Pure high-order dispersion dissipative Kerr solitons in optical cavitieSs

4262 Vol. 50, No. 13 / 1 July 2025 / Optics Letters Letter Pure high-order dispersion dissipative Kerr solitons in optical cavities Carlo Silvestri,1,2,∗Y. Long Qiang,1,2 Krupamaya Panda,1,2,3 Justin Widjaja,1,4 Stéphane Coen,5,6 C. Martijn de Sterke,1,2 AND Antoine F. J. Runge1,2 1Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, NSW 2006, Australia 2ARC Centre of Excellence for Optical Microcombs for Breakthrough Science (COMBS), School of Physics, University of Sydney 2006, Australia 3Ecole Centrale de Lyon, INSA Lyon, CNRS, Université Claude Bernard Lyon 1, CPE Lyon, INL, UMR5270, Ecully, 69130, France 4Current address: Department of Applied Physics, California Institute of Technology, Pasadena 91125, California, USA 5Department of Physics, University of Auckland, Auckland 1010, New Zealand 6The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand *[email protected] Received 11 April 2025; revised 28 May 2025; accepted 1 June 2025; posted 2 June 2025; published 20 June 2025 Through numerical simulations, we demonstrate the existence of an infinite family of temporal cavity solitons (CSs), which balance arbitrary negative pure, even-order dispersion kand self-phase modulation, as well as loss and parametric gain. These correspond to frequency combs with increasingly flatter spectra as kincreases. We determine the analytic forms of these solitons at high pump power and detuning and derive that their energy is related to the pulse duration ∆τas ∆τ−(k−1).© 2025 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved. https://doi.org/10.1364/OL.564975 Optical frequency combs have revolutionized fundamental science by allowing frequency measurements with unprecedented precision [1]. Originally developed using complicated, bulky, and expensive lasers, subsequent investigations led to their generation in simple, compact, passive microresonators [2]. In current state-of-the-art systems, the generation of octave spanning frequency combs is associated with the excitation of cavity solitons (CSs), pulses that maintain their shape by balancing loss with parametric gain, along with the balance of Kerr nonlinearity and negative dispersion [3,4]. CSs are stationary solutions of the Lugiato–Lefever equation (LLE), which for high-Qresonators is equivalent to the nonlinear Schrödinger equation (NLSE) with external driving and damping [5,6]. It is thus not surprising that CSs exhibit similar properties to NLSE solitons in conservative systems in the appropriate limit [5,7]. In modeling the formation of CSs, the dispersion is usually taken to be purely quadratic, as it is the dominant effect in standard resonators [3,4], although combinations of quadratic and cubic orders of dispersion have also been considered [8,9]. However, recent studies showed that soliton pulses can also arise from the balance between the Kerr effect and negative pure-quartic dispersion [10,11]. These pure-quartic solitons (PQSs) have different intensity profiles and, more interestingly, follow a different scaling with their energy varying as ∆τ−3, where ∆τis the pulse duration. PQSs have been experimentally observed in integrated photonic waveguides [10,12] and lasers [13]. Cavity-PQSs in passive dissipative resonators are described by the LLE with only negative quartic dispersion and have also been investigated theoretically and numerically [14–18]. These studies confirmed that the advantageous scaling property carries through, making cavity-PQSs attractive for generating high-energy, microresonator-based frequency combs [15]. Solitons with negative quadratic or quartic dispersion are only the first two members of a more general family of shapemaintaining nonlinear pulses [19]. In fact, the Kerr nonlinearity can be balanced by any negative even order of dispersion k, and the energy of these pulses scales as ∝∆τ−(k−1)[19]. These results were confirmed in laser experiments in which the netcavity dispersion could be arbitrarily adjusted [13,19]. However, this more general family of solitons has not been investigated in the context of passive resonators with coherent driving. In this Letter, we theoretically and numerically study the generation and the properties of CSs in passive optical Kerr resonators with pure, high, even-order dispersion, by solving a generalized LLE. We show that, consistent with their conservative soliton counterparts, they form a family that has quadratic CSs and pure-quartic CSs as their two lowest-order members [5,15]. The quartic case was studied in [15], so here we focus on k≥6. We show that the typical dynamical behavior of modulation instability (MI)-unstable MI-CSs, observed in Kerr cavities with quadratic dispersion, is independent of the dispersion order. However, we find that higher-order dispersion favors the formation of perfect soliton crystals, characterized by equally spaced solitons in the cavity. We determine the scaling properties of these CSs and show that they also scale as the inverse pulse width ∆τ−(k−1). Finally, we show that the spectral shape becomes flatter as the dispersion order increases. Our results could be verified experimentally using existing approaches, such as passive fiber cavities with dispersion tailored by a spectral wave shaper [20], or integrated microresonators, dispersion engineered via a Bragg mirror design [21]. We expect our results to stimulate 0146-9592/25/134262-04 Journal © 2025 Optica Publishing Group Letter Vol. 50, No. 13 / 1 July 2025 / Optics Letters 4263 Fig. 1. LLE simulated dynamics when scanning the driving laser over a resonance for k=6 (top row), k=8 (middle), and k=12 (bottom). (a), (e), and (i) Average intracavity power versus normalized detuning. Temporal intracavity intensity profiles for selected detuning corresponding to different regimes: (stable MI) ∆=−1.5 (b), (f), and (j), (k) (unstable MI) ∆=2 (c)–(g), ∆=3, and (d), (h), and (l) perfect soliton crystal ∆=8 . The insets show the associated field spectra. The pump power is X=12 for all simulations. follow-up investigations of nonlinear pulse dynamics in driven dissipative systems with high orders of dispersion in photonics and applied mathematics. We begin by considering the evolution of the intracavity field envelope E(t,τ)in high-Qring resonators, which can be described by the normalized LLE: ∂E(t,τ) ∂t =[︃−1+i(|E|2−∆)+sign (βk)ik+1∂k ∂τk]︃E+S,(1) where βkis the kth order dispersion coefficient and tis the slow time describing the evolution of E(t,τ)at the scale of the cavity photon lifetime, while τis a normalized fast time defined in the reference frame traveling at the group velocity of the light in the resonator. The terms on the right-hand side of Eq. (1) correspond to the cavity losses, Kerr nonlinearity, pump-cavity detuning, kth order dispersion, and external pumping, respectively. In our normalization, the scaled τis expressed as τ=˜τ(︃αk! |βk|L)︃1/k ,(2) where ˜τ,α, and Lare the unscaled fast time, loss coefficient, and resonator length, respectively. We take anomalous dispersion, so sign (βk)=−1. Parameters ∆,S, and the slow time tfollow the scaling in Refs. [22,23]. We consider the experimental parameters for a 35.2 GHz MgF2resonator with L=6.2 mm,α=π/F with F=73 ×103, driven with Pin ≈9.3 mW, corresponding to a normalized pump power X=|S|2=12 [3,6]. In order to compare our results with the quadratic dispersion case, we define βk=β2k! 2∆ωk−2, with ∆ω=1.6 THz,β2=−5.9 ps2·km−1[3]. We numerically solve Eq. (1), using a sixth-order split-step Fourier method [24], for three different orders of dispersion k. The calculated average intracavity powers against normalized detuning ∆for k=6, 8, and 12 are shown in Figs. 1(a)–1(c), respectively. For all orders k, the average intracavity power exhibits somewhat similar dynamics, corresponding to different types of LLE solutions [23], as the pump frequency is scanned across the cavity resonance [6,15]. MI is observed when the pump is blue detuned with respect to the cavity mode (∆<0), and the total intracavity power increases with increasing detuning. This is illustrated in Figs. 1(b), 1(f), and 1(j), which show the spectral intensity profiles for the three dispersion orders kat ∆=−1.5. They display several spectral lines separated by multiple free spectral ranges (FSRs) [6,23]. The number and relative intensities of the sidebands decrease as kincreases due to increasingly poor phase-matching between the pump and higher sidebands, limiting four-wave mixing frequency generation [25]. Effectively, for k=12, only the pump and the ±1 sidebands resonate, as seen in Fig. 1(j). Note that the normalized fast time interval (corresponding to the scaled round trip time) varies with k, as defined by Eq. (2). As the pump frequency is further detuned, the system crosses a region of unstable MI (Figs. 1(c), 1(g), and 1(k)), followed by the emergence of soliton crystals [26] (Figs. 1(d), 1(h), and 1(l)), before decaying back to continuous wave (CW) [3]. For k=6, the unstable MI states exhibit a periodic intracavity profile that varies over different values of the slow time while keeping its periodicity. An example is shown in Fig. 1(c), where the intensity profile consists of three replicas of the same pattern. However, we verified that for higher pump power values, spatiotemporal chaos occurs for k=6. Although purely chaotic states emerge for k=8 and k=12 (Figs. 1(g)–1(k)) at X=12, we overall observe that the detuning range associated with chaos is narrower than for quadratic dispersion at the same pump power [6]. Moreover, in contrast to quadratic dispersion, with high-order dispersion, most reported soliton crystals are defect-free: they consist of equidistant, identical solitons, resulting in comb spectra with intermode spacings that are multiples of the cavity free spectral range (FSR). However, soliton crystals with different periodicities can emerge at the same values of ∆and Xwhen using different detuning step sizes in the scans. These regimes are relevant for chip-scale optical frequency combs with repetition rates beyond several terahertz [27]. Moreover, the intracavity power is concentrated in a reduced number of optical lines, so that each line carries a power N2times greater than that of a single-soliton state generated under the same detuning and pump 4264 Vol. 50, No. 13 / 1 July 2025 / Optics Letters Letter Fig. 2. Temporal (left) and spectral (right) intensity profiles of the phase-locked CSs for k=6 (top), k=8 (middle), and k=12 (bottom), by numerically solving Eq. (1) for X=12 and ∆=11 (solid blue). Red dashed curves correspond to the analytic solutions in the self-pumping limit. Insets show temporal profiles on a logarithmic scale. power, where Nis the number of identical solitons in the crystal [27,28]. A possible explanation for the formation of perfect soliton crystals is linked to the oscillating tails of solitons in the presence of high-order dispersion [11,19]. It has been recently shown that the interactions within a chaotic soliton bunch in a mode-locked laser could be regulated using quartic dispersion [29]. We also note that chaotic and unstable regions can undergo different dynamics for large kcompared to the lowerorder dispersion cases [18,30]. However, a detailed stability and bifurcation analysis is beyond the scope of this work. Single CSs can be induced in simulations by using an appropriate single-pulse initial condition (e.g., with a Gaussian profile). The temporal and spectral intensity profiles of these phase-locked CSs, for the same three different dispersion orders k, are shown in Fig. 2. As for conservative solitons arising from high-order dispersion, the temporal profiles exhibit oscillations in the tails, which become more prominent with increasing dispersion order [11,19], as seen in the insets of Fig. 2. The corresponding CS spectra, shown in the right column, become increasingly flat with dispersion order k. This can be understood by recalling that the second derivative of a function is associated with the second moments of its Fourier transform [11]. Therefore, the increasing oscillations in the temporal domain lead to increasingly lower second-order moment, and the associated Fourier transform is thus increasingly flat [11,15,19]. For quadratic dispersion (k=2), the shape and properties of the CSs were derived using the perturbation theory of Eq. (1) [7,23], while for k=4, Taheri and Matsko used the Lagrangian variation method [15]. Here, we use a method to generate analytic, stationary solutions, of nonlinear wave equations in the form of a rapidly converging sum of functions [31]. An outline of this method is provided in Supplement 1. We apply this method to solve Eq. (1) under the self-pumping condition S=E, so the driven problem then maps to the conservative case [31]. The analytic solutions (red dashed curves) for the same parameters as Fig. 3. Normalized cavity soliton peak power versus normalized detuning (left) and kth power of the pulse width ∆τversus normalized inverse detuning (right), for X=10 (blue markers), X=30 (black), and X=100 (red) and for dispersion orders (a), (b) k=6, (c), (d) k=8 , and (e), (f) k=12. Dashed purple: analytic results. Continuous lines are best fits. the numerical ones are shown in Fig. 2, demonstrating excellent agreement with the numerical results. To find the scaling properties of high-order CSs, we determine how the evolution of the pulse peak power Ypeak and pulse duration ∆τdepend on the detuning ∆by numerically solving Eq. (1) [7,15,23] for different pump powers. These parameters can be directly linked to the pulse energy as Ek≈Ypeak ×∆τ. For all dispersion orders, the peak power increases linearly with the detuning, as seen in Figs. 3(a), 3(c), and 3(e), consistent with previous studies for k=2 and 4 [15,23]. The kth power of the pulse width ∆τscales linearly with the inverse detuning (Figs. 3(b), 3(d), and 3(f)). Note that we have adopted a normalization of the axes with respect to Xanalogous to [23], which reveals universal scaling trends analogous to the quadratic case. The combination of these scaling properties reveals a generalized scaling law for the energy of these CSs, which varies as ∝∆τ−(k−1), as for the conservative case [19]. We further observe that, for sufficiently large detunings, as the pump power Xincreases, all curves approach those obtained from the analytic solutions in the self-pumping limit. This is also evident when observing how the normalized intensity profile of CS solutions evolves for different values of k as Xincreases while keeping the ratio X/∆constant (see Fig. 4). Therefore, analogously to quadratic dispersion, for X≫1, the CS solutions tend toward the self-pumping case, which maps to the conservative limit. In fact, in Fig. 4, the red curves, corresponding to X=100, are indistinguishable from the analytical solutions (purple dash). The results discussed above correspond to the case of microresonators, where the temporal waveform has a time duration comparable with the cavity round trip time. We also performed numerical integration of Eq. (1) for parameters corresponding to long passive fiber-loop cavities (not shown here), similar to the ones reported in Refs. [22,32], and obtained similar results. Letter Vol. 50, No. 13 / 1 July 2025 / Optics Letters 4265 Fig. 4. Normalized soliton intracavity profiles for (a) k=6, (b) k=8, and (c) k=12, at different pump powers: X=10 (blue), X=30 (black), and X=100 (red). In all cases, ∆/X=0.7. The analytic solution is shown as a dashed purple curve. In conclusion, we have theoretically and numerically investigated the existence and properties of a general family of CSs in passive optical Kerr cavities. These pulses arise from the double balance between loss and parametric gain, as well as self-phase modulation and any negative, pure, even-order dispersion k. In analogy with the quadratic dispersion case [23], we identified universal trends in the solutions, showing that the energy of these novel CSs scales as ∝∆τ−(k−1), like conservative highorder dispersion solitons [19]. Moreover, the spectrum of the CSs becomes increasingly flat with increasing k. This could be used to generate broadband frequency combs with small tooth power variations, for telecommunication applications [33,34]. We expect our results to encourage future experimental studies of dissipative solitons in physical systems with high-order dispersion. Novel design approaches could be used to achieve the required dispersion in integrated Kerr resonators [21,35,36]. Alternatively, these novel CSs could also be generated in passive fiber cavities, where the net-cavity dispersion can be arbitrarily modified using a spectral pulse-shaper [20]. Funding. Australian Research Council (CE230100006, DE220100509, DP230102200); International Associated Laboratory in Photonics between France and Australia (LIA ALPhFA); Australia-France Network of Doctoral Excellence (AUFRANDE); European Union’s Horizon Europe Framework Programme; Marie Sklodowska-Curie HORIZON-MSCA-2021-COFUND (101081465). Acknowledgment. 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