Optical Resonators With Whispering-Gallery applications(2)

whereTdescribestheamplitudetransmission,γ,γc,andω0aretheabsorptionandcouplinglinewidth,andresonancefrequencyofamodeoftheresonator,respectively[weassumethat|ω ω0|ismuchlessthanthecavityfreespectralrange(FSR)].Thepowertransmission|T|2throughtheresonatorisLorentzian.Conditionγ=γccorrespondstocriticalcouplingoftheresonator[12],[13].

The lterdescribedby(1)isastop-band lterbecauseitischaracterizedbytheabsorptionresonance.AWGRwithtwoinputandoutputcouplersischaracterizedbyatransmissionres-onance.Thisisanexampleofapassband lter.Thetransmissionandre ectioncoef cientsthroughtheresonatorare

T=

γc

γω ω,R=i(ω ω0)i(ω ω(2)c+i(0)γc+0)

whereTandRdescribetheamplitudetransmission(lightgoesintoonecouplerandexitstheothercoupler)andre ection(lightgoesandexitsthesamecoupler),respectively,andγc γisassumedforsimplicity.

Singlering-shapedWGR-based lterswerestudiedin[14]–[19],seealso[8],[20]forareview.Anall-opticalpassivefour-portsystemincludingafusedsilicamicrosphereandtwota-pered berswasusedasachanneladding–droppingdevice[21].The lterresponseofsingle-ringresonatorswithintegratedsemiconductoropticalampli ersbasedonGaInAsP–InPispre-sentedin[22].Achanneldropping lterbasedonadielectricmi-crosphereintegratedtoasiliconphotodiodewasstudiedin[23].Twodielectricwaveguidesthatareevanescentlycoupledtoafewmicronsizedsquareorrectangularregionofincreasedrefractiveindexcanserveasaverycompactintegratedopticalmicroresonator,similartoaringWGR.Applicationsofthedevicefor lteringarediscussedin[24].

Unfortunately,theLorentzianlineshapeofthe lterfunctionassociatedwithasinglemicroresonatorrepresentsalimitationforitsapplicationinmanysystemsthatrequirelargesidemoderejection,inadditiontoanarrowbandpassandalargetuningrange.

B.High-OrderFilters

Cascadedresonators,suchascoupledoptical berresonators,arewidelyusedasopticalandphotonic lters[25],[26].WGRsoffernewpossibilitiesformultipole lteringbecauseoftheirsmallsize,lowlosses,andintegrabilityintoopticalnetworks.Multipole ltersbasedoncascadedintegratedmicroringresonatorsfabricatedwithsilicahavebeendemonstratedincompactandrobustpackages.The ltershave10–100GHzbandwidthsandcorrespondingopticalQsontheorderof105–104[15],[27]–[31],andare,infact,commerciallyavailable.These ltersprovidepassbandswith attopsandsharpskirts,suitableforhighperformanceapplications,especiallyinopticalWDM.Asecond-orderoptical lterwithaMHzbandwidthwasrealizedwithtwocoupledhigh-Q(108)microsphereresonators,oneofwhichwastunable.ThetunableWGRwasmadeofgermanateglass[32].

Atunablethree-resonator ltermadeofLiNbO3WGRswasdemonstratedin[33].The lterhasthefollowingdistinctivefeatures/advantagesoverotherWGM lters:1)agiletunabilityaccompaniedbyahigh-order lterfunction;2)narrowlinewidth(≤20MHz);and3)low ber-to- berloss.Acombinationofthethreefeaturesmakesthis lterauniquedeviceforawiderangeofapplicationsinoptics.Sincethemicrowavesignalsinphotonicsystemsaresidebandsofanopticalcarrier,these lters,inprinciple,canbeusedatanymicrowavefrequency,providingthesamecharacteristicsthroughouttheband,from1to100GHzandhigher.

Multiresonator ltershavesigni cantlymoresparsespec-tracomparedwithastandaloneWGMresonator.ThisfeatureisduetothesocalledVerniereffect[34],andissimilartothefeatureobservedincoupled ber-ringresonators[25],[26]whicharenotedforararespectrum.Anef cient nesseofsuchmultiresonatorsystems,introducedasaratiooffrequencydif-ferencebetweenthetransmissionbandsofthe lterdividedoverthefrequencywidthofaband,isverylarge;e.g.,theFSRofthe lterreportedin[33]exceedsoneterahertz.Polymerdoublemicroring lterswiththermoopticaswellaselectrooptictuningweredemonstratedandreportedin[35].

Itwasnotedthatoneofseveraladvantagesofusingcou-pledmicroringsfor lteringisthepossibilityofahightuningenhancementfactorMgivenby

M=

1

1 a(3)2/a1

wherea1anda2aretheradiiofthetworings.Thetuningrangeofthedoublemicroring lterisMtimesthetuningrangeofasinglering[36].SinceMcanbeverylargeindouble-WGRstructures,thefasterbutmuchsmallerelectro-opticeffectcanbeusedfortuning.Itisimportanttonotethatthethederivationoftheparameterisvalidfortheresonatorswithsucharadiusratio,andthatMstayslessthanthe nesseoftheresonators.EachringresonatorhasasetoftransmissionspectraandthewavelengthperioddeterminedbyitsFSRfFSR.Thetworingshaveslightlydifferentradii(oreffectiveindices).Therefore,thetwosetsoftransmissionpeakcombshavesmalldifferentpeakspacing.Wavelengthtuningisachievedbyaligningthepeaksinthetwosetsofcombswiththeadjustmentofindexinoneorbothringresonators.The ltertransmissiondiscreetlyjumpsfromtransmissionatwavelengthf0tof0+fFSR,withMtimeslessvoltageappliedtooneoftheringscomparedwithvoltagenecessarytocontinuouslyshiftthespectrumoftheringbyfFSR.AtuningenhancementfactorofM=40inadouble-ring lterwasachievedatwavelengthsnear1.55µm[35].Thetuningrateforthethermoopticdeviceis120GHz/mW,andfortheelectroopticdeviceis120GHz/12V.Atunablelaserwithaside-modesuppressionratiogreaterthan30dBwasdemonstratedusingthis lteranderbium-doped berampli ergain.Thermaltuningover35nmwasachieved[35].

WGM谐振腔综述

ILCHENKOANDMATSKO:OPTICALRESONATORSWITHWGMs—PARTII:APPLICATIONS17

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