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Abstract
Pure quartic soliton (PQS) is a new class of solitons demonstrated in recent years and provides innovations in nonlinear optics and its applications. Generating PQSs in micro-cavities offers a novel way to achieve coherent microcombs, presenting a promising application potential. Here we numerically investigate the PQS generation in a dispersion-engineered aluminum nitride (AlN) micro-cavity. To support PQS, a well-designed shallow-trench waveguide structure is adopted, which is feasible to be fabricated. The structure exhibits a dominant fourth-order dispersion reaching up to -5.35×10−3 ps4/km. PQSs can be generated in this AlN micro-cavity in the presence of all-order-dispersion and stimulated Raman scattering. Spectral recoil and soliton self-frequency shift are observed in the PQS spectrum. Furthermore, we find that due to the narrow Raman gain spectrum of crystalline AlN, the PQS evolves directly to chaos rather than turning into a breather. The threshold pump power with which the PQS turns into chaos is also theoretically calculated, which squares with the simulation results.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Solitons are particle-like localized structures existing in various nonlinear systems and have been widely studied in decades [1]. In optical fibers and other waveguides with cubic nonlinearity, solitons usually result from single balance between group velocity dispersion (GVD) and self-phase modulation (SPM) [2]. Furthermore, the concept of dissipative optical solitons emerged since solitary waves were observed in non-conservative optical systems such as fiber lasers [3], continuous wave pumped passive fiber cavities [4] and micro-cavities [5]. Those solitons rely on the complex balances between GVD, SPM, dissipation and (parametric) gain. Particularly, dissipative solitons existing in micro-cavities, known as dissipative Kerr solitons (DKS), have attracted widespread interest over the past decade. They are temporal interpretation of coherent Kerr frequency combs, and have been widely used in applications such as arbitrary waveform generation [6], mid-infrared comb generation [7], optical communication [8], optical atomic clocks [9], dual-comb spectroscopy [10], laser ranging system (LiDAR) [11,12] and photonics microwave generation [13], due to their high compactness, broad bandwidth, high comb line power, complementary-metal-oxide-semiconductor (CMOS) compatibility and high flexibility in dispersion engineering.
Recently, the experimental realization of a new class of solitons, known as pure-quartic solitons (PQS) has opened up an avenue for the exploration of nonlinear optics [14,15]. In contrast to conventional solitons, PQSs arise from balance between negative fourth order dispersion (FOD) and SPM. Therefore, they only exist at wave band where FOD is negative, while GVD and third order dispersion (TOD) are negligible [14]. Aside from the physical origin, PQSs show distinct features from conventional solitons, e.g., a Gaussian temporal shape and the advantageous energy scaling [16], presenting a promising application potential. In fact, the concept of “quartic solitons” has been suggested in early 1990s from nonlinear Schrödinger equation (NLSE) [17]. Nonetheless, it was considered only as a mathematical curiosity, and has been largely ignored during the past two decades until recent years [18]. This is mainly attributed to the fact that soliton self-frequency shift (SSFS) induced by stimulated Raman scattering (SRS) in silica fiber continuously shifts the pulse spectrum to the red wavelength upon propagation, moving solitons away from the wave band where PQS is supported [19]. Another challenge lies in the difficulty of providing a proper dispersion profile for PQS generation.
Conversely, those obstacles can be overcome in micro-cavities. Firstly, the cavity dispersion can be flexibly engineered via tailoring the geometry of the waveguide [20,21]. Secondly, SSFS for coherent combs in micro-cavities has been proved stable and controllable both by simulations and experiments [22], making it possible for PQS to exist in micro-cavities even with the SRS effect. Furthermore, those distinct features of PQSs, such as the flat-top spectral envelope and the scaling relationship between energy U and pulse width τ0 (i.e., U∝τ0−3) make PQS microcombs more attractive for applications such as spectroscopy, optical communication, LiDAR, etc. The foregoing factors make micro-cavities an ideal platform for PQS investigation, as a micro-cavity with slot waveguide supporting PQS was demonstrated [23], related theoretical studies [24–26] and a preliminary experimental result [27] have been reported. However, studies for PQS in micro-cavities are still insufficient, e.g., whether PQSs are supported in micro-cavities with the effect of higher order dispersion and higher order nonlinearity, such as SRS, remains an open question.
In this paper, we numerically investigate the PQS dynamics in a dispersion-engineered micro-cavity. Aluminum nitride (AlN) on sapphire is adopted as the platform for the investigation, as it has attracted wide interest in microcombs generation due to its strong Kerr and Pockels nonlinearity as well as a wide transparency window [28–30]. Furthermore, the intrinsic electro-optic and piezoelectric effect of AlN make it possible to manipulate light field within an on-chip system, which is quite promising in future applications with PQS microcombs such as wavelength-division-multiplexing data communications [31]. To support PQSs in the AlN micro-cavity, we propose a shallow-trench waveguide structure which is feasible to be fabricated. Besides, a dominant negative FOD reaching up to -5.35×10−3 ps4/km is achieved in the proposed structure, whose absolute value is one order of magnitude larger than that of the previously designed structure [23]. The possibility of PQS generation in the designed micro-cavity is further numerically examined with Lugiato-Lefever equation (LLE) [4]. We found that PQSs survive in the cavity in the presence of all-order-dispersion (AOD) and SRS, exhibiting spectral recoil and SSFS, which is similar to conventional DKS. Moreover, we note that when the pump power increases (or the detuning descends), the PQS in the proposed cavity evolves directly to spatiotemporal chaos rather than turning to breather, which is quite different from conventional DKS dynamics. This absence of breather dynamics is attributed to the narrow Raman gain spectrum of AlN. We theoretically predict the threshold pump power when the PQS is destabilized and turns into chaos, which is in good agreement with the numerical simulations. Our work provides a feasible scheme to generate PQS in micro-cavity and enriches the soliton dynamics.
2. Design of the micro-cavity
To support PQS generation in AlN micro-cavity, the dispersion parameters around pump wavelength must satisfy: β2 ≈ 0, β3 ≈ 0, β4 < 0, where β2 is GVD, β3 is TOD and β4 is FOD. We design a specific AlN waveguide geometry to meet the dispersion requirement. The proposed structure, as depicted in Figs. 1(a) and 1(b), is composed of AlN-on-sapphire film. An extra trench is etched within a rectangular structure. The cladding is air. The dispersion profile of the waveguide is calculated using a finite element mode (FEM) solver, with material dispersion [32] and bending effect taken into account (the radius of the cavity is 200 μm here). The calculated GVD of the quasi-TM10 mode (see the optical mode distribution in Fig. 1(c)) curve is shown in Fig. 1(d), exhibiting three extrema. Thus, the dispersion condition for the possible existence of PQS could be obtained around the local maximum of the GVD, where TOD is negligible and FOD is negative. It is notable that dispersion curves with a similar shape can be find in slot waveguides [23,33] and bilayer waveguides [34]. However, those structures introduce additional challenges in fabrication, such as depositing and etching different materials with accurate control of dimension. In comparison, the proposed structure in our work can be fabricated by introducing an additional etching process, which is relatively easier to achieve [35]. Certainly, a waveguide with similar structure has been reported recently for supercontinuum generation [36], indicating the possibility of the structure to be applied in other nonlinear optics, e.g., PQS generation, despite the additional loss induced by the trench surfaces. Furthermore, there are more degrees of freedom to engineer the dispersion in the structure proposed here, which offer the possibility to achieve larger |β4|. A larger |β4| is able to compensate not only SPM, but also the effect of large normal GVD when deviations are introduced during fabrication [37] (also see details in Part 3). Therefore, PQSs are more likely to be supported when |β4| is larger.
Fig. 1. Structure and dispersion of the AlN micro-cavity for PQS generation. (a) Schematic of the AlN micro-cavity. (b) Cross-section view of the trenched AlN waveguide, with Al2O3 substrate and air cladding. (c) Optical power distribution near the zero-dispersion wavelength. (d) GVD profile of the designed structure. Inset in (d), zoomed in GVD around the local maximum. (e) Calculated GVD value at 1555.7 nm with different element size in the etched part.
We choose the following structural parameters: the thickness of the waveguide h = 935 nm, the trench depth hetch = 535 nm, the trench width W2 = 903 nm, the inner and outer part of the ring are W1 = 1009 nm, W3 = 759 nm, respectively. The calculated dispersion profile around 1555.7 nm for this particular design is detailed in the inset in Fig. 1(d), showing a small GVD maximum value. Similar to the slot waveguide, we found that the element size of the trench part of the waveguide in our FEM solver significantly affect the calculated dispersion value [33]. To guarantee the accuracy of the dispersion profile, we calculated the GVD value at 1555.7 nm using increasingly reduced element size of the FEM solver from 50 nm to 1 nm in the trench part (with a fixed element size of 5 nm in the AlN waveguide part), as is shown in Fig. 1(e). The results show good convergence, also indicating that an element size of 5 nm is good enough to produce an accurate dispersion value.
3. Dispersion tailoring and robustness
Here we look at how variations of the designed structure affect the dispersion profile, in order to understand the principle of tailoring the dispersion profile and to estimate the robustness to fabrication tolerances. We calculate the dispersion for structures where all the parameters from Fig. 1(a) are tuned independently by 1%. Dispersion for structures with different radiuses is also calculated. The GVD curves are shown in Fig. 2. It is shown that small increases (decreases) in the size of W2, W3 and h lead to combined effects of shifting the GVD curve to higher (lower) values and to longer (shorter) wavelengths, variations for W1 and hetch lead to effects to the contrary. In contrast, according to the calculation, FOD is not significantly changed by these variations (also can be indicated by the unchanged shape of different GVD curves). Besides, the radius of the cavity has a minor impact on the dispersion profile, as only a deviation of tens of microns will cause an obvious difference, see Fig. 2(f). Hence, when deviations of parameters are introduced during the fabrication process, e.g., the GVD curve is shifted to higher values and longer wavelengths, PQSs can be supported if pump wavelength is carefully selected at the wave band where GVD is weak enough for FOD to dominate. In addition, a series of micro-cavities can be fabricated, with a single parameter (e.g., W2) swept around the required value, in order to obtain a proper dispersion profile for PQS generation. Considering the well-developed fabrication technology [38], it is feasible to generate PQS in the proposed micro-cavity structure.
Fig. 2. GVD curves for the designed structure with 1% variations in (a) W1, (b) W2, (c) W3, (d) h, (e) hetch. (f), GVD curves for the designed structure with different radiuses.
4. PQSs in the AlN micro-cavity
To examine the possibility of PQS generation in the designed micro-cavity, we perform numerical simulations of the optical field propagation with LLE [4]:
To minimize the effect of TOD, the pump wavelength is chosen at 1555.7 nm, where TOD is negligible. The dispersion parameters used in the equation are calculated in Section 2. 2th to 5th order dispersion around pump wavelength is detailed in Figs. 3(a) and 3(b). Note that owing to the multi-parameters-engineering, we obtain a FOD of -5.35×10−3 ps4/km, about one order of magnitude larger than the value in the previous work [23,37]. Other parameters used in the simulation are as following: the cavity length is L = 1.257 mm, the free spectral range is 116.4 GHz, the nonlinear coefficient of the waveguide γ = 2πn2 / (λAeff) is 0.9 W-1 m-1, with Aeff calculated in the FEM solver, and the nonlinear refractive index of AlN n2 = 2.5×10−19 m2/W [39]. The waveguide loss αi is set to be 0.18 dB/cm, while the power coupling coefficient between the ring and straight waveguide κ = 5.2×10−2, indicating a micro-cavity with a loaded Q factor of 1×106 and a critical coupling condition.
Fig. 3. (a) GVD and TOD (b) FOD and fifth order dispersion (FiOD) around 1555.7 nm for the designed structure. The maximum value of GVD is 0.37 ps2/km, and TOD is close to 0 in this band. FOD around 1555.7 nm is -5.35×10−3 ps4/km. (c) Temporal waveforms and (d) optical spectra of the PQSs in the micro-cavity. Inset in (c), comparison of the oscillatory tails of the PQS when AOD (black solid line) and only up to FOD (red dashed line) are considered in the LLE. Inset in (d), overall spectrum of the PQS when AOD is considered.
Considering the rapid fluctuation of GVD profile caused by higher order dispersion (see Fig. 1(d)), all-order-dispersion (AOD) is included in equation, for the sake of accuracy. We first neglect the Raman term in the LLE. With the pump tuning method similar to Ref. [24], we obtain PQSs in the cavity. Figure 3(c) shows the temporal waveform of a single PQS supported in the cavity when pump power is 240 mW and the pump-cavity detuning is 0.037. Similar to conventional DKSs in optical Kerr cavities, the PQS is situated on continuous background with dips on both sides of the main peak. However, the dips here are mainly caused by exponentially decaying oscillatory tails, which is a distinctive feature of PQSs and has been reported previously [16]. The formation of the oscillatory tails can be theoretically understood by an analysis of LLE as a dynamical nonlinear system [25]. The solid black line in the inset of Fig. 3(c) shows the zoomed-in bottom of the PQS, showing a clear oscillatory behavior. Besides, an asymmetry in temporal profile is observed. This asymmetry is generally associated with the recoil in the spectrum [40], as is shown in Black solid line in Fig. 3(d). The spectral recoil is usually caused by the emission of dispersive waves, when phase-matching condition is satisfied by higher order dispersion [2]. To verify this, we perform another simulation on PQS generation with only up to FOD is considered. As TOD is close to 0 at pump wavelength, PQS generated in this case shows good symmetry in both temporal and spectral profiles, as shown in red dashed line in the inset of Fig. 3(c) and in Fig. 3(d). We further simulated the spectrum of the PQS when up to fifth order dispersion (FiOD) is taken into account, the result is shown in blue dash-dotted line in Fig. 3(d). When FiOD is considered, the spectrum of PQS shows similar recoil to the AOD condition. Thus, we believe the spectral recoil of the PQS in our simulation is dominated by FiOD. The inset in Fig. 3(d) shows the overall spectrum of the PQS when AOD is considered. The spectrum shows a flat-top feature [24], indicating that though spectral recoil is introduced by higher order dispersion, the spectrum is not critically affected.
We further investigated the impact of SRS on PQS generation by considering the Raman effect in the simulation. The SRS term is calculated in the frequency domain with a Lorentzian gain spectrum which is the same as in Ref. [32]: $\tilde{H}$(Ω) = ΩR2 / (ΩR2 - Ω2 - 2iΓRΩ) where $\tilde{H}$(Ω) is Raman response in frequency domain, ΩR / 2π = -18.3 THz, ΓR / 2π = 138 GHz denote the central and half-width-half-maximum angular frequency of the Raman gain spectrum, respectively [41]. Note that due to the narrow Raman gain spectrum of AlN and large radius of our cavity, single mode Raman lasing is first stimulated in simulation. To obtain PQSs, we tune the detuning after Kerr-Raman combs are generated via Raman assisted four wave mixing (RFWM) process [39]. Figure 4(a) shows the temporal waveform of the PQS when SRS is considered in simulation. The corresponding spectrum is shown in black solid line in Fig. 4(b). Compared to the spectrum of the PQS when only AOD is considered (see red dashed line in Fig. 4(b)), there is a redshift of the central frequency when the SRS term is added into the equation. This can be attributed to SSFS induced by SRS [22]. The overall spectrum of the PQS is depicted in the inset in Fig. 4(b), showing a flat-top feature despite the effect of SSFS, confirming the formation of the PQS. Besides, there are spikes around 1719 nm and 1421 nm in the spectrum of the PQS. This can be explained as follows: as the Raman gain spectrum is quite narrow for AlN (138 GHz), modes around Stokes mode (1719 nm) experience extra phase shift because of Raman-induced index changes [2]. Modes around anti-Stokes mode (1421 nm) also experience extra phase shift induced by four-wave-mixing. Those extra phase shifts cause the spikes, which are similar to the spikes induced by avoided mode crossing [42].
Fig. 4. PQS in the micro-cavity when AOD and Raman effect is considered. (a) Temporal waveform of the PQS. (b) Comparison of the optical spectrum of the PQS when only AOD (red dashed line) and AOD as well as Raman effect are considered. Inset in (b), overall spectrum of the PQS when AOD and Raman effect is considered.
The influence of the pump power on PQS dynamics is also investigated. It is well established that conventional DKSs in micro-cavities can be destabilized by a Hopf bifurcation, leading to time-oscillating pulse, i.e., breather, if pump power is boosted (or pump detuning decreases) [43]. This phenomenon is associated to the Fermi-Pasta-Ulam recurrence which is quite common in nonlinear systems [44]. For dissipative solitons in other nonlinear cavities, such as in fiber lasers, similar transition to period-N pulsations via bifurcation can occur when energy pumped into the fiber cavity is increased [45]. For single PQS generated in our designed cavity, it is interesting to know whether similar transition could happen when pump power increases.
A stable PQS is first generated in simulation with a pump power of 3 W and a detuning of 0.09. Then we increase the pump power gradually with the detuning fixed to 0.09, and the peak power is recorded after the operation mode of the intracavity field is stable at each pump power. The data is plotted in the bifurcation diagram in Fig. 5. We first neglect Raman effect, the result is plotted in Fig. 5(a). Clearly, as the pump power increases above 10.5 W, the PQS in the micro-cavity undergoes a bifurcation, that is, evolves to a breather soliton before turning into chaos, which is similar to a conventional DKS. However, when SRS is added in the simulation, the PQS evolves directly to chaos as the pump power is above 5.3 W, as shown in Fig. 5(b).
Fig. 5. Bifurcation diagram showing intracavity peak power with different pump power, when Raman effect is (a) neglected and (b) considered. The pump detuning is 0.09 in both cases. Each dot on the diagram represents the intracavity peak power at each roundtrip. For each pump power, 5000 data are recorded.
To understand the underlying principle of this absence of PQS breathers when SRS is included, we investigate the transition process of a single PQS to chaos when pump power is above 5.3 W. We seed the simulation with a PQS which exist stably with a pump power of 5.2 W. The pump power is set to be 6 W, and the pump detuning is fixed to 0.09. The representative temporal waveforms and corresponding spectra during the transition are depicted in Fig. 6. During the first 48.2 ns, the PQS propagates in the cavity without an obvious distortion in shape, see Fig. 6(a) I-III. However, as the intracavity continuous background (also referred as lower branch homogenous steady state (HSS) in this bistable system) is strong enough to provide sufficient Raman gain, a Stokes peak around 1719 nm rises, as shown in Fig. 6(b) I-III. Meanwhile, oscillating tails in time domain start to arise due to interference between the pump and the Stokes peak. When the Stokes peak increases further, anti-Stokes peaks and the second order Stokes peak emerge, and the PQS submerges because of the perturbation of the enhanced oscillating tails, see Figs. 6(a) IV and 6(b) IV. Finally, comb lines produced by RFWM fill in the spaces between the Raman peaks, and the intracavity field features a chaotic state, as shown in Figs. 6(a) V and 6(b) V. In short, the PQS turns to chaos as long as the Stokes peak is stimulated by HSS, owing to the narrow Raman gain of AlN. Notably, the generation of a Stokes peak, or the Raman laser does not necessarily cause the chaotic state [46]. Nonetheless, for the case when PQS is formed, the background field is located at the red-detuned regime to the cavity resonance. Then the growth of the background field induced by Raman peaks shifts the cavity resonance towards longer wavelength via Kerr nonlinearity, moving the background field to a nonstationary position. Consequently, the intracavity field is destabilized and become chaotic.
Fig. 6. Transition process of a single PQS to chaos when the pump power is 6 W and the detuning is 0.09. (a) Temporal waveforms and (b) corresponding optical spectra of different times during the transition.
The critical value of the pump power with which a Stokes peak is stimulated can be theoretically calculated as follows: The threshold power of the background field providing adequate Raman gain can be obtained as [2]:
where Aeff is the effective mode area of the cavity waveguide, gR(Ω) is the Raman gain at the resonance near to the Raman gain peak, which is calculated as:Fig. 7. Phase diagram in the (pump power, detuning) plane representing different pump parameters leading to various dynamics. The blue diamond and the red triangle represent stable PQS and chaos respectively. The dashed black line illustrates the theoretical prediction of the threshold pump power.
5. Conclusion
In conclusion, we have demonstrated numerically the possibility of PQS generation in a dispersion-engineered AlN micro-cavity. We designed a shallow-trench AlN micro-cavity structure with dominant negative FOD. PQSs are supported in the cavity with the impact of AOD and SRS. Spectral recoil and SSFS of the PQS are observed. We also found the absence of breathing state in PQS dynamics in our simulation, which is caused by the narrow Raman gain spectrum of AlN. The critical value of pump power with which a PQS evolves into chaos is theoretically predicted. Our results verify the possibility of generating PQSs in micro-cavities and reveal soliton dynamics with Raman gain.
Funding
National Key Research and Development Program of China (2017YFF0206104).
Acknowledgments
The authors thank Dr. Pan Wang for insightful discussions.
Disclosures
The authors declare no conflicts of interest.
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