## Abstract

We theoretically and numerically investigate the effects of high-order dispersion (HOD) on microcavity solitons, both in time and frequency domain with an extended normalized Lugiato–Lefever equation (LLE). The observed temporal drift of bright and dark solitons is shown to originate from high-odd-order dispersion, while the sign determines the direction of soliton movement and the amplitude decides the drift speed. HOD can also be introduced to stabilize the breathing bright and dark cavity solitons. In spectral domain, the nonlinear symmetry breaking is mainly introduced by third-order dispersion, whereas both third- and fourth-order dispersion can introduce dispersive wave accompanied by soliton tail oscillation. This work could give insight for exploring detailed intracavity pulse dynamics and spectral characteristics of Kerr combs influenced by HOD, as well as provide a viable route to delicate control of Kerr comb generation through tailoring the dispersion parameters.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microresonator-based Kerr frequency comb has opened great prospects for nanophotonics and chip-scale optical frequency synthesis [1, 2]. Kerr combs generated in platforms such as silicon nitride [3–5] and silicon resonators [6, 7], CaF_{2} and MgF_{2} crystallines [8, 9] have attracted significant attention due to advantages of their high repetition rates, octave-spanning operation and high compactness [10, 11]. Optical frequency combs (OFC) are promising candidates for a number of applications including optical clocks [12], coherent communications [13] and high precision spectroscopy [14, 15]. Stable soliton state combs that have outstanding features of low frequency and amplitude noise are essential for these applications, and both bright solitons and dark pulses in microresonators have been reported [16, 17].

The instabilities of Kerr frequency combs, usually originate from thermal effects, cavity loss [5, 18] and some other factors such as dispersion, in particular HOD [19–22]. Importantly, M. Tlidi *et al*. first derived the LLE equation for fiber resonators taking up to fourth-order dispersion effects into account [23], and the stabilization of dark cavity solitons thanks to the fourth order dispersion has been analyzed [22]. M. Tlidi *et al*. investigate temporal cavity solitons exhibit a motion with a constant velocity which induced by third-order dispersion in photonic crystal fiber resonator focusing the analysis on dark temporal cavity solitons [24]. L. Bahloul *et al*. investigate formation of temporal localized structures influenced by third-order dispersion and fourth-order dispersion in an all photonic crystal fibre resonator [25]. Researchers have also analyzed the stability of bright solitons and snaking structure in anomalous dispersion microresonators [26, 27]. Theoretically, the intracavity pulse can form either bright or dark solitons when the cavity shows anomalous or normal dispersion. In the presence of HOD, it has been proved that bright solitons can emit soliton Cherenkov radiation (i.e., dispersive wave) in optical fibers [28, 29], which is a key nonlinear frequency conversion mechanism in coherent supercontinuum generation [30]. Such solitons with dispersive wave (DW) could provide a path to broadband frequency comb generation [31, 32]. In normal dispersion regime, dark solitons have been analyzed by the prevailing Lugiato–Lefever model, whereas dark solitons with HOD and their spectra have not been studied. Furthermore, the DW generation and temporal evolution of both bright and dark solitons in microresonators affected by HOD need to be uncovered.

In this paper, we introduce an extended normalized LLE incorporating HOD and theoretically investigate the influences of HOD on temporal and spectral characteristics of both dark and bright solitons. The temporal delay of microcavity solitons induced by high-odd-order dispersion are observed. In addition, the sign of odd-order dispersion determines the direction of soliton movement and the phase distribution while its value decides the drift speed. Breathing bright and dark cavity solitons are stabilized in the presence of HOD. In frequency domain, HOD leads to DW excitation and concomitant advanced or delayed soliton tail oscillation, whereas the nonlinear symmetry breaking is mainly introduced by third-order dispersion. As a result, all HOD parameters jointly tailoring the dispersion profile can directly lead to different locations of DWs, and finally affect the entire comb spectrum envelope as well as temporal stability. These phenomena captured well by the proposed extended LLE are in accordance with the theoretical moment analysis and dispersive radiation theory reported in other literatures. Our work could deepen the understanding of microcavity soliton dynamics and their spectral properties influenced by HOD, and thus provide a viable route to delicate control of microresonator-based broadband comb generation.

## 2. Theoretical model and numerical calculation

The total intracavity field evolution and the corresponding Kerr frequency comb generation can be described by a normalized LLE containing damping, detuning, driving and HOD [33–35]; that is,

*Ψ*(

*τ*,

*θ*) is the slowly varying field envelope,

*τ*is the slow time,

*θ*∈(-

*π*,

*π*] is the azimuthal angle along the circumference, dimensionless frequency detuning

*α*= −2(

*Ω*

_{0}-

*ω*

_{0})/Δ

*ω*

_{0}describes the detuning between the pump and cavity resonance frequencies where

*Ω*

_{0}and

*ω*

_{0}are the angular frequencies of the pump laser and cold cavity resonance.

*β*

_{n}= −2

*ζ*

_{n}/Δ

*ω*

_{0}is the dispersion term and

*ζ*

_{n}=

*d*

^{n}

*ω/dl*|

^{n}

_{l}_{=}

*is the n-order dimensionless dispersion coefficients, where*

_{l0}*l*represents the eigennumber of the pumped mode.

_{0}*F*is the dimensionless external pump and can be related to laser pump power

*P*through [33]Δ

*ω*

_{ext}and Δ

*ω*

_{0}represent the coupling and total linewidths, respectively. The nonlinear gain

*g*

_{0}iswhere

*n*

_{0}and

*n*

_{2}are the linear and nonlinear refractive index of the material, respectively.

*V*

_{0}is the mode volume of pump mode, and $\hslash $ is reduced Planck’s constant.

*F*is real valued and positive here, and the optical phase reference is arbitrarily set by this pump radiation for all practical purpose. In this model, the self-steepening and Raman effect have not been considered since self-steepening has little influence on the soliton comb generation, and Raman effect depends on the property of material and pump condition. We mainly focus on the general case of comb generation influenced by high-order dispersion, so these two effects are neglected in our calculation for simplification and a more clear analysis of dispersion.

To analyze HOD influence on bright and dark soliton, moment analysis is employed in this work, which has been widely used for analyzing pulse propagation in fiber optics and microcavity [36, 37] as well as for calculating timing jitter in lightwave systems without Raman effect [38, 39]. Assuming that the pulse can maintain a particular shape when it is transmitted in a nonlinear medium, it can be solved approximately by the moment method despite the amplitude and pulse width change in a continuous manner. Essentially, classical solitons rely on the double balance between anomalous Kerr nonlinearity and cavity dispersion on the one hand, and cavity loss and parametric frequency conversion of the pump light on the other hand, which can be analyzed by this method. The basic principle of moment method is to treat the pulse as a particle. Pulse energy *E*, position *φ*_{c} and frequency shift *μ _{c}* can be defined as:

Applying derivations with respect to slow time (*τ*) in Eq. (1) on these moment and after some calculations, the following equation can be obtained

It can be seen that the energy *E* built up in the cavity depends on the external pump and cavity loss. Hence the energy must be constant for a stable soliton formation, that is, $\partial E/\partial \tau =0$is fulfilled. The position and frequency shift moment can be written as

It is well known that

Therefore the temporal position shift of the solition is caused by high-odd-order dispersion. The frequency shift *μ*_{c}, mainly related to the Raman effect and detuning, is not observed here for the absence of Raman effect. In order to validate the theory analysis, numerical simulations are implemented by solving Eq. (1). The bright and dark solitons are both taken into account. The common approaches are introduced an exponential shape pulse as the initial condition, as employed in [33, 40]

*+*and - are often used for bright and dark solitons, respectively. Here a = 0.5, b = 1, c = 0.1 is employed for bright soliton and a = 1.8, b = 1, c = 0.1 for dark soliton.

Figures 1 and 2 show temporal evolution of bright and dark solitons with slow time *τ*, respectively. Specifically, solitons evolve without any temporal drift if only even-order dispersion (e.g., *β*_{2} or *β*_{4}) is taken into account whether in positive or negative chromatic dispersion regime, as can be seen in Figs. 1(a) and 1(c) for bright solitons (anomalous dispersion) as well as Figs. 2(a) and 2(c) for dark solitons (normal dispersion). Whereas obvious temporal shifts can be observed when *β*_{3} or *β*_{5} is considered, as shown in Figs. 1(b) and 1(d) for bright solitons and Figs. 2(b) and 2(d) for dark solitons. Meanwhile, soliton tail oscillation seen in pulse trailing edge is due to the DW excitation via Cherenkov radiation process [32]. This means that different dispersion will lead to different pulse shapes, time domain evolution and tail shock. These results match well with theoretical analysis in Eq. (11). In addition, *β*_{3} and *β*_{5} show different degrees of influence on drift speed of solitons since third-order dispersion disturbance is stronger than that of fifth-order dispersion.

In order to further study the effects of third-order dispersion on bright soliton, the results of solitons drift with different speed and direction are shown in Fig. 3. Figures 3(a) and 3(b) depict the temporal evolution with the same *β*_{2} while with the opposite sign of *β*_{3}. The upper graphs show the temporal profile and phase distribution. It can be seen that the solitons shift in opposite directions and thus leading to different phase profiles of solitons. Advanced and delayed oscillations are attributed to different group velocity. Figures 3(b) and 3(c) show the results with the same *β*_{2} while Fig. 3(c) with a smaller *β*_{3}. A more detailed image of soliton drift speed is shown in Fig. 3(d). *V* is the soliton temporal drift velocity in a single roundtrip due to high-odd-order dispersion. Soliton in Fig. 3(b) drifts faster than that in Fig. 3(c), marked as point b and c in Fig. 3(d), respectively. It can be observed in Fig. 3(d) that the *V* first increases with *β*_{3} whereas almost keep unchanged when absolute value of *β*_{3} exceeds 8 × 10^{−4}. This indicates the drift speed increases with third-order dispersion and then becomes constant when third-order dispersion surpasses a certain value. Obviously, it can be concluded that the sign of *β*_{3} just determines the direction of soliton movement, which results in different phase relationship, while the amplitude of *β*_{3} decides the drift speed. Similarly, L. Bahloul *et al*. investigate the linear velocity and the nonlinear velocity of temporal localized structures in an all photonic crystal fiber resonator [25]. Theoretically, this regular drift of bright and dark solitons is induced by a broken reflection symmetry mediated by high-odd-order dispersion.

An interesting solution of the LLE is the breathing solitons exhibiting a periodical amplitude varies as reported by Cyril Godey *et al*. [33]. As displayed in Figs. 4(a) and 4(c), for bright and dark breathers respectively, the breather solitons oscillate in time and approximately keep the same pulse width. The breathing temporal cavity solitons will degrade the coherence of the final comb profile. However, such cavity breathing solitons can evolve into a stable single soliton in the presence of high-order dispersion. It can be clearly seen from Figs. 4(b) and 4(d), for stable bright and dark solitons respectively, that the breathing feature vanishes if the third-order dispersion is incorporated. Specifically, just taking the third-order dispersion as an example here, high-order dispersion like fifth-order dispersion can also be taken into account to stabilize the breathing bright and dark solitons. It can be concluded that high-order dispersion can be introduced to stabilize breathers both in anomalous and normal dispersion regime.

In the following section, the influences of HOD on comb spectrum are investigated. The commonly known DW has a close relationship with HOD. According to the dispersive radiation theory reported in [41, 42], and considering the fourth-order dispersion, after some calculations the explicit expression for frequency of DW is:

*P*is the background power.

_{0}*V*is the soliton temporal drift velocity in a single roundtrip due to high-odd-order dispersion. Evidently, the roots for DW(

*μ*) = 0 are the frequency location of the generated DWs.

Some representative results of generated spectrum with DWs of bright solitons are shown in Fig. 5. Figure 5(a) illustrates two asymmetry DWs when *β*_{2} and *β*_{3} are included in the simulation, while Fig. 5(b) shows two symmetry DW when *β*_{2} and *β*_{4} are considered. A. G. Vladimirov *et al*. reported Cherenkov radiation emitted in the presence of third-order dispersion breaks the symmetry of the localized structures interaction [43]. This difference can be clearly derived from Eq. (13). Both third- and fourth- order dispersion can introduce DWs and the accompanying soliton tail oscillation in time domain but third-order dispersion often leads to asymmetry of the spectrum, that is, nonlinear symmetry breaking which is similar to the phenomenon in optical fiber cavities [44]. Figures 5(c) and 5(d) correspond to spectra with three and four DWs respectively, when *β*_{2}, *β*_{3} and *β*_{4} are taken into account. A relative larger *β*_{4} is used in Fig. 5(d) than that in Fig. 5(c). Since fourth-order dispersion contributes to the sideband phase matching, which becomes important for four-wave-mixing process over broader bandwidth [45], the bandwidth in Fig. 5(d) is narrower than that of Fig. 5(c). This suggests small fourth-order dispersion is necessary for further extending the comb bandwidth. This results imply that HOD play critical role in dispersion engineering for different location of DW generation and broadband comb generation.

As a comparison, Fig. 6 displays four kinds of typical spectra for dark solitons affected by HOD. Figures 6(a) and 6(b) show one and two DWs with different *β*_{2} and *β*_{3}, while Figs. 6(c) and 6(d) depict another two representative results with the same *β*_{2} and *β*_{3} but different *β*_{4}. A relatively larger *β*_{4} is used in Fig. 6(d) than that in Fig. 6(c), so the bandwidth is narrower than that of Fig. 6(c), due to similar reason mentioned above for anomalous dispersion regime. It should be noted that the fourth-order dispersion could modulate the phase match condition and result in the disappearance of DW [46], which causes the absence of time domain oscillation in temporal evolution of Fig. 6(d). All the results of DW through solving LLE are consistent with the radiation theory analysis. Therefore, it can be concluded that through optimal design of the waveguide geometry, the dispersion parameters can be carefully tailored and thus, delicate control of the stability and bandwidth for Kerr frequency combs can be achieved via the precise management of the soliton drift, emitted DW numbers and locations, etc.

## 3. Summary

In conclusion, the effects of HOD on both dark and bright solitons generated in normal and anomalous dispersion regime have been theoretically investigated, based on an extended normalized LLE. It is found that HOD can give rise to the intracavity soliton drift, as well as affect the DW generation properties accompanied by soliton tail oscillation. In temporal domain, the drift direction and speed of microcavity soliton are determined by the sign and amplitude of high-odd-order dispersion, respectively. HOD also tends to suppress dynamical instabilities of breathing bright and dark cavity solitons. In spectral domain, the asymmetric distribution of spectrum envelop mainly originate from the third-order dispersion, while all dispersion parameters together contribute to the emitted DW locations and finally decide the entire comb spectra. These phenomena revealed well by the proposed extended LLE are in good accordance with the theoretical moment analysis and dispersive radiation theory. Our work could help to a deeper understanding of the influence of HOD on the detailed soliton dynamics and characteristics, as well as give insight for exploring the path towards more precise control of microresonator-based frequency comb generation.

## Funding

National Natural Science Foundation of China (Grant Nos. 61635013; 61675231; 61475188; and 61705257) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB24030600).

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