Dynamics of the Modulational Instability in Microresonator Frequency Combs
DDynamics of the Modulational Instability in Microresonator Frequency Combs
T. Hansson, ∗ D. Modotto, and S. Wabnitz Dipartimento di Ingegneria dell’Informazione, Universit`a di Brescia, via Branze 38, 25123 Brescia, Italy
A study is made of frequency comb generation described by the driven and damped nonlinear Schr¨odingerequation on a finite interval. It is shown that frequency comb generation can be interpreted as a modulationalinstability of the continuous wave pump mode, and a linear stability analysis, taking into account the cavityboundary conditions, is performed. Further, a truncated three-wave model is derived, which allows one to gainadditional insight into the dynamical behaviour of the comb generation. This formalism describes the pumpmode and the most unstable sideband and is found to connect the coupled mode theory with the conventionaltheory of modulational instability. An in-depth analysis is done of the nonlinear three-wave model. It is demon-strated that stable frequency comb states can be interpreted as attractive fixed points of a dynamical system.The possibility of soft and hard excitation states in both the normal and the anomalous dispersion regime isdiscussed. Investigations are made of bistable comb states, and the dependence of the final state on the way thecomb has been generated. The analytical predictions are verified by means of direct comparison with numericalsimulations of the full equation and the agreement is discussed.
PACS numbers: 42.60.Da, 42.65.Hw, 42.65.Ky, 42.65.Sf
I. INTRODUCTION
An optical frequency comb consists of a large number ofhighly resolved and nearly equidistant spectral lines. Fre-quency combs show great promise for many current andemerging applications. This includes numerous applica-tions to spectroscopy, precision frequency metrology, opti-cal clocks, synthesis of optical waveforms as well as chan-nel generation for wavelength division multiplexing telecom-munication systems [1]. Many of these applications rely oncreating radio-frequency beatnotes between known comb linefrequencies and other unknown optical frequencies, enablingradio-frequency measurements of optical phenomena. Thisrequires knowledge about the absolute frequency of the comblines, which can be determined with great precision by theself-referencing method if the optical comb spans a full oc-tave.Frequency combs can be generated either using mode-locked femtosecond lasers or using continuous wave (CW)pumped microresonator cavities. The light is confined withina small mode volume in microresonators, which enhances theintensity dependent nonlinear interaction, thereby enabling ef-ficient frequency conversion. Comb generation in microres-onators relies fundamentally on the parametric four-wavemixing process. This nonlinear process is responsible fortransferring energy from the pump mode and redistributing itamong the frequency sidebands. A special degenerate caseof four-wave mixing is the modulational instability, wheretwo pump photons are annihilated to create a signal and idlerpair at an equidistant frequency spacing from the pump. Thismechanism is responsible for creating the primary sidebands,and is the most important frequency conversion process whichoccurs in microresonators.In this paper we consider microresonator based frequencycombs in the formalism of a driven and damped nonlinear ∗ E-mail address: [email protected]
Schr¨odinger (NLS) equation. Our aim is to get a better un-derstanding of the generation of Kerr frequency combs andtheir long-term dynamical behaviour. The principal path tofrequency comb generation is to start with an empty cavity anduse the modulational instability to create sidebands from theunstable steady-state CW solution of the pump mode. Sincea given comb state can be represented as a point in an infi-nite dimensional phase space, this can be interpreted to implythat a trajectory exists which connects the modulationally un-stable CW solution with the final comb state. A comb that isgenerated in this manner is known as a soft excitation [2]. Alinear modulational instability analysis can be used to deter-mine when the steady-state CW pump solution becomes un-stable, but it cannot make any predictions about the final combstate. Moreover, stationary comb states may also exist whichare isolated in phase space in the sense that their trajectoriesare not connected with the CW solution. These are converselyknown as hard excitations. In this paper we use a finite modetruncation to limit the size of the phase space to four dimen-sions. This is accomplished by using a three-wave mixingmodel that takes into account the pump mode and the domi-nant sideband pair. This approximation can be justified by thefact that a substantial part of the total energy will be containedwithin these three modes [3]. The location of different solu-tions representing either stationary or breather type states canthus be determined by studying the location and stability offixed point curves of the three-wave model.An analysis using a similar three-mode model was previ-ously presented in [2], c.f. also [4], where the concept of softand hard excitations was introduced. In this paper we make amore detailed analysis, which is neither limited to the anoma-lous dispersion regime nor assumes a particular physical sys-tem with a fixed dispersion.The driven and damped NLS equation and the cavityboundary condition are introduced in section II together withthe normalization used throughout the rest of the article. Sec-tion III contains an analysis of the modulation instabilitywhich is used for later comparison with the three-wave modeland to determine the dominant sideband pair. The truncated a r X i v : . [ phy s i c s . op ti c s ] J un three-wave mixing model is derived in section IV togetherwith a set of equations for the location of its fixed points. Theinfluence of the comb parameters on the dynamics is consid-ered in section V, which describes different regimes of combgeneration. Section VI discusses the dependence of the finalcomb state on the route used to generate the comb. The finalsection contains conclusions and a discussion about the agree-ment of the model with numerical simulations. II. THE DRIVEN AND DAMPED NLS MODEL
Most theoretical descriptions of microresonator frequencycombs to date have been carried out using a modal expan-sion approach, which describes the slow evolution of the combspectrum using coupled mode equations [4, 5]. This approachis not very suitable for direct modelling of temporal struc-tures and may be cumbersome to apply to broadband combs,which in the case of octave spanning combs can comprise hun-dreds or even thousands of resonant modes. An alternative de-scription of microresonator frequency combs was proposed byMatsko et al. in [6], where the intracavity field is modelled inthe time-domain using a mean-field driven and damped non-linear Schr¨odinger (NLS) equation, viz. τ ∂ A ∂ τ + i β ∂ A ∂ t − i γ | A | A = − (cid:18) α + T c + i δ (cid:19) A + √ T c A in . (1)Where t is the ordinary time that describes the temporal struc-ture of the field inside the cavity, and τ is a slow time describ-ing the evolution of this structure over successive round-trips.This equation is also known in the literature as the Lugiato-Lefever equation [7, 8], and has previously been used to modelthe nonlinear dynamics of dispersive fiber ring cavities [9],where the equation is obtained as the mean field limit of aninfinite-dimension Ikeda map [10].In order that only resonant frequencies should contribute tothe total field inside of the cavity, it is necessary to subjectEq.(1) to the periodic boundary condition A ( t + τ , τ ) = A ( t , τ ) (2)where τ is the cavity round-trip time. This condition ensuresfrequency selectivity and is appropriate for resonators pos-sessing a high quality factor. The spectral location of the cav-ity eigenfrequencies are assumed to be given by the Taylor ex-pansion ω µ = ω + D µ +( D / ) µ , c.f. [11], and the bound-ary conditions enable the frequency sampling of the resonancespectrum to be made using an equidistant frequency step witha spacing corresponding to the free-spectral-range ( D ). Cav-ity dispersion arises from the second term of Eq.(1) with thedispersion coefficient β = − D ( τ / D ) . The generalizationto higher-orders of dispersion is easily accomplished, see [8].The remaining parameters in Eq.(1) are defined as follows: γ is the nonlinear coefficient, α is the loss per cavity round-trip, T c is the coupling coefficient, δ is the detuning of the pump frequency and A in is the external pump field. In deriv-ing Eq.(1) it is implicitly assumed that the cavity modes aredegenerate so that light is localized in only one spatial mode.One must further neglect the frequency dependence of bothabsorption and coupling coefficients.Eq.(1) can be simplified by normalizing it to reduce thenumber of parameters. It is convenient to choose the normal-ization so that the complete cavity loss ¯ α = α + T c / = τ = π , giving ∂ A ∂ τ + i β ∂ A ∂ t − i | A | A = − ( + i δ ) A + f (3)where ˜ A = (cid:112) γ / ¯ α A , ˜ τ = ( ¯ α / τ ) τ , ˜ t = ( π / τ ) t , ˜ β =( β / α )( π / τ ) , ˜ δ = δ / ¯ α and f = ( √ γ T c / ¯ α / ) A in , withthe tilde which has been dropped in Eq.(3) referring to thenew system. Eq.(3) has three independent parameters, con-trary to the case of the driven and damped NLS equation onthe infinite line which has only two [12]. The reason forthis difference is that an extra parameter appears due to theboundary condition. We have taken these parameters to bedispersion ( β ), detuning ( δ ) and the external pump field ( f ).The boundary condition in Eq.(2) is changed to A ( t + π , τ ) = A ( t , τ ) , which corresponds to a unit free-spectral-range. III. MODULATIONAL INSTABILITY
Eq.(3) is well known to have a bistable behaviour [7], andcan have either one or three steady-state continuous wave(CW) solutions with amplitude A for a given pump intensity | f | . These solutions satisfy the equation | f | = | A | (cid:104)(cid:0) δ − | A | (cid:1) + (cid:105) (4)which is single valued for detunings δ ≤ √
3, while exhibitingbistability for detunings δ > √
3. The middle, negative slopebranch between | A | = ( δ ± (cid:113) δ − ) / A = A ( t , τ ) , viz. A ( t , τ ) = A ( τ ) + ∑ k A k ( τ ) exp ( − ikt ) (5)where k is an integer, and project the resulting equation ontoa Fourier basis by multiplying with exp ( i µ t ) / π and integrat- ing over t . This produces a system of coupled mode equationsequivalent to those used in the modal expansion approach[4, 5]. In this paper we consider the dynamics of a finitemode truncation of this infinite dimensional coupled systemof equations. Specifically, a three-wave model composed ofthe pump mode ( A ) and one pair of sidebands at integer fre-quency µ ( A µ , A − µ ). The coupled mode equations then takethe form ∂ A ∂ τ = i (cid:0) | A | + | A − µ | + | A µ | (cid:1) A − ( + i δ ) A + i A ∗− µ A ∗ µ A + f , (6) ∂ A − µ ∂ τ = − i β µ A − µ + i (cid:0) | A | + | A − µ | + | A µ | (cid:1) A − µ − ( + i δ ) A − µ + iA ∗ µ A , (7) ∂ A µ ∂ τ = − i β µ A µ + i (cid:0) | A | + | A − µ | + | A µ | (cid:1) A µ − ( + i δ ) A µ + iA ∗− µ A . (8)Assuming that the amplitudes of the sidebands are small,we follow the classical procedure and linearize the above sys-tem of equations around the steady-state CW solution whilelooking for plane-wave perturbations, satisfying A µ = ae λτ , A − µ = A ∗ µ . The instability growth rate λ ( µ ) is then found tobe given by the following dispersion relation: λ = − ± (cid:113) I − ∆ k (9)where we have introduced the pump mode intensity I = | A | and the wave vector mismatch ∆ k = β µ + I − δ . It is eas-ily seen from Eq.(9) that the maximum gain occurs when thismismatch is zero, i.e. for frequencies β µ = δ − I , or inthe event that this equations lacks real solutions for µ = λ max = I − I < λ =
0, is obtainedwhen β µ = δ − g ± (10)where we have defined g ± = I ± (cid:113) I −
1. This equationmust have real solutions, i.e. there must exist a µ correspond-ing to a real frequency, in order for an instability to occur.For anomalous dispersion (i.e. β < δ < g + , while for normal dispersion ( β >
0) the required de-tuning is δ > g − . The solution will additionally be unstableto CW perturbations in the range g − < δ < g + .The comb spectrum will consist of a discrete set of frequen-cies, due to the periodic boundary condition. This implies thatthe above gain spectrum must overlap with the location of atleast one cavity resonance. In certain cases it may happen that the gain spectrum falls in between two resonances for aparticular pump power level, in which case there is a stabilitywindow where the steady-state CW solution is stable [13].Fig. 1a shows an example of the MI growth-rate for the firstthree sidebands as a function of the pump mode amplitude.The different gain curves correspond to increasing integers of µ with the first gain curve representing the CW instability, i.e. µ =
0. In Fig. 1b, we see a typical gain curve for a fixed pumpmode intensity. Only the first sidebands is unstable in thiscase since the only frequencies that are allowed correspond tointeger µ ’s. (a) Vs pump mode amplitude: δ = , β = − , µ = , , , δ = , β = − , | A | = . FIG. 1: Modulational instability growth rate for pump modeand frequency sidebands.
IV. THREE-WAVE MIXING
Although the linear MI analysis can be used to predict whenthe CW solution becomes unstable, and the initial growth rateof the instability, it does not provide any additional informa-tion about the subsequent dynamics or the stability of the fi-nal state. However, important insight into the dynamics canbe obtained by considering finite mode truncations. The sim-plest such truncation is the three-wave model: consisting ofthe pump mode and the dominant sideband pair. This allowsus to reduce the infinite dimensional phase space to one ofonly four dimensions. The approximation is motivated by thefact that a substantial part of the comb energy will be con-tained in these three modes [3]. While this procedure neglectsany additional sidebands and can thus only provide an approx-imate description of comb generation, it is nevertheless usefulfor predicting the range of comb stability and the location offixed point curves.We consider the truncated three-wave model Eqs.(6-8) andintroduce a new set of dynamical variables, viz. η = | A | / P , φ = φ − µ + φ µ − φ , P = | A − µ | + | A | + | A µ | , θ = φ f − φ . (11) These correspond to the normalized pump mode intensity ( η ),the relative phase between pump mode and the sidebands ( φ ),the total intensity ( P ), and the pump phase detuning ( θ ), re-spectively. The normalized pump mode intensity η , providesa single scalar parameter that has a constant value differentfrom one, whenever a stable comb state is reached.We make the assumption that the sideband amplitudes areequal, i.e. | A − µ | = | A µ | = | a | . The difference between theamplitudes can in general be shown to be invariant. The trun-cated model is then found to be governed by the following setof dynamical equations: ∂ η∂ τ = − η ( − η ) (cid:20) P sin φ − | f |√ P η cos θ (cid:21) , (12) ∂ φ∂ τ = ( β µ − P ) + P η − P ( − η ) cos φ − | f |√ P η sin θ , (13) ∂ P ∂ τ = − P + | f | (cid:112) P η cos θ , (14) ∂ θ∂ τ = ( δ − P ) + P η − P ( − η ) cos φ − | f |√ P η sin θ . (15)The limit of the system without pumping and loss was in-vestigated in [3]. In this limit the system is conservative andreduces to two dimensions ( η , φ ) so that the dynamics fol-lows closed orbits in this plane. Seen as a projection onto the( η cos ( φ ) , η sin ( φ ) ) plane, the dynamics takes place withinthe unit circle, with η = η (cid:54) = , θ and φ , to obtain the followingtwo coupled equations for the absolute pump mode intensity I = | A | = P η and the normalized pump mode intensity η : | f | I = (cid:20) ( δ − I ) − ( − η ) η g ± (cid:21) + η , (16) β µ = δ − g ± − I ( − η ) η . (17)Eqs.(16-17) are generalizations of Eq.(4) and Eq.(10), towhich they reduce in the limit of η →
1. It is possible toobtain a lower limit for the normalized pump mode intensity η corresponding to the threshold intensity from Eq.(16), whichshows that 1 / | f | ≤ η ≤ , ≤ I ≤ | f | η . (18) With the limits for the total intensity of the comb being givenby P = I / η . Note that the analysis predicts that η maybecome less than 1 /
3, at which point the sidebands becomelarger than the pump mode.Eq.(17) gives the same result as before for the stability ofthe pump mode when there is no power initially in the side-bands, i.e. when η =
1. However, in the case when η < µ is real in the anomalous dis-persion regime ( β <
0) whenever δ − g ± < I ( − η ) / η .The parameter range where solutions can exist is determinedby the right hand side of the inequality, which has a mini-mum of zero for η = I ( | f | − ) / η = / | f | . Note that the parameter range for solution in thenormal dispersion regime ( β >
0) is smaller when the side-bands are initially excited since the solutions must now satisfythe inequality δ − g ± > I ( − η ) / η .Eqs.(16-17) can also be combined to yield a single implicitequation for I , viz.32 | f | h = (cid:20) ˆ δ − (cid:18) g ± − I (cid:19) h ˆ κ (cid:21) + h ˆ κ (19)where ˆ δ = δ + g ± − I , ˆ κ = κ + g ± − I / κ = β µ and h = I / + ( g ± − I / ) . (20)The implicit equation is valid under the additional restrictionthat ˆ δ − ˆ κ ≥ I /
2, corresponding to the requirement that 0 < η < I > V. REGIMES OF COMB GENERATION
The three-wave model shows that comb generation in mi-croresonators can display qualitatively different dynamics, de-pending on the values assumed by the free parameters and theapplied initial conditions. Stable frequency comb states can,as we have argued using the phase space interpretation, beclassified as either soft or hard excitations [2]. A soft exci-tation is a comb state which can be reached in an adiabaticmanner when starting from zero initial conditions. With a sta-ble comb state, which in contrast is not adiabatically reachablefrom zero initial conditions, being known as a hard excitation.To reach a hard excitation state normally requires initial con-dition where the sidebands are already excited but they maysometimes also be reached by abrupt changes in pump inten-sity or detuning.The maximum growth rate for the modulational instabilitygives a minimum threshold intensity for the pump mode, viz. I =
1. However, comb generation will usually not initiate atthis point since the solution is part of a curve of fixed pointsthat lies outside of the region that is modulationally unstable,c.f. [5]. This can be seen in Fig. 2a which shows the am-plitude of the mode pump as a function of the detuning. Theblue curve shows the amplitude variation of the CW solution,while the red and green curves are the fixed point solutionsof Eqs.(16-17), with green color denoting stable solutions andred color unstable solutions. The shaded area shows the re-gion where the CW solutions are predicted to be modulation-ally unstable. The threshold intensity is indicated by the line | A | = √ I =
1. The stability of the solutions is determined by considering the eigenvalues of the linearization of Eqs.(16-17) around the steady-state solution. The first fixed pointwhich overlaps with a modulationally unstable CW solutionof the system Eqs.(16-17) appears when η = | f | = ˆ I (cid:2) ( δ − ˆ I ) + (cid:3) with ˆ I = (cid:104) ( δ − κ ) + (cid:112) ( δ − κ ) − (cid:105) / δ − κ > √
3. The minimum pump inten-sity for this fixed point curve (upper green curve in Fig. 2a)is obtained when the detuning δ = κ + √
3, which impliesthat ˆ I = / √ | f | = ( / √ ) (cid:104)(cid:0) κ + / √ (cid:1) + (cid:105) , cor-responding to the critical threshold power found in [5]. Thesolution at the threshold intensity I = | f | = (cid:18) δ − + κ (cid:19) + ( + κ ) . (21)We now consider frequency comb generation in the anoma-lous dispersion regime for a fixed pump intensity. Figure 2ashows a case similar to that considered by Matsko et al. in[2]. The upper green curve in Fig. 2a is a fixed point curve ofsoft excitation solutions, which can be reached adiabaticallyby slowly increasing the frequency detuning of the pump un-til the curve of the CW solution intersects the modulationallyunstable region. The lower fixed point curve of stable combstates lies outside of the MI region, and can therefore not bereached directly from the CW pump mode. However, it is stillpossible to generate a solution laying on this curve by slowlychanging the detuning, at least for the parameter values con-sidered in the figure. This is accomplished by first traversingthe upper fixed point curve, after which the solution will jumpto the lower fixed point curve. Consequently we have a casewhere a phase space trajectory exists which connects one sta-ble solution set with another. The lower curve of fixed points,which is considered a hard excitation in [2], can in this waybe reached in an adiabatic manner without the need for abruptchanges in either detuning or pump power.In Fig. 2b we consider the same parameters as in Fig. 2a,with the exception of the dispersion which is taken to be twiceas large. It is seen that the upper fixed point curve startsat a smaller detuning than before, with the discontinuity be-tween the two curves happening outside of the bistable region.The lower fixed point curve can therefore be reached directlyfrom the CW solution. E.g. choosing the pump detuning tobe δ = (a) | f | = , β = − | f | = , β = − | f | = , β = − | f | = , β = − FIG. 2: Intracavity pump mode amplitude as a function ofdetuning for fixed external pump amplitude. The CWsolution is shown by the blue curve. Green curves are stablefixed point curves while red curves are unstable. The shadedarea is the region where the CW solution is modulationallyunstable.The three-wave model suggests that stable comb statescould also exist for the zero dispersion case ( κ =
0) as long asthe system is bistable, i.e. δ > √
3. These states are not pre-dicted by the linear analysis, c.f. [5], and can only be reachedby hard excitations. Comb generation from MI would nor-mally not be expected to occur for zero dispersion since MIis fundamentally an interplay between dispersion and nonlin-earity which cannot take place without the presence of botheffects. However, the detuning provides an extra degree offreedom which enables Eq.(3) to exhibit MI in the normal dis-persion regime [15] and also for the zero dispersion case. Thetheory does not predict stable soft excitation comb states closeto the zero dispersion point, which may seem contradictoryto the experimental observations which have been made ofhighly equidistant frequency combs [16]. But one should re-member that mode pulling through thermal and nonlinear ef-fects can change the path length and the refractive index prop-erties of the resonator so that the spectrum becomes nearly equidistant. The three-wave model thus only shows that combgeneration cannot be initiated in a cold resonator through MIunless the dispersion is non-zero, while the final comb state ofthe warm and populated resonator may correspond to a hardexcitation state. Moreover, close to the zero dispersion point itwill also become necessary to include the effects of higher or-ders of dispersion, which will limit the attainable comb width.It should furthermore be noted that the zero dispersion caseis a limit where the assumptions behind the three-wave modelbreak down, since no particular pair of sidebands can be con-sidered to be dominant. It is in fact easily seen that Eq.(3)does not have any continuous steady-state solutions for zerodispersion, except for flat-locked CW solutions, c.f. [12].Regardless, numerical simulations of the driven anddamped NLS equation show that ultra wideband frequencycombs can indeed be generated near the zero dispersion pointeven if a very small anomalous dispersion is present. It is nec-essary to include a small amount of dispersion not only be-cause the zero dispersion limit is unstable but also because ofnumerical limitations which require the comb spectrum to befinite. E.g. using parameters δ = , | f | = . , β = − − and a starting value of | A ( ) | = .
2, produces a frequencycomb consisting of nearly 40000 resonant modes. Decreas-ing the dispersion further would allow the generation of evengreater combs.Comb generation occurs also in the normal dispersionregime. However, most of these frequency comb states areonly hard excitations, c.f. [17]. A typical example is foundin Fig. 3, which shows the stable fixed point curve laying in-side the region exhibiting bistability of the CW solution, whileoutside of the MI region and not connected in any way withthe CW solution curve. This fixed point curve represents anisolated solution set in phase space, whose trajectories do notconnect with the state corresponding to zero initial conditions.They can therefore not be reached in an adiabatic manner andare hard excitation states. For the system to reach these so-lutions the initial condition need to be non-zero, which willgenerally require the sidebands to already be present insidethe resonator. However, it may sometimes be possible to ex-cite these states from MI of the CW solution by some abruptchange in either pump intensity or detuning: a possible routefor exciting these solutions could e.g. be to rapidly change theexternal pump intensity in a manner so that the internal pumpmode power falls into the MI region when starting from aninitial state on the stable steady-state CW solution above thefixed point curve.Using the three-wave model it is found that it is also pos-sible to generate soft excitation frequency combs in the nor-mal dispersion regime. Such a case is demonstrated in Fig.5a, which shows a stable fixed point curve located in the re-gion where the CW is modulationally unstable. It can berigourously proved that stable comb generation by means ofsoft excitations is possible in the normal dispersion regimewithin the range defined by √ + κ < δ < (cid:16) κ + (cid:112) κ + (cid:17) (22)FIG. 3: Hard excitation in the normal dispersion regime;intracavity pump mode amplitude as a function of externalpump amplitude for fixed detuning: δ = , β = . δ > / √ κ > / √
3, see Fig. 4.FIG. 4: Different parameter regions where soft and hardexcitations are possible.However, it should be noted that the solution need not al-ways converge to a stable comb state, even when such a stateis predicted to exist by the three-wave model. The three-wavemodel is only capable of predicting local and not global sta-bility. As there is often a competition between two differentstable states, one being the frequency comb state and the otherthe steady-state CW solution, it is usually required that the MIbe able to act under a sufficient long time for the system toend up in the comb state, allowing the amplitude of the side-bands to build-up. This will usually only be the case if theparameters are sufficiently large, c.f. Fig. 5a. Fig. 5b is theresult of a numerical simulation of a soft excitation in the nor-mal dispersion regime, showing the intensity and spectrum ofthe frequency comb. The comb was generated from noise us- ing MI, for the parameters of Fig. 5a. A closer look at theMI gain spectrum for this case reveals that the gain for thefirst sideband is separated from the gain of the CW instabil-ity, by a sizable stability window which inhibits the systemfor jumping to the upper CW solution. The parameter rangefor which soft and hard excitations are possible is shown inFig. 4. Soft excitations have been found to always be possiblein the anomalous dispersion regime, and also within a part ofthe normal dispersion region. These soft excitations were notfound by Matsko et al. in [17] as they only considered a spe-cific set of resonator parameters laying outside of the rangegiven by Eq.(22). (a) Intracavity pump mode amplitude as a function of external pumpamplitude for fixed detuning: δ = , β = | f | = FIG. 5: Example of soft excitation frequency combgeneration in the normal dispersion regime.
VI. DEPENDENCE ON ROUTE IN COMB GENERATION
Frequency combs can have either one or two stable curvesof fixed points for a particular set of parameters. The res-onator thus exhibits bistability and hysteresis not only for theCW solution but also for comb states when the sidebands areexcited. It therefore becomes important to consider the waythat frequency combs are generated, since variations in theroute may produce different final states.Fig. 6a shows an example where two different fixed pointcurves exist for the same set of comb parameters. We con-sider a case where the field inside the resonator is initially zeroand the pump intensity is set above the threshold intensity forbistability of the CW solution. The pump mode amplitudewill then first rise to a level corresponding to the CW solutionon the upper curve, but since this solution is in the MI region,it will be unstable and the system will instead approach thestable upper fixed point curve (point 1). However, if the ex-ternal pump intensity is later slowly reduced beyond the pointwhere the upper curve ends, then the system will jump downto the lower fixed point curve (point 2). If the pump intensityis now slowly increased again to its original value, then wefind that the system will stay on the lower curve (point 3). Asimilar procedure involving the detuning would also allow usto reach different comb state corresponding to identical combparameters in an adiabatic manner.A numerical simulation showing this is found in Fig. 6b,which is a plot of the spectral intensity as a function of theevolution time. The figure was obtained by solving Eq.(3)with a time varying pump intensity | f | as illustrated by theblue line in the figure (right hand scale). The frequency combis clearly seen to display qualitatively different dynamics forthe first half of the figure as compared to the second half. E.g.even though the external parameters are the nearly same, wefind the comb state at τ (cid:39)
60 to correspond to a stable soli-ton state with a stationary intensity profile while the comb at τ (cid:39) VII. CONCLUSIONS
While the three-wave model can only give an approxima-tion of the full comb dynamics, it often produces remarkablygood agreement. The three-wave model is the lowest ordernon-trivial finite mode truncation that can be made. The in-clusion of additional modes could provide an even better ap-proximation, but requires a more involved analysis since thephase space dimensionality grows quickly. Comparisons be-tween numerical simulations of both of the three-wave mix-ing model and the full driven and damped NLS Eq.(3), haveshown that the three-wave model is useful in finding the lo-cation of frequency comb states as well as helping to predictwhere comb generation may be stable. Unfortunately, it is notpossible to use the model to predict absolute comb stabilitysince higher order sidebands are neglected. Pump mode andsidebands amplitudes are also not always accurate, especiallyfor large frequency sidebands.The three-wave model could be useful as a tool, togetherwith numerical simulations of Eq.(3), to help design microres-onator devices capable of generating specific comb states. Itcould e.g. be applied to finding specific routes which gener- (a) Bistable behaviour of comb states; intracavity pump modeamplitude as a function of external pump amplitude for fixeddetuning: δ = , β = − . FIG. 6: Simulation results showing bistable behaviour anddependence of route in frequency comb generation.ates soliton trains or octave spanning frequency combs. How-ever, it should be remembered that if mode pulling due to ther-mal and nonlinear effects is significant, it may be necessary toalso consider temporal changes in the dispersion, in order toaccurately model the complete excitation dynamics observedunder experimental conditions.In this article we have made a study of the dynamics of themodulational instability of microresonator based frequencycombs in the context of a formalism provided by the drivenand damped NLS equation. We have demonstrated that theprimary path to comb generation relies on the modulationalinstability of the CW pump mode, although other comb exci-tation routes are also possible and indeed necessary to reachcertain comb states. A linear stability analysis has been made,which has taken into account the proper boundary conditionsof frequency selective, high Q-factor, microresonators. Ad-ditionally, we have derived a truncated three-wave mixingmodel which describes both the dynamics and long-term be-haviour of different frequency comb states in a reduced, fourdimensional, phase space. The fixed points of the dynamicalsystem have been identified, with stable states correspondingto either stationary or chaotic frequency combs. A discussionof different regimes of frequency comb generation has beenmade. This discussion has highlighted the role of the combparameters in determining the excitation dynamics and shownthat not only the sign of the dispersion but also its magnitudeis crucial in determining the dynamical behaviour and differ-ent excitation routes. We have found that soft excitation combgeneration is possible even in the normal dispersion regime,and derived a range which shows where such excitations can occur. Finally we have considered the dependence on routein comb generation, and the fact that resonators may exhibitbistable behaviour not only for the CW solution but also fordifferent frequency comb states.
ACKNOWLEDGEMENTS
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