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