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Abstract
We demonstrate an emergent solitonic pulse generation approach exploiting the externally introduced or intrinsic loss fluctuation effects. Single or multiple pulses are accessible via self-evolution of the system in the red, blue detuning regime or even on resonance with loss perturbation. The potential well caused by the loss profile not only traps the generated pulses, but also helps to suppress the drift regarding high-order dispersion. Breathing dynamics is also observed with high driving force, which can be transferred to stable state by backward tuning the pump detuning. We further investigate the intrinsic free carrier absorption, recognized as unfavored effect traditionally, could be an effective factor for pulse excitation through the time-variant loss fluctuation in normal dispersion microresonators. Pulse excitation dynamics associated with physical parameters are also discussed. These findings could establish a feasible path for stable localized structures and Kerr microcombs generation in potential platforms.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical solitons are intensity and phase double-balanced waveforms that ubiquitously exist in many nonlinear dynamical systems, usually classified as bright solitons and dark pulses formed in the anomalous and normal group velocity dispersion (GVD) regime, respectively. In particular, their observations in microresonators are relevant to mode-locked frequency combs, which leverages numerous applications in precision ranging [1–3] and imaging [4], astronomy [5,6], spectroscopy [7] and microwave generation [8]. Recently, research has been largely focused on the paradigm of the bright single soliton. Since its formation undergoes the high noise modulation instability state and requires intracavity thermal equilibrium, non-trivial technics have been developed [9–15]. As the counterpart of bright solitons, dark pulses feature higher conversion efficiency from the CW pump to the microcombs on the one hand [16], and relax the requirement of dispersion engineering in intrinsic normal GVD materials on the other [17].
The excitation of dark pulses, also referred as "platicons" [18], "switching waves" [19,20] or "domain walls" [21,22], remains a challenging topic to be resolved. As such, pump amplitude modulation [23], mode coupling [16,17,24], negative thermal effects [25], free carrier effects [26] or self-injection locking scheme [27–29] have been introduced. Consequently, dark pulses generated through these methods mainly exist in the red detuning region. Whether dark pulse can be triggered in the blue detuning region or on resonance is still unrevealed. Optical saturable absorption has been extensively exploited in mode-locked laser for bright solitons formation via loss modulation [30,31]. Microresonators embedded with saturable absorption are also developed for soliton generation [32,33]. A novel single-layer graphene assisted Fabry–Pérot fiber resonator has been proposed and the possibility of deterministic single-soliton generation without frequency tuning via a pulse pump is demonstrated [33]. Y. Chen et al. demonstrate dissipative soliton generation in optical nonlinear cavities in the presence of loss fluctuations. The effect of the optical saturable absorption, recognized as an effective approach to the transient loss fluctuation in the cavity, is also investigated in detail [34]. This artificially introduced loss could induce loss switching and symmetry breaking in a cavity, which constitutes a new way to excite dark pulses in the normal GVD regime. Concerning the possibility of using losses for optical frequency combs generation in the normal GVD, some other approaches have also been studied and demonstrated [35–37]. Coincidentally, many materials exhibit absorption of light in certain wavelength ranges due to their special energy band structure, including Si [38], Ge [39], GaAs [40], AlGaAs [41], SiC [42] and chalcogenide glasses [43] such as As$_{2}$S$_{3}$, As$_{2}$Se$_{3}$ [44] etc. This will introduce an additional time-varying, intensity-dependent free carrier absorption, inherently existing in resonators while is usually considered to be avoided. Reports on the utilization of this nonlinear loss remain to be revealed.
In this work, we investigate the excitation of dark pulse in resonators with normal GVD via different loss modulation effects. Emerging of single- or multi- dark pulses depends on the modulation period and depth, which circumvents the noisy state in the anomalous GVD regime. Turn key generation of dark pulse is feasible by self-evolution of the system when the pump is in the blue, red detuning region or on cavity resonance. The shift of pulse with regard to high-order dispersion (HOD) is restrained by the potential well caused by the loss profile. Breathing dynamics are also observed at the end of laser sweeping with strong pump. Intriguingly, we further employ the free carrier absorption effect that will initiate a time-variant loss along with the evolution of the intracavity energy from the homogeneous state. Such nonlinear loss, intrinsically existing in group IV photonic materials and traditionally considered to be detrimental to nonlinear processes, could be exploited for pulses generation. The number and position of dark pulses are related to carrier lifetime, scan speed of detuning and pump power. These findings may provide new insight in dark pulse initiation and turn-key microcomb generation in normal GVD media.
2. Theory
The system concerning dark pulse evolution with loss modulation can be described by the normalized Lugiato-Lefever equation (LLE) [34,45–47],
3. External loss modulation scheme
We first consider the case of external loss modulation while the multiphoton absorption effect is absent. Simulations have been performed on dark pulse excitation with pump laser sweeping scheme. Generally, pulse cannot form with only laser frequency sweeping in the absence of loss modulation. In contrast, platicon can smoothly emerge from a modulated background wave via loss fluctuation as shown in Fig. 1. Figure 1(a) depicts the temporal evolution of intracavity pulse. Unlike the anomalous dispersion regime, platicon directly evolve with the increasing of detuning without experiencing the chaotic states [34]. Loss fluctuations not only act as a necessary mechanism for pulse excitation, but also improve the performance of generation process. The pulse deterministically evolves from the peak of modulated pump (loss valley) due to spontaneous symmetry breaking, with a peak power much larger than the background wave (Fig. 1(b)). The generated spectrum exhibits two pronounced wings which is similar to the case reported previously [23,46].
Fig. 1. Platicon excitation via loss modulation. (a) Intracavity energy evolution versus detuning with driving force $F^{2}$ = 10, the detuning is linearly tuned from −0.001 to 8 with a step of 0.02. (b) Corresponding temporal profile (purple curve) marked with white dashed line in (a) and loss profile (red curve). (c) Spectral evolution. (d) Spectrum corresponds to the location marked with white dashed line in (c).
The modulation period of the transient loss profile determines the number of platicon generated per roundtrip. As the maximum power of the pulse is always allocated at the maximum (minimum) of the modulated pump (loss profile). Two, three, and four platicons are accessible via self-evolution of the system with a homogeneous CW driving force as depicted in Fig. 2. The number of platicon is closely related to loss modulation period $N$. Here, the modulation depth is fixed as $l_{0}$ = 0.5, $N$ = 2, 3, 4, and the driving intensity $F^{2}$ = 10. In spectral domain, the comb lines are also an integer multiple of the free spectral range. Platicons exhibit a higher pump to comb conversion efficiency [16] compared with bright solitons in anomalous dispersion regime. The evenly distributed platicons in cavity have relatively high power which is analogous to the perfect soliton crystal [51,52]. Although we mainly consider the linear losses to be modulated at integer multiples of the cavity free spectral range, the system has a good tolerance to the detuned modulation period.
Fig. 2. Multi-platicon state generation. (a)-(c) 2-, 3-, 4- platicons excitation and corresponding spectra evolution (lower panel) with the loss modulation period is $N$ = 2, 3, 4, and the modulation depth is fixed as $l_{0}$ = 0.5, and the driving intensity $F^{2}$ = 10.
Fundamentally, loss fluctuations force a symmetry breaking and trigger the formation of localized structures. Bringing an advantage that it is unnecessary to excite the pulse at the red detuning side. This indicates that different platicon state can be obtained at the zero-detuning or even blue detuning region. With the assistance of loss fluctuations, platicons can be deterministically accessed by self-evolution of the system, from a noise-like background and driven by a CW source. Figure 3 illustrates the turn-key generation of platicon with fixed detuning $\Delta$ = −1, 0 and 4. These results greatly broaden the detuning range for pulse excitation and simplify the required controlling scheme for comb generation. Similarly, the self-starting number of platicon is also adjustable via altering the loss modulation period $N$. Figure 4 shows the on-resonance platicon state in the condition of multiple loss modulation periods, i.e. $N$ = 2, 3, and 4.
Fig. 3. Self-starting single-platicon state at different detuning region. (a-c) Evolution of the intracavity field pattern over cavity round trips, where $\Delta$ = −1, 0, and 4, respectively. (d) The final stable pulse profile (blue curve) and the preformed background field (red curve). The modulation depth is set as $l_{0}$ = 2, 5, 50, respectively.
Fig. 4. Self-starting multi-platicon state. (a-c) The evolution of the intracavity field pattern (upper panel) and corresponding spectral evolution (lower panel), the loss modulation period is $N$ = 2, 3, 4, and the modulation depth is fixed as $l_{0}$ = 2, the detuning is always $\Delta$ = 4.
As discussed above, platicons tend to emerge at the top (bottom) of the modulated pump (loss profile). HOD [53,54] and Raman induced self-frequency shift [19,55] will cause the drift of temporal structure and influence the stability of the repetition rate. Whether the influence of HOD could separate the location for peak power of the pulse from the maximal modulated pump, thus affecting the generation dynamics, needs to be discussed. Figure 5(a) reveals the platicon evolution when considering HOD, the platicon does not evolve in the cavity in a drifting way as reported in [54]. It can be further observed in Fig. 5(b) that the platicon profile is asymmetric and slightly shifted by HOD. This phenomenon is understood as the generated pulse is trapped by the potential well caused by the artificial loss modulation. Providing a route to broadband frequency comb generation with HOD engineering.
Fig. 5. Platicon excitation with HOD. (a) Intracavity energy evolution versus detuning with $F^{2}$ = 30, the detuning is linearly tuned from −0.01 to 12.8 with a step of 0.03 and then kept at 12.8 for stable evolution. (b) Corresponding final temporal profile (purple curve) and loss profile (red curve). (c) Spectral evolution. (d) Final stable spectrum with dispersive wave and the corresponding dispersion curve.
Interestingly, the breathing platicon is also observed when driven with a larger pump power as depicted in Fig. 6. Such breathing dynamics emerges at a relatively larger detuning region and attributed to the bistable of the cavity solution. In detail, third order dispersion change the phase diagram and accelerates the process to Hopf instability threshold [56], thus transiting the pulses from stable to breathing states as the detuning increasing over a certain value ($\Delta$= 15.9 here, see energy traces in Fig. 6). Similarly, the number of breathing platicon is determined by the loss fluctuation period. The critical transition value of TOD depends on combination of detuning and pump power. The two states can be switched between each other only by backward or forward tuning the laser detuning.
Fig. 6. Different breathing platicon states. (a) Intracavity energy, temporal evolution and spectral evolution of single breathing platicon. (b) Intracavity energy, temporal evolution and spectral evolution of dual breathing platicon. The modulation depth $l_{0}$ = 1 and driving force $F^{2}$ = 45.
4. Internal loss fluctuation scheme
Apart from the artificially introduced loss, many materials show intrinsic nonlinear absorption of light due to their different bandgap structure. The group IV materials present the multi-photon absorption and the concomitant free carrier absorption (FCA) and free carrier dispersion (FCD) effects, including silicon [57], germanium [58] and their derivatives. Generally, these two nonlinear effects coexist in microcavities. FCD will induce an effective nonlinear detuning, which has been studied theoretically and experimentally in previous works [59,60]. It is ignored here to clarify the benefit of FCA. The strong nonlinear loss associated with two-photon absorption inhibits pulses generation, thus platicon can not be generated in this wavelength range. The three- and four-photon absorption are less detrimental and dominated in the wavelength range of 2.2-3.3 $\mu$m and beyond 3.3 $\mu$m in silicon, respectively. Here, we focus on three-photon absorption and find the appropriate free carrier absorption is beneficial for platicon generation. Three-photon absorption coefficient is constant during light circulating in the cavity, which can not assist platicon generation. Although FCA remains a constant within a single cycle (in the fast time frame), it fluctuates each round-trip and contributes to the symmetry breaking and triggering the formation of platicon. The normalized intensity-dependent loss factor -$\phi _{c} \psi$/2 is time-varying with the power change in the cavity, inducing the background amplitude modulation thus assisting pulse excitation. As the impact of free-carrier dispersion effect has been studied in previous works [59,60], it is ignored here to clarify the problem. Pulses can be generated in the sole presence of free carrier absorption via laser detuning sweeping with $A_{3}$= 0.0014, $\theta _{3}$=0.0472, $K$ = 0. Figure 7 illustrates the generated platicon with different FC lifetime and pump power. When the pump power is less than a certain value, the pulse can not form (number of platicon is zero, see Fig. 7(a)). The number of platicon is related to the speed of laser sweeping. Two or three platicons are observed when detuning step is 0.03. While a smaller detuning step tends to produce diverse evolution dynamics (compare Figs. 7(a)-(b)). One to four platicons can be observed with appropriate parameters. Furthermore, the intracavity platicon appears randomly and will disappear one by one as the increasing of detuning (see Fig. 7(c)). Nonlinear loss induced by free-carrier absorption is also changing with detuning. There is a jump in the loss curve for each reduced pulse (purple curve in Fig. 7(c)). Such internal loss fluctuation is quite different in contrast to the abovementioned scheme and is a potential method for platicon exaction in group IV materials.
Fig. 7. Platicons excitation via FCA effect. (a)-(b) Excitation number of platicon versus different FC lifetime and pump power with a detuning step of 0.03 and 0.01, respectively. (c) Representative results for 1-, 2-, 3-, 4- platicons. The corresponding intracavity FC loss (purple curve) of 4- platicons is also given.
5. Conclusion
In summary, we have proposed a promising route to platicon excitation by means of the loss fluctuation effects that can be introduced artificially or intrinsically exist in media. Single or multiple pulses are accessible via self-evolution of the system in the red, blue detuning regime or even on resonance. The temporal shift associated with high-order dispersion can be restrained by the potential well caused by the loss profile. Breathing dynamics is found to be alternate between stable state by forward or backward tuning the laser detuning. Furthermore, the internal loss of free carrier absorption, recognized as undesired influence generally, could be an effective factor of platicon excitation thanks to the time-variant loss fluctuation in normal GVD cavity. These findings could promote to find alternative path that can be implemented to realize such localized structures.
Funding
Natural Science Basic Research Program of Shaanxi Province (No.2020JQ-280, No.2022JQ-066, No.2022JQ-688); National Natural Science Foundation of China (NO.52002331).
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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