Generation of Kerr Frequency Combs in Resonators with Normal GVD
aa r X i v : . [ phy s i c s . op ti c s ] N ov Generation of Kerr Frequency Combs in Resonators with Normal GVD
A. B. Matsko, A. A. Savchenkov, and L. Maleki
OEwaves Inc., 465 N. Halstead St. Ste. 140, Pasadena, CA 91107
We show via numerical simulation that Kerr frequency combs can be generated in a nonlinearresonator characterized with normal group velocity dispersion (GVD). We find the spectral shapeof the comb and temporal envelope of the corresponding optical pulses formed in the resonator.
Kerr optical frequency combs have recently attractedconsiderable attention because of their rich physics andtheir potential for multiple applications in science andtechnology [1]. Kerr combs are spontaneously generatedfrom quantum fluctuations in solid state nonlinear opti-cal microresonators pumped with continuous wave light.Since the first demonstration of a comb observed in afused silica microresonator [2] the possibility of generat-ing combs in other types of resonators have been reportedin numerous experiments. In accordance with experimen-tal observations, the observed combs preferably occur inresonators characterized with anomalous group velocitydispersion (GVD). This conclusion has also been con-firmed theoretically [3, 4]. Recent numerical simulationsconclude that generation of Kerr combs is also possibleand stable at all but zero GVD [5].Generation of the Kerr comb results from modulationinstability (MI) of the continuous wave light propagat-ing in a nonlinear host material of an optical resonator.The physical origin of the instability is the same as thoseobserved in optical fibers, where MI is observed underconditions of anomalous GVD and positive cubic non-linearity. It is natural to expect that the resonant MIhas properties similar to the non-resonant MI. However,it was noted that MI also takes place in a fiber ring res-onator with a net normal GVD [6]. The reason is that theresonant configuration introduces an additional degree offreedom in the system, namely frequency detuning of thepumping light from the eigenmode of the nonlinear res-onator [7]; this can shift the MI instability point to theregion of normal GVD.There have been several experiments in which thecomb formation was achieved in microresonators withnormal GVD. For instance, a comb was observed in aCaF whispering gallery mode (WGM) resonator with adiameter of 0.78 cm, pumped with 1320 nm light, andcharacterized with β ≃ . / km normal GVD [8]. Acomb was also generated in a CaF WGM resonator fea-turing a diameter of 0.255 cm, pumped with 1550 nmlight, and having β ≃ . / km [9].The experimental results were explained theoretically.It was shown that the resonant hyper-parametric oscilla-tion can occur at any GVD (the term ”hyper-parametricoscillation” was introduced to describe the underdevel-oped Kerr frequency combs, containing only two sym-metric sidebands with respect to the carrier) [10, 11].The excitation dynamics represents the basic differencein Kerr combs generated in resonators with normal andanomalous GVD. Kerr combs can have both hard and soft oscillation onset in the case of anomalous GVD, butonly the hard oscillation onset takes place in the case ofnormal GVD, as noted in [11]. Nevertheless, a more rig-orous study of the subject is still required. The theory in[11] was developed for a hyper-parametric oscillator only,and does not cover generation of Kerr frequency combs.The numerical simulations [5] do not describe hard exci-tation of the comb and do not allow finding any steadystate solution for the comb, since it is assumed that thegenerated sidebands are always smaller compared to theoptical pump.In this Letter we present results of our numerical sim-ulations of Kerr frequency combs produced in a non-linear microresonator with normal GVD. For the sakeof specificity, we present a steady state solution for thecomb generated in a CaF WGM resonator pumped at1554 nm. The resonator is characterized with 667 µ mdiameter, 100 GHz FSR, n = 1 .
43 refractive index,and β ≃ .
055 ps / km GVD, corresponding to 110 kHzfrequency difference between adjacent FSRs ( ν + + ν − − ν ≃ −
110 kHz, where ν is the linear frequency of thepumping light, ν ± are the frequencies of the optical side-bands, ν + − ν ≃ ν − ν − ≃
100 GHz). The resonator hostmaterial has cubic nonlinearity n = 1 . × − cm / W,and mode volume V = 1 . × − cm . We assume thatthe full width at the half maximum of the overloadedWGMs is 2 γ = 220 kHz, so that ν + + ν − − ν = − γ .To find the GVD we used CaF Sellmeier equation[12] and an asymptotic expression describing the spec-trum of a dielectric spherical resonator [13]. The over-all dispersion β ( ν ) was calculated using the formula ν + + ν − − ν = − πcβ ( ν )( ν + − ν − ) / n ( ν ) [14]. Theselected value of GVD ( β ) seems to be small, since it ismuch smaller compared with the dispersion of a conven-tional optical fiber, however the significance of GVD isdetermined by the ratio | ν + + ν − − ν | /γ in a resonator[10]. The GVD is considered small if this parameter ismuch less than unity. Since the difference ν + + ν − − ν depends on the FSR, larger resonators have smaller rel-ative GVD than smaller resonators. Resonators withhigher Q-factor have larger relative dispersion comparedwith lower Q resonators.To study the behavior of the comb numerically itis convenient to introduce a coupling constant [10] g = ¯ hω cn / ( V n ) = 3 . × − s − , where ω =2 πν . Selecting the pump power P = 36 . µ W,we also introduce the dimensionless pumping constant f = ( F / πγ )( g/ πγ ) / = 2 . F =(4 πγ P/ (¯ hω )) / describes the amplitude of the contin-uous wave external pump. The selection of the abovelisted parameters is an arbitrary one, simply serving thegoal of comparison of the results obtained in our numer-ical simulations with parameters of a realizable system.In our model we numerically study the nonlinear in-teraction of 21 optical modes. The selected number ofmodes is limited by the available computational capac-ity. We expect that all the modes are identical and com-pletely overlapping in space. The external continuouswave pump is applied to the central mode of the modegroup, so the simulated Kerr comb is expected to haveten red- and ten blue-detuned harmonics with respect tothe frequency of the pumped mode. We take into ac-count only the second order frequency dispersion that isrecalculated to the frequency of the modes [5].To write the nonlinear equations we introduce an in-teraction Hamiltonian ˆ V = − g (ˆ e † ) ˆ e /
2, where ˆ e = P j =1 ˆ a j . The equations of motion are [5, 10]˙ˆ a j = − ( γ + iω j )ˆ a j + i ¯ h [ ˆ V , ˆ a j ] + F e − iωt δ ,j , (1)where δ ,j is the Kronecker’s delta. The set of nonlin-ear forced equations, (1), is solved numerically withoutany further assumptions. The evaluation was interruptedwhen the solution reached its steady state ( γ t s ≈ -20 -10 0 10 200.00.51.01.52.02.5 -20 -10 0 10 20-1.5-1.0-0.50.00.51.01.5 -4.2 -4.0 -3.8-0.34-0.32-0.30-4.2 -4.0 -3.81.321.34 N o r m a li z ed a m p li t ude Normalized detuning P ha s e , r ad Normalized detuning
FIG. 1: Normalized slow amplitude ( | A | ( g/ πγ ) / ) andphase of the intracavity field of the pumped mode ver-sus normalized detuning (( ω − ω ) / πγ ). The region ofthe maximum field accumulation is shifted to smaller fre-quencies due to the self-phase modulation effect (the effec-tive index of refraction increases with intensity of light cir-culating in the mode). The pump power is fixed, f =( F / πγ )( g/ πγ ) / = 2 . The results of simulations are summarized in Figs. (1)-(4). To present the results we introduce slow amplitudesof the field in the modes ˆ a j = A j exp( − iω j t ). The slowamplitudes are further normalized as | A j | ( g/ πγ ) / .The frequency detuning is also normalized to the halfwidth at half maximum of the resonator mode, (( ω − ω ) / πγ ). As a rule, the solution followed stablebranches (shown by solid blue line) and then jumped to the attractor corresponding to comb generation (shownby the red line). The jump was observed only for thecase of nonzero initial conditions. Naturally, the solutiondoes not converge to unstable solutions.We find that a Kerr comb can be generated in theresonator in the case of normal GVD. For the resonatordescribed above we obtain the power and the frequencyof the pumping light that result in comb excitation. Thefrequency comb is dynamically stable, and the genera-tion onset is hard in its nature, i.e. there is a discontinu-ous jump in parameters of the comb harmonics when thepump power exceeds a certain threshold. N o r m a li z ed a m p li t ude Normalized pump rate
FIG. 2: Normalized slow amplitude ( | A | ( g/ πγ ) / ) andphase of the intracavity field of the pumped mode ver-sus the normalized amplitude of the external pump ( f =( F / πγ )( g/ πγ ) / ). The frequency detuning is fixed( ω − ω ) / πγ = −
4. Stable solutions of the set (1) thatdo not reveal Kerr comb generation are shown by blue lines.The stable solution that reveals comb generation is shown bythe red line. The solution is localized. The inset shows theregion of parameters where the comb is generated.
The behavior of the field of the mode pumped opticallyis illustrated by Figs. (1) and (2). If the comb sidebandsare small the stable oscillation occurs in the region that is unstable . This stability region was not found in previousresearch [5] since only solutions with adiabatically stableamplitude of the pump mode were analyzed there. Thecorresponding behavior of amplitudes of the first two op-tical sidebands ( A and A ) is shown in Fig. (3). Thestability region where the sidebands are generated is lo-calized with respect to the frequency and power of thepumping light.We found the time dependence of the overall field am-plitude in the resonator (Fig. 4) by selecting specific val-ues of power and frequency of the pump light. Appar-ently, the modes of the frequency comb create a singleoptical pulse traveling in the resonator. The shape ofthe pulse is rather complex and more effort is requiredto derive an analytical expression for its description. Weverified that if the power of the external pump is changeda similar pulse shape, but with different amplitude andduration, is generated in the resonator .The initial conditions and parameters of equations are N o r m a li z ed s i deband a m p li t ude Normalized pump rate Normalized detuning
FIG. 3: Normalized slow amplitude ( | A | ( g/ πγ ) / and | A | ( g/ πγ ) / are the same) of the first sidebands of thepump mode for the same parameters of the system as used inFigs. (1) and (2). -4 -3 -2 -1 (a) N o r m a li z ed a m p li t ude Time, ps (b) N o r m a li z ed a m p li t ude Mode number
FIG. 4: Normalized slow amplitude (( g | ˆ e † ˆ e | / πγ ) / ) of theintracavity field versus time, and the corresponding opticalcomb found for a fixed pump power f = 2 .
244 and fre-quency detuning of the pump light from the selected mode( ω − ω ) / πγ = −
4. The 100 GHz FSR of the resonator isexpected. varied to verify the stability of the solution. We also re-duced and increased the number of modes in our codeand confirmed that the spectrum does not change signif-icantly. It is worth noting that the method is suitable forthe simulation of a resonator with arbitrary GVD. Thelimited number of modes used in the simulation imposesa practical restriction on the values of the pump powerand GVD values on the outcome of the simulations. Weintentionally selected a large value of the relative GVDto suppress this undesirable effect.To conclude, we have studied Kerr frequency combgeneration in microresonators made out of a transparentoptical material possessing positive cubic nonlinearity, aswell as normal group velocity dispersion. We find thatthere exists a stable attractor for oscillation that resultsin comb generation. Stable Kerr frequency combs canbe produced in such resonators if the proper power andfrequency for the continuous wave light pumping a modeof the resonator is selected.This work was supported in part by DARPA MTO(IMPACT program). [1] T. J. Kippenberg, R. Holzwarth, and S. A. Diddams,Science , 555 (2011).[2] P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R.Holzwarth, and T. J. Kippenberg, Nature (London) ,1214 (2007).[3] I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping,and A. L. Gaeta, Phys. Rev. A , 043837 (2007).[4] I. H. Agha, Y. Okawachi, and A. L. Gaeta, Opt. Express , 16209 (2009).[5] Y. K. Chembo and N. Yu, Phys. Rev. A , 033801(2010).[6] M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. ,745 (1992).[7] S. Coen and M. Haelterman, Phys. Rev. Lett. , 4139(1997).[8] A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S.Ilchenko, and L. Maleki, Opt. Express , 4130 (2008). [9] A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solo-matine, D. Seidel, and L. Maleki, Phys. Rev. Lett. ,093902 (2008).[10] A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S.Ilchenko, and L. Maleki, Phys. Rev. A , 033804 (2005).[11] A. Matsko, A. Savchenkov, W. Liang, V. Ilchenko, D.Seidel, and L. Maleki, paper NWD2 in Nonlinear Optics:Materials, Fundamentals and Applications, OSA Techni-cal Digest (CD) (Optical Society of America, 2011).[12] M. Daimon and A. Masumura, Appl. Opt. , 5275(2002).[13] C. C. Lam, P. T. Leung, and K. Young, J. Opt. Soc. Am.B , 1585 (1992).[14] A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, L.Maleki, J. Opt. Soc. Am. B20