soliton repetition rate in a silicon-nitride microresonator 2017 chengying bao参考.pdf

该【soliton repetition rate in a silicon-nitride microresonator 2017 chengying bao参考 】是由【小舍儿】上传分享，文档一共【4】页，该文档可以免费在线阅读，需要了解更多关于【soliton repetition rate in a silicon-nitride microresonator 2017 chengying bao参考 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。,-nitridemicroresonator1,1,211,31CHENGYINGBAO,*YIXUAN,CONGWANG,-VILLEGAS,,1,21,2MINGHAOQI,,PurdueUniversity,465NorthwesternAvenue,WestLafayette,Indiana47907-2035,USA2BirckNanotechnologyCenter,PurdueUniversity,1205WestStateStreet,WestLafayette,Indiana47907,USA3FacultaddeIngenierías,ológicadePereira,Pereira,Risaralda66003,Colombia*Correspondingauthor:******@;revised3January2017;accepted19January2017;posted23January2017();published9February2017posedofasinglesol-therepetitionratechangeina22GHzsilicamicrocavity[13].Initoninananomalousgroupvelocitydispersionsilicon-additiontosilica,silicon-nitride(SiN)--enceofthermaleffectsandRamaneffectbetweensilicaandSiN,itsolitonstates,thecontributionsfromtheRamansolitonself-paretherelativeimportanceoftheSSFSandfrequencyshift(SSFS)andthethermaleffectsareevaluated;thermalcontributionstosolitonrepetitionratechanges(denotedΔtheSSFSisfoundtodominatethechangesintherepetitionfr),weusefrequencyrate,-assisted-laserdiodespectroscopy[15,16]tomeasurethechangesintherepetitionrateandthepumpfrequencyΔbgeneratedfromaSiNdetuningisfoundtobeindependentofthenonlinearityco-～-frequencyistuned,–etitionratechangebyusingthegeneralizedLugiatoLefevercontributionsfromtheSSFSandthermaleffectsareisolatedandequationisdiscussed;pared,-TheLugiato–Lefeverequation(LLE)isnowwidelyusedto～fstiondownto50.?bs[17,18].However,ithasrarelybeenusedOCIScodes:()Pulsepropagationandtemporalsolitons;forthemodelingofchangesinrepetitionrateandpulsetiming.()Nonlinearoptics,,sincepulsepropagationisaveragedinthehttps:///.[19].Here,wefurthershowLLEiscapablebsconsistofaseriesofdiscrete,evenlyofmodelingofthechangesinfrthatweobserveinexperimentsν??,whosefrequencyisnnff,wherer0giveinsightsintotheinfluenceofKerrshock,andweshowthatfristherepetitionrateandf0isthecarrierenvelopeoffsetfrequency[1].Microresonator-bs[2]showpoten--lockedlaserFormode-lockedlasers,thecenterfrequencyisconstrainedbycombs,;aweakmodulationofthepumppowercanintegernisgenerallylarge,bonlychangethecenterfrequencyslightlybyseveraltensofgiga-hertz[20,21].Hence,,-lockedlasersandnon-solitoncoher-,thecenterfrequencyofCSsisstronglystretchinghavebeenexploitedtostabilizethefroftheKerrinfluencedbytheinteractionwiththewaveguideviaSRS[12].combs[3–6].Recently,cavitysolitons(CSs)havebeendemon-TuningthepumpfrequencycanvarythisSSFSoverseveraltera-–stratedinmicroresonators[711].Unlikeothernon-solitoncoher-hertz[12],,wemeasuretheentoperationregimes,CSsexhibitasolitonself-frequencyshiftdependenceoffrontheSSFSviatuningthepumpfrequency(SSFS)inducedbystimulatedRamanscattering(SRS)[9–14].,viadispersion,constitutinganewmechanismforchangesinthe800×2000nmwaveguidegeometry,andaloadedQ-×106,b0146-9592/17/040759-04Journal?,-cavitysolitonsata227GHzrepetitionrate,,(×10?4)withthepumpfrequency[12][(d)],havingaslopeof[10],,wearealsoabletomea-powerwithoutthedirectlytransmittedpumpthatispresentatsuretheaverageintracavitypower;-assistedlaserdiodespectros-linearlywiththepumpfrequency(increaseswithpumpwave-blengthanddetuning),withaslopeof?∕GHz.[15,16].b,ThemeasurementoftheSSFSandintracavitypowerallowsparisonbetweentheinfluenceoftheSSFSandthermalef--,denotedfr,canbecalculatedasuracysimilartothat1LΔβachievedusingtheelectro-opticmodulation-assistedmeasurementΔs?Δ??1??2βΔΩ(1)frβ?β?2Lfr2cs;ofthelinespacing[15,16].1L1L∕ββAsanexample,bwhereListhelengthofthecavity,11isthegroupvelocity,(a).Whenwezoomintoasinglelineisthegroupvelocitydispersion,andcsistheSSFSinangularintheCSregime,wecanseetwonarrowpeaks,[23].FromEq.(1),weΔsbline[inset(i)].Incontrast,,wemeasurethedispersionusing(pumpedatashorterwavelength),b-assistedspectroscopy[15].Weshowthede-?broadandstructured[inset(ii)(a)].Wecanfittheviationoftheresonancefromtheequidistantspacing,dintν?ν?μ≈μ2∕νblinestoalineμ0d1d22(μistheresonantfrequencyofmodeμtogetfr[(b)].ThepowerandSSFSoftheCS,d1isthefreespectralrange,andd2isthedispersioncoef-hepumpfrequency[7,12].Forficient),(a).Fromaquadraticfit,d2isfoundtobeβ??2∕ourdevice,,equivalentto266pskm[24],closetothe～?2∕rangeofpumpfrequency;frchangesby25MHzinthissimulateddispersionof51pskm[10].Δrange,[(b)].(b)isdrivensolelybytheSSFS,weβthisslopefora18GHzSiNmicroresonator-bcanextract2byusingEq.(1).Foranon-chippumppowerofβ?2∕wasmeasuredtobe57kHz/GHzforanon-solitonstatein[5].800mW,,theob-WealsomeasurethespectrumofthegeneratedCS,whichtainedvalueremainsnearlythesameunderdifferentpumppowersfitswelltoasech2function,exceptsomespikesfromthemode[(b)].Thesmallfluctuation(55∕km)isbecauseinteraction[(c)][22].Thespikescanleadtospectralre-,thesevaluescoiling[23].Thisrecoilingisrelativelyweakinourdevice,,theSSFS(cs)canbeapproximatedas[24]320τβτβ(i)soliton8cQ8Q(a)(b)Ω?R2?R2pumpcs;(2)(ii)chaotic1GHz15ngrs15r1sInten.(.),R,ng,c,Q,andsaretheresonancefrequencyof-(.)-227515(MHz)rthecavity,Ramantimeconstant,effectivegroupindex,speedoff216ModeNumber0-200123450123456789RelativeFrequency(THz)RelativeWavelength(pm)(a)~(c)spectrum(d)2-(MHz)~--70-8-6-4-(W)Intensity(dB)Slope-)-40-(THz)(b)148015201560160016400123456789-50Wavelength(nm)RelativeWavelength(pm)ΔΩ2/km)r2β-60measured2/(.(a)b(psfΔ-(i)showsthezoominasinglefre--300400500600700800quencymarkerintheshadedboxinthesolitonstate,whileinsetPumpPower(mW)(ii).(a)Resonancedeviationfromevenspacing(circles)mea-axisiscalibratedbythefrequencymarkersbeatingwiththefemtosec-b-assistedspectroscopyandquadraticfit?2∕b.(b)Dependenceoffronthepumpwavelength;(line),yieldingadispersionof66pskm.(b)Dispersion(circles)bfrequencyextractedfromEq.(1),basedonthemeasurementsoftheSSFSandΔmarkerstogetfr.(c)Solitonspectrumfromthedropportanditsfratdifferentpumppowers,assumingthattheobservedchangesinsech2fit.(d)Measureddependenceofthecenterfrequencyshift(-dots)andtheintracavitypower(greendots)b-,,qualityfactor,andpulsewidthoftheCS,respectively.(a)γ?(W)(P0)ofCSsfollowsthesolitonlaw,∕τ2γ02s(isthenonlinearcoefficient);P0alsoscalesnearly12thlinelinearlywithpumpfrequencydetuning[9,25],-20pump2?ω?ω?tslope-608MHz/VrpR-40γP≈≈2?ω?ω?β;(3)(MHz)10dB/div0(pump)0rp112th0,12LfΔω-60Intensitywherepisthepumpfrequency,andtRistheround-(b)Wavelength(nm)slope-643MHz/V-80BasedonEqs.(2)and(3),(V)(d)arisesbecausethedetuningvariesbyonly33%,.(a)Measuredintracavitypowerchangeanditslinearfitver-comparedtoitsmeanvalueoverthetuningrange.(Theband-susthevoltagetocontrolthepiezoofthepumplaser.(b)Measured–;frequencychangeandthelinearfitforthepumplineandthe12ththedetuningchangecanbeestimatedbasedonEq.(3).)CombiningEqs.(1)–(3)?ω?ωfrequencydetuningasdrp,≈?RrΔω2≈?RdΔω(4)frωdωfrd:∕ΔωThisrelationshipmeansfrdisnotdirectlydepen-,-optic(TO)blinesandtheaverageΔteffecttothechangeinrepetitionrate,bpowerwithrespecttothevoltageusedtocontroltheThiscontributioncanbeexpressed?Δ??Δ??rΔ(5)fr22ngng;,wefindthatthecoefficientofthegLcrepetitionratechangewithintracavitypoweris?∕,(d),,whichdnτ?α??gθfθPng:(6)SinceΔfisdominatedbytheSSFS,whichcanbemodeleddTCθρAreffbythegeneralizedLLE[9,12,24],weperformsimulationsus-∕Here,dngdTisthederivativeoftheeffectivegroupindexwithingthegeneralizedLLEtolookintothepulsetimingdynamics.∕,weperformWewritethegeneralizedLLE[12,18,22,28]aswaveguidesimulationstoobtaintheeffectivephaseindex(n)ofp?κ?κβL?2p?????themicroresonatorfordifferentwavelengthsandtemperatures;thet?01?iδ?i2E?κE??λ∕λR?t202?τ21ingroupindexisngnpdnpd[16].UsingthesameTOco-Z∕??∞efficientforSiNandSiO2as[16],wefinddngdTi?000?5?1?γ??τ?j?τ?τ?j2τ?×?760J∕?kg·K?istheheatcapacityofiL1EREt;d0;(7)ω?τ?∞SiN,ρ?×103kg∕m3isthedensityofSiN[26],thetime0κκconstant(τθ)-where0,1,andEinaretheintrinsicloss,externalcouplingωnatorinourgroup’spreviouswork[27],αisthepropagationlosscoefficient,andpumpfieldatthefrequency0,respectively.?3?1?τ????Θ?δ?τ??Θ?τ?coefficient(×10∕628μm),fθisthefrac-R1RRhRisthenonlinearresponse,in-tionoftheabsorbedenergythatisconvertedtoheat,Aeffisthecludingboththeelectronicandthedela