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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 (C) 2021 American Physical Society 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 jµj 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 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: = µ µ(t)e . Here µ and µ are the mode arXiv:2101.02247v1 [physics.optics] 6 Jan 2021 spectral non-equidistance and of the resonance density in numbers and amplitudes. # 2 [0; 2π) is an angle varying the rotating frame, that play a critical role in bringing along the ring. Resonant frequencies, !µ, are counted the microresonator Arnold-tongues to the physical realm. from !0 and approximated as The Lugiato-Lefever equation (LLE) [21] is a key 1 2 model in the microresonator area [15]. Originally pro- !µ = !0 + D1µ + 2 D2µ ; µ = 0; ±1; ±2;:::: (1) posed in the context of pattern formation in an opti- cal 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, jD2j=2π = 1kHz are good tion of LLE (mLLE) describes the interaction of many estimates for CaF2 resonators for !0=2π being close to longitudinal modes and therefore should reflect on the 200THz pump, !p, [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 !µ±1 − !µ 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 ! = (! +µD +Imλ )+ (! −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 = !0/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 jFdj . 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 jFdj 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 = ! − !p , i@t = δ0 − 2 D2@θ − i 2 κ ( − H) − γj j : (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 = !0 − !p is detuning 2 of the pump laser frequency, !p, from !0. Eliminating µ one finds Transformation to the rotating reference frame, θ = 2 1 2 (µF )2 # − D1t, replaces the !µ-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.

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