Dynamics of dark breathers and Raman-Kerr frequency combs influenced by high-order dispersion

Mulong Liu; Huimin Huang; Zhizhou Lu; Yuanyuan Wang; Yanan Cai; Wei Zhao

doi:10.1364/OE.427718

1. Introduction

Bright dissipative solitons (DS) in anomalous dispersion microresonators, due to their compact, stable and low-cost potential towards ultrashort pulse sources in time domain as well as frequency combs with low amplitude and phase noise in spectral domain, have been widely studied in recent years [1–3]. Bright DS microcombs are promising candidates for numerous applications including high precision spectroscopy [4–6], ranging [7–9], optical frequency synthesizer [10], coherent communications [11] and astronomy [12,13]. Generation of bright DS in platforms like Si₃N₄ [14–17], silica disks [18–21], MgF₂ [22], mainly in the anomalous dispersion region achieved by elaborate waveguide cross-section engineering. However, tailoring the geometry to obtain anomalous dispersion in arbitrary center wavelength is still challenging since most nonlinear materials possess normal dispersion, especially in the visible and near infrared wavelength range due to ultraviolet absorption [14]. As the counterpart of bright DS, dark pulses existing in normal dispersion regime are less understood from a fundamental point of view. Such localized dissipative structure, also called platicons [23], are naturally formed by interlocked switching waves (SW) or fronts connecting the high and low homogeneous steady-states of the bistable Kerr microresonators [24–26]. Dark pulses not only possess higher conversion efficiency compared with bright DS [27], providing an important tool for high-power comb line generation, but also increase the freedom in microresonators design and fabrication thus making it possible to acquire microcombs in an extended wavelength range.

To date dark pulses and frequency combs have been demonstrated in normal dispersion microresonators including MgF₂ [28], CaF₂ [29], and Si₃N₄ [30–32]. The switching mechanism of dark pulses ascribed to new resonance have been investigated [24]. Breathing dark pulses and their transition are also observed with strong pump power and small detuning [33]. The discovery of zero-dispersion dissipative soliton in the regime of nearly vanishing second-order dispersion presents a new perspective on SW [34]. Silicon exhibits large nonlinearity and strong Raman gain compared with abovementioned materials and is transparent over a large wavelength range (1.2-8 µm), as a promising candidate for broadband MIR microcomb generation via Cherenkov radiation (CR). It is also meaningful for molecular spectroscopy motivated by new approaches to absorption spectroscopy, as one of the major techniques of non-intrusive analysis of matter [35]. While detailed understanding of dark pulses affected by strong Raman gain and HOD in silicon microresonators has largely unexplored. Report about the dynamics of SW and their breathing properties is even scarce.

In this work, we predict novel dark breathers in the presence of HOD and Raman effects within normal dispersion silicon microresonators by a full-term Lugiato-Lefever equation (LLE). Theoretical and numerical analysis reveal that there exists minimum microresonator FSR for mode-locked dark pulses generation with suppressing Raman gain. Moreover, HOD induced dark breathers and their transitions in the relatively large red detuning region have been discovered and analyzed. Specifically, strong third-order dispersion (TOD) mainly modifies the route to Hopf instability threshold [36]. Such breathing structures with SRS can exist in a broader detuning region than that without SRS during laser sweeping. Dark breathers represent an intrinsic source of bi-stability for SW, characterization and modeling of such localized dissipative structure may offer better opportunities for their control, and trigger the search for dark breathers in other physical systems. Correspondingly, octave spanning microcombs with alterable line spacing is obtained via CR generation without experiencing noisy chaotic and multi-soliton states compared with their anomalous dispersion-based counterpart. These results constitute an important addition to the study of breathers as well as MIR Raman-Kerr microcomb generation and could have potential applications in molecular spectroscopy, chemical and biological sensing.

2. Theoretical model

In order to model the physical process in normal dispersion silicon microresonators, the modified LLE including HOD, three- and four-photon absorption (3PA and 4PA), free-carrier (FC) effects, self-steepening and SRS is used to describe the spectral-temporal dynamics of frequency combs [37,38]. The generalized equation can be written as,

(1)$$\begin{aligned} {T_R}\frac{{\partial E({t,\tau } )}}{{\partial t}} &= \left[ { - \frac{\alpha }{2} - \frac{\kappa }{2} - i{\delta_0} + iL\sum\limits_{n \ge 2} {\frac{{{\beta_n}}}{{n!}}{{\left( {i\frac{\partial }{{\partial \tau }}} \right)}^n}} } \right. + \left( {1 + \frac{i}{{{\omega_0}}}\frac{\partial }{{\partial \tau }}} \right) \times \left( {\int_\textrm{0}^\infty {R({\textrm{t}^{\prime}} )} {{|{E({t - \textrm{t}^{\prime},\tau } )} |}^2}d\textrm{t}^{\prime} - } \right.\\ &\textrm{ }\left. { - \frac{{{\beta_{3PA}}L}}{{3A_{eff}^2}}{{|{E({t,\tau } )} |}^4} - \left. {\frac{{{\beta_{4PA}}L}}{{4A_{eff}^3}}{{|{E({t,\tau } )} |}^6}} \right) - \frac{{\sigma L}}{2}({1 + i\mu } )\left\langle {{N_c}({t,\tau } )} \right\rangle } \right]E({t,\tau } )+ \sqrt \kappa {E_{in}}, \end{aligned}$$

(2)$$\frac{{d\left\langle {{N_c}(t )} \right\rangle }}{{dt}} = \frac{{{\beta _{3PA}}}}{{3\hbar \omega }}\frac{{\left\langle {{{|E |}^6}} \right\rangle }}{{A_{eff}^3}} + \frac{{{\beta _{4PA}}}}{{4\hbar \omega }}\frac{{\left\langle {{{|E |}^8}} \right\rangle }}{{A_{eff}^4}} - \frac{{\left\langle {{N_c}(t )} \right\rangle }}{{{\tau _{eff}}}},$$

(3)$${E_{in}} = \sqrt {{P_{in1}}} + \sqrt {{P_{in2}}} \exp ({ - i2\pi f\tau } ), $$

(4)$${H_R}(\Omega )= \frac{{\Omega _R^2}}{{\Omega _R^2 - {\Omega ^2} - 2i{\Gamma _R}\Omega }},$$

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, $\alpha ^{\prime} = ({\alpha + \kappa } )/2$. δ₀ 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, β_4PA are the three-, and four-photon absorption coefficients, σ the free-carrier absorption (FCA) cross-section and μ free-carrier dispersion (FCD) parameter. Considering the bandgap energy of silicon (i.e., Eg ∼ 1.1 eV), two photon absorption (2PA) is significant at the operating wavelength λ < 2.2 µm, which is ignored in calculations since the pump and output spectra are mainly located in the wavelength range beyond 2.2 µm in this work. 3PA and 4PA are dominated in the wavelength range of 2.2-3.3 µm and beyond 3.3 µm, respectively, which will introduce additional nonlinear loss and FC generation. FC induces FCA losses to the system and is associated with the FCD as free carriers will alter the refractive index of the medium. In suitable conditions such as low pulse energy or sweeping the FC with external bias that the effects of FCA and FCD can be minimized. The Self-steepening effect contributes to drift of temporal pulses and asymmetrical distribution of the spectrum. In Eq. (2), the averaged FC density $\left\langle {{N_c}(t )} \right\rangle = ({{1 / {{T_R}}}} )\int_{ - {T_R}/2}^{{T_R}/2} {N({t,\tau } )} d\tau $ describes the buildup of carriers within the cavity over successive round trips [37]. FC generation is governed by multi-photon absorption and the recombination rate determined by the effective FC lifetime τ_eff, which can be controlled by a positive intrinsic negative (PIN) diode [39,40]. E_in is the pump field with P_in₁ and P_in₂ denote the power of two pumps, f is the frequency spacing between the two pumps. Here, single pump combining with the pump mode eigenfrequency shifted or alternatively bichromatic pump scheme are adopted [23,41]. The Raman effect is calculated in the frequency domain with a Lorentzian gain spectrum in the form of Eq. (4), 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.

3. Influence of SRS on dark pulses

In order to generate comb in normal dispersion silicon microresonators, the easily accessible waveguide cross-section is 500×2450 nm, with a propagation loss of 0.7 dB/cm, loaded quality factor is 220,000 at a wavelength of 2700 nm. Up to seventh order dispersion is included in the simulations. These parameters are based on the present production technology and can be achieved in reality [42]. Other parameters are set as: P_in₁ = 130 mW and P_in₂ = 0 around 2700 nm, β_3PA= 2×10⁻²⁶ m³/w², β_4PA= 3×10⁻⁴² m⁵/w³ [43,44], n₂= 6×10⁻¹⁸ m²/w, κ = 0.0083, σ = 8.26×10⁻²¹ m², µ = 3.16 [45] and τ_eff = 5 ps [42].

The Raman effect is firstly considered and the Stokes and pump mode are assumed to be co-polarized. Single continuous wave pump combining with 130 MHz mode eigenfrequency shift of the pump is applied as in experimental [32,30] and theoretical works [23,41]. Simulations are started with a weak exponential shape perturbation. Figure 1 shows the spatiotemporal results for two different FSRs, 70.7 GHz [Figs. 1(a) and 1(b)] and 129 GHz [Figs. 1(c) and 1(d)]. For Figs. 1(c) and 1(d), the detuning δ₀ is linearly scanned from −0.001 to 0.1 with a step of 0.0001 and kept at 0.1 after reaching the stable state. Different comb states can be observed depending on the FSR. Pure Raman oscillation comb with the Raman frequency shift marked by green triangles is shown in Fig. 1(a). Such strong and narrow Raman gain can initiate platicons [46] and combs [47] in microresonators. Stable microcomb state is given in upper graph of Fig. 1(c), and its spectral evolution manifests a route of Raman comb (red arrows) to Kerr comb as detuning increases. The phase-matching condition governing the resonant dispersive wave (DW) vis CR is,

(5)$$\beta ({{\omega_{\textrm{DW}}}} )- {\beta _1}({{\omega_S}} ){\omega _{\textrm{DW}}} = \beta ({{\omega_S}} )- {\beta _1}({{\omega_S}} ){\omega _S},$$

where ω_DW and ω_S are the central angular frequencies of the DW and the dark pulses, respectively [48]. Figure 1(d) shows the temporal evolution of interlocked SW, where the drift is attributed to HOD and Raman effect. The pulse tail becomes indented and asymmetric with a pronounced one-sided oscillating tail due to HOD, keeping shape-invariant over slow time. However, the stable SW evolution is inhibited by strong Raman effect with a smaller FSR of 70.7 GHz. In detail, the Raman gain should be suppressed by exceeding the ratio of g_R/g_K, where g_R is the Raman gain and g_K is the Kerr parametric gain. Assuming the Raman gain profile with a Lorentizan g_R(ε) = g_R/(1 + 4ε²/Γ_R²) [49]. Theoretically, selecting a cavity radius so that Raman gain is located between two cavity resonances, the Stokes fields will be restrained by over g_R/g_K if $2{\varepsilon }/{\mathrm{\Gamma }_\textrm{R}} > \sqrt {{g_R}/{g_K}} $ [49] as scheme of Raman gain depicted in Fig. 1(e). In silicon microresonators, Kerr gain coefficient g_K = 4πn₂/λ = 2.8×10⁻¹¹ m/W with n₂ = 6×10⁻¹⁸ m²/W at wavelength of 2700 nm, Raman gain coefficient g_R = 7×10⁻¹¹ m/W. To achieve pure comb generation in normal dispersion silicon microresonators, ε/2π = 83 GHz according to the analysis above which indicates the FSR should be more than 166 GHz. This well explains the simulations for the FSR of 70.7 GHz. While with the FSR of 129 GHz, the Raman and Kerr oscillation can be observed for different detunings, resulting from the Raman assisted four wave mixing process which enables stable comb formation. Therefore, relatively larger FSR or smaller radius of silicon microresonators are demanded to inhibit strong Raman effect and achieve mode-locked dark pulses in the normal dispersion regime. Note that S. Yao et al. theoretically and numerically investigates the generation of dark pulses with Raman assisted four wave mixing (RFWM) process in AlN microresonator [46]. Raman-Kerr frequency combs and platicon are also studied in crystalline microresonators [41] while Raman-induced platicon excitation is not observed. Such difference mainly originate from different Raman parameters of various materials. As the Raman parameters of silicon are comparable with AlN (Γ_R/π = 138 GHz, Ω_R/2π = 18.3 THz), dark pulse generation is also possible with proper microresonator structure and pump conditions although in the absence of mode shift and without dual-pump.

Fig. 1. Spectral (a) and temporal (b) profile generated with the FSR of cavity is 70.7 GHz. The green triangles indicate the Raman frequency shift. (c) Evolution of spectral power generated with the FSR of cavity is 129 GHz. Upper graph shows the corresponding stable states. DW labels the resonant radiation roots. FWM peaks resulting from the interaction of pump and left wing are also labeled. The red arrows indicate the Raman frequency shift. (d) Corresponding temporal intensity evolution of (c). Inset in (d) shows the oscillatory SW field. (e) Schematic diagram for Raman gain in silicon microresonators.

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4. HOD induced breathing dark pulses

Apart from shape-invariant stationary DS and dark pulses in anomalous and normal dispersion regime, respectively, Kerr-bistable microresonators can also support breathing DS exhibiting a periodic oscillatory behavior. So far, optical breathers have been mainly explored in anomalous dispersion regime, often appearing before stable DS during pump-detuning sweeping [50–52]. Intriguingly, a different situation occurs in normal dispersion microresonators. Breathing dark pulses induced by HOD can be observed when the detuning is tuned over the stable region. To illustrate this phenomenon, the main LLE [Eqs. (1)–(2)] can be normalized [53] to Eqs. (6)–(8) with regardless of self-steeping while incorporating Raman effects here.

(6)$$\begin{aligned} \frac{{\partial \psi }}{{\partial \zeta }} &= - ({1 + i\Delta } )\psi - \frac{1}{2}({1 + \textrm{i}K} ){\phi _c}\psi - i\frac{{{d_2}}}{2}\frac{{{\partial ^2}\psi }}{{\partial {\eta ^2}}} + i\sum\limits_{n \ge 3} {\frac{{{d_n}}}{{n!}}\frac{{{\partial ^n}\psi }}{{\partial {\eta ^n}}}} + i{|\psi |^2}\psi \\ &\textrm{ } - \frac{{{A_3}}}{3}{|\psi |^4}\psi + i({1 - {f_R}} ){|\psi |^2}\psi + i{f_R}({\Re \otimes {{|\psi |}^2}} )\psi + F, \end{aligned}$$

(7)$$\frac{{\partial {\phi _\textrm{c}}}}{{\partial \zeta }} = {\theta _3}{|\psi |^6} - \frac{{{\phi _\textrm{c}}}}{{{\tau _c}}}, $$

(8)$$\Re (\eta )\textrm{ = }H(\eta )\cdot \textrm{a}{\textrm{e}^{ - b\eta }}\sin ({c\eta } ), $$

where $\psi = E/\sqrt {\alpha ^{\prime}/\gamma L} $, $\zeta = t\alpha ^{\prime}/{T_R}$, $\eta = \tau \sqrt {\alpha ^{\prime}/|{{\beta_2}} |L} $, $\mathrm{\Delta } = {\delta _0}/\alpha ^{\prime}$, $K = \mu $, ${d_2}$, ${d_n} = {i^{n + 1}}{(\frac{{\alpha ^{\prime}}}{L})^{\frac{n}{2} - 1}}\frac{{{\beta _n}}}{{{{|{{\beta_2}} |}^{n/2}}}}$, ${A_3} = \frac{{\alpha ^{\prime}{\beta _{3PA}}}}{{{\gamma ^2}LA_{\textrm{eff}}^2}}$, $F = {E_{in}}\sqrt {\kappa \gamma L/{{\alpha ^{\prime}}^3}} $, ${\phi _c} = {N_c}\sigma L/\alpha ^{\prime}$, ${\theta _3} = \frac{{\sigma {{\alpha ^{\prime}}^2}{\beta _{3PA}}}}{{3h\upsilon {\gamma ^3}{L^2}A_{\textrm{eff}}^3}}\sqrt {|{{\beta_2}} |L/\alpha ^{\prime}} $, ${\tau _c} = {\tau _{\textrm{eff}}}\sqrt {\alpha ^{\prime}/|{{\beta_2}} |L} $, are the normalized field amplitude, slow time, fast time, detuning, FCD coefficient, sign of second-order dispersion, HOD, three-photon absorption (3PA), pump amplitude, FC density, FCA coefficients, and carrier lifetime. Note that $\Re (\eta )$ is the normalized Raman kernel function and H is the Heaviside function with $a = {T_0}({\tau_1^2 + \tau_2^2} )/({{\tau_1}\tau_2^2} )$, $b = {T_0}/{\tau _2}$, $c = {T_0}/{\tau _1}$, ${T_0} = \sqrt {|{{\beta_2}} |L/\alpha } $ is the normalization factor. The relationships between ${\tau _{1,2}}$ and Raman gain parameters are τ₁= 1/$\sqrt {\mathrm{\Omega }_R^2 - \mathrm{\Gamma }_R^2} $, τ₂= 1/Γ_R. The Raman effect is calculated by the convolution theorem which states ${\Re} \otimes {|\psi |^2} = {{\cal F}^{ - 1}}({{\cal F}[{\Re } ]\cdot {\cal F}[{{{|\psi |}^2}} ]} )$.

The CW steady state solutions of Eq. (6) without SRS satisfy the following well-known cubic polynomial,

(9)$$X = ({1 + {\Delta ^2}} )Y - 2\Delta {Y^2} + \left( {1 + \frac{2}{3}{A_3}} \right){Y^3} + \frac{{A_3^2}}{9}{Y^5}, $$

where X = |F|² the normalized pump power, Y = |ψ|² the intracavity power. Figure 2(a) shows the bistable hysteresis curve of Y manifesting negligible influence with 3PA. Thus the influence of the weaker four-photon absorption can be neglected. The bistability versus normalized phase detuning is given in Fig. 2(b). The parameters are normalized according to the realistic values of silicon microresonators with A₃=0.0015, Δ=7.6. In the absence of HOD and SRS, the top branch ${\psi _t}$ (solid line) is always stable and the middle branch is always unstable (dashed line), while the bottom branch ${\psi _b}$ undergoes modulational instability. Generally, ${\psi _t}$ tends to move outward when X exceeds a certain value while move inward when X below the certain value, termed as Maxwell point [26]. SWs connecting the top and bottom branches may lock around Maxwell point [26] with certain pulse width. The locked dark pulse moves inward and finally annihilate as the detuning increases.

Fig. 2. (a) Intracavity power Y versus input pump power X with (blue curve) and without (red dashed curve) TPA. (b) Contour of the bistable intracavity solutions versus normalized phase detuning. (c) Simulated group velocity dispersion of the fundamental TE mode with different waveguide cross-section. (d) Evolution of intracavity energy via continuously tuning detuning with Raman (red curve) and without Raman (blue curve) effect. Δ is linearly tuned from −0.01 to 14 with a step of 0.002.

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Notably, the intracavity pulses exhibit a different route when HOD and SRS are taken into account in the normal dispersion regime (${d_2} = 1$). Figure 2(d) compares the intracavity energy evolution with HOD in the presence (red curve) and in the absence of (blue curve) Raman effect. Both cases clearly show the asymmetrical unfolding of the characteristic step feature. The energy trace with Raman effect presents a peak area around Δ=4.2 [see light red area in Fig. 2(d)], which is attributed to the competition between Raman and Kerr effects and is similar to the case in Figs. 1(c) and 1(d). The dark pulses also survive longer (annihilate around Δ = 13.9) than the one without Raman Effect (annihilate around Δ = 13.2) due to the Raman assisted four-wave mixing (FWM) process. Both cases will transit from stable to breathing states [see light blue area in Fig. 2(d)] as the detuning increasing over a certain value (Δ = 10.3 here). Such transition mainly due to large third-order dispersion (TOD, with d₃/d₂ = 0.345), which accelerates the process to Hopf instability threshold [36,54]. The critical transition value of TOD depends on combination of detuning and pump power. For instance, the threshold value of d₃/d₂ is 0.3 when Δ = 10.3 while d₃/d₂ is 0.29 when Δ = 11. The breathing depth also depends on the detuning as can be seen from Fig. 2(d). It can be as high as 33.9% with Δ = 10.8 but only 3.1% with Δ = 12.8. For the negative TOD (d₃/d₂ = −0.345), similar breathing dynamics can also be observed. However, dark breathers cannot be observed when d₃/d₂ = −0.28 (see Section 5 when d₃/d₂ = 0.28). Here, one can return to the stable dark pulse state only by backward tuning the laser detuning, suggesting that the two states can be switched between each other. Dark pulse dynamics in the presence of TOD was studied in several works [55–57], yet, dark breather was not observed or less understood. Breathing dark pulses form an important part of different classes of nonlinear wave systems, manifesting themselves as oscillatory localized temporal structure. Material with narrow Raman gain can also induce breathing dynamics [46]. Moreover, it is found that such breathing states can also be observed in conventional Lugiato-Lefever equation without FC effects and nonlinear absorption.

The detailed discussion about dark breathers is depicted in Fig. 3. Figure 3(a) shows the spectral evolution of breathing pulses obtained by Eq. (6) with the abovementioned parameters. In this calculation, Raman and self-steeping effects are neglected and it is found that the large TOD mainly determine the breathing properties of the spectrum. TOD breaks the time-reversible symmetry with a strong dispersive wave-like peak on the long wavelength side of the spectrum. A notable feature is the broader resonant radiation comb caused by dispersive wave as clearly seen from Fig. 3(e). This can be understood from oscillations of a breathing frequency in Fig. 3(b) and its Fourier spectrum separated by Hopf frequency Ω_r in Fig. 3(c). Each of the component serves as a source of Cherenkov radiation with the phase matching condition expressed as Eq. (10) [54,58]. The graphical solution of Eq. (10) is illustrated in Fig. 3(d).

(10)$$RR(\omega )= Z{\Omega _\textrm{r}} + \frac{{{d_5}}}{{120}}{\omega ^5} + \frac{{{d_3}}}{6}{\omega ^3} - V\omega \pm \sqrt {{{\left( {\frac{{{d_2}}}{2}{\omega^2} + \frac{{{d_4}}}{{24}}{\omega^4} + 2P - \Delta } \right)}^2} - {P^2}} ,$$

where Ω_r is the frequency of dark breather component and Z = 0, ±1, ±2,…V is the temporal drift velocity of dark pulses in a single roundtrip due to high-odd-order dispersion. P is the background power. Evidently, the roots for RR(ω) = 0 are the multifrequency location of the resonant radiation (RR) corresponding to the dark breathers. The theoretical result does not match exactly with simulations in Fig. 3(d), which still can deepen the understanding of the resonant radiation process.

Fig. 3. (a) Evolution of the spectrum with only HOD. (b) Oscillations of a breathing frequency and (c) its Fourier spectrum separated by Hopf frequency Ω_r. (d) Phase matching diagrams of resonant radiation condition. (e) Spectrum at ζ = 880 in (a) with FSR of 129 GHz.

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The stable dark pulses in normal dispersion cavities are preferable for the lack of noisy chaotic and multi-soliton states formation. A more explicit map of their transition is depicted in Fig. 4. One can obtain Figs. 4(a) and 4(b) by linearly adjusting detuning δ₀ from −0.001 to 0.09 and keeping it from 214 ns to 250 ns, then increasing detuning to 0.107 to reach the breathing state. The individual frequency and time domain field are shown in Figs. 4(c) and 4(d), corresponding to the locations marked with arrows in Fig. 4(b). The observation of CR in Fig. 4(c) provides a novel ingredient to realize broadband MIR frequency combs.

Fig. 4. (a) Evolution of intracavity energy. Transition from stable state to breathing state is marked. (b) Corresponding evolution of spectral power. (c) Spectrum and (d) temporal waveform for different locations (marked with arrows) in (b).

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5. Dark breathers in bichromatic pump regime

Alternatively, dark pulse Kerr comb can be generated by bichromatic pump scheme [37,59] to further confirm and investigate the SW dynamics with HOD. The slightly different waveguide cross-section used here is 500×2500 nm resulting in d₃/d₂ = 0.28. P_in₁ = P_in₂ = 130 mW around 2600 nm, f = 129 GHz, n₂= 7×10⁻¹⁸ m²/w and other parameters of material are the same as single pump regime. In order to quickly reach the breathing state, discrete detuning is applied in the first stage and the detuning is kept at certain value after reaching the desired results. Similar tuning method is used for Figs. 5–8. Figures 5(a)–5(c) show three representative spectra in an oscillation cycle of the breathers in Figs. 5(d)–5(f), respectively. The spectral envelope changes regularly with varying pulse width. This can be clearly seen from the temporal evolution in Fig. 5(g), where the pulse width periodically increases or decreases over slow time. Pulse drift in temporal evolution originated from HOD disappear due to dual-pump gradient [60]. Accompanying pulse energy also shows periodic changes as plotted in Fig. 5(h), where the energy peak before breathing state is caused by discretely increasing of laser detuning. Similar phenomenon was also observed with pump amplitude modulation [60]. These results reveal the transition to dark breathers is determined by HOD and not related to the excitation method.

Fig. 5. (a)-(c) Spectra and (d)-(f) temporal profile of dark breathers in one cycle. (g) Temporal evolution of dark breathers. (h) Intracavity energy evolution of (g).

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Fig. 6. (a) Intracavity energy evolution. Different colors correspond to different pump power. (b) Temporal intensity evolution. The stable and breathing states are also marked.

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Fig. 7. (a)-(c) Spectra with frequency separation of dual-pump is 2-, 3- and 4 FSR, respectively. (d)-(f) Corresponding temporal waveform evolution.

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Fig. 8. (a) Temporal evolution of breathing multi-dark pulses. (b) Evolution of intracavity energy. (c) Representative temporal profile and (d) corresponding spectrum.

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To further elucidate dark breathing cycle affected by different driving power, Fig. 6(a) shows intracavity energy evolution with varying pump power, divided into two parts marked as stable and breathing. In the stable region, dark pulses manifest stable evolution and will not show any profile change just resembles the solitons in anomalous dispersion case. The breathing region can be reached after the stable region, that is, continuously increasing detuning when intracavity pulses reach the steady state, pulses will enter the next breathing state. The HOD induced dark breathers exhibit a smaller breathing frequency (49.8 MHz, defined as the reciprocal of ${\tau _{\textrm{bre}}}$ with P_in= 140 mW, δ = 0.128) than that of the Kerr effect and second-order dispersion dominated dark soliton breathers [33] and bright soliton breathers [50]. Secondly, the HOD induced breathers require a larger red detuning while the latter two breathers exist in a regime with relatively smaller detuning. The bistable phase diagram are highly related to HOD and SRS effects, which change the process to Hopf instability threshold. Figure 6(b) depicts one of the temporal evolutions in Fig. 6(a). Both the stable and breathing states possess stability. The final energy and breathing cycle of SW are distinctive under different pump power and detuning, indicating that the breathing period is dependent on the pump-cavity conditions.

Naturally, we alter the frequency separation of dual-pump to achieve MIR microcombs with different comb line spacing. Figures 7(a)-(c) show spectra with frequency separation of 2-, 3- and 4- FSR, respectively. These spectra can be acquired by only changing pump frequency separation under the same detuning condition. Figures 7(d)–7(f) show corresponding temporal profile, where the number of dark pulse is equal to the mode spacing of the two pumps. Such multi-dark pulses are analogous to soliton molecule in the anomalous dispersion case. Assuming f to be equivalent to N FSR, then N dark pulses can equidistantly exist per round trip in the temporal profile. Correspondingly, a spectrum with mode spacing equivalent to N FSR will form. This proposed approach for achieving comb with varying line spacing could find potential applications in optical metrology and MIR spectroscopy.

The frequency separation f of two pumps is chosen as 3-FSR to acquire the breathing state by applying a large red detuning. Temporal evolution of breathing multi-dark pulses and intracavity energy evolution are shown in Figs. 8(a) and 8(b), respectively. The pulse indicates a periodic oscillation feature which is similar to single dark breathers in Figs. 5(g) and 6(b). The temporal and spectral profiles shown in Figs. 8(c) and 8(d) possess top oscillation and DW, respectively, also attributed to HOD as discussed above. Our investigation reveals that microresonators with normal dispersion can also be an ideal test bed for fundamental theories of nonlinear wave dynamics.

6. Conclusion

In conclusion, the influences of HOD and SRS on breathing and stable dark pulses as well as their transition are theoretically studied in normal dispersion silicon microresonators. The reported dark pulses can only exist in microresonators with relatively large FSR in comparison with Raman gain linewidth. Breathing dark pulses forming through HOD-perturbed interlocked SW, have been obtained in a relatively large red detuning region. Their breathing characteristics mainly depend on the value of TOD coefficient and Raman effect. Knowledge of such an entity could deepen the understanding of instabilities related to Raman-Kerr combs formation in normal dispersion microresonators and avoid these instabilities in practical applications. Correspondingly, the generated broadband MIR microcombs via DW increase the freedom in microresonators fabrication and also make it possible to acquire frequency combs in platforms where normal dispersion is dominant. These results provide a novel ingredient for broadband MIR comb generation and could have potential applications in molecular spectroscopy, chemical and biological sensing.

Funding

Northwest A and F University (Z1090121016, Z1090220308); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB24030600); National Natural Science Foundation of China (52002331, 61675231).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at the time but may be obtained from the author upon reasonable request.

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Dynamics of dark breathers and Raman-Kerr frequency combs influenced by high-order dispersion

Abstract

1. Introduction

2. Theoretical model

3. Influence of SRS on dark pulses

4. HOD induced breathing dark pulses

5. Dark breathers in bichromatic pump regime

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (10)

Optics Express