Transient Regime of Kerr Frequency Comb Formation
Anatoliy A. Savchenkov, Andrey B. Matsko, Wei Liang, Vladimir S. Ilchenko, David Seidel, Lute Maleki
aa r X i v : . [ phy s i c s . op ti c s ] N ov Transient Regime of Kerr Frequency Comb Formation
A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki
OEwaves Inc., 465 N. Halstead St. Ste. 140, Pasadena, CA 91107
Temporal growth of an optical Kerr frequency comb generated in a microresonator is studied both experi-mentally and numerically. We find that the comb emerges from vacuum fluctuations of the electromagnetic fieldon timescales significantly exceeding the ringdown time of the resonator modes. The frequency harmonics ofthe comb spread starting from the optically pumped mode if the microresonator is characterized with anoma-lous group velocity dispersion. The harmonics have different growth rates resulting from sequential four-wavemixing process that explains intrinsic modelocking of the comb.
PACS numbers: 42.62.Eh, 42.65.Hw, 42.65.Ky, 42.65.Sf
Kerr combs excited in nonlinear optical microresonatorshold promise as chip scale generators of octave spanning op-tical frequency combs with unique characteristics [1]. Theyalso reveal a rich, and as yet not well understood, variety ofnonlinear dynamical phenomena. Different regimes of combgeneration have been observed and several theoretical mod-els developed. While the mainfocus of the research in thesestudies is related to understanding the spectral properties ofthe Kerr comb, less work has been devoted to its time do-main behavior. Several experimental [2–5] and theoretical [6–9] studies of mode locking regimes of this nonlinear processhave been published very recently, but the problem of tem-poral growth of comb harmonics was addressed only theoret-ically [7, 8]. In this Letter we report on experimental studyof transient dynamics of Kerr frequency combs and also pro-pose a theoretical explanation for the observed results. Wealso suggest a numerical model, backed by the experiment,that predicts a different dynamics for Kerr comb growth com-pared with earlier theoretical predictions. Our research clearlyexplains why harmonics of the comb are modelocked.Kerr combs are generated from electromagnetic vacuumfluctuations due to modulation instability of a continuouswave (cw) light confined in an externally pumped nonlineardispersive resonator. When the power of the cw optical pumpthat is nearly resonant with one of the modes of the resonatorexceeds a certain threshold, the cw field inside the resonatorbecomes unstable, and multiple frequency harmonics are gen-erated in the modes. The harmonics are equally spaced dueto energy and photon number conservation laws, imposed byfour-wave mixing process (FWM), which is responsible forcomb generation [10, 11].The growth of the comb is not instantaneous. It was found[7, 8] that formation of a fully developed comb can take upto a hundred ring-down periods of the resonator mode. Thegoal of the present contribution is to measure this time inter-val directly. We performed a numerical simulation and foundthat the DC power of light exiting the resonator depends onthe degree of comb formation. The effective intrinsic qual-ity factor of the pumped optical mode suddenly drops longafter the steady state amplitude of the circulating light in themode is reached. The reduction of the intrinsic quality fac-tor changes the attenuation of the pump mode since the bal-ance between intrinsic loss and coupling loss in the resonatorchanges. Hence, the time delay between the moment the light enters the pump mode and the moment when the frequencycomb is generated can be directly measured by detecting thelight escaping the resonator. It is also possible to track thetime dependence of the power of the RF signal generated bythe comb to measure the delay. We performed such a mea-surement with a Kerr frequency comb produced in a high-QCaF whispering gallery mode (WGM) resonator, and experi-mentally confirmed the theoretical prediction of [7, 8] as wellas the result of our numerical simulations. In what follows wedescribe our simulations and the experiment in detail.To simulate the transient regime of the Kerr comb we usethe theoretical model developed in [8, 12]. We introduce aninteraction Hamiltonian ˆ V = − ¯ hg e † ) ˆ e , (1)where g = ¯ hω cn V n (2)is the coupling parameter obtained under assumption of com-plete space overlap of the resonator modes, ω is the value ofthe optical frequency of the externally pumped mode, c is thespeed of light in vacuum, n is the cubic nonlinearity of thematerial, V is the effective geometrical volume occupied bythe optical modes in the resonator, and n is the linear indexof refraction of the resonator host material. The operator ˆ e isgiven by the sum of annihilation operators of the electromag-netic field for 41 interacting resonator modes that we took intoconsideration: ˆ e = X j =1 ˆ a j . (3)The external cw pump is applied to the central mode of thegroup, so the Kerr frequency comb is expected to have twentyred- and twenty blue-detuned harmonics with respect to thefrequency of the pumped mode. We take into account onlythe second order frequency dispersion that is recalculated forthe frequency of the modes.Equations describing the evolution of the field in the res-onator modes are generated using Hamiltonian (1) and input-output formalism developed for ring resonators [13] ˙ˆ a j = − ( γ + iω j )ˆ a j + i ¯ h [ ˆ V , ˆ a j ] + F e − iωt δ ,j , (4)where δ ,j is the Kronecker’s delta; γ = γ c + γ i is the halfwidth at the half maximum for the optical modes, assumed tobe the same for the all modes involved; and γ c and γ i standfor coupling and intrinsic loss. The external optical pumpingis given by F = r γ c P ¯ hω , (5)where P is the value of the cw pump light. We neglect thequantum effects and do not take into account correspondingLangevin noise terms.The set (4) should be supplied with an equation describingthe light leaving the resonator. Assuming the pump powerdoes not depend on time, the relative amplitude of the outputfield is given by [13] ˆ e out ˆ e in = p − γ c τ − p − γ i τ γ c γ c + γ i γ ˆ eF , (6)where τ = 2 πRn /c is the light round trip time for the res-onator, and R is the radius of the resonator. It is assumed that ˆ e ( t − τ ) ≈ ˆ e ( t ) . -4 -3 -2 -1 R e l a t i v e a m p li t ude o f t he s i deband s , d B Normalized time, t(b)(a) N o r m a li z ed a m p li t ude Mode number
FIG. 1: (a) Optical frequency comb generated in the WGM res-onator for selected numerical parameters of the system. (b) Timedependence of the normalized amplitudes of the first five harmon-ics of the optical frequency comb generated in the WGM resonatorcharacterized with ( g/γ ) / = 5 × − . We solved Eqs. (4) and (6) numerically taking into accountthe interaction of 41 optical modes. Since the number ofmodes in our numerical simulation is limited, we selected aresonator with large GVD to be sure that the spectral bound-ary conditions do not impact the nonlinear process. We as-sumed ω − ω − ω = − γ , ( F /γ )( g/γ ) / = 1 . ,and ω = ω − . γ , and found that the resonator generatesthe optical frequency comb with spectrum shown in Fig. (1a).The comb has a frequency harmonic in each second opticalmode (for the selected values of the parameters of the sys-tem), which sometimes is observed experimentally (see Fig. 5 in [14]). The different regimes of Kerr comb generation willbe studied elsewhere, while in this work we focus on the tran-sient processes.We performed a numerical simulation of the growth of thefirst five harmonics of the frequency comb Fig. (1b) revealingseveral important features of comb generation. The sidebandsgrow exponentially with different growth rates. The first har-monic ((1) in Fig. 1) has the slowest growth rate of γ , thesecond harmonic ((2) in Fig. 1) has twice faster growth rate, γ , the third harmonic – γ . The first harmonic starts togrow much earlier than others, the second harmonic starts be-fore the third, etc. Such a temporal behavior shows that theparticular realization of the Kerr comb is initiated by hyper-parametric oscillation [12] that involves only two optical side-bands closest in frequency to the pump. The next order ofoptical harmonics is generated in the stimulated comb due toFWM of the already generated sidebands and the pump light[14]. This stimulated process does not have a threshold. Inother words, the threshold of comb generation coincides withthe threshold of the hyper-parametric oscillation. The lowestorder harmonics also determine the phase as well as the fre-quency of the rest of the harmonics. Such a comb is alwaysphase locked and optical pulses are formed in the resonator (Fig. 2). R e l a t i v e po w e r P ha s e , r ad Normalized time, t/ FIG. 2: Relative power and phase for the optical pulses leaving theresonator. The shape of these pulses is similar to the shape of thepulses reported in [2].
Another important observation is related to the temporal be-havior of the pump light. Since the growth of the comb har-monics is not instantaneous, the pump light is not influencedby the comb growth initially. The pump power is impactedonly after the frequency harmonics approach their saturationvalues. This behavior is clearly seen at the phase diagram(Fig. 3a), where the pump mode has two attractors. The firstone ((I) in Fig. 3a) corresponds to the steady state solution forfor the pump light in the nonlinear resonator with no harmon-ics generated, and the second attractor ((II) in Fig. 3a) corre-sponds to the steady state solution with the saturated comb.The duration of the transition process depends on the non-linearity parameter g/γ which defines the maximal numberof photons generated in the comb harmonics. The larger g is,the smaller is the number. On the other hand, the initial pho-ton number is equal to unity. The growth rate of the comb ison the order of γ . Therefore, the transient process is longerwhen g/γ is smaller. We have calculated the delay for twocases: ( g/γ ) / = 5 × − (estimated for a small resonator,e.g. [15]) and ( g/γ ) / = 10 − (estimated for a much largerresonator). The result of the calculation is shown in Fig. (3b).The behavior of the DC power of light leaving the resonatorhas a certain peculiarity. The value of the power exiting theresonator initially decreases and then increases (Fig. 3b). Thephenomenon can be explained from the stand point of criti-cal coupling [13]. There is no light at the output of a linearresonator if γ i = γ c and the steady state is reached; all thepump light is absorbed in the resonator. That is why the ampli-tude of the exiting light drops after the pump is on. A numberof the pump photons is redistributed between harmonics of thecomb, as the comb is generated. Those harmonics leave theresonator reducing the absorption of the pumping light. Theinterference phenomenon resulting in the critical coupling isalso deteriorated since the pump light confined in the corre-sponding mode changes its amplitude and phase due to thenonlinear process. (b)(a)II I I m ag i na r y pa r t o f t he pu m p a m p li t ude Real part of the pump amplitude R e l a t i v e ou t pu t DC po w e r Normalized time, t FIG. 3: (a) Transient behavior of the normalized amplitude ofthe electromagnetic field of the externally pumped optical mode, h ˆ a ( t ) i . Since the mode is initially empty, its amplitude increasesand reaches the attractor (I) approximately during resonator’s ringdown time, after the cw pump light is on. Then, after a certainperiod of time equal to the time interval needed for the comb togrow, the amplitude drops to a certain level and changes its phaseapproaching another attractor, (II). (b) The transient behavior ofthe light exiting the resonator is calculated for slightly overcoupledmodes, γ c = 1 . γ i , and for different values of nonlinearity g .The oscillator reaches its steady state faster for larger nonlinearity( ( g/γ ) / = 5 × − , solid blue line), and slower for smallernonlinearity ( ( g/γ ) / = 10 − , dashed red line). The temporal dependencies shown in Fig. (1b) and Fig. (3b)can be verified experimentally if one measures the power ofthe comb harmonics exiting the resonator. Instead of the directmeasurement of the optical harmonics, the power of the radio frequency (RF) signal generated on a fast photodiode by thecomb can be measured. We performed such experiments.We used a calcium fluoride (CaF ) whispering gallerymode (WGM) microresonator with loaded Q-factor . × (full width at the half maximum of the mode is 90 kHz, andcorresponding ring down time 1.75 µ s). The intrinsic Q-factorof the resonator was . × ( γ c = 1 . γ i ), which meansthat the attenuation of light in the modes was primarily givenby the interaction with the evanescent field coupler (a glassprism), and not by the scattering and loss of the resonator hostmaterial. The resonator was pumped using a 1545 nm dis-tributed feedback (DFB) semiconductor laser, self-injectionlocked to a selected resonator mode [16]. The laser was emit-ting approximately mW of power, 30% of which entered theresonator. FIG. 4: Schematic of the experimental setup. The WGM resonatoris pumped with a DFB laser, self injection locked to the selectedmode. The output light is analyzed with an optical spectrum ana-lyzer (OSA) showing the spectrum of the generated Kerr comb. Partof the light is sent to a fast photodiode (PD). The photocurrent, mod-ulated with a frequency equal to the comb repetition rate, is directedto an RF power detector (RFD), and a fast oscilloscope. This sig-nal is proportional to the convolution of the harmonics of the opticalfrequency comb. Some of the light is also detected with a slow pho-todiode (PD) and the photocurrent from this photodiode is forwardedto another channel of the same oscilloscope. This signal shows theintegral DC power leaving the resonator. Comparing the signals withthe fast oscilloscope we are able to measure the time delay betweenthe generation of the Kerr comb and the moment the pump light en-ters the corresponding mode.
We tuned the laser frequency by changing the laser current,and observed the temporal behavior of the signals at the OSAand the oscilloscope (Fig. 4). We noticed a significant de-lay between the start of the optical pumping of the mode, andthe moment of comb generation (Fig. 5). Some experimen-tal data are very similar in shape to the theoretical predictions(Fig. 5a,b), while other data are not (Fig. 4c,d). However thedelay is always there. The different shapes of the signals aremeasured since we selected different modes in the resonator.It was shown that the GVD of the modes change dependingon the resonator morphology [11] and depending on the ex-ternally pumped mode. In addition to the GVD, the couplingefficiency is also different for different modes. (d)(c)(b)(a)
Time delay, s
Time delay, s V o l t age a t t he P D , a r b . u . FIG. 5: Experimentally observed transient behavior of the Kerr fre-quency comb measured via monitoring the power of the DC signalon a slow photodiode generated by the light exiting the resonator(curves (a) and (c)) as well as the power of the RF signal generatedby the comb on a fast photodiode (curves (b) and (d)). Two modesare considered. The coupling to the mode resulting in the comb thatproduces in curves (a) and (b) is much lower compared with the cou-pling to the mode responsible for curves (c) and (d). The modes arealso characterized with different GVD values.
It may appear that conditions of the numerical simulationand the experiment described above are not exactly the same.In the experiment we used a large CaF resonator ( . mmin diameter) with free spectral range of 10 GHz. The GVDof modes of an ideal spheroidal resonator of this size is muchsmaller compared with the value of GVD we utilized in thenumerical simulations. Moreover, GVD is normal for the fun-damental mode sequence of the CaF resonator (assuming itsideal spheroidal shape). Nevertheless, the optical frequencycomb observed in the experiment has a spectral shape similar to the one obtained with the simulation, and the transient pro-cesses measured in the experiment have good correspondencewith the theory. This contradiction is resolved if we take intoaccount the notion that the morphology of a monolithic res-onator allows changing the sign and value of GVD [11]. Inthe experiment we did not use the ideal spheroidal resonatoror the fundamental mode sequence. We rather sent the lightto a higher-order mode of the resonator. The observed Kerrfrequency comb was generated for light tuned nearly at thetop of the WGM resonance ( ω ≃ ω ). This is possible onlyif GVD is anomalous and large for a locally selected modefamily, in accordance with the theory of hyper-parametric os-cillation [12]. That is why the numerical simulation for a res-onator characterized with large anomalous dispersion is appli-cable for the description of the experiment performed with aresonator seemingly having normal dispersion.To conclude, we have studied the transient regimes of Kerrfrequency comb formation in a nonlinear monolithic opticalresonator. We found that the well developed comb genera-tion is delayed by tens of ring down intervals of the resonator.Kerr combs created in larger and/or less nonlinear resonatorshave longer transient period compared with those generated insmaller and/or more nonlinear resonators. We noted that Kerrcombs generated in resonators possessing comparably largeanomalous GVD are always mode-locked. Finally, we ex-perimentally found that there exist mode families with largeanomalous GVD even in WGM resonators made out of a ma-terial with normal GVD. We experimentally validated the re-sults of our numerical simulations with a suitable mode fam-ily. [1] T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science , 555 (2011).[2] O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T.J. Kippenberg, in Practical Applications of Microresonators inOptics and Photonics,
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