Universal Optical Frequency Comb
A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, L. Maleki
aa r X i v : . [ phy s i c s . op ti c s ] S e p Universal Optical Frequency Comb
A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki
OEwaves Inc., 2555 E. Colorado Blvd., Ste. 400, Pasadena, California 91107 (Dated: September 5, 2018)We demonstrate that whispering gallery mode resonators can be utilized to generate opticalfrequency combs based on four wave mixing process at virtually any frequency that lies in thetransparency window of the resonator host material. We show theoretically how the morphology ofthe resonator can be engineered to produce a family of spectrally equidistant modes with anomalousgroup velocity dispersion appropriate for the comb generation. We present experimental results fora frequency comb centered at 794 nm to support our theoretical findings.
Since their introduction over a decade ago, opticalfrequency combs have revolutionized many applicationsacross the fields of science and metrology, and haveopened new vistas in sensing and spectroscopy. Theconventional approach for optical comb generation withfemtosecond lasers was recently augmented with opticalcombs generated using whispering gallery mode (WGM)resonators made of materials possessing Kerr nonlinear-ity [1–12]. These small structures address one of theshortfalls of femtosecond laser frequency combs, namelytheir size and relatively complex architectures that limitstheir range of applicability outside research laboratories,and thus restricts their applications. The second shortfallof optical frequency combs related to the frequency of op-eration and has not been effectively addressed. Though anumber of optical combs covering selected spectral rangefrom UV to IR have been demonstrated, there is no uni-versal approach that can provide an optical frequencycomb centered at any desired wavelength. We propose asolution of the problem in this paper.The generation of a phase locked frequency comb withan optically pumped WGM microresonator is mediatedby hyper-parametric oscillation (modulational instability[16–18]) resulting from four wave mixing (FWM), andby group velocity dispersion. In the FWM process, twopump photons and two sideband photons are involved.The generation of two sidebands symmetrically aroundthe excited mode starts when the power of a continuouswave laser pumping the mode exceeds some thresholdvalue [15]. Increasing the pump power results in gen-eration of multiple sidebands via the hyper-parametricprocess, producing a phase locked group of tines in theoptical comb. Depending on the dispersion of the WGMspectrum, the initial generation of sidebands separatedby multiple free spectral range (FSR) could have a lowerthreshold compared with that of closely separated side-bands [3, 12].Group velocity dispersion (GVD) also plays a criticalrole in the generation of frequency combs. To explainthis, let us recall that the threshold condition for theaverage number of photons, ¯ N , in the pumping modenecessary to start the FWM oscillation process, whichgenerates the Kerr frequency comb in a WGM resonator, reads as [15]4( g ¯ N − γ ) = (cid:2) ω − ω ) + ( ω + + ω − − ω ) − g ¯ N (cid:3) , (1)where γ is the full width at the half maximum of theoptical modes (we assume that the modes are identicalin loss), ω and ω ± are the frequencies of the opticallypumped mode and the first pair of the modes whereFWM sidebands are generated at, ω is the carrier fre-quency of the external monochromatic pump, g is thenonlinearity parameter g = ω ¯ hω c V n n n , (2) c is the speed of light in the vacuum, V is the modevolume, n and n are the linear and nonlinear refractiveindices of the resonator’s host material.In accordance with Eq. (1) oscillation is allowed ifat least one of the following two conditions is satisfied:i) the optical pump is detuned to the lower frequencythan the corresponding optical resonance (red detuned), ω > ω ; and ii) the resonator modes are characterizedwith anomalous dispersion, β < ω − ω + − ω − ≃ cβ ω F SR /n , where ω F SR is the free spectral range ofthe resonator). The condition (ii) is usually strongerthan (i). Even though oscillation can occur in a res-onator possessing arbitrary frequency dispersion in themode family participating in the comb generation [16–18], it generally starts when the dispersion is smalland anomalous. Resonators with normal dispersion ex-hibit much weaker FWM process, which frequently com-petes with stimulated Raman scattering (SRS), whichhas a lower threshold for oscillation [19]. The red de-tuning of the pump light with respect to the corre-sponding resonator mode is required for phase match-ing and compensation of the dispersion, though it in-creases the oscillation threshold power, so the SRS pro-cess starts first. By contrast, anomalous dispersion sim-plifies the phase matching of the FWM process [5] andremoves the burden away form laser detuning. Theseassertions have been confirmed experimentally. For in-stance, for combs observed with WGM resonators, avalue of (2 ω − ω + − ω − ) / (2 π ) ≃ − . µ m fused silica microresonator at 1,550 nm[1]. The dispersion of a 0.7 cm CaF resonator was(2 ω − ω + − ω − ) / (2 π ) ≃ −
300 Hz [20], and a 0.255 cmCaF resonator had (2 ω − ω + − ω − ) / (2 π ) ≃ −
680 Hz[3].The dispersion β includes both geometrical and mate-rial parts. The geometrical dispersion is usually normal.To obtain mode spectra characterized with small anoma-lous dispersion one needs to produce a resonator out ofa material with anomalous dispersion for the given laserwavelength. The shape and size of the resonator mustbe properly selected to have an overall small and anoma-lous dispersion ( | ω − ω + − ω − | ≪ γ ). These require-ments prohibit generation of Kerr frequency combs withan arbitrary repetition rate at an arbitrary wavelength,since the repetition rate is determined by the resonatorsize, and the wavelength is determined by the materialdispersion. For instance, a fused silica resonator is notexpected to generate Kerr frequency comb if pumped ata wavelenght shorter than 1.3 µ m.Now, consider a spheroidal resonator with two nearlyequidistant mode sequences characterized with frequencyintervals (free spectral range, FSR) F SR l = c/ (2 πna )and F SR p = F SR l ( a − b ) /b , where l and p are the az-imuthal and the vertical numbers of the mode, c is thespeed of light in the vacuum, n is the refractive indexof the material, and a and b are the semi-axes of thespheroid [14]. Modes belonging to a basic sequence char-acterized with F SR l are generally utilized for generationof Kerr combs. However, for the purpose of comb gener-ation at any arbitrary frequency within the transparencywindow of the resonator material, we propose to use thevertical mode family characterized with F SR p , which de-pending on the ratio a/b can have an FSR significantlydifferent from F SR l . An important feature of a verticalmode family is that the mode spacing strongly dependson the shape of the spheroid, and the corresponding GVDcan be anamoulous. FIG. 1: Illustration of the cross sections of a spheroid and acylinder, and a WGM resonator being a convolution of thoseshapes. A spheroidal WGM resonator has normal geomet-rical group velocity dispersion in vertical modes, and nearlyequidistant spectra. A cylindrical resonator has anomalousgroup velocity dispersion for its modes but not equidistantspectra. The resonator designed as a convolution of thosetwo shapes has a nearly equidistant spectrum and anoma-lous group velocity dispersion necessary for achieving phasematching in the FWM process for any resonator host mate-rial.
The morphological dependence of the dispersion canbe understood as follows. Consider the convolution ofa cylindrical and a spheroidal resonator (Fig. 1). The frequency spectrum of the cylinder is given by ω p,l,q = cn r k l,q + (cid:16) pπL (cid:17) , (3)where k l,q a ≈ l + α q ( l/ / is the azimuthal wave num-ber given by the spherical Bessel functions, q is the radialwave number of the mode, α q is q th root of Airy func-tion ( Ai ( − α q ) = 0), and L is the hight of the cylinder.In the case of l ≫ p the spectrum is significantly non-equidistant with p , but the group velocity dispersion isanomalous 2 ω p,l,q − ω p +1 ,l,q − ω p − ,l,q <
0. The nearlyequidistant frequency spectrum of a spheroid can be ap-proximated by [14] ω p,l,q ≃ cn (cid:20) k l,q + ( a − b )(1 + 2 p )2 ab − ap b l (cid:21) , (4)so that the dispersion is normal, 2 ω p,l,q − ω p +1 ,l,q − ω p − ,l,q >
0. A resonator being a convolution of thecylinder and spheroid has a frequency spectrum given by ω p,l,q ≃ cn (cid:20) k l,q + ( a − b )(1 + 2 p )2 ab + ξp (cid:21) , (5)where ξ is a numeric parameter depending on L , a , and b . The parameter ξ can be positive in the case of compa-rably small L < b , which means that the dispersion ofthe vertical resonator modes is anomalous. In this way,the anomalous dispersion can be used to compensate forthe normal material dispersion in the Kerr comb.There are no measurements of the group velocity dis-persion for the majority of optical materials, and rigorouscalculations confirming these prediction are rather math-ematically involved. So we performed an experiment toverify these estimates. We designed and fabricated atruncated spheroidal crystalline CaF WGM resonatorand pumped it with 794 nm light emitted by a semi-conductor laser that was self-injection locked [13] to aselected WGM. The light was sent in and retrieved outof the resonator using a glass coupling prism. The exit-ing light was collimated and injected into a single modeoptical fiber to be further analyzed. The spheroidal res-onator with 3 mm and 3 / a/b ≈ wafer by mechanicalpolishing. The loaded quality (Q-) factor of the resonatorexceeded 3 × (126 kHz full width at the half maximumof the corresponding WGMs).We observed the onset of hyperparametric oscillationwith 0.1 mW of optical power. Increasing the power to2 mW resulted in the formation of a well pronounced op-tical frequency comb shown in Fig. (2). The frequencyenvelope of the comb suggests that the vertical modefamily is involved in this process. The asymmetry ofthe spectrum results from the significant spatial spreadof the light emitted from the resonator that is selec-tively collected with the fiber. Since the group velocitydispersion of the material is normal at this wavelength( β = 28 ps / km) and the difference between two consec-utive FSR’s is positive and large (2 ω − ω + − ω − ) / (2 π ) ≃
21 kHz, but the comb is still generated, we conclude thatthe anomalous dispersion of the resonator compensatedthe material dispersion. The experiment proves our the-oretical prediction.
792 793 794 795 796-60-50-40-30-20 -100 -50 0 50 100-80-60-40-20 O p t i c a l po w e r , d B m Wavelength, nm R F po w e r , d B m Detuning from 23.8GHz, kHz
FIG. 2: The output spectra of a critically coupled CaF res-onator pumped with 2 mW of 794 nm light. No stimulatedRaman scattering is observed. The resonator has 8 GHzfree spectral range, while the optical frequency comb has23.78 GHz repetition rate. The comb harmonics are phaselocked, so they generate a coherent RF signal (inset) on afast photodiode. Note that the repetition frequency of the observedcomb is approximately three times as large as the ba-sic mode frequency FSR. In our case a/b ≈
4, so that
F SR p ≈ F SR l , which is close to experimentally mea-sured value F SR p /F SR l = 23 . GHz/ GHz = 2 . p and the same orbital momentum l .The maximum number of Stokes and anti-Stokes com-ponents of a conventional Kerr comb generated in thebasic mode family [1–12] is limited by the dispersion andnonlinearity of the resonator and can be very large. Onthe contrary, the number of the Stokes comb componentsobserved in our system is limited by the vertical index p of the mode that is selected for optical excitation. In-deed, figure (2) represents an optical comb produced bypumping the mode with p = 42. The saturated combhas a flat top and a sharp edge at the lower frequencyside, because there are simply no modes with negativetransverse index. This type of Kerr comb affords muchmore flexibility compared with a comb generated at the basic mode sequence: its repetition frequency is large inan oblate spheroid, as observed in our experiment, anda nearly spherical resonator would generate a comb withrepetition frequency much lower than the azimuthal FSRof the resonator. In general, the repetition frequency ofa transverse comb depends symmetrically on both theresonator semi-axes a and bν comb = cπn (cid:12)(cid:12)(cid:12)(cid:12) b − a (cid:12)(cid:12)(cid:12)(cid:12) . (6)This property results in the ability for generation of lowfrequency combs even in very small microresonators ofvery low mode volume, and therefore, very low threshold.It also ultimately leads to a new kind of RF photonicoscillator [21] with extremely low phase noise.The asymmetry of the observed comb envelop requiresa special explanation. A WGM of vertical index p is de-scribed by an eigenfunction which has p + 1 maxima inthe electric field along the vertical axis of the resonator[14]. The overlap of the evanescent field of the mode onthe coupling prism determines the emission pattern. Theemission pattern of the transverse mode consists of twosymmetrical lobes with respect to the horizontal plane.Since the angular space between the lobes and the fre-quency of the comb components grows with index p , com-ponents of lower index p are concentrated closer to theequatorial plane of the resonator. The optical fiber cou-pler that we use is aligned in the equatorial plane of theresonator and thus has much higher efficiency for col-lecting the modes with smaller p and, therefore, lowerfrequency. This is why the comb as observed has the ap-pearance of being asymmetrical. We note that the spatialpattern of the comb emission provides a simple way tophysically separate strongly correlated Stokes and anti-Stokes components of the comb for applications wherecorrelated photons are desired.In conclusion, we have proposed and experimentallyvalidated a universal method for the generation of Kerrfrequency combs at any desirable wavelength, and withany desirable repetition rate. The method is based onthe management of the geometrical group velocity disper-sion of whispering gallery modes participating in the pro-cess through the proper engineering of the shape of theresonator. We have found that the vertical sequence ofthe modes belonging to a specially shaped resonator hasnearly equidistant spectrum characterized with anoma-lous group velocity dispersion. Using these modes wewere able to generate a frequency comb at 794 nm centerwavelength in a properly designed calcium fluoride res-onator. The conventional Kerr comb produced with thebasic sequence of the azimuthal modes was absent in ourobservations. [1] P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R.Holzwarth, and T. J. Kippenberg, ”Optical frequency comb generation from a monolithic microresonator,” Na- ture (London) , 1214-1217 (2007).[2] P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, andT. J. Kippenberg, ”Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. ,053903 (2008).[3] A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I.Solomatine, D. Seidel, and L. Maleki, ”Tunable OpticalFrequency Comb with a Crystalline Whispering GalleryMode Resonator,” Phys. Rev. Lett. , 093902 (2008).[4] I. S. Grudinin, N. Yu, and L. Maleki, ”Generation ofoptical frequency combs with a CaF resonator,” Opt.Lett. , 878-880 (2009).[5] I. H. Agha, Y. Okawachi, and A. L. Gaeta, ”Theoreticaland experimental investigation of broadband cascadedfour-wave mixing in high-Q microspheres,” Opt. Express , 16209-16215 (2009).[6] J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, ”CMOS-compatiblemultiple-wavelength oscillator for on-chip optical inter-connects,” Nature Photonics , 37-40 (2009).[7] L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S.Chu, B. E. Little, and D. J. Moss, ”CMOS-compatibleintegrated optical hyper-parametric oscillator,” NaturePhotonics , 41-45 (2009).[8] D. Braje, L. Hollberg, and S. Diddams, ”Brillouin-enhanced hyperparametric generation of an optical fre-quency comb in a monolithic highly nonlinear fiber cav-ity pumped by a cw laser,” Phys. Rev. Lett. , 193902(2009).[9] P. Del’Haye, T. Herr, E. Gavartin, R. Holzwarth, and T.J. Kippenberg, ”Octave spanning frequency comb on achip,” arXiv:0912.4890 v1 [physics.optics] 24 Dec 2009.[10] O. Arcizet, A. Schliesser, P. DelHaye, R. Holzwarth, andT. J. Kippenberg, ”Optical frequency comb generation inmonolithic microresonators,” in Practical Applications ofMicroresonators in Optics and Photonics , A. B. Matsko,ed. (CRC Press, 2009), Chap. 11.[11] A. B. Matsko, A. A. Savchenkov, W. Liang, V. S.Ilchenko, D. Seidel, and L. Maleki, ”Whispering gallerymode oscillators and optical comb generators,” Proc. of7 th Symp. Frequency Standards and Metrology, ed. L. Maeki, pp. 539-558 (World Scientific, New Jersey, 2009).[12] Y. K. Chembo, D. V. Strekalov, and N. Yu, ”Spectrumand dynamics of optical frequency combs generated withmonolithic whispering gallery mode resonators,” Phys.Rev. Lett. , 103902 (2010).[13] W. Liang, V. S. Ilchenko, A. A. Savchenkov, A.B. Matsko, D. Seidel, and L. Maleki, ”Ultra-narrowlinewidth external cavity semiconductor lasers using crys-talline whispering gallery mode resonators,” to be pub-lished in Optics Letters (2010).[14] M. L. Gorodetsky, A. E. Fomin, ”Geometrical theory ofwhispering gallery modes,” IEEE J. Sel. Top. QuantumElectron. , 33-39 (2006).[15] A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S.Ilchenko, and L. Maleki, ”Optical hyper-parametric oscil-lations in a whispering gallery mode resonator: thresholdand phase diffusion,” Phys. Rev. A , 033804 (2005).[16] M. Haelterman, S. Trillo, and S. Wabnitz, ”Additive-modulation-instability ring laser in the normal dispersionregime of a fiber,” Opt. Lett. , 745-747 (1992)[17] S. Coen and M. Haelterman, ”Modulational instabilityinduced by cavity boundary conditions in a normallydispersive optical fiber,” Phys. Rev. Lett. , 4139-4142(1997).[18] S. Coen and M. Haelterman, ”Continuous-waveultrahigh-repetition-rate pulse-train generation throughmodulational instability in a passive fiber cavity,” Opt.Lett. , 39-41 (2001).[19] S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, ”Ul-tralowthreshold Raman laser using a spherical dielectricmicrocavity,” Nature (London) , 621-623 (2002).[20] A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S.Ilchenko, and L. Maleki, ”Phase noise of whisperinggallery photonic hyper-parametric microwave oscilla-tors,” Opt. Express , 4130-4144 (2008).[21] A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mo-hageg, V. S. Ilchenko, and L. Maleki, ”Low thresholdoptical oscillations in a whispering gallery mode CaF resonator,” Phys. Rev. Lett.93