OpticalKerrFrequency CombGenerationinOvermoded Resonators A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki OEwaves Inc., 465 N. Halstead Street, Pasadena, CA 91107 Weshow that scattering-based interactionamong nearlydegenerate optical modes isthe key factor inlow thresholdgenerationofKerrfrequencycombsinnonlinearopticalresonatorspossessingsmallgroupvelocity dispersion (GVD). The mode interaction is capable of producing drastic change in the local GVD, resulting ineitherasignificantreductionorincreaseoftheoscillationthreshold. Itisalsoresponsibleforthemajority of observed combs in resonators characterized withlarge normal GVD. Wepresent results of our numerical simulationsaswellassupportingexperimentaldata. 2 1 PACSnumbers:42.62.Eh,42.65.Hw,42.65.Ky,42.65.Sf 0 2 There are several experimentally observed phenomena in spectrum so much that the local GVD significantly changes n opticalfrequencycombsgeneratedwithmonolithicmicrores- itsvalue, aswellasitssign. We arguethatmodecrossingis a J onators (Kerr combs) that have been studied by multiple re- the chief reason for observation of low threshold frequency searchers (see [1] for review), but are not as yet explained combsinlargeenoughmonolithicresonators. 0 1 theoretically. An example of the observed phenomenalack- Very recently performeda study of universaldynamicsof ingatheoreticalexplanationisthestrongdependenceonthe Kerrcombformationindielectricmicroresonators[11]. This ] selection of the externally pumped optical mode of both the study explained the anomalously large linewidth broadening s c spectral envelope and the generation threshold of the comb. in octave spanning frequencycombs[12] and multiple beat- i t The purpose of this contribution is to provide a theoretical notesobservedinlowrepetitionratecombsystemswhenthe p frameworkfortheseobservations. comb was demodulated on a photodetetor [13]. Our current o studyshowsthatthetheoreticalconsiderationofthelowrep- . Generationofopticalfrequency(Kerr)combsresultsfrom s etitionratecombsystemsshouldbeadjustedtotake intoac- c modulationinstabilityofthecontinuouswave(cw)lightcon- i finedwithinamodeoftheresonator[2]. Thecombisgener- counttheinteractionbetweenthemodesofthenonlinearres- s y atedwhenthepowerofthecwpumpexceedsacertainthresh- onator. h old.Thefrequencyharmonicsofthecombareproducedinthe Torevealtheimportanceofmodecrossingwenumerically p modes of the resonator adjacent to the pumped mode; these simulate generation of combs in 21 identical optical modes [ harmonicsarephaselocked[3]. coupledthroughcubicnonlinearity. Themodeshaveunequal 1 separationresultingfromnonzeroGVD of the resonator. To Theoretical studies dealing with idealized nonlinear res- v performthesimulationwenumericallysolveasetofnonlinear 9 onatorssuggestthatthecombspectraaresymmetricwithre- equationsderivedusingthephysicalmodelpresentedin[2,7]: 5 spect to the frequency of the optical pump [4–7]. Further- 9 more, the frequencyspacing of the generatedharmonicscan i 1 vary from a single free spectralrange (FSR) (the fundamen- aˆ˙j =−(γ0+iωj)aˆj + [Vˆ,aˆj]+F0e−iωtδ11,j, (1) ¯h . 1 talcomb)tomultipleFSRsoftheresonator,dependingonthe 0 power and frequency of the pump as well as group velocity where Vˆ = −¯hg(eˆ†)2eˆ2/2 is the interaction Hamiltonian, 2 dispersion(GVD)oftheresonatormodes[8,9]. eˆ = P21 aˆ (aˆ is the annihilation operator of the field in 1 j=1 j j v: Theoretical predictions [4–7] describe results of experi- the mode), g = h¯ω121cn2/(Vn20) is the interaction constant, i ments performed with resonators having anomalous GVD ω11 isthefrequencyofthepumpedmode,n0 andn2 arethe X [14], while multiple experimental observation with normal linearandnonlinearrefractiveindexesofthematerial,Visthe r or nearly-zero GVD resonators [8, 15] evidently contradict modevolume,δ11,j istheKronecker’sdelta;γ0 = γ0c+γ0i a thetheory. InresonatorscharacterizedwithanomalousGVD isthe half widthatthe halfmaximumfor theopticalmodes, thepower-dependentfrequencyshiftofthemodesiscompen- assumed to be the same for the all modes involved; and γ0c sated by the dispersion, leading to extended dynamic range andγ0istandforcouplingandintrinsicloss. Theexternalop- forcombformationaswellasensuringexcitationofthecomb tical pumping is given by F0 = (2γ0cP/(h¯ω11))1/2, where from zero field fluctuations (soft excitation regime). Theo- P isthevalueofthecwpumplight. Weneglectthequantum retically, there is no soft excitationregime for combsof res- effectsand donottake into accountcorrespondingLangevin onatorswithnormalGVDandthedynamicrangeforthecomb noiseterms. formationislimitedthere[10].Practically,asymmetriccombs Wesolved(1)numericallytakingintoaccountonlythesec- andthesoftexcitationregimeofthecombarefrequentlyob- ondorderfrequencydispersionrecalculatedforthefrequency served. Inthisworkweshowthatthediscrepancyoriginates ofthemodesωjasω12+ω10−2ω11 =−β2cωF2SR/n0,where fromtheinteractionofresonatormodesoverlappingthesame β2istheeffectiveGVDofthemodeswithnointeractiontaken modevolumeandbelongingtodifferentmodefamilies. The intoaccount,cisthespeedoflightinthevacuum,ω isthe FSR mode interaction disturbs the regular GVD of the resonator freespectralrangeoftheresonator,e.g.2ωFSR ≃ω12−ω10. 2 TheeffectiveGVDvalueisnegligibleinalargeWGMres- onatorthathasnomodeinteraction,γ0 ≫|D|=|ω12+ω10− 2ω11|. For example, the mode non-equidistancecan be esti- de u Tt(Fmy2ShpaγReti0ecvdaC>alaalbusFa2eD2nπdc(h≃×wMaingd−2gFt0h2e20πs)okfWr×HatthGzh3e,e9MrmoHriornzedQss(eioDgs<nnuai≃sfiteo1cd2r0aπ9nipn)tu×lyimrne1pcpsoeokmdmrHteaepzdldla)eraierentxdrp1aeew5s1ro5ii0nt0mhaGenttoHnmhrtezss. separation, /01234500000 alized field amplit 0000....3589 (O3n5tGheHozthFeSrRh)a:nDd,D≃≈−22πγ0×is5r.e2qukiHrezd(fDor≃gen2eπra×tio8nokfHtzh)e. uency -100 Norm 0.1 q fundamentalfrequencycombwithsoftexcitation[2]. Thisis e Fr-20 possibleinasmallresonator. Forinstance,forafundamental -30 TEmodeofa100GHzFSRMgF2 WGMresonatorpumped -40 with1721nmlightD ≈ 2π×200kHz. Anyobservationof soft excitation of the comb in a larger resonator is the evid- -75 -50 -25 0 25 50 75 100 ance thatthe spacingofthe modesis somehowaltered com- Frequency shift, /0 paredwiththevalueofanidealresonator.Onewayforsucha modificationisrelatedtotheusageofadifferentmodefamily, FIG. 1: Density plot illustrating frequency and amplitude depen- the GVD of which can be tunedwith the morphologyof the dence of theforced response of alinear mode a12 on thedetuning resonator[16].Anotherwayisrelatedtothemodeinteraction ∆ = ωc−ω12. Modea12 ispumpedexternallywithamonochro- describedbelow. maticforceofaconstantamplitude(|F0/γ0| = 1). Thefrequency Interactionbetweenresonatormodeshasbeenencountered separationbetweentheforcefrequency(ω)andthefrequencyofthe inWGMresonators. Itisdirectlyobservedasadisruptionof mode(ω˜12)isdeterminedasδ=ω˜12−ω12,whereω12corresponds to the frequency of the free mode. The calculations are made for the continuity of dispersion in smaller resonators [17], and κ/γ0 =20.Thedependenceshowsthatforlarge|∆|theinteraction is indirectly indicated by the asymmetry of the Kerr comb primarilyresultsinfrequencyshiftofthemodeofinterest.Themode spectra[8,15](thespectrallynarrowcombsgeneratedinres- splitsintoasymmetricdoubletwhen∆approacheszero. onatorswithnomodeinteractionhastobesymmetricbecause of energy conservation). A spectrum of a microresonator takeninabroadfrequencyrange[18]clearlyshowsthepres- modec leadsto the modificationofthe effectiveGVD value ence of spurious modes changing their position with wave- forthepumpedmodeandthefirst twosidebandmodes. Re- lengthwithrespecttothefundamentalmodefamily.Sincethe ally,usingEq.(2)wefind spuriousand the fundamentalmodesare able to interactdue cω2 κ2 tounavoidableimperfectionsoftheresonatorshape,theyalso ω˜12+ω10−2ω11 =−β2 FSR − . (3) canchangethedispersivepropertiesoftheresonator. Evenif n0 ∆ thespuriousmodesarenotseenintheexperimentwithapar- According to Eq. (3), it is enough to have κ2 ≈ −2∆γ0 to ticularselectionofthecouplingtechnique,theystillcanexist achievethedesirablevalueoftheGVDinanyresonatorwith in the resonator[19, 20]. Only a resonatorsupportinga sin- asmallintrinsicdispersion. glemodefamily(see,e.g.,[21])canbeconsideredfreefrom Tovalidatetheanalyticalcalculationsweperformedasim- modeinteraction. Therefore,itisnaturaltoexpectthatmode ulation for the 21 mode at the above conditions but also interactionisessentialinthemajorityofobservationsofKerr took into account the interaction with a mode c. Selecting frequencycombsinopticalresonators. κ/γ0 = 20 we found that the soft excitation of the Kerr Letusassumethatthemodecontainingthefirsthigherfre- comb is possible for a wide range of frequency detunings: quencyharmonic(a12)interactswithamodeoftheresonator −0.2γ0 > ∆ > −2.7γ0 (Fig.2). TheKerrcombforspecifi- (c)havingthesameloadedQ-factorandnotbelongingtothe callyselectedparametersofthesystemisshowninFig.(3a). mode family the comb is generated in. The interaction, de- Ithasa slightly asymmetricspectrumand fast roll-offof the scribedbyHamiltonian¯hκ(a†12c+c†a12),resultsinthewell higher order harmonics. The comb starts from zero fluctua- known splitting of the resonance for a12 (see Fig. 1). The tionsofthefieldwithessentiallyzeroinitialconditions.Since splitting may be considered as a pure frequency shift of the suchexcitationregimeofthecombisabsentinthecasewhere modea12expressedas nomodeinteractionisavailable,weconcludethatmodeinter- actionisthecause,aspredictedbythereasoningabove. 2 κ ω˜12 =ω12− , (2) Accordingtothenumericalsimulationsthefrequencycomb ∆ based on mode interaction has two distinct features: (i) the intheasymptoticcaseoflargedifferencebetweentheeigen- dynamic range of the soft excitation of the comb is limited frequencies of the interacting modes, ∆ = ωc −ω12, com- with respect to the power of the external pump and (ii) the paredwiththeinteractionconstantκ,|∆|≫κ. repetitionrate of the combdependson the pumppower. We The shift of mode a12 resulting from the interaction with evaluated the power of the first comb sideband generated in 3 the shifted resonator mode (a12) as a function of the pump amplitude and frequency detuning for the fixed interaction d value with another mode (Fig.4). The limitation occurs due n ba to clamping of the value of the nonlinear frequency shift of e d 0.84 the resonator modes that can be compensated by the inter- si on, -75 e first 0.65 accotmiobnforefqau1e2ncayndiscillmusotdraetse.d bTyhe(Fpiogw.5e),rsdheopwenindgenthcee doiffftehre- eparati e of th 00..2487 eqnuceenc[y(ωo1f2th−eωge1n1)er−ate(dωs′1id2e−baωnd)]./γ0, where ω′12 is the fre- s-150 d ncy plitu 0.10 e m qu A d e-225 n Fr F|/008.4 st sideba 11..59 -300-3.3 -2.7 -2.1 -1.5 -0.9 -0.3 mp, |6.9 he fir 1.2 tFhIeGfi.r2s:thDigehnesritfyrepqlouDteneschtuyonwsiniidngeg bonaf onthrdmeo apfluitzmheedpK, aemrrplciotumdbedgiesntreirbautetidonduoef plitude of the pu35..94 Amplitude of t 000...369012 tointeractionoftheresonatormodes. Thedistributionisplottedas m A a function of the detuning between the pumping laser and the cor- 2.4 respondingmodeoftheresonator(∆0/γ0),andthefrequencysepa- rationbetweentheopticalmodecontainingthecombharmonicand -7.18 -4.41 -1.64 the service mode (c). Coupling parameter κ/γ0 = 20 isselected. The GVD of the resonator is selected to be D/γ0 = −0.02. The Detuning of the pump, 0/0 amplitude of the external cw pump is |F0|/γ0 = 2. Mode a11 is pumped. FIG.4: Normalizedamplitudeofthefirstcombsidebandgenerated inthemodea12shiftedduetotheinteractionwiththeservicemode. NearlyzeronormalGVD(D/γ0 =−0.02)andcouplingparameter κ/γ0 =20areselected. 0 10 e (a) d -3 u10 plit -6 m10 d zed a1100-03 (b) F|/008.4 sideban 2.0 Normali11110000---0663 -18 -12 -6 0 6 12 (c1)8 plitude of the pump, | 356...949 Detuning of the first 22333.....37037 m Mode number A 2.4 FIG. 3: Spectra of the frequency combs generated in a resonator -7.18 -4.41 -1.64 havingnearlyzeronormalGVD(D/γ0 = −0.02)whenoneofthe Detuning of the pump, 0/0 resonatormodesisshiftedduetotheinteractionwithanothermode. Coupling parameter κ/γ0 = 20 is selected. The amplitude of the FIG.5: FrequencydifferencebetweenthelocalFSRoftheresonator externalcwpumpofthecentralmode(a11)is|F0|/γ0 = 2. Three (notmodifiedbytheinteractionwiththeservicemode)andtherep- casesareshown: (a)modea12 isshiftedduetotheinteraction; (b) etitionrateofthecombasafunctionoftheamplitudeandfrequency modea13isshifted;and(c)modea14isshifted.Becausetherelative ofthepump. GVD of the resonator is small it does not influence the nonlinear process,soshapesofthecombenvelopessimplyscalefrompicture topicture. Interaction of the modes also explains experimentallyob- servedgenerationofcombshavingdifferentrepetitionratesin 4 resonatorswithnormalGVD[8]. Thecombstepchangesde- pendingwhichofthecomb-generatingmodeinthesequence isshiftedduetotheinteraction.TheintrinsicGVDoftheres- 0 (a) onatorsistoosmallto influencethe process. We havesimu- -20 latedgenerationofhigherrepetitionratecombsandvalidated -40 theconcept(Fig.3b,c). Thesimulationshowsthatthedetun- m-60 ingofthepumplightfromthecorrespondingmodeoftheres- dB 1544.8 1545.2 1545.6 1546.0 1546.4 onator, discussed in [8], is of less importancefor generation r, 0 ofhighrepetitionratefrequencycombshavingsoftexcitation we-20 (b) o regime,comparedwithmodeinteractionconsideredhere.The p-40 changeofthecombstepinnormalGVDresonatorsisdiffer- al -60 c entfromthemechanismofthestepchangeinresonatorswith Opti 0 1547.5 1548.0 1548.5 1549.0 anomalous GVD, where repetition rate strongly depends on -20 (c) theintracavitypowerofthepumplight[9,14]. -40 WeperformedanexperimentwithalargeWGMresonator -60 having nearly zero relative GVD (D/γ0) and observed gen- eration of frequency combs with envelopes similar to those 1547.5 1548.0 1548.5 1549.0 predicted by the theory. In our experiment we used a CaF2 Wavelength, nm resonator having 6721 µm in diameter. The resonator had approximately 9.9 GHz FSR with loaded quality factor ex- FIG.6: Experimentalspectraofthreeopticalfrequencycombsgen- ceeding109. Wepumpedtheresonatorwith1545.5nmlight erated in the same overmoded WGM resonator when light is cou- emitted by a distributed feedback semiconductor laser. The pled to three arbitrary modes having strong interaction with other lightwascoupledtotheresonatorviaacouplingprism. The modes. The wavelength of the pumping light was changed to se- lect the modes, and depending on the mode selection we observed opticalpoweremittedbythelaserwas15mW,and3.2-1mW (a)single-FSR,(b)dual-FSRcomb,and(c)triple-FSRcombs. Gen- of the light entered the selected modes of the resonator (the erationofhigher orderfrequency combs, notshownhere, wasalso value depends on the selected mode). The output light was recorded. collectedusingaPMPandafiberandintroducedtoanoptical spectrumanalyzer. Theresultantopticalspectraareshownin Fig.(6). (2007). Frequency combs simulated numerically have frequency [4] I.H.Agha,Y.Okawachi,M.A.Foster,J.E.Sharping,andA. shapes similar to the combs observed experimentally (com- L.Gaeta,Phys.Rev.A76,043837(2007). [5] I. H. Agha, Y. Okawachi, and A. L. Gaeta, Opt. Express 17, pareFig.3and6). Thecombpropertieschangesignificantly 16209(2009). when we pump different modes of the resonators. Such a [6] Y.K.Chembo,D.V.Strekalov,andN.Yu,Phys.Rev.Lett.104, modification cannot be explained by the change in the geo- 103902(2010). metricalpartoftheGVDofthemodes. Theinteractionwith [7] Y.K.ChemboandN.Yu,Phys.Rev.A82,033801(2010). the degenerate modes, on the other hand, perfectly explains [8] A.A.Savchenkov,A.B.Matsko,V.S.Ilchenko,I.Solomatine, theobservation. D.Seidel,andL.Maleki,Phys.Rev.Lett.101,093902(2008). [9] O.Arcizet,A.Schliesser,P.DelHaye,R.Holzwarth,andT.J. 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