Alternative Formats If You Require This Document in an Alternative Format, Please Contact: [email protected]

Alternative Formats If You Require This Document in an Alternative Format, Please Contact: Openaccess@Bath.Ac.Uk

Citation for published version: Skryabin, D, Fan, Z, Villois, A & Puzyrev, D 2021, 'Threshold of complexity and Arnold tongues in Kerr-ring microresonators', Physical Review A, vol. 103, no. 1, L011502. https://doi.org/10.1103/PhysRevA.103.L011502

DOI: 10.1103/PhysRevA.103.L011502

Publication date: 2021

Document Version Peer reviewed version

Link to publication

University of Bath

Alternative formats If you require this document in an alternative format, please contact: [email protected]

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Download date: 11. Oct. 2021 Threshold of complexity and Arnold tongues in Kerr ring microresonators

D.V. Skryabin,∗ Z. Fan, A. Villois, and D.N. Puzyrev Department of Physics, University of Bath, Bath BA2 7AY, UK (Dated: submitted 04 April, 2020; published January 2021) We show that the threshold condition for two pump photons to convert into a pair of the sideband ones in Kerr microresonators with high-quality factors breaks the pump laser parameter space into a sequence of narrow in frequency and broad in power Arnold tongues. Instability tongues become a dominant feature in resonators with the ﬁnesse dispersion parameter close to and above one. As pump power is increased, the tongues expand and cross by forming a line of cusps, i.e., the threshold of complexity, where more sideband pairs become unstable. We elaborate theory for the tongues and threshold of complexity, and report the synchronisation and frequency-domain symmetry breaking eﬀects inside the tongues.

Arnold tongues [1, 2] is a well-known phenomenon in aforementioned connection between finesse and disper- parametrically driven and coupled oscillator systems hav- sion. Exploring an interplay of the two, we have found ing cross-disciplinary applications [3]. The tongues ap- a broad range of parameters where a transition from the pear as a sequence of the expanding instability and syn- single-mode, i.e., continuous-wave (cw), operation to fre- chronisation intervals in the parameter space of the exter- quency conversion happens via the Arnold-tongue route. nal drive frequency and amplitude. The Arnold-tongues The mLLE Arnold-tongues structure emerges across the concept established itself in neuroscience [4], structural pump frequency range from the limit of the infinitesimal [5] and nano-mechanics [6], quantum engineering [7–9] pump powers. Therefore a focus of this work is outside and in other areas. Periodicity embedded into the model the hard excitation regimes, i.e., proximity of the upper equations is the key property underpinning the formation state of the bistability loop, giving rise to the mLLE soli- of Arnold tongues [1, 2]. Nonlinear effects in periodic op- ton modelocking [15, 23–28]. tical systems have traditionally attracted considerable at- We show that the tongue tips are well separated if the tention [10–12]. In particular, resonators that recirculate finesse dispersion is the order of one, which implies that light and reinforce light-matter interaction are naturally the regions of stable single-mode operation are first inter- suited to explore the interplay of periodicity, dissipative leaved with the instability intervals where only one pair and nonlinear effects [12–18]. of sidebands with the mode numbers ±µ kicks off the Arnold tongues emerge through the transformation of frequency conversion process. As the pump is increased resonances of the underlying linear oscillator under the the tongues expand, at some point they start overlapping influence of dissipative and nonlinear effects. Hence, and form the complexity threshold. Above this thresh- practical utilisation of this concept in resonators requires old, two and more pairs of modes with different |µ| are a robust control over the linewidth and mode-density. becoming unstable simultaneously. We elaborate a the- High-quality-factor microresonators are currently used ory of the threshold of complexity and report nonlinear in the state-of-art frequency conversion, precision spec- effects in its proximity. The term - threshold of complex- troscopy, comb generation and single quanta manipula- ity is inspired by the cellular automata related concepts tion [15–19]. These devices are few mm to few hundreds and terminology [29]. of nm long with the tens of GHz to THz resonance sep- We now introduce a model and explain how it maps arations, i.e., free spectral ranges (FSRs), and finesses onto physical devices, while more cross-area context is 3 6 2 10 -10 , that should be compared to the finesses . 10 provided through the text and before summary. Am- in fibre resonators [13, 14] and other modelocking devices plitude ψ of the electric field in a ring microresonator [20]. High finesse is naturally accompanied by the rela- can be expressed as a superposition of angular harmon- P iµϑ tively large finesse dispersion, which is a measure of the ics: ψ = µ ψµ(t)e . Here µ and ψµ are the mode arXiv:2101.02247v1 [physics.optics] 6 Jan 2021 spectral non-equidistance and of the resonance density in numbers and amplitudes. ϑ ∈ [0, 2π) is an angle varying the rotating frame, that play a critical role in bringing along the ring. Resonant frequencies, ωµ, are counted the microresonator Arnold-tongues to the physical realm. from ω0 and approximated as The Lugiato-Lefever equation (LLE) [21] is a key 1 2 model in the microresonator area [15]. Originally pro- ωµ = ω0 + D1µ + 2 D2µ , µ = 0, ±1, ±2,.... (1) posed in the context of pattern formation in an optical resonator supporting a single longitudinal and many D1/2π is the resonator FSR, and D2 is the second order transverse modes, it has since gained broad interdisci- dispersion, characterising how FSR is changing with µ. plinary significance [22]. A microresonator implementa- For example, D1/2π = 15GHz, |D2|/2π = 1kHz are good tion of LLE (mLLE) describes the interaction of many estimates for CaF2 resonators for ω0/2π being close to longitudinal modes and therefore should reflect on the 200THz pump, ωp, [30]. If κ/2π is the linewidth, then 2 the resonator finesse is Thus, g is simultaneously responsible for the nonlinear shifts of the resonances and the anti-Hermitian coupling ωµ±1 − ωµ D1 F = ± = F + α , F = , (2) λµt µ κ µ κ between the ±µ sidebands. Setting ~qµ ∼ e gives [21] (1) (2) where we take +1 for µ > 0 and −1 for µ < 0. Here, λµ λµ + κ = 3 gµ − g g − gµ , (8) F is the dispersion free finesse, and αµ = Fµ − F is the p g(1),(2) = 2 ∆ ± 1 ∆2 − 3 κ2. (9) residual finesse, µ 3 µ 3 µ 4 Degenerate four-wave-mixing (FWM) process de- 1 D2 αµ = µ ± Fd ≈ µFd, Fd = . (3) 2 κ scribed by Vbµ corresponds to the following photon energy conservation: 2 ω = (ω +µD +Imλ )+ (ω −µD − F is the finesse dispersion, which is a key parameter ~ p ~ p 1 µ ~ p 1 d Imλ ). CW stability is neutral at Reλ = λ = 0, in what follows. It can be either positive (anomalous µ µ µ dispersion, D2 > 0) or negative (normal dispersion, D2 < g = g(1), g = g(2). (10) 9 µ µ 0). Q = ω0/κ is the resonator quality factor. Q = 3 · 10 , which is a conservative number for CaF2 resonators, gives Hence, FWM sidebands are exponentially amplified for 4 −2 (2) (1) κ/2π = 67kHz, F = 22 · 10 , and |Fd| . 1.5 · 10 for gµ < g < gµ . Lower boundary of the cw stability in 11 these samples [30, 31]. For the high Q = 10 samples (δ0, H)-plane, i.e., the FWM-threshold, is made by the 4 [32], κ/2π = 2kHz, F = 750 · 10 , and |Fd| 0.5. (2) (2) . minima of gµ in δ0, which are found at ∆µ = κ, gµ = The Kerr mLLE is [23, 24, 33] κ/2 for every µ. Explicitly, the corresponding intracavity 1 2 1 2 2 (µF ) (µF ) pump power, H , and detuning, δ = ω − ωp , i∂tψ = δ0ψ − 2 D2∂θ ψ − i 2 κ (ψ − H) − γ|ψ| ψ. (4) µF 0 0 along the FWM threshold is Here, H2 = η FW is the intracavity pump power, where π 2 1 2 2 W is the laser power. η = κc/κ < 1 is the pump coupling γHµF = 2 κ 1 + (1 − µ Fd) , (11) efficiency and κc is the coupling rate. γ/2π = 10kHz/W 1 (µF ) 1 2 κ δ0 = 1 − 2 µ Fd. (12) is the nonlinear parameter [33]. δ0 = ω0 − ωp is detuning 2 of the pump laser frequency, ωp, from ω0. Eliminating µ one finds Transformation to the rotating reference frame, θ = 2 1 2 (µF )2 ϑ − D1t, replaces the ωµ-spectrum with the residual, i.e., γHµF = 2 κ 1 + 1 − κ δ0 (13) mLLE, spectrum, For κ = 2, Eqs. (12) and (13) match equations for a(nc) 1 1 1 2 κ ∆µ = κ δ0 + 2 µ Fd. (5) and EIc from Ref. [21]. Here and below, the sub- and super-scripts F,C stand for the ’FWM’ and ’Complexity’ Respectively, the finesse Fµ is replaced with the resid- thresholds, respectively. ual finesse, αµ, Eq. (3). The latter is also expressed as Eqs. (11), (12) represent a discrete set of values, while αµ = ±(∆µ±1 − ∆µ)/κ and can be interpreted as the the laser power and frequency can be tuned continu- inverse mode density. Fig. 1(c) illustrates how the den- ously. Hence, the instability threshold in (H, δ0) also sity of states in the residual spectrum is reduced with exists between the discrete points with the coordinates Fd increasing, and also shows that the linear resonances, (µF ) 2 specified by δ0 and HµF . To demonstrate if and when ∆µ = 0, are located at δ0 < 0 for Fd > 0, and at δ0 > 0 their separation is important, we substitute g = g(1) for Fd < 0. Residual spectrum is non-equidistant, and µ (2) 2 hence, even for very small Fd’s there always be a suf- and g = gµ directly to Eq. (6) and compute H vs ficiently large |µ| making its resonances well separated. δ0 for all µ. This procedure returns, in general, an infi- Thus, |αµ| 1 and |αµ| 1 correspond to the quasi- nite number, one for each |µ|, of threshold lines that are 9 continuous and sparse limits of the residual spectrum, shown in Figs. 1(a),(b) for |Fd| = 0.005 and 0.5 (Q = 10 11 respectively. Note, that the finesse itself is maintained and 10 ). For relatively small |Fd| and quasi-continuous arbitrarily large in any case, Fµ |αµ|. residual spectra, Fig. 1(c), the individual instability lines in Fig. 1(a) overlap tightly and form a visibly single We now√ define the single-mode cw-solution of Eq. (4) iφ0 (µF ) as ψ = g/γ e , where g > 0 solves threshold [24] reproduced by Eq. (13) with HµF vs δ0 2 2 2 considered as a continuous function. γH = g + 4g (δ0 − g) /κ , (6) As |Fd| starts approaching 1 and if it goes above it, and recapture key steps in its stability analysis [21]. Per- then resonances in the residual spectrum, Fig. 1(d), and the respective thresholds for the neighbouring |µ| √turbing the cw with a pair of sideband modes, ψ = iφ0 iµθ ∗ −iµθ separate, so that the pump frequency and power range g/γ e + ψµe + ψ−µe , we find i∂t~qµ = Vbµ~qµ, T with FWM gain reshapes profoundly and forms instabil- where ~qµ = (ψµ, ψ−µ) , and ity tongues, Fig. 1(b). Now, Eq. (13) describes only 1 i2φ0 ∆µ − 2g − i 2 κ −ge the lowest power limit at which instability becomes pos- Vbµ = −i2φ0 1 . (7) ge −∆µ + 2g − i 2 κ sible, with the large areas of stability present above it, 3

FIG. 1. (a,b) Arnold tongues and interplay of the FWM, W = WµF , and complexity, W = WµC , power thresholds for the 9 11 quasi-continuous (|Fd| = 0.005, Q = 10 ) and sparse (|Fd| = 0.5, Q = 10 ) residual spectra; |D2|/2π = 1kHz, η = 0.5, residual ﬁnesse Fd = D2/κ = D2Q/ω0. The color scheme shows the number of the simultaneously unstable ±µ sideband pairs. Lines dropping below W = WµF and reaching W = 0 show a pair of the synchronisation tongues. (c,d) Resonances in the residual 2 2 spectrum for small (c) and large (d) |Fd|. Plots show Lorentzian lines, 1/(1 + 4∆µ/κ ), for µ starting from 0. (e) Distribution of the dynamical regimes across the |µ| = 35 instability tongue and its neighbourhood for Fd = 0.5. Black color corresponds to the repetition rate locking to |µ|D1, see Fig. 2(a). Yellow marks the ’unlocked’ regimes with the |dµ| = const symmetry breaking, see Fig. 2(b). Grey is the weak chaos with the symmetry breaking, see Fig. 2(c). Blue is the multimode complexity (2) with dense continuous spectra, see Fig. 2(d). The intra-resonator cw power at the tongue minima g35 /γ = κ/2γ = 0.1W (1) (2) (FWM threshold), and g35 /γ = g36 /γ ' α35κ/2γ = 1.75W at the threshold of complexity.

see dashed black line in Fig. 1(b). We note, that ωp, tions, see red line in Fig. 1(b). We call this boundary in Figs. 1(a,b), is scanned across the narrow interval of - the threshold of complexity. Reaching this threshold −4 10 D1 around ω0 so that ωµ6=0 are not approached even signals two events. First is that the inter-tongue stabil- remotely. ity intervals cease to exist. Second is that the resonator The instability tongues, when they are formed, provide is brought into the regime where ±µ and ±(µ + 1) side- selective excitation conditions for the sideband pairs with bands can become simultaneously unstable. As the pump a given |µ|. The tongue tips have been found along the is increased further a sequence of thresholds, where more √quasi-linear tails of the g vs δ0 solution of Eq. (6) for δ0 < modes become unstable, can be seen in Figs. 1(a),(b). 3κ/2 = δb if Fd > 0, and for δ0 > δb if Fd < 0. The For smaller |Fd| the residual spectrum is dense and all tips are thus directly connected to the ∆µ = 0 resonances the thresholds tend to merge, while, for larger |Fd|, their in the residual spectrum of the linear, g → 0, resonator, separation is pronounced. see Fig. 1, Eq. (5), and more details below. The limiting Thus, the complexity threshold consists of the points value of δ0 = δb, where the tongue structure diminishes, where bifurcation lines corresponding to the excitations (1) (2) of the ±µ-sidebands intersect with the ones for ±(µ + 1). is found from g0 = g0 , which is the condition for the 2 onset of bistability and the soliton regime (Fd > 0, δ0 > Since ∆µ is a function of µ , assuming µ > 0 does not δb) [15]. Tongues have also been found by us in Si3N4 restrict the generality. Now, the intersection points, for 6 7 resonators [34] with Q ∼ 10 − 10 [35, 36], and D2 ∼ Fd > 0, can be found applying 4 (1) 10 kHz. The g = gµ (green squares in Fig. 1(e)) and (2) (2) (1) (2) (2) g = gµ = gµ+1, and g = gµ = gµ+1, (14) g = gµ (green full lines) conditions shape the edges of (2) the tongues, while their tips always belong to g = gµ . cf., Eq. (10). If Fd < 0, then the second condition be- (2) (1) Importantly, there is also a well-defined line that limits comes g = gµ = gµ+1. Equations (14) is the mathe- the area above the tongue tips and below their intersec- matical quintessence of the complexity threshold. Each 4 of Eqs. (14) is a double, i.e., codimension-2, condition Tongues are classified by their synchronisation order which marks a sequence of the instability points along n : m, where n is the number of nonlinear pulses coming the threshold line, see Fig. 1(b). Equations (14) are the through the system per m periods of the linear oscilla- cusp conditions and hence are non-differentiable in δ0, tor [3]. In our case, the linear round trip time is 2π/D1 (2) unlike g = gµ , Eqs. (10). For |αµ| 1, |Fd| 1, and periods of the synchronised waveforms are 2π/|µ|D1, the cusps become very shallow and the tongue structures thus the corresponding orders are |µ| : 1. As for the clas- disappear, while F is still 1. sic Arnold tongues [3], the synchronization intervals start Resolving Eqs. (14) gives for infinitesimal pumps at the resonance points, ∆µ = 0, in the linear, g → 0, spectrum. The upper and middle p 1 2 branches of the g vs δ curve inside the bistability in- ∆µ = κ − 2 αµ + 1 + αµ . (15) 0 terval (unlike its quasi-linear tails) do not withstand the The respective pump detunings are found applying g → 0 limit, and therefore their stability maps do not Eqs. (5), (15), reveal the tongues.

(µC) p Though the linear theory predicts |µ| : 1 syn- 1 δ = − 1 α + 1 + α2 − 1 µ2F , (16) κ 0 2 µ µ 2 d chronisation across the whole tongues area, the nonlin- cf., Eq. (12). ear processes make other regimes possible. The time- 1 If, e.g., α 1 (µ 1, F > 0), then − α + average mode frequencies are ωµ = ωp + D1µ + h∂tφµi, √ µ d 2 µ e 2 1 −1 where φµ(t) = argψµ [37]. Beating ψµ against ψ−µ pro- 1 + αµ = 2 αµ + O αµ . Hence, Eq. (9) gives g = (1) 1 vides a measure of the average nonlinear repetition rate gµ ' καµ along the threshold of complexity. Si- 1 2 De1µ = (ωµ − ω−µ). The difference between the two multaneously, the relatively large detunings, |δ | g, 2 e e 0 rates, D − |µ|D = 1 (h∂ φ i − h∂ φ i) ≡ |µ|d , is imply dispersive quasi-linear resonator response. E.g., e1µ 1 2 t µ t −µ µ a function of ω , and g. Hence, d 6= 0 implies de- for κ/2π = 2kHz (Q = 1011) and µ = 35, we have p µ (1) synchronisation, i.e., ’unlocking’ of the repetition rate δ0/2π ' 0.5MHz and g = g35 ' 2π × 17kHz at the from |µ|D . complexity threshold. Therefore, Eq. (6) can be approx- 1 2 2 2 Numerical simulations of Eq. (4) conducted for 20000 imated by γH ' 4gδ0/κ and, hence, the laser power is parameter points across the tongues have revealed that the locking, dµ = 0, range starts from the tongue tips and 2 2 π κµ 3 πD2 |µD2| 4 (1) WµC ' |αµ| = µ , |αµ| 1. (17) spreads along their g = gµ edges upwards, see black ηF 2γ 2γD1 ηκ areas in Fig. 1(e), and Fig. 2(a). Another expressed feature of the dynamics inside the tongues is the vio- ∆ = κ and g = g(2) = κ/2 at the FWM thresh- µ µ lation of the repetition rate locking regimes producing old, see Eq. (13), and hence γH2 ' 4gδ2/κ2 now gives 0 d 6= 0, see yellow and grey areas in Fig. 1(e). In the W = W /|α | = κW /|µD |. The prefactor that µ µF µC µ µC 2 coordinate space, the d 6= 0 regimes correspond to a makes the difference between the powers at the FWM and µ pair of the coexisting rolls rotating with the unlocked complexity thresholds and hence measuring the relative rates, |µ|(D ± d ). Two rolls can either coexist inde- depth of the instability tongues is exactly the residual 1 µ pendently (yellow areas), see one of them in Fig. 2(b), or finesse, α . µ generate chaotic switching dynamics (grey areas) associ- The proximity of each tongue provides a param- ated with the weak modes emerging around the dominant eter range where the mLLE dynamics is dominated ones, see Fig. 2(c). Here we deal with the frequency- by the competition between the two normal modes (±) (±) (±) domain symmetry breaking [42], when symmetry is bro- of Vbµ, Vbµ~qµ = iλµ ~qµ . Rearranging Eq. (8) as ken not only for the sideband powers, see mode-number (λ + 1 κ)2 = (∆ − g)(3g − ∆ ), we find Imλ(±) = µ √2 µ µ µ spectra in Figs. 2(b), but also for their spectral content, 1 (±) see and compare the radio-frequency (RF) spectra of the ± Im 3(∆µ − g)(g − 3 ∆µ), and hence Imλµ = 0 is 1 dominant modes in Figs. 2(a) and 2(b). As the thresh- satisfied for 3 ∆µ < g < ∆µ. This is the normal mode synchronisation condition implying that the repetition old of complexity is approached and crossed, the few- rate of the emerging quasi-harmonic in θ, i.e., roll, pat- mode dynamics is replaced by the transition to the de- terns locks to |µ|D1. While D1 is a property of the lin- veloped multimode chaos, when every consecutive mode ear resonator and is independent of the pump parame- in the spectrum is excited forming very dense continua, ters, ωp and g, the normal-mode synchronisation implies see Fig. 2(d). that the nonlinear, g 6= 0, repetition rate locks to an Before concluding, we make further contextual connec- integer-multiple of D1. The instability and synchroni- tions. Refs. [38, 39] looked at the similar to our problem sation tongues merge together with |Fd| increasing, and of the Arnold tongue overlaps in dissipative maps and bi- (1) (2) 1 gµ → ∆µ, gµ → 3 ∆µ, cf., Figs. 1(a) and (b) where ological systems. Refs. [40, 41] measured synchronisation two synchronisation tongues are contrasted against the and tongues for the soliton sequences flowing across a fi- instability ones. bre linking two micro-rings. Tongue formation studied by 5

FIG. 2. (a) Repetition rate locking (dµ = 0, black area in Fig. 1(e)); (b) Frequency-domain symmetry breaking (dµ = const, yellow area in Fig. 1(e)); (c) Few-mode chaos (chaotic variations of dµ, grey area in Fig. 1(e)) (d) Multi-mode complexity 2 (blue area in Fig. 1(e)). Top two rows show the µ = ±35 sideband RF spectra |Sµ(∆)| [37]. 3rd row shows the mode powers at t = 16ms. 4th row is the space-time dynamics. us should be compared with the multiple instability do- [5] G. W. Hunt and P. R. Everall, Arnold tongues and mode- mains containing continua of modes reported in the zero jumping in the supercritical post-buckling of an archety- finesse diffractive feedback systems [43], and in the low- pal elastic structure, Proc. R. Soc. Lond. A. 455, 125 finesse resonators [13, 14, 44]. Refs. [45–49] reported sta- (1999). [6] S.B. Shim, M. Imboden, and M. Pritiraj, Synchronized tionary and breathing rolls with no symmetry breaking in oscillation in coupled nanomechanical oscillators, Science mLLE and alike systems. Refs. [42, 50, 51] studied sym- 316, 95 (2007). metry breaking of the counter-propagating single-mode [7] T. E. Lee, C.-K. Chan, and S. Wang, Entanglement and soliton regimes in bi-directional microresonators. tongue and quantum synchronization of disordered os- In summary: We elaborated theory for the threshold of cillators, Phys. Rev. E 89, 022913 (2014). complexity and nonlinear effects within Arnold tongues [8] M. R. Jessop, W. Li, and A.D. Armour, Phase synchronization in coupled bistable oscillators, Phys. Rev. Re- in Kerr microresonators. Our results hold potential for search 2, 013233 (2020). applications in frequency conversion and RF-photonics [9] N. Es’haqi-Sani, G. Manzano, R. Zambrini, and R. Fazio, areas relying on a range of existing and emerging high-Q Synchronization along quantum trajectories, Phys. Rev. resonators. Research 2, 023101 (2020). This work was supported by the EU Horizon 2020 [10] I.L. Garanovich, S. Longhi, A.A. Sukhorukov, and Framework Programme (812818, MICROCOMB). We Y.S. Kivshar, Light propagation and localization in mod- are deeply indebted to W.J. Firth and L.A. Lugiato for ulated photonic lattices and waveguides, Phys. Rep. 518, 1 (2012). invaluable comments and interest in our work. [11] K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, Ob- servation of Geometric Parametric Instability Induced by the Periodic Spatial Self-Imaging of Multimode Waves, Phys. Rev. Lett. 116, 183901 (2016). ∗ [email protected] [12] D. W. Mc Laughlin, J. V. Moloney, and A.C. Newell, [1] V. I. Arnold, Remarks on the perturbation problem for Solitary Waves as Fixed Points of Infinite-Dimensional problems of Mathieu type, Usp. Mat. Nauk, 38 189–203 Maps in an Optical Bistable Ring Cavity, Phys. Rev. (1983); Russ. Math. Surveys, 38, 215–233 (1983). Lett. 51, 75 (1983). [2] V. I. Arnold, Small denominators and mappings of the [13] F. Bessin, F. Copie, M. Conforti, A. Kudlinski, A. Mus- circumference onto itself. Izv. Akad. Nauk Ser. Mat. 25, sot, and S. Trillo, Real-Time Characterization of Period- 21–86 (1961); AMS Transl. Ser. 2, 46, 213–284. Doubling Dynamics in Uniform and Dispersion Oscillat- [3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- ing Fiber Ring Cavities, Phys. Rev. X 9, 041030 (2019). tion: A universal concept in nonlinear sciences (Cam- [14] N. Tarasov, A. M. Perego, D. V. Churkin, K. Staliunas, bridge University Press, 2001). and S.K. Turitsyn, Mode-locking via dissipative Faraday [4] M.M. Ali, K.K. Sellers, and F. Frohlich, Transcranial instability, Nat. Comm. 7, 12441 (2016). Alternating Current Stimulation Modulates Large-Scale [15] T.J. Kippenberg, A.L. Gaeta, M. Lipson, and Cortical Network Activity by Network Resonance, J. M. Gorodetsky, Dissipative Kerr solitons in optical mi- Neuroscience 33, 11262 (2013). 6

croresonators, Science 361, eaan8083 (2018). press 27, 35719-35727 (2019). [16] S.A. Diddams, K. Vahala, and T. Udem, Optical fre- [36] H. El Dirani, L. Youssef, C. Petit-Etienne, S. Kerdiles, quency combs: Coherently uniting the electromagnetic P. Grosse, C. Monat, E. Pargon, and C. Sciancale- spectrum, Science 369, eaay3676 (2020). pore, Ultralow-loss tightly confining Si3N4 waveguides [17] H. Kimble, The quantum internet, Nature 453, 1023 and high-Q microresonators, Opt. Express 27, 30726- (2008). 30740 (2019). R 2 R 2 [18] M. Kues, C. Reimer, J.M. Lukens, W.J. Munro, [37] h∂tφµi = ∆|Sµ| d∆/ |Sµ| d∆, where Sµ(∆) = −1 R τ1+τ i∆t A.M. Weiner, D.J. Moss, and R. Morandotti, Quantum τ ψµ(t)e dt. The values of dµ used to separate τ1 optical microcombs, Nat. Phot. 13, 170 (2019). the yellow and grey areas in Fig. 1(e) were also averaged [19] Z. Vernon and J. E. Sipe, Strongly driven nonlinear quan- over τ1. tum optics in microring resonators, Phys. Rev. A 92, [38] M. H. Jensen, P. Bak, and T. Bohr, Transition to chaos 033840 (2015). by interaction of resonances in dissipative systems. I. Cir- [20] U. Keller, Recent developments in compact ultrafast cle maps, Phys. Rev. A 30, 1960 (1984). lasers, Nature 424, 831 (2003). [39] M. Heltberg, R. A. Kellogg, S. Krishna, S. Tay, and [21] L. A. Lugiato and R. Lefever, Spatial Dissipative Struc- M.H. Jensen, Noise Induces Hopping between NF-kB En- tures in Passive Optical Systems, Phys. Rev. Lett. 58, trainment Modes, Cell Systems 3, 532–539 (2016). 2209 (1987). [40] J.K. Jang, A. Klenner, X.C. Ji, Y. Okawachi, M. Lip- [22] L. Lugiato, F. Prati, and M. Brambilla, Nonlinear Optical son, and A.L. Gaeta, Synchronization of coupled optical Systems (Cambridge University Press, 2015). microresonators, Nat. Photon. 12, 688 (2018). [23] T. Herr, V. Brasch, J.D. Jost, C.Y. Wang, N.M. Kon- [41] J.K. Jang, X. Ji, C. Joshi, Y. Okawachi, M. Lipson, and dratiev, M.L. Gorodetsky, and T.J. Kippenberg, Tempo- A.L. Gaeta, Observation of Arnold Tongues in Coupled ral solitons in optical microresonators, Nat. Photon. 8, Soliton Kerr Frequency Combs, Phys. Rev. Lett. 123, 145-152 (2014). 153901 (2019). [24] C. Godey, I.V. Balakireva, A. Coillet, and Y.K. Chembo, [42] D.V. Skryabin, A.G. Vladimirov, and A.M. Radin, Spon- Stability analysis of the spatiotemporal Lugiato-Lefever taneous phase symmetry-breaking due to cavity detuning model for Kerr optical frequency combs in the anomalous in a class-A bidirectional ring laser, Opt. Commun. 116, and normal dispersion regimes, Phys. Rev. A 89, 063814 109-115 (1995). (2014). [43] W. J. Firth, I. Kresić, G. Labeyrie, A. Camara, and [25] Y. H. Wen, M. R. E. Lamont, S. H. Strogatz, and T. Ackemann, Thick-medium model of transverse pat- A.L. Gaeta, Self-organization in Kerr-cavity-soliton for- tern formation in optically excited cold two-level atoms mation in parametric frequency combs, Phys. Rev. A 94, with a feedback mirror, Phys. Rev. A 96, 053806 (2017). 063843 (2016). [44] Y.V. Kartashov, O. Alexander, and D.V. Skryabin, [26] H. Taheri, P. Del’Haye, A. A. Eftekhar, K. Wiesen- Multistability and coexisting soliton combs in ring res- feld, and A. Adibi, Self-synchronization phenomena in onators: the Lugiato-Lefever approach, Opt. Express 25, the Lugiato-Lefever equation, Phys. Rev. A 96, 013828 11550-11555 (2017). (2017). [45] L.A. Lugiato, C. Oldano, and L.M. Narducci, Coopera- [27] D. V. Skryabin and Y. V. Kartashov, Self-locking of the tive frequency locking and stationary spatial structures frequency comb repetition rate in microring resonators in lasers, J. Opt. Soc. Am. B 5, 879-888 (1988). with higher order dispersions, Opt. Express 25, 27442- [46] G.L. Oppo, M. Brambilla, and L.A. Lugiato, Formation 27451 (2017). and evolution of roll patterns in optical parametric oscil- [28] D.C. Cole, A. Gatti, S.B. Papp, F. Prati, and L. Lu- lators, Phys. Rev. A 49, 2028 (1994). giato, Theory of Kerr frequency combs in Fabry-Perot [47] P. Parra-Rivas, D. Gomila, L. Gelens, and E. Knobloch, resonators, Phys. Rev. A 98, 013831 (2018). Bifurcation structure of periodic patterns in the Lugiato- [29] S. Wolfram, A New Kind of Science (Wolfram Media, Lefever equation with anomalous dispersion, Phys. Rev. 2002). E 98, 042212 (2018). [30] A.A. Savchenkov, A.B. Matsko, V.S. Ilchenko, I. Solo- [48] A. Coillet, Z. Qi, I.V. Balakireva, G. Lin, C.R. Menyuk matine, D. Seidel, and L. Maleki, Tunable Optical Fre- and Y.K. Chembo, On the transition to secondary Kerr quency Comb with a Crystalline Whispering Gallery combs in whispering-gallery mode resonators, Opt. Lett. Mode Resonator, Phys. Rev. Lett. 101, 093902 (2008). 44, 3078 (2019). [31] I.S. Grudinin, N. Yu, and L. Maleki, Generation of op- [49] Z. Qi, S. Wang, J. Jaramillo-Villegas, M.H. Qi, tical frequency combs with a CaF2 resonator, Opt. Lett. A.M. Weiner, G. D’Aguanno, T.F. Carruthers, and 34, 878-880 (2009). C.R. Menyuk, Dissipative cnoidal waves (Turing rolls) [32] A. A. Savchenkov, A. B. Matsko, V.S. Ilchenko, and and the soliton limit in microring resonators, Optica 6, L. Maleki, Optical resonators with ten million finesse, 1220-1232 (2019). Opt. Express 15, 6768-6773 (2007). [50] M.T.M. Woodley, J.M. Silver, L. Hill, F. Copie, [33] D.V. Skryabin, Hierarchy of coupled mode and envelope L. Del Bino, S. Zhang, G.L. Oppo, and P. Del’Haye, models for bi-directional microresonators with Kerr non- Universal symmetry-breaking dynamics for the Kerr in- linearity, OSA Continuum 3, 1364-1375 (2020). teraction of counterpropagating light in dielectric ring [34] D.N. Puzyrev, Z. Fan, and D.V. Skryabin, Modelling RF resonators, Phys. Rev. A 98, 053863 (2018). spectra of frequency combs (presented at Photon 2020, [51] W. Weng, R. Bouchand, E. Lucas, and T. J. Kippenberg, UK, 1-4 September 2020). Polychromatic Cherenkov Radiation Induced Group Ve- [35] Z. Ye, K. Twayana, P. A. Andrekson, and V. Torres- locity Symmetry Breaking in Counterpropagating Dis- Company, High-Q Si3N4 microresonators based on a sub- sipative Kerr Solitons, Phys. Rev. Lett. 123, 253902 tractive processing for Kerr nonlinear optics, Opt. Ex- 7

(2019).