nonetheless,wenotethatthemicrowavestabilityfor100ms .τ . 1s is more than two orders of magnitude better that similar work on microcomb stabiliza- tion [32,88]. This could be due to the high environmental temperature stability inside thevacuumchamber,and/ortheRFnoisereductionaffordedbyPDHlockingtoagiven offset. 155 10 −2 10 −1 10 0 10 1 10 2 10 −11 10 −10 10 −9 10 −8 τ [s] Fractional Instability σ (τ) Δ(Δ f ) / f opt Δ f opt / f opt Δ f rep / f rep Time [s] Δ(Δ f ) [MHz] 0 500 1000 −0.5 0 0.5 1 1.5 FIGURE 5.11 Allan deviation representing the stability of down- conversion from optical (PDH-locked pump laser) to microwave frequency, i.e., the extent to which the two frequencies maintain a fixed ratio l. Also shown are the absolute stabilities of optical and microwave frequen- cies. Inset: time-domain data as calculated by (5.4). Black dashed line: 5×10 −11 τ 1/2 . Atabout10s,thestabilitiescrossover,andweseeevidenceofthedown-conversion stability. Roughly speaking, the instability of ∆f is 5×10 −11 τ 1/2 for τ > 10 s, indi- cating a random walk at these time scales. Although the coefficient of this slope is respectablebycomparisontocrystal-basedfrequencyreferences,therandomwalkchar- acteristicsuggeststhepresenceofaphysicalmechanismdestabilizingtheratiof opt /f rep . One possibility is a drift of lockpoint (input-offset bias) in the analog lock-box. How- ever, noting that the value of σ at 100 s would correspond to about 100 kHz in optical instability–approximatelytheWGMlinewidth–suggeststhatthesystemwouldcome out of lock if the bias drifted this far. Another possibility is that, for a comb generated 156 by an actively locked pump, a small drift in input-offset changes the microwave beat frequency incommensurately through the Kerr or thermal affects; based on the previ- ous section and other work [56], it’s not surprising that small changes in offset could considerably change f rep . Other researchers have reported frequency down-conversion from a microcomb at the parts in 10 13 level [96], however after much longer averaging timesand foran injection(rather than PDH)lockedlaser. Investigationof theobserved down-conversioninstabilityisanareaofongoingwork. 5.4 Summary Frequency comb generation from a whispering gallery mode resonator is described, along with how such combs are used for frequency stability transfer and down- conversion. Itisshownthatthegeometricdimensionsofamicrocombcanbeengineered such that the ratio of cavity dispersion to resonance linewidth is close to unity, and that the resulting comb is “native.” A microcomb generated by PDH locking of the pump lasertoamulti-modecavitypresented. Theresultingmicrowavebeatnotedemonstrates three distinct, qualitative regions which can be accessed via changing the input-offset biasinthepumplock. 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Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detec- tion. J.Opt.Soc.Am.B2,9(Sep1985),1527–1533. 168 AppendixA FabricationofWGMresonators A.1 WGMR BasicfabricationofacrystallinewhisperinggallerymoderesonatorisshowninFig.A.1 anddescribedasfollows. Awafer/windowofhighqualitymaterial–usually“excimer” grade–isobtained. Forwellpolishedresonators,thequalityfactorbecomeslimitedby materialabsorption, thususingacrystalofutmostpurityisparamount. Next, thewafer ismountedonapieceofaluminumandacylindricalpreformiscutusingacommercially available diamond-coated hole saw, or a home-made brass device and diamond slurry (panel(a)inFig.A.1). Next,thepreformismountedonabrasspost,eitherwithmount- ingwaxorhigh-densitythermalepoxy. Then,thepreformcanbepolishedimmediately withdiamondlappingfilmsandslurries,orajigcanbesetuptoholdapieceoflapping film firmly (usually using an x-y stage) in order to make the preform concentric. This can be seen in Fig. A.1(b) where a thin cylinder (∼3 mm diameter) of MgF 2 is almost in contact with lapping film taped to a microscope slide. The resonator/post assembly is mounted in an air-bearing lathe, and a stereoscope used to view the resonator during polishing,showninFig.A.1(c). Successively finer grits of diamond lapping film are used to polish and shape the resonator. For the larger, multimode resonators described in this work, shaping was 169 FIGURE A.1 Basic WGM fabrication steps. (a) A cylindrical preform of high-quality optical crystal is cut on a mill or drill-press. (b) This preform is then mounted on a brass post for polishing and shaping. Here, a piece of diamond film is held against the preform, making it concentric. (c) All polishing and cleaning is done on an air-bearing lathe, with microscope. (d) Acompletedmultimoderesonator. typicallydonewithapieceof6µmfilm,handheldattheproperangletoforma“sharp” ridge around the center of the disc. After shaping, finer grits of lapping film – down to 1µm – were used, followed by diamond suspension slurries ranging in size from 1µm downto250nmor100nm,dependingonhowhighaQisdesired. Afinishedresonator ofthistypeisshowninFig.A.1(d). Finally, when abrasive polishing is complete, the disc must be cleaned very care- fully with isopropanol. It is imperative that the alcohol be free of water, oil, and other 170 contaminants, as even a small amount of impurities on the disc surface can degrade the Q. 171 AppendixB Detailsandschematicsoflaserlock electronics B.1 Photodetector For the second half of this work, we used a photodetector consisting of a homemade transimpedanceamplifierandInGaAsphotodiode. Thetransimpedanceampisbasedon acircuitdesignbyPhilipHobbs[59]. Hecallsitthe“bootstrappedcascode,”whichuses twotransistorstopartiallycircumventthetypicaltrade-offbetweengainandbandwidth in amplifier design. The opamp used in the original design is no longer made; our modificationstothecircuitinclude±5Vregulatorssothatasuitablesubstituteopamp can be used – but one that needs 5 V power supply. The circuit was then laid out and printed to a PCB, then packaged in a 1”×1”×2” enclosure with an SMA output connector. TheschematicisshownbelowandphotographsareshowninFigB.1. B.2 Laserlockelectronics Several versions of lockbox were built and optimized during this work. Initially, lock- boxes contained both the RF portion of the PDH lock (modulation frequency genera- 172 FIGURE B.1 Photographs of the lock electronics used in this work. (a) Thecompletedphotodetectorand(b)withtopremoved. (c)Thetwo-channel PDH error signal generation board is mounted, together with two PI servos andapowersupply,inarack-mountenclosureforthedual-modeexperiment. (d)Thefinalversionofthepiezo+AOM+amplitudestabilizationlockbox. tor, mixer, phase shifter, and low-pass filter) and the PID servo. This type of box was usedthroughoutthedual-modeexperiment,culminatingwithatwochannel,modulation demodulationprintedcircuitboard(PCB).Thiswasusedtosimultaneouslygeneratethe errorsignalforfrequencylockingofthelaser,andtemperaturelockingoftheresonator. Subsequently, RF generation of the PDH error signal was accomplished with dis- creet components and a separate signal generator. This yielded a less noisy error signal because the oscillator chip used in the original versions produced a square wave, and it was difficult to effectively filter out the resulting high harmonic content. Moreover, it wasmuchmoreconvenienttohaveamodulationfrequencythatiseasilyadjustable. The next four pages show schematics of the aforementioned lock electronics in the 173 following order. (1) Transimpedance amplified photodetector based on Hobbs design. (2) Two channel PDH error signal modulation/demodulation board used for dual-mode experiment. (3) Parts list for (2). (4) Lockboxes used to stabilize the laser frequency via AOM and piezo feedback. They are designed to work with Isomet VCOs, and pro- videthenecessarytuningvoltagebiasesand“MOD”output–theTTLsignalnecessary to turn the RF on. Both circuits provide for stabilizing the amplitude of the laser by incorporating an auxiliary detector. One of the circuits can also stabilized the RAM withanotherauxiliarydetectorandmixer(5). Notethatthepiezocorrectionoutputhas a relative negative sign between the two boxes, owing to the fact that one Koheras has oppositepiezotuningtotheother. 174 !" # " # " # $ %&'() *+"", &- * *, . /*0 1 2 *3 . /*041 2 *3 # *. * 1 2 * *. *0 1 2 * # *. / 1 2 * *. * 1 2 * *. * #12 * 4 *. * #12 * #4# *. * 12 *# ## *. *412 *4 #44 5 2 6 7 1 )* 45 2 6 7 4)* 40#5 2 6 7 40#)* 5 2 6 7 )* 5 / & 7 $* 85 / & 7 $* +/3. # & 2 " 4#4 !" &*//.555 5/ 6 # 9 .5 '// &76(: 7 ; ( #""$ '9 < 6 # 4 $)+*9/<)+1 6 **& AppendixC Short-termstabilityoftheMenlo frequencycomb C.1 Measurement A medium finesse (F ≈ 15,000) FP cavity was assembled using a ULE spacer and placed in a vacuum chamber with heat shields. A fiber laser + AOM fast frequency shifter identical to the one previously described was locked to the cavity. Locking to the error signal noise density on the order of 1 Hz/ √ Hz is expected from this system as well. It’s not known what the contribution from thermal noise in the mirror coatings and cavity spacer will be, though it’s expected to be below the phase noise of the fs frequencycomb,whichisthegoalofthismeasurement. AsmallportionofthelightfromtheFPstabilizedlaserispickedoffandbeatagainst acombtooth. Theresultingbeatnoteiscountedwithgatetimesof10ms. Themeasure- ment is repeated for several different values of gain on the comb’s rep-rate PI stabiliza- tion loop. This gain is significant because, if too high, phase noise from the hydrogen maser to which the comb is locked will be up-converted to the optical domain, further broadeningtheintrinsicallybroadlinewidthofthepulsedfiberlaser. ResultsareshowninFig.C.1alongwithlinesrepresentingtwodifferentσ-τ slopes. 180 10 −2 10 −1 10 0 10 1 10 −13 10 −12 10 −11 10 −10 τ [s] σ(τ) [Δ f/f] G 1 Raw G 1 ; 23 Hz/s removed G 2 Raw G 2 ; 66 Hz/s removed 2× 10 −13 τ −1/2 4× 10 −13 τ −1 FIGUREC.1 BeatnotecomparisonoftheFP-stabilizedlaserandthecomb whichisstabilizedtothehydrogenmaser. AllanDeviationisshownfortwo differentvaluesofcombrep-ratelockgain(G 1 andG 2 ). Alsoshownarethe datawithlineardriftsremoved,andtwolinesrepresentingtheapproximated combstability. These lines give a visual indication of the minimum expected instability from, in prin- ciple,eitherthecwlaserorthecomb. However,becausetheAllandeviationforτ <1s isverysimilartopreviousresultswithaWGM-stabilizedlaser(see§3.1.5andFig.3.7), wehypothesizethatthemeasurementislimitedatthistimescalebycombphasenoise. Tofurtheranalyzetheshort-termstabilityofthecomb,itisusefultoemployastabil- ity analysis which can distinguish between white and flicker phase modulation (unlike “regular” Allan deviation, in which they both have slope τ −1 on a σ-τ plot). We show such an analysis below, where the Modified Allan Deviation (capable of distinguishing thesenoises[91])isplotted. Basedonthesemeasurementsandthehypothesisthatforτ <4s,stabilityislimited 181 10 −2 10 −1 10 0 10 1 10 −13 10 −12 10 −11 τ [s] σ(τ) [Δ f/f] G 2 ; 66 Hz/s removed 3×10 −14 τ −3/2 1×10 −13 τ −1 FIGURE C.2 Modified Allan Deviation plots of one of the comb vs FP- stabilizedlasercomparisonsshownabove. Alsoshownarelinesrepresenting two different colors of phase noise: white phase noise (∼ τ −3/2 ) and flicker phasenoise(∼τ −1 ). by comb phase noise, we can conclude that the measurement floor for comparisons between a cw laser and the H-maser disciplined comb is≈4×10 −13 τ −1 for 10 ms < τ <4sand≈2×10 −13 τ −1/2 for4s<τ <10,000s. Theassumptionforτ >4sisthat the comb continues averaging down as τ −1/2 from it’s value at 4 s, which likely over estimates the noise. It’s probable that in fact it averages down faster to meet with the maserstability,afterwhichtimeitfollowsthelatterto10,000s(andbeyond). Aplotof atypicalmaserstabilityisshownonthefollowingpage. 182
Abstract (if available)
Abstract
This dissertation describes an investigation into the use of whispering gallery mode (WGM) resonators for applications towards frequency reference and metrology. Laser stabilization and the measurement of optical frequencies have enabled myriad technologies of both academic and commercial interest. A technology which seems to span both motivations is optical atomic clocks. These devices are virtually unimaginable without the ultra stable lasers plus frequency measurement and down-conversion afforded by Fabry Pérot (FP) cavities and model-locked laser combs, respectively. However, WGM resonators can potentially perform both of these tasks while having the distinct advantages of compactness and simplicity. This work represents progress towards understanding and mitigating the performance limitations of WGM cavities for such applications. ❧ A system for laser frequency stabilization to a the cavity via the Pound-Drever-Hall (PDH) method is described. While the laser lock itself is found to perform at the level of several parts in 10¹⁵, a variety of fundamental and technical mechanisms destabilize the WGM frequency itself. ❧ Owing to the relatively large thermal expansion coefficients in optical crystals, environmental temperature drifts set the stability limit at time scales greater than the thermal relaxation time of the crystal. Uncompensated, these drifts pull WGM frequencies about 3 orders of magnitude more than they would in an FP cavity. Thus, two temperature compensation schemes are developed. An active scheme measures and stabilizes the mode volume temperature to the level of several nK, reducing the effective temperature coefficient of the resonator to 1.7 × 10⁻⁷ K⁻¹
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Baumgartel, Lukas
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Whispering gallery mode resonators for frequency metrology applications
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Doctor of Philosophy
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Physics
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11/06/2013
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frequency down-conversion,frequency reference cavity,laser stabilization,micro resonator,microcomb,OAI-PMH Harvest,temperature coefficient,temperature stabilization,thermal compensation,whispering gallery mode resonator
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frequency down-conversion
frequency reference cavity
laser stabilization
micro resonator
microcomb
temperature coefficient
temperature stabilization
thermal compensation
whispering gallery mode resonator