Dynamics of Dark Pulse Affected by Higher-Order Effects in Microresonators

We theoretically investigate dynamics of dark pulse and Raman-Kerr microcombs generation influenced by higher-order effects, including high-order dispersion (HOD), stimulated Raman scattering (SRS) and self-steepening (SS) effects in silicon microresonators. These three effects cause the delay of dark pulse individually, or interact with each other to alter the drift velocity and direction of pulses. HOD effect can change pulse shift direction and even cause bifurcation. The temporal drift induced by SS or SRS effects could be balanced by the simultaneous third-order dispersion (TOD) engineering. In spectral domain, stable Raman-Kerr frequency comb will be generated due to the competition between strong SRS and Kerr effects. The Raman comb components are suppressed when HOD effect coexists, while SS effect has ignorable effect on the distribution of the Raman comb. Furthermore, the SS effect will increase the total energy of the spectrum by shifting the dispersive wave (DW) generation to the longer wavelength side. Our findings could deepen the understanding of intracavity nonlinear dynamics and provide theoretical guidance to precisely control the stabilization of dark pulse and the generation of broadband mid-infrared (MIR) microcomb.


I. INTRODUCTION
M ICRORESONATOR-BASED soliton microcombs have attracted significant interest among researchers due to their prospective applications in many fields, including high precision spectroscopy [1], [2], [3], coherent communication [4], [5], optical clocks [6], [7], microwave signal synthesis, imaging and ranging [8], [9], [10], [11]. This field are currently studied in two branches, namely bright and dark solitons generated in the anomalous group velocity dispersion (GVD) and the normal Manuscript  group velocity dispersion regime, respectively. In particular, bright solitons have been explored actively in recent years and have been observed in many platforms like MgF 2 [12], [13], diamond [14], [15], silica [16], [17], aluminum nitride [18], silicon nitride [19], [20], [21], [22], [23], [24], [25], [26], and silicon [27], [28] microresonators, achieved by elaborate waveguide cross-section engineering. Actually, many optical materials present normal GVD in the visible and near-infrared range due to ultraviolet absorption, which makes it still challenging to obtain anomalous dispersion at arbitrary center wavelengths [26]. What's more, dark pulses are more efficient than bright solitons sequences for the same absolute value of anomalous GVD in terms of the pump-to-comb conversion [29], [30]. Excitation of dark pulse in the normal GVD can also increase the degree of freedom in the design and fabrication of microresonators, thus enabling access to microcombs generation in more materials and extended wavelength ranges. Furthermore, compared with their anomalous dispersion-based counterpart, the generation of dark pulse is also more stable due to the avoidance of chaotic and multi-soliton states [31]. To date, dark pulses have been observed experimentally in silica [32], AlN [33], [34], Ge [35], [36], [37], and Si 3 N 4 [38], [39], [40] microresonators with normal GVD. Extensive technics have been introduced for their excitation, including mode coupling [41], [42], self-injection locking [43], [44], [45], negative thermal effects [46], free carrier effects [47], bichromatic or amplitude-modulated pump schemes [29], [48]. In general, the dark pulse microcombs currently are mainly studied in the communication band. Silicon has a high nonlinear coefficient and a wide transparent window, as an excellent platform for generation of microcombs in the MIR. The MIR band is the fingerprint region of many gas molecules (e.g., pollutants and greenhouse gases) [49], [50], and the high conversion efficiency of dark pulse could provide an effective support for MIR spectroscopy. However, reports on silicon microresonators with normal GVD are limited. The influences of TOD and SRS effects on dark pulse have been solely investigated. TOD induces the temporal drift of dark pulse [41], [51], SRS causes strong instabilities of dark pulse associated with the Kerr effect [52]. In addition, SS effect within the cavity also impose non-negligible influences on pulse stability and spectral bandwidth while its role in microresonators still remain unrevealed. Due to complex dynamic processes within the microcavity, the threshold of complexity and nonlinear effects within Arnold tongues in Kerr microresonators is studied [53]. Higher-order effects such as HOD, SRS, and SS effects generally exist simultaneously, and the interaction mechanism among them has not been reported. Studying the interaction of these effects contributes to further control the stabilization of dark pulses and their spectral bandwidths.
In this work, we theoretically investigate the dynamics of dark pulse and the generation of Raman-Kerr microcombs in silicon microresonators with normal GVD in the presence of HOD, SRS and SS effects. It is found that higher-order effects can cause pulse bifurcation and drift direction changes. HOD effect attenuates the temporal shift and perturbation caused by SRS effect, while drift velocity increases significantly when HOD and SS effects exist simultaneously. Pulse delay associated with the SS or SRS effects can be balanced by TOD design. In spectral domain, strong SRS effect competes with the Kerr effect during the four-wave mixing (FWM) process, eventually resulting in stable Raman-Kerr frequency comb that reach equilibrium. Raman comb components and spectral broadening due to SRS effect can be suppressed by HOD effect but are less affected by SS effect. It is revealed that SS effect will shift the generation position of the DW to the longer wavelength side and thus enhance the total energy of spectrum. These studies are important theoretical complements to the dynamics of dark pulse and the generation of Raman-Kerr microcombs, not only providing novel ideas for the further stabilization of dark pulse, but also contributing to the acquisition of broadband MIR microcomb.

II. THEORETICAL MODEL
In order to model the physical process in normal dispersion silicon microresonators, the modified Lugiato-Lefever equation (LLE) including HOD, three-photon absorption (3PA), freecarrier (FC), SS and SRS effects is used to describe the spectraltemporal dynamics of frequency combs [54]. This model is a variant of the nonlinear Schrödinger equation widely used to describe nonlinear effects in pulse transmission [55], [56], [57], [58].
where E(t, τ ) is the field in the resonator, t and τ correspond to the slow and fast time, respectively, T R is the round-trip time, L is the total cavity length, α is the round-trip loss, κ is the power transmission coefficient between bus-waveguide and microresonator. δ 0 is the cavity detuning, β n is the n-th order dispersion parameter, γ is the nonlinear coefficient, R(t) is the Raman response, β 3PA is the three-photon absorption coefficient, σ is the free-carrier absorption (FCA) cross-section and μ is the free-carrier dispersion (FCD) parameter. E in is the pump field. The term contains i/ω 0 is related to SS effect. In (2), dτ describes the buildup of carriers within the cavity over successive round trips [29]. Free-carrier generation is governed by 3PA and the recombination rate is determined by the effective FC lifetime τ eff , which can be controlled by a positive intrinsic negative (PIN) diode and is set as 5 ps [59], [60]. Here, single pump combining with the pump mode eigenfrequency shifted is adopted [52]. The Raman effect is calculated by the convolution theorem which states . The Fourier transform of the Raman response used here is a Lorentzian gain spectrum in the form of (3), where the full width at half maximum (FWHM) of Raman gain spectrum is Γ R /π = 105 GHz at room temperature and the peak gain frequency shift Ω R /2π = 15.6 THz. The Raman part of the material susceptibility in the classical approximation can also be described by the oscillator equation, which allows a straightforward theoretical study of instabilities as done in [61].

III. THE INFLUENCE OF HOD, SRS AND SS ON DARK PULSE, RESPECTIVELY
We first consider the dark pulse generation in silicon microresonators with only Kerr effect. A dual-pulse can be excited with mode-coupling method as shown in Fig. 1, where the pump detuning is kept at fixed value of 0.065 after 30 ns. Fig. 1(a)-(b) show the temporal and spectral profiles of the generated dark pulse exhibiting a two-soliton-like property as in the anomalous dispersion regime. Such pulses can stably evolve in the cavity as can be seen from the temporal evolution [ Fig. 1(c)] and its energy evolution diagram [ Fig. 1(d)]. While dark pulses are largely affected by higher-order effects, including HOD, SRS and SS effects. Fig. 2 shows the spatiotemporal results with HOD [ Fig. 2 In detail, dark pulses have asymmetric bilateral oscillatory tails and weak top oscillations with HOD [ Fig. 2(a)]. The breaking of nonlinear symmetry in time domain is mainly due to the DW introduced by the HOD effect through the Cerenkov radiation (CR) process. Emission of the DW will induce the spectral recoil on the soliton core shifting the soliton carrier frequency away from the zero group-velocity dispersion (GVD) point. Theoretically, the drift direction and velocity of dark pulses mainly depends on the sign and amplitude of TOD and the detuning [62]. The phase-matching condition governing the  resonant DW vis CR is [31], where ω DW and ω S are the central angular frequencies of the DW and the dark pulses, respectively [31]. The location of DW is marked with red arrows in Fig. 2(b). The strong and narrow Raman gain in silicon microresonators exerts different influence on dark pulse. The energy balance is among drive, loss and the intra-pulse Raman scattering induced pulse energy transfer. The dark pulse profile is consequently distorted and reveals some new spectral components located at specific wavelengths around the pump. Crystalline materials (e.g., Silicon) with narrow Raman gain bandwidths typically exhibit negligible Raman self-frequency shift. SRS can interfere the FWM process and thus perturbing the temporal and spectral profiles distribution of dark pulse [ Fig. 2(d) and (

IV. THE INTERACTION BETWEEN HOD, RAMAN AND SS EFFECTS
Physically, these higher-order effects often exist simultaneously in microcavities, but their interaction have not yet been revealed. In order to study the interaction between SRS and  SS effects, we set HOD terms as 0 and the results shown in Fig. 3 are obtained. Obviously, the temporal profile also shows a severe top oscillation [ Fig. 3(a)] and the spectrum is broadened [ Fig. 3(b)] due to SRS effect, which means that SS effect has ignorable effect on the distribution of Raman comb components. In addition, the increased SS effect also intensifies the drift of pulse [compare Figs. 3(c) and 2(f)]. To illustrate the additional velocity of the dark pulse envelope in time domain in detail, the slope is described by k (fs/ns). Specifically, k is 0 for the dark pulses in Fig. 1(c) (only Kerr), while it is equal to −3.51, −5.81 and −12.99 in Fig. 2(c), (f) and (i) (only HOD, SRS or SS), respectively. The situation becomes different when interaction between higher-order effects is considered in the system. Fig. 4(a) [63]. Since dark pulses are formed by binding of two counter-propagating SWs, the bifurcation discussed here results from different relative  velocities of these fronts caused by higher-order effects, which frustrates robustness of dark pulses. Their spectra are given in Fig. 4(d). The Raman frequency comb components and the broadening effect of SRS effect (shown in Fig. 2(e)) will be suppressed by HOD effect as the spectral profiles without (purple curve in Fig. 4(d)) and with SRS effect (yellow curve in Fig. 4(d)) are roughly coincided in the bandwidth. Furthermore, SS effect can also broaden the spectral range of the frequency combs in the presence of HOD effect, since the spectrum with SS effect (red curve in Fig. 4(d)) is wider than that without SS effect (yellow curve in Fig. 4(d)) after 3800 nm (also seen by comparing green curve with purple curve). Meanwhile, SS effect changes the position of the DW, moving it towards the longer wavelength and thus increasing the energy of the spectrum.

V. THE BALANCE OF PULSE DRIFT WITH TOD
In this section, the possibility of balancing the pulse drift induced by TOD, SS and SRS effects is investigated. As discussed above, higher-order effects can individually cause temporal drift. Especially, the temporal drift caused by HOD mainly depends on the sign and amplitude of TOD values [41]. Here, TOD effect is considered to balance such delay introduced by SS or SRS effects in simulations. Fig. 5(a)-(d) show the temporal evolution with same SS while different TOD coefficient β 3 (ps 3 /km). Specifically, the pulses in Fig. 5(a) and (c) drift in opposite directions and they are split into two parts with a larger β 3 . Thus the pulse evolves without any temporal drift if a suitable β 3 is taken into account, as shown in Fig. 5(b). This means that TOD balances the temporal drift caused by SS effect of the dark pulse. In addition, dark pulse in Fig. 5(d) undergoes a turning point during evolution due to the influence of the added TOD.
Besides SS effect, the temporal drift of dark pulse resulted from SRS effect can also be balanced by TOD. An exact balance between the DW-induced spectral recoil and the Raman effect can cause the existence of a quiescent soliton [61]. To illustrate, a more detailed image about pulse drift velocity due to the interaction of TOD and SRS effects is shown in Fig. 6(a). Fig. 6(b)-(d) depict the temporal evolution at points i, ii and iii in Fig. 6(a), respectively. It can be observed in Fig. 6(a) that there is a parabolic-like relationship between k and β 3 . As the increase of β 3 , the drift direction and velocity of dark pulses varies since k first changes from negative to positive, and then starts to decrease after β 3 exceeds 0.462. In particular, there are two equilibrium points G 1 and G 2 (values of β 3 are 0.446 and 0.474, respectively), where the drift of the pulse will vanish (see Fig. 6(c)). This means that the direction and magnitude of the pulse drift caused by SRS effect can also be modulated by TOD effect, just like the pulse drift by SS effect. TOD effect has a regular influence on the pulse delay caused by SRS effect, and we found two β 3 values where the pulse drift is balanced out. Overall, these results indicate that different TOD will result in different drift directions, drift velocities and pulse shapes in time domain.

VI. CONCLUSION
In conclusion, dark pulse dynamics and Raman-Kerr microcombs generation in silicon microresonators affected by higherorder effects are investigated theoretically. In time domain, HOD effect can introduce changes of pulse drift direction or even bifurcation. Temporal drift caused by SS or SRS effects could be balanced by the design of TOD effect. In spectral domain, strong SRS effect competes with Kerr effect, resulting in stable Raman-Kerr frequency comb. Raman comb components and spectral broadening of SRS effect are suppressed by the HOD effect but are slightly influenced by the SS effect. The SS effect increases the total energy of the spectrum by shifting the DW generation to the longer wavelength side. These studies provide new ideas for stabilizing of dark pulse, and also facilitate the acquisition of stable broadband microcomb in the MIR. Our work could also constitute supplements to deeper the understanding of impact of higher-order effects and their interaction dynamics.