Oneofapparentapplicationsoftheoptical ltersisinopticaldelaylines.OpticaldelaydevicesbasedonchainsofcoupledWGRshavebeenstudiedin[37],[38].ItwasshownthattheQ-factorofthecoupling-splitmodesforasystemofNidenticalcoupledresonatorsisgreaterthanthatofasingleresonatorinthechainbyafactorofN,andevenmoreinthecaseofoptimumcoupling[39].Stoppinglightallopticallywithachainofinteractingtunableopticalresonatorswasdiscussedin[40].Anotherconceptofcoupledresonatoropticalwaveguideswasdevelopedin[41],[42].Inparticular,suchwaveguidescanberealizedinchainsofcoupledWGRs[44].Anumericalsimu-lationoflightpropagationinmicrocylindercoupledresonatorwaveguideswasreportedin[43].ThesimulationsshowthatlightpropagatesslowerintheWGMchainsformedbycouplingofthemodeshavingbiggerazimuthalnumbers.Thelightprop-agationbyWGMsofthesameazimuthalnumberhavethesamespeedregardlessofthesizeandthematerialoftheresonators.Generally,WGRtunable ltersallowshiftingthespectrumoftheresonators;however,theydonotprovidelinewidthtuningcapabilities.Cascadedresonatorswereproposedtobeusedforreal-timeshapingoftheirmodalstructures[45],[46].Akeyfeatureoftheapproachisthatitpointstoasimpletuningofthefrequencyandthewidthofthe ltertransmissionwindow,resultinginthetuningofthegroupdelayofopticalsignals—ahighlydesirablefeatureforsignalprocessingapplications.Thetransmissionspectralwindowofthe ltercouldbecuri-ouslynarrow.Theoretically,inthecaseofresonatorswithoutab-sorption,thewidthofthewindowcanbearbitrarilynarrow[45];however,inreality,theminimumwidthoftheresonanceisde-terminedbythematerialabsorption.Thephysicalprincipleofthe lteroperationthatresultsinthenarrowspectralwindowhasbeenrecognizedin[15],[25],[26].Theexistenceofthewindowhasalsobeendemonstratedexperimentally[47].C.TunableFilters
Tunabilityisahighlydesirablepropertyofanyapplicationofresonators.ThoughWGRsaresolidstatedevicesandtheirtun-abilityisnotreadilyconceivable,tunabilitycan,infact,achievedbyseveralmethods.MechanicaltrimmingofWGMswithap-pliedstrain[48]–[50]andtemperaturetuning[51],[52]havebeenpreviouslyused.Thoughthemechanicalaswellastem-peraturetuningrangesarerelativelylarge;e.g.,ontheorderofafewtoseveraltensofnanometersforthermaltuning,thesemethodsarenotveryconvenientformanyapplicationsbecauseofsmalltuningspeedsandlowtuningaccuracy.Thetuningaccuracyisespeciallyimportantforhigh-Qresonatorswithnar-row lterbandwidth.Anall-opticaltunable lterdesignbasedon“discontinuity-assistedringresonators”thatdoesnothavethepreviouslymentioneddisadvantageshasalsobeenproposedtheoretically[53],but,toourknowledge,noexperimentalim-plementationofthecon gurationhasbeenreported.
AtechniqueforWGMresonancetuningwasdemonstratedusingmicroringresonatorswithaphotosensitivecoating.Inthatstudy,glassmicroringsweredippedinapolymercoatingmaterialandwereexposedtoUVlight.ThismethodproducedresonatorswithrelativelysmallQ(about800)becauseofthe
polymer-inducedabsorption;yetitstillallowedlargetunabilityoftheopticalresonanceofthemicroring,enoughforwavelengthselectiveapplications[47].
Amethodforthetrimmingofpolymeropticalmicrores-onatorswasproposedin[54].Themethodisbasedonphoto-bleachingCLD-1chromophores.Amaximumwavelengthshiftof8.73nmwasobservedat1.55µm.Theresonatorshada3dBbandwidthof0.12nm,anFSRof1.11nm,anintrinsicQvalueof～2×104,anda nesseof～10.
Anotherapproachfortrimmingthefrequencyofmicrores-onatorsexploitsthephotosensitivityofthegermanatesilicaglass.WhenexposedtoUVlight,thismaterialundergoesasmallpermanentchangeinstructurethataltersitsindexofre-fraction.InthecaseofaWGR,thespatiallyuniformchangeintheindexofrefractionresultsinauniformtranslationoftheresonantfrequencies.Suchatunableresonator,aswellasasecond-orderoptical lterbasedontwocoupledresonators,oneofwhichwastunable,wasexperimentallyrealizedforopticalhigh-Q(108)WGMs[32],[55],[56].
Recently,fabricationofopticalWGMresonatorswithlithiumniobate[57]hasledtothedemonstrationofahigh-Qmicrowave lterwithalinewidthofabout10MHzandfastelectrooptictuningwithatuningrangeinexcessof10GHz[58].ThebesttunabilityforaLiNbO3singleresonator lterwas±20GHzbyapplyingdcvoltageof±50Vtoanelectrodeplacedovertheresonator.
ThefrequencyshiftoftheTEandTMmodesinaWGRmaybefoundfromthetheoryoftheelectroopticeffect[59].FortheLiNbO3WGR lterdiscussedin[58],we nd
ωTE
=ωn2
0e2
r33EZ,
ωTωn2
M
=0o2
r13EZ
(4)
whereω0=2π×2×1014Hzisthecarrierfrequencyofthelaser;r33=31pm/Vandr13=10pm/Varetheelectroopticconstants;ne=2.28andno=2.2aretherefractiveindicesofLiNbO3;andEZistheamplitudeoftheelectric eldappliedalongthecavityaxis.TMmodeswereusedintheexperimentreportedin[58]becausetheyhavelargerqualityfactorsthantheTEmodes.Ifthequalityfactorisnotveryimportant,itisbettertousetheTEmodes,becausetheirelectroopticshiftsarethreetimesaslargeasthoseofTMmodesforthesamevaluesoftheappliedvoltage.
Theoretically, ωTEand ωTMdonotdependontheres-onatorproperties,andarerelatedtothefundamentallimita-tionsofopticalresonator-basedhighspeedelectroopticmod-ulators[60].Forexample,thedomainreversalinacongruentLiNbO3crystaloccursatEZ 20kV/mm,whichcorrespondstoarelativeWGRfrequencyshiftof ωTE/ω0 1.6×10 3and ωTM 5×10 4,whichiswellabovetheobservedshifts.Lithiumniobate ltersareconvenient;however,theirlinewidthisrestrictedbyafewMHzbecauseoftheresidualabsorptionofthematerial.CrystallineWGRscouldpossessmanyordersofmagnitudenarrowerlines.Forinstance,opti-cal lterswithbandwidthsofabout10kHzusingCaF2WGMresonatorswasdemonstratedin[61].TheCaF2resonatorshavestableultrahighQ-factorscomparedwithfusedsilicaresonators,whereQdegradeswithtime.LimitedtuningoftheCaF2 lters