Oscillatory dynamics in nano-cavities with non-instantaneous Kerr response Andrea Armaroli,∗ Stefania Malaguti, Gaetano Bellanca, and Stefano Trillo Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy Alfredo de Rossi and Sylvain Combri´e Thales Research and Technology, Palaiseau cedex, 91767 France (Dated: February 1, 2012) We investigate the impact of a finite response time of Kerr nonlinearities over the onset of spon- taneous oscillations (self-pulsing) occurring in a nanocavity. The complete characterization of the underlyingHopfbifurcationinthefullparameterspaceallowsustoshowtheexistenceofacritical value of the response time and to envisage different regimes of competition with bistability. The transitionfromastableoscillatorystatetochaosisfoundtooccuronlyincavitieswhicharedetuned far off-resonance, which turns out to be mutually exclusive with the region where the cavity can 2 operate as a bistable switch. 1 0 PACSnumbers: 42.65.Pc,42.65.Sf,42.55.Sa 2 n I. INTRODUCTION been recently predicted, owing to the free-carrier disper- a sion induced by two-photon absorption [13]. The main J 1 Self-pulsing(SP),theonsetofspontaneousoscillations, features of such mechanism is the existence of a critical 3 is a universal feature of nonlinear structures with feed- value of the time relaxation constant τ, as well as a wide back. As long as passive systems are concerned SP has regionoftheparameterspacewherestable(non-chaotic) ] been investigated theoretically in settings ranging from SPcanbepotentiallyobserved. Inthispaperweanalyse D isolated ring cavities [1–3] and parametric intracavity thedynamicsofanano-resonatorinthegood-cavitylimit C mixing [4–6] to Bragg gratings [7, 8] or grating-assisted when the underlying nonlinear mechanism is a Kerr-like n. backward frequency conversion schemes [9, 10], and it is nonlinearity with finite response time. Our analysis is i still a subject of active research [11–13]. In particular, based on the differential model proposed in Ref. [2]. In nl the dynamics of nonlinear passive cavities, whose study spite of the simplicity of such model, which makes it an [ hasbeenpioneeredintheeighties[1–3],isextremelyrich ideal prototype for understanding the role of relaxation encompassingstableaswellaschaoticSPwhichcancom- processes, a full characterization of SP and its competi- 1 pete with bistabilities and transverse effects [14]. His- tionwithbistabilityinthefullparameterspacewasnever v 1 torically, SP and chaos (the optical equivalent of strong reported(tothebestofourknowledge)afterRef.[2]. We 2 turbulence) have been first analyzed by means of delay- proposesuchanalysis,adoptingadifferentnormalization 6 differentialmodelsaccountingfortheround-tripdelayat with respect to that employed in Ref. [2], better aimed 6 each passage, which can be large in ring cavities. An at capturing the key role of the relaxation time. This is 1. instability named after Ikeda occurs in this framework especially important nowadays in view of assessing how 0 when the relaxation time of the nonlinear response is a given designed nano-cavity may be expected to behave 2 much shorter than the transit time [1] and has been bychangingthecharacteristicrelaxationtimeofthenon- 1 tested experimentally [15]. However SP and so-called linearity as a consequence of choosing different materi- v: weak turbulence occurs also in the opposite (say short als and/or adopting techniques for fine tuning their re- i cavity)limit,wherethedelaycanbeaveragedouttoend sponse time. We propose an analytical characterization X up with a differential mean-field model [2]. This regime of the SP instability and its competition with bistability r becomes important nowadays where nano-cavities are in the full parameter space, pointing out the qualitative a employedformanymodernphotonicapplications[16],in- similarity with the features observed for a nonlinearity cludingbistability[17],demonstratedinphotoniccrystal dominated by free-carrier dispersion [13]. Nonetheless, (PhC) membranes which offer great flexibility of design wefurtherinvestigatealsothedestabilizationmechanism as well as high nonlinear performances in semiconduc- of the oscillatory states initially described in Ref. [2], tors [18–25]. In these cavities, transverse effects are ab- showing that the chaotic regime, being confined to far sent and their dimensions are so small that the response off-resonance cavities, is indeed mutually exclusive with time of the medium can be much larger than the light bistable switching. transittimeinthecavity, yetbeingcomparablewiththe photonlifetimewhichisstronglyenhancedonaccountof the large quality factor Q. SP in such nano-cavities has II. MODEL DEFINITION AND LINEAR STABILITY ANALYSIS We start from the following dimensionless coupled- ∗ [email protected] modemodelthatrulesthetemporalevolutionofthenor- 2 malized intra-cavity field a(t) coupled to the frequency and the corresponding input powers P± = P(E±) can b b deviation n(t), owing to the intensity-dependent refrac- be calculated by means of Eq. (2). tive index change The stability of the solution (2) can be investigated da √ by plugging into Eqs. (1) the ansatz a(t) = A+δa(t), = P +i(δ+χn)a−a, (1a) n(t) = N + δn(t), while retaining linear terms in the dt perturbations δa,δn. dn τ +n=|a|2. (1b) The perturbation column array ε ≡ (δa,δa∗,δn)T is dt found to obey the following linearized equation Noteworthy Eqs. (1) describe a photonic crystal nano- dε cavity with high Q coupled to a line-defect waveguide =Mε; (4a) [18, 20, 24]. They implicitly assume that the nonlinear- dt ityisdominatedsolelybytheKerreffectwithrelaxation iδˆ−1 0 iχA  time τ, while other possible nonlinear contributions, e.g. M = 0 −iδˆ−1 −iχA∗, (4b) two-photonabsorptionalongwiththefree-carrierdisper- A∗/τ A/τ −1/τ sion [13, 18, 20, 24]), are neglected. Here P is the nor- malized power injected in the cavity through coupling where δˆ≡δ+χE. with the waveguide, and |a|2 is the normalized intra- The characteristic equation of M reads as cavityenergy,whichcanbeeasilyrescaledintoreal-world units by comparison with widely used dimensional mod- λ3+a λ2+a λ+a =0, (5) 2 1 0 els (see, e.g., Ref. [24]). It is worth emphasizing that (cid:16) (cid:17) a unit coefficient in front of the loss term in Eqs. (1) wherethecoefficientsarea =2+1,a = 1+δˆ2+ 2 , 2 τ 1 τ implies that the time t is measured in units of the in- (cid:16) (cid:17) and a = 1 1+δˆ2+2χEδˆ . verse damping coefficient 1/Γ0 =2Q/ω0 =2tc, where Q, 0 τ ω , and t stand for the overall quality factor, the res- SP occurs when the system undergoes a Hopf bifurca- 0 c onant frequency, and the photon lifetime of the cavity, tion,i.e.apairofcomplexconjugateeigenvaluesλ ±iλ R I respectively. In these units, the two key (normalized) crossesintotherighthalfcomplexplane,entailinganex- parameters are the detuning δ = (ω −ω)/Γ , and the ponentialgrowthofapulsatingperturbationwithperiod 0 0 time constant τ = τ Γ , where τ is the response time T =2π/|λ |. Thebifurcationpoint(λ =0)corresponds p 0 p I R of the nonlinearity in real-world units, while χ=±1 ac- totheconstrainta a =a , whichcanbesolvedtoyield 1 2 0 counts for the sign of the nonlinear Kerr coefficient. We thefollowingexplicitexpressionfortheSP(Hopf)thresh- point out that our model differs from [2], inasmuch as old values E± H the time scale is referred to the cavity lifetime instead of (cid:113) the response time of the medium. Indeed Eqs. (1) can −χδ(cid:0)2− 1(cid:1)± δ2 −4(cid:0)1+ 1(cid:1)2(cid:0)1− 1(cid:1) be reduced to the model analyzed in Ref. [2] by means E± = τ τ2 τ τ , (6) √ H 2(cid:0)1− 1(cid:1) of the substitution a,n,t → a/ τ,n/τ,τt. The effect of τ such transformation is however to rescale the detuning and the corresponding injected power threshold P± = and the injected power in such a way that they become H E±(cid:2)(1+(δ+χE±)2(cid:3). dependent on the response time of the medium, which is H H The analysis reported above shows that the bistable not suitable for our purpose of investigating the impact response depends only on the detuning, while the time of the relaxation time on the dynamics of a given cavity constant τ can affect qualitatively the onset of SP due with fixed characteristics. to the Hopf bifurcation. In fact different scenarios are For a cw driving P = constant, Eqs. (1) have the fol- possible depending on the existence of one or both Hopf lowing steady-state solution a(t) = A, n(t) = N = |A|2, thresholds(inturncorrespondingtotherootsinEq.(6)) where beingreal),andwhethersuchthresholdoccursatpowers P =E(cid:2)(1+(δ+χE)2(cid:3), (2) above or below the bistable knee for up-switching. Four possible scenarios are displayed in Fig. 1, where we show E = |A|2 being the stationary intra-cavity energy. It √ the bistable stationary response along with the unsta- is well known that bistability occurs for δ > 3 when √ ble eigenvalues responsible for instabilities. First, Fig. 1 χ=−1,andδ <− 3whenχ=1[14]. Inthediscussion shows the well-known fact that a purely real and pos- below, we will focus on the latter case, where the cavity itive eigenvalue of M leads to instability of the steady resonance is blue-shifted due to the nonlinearity, a case state solution along the negative slope branch, which which is directly comparable with the net effect of free- turns out to be a saddle point in phase-space. Con- carrierdispersioninducedbytwo-photonabsorption[13]. versely, SP is characterized by a pair of complex con- Alltheconclusionsofthispaperremainvalidalsoforχ= jugate eigenvalues, and the relative threshold are high- −1, provided δ →−δ. The values of intracavity energies lighted by empty triangles. We find such threshold to corresponding to the knees of the bistable response are occuralwaysontheupperbranchofthebistableresponse √ −2χδ± δ2−3 (orwhentheresponseismonotone,seebelow). However, Eb± = 3 , (3) depending on the value of τ, the lower Hopf threshold 3 40 35 30 y g old Ener2205 EH− EH+ τc h es15 hr SP T10 E+ b 5 0 0 0.5 1 1.5 2 τ FIG.2. (Coloronline)SPthresholdenergiesE± asafunction H ofτ forfixeddetuningδ=−4. Theshadedareacorresponds to the domain E− ≤ E ≤ E+ where SP occurs, which lies H H above the upper knee level of energy E+ (red dashed line). FIG.1. (Coloronline)Steady-stateresponseE vs.P forδ= b −4 (a-b-c) and δ = −10 (d) and different τ (χ = 1). Stable andunstablebranchesarereportedassolidanddottedlines, respectively. The blue and red curves superimposed on the right side show how the real part Re(λ) and imaginary part Im(λ)(ofthedominanteigenvalueunderlyingtheinstability) change with E. Bifurcation points are highlighted over with the dashed line Re(λ) = 0. The shaded regions labeled BI (light blue) and SP (light yellow) correspond to the negative slope branch of the bistable response (real eigenvalue) and SPinstability(pairofconjugateeigenvalueswithpositivereal part),respectively. Fourdifferentscenariosareshown: (a)SP occurs at P = P− values above the bistable knee value P+, H b andisunboundedforincreasingP;(b)SPoccursatP =P− H values below the bistable knee P+, still being unbounded; b (c)asin(a),exceptSPoccursinafiniterangebelowagiven valueP =P+;(d)asin(b),exceptSPoccursinafiniterange H belowagivenvalueP =P+. Theshadedgreenregionsyield H the range of power where bistable up-switching to a stable state is permitted, while red ones identify the coexistence of a SP and an unstable saddle branch. FIG. 3. (Color online) Color level plots of the (a) ”on” E− H and(b)”off”E+ valuesofthresholdenergyforSP,inthepa- H rameterplane(τ,δ). Bistabilityoccursbelowthelineδ=δ . b P(E−) can take place above [as in Fig. 1(a,c)] or be- ThecurveP(E−)=P(E−)(blacksolid)dividesthebistable H H b low [see Fig. 1(b,d)] the knee P(E−) which characterize region into a domain labeled BI+SP where SP sets in only b for powers above the bistable knee for up-switching, and a the bistable up-switching. In the former case the cav- domainlabeledSP,wherestableup-switchingisnotpossible, ity can exhibit bistable up-switching to a stable steady being hampered by SP, which dominates the dynamics. state, whereas in the latter case up-switching occurs in- evitably towards a SP-unstable state: stabilization re- quires to move along the hysteresis cycle below the Hopf as a critical value τ = τ is reached. For τ = τ , SP no c c threshold. Moreover, also depending on the value of τ, longer takes place and the entire upper branch becomes the upper branch can be indefinitely SP-unstable above stable. From Eq. (6), we find such critical value τ to be the first threshold P(E−) [see Fig. 1(a,b)], or, viceversa, c H given by the positive real root of the cubic polynomial exhibitaSPswitch-offenergyorsecondarythresholdbe- (its explicit expression is too cumbersome) yond which SP ceases to take place [see Fig. 1(c,d); note that we change the detuning in Fig. 1(d) only to make (cid:18) δ2(cid:19) |δ| τ3+τ2− 1+ τ −1=0, τ ≈ , |δ|(cid:29)1. (7) thepictureclearer,thoughthesamequalitativebehavior c c 4 c c 2 occurs at δ = −4]. Indeed, if τ ≥ 1, Eq. (6) may also admit a real solution E+, and hence SP occurs only in The behavior discussed above can be clearly seen by H a finite range of energies (and powers) E− < E < E+. reportingtheSPthresholdenergiesE± versusτ, atcon- H H H When τ grows, this finite interval shrinks, and vanishes stantdetuning. Suchplot,displayedinFig.2forδ =−4, 4 shows a shaded region (E− < E < E+), which corre- to have a response time of the nonlinearity of the same H H spondtoSP.WeclearlyseethatnoSPoccursforτ >τ . order of magnitude is naturally met in semiconductors c Conversely, decreasing τ below τ results into widening withnonlinearresponsedominatedbyfree-carrierdisper- c the portion of the upper branch that exhibits SP, until sion, the same constraint in the framework of the Kerr below τ = 1 the whole upper branch above the switch- model rule out the possibility to observe SP dynamics in on threshold E− turns out to be unstable. In this case, media with nonlinearities of electronic origin since they H the SP switch-off energy E+ diverges as the asymptote are too fast (fs range). Nevertheless the predictions of H τ = 1 (dashed vertical line) is approached. Importantly our Kerr model become interesting for Kerr-like mate- for τ → 0 also the first threshold E− diverges, which rials with response time in the ps range such as, e.g., H means that a finite response time is a key ingredient for soft matter, metal films [26], or more traditional liquids SPtobeobservable. Thisisconsistentwiththefactthat with reorientational nonlinearity, which are still the ob- Kerr instantaneous nonlinearities yield no SP at all. In ject of recent studies [27, 28]. In particular, for instance, fact, in this case, the eigenvalues are easily found to be highlynonlinearliquidssuchasnitrobenzene[29]orCS2 (cid:112) λ± =−1± E2−(δ+2χE)2, which rule out the possi- could be easily employed to fill a photonic crystal ma- bilitytohaveacomplexconjugatepairwithpositivereal trix (as also recently proposed for microstructured fibers part. [28]),whilemetalfilmscouldbeemployedinconjunction with dielectrics to form a single cavity or cavity arrays In order to have a complete picture and further show [30]. Assuming, for instance, a response time τ ∼30 how the onset of SP changes with detuning, we have phys ps [29], which yield τ = (τ /2t ) ∼ 0.75 in a cavity drawn in Fig. 3 a color map of the level curves of SP phys c threshold energies E±, in the parameter plane (τ,δ). A with Q ∼ 25000 (tc ∼ 20 ps at λ = 1.55µm), assum- H ing n ∼ 10−17m2/W and a nonlinear modal volume number of interesting observations can be drawn. The 2I V = 3(λ/n)3, the threshold power P = 10 in Fig. 4 SP region is bounded by the border τ = τ , and in the c corresponds to a real-world power P = (γ/Γ2)P ∼ 10 bistable region (δ <δb) τc decreases with decreasing val- in 0 mW in the waveguide coupled to the nanocavity, where ues of absolute detuning |δ|. Interestingly enough, the γ = ω n c/(n nV) is the overall nonlinear coeffi- scenarioillustratedinFig.2remainsvalidalsofordetun- 0 2I eff cient [24]. Here we have assumed a refractive index ings δ > δ , where bistability disappears. Finally in the b n ∼ n ∼ 1.5 and Q to be essentially determined by region of positive detunings, we are left with the upper eff branch being fully unstable for all energies E > E−. In the coupling itself. H thisregion,thethresholdenergyE− diverges,notonlyin Having characterized so far the threshold for SP and H itscompetitionwithbistability,sinceIkedaandAkimoto theinstantaneouslimitτ =0,butalsoforτ =1,whereas relatively low values of E− are found for τ ∼ 1/2, i.e. haveshownthatthelimitcyclesdestabilize,leadingeven- H tually to chaos [2], in the next section we deepen this whentheresponsetimeofthenonlinearityisnearlyequal point with the aim of determining the domain of the pa- to the photon lifetime. Furthermore, the bistable region rameterplanewherethetransitiontochaoscouldbeob- is divided into two distinct domains by the (solid black) curve which arises from the condition P(E−) = P(E−) served. H b (its explicit expression is too cumbersome). In the do- mainBI+SPabovesuchcurve(boundedfromabovealso by the line δ = δ ), one has that the cavity can work as III. THE TURBULENT REGIME b a bistable switch since the upper branch right above the kneeforup-switchingisstable[asinFig.1(a,c)],whereas In Ref. [2] Ikeda and Akimoto have studied the transi- in the domain labeled SP below the curve, up-switching tiontochaos,identifyingaperioddoublingcascadeupto is no longer allowed, since the upper branch above the 22P (i.e. oscillation with period-four) at a fixed value of knee is SP-unstable [as shown in Fig. 1(b,d)] The reader detuning. Herewereportfurtherdetailsabouttheemer- caneasilyrecognizeaqualitativesimilarityofthepicture gence of chaos in a wide domain of parameters. We em- discussedherewiththedynamicsofSPruledbyfreecar- ploy different tools, ranging from Poicar´e section and its rier dispersion, recently discussed in Ref. [13]. Although corresponding bifurcation diagram to the calculation of a detailed analytic investigation of the stability of the Lyapunov exponents. Our principal purpose is to assess SP-oscillating state (limit cycle) is beyond the scope of whether a nano-cavity described by the model (1) can this paper, similarly to the case discussed in Ref. [13], work as a reliable bistable switch, and hence whether our numerical simulations of Eqs. (1) suggest that sta- the onset of chaos should be expected when the cavity blelimitcycles,workingasattractorsfromalargebasin, operates progressively off-resonance, especially in the re- exist in a wide domain of the parameter plane (witness- gion labeled BI+SP in Fig. 3. To begin with, it is in- ing the supercritical nature of the Hopf bifurcation). An structive to report about the dynamics ruled by Eqs. (1) example of such stable dynamics is shown in Fig. 4. when, starting above the threshold P−, the input power H LetuscommentontheobservabilityoftheSPdynam- P isadiabaticallydecreased. Infact,thisisthesituation ics. In nanocavities with high Q (Q = 103 −105) the where the onset of chaos is expected according to Ref. photon lifetime in the near infrared ranges from few pi- [2]. We start at moderately low detuning (δ = −4) and coseconds to tens of picoseconds. While the constraint for τ =0.45, which corresponds to SP being unbounded 5 30 80 (a) n(t)20 δ=−4, τ=0.75 70 2, a(t)|10 60 | 0 50 0 10 20 30 40 50 t E40 10 (b) 5 (c) 30 E 5 m a 0 20 I −5 10 0 0 10 20 −5 0 5 0 P Re a 500 1000 1500 2000 2500 t FIG. 4. (Color online) Dynamics of SP ruled by Eqs. (1a- FIG. 6. (Color online) As in Fig. 5, with δ = −15. The 1b), with δ = −4, and τ = 0.75: (a) temporal evolution chaotic region is highlighted in yellow. of the intra-cavity energy and carrier density corresponding to the rightmost blue circle in (b); (b) steady response with superimposed peak energy of the periodic oscillations (blue opencircles);TheredfilledcirclemarkstheHopfbifurcation plexsequenceofperioddoublingbifurcationsandchaotic point. The black filled circle marks the SP-unstable steady motion are detected at higher detunings when the input state which gives rise to the dynamics shown in (a,c); (c) power approaches the knee value P−. In Fig. 6, where b phase-spacepictureoftheopticalfieldshowingtheattracting δ = −15, oscillations with several different periods are limitcyclefromtwodifferentinitialconditions(opencircles); evident. Moreoverachaoticregimeappears,intherange √ P ≈17−20 (P ≈280−400). In phase space this cor- responds to the appearance of a strange attractor (not 15 P− PH− (a) 60 (b) shown because its structure is already illustrated in Ref. b [2]). 10 40 Since following simply the adiabatic dynamics could E E 5 P+ 20 be possibly misleading (e.g. because of critical slowing b down), as it neglects the rich variety of phenomena that occurs over the small scale, we have drawn also bifurca- 0 0 2000 2200 2400 1800 2000 2200 2400 tion diagrams calculated by collecting trajectory points t t on a Poincar´e section for different powers. A typical ex- ample,usingthesameparametersasinFig.6,anddefin- FIG.5. (Coloronline)DynamicalevolutionruledbyEqs.(1) √ ing the Poincar´e section on the fixed phase ∠a−π = 0 as the forcing term P varies adiabatically in time (dashed of the intra-cavity field, is reported in Fig. 7. We can line). Here τ =0.45, and (a) δ =−4; (b) δ =−10. The ver- clearly identify a period doubling cascade (up to 23P), tical lines mark the time instants at which the input power cross the main bifurcation points: bistable knees P± (dotted as well as chaotic regimes. The onset of chaos follows b blueandblack,respectively)andHopfthresholdP− (dashed a non-trivial scenario where narrow windows of period- H red). The insets zoom over characteristic time intervals, in- threesolutions(3P)areinterspersedbetweentworanges dicated by arrows. Notice that in (b) the two thresholds P± ofpowerswherethemotionturnsouttobechaotic. The b and P− almost overlap. verticaldashedlinesinFig.7markindeeda3P window. H ThisisanalogoustotheFeigenbaum’sroutetochaosand confirms the observation of chaos for P ≈220−380, al- on the upper branch. As shown in Fig. 5(a), in this ready drawn above from Fig. 6. case, the Hopf bifurcation is clearly supercritical, since The bifurcation diagram is computed up to P ≈ 150 thesystemsettlesonalimitcycle,whoseamplitudevan- because lower power levels do not result into any limit ishes approaching the bifurcation point (approximately cycle. Vice-versa the solution is observed to collapse to- as [E−E−]1/2). However, at larger (in modulus) detun- ward the stable node represented by the lower branch H ings, we observe a jump in |a|2 as the limit cycle (1P) of the bistable response. This phenomenon is indepen- looses its stability and the system settles on a period- dent from chaos, as mentioned above with reference to two (2P) oscillation, as shown in Fig. 5(b) for δ = −10. detuning δ = −10. It occurs at large negative detun- The 2P solution does not visit anymore the simplest 1P ings for input powers above the SP threshold (P >P−). H limit cycle. Conversely it abruptly switches to the sta- This can be qualitatively explained by the coexistence ble low-branch steady state. Remarkably this happens of a stable fixed point (lower branch solution), a saddle still above the Hopf bifurcation point E−. A more com- (negative-slope branch), and an unstable limit cycle. As H 6 40 −10 0.6 30 35 25 −11 0.5 30 20 E− 0.4 25 H 400 450 −12 2 a|20 δ 0.3 | −13 15 0.2 10 −14 0.1 5 0 −15 0 200 400 600 800 0.2 0.4 0.6 0.8 P τ FIG. 7. (Color online) Bifurcation diagram, τ = 0.45, δ = FIG.8. (Coloronline)ColorlevelmapofmaximalLyapunov −15. The vertical dashed lines highlight a 3P window and exponent in the parameter plane (τ,δ). the collapse at small P. Inset shows a detail in which period doublingandperiodicwindowscanbeidentifiedmoreclearly. scale in Fig. 4 and Fig. 7). As a final remark about the existence of 2nP periodic the 2P limit cycle spans the phase space with wide os- solutions, we point out that they can be detected only cillations around the upper branch, it can approach the whenPH− <Pb−[asintheexamplesshowninFig.1(b,d)]. saddle point (which near the first bistable knee is closer As discussed with reference to Figs. 1-3, this may occur in phase space to the center of the oscillations) being not only for both τ <1 (as shown explicitly above), but forced away, until eventually it can be captured by the also for τ > 1, where the Hopf bifurcation is bounded lowest energy (stable) solution. fromabove. Morespecifically2P solutionsareeasilyseen in a small subset of the region marked as SP in Fig. 3. While the bifurcation map is a useful visual tool to In this case two unstable solutions, namely a repulsive characterize the onset to chaos and the full dynamics at (negative slope) branch and a SP branch, coexist. This fixed parameters, in order to explore in which region of seems to be a key ingredient for the limit cycles to loose the parameters one should expect to observe the chaotic their stability. dynamics, we have resorted to compute the dominant (maximal)Lyapunovexponent. Wehaveexploredawide regionoftheparameterplane(τ,δ),where,ineachpoint of such plane, we have iterated over the values of power IV. CONCLUSIONS P to find the largest exponent. We recall that a posi- tive Lyapunov exponent (within numerical inaccuracies) In this work we have revisited the model that rules entails that the system exhibits a chaotic behavior (here thebehaviorapassivesmallcavitywithKerrdelayedre- quasi-periodicmotionisexcludedbythedissipativechar- sponse, pioneered in Ref. [2]. We have reported a full acter of our model (1a)). From the map displayed in characterizationofSPinstabilitiesandtheircompetition Fig.8,wenoticethatchaoticmotionmanifestsitselfonly with bistability, outlining the existence of different pos- when the cavity is detuned far off-resonance (i.e. at very sible scenarios. Importantly we have found a maximal large values of |δ|), provided that the SP-unstable range critical value for the relaxation time that allows SP to is not finite, or in other words that the Hopf bifurcation occur, and have shown that SP can have two bifurcation is not bounded from above (E+ → ∞) which requires points, while it can occur also in the absence of bista- H τ < 1. Therefore we can draw the important conclu- bility. We have further characterized the destabilization sion that the region where the cavity could work as a mechanismofthelimitcycleinthefullparameterspace, bistable switch is mutually exclusive with chaos. There- finding that chaos is mutually exclusive with the domain fore the onset of chaos cannot spoil the behavior of the wherethecavitycanbeemployedasabistableswitching cavity as a switch, once the latter is used in the region element. In particular the chaotic regime predicted by labeledBI+SPinFig.3. Wepointoutthat,intermsof Ikeda [2] in this system requires to be detuned strongly power,theobservationofchaosismuchmorechallenging off-resonance, inturnimplyingtheuseofextremelyhigh than stable SP since power levels leading to the former powers, thus making its observation rather challenging. turn out to be much larger than those leading to the lat- Viceversa, in contrast with Kerr instantaneous nonlin- ter; indeed at very large detuning, which corresponds to earities, the observation of stable SP appear feasible in several times the cavity linewidth, bistability is observed high-Q nanocavities filled with Kerr-like media with re- at much higher power level (compare the horizontal axis sponse time in the range of picoseconds. Future work 7 will be devoted to study the effect of coupled cavity sys- (no. 249012). temsandtheinterplayofdifferentnonlinearmechanisms. ACKNOWLEDGMENTS This work was supported by the European Com- mission, in the framework of the Copernicus project [1] K. Ikeda, Opt. Commun. 30, 257 (1979); K. Ikeda, [15] H.M.Gibbs,F.A.Hopf,D.L.Kaplan,andR.L.Shoe- H. Daido and O. Akimoto, Phys. Rev. Lett. 45, 709 maker, Phys. Rev. Lett. 46, 474 (1981); L. A. Orozco, (1980). A. T. Rosenberger, and H. J. Kimble, Phys. Rev. [2] K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 Lett. 53, 2547 (1984); B. Segard and B. Macke, Phys. (1982). Rev. Lett. 60, 412 (1988). [3] L. A. Lugiato, L. M. Narducci, D. K. Bandy, and [16] K. J. Vahala, Nature 424, 839 (2003). C. A. Pennise, Opt. Commun. 43, 281 (1982). [17] M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and [4] C.M.SavageandD.F.Walls,Opt.Acta30,557(1983). J.D.Joannopoulos,Phys.Rev.E66,055601(R)(2002). [5] L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and [18] P. E. Barclay, K. Srinivasan, and O. Painter, Opt. Ex- R. J. Horowicz, Il Nuovo Cimento 10D, 959 (1988). press 13, 801 (2005). [6] S.TrilloandM.Haelterman,Opt.Lett.21,1114(1996); [19] M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Ku- S.TrilloandM.Haelterman,inSpatialsolitons(Springer, ramochi, and T. Tanabe, Opt. Express 13, 2678 (2005). Berlin, 2001), S. Trillo and W. Torruellas, eds. [20] T. Uesugi, B.-S. Song, T. Asano, and S. Noda, Opt. Ex- [7] H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. press 14, 377 (2006). 40, 298 (1982). [21] E. Weidner, S. Combri´e, A. de Rossi, N. V. Q. Tran, [8] A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Lo- S. Cassette, A. Talneau, and H. Benisty, Appl. Phys. catelli, C. De Angelis, J. Opt. Soc. Am. B 24, 2229 Lett. 89, 221104 (2006). (2007). [22] E.Weidner,S.Combri´e,A.deRossi,N.V.Q.Tran,and [9] G. D’Alessandro, P. St. J. Russell, and A. A. Wheeler, S. Cassette, Appl. Phys. Lett. 90, 101118 (2007). Phys. Rev. A55, 3211 (1997). [23] S. Combri´e, N. V. Q. Tran, A. de Rossi, and H. Benisty, [10] M. Conforti, A. Locatelli, C. De Angelis, A. Parini, Opt. Lett. 33, 1908 (2008). G. Bellanca, and S. Trillo, J. Opt. Soc. Am. B 22, 2178 [24] A. de Rossi, M. Lauritano, S. Combri´e, N. V. Q. Tran, (2005). and C. Husko, Phys. Rev. A 79, 043818 (2009). [11] B. Maes, M. Fiers, and P. Bienstman, Phys. Rev. A 80, [25] X. Yang, C. Husko, C. W. Wong, M. Yu, and 033805 (2009). D. L. Kwong, Appl. Phys. Lett. 91, 051113 (2007). [12] V.GrigorievandF.Biancalana,Phys.Rev.A83,043816 [26] M. Conforti and G. Della Valle, arXiv:1109.5875. (2011). [27] G. Fanjoux, J. Michaud, H. Maillotte, and T. Sylvestre [13] S. Malaguti, G. Bellanca, A. de Rossi, S. Combri´e, and Phys. Rev. Lett. 100, 013908 (2008). S. Trillo, Phys. Rev. A 83, 051802(R) (2011). [28] C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Bian- [14] L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 calana, Phys. Rev. Lett. 105, 263902 (2010). (1987). [29] M. A. Duguay and J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969). [30] R. S. Bennink, Y.-K. Yoon, and R. W. Boyd, and J. E. Sipe, Opt. Lett. 24, 1416 (1999).