Open Access
Add to BibSonomy
Abstract
We numerically investigate the excitation of vector solitonic pulse with orthogonally polarized components via free-carrier effects in microresonators with normal group velocity dispersion (GVD). The dynamics of single, dual and oscillated vector pulses are unveiled under turn-key excitation with a single frequency-fixed CW laser source. Parameter spaces associated with detuning, polarization angle, interval between the pumped orthogonal resonances and pump amplitude have been revealed. Different vector pulse states can also be observed exploiting the traditional pump scanning scheme. Simultaneous and independent excitation regimes are identified due to varying interval of the orthogonal pump modes. The nonlinear coupling between two modes contributes to the distortion of the vector pulses’ profile. The free-carrier effects and the pump polarization angle provide additional degrees of freedom for efficiently controlling the properties of the vector solitonic microcombs. Moreover, the crucial thermal dynamics in microcavities is discussed and weak thermal effects are found to be favorable for delayed vector pulse formation. These findings reveal complex excitation mechanism of solitonic structures and could provide novel routes for microcomb generation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The recent discovery of different types of solitonic pulses in optical microresonators, including bright solitons and dark pulses in the anomalous [1–6] and the normal [7–12] group velocity dispersion (GVD) regime, respectively, has facilitated the development of fully coherent, chip scale microcombs. Such phase-locked and low-noise multiwavelength laser sources have promoted plenty of applications in astronomy [13,14], spectroscopy [15–19], metrology [20–22], coherent communications [23–25] and optical clock [26–28]. Successes in molecular spectroscopy with unprecedented precision via dual-comb systems have triggered intensive explorations for obtaining microresonator-based two isolated soliton states. To date, a pair of mode-locked solitons generation with dual silicon microresonators [16] or two counter-propagating solitons in a single microcavity [29] have been demonstrated in the anomalous GVD region. The report about bright-dark soliton bound states further broadens the dispersion range for complex soliton state generation [30]. Realizing complex solitonic structures by nonlinear coupling between two orthogonally polarized modes has also been experimentally discovered [31,32] and theoretically studied [33–36]. As counterparts of bright solitons, dark pulses exhibit higher efficiency than bright solitons in terms of the pump-to-comb conversion efficiency [9,37]. The generation of multicomponent dark pulses seems promising for many actual applications, such as coherent optical communications and dual-comb spectroscopy. However, solitonic pulses generation is conventionally inaccessible in normal dispersion resonators due to the lack of modulational instability (MI) on the upper branch of the bistable system. It is proved that MI can occur in the normal dispersion regime with additional techniques such as mode interaction [8,38,39] and self-injection locking [7,40–43]. Intriguingly, generation of dark pulses is possible via interplay of Kerr and thermal nonlinearities [44]. The modulated pump scheme is also proposed for dual flat-top dark pulses generation in Ref. [36], which can be hard to implement in the mid-infrared (MIR) for the lack of high-speed modulators.
Most molecules are responsible for absorption at specific wavelengths in the MIR, facilitating significant applications such as spectroscopy and biochemical-sensing [16,45–47]. Broadband light sources in the MIR thus constitutes promising basis for their practical implementation. The transparency window of Si and Ge ranges from 2.2 to 8.5 $\mu$m [48] and 1.5 to 14.3 $\mu$m [45], respectively. SiGe alloy expands more deeply into the MIR (potentially up to 15 $\mu$m), covering the functional group (2.5 $\sim$ 7.7 $\mu$m) and molecule "fingerprint" regions (7.7 $\sim$ 16.7 $\mu$m) [49]. A recent report about supercontinuum generation from 3 to 13 $\mu$m further accelerates its utilization [47]. On one hand, these group IV materials intrinsically present multi-photon absorption and the accompanying free-carrier effects, e.g. free-carrier absorption (FCA) and free-carrier dispersion (FCD), regarding as detrimental generally. On the other hand, the FCD effect induces a blue-shifted cavity resonance since the refractive index of silicon has a weak dependence on the electron–hole pair concentration [50], which has been successfully utilized for bright soliton generation [51]. Such nonlinear effects could also introduce symmetry-breaking and pulse excitation in the normal GVD regime, yet, they have not been thoroughly investigated.
In this work, we theoretically demonstrate the generation of vector solitonic pulses with orthogonally polarized components via free-carrier effects in normal GVD microresonators. Turn-key excitation of dual, single and oscillating vector pulses is feasible with a red-detuned CW pump. Generation conditions related to detuning, interval between pumped modes and polarization angle are uncovered. Different vector pulse states can also be observed using the traditional pump scanning scheme. Simultaneous and independent excitation regimes are identified due to varying deviation of the two pumped orthogonal resonances. The nonlinear coupling between two modes leads to the distortion of the wider vector pulses. The pulse width, number and dynamics of the vector solitonic structures can be controlled by the pump polarization angle and the FC life time. Furthermore, the weak thermal-optical effect in microresonators will trigger delayed spontaneous solitonic pulse generation while a strong one could suppress microcomb formation due to the blueshift of resonance associated with intracavity power drop. Our work will deepen the understanding of FC related complex nonlinear process and promote performance of the vector solitonic microcomb sources.
2. Theory
Here, numerical simulations are performed based on the coupled, generalized Lugiato-Lefever equations (GLLEs) incorporating cross-phase modulation (XPM) of the orthogonally polarized fields, three-photon absorption (3PA), FCA, FCD, and thermal effects [52,53]. The intracavity field does not vary significantly between successive round-trips, thus the mean-field model can be used and also explains experimental results well [30,31]. HOD is not the motivation for pulse excitation and thus neglected here for simplicity.
Table 1 lists the normalized variables and their corresponding original symbol used in Eqns. (1)–(4). Some other parameters: $L$, $\delta _{10}$, $\alpha$ and $T_{R}$ are the total cavity length, external pump detuning, linear loss coefficient and single round-trip time, respectively. $\alpha =\omega _{10}T_{R}/(2Q_{L})$ with $Q_{L}$ being the loaded quality factor. $T_{R}=n_{g} L/c$, where $n_{g}$ is the group refractive index and $c$ is the speed of light in vacuum. $\kappa _{1,2}$ are the power transmission coefficients for the first (Hereinafter also referred to TE mode) and second (also termed TM mode) pumped modes, and we assume $\kappa _{1}$/$\kappa _{2}$ = 1. $\delta =(\omega _{10}-\omega _{20})T_{R}/\alpha$ is the normalized spectral interval between two orthogonally polarized modes, where $\omega _{10,20}$ are the frequencies of two modes. $\epsilon$ is the normalized group velocity mismatch between two pumped modes. $v_g^{TE}$ and $v_g^{TM}$ are the group velocity of two polarization modes. $\theta$ is the polarization angle of the pump source defined relatively to the polarization directions of the pumped modes ($\theta$ = 0 corresponds to the driving of the first mode only, while $\theta$ = $\pi$/2 refers to the second mode pumping). The effective pump power of the first and second modes can be controlled by varying $\theta$. The coefficient 2/3 in the XPM term due to the orthogonal polarization of the interacting fields [54]. The linear coupling effects and the corresponding modifications of the dispersion terms are ignored. In Eq. (3), the averaged power $\left \langle |\psi |^6\right \rangle =\left (1 / T_R^{'}\right ) \int _{-T_R^{'} / 2}^{T_R^{'} / 2}|\psi |^6 d \eta$ where $T_{R}^{'}$=$T_{R}$ $\sqrt {(\alpha /|\beta _{2}|L)}$ is the normalized round-trip time. After normalization, dispersion parameters $d_{21, 22}$ = $\pm$1, which represents the normal and anomalous dispersion regime, respectively. The FC lifetime $\tau _{c}$ is within the experimental capability depending on the value of applicable reverse bias voltages [55]. Here, we consider a fast time range of 143.5, which corresponds to a cavity radius of 100 $\mu$m. The $\tau _{c}$ varies from 0.56 to 5.66 corresponding to FC lifetime between 500 ps and 5 ns. Thermal dynamics is considered in Eq. (4). $\delta _{1 T, 2 T}=T_R\left (\omega _{1 T, 2 T}-\omega _{10,20}\right )$ are the thermal detuning and $\omega _{1T,2T}$ are the cavity resonances shifted by thermal effects of the two fields. We assume $\delta _{1T,2T}$=$\delta _{T}$ for simplicity. $\overline {P}$ is the average total intracavity power, and $\kappa$ is the coefficient representing the shift of the detuning in response to $\overline {P}$, which depends on the thermo-optical coefficient of the material and the absorbed intracavity power converted to heat. $\tau _{T}$ is the normalized thermal response time determined by the material and the practical thermal dissipation. These coupled nonlinear partial differential equations can be numerically solved by the Runge-Kutta algorithm.
To illustrate the mechanism of vector pulse excitation, thermal effects are ignored in Eqn. (1) which then can be modified as,
Fig. 1. Schematic process for vector pulse generation. (a) A laser output can be changed by a PC. (b) The hybrid CW pump is injected into a microresonator with positive intrinsic negative (PIN) structure. (c) The generated vector pulses.
3. Turn-key excitation
Reliable self-starting oscillation of soliton microcombs that are naturally robust to perturbations makes them attractive for optical memories [58]. The turn-key generation approach, which can get rid of the complex controlling technics designed for thermodynamic equilibrium [2], is first considered here. In this part, thermal dynamic is neglected and will be discussed separately. Figures 2–4 show the calculated results with a single CW laser, of which the wavelength is kept at a certain value for each case. Vector pulses form from a noise-like initial condition by self-evolution of the system. In our case, an exponential shape noise in the form of $\psi _0(0, \eta )=a+b * \exp \left (-(\eta / c)^2\right )$ is used as the initial input for both fields, where $a$ = 0.01, $b$ = 0.01, $c$ = 0.3. Vector pulse can be generated deterministically with varied noise seed. It should be noted that the FCD and FCA effects could cause pulse generation individually or together (please see Figs. 2(a)–2(c), 2(g)–2(i) and 2(j)–2(l)), while pulse can not be generated in the absence of both effects (see Figs. 2(d)–2(f)).
Fig. 2. Single vector pulse excitation. Vector pulse excitation in the presence (a)-(c) and absence (d)-(f) of FC, with only FCD (g)-(i) and only FCA (j)-(l) effects. Columns from left to right: temporal evolution, intracavity energy, final temporal profile and spectra of TE (blue curve) and TM (red curve) modes with $\Delta _{1}$=7, $\delta$=2. In all cases $\epsilon$ = 0, $\theta$ = 0.25$\pi$, $F$ = 6.
Fig. 3. Dual vector pulse excitation. (a)-(b) Temporal evolution and final profile of TE (left column) and TM (right column) modes with $\Delta _{1}$=6, $\delta$=1. (c)-(d) Temporal evolution and final profile of TE (left) and TM (right) modes with $\Delta _{1}$=6, $\delta$=4. In both cases $\theta$ = 0.25$\pi$, $\epsilon$ = 0, $\tau _{c}$=1.2, $F$ = 6.
Fig. 4. Oscillation vector pulse excitation. (a)-(d) Intracavity energy, temporal evolution and representative temporal profile of TE (blue curve) and TM (red curve) modes with $\Delta _{1}$=6.5, $\delta$=3. (e)-(h) Intracavity energy, temporal evolution and temporal profile of TE (blue curve) and TM (red curve) modes with $\Delta _{1}$=7.3, $\delta$=4. In both cases $\theta$ = 0.25$\pi$, $\epsilon$ = 0, $\tau _{c}$=1.2, $F$ = 6.
There are mainly three possible cases under different combinations of interval between two orthogonal resonances and polarization angle of the pump: single (Fig. 2), dual (Fig. 3) and oscillation (Fig. 4) vector pulses dynamics. For the former two cases, one (Fig. 2) or two (Fig. 3) flat-top components of vector soliton simultaneously emerge directly (see Figs. 2(a)–2(c) and Figs. 3(a)–3(b)) or after an oscillatory transient process (see Figs. 3(c)–3(d)), then they keep a stable evolution manner. The main parameter for states transition is the linear detuning with a given polarization angle and a chosen interval between two orthogonal resonances. The transition between different states is not continuous. For instance, vector pulse with two peaks can be generated when $\Delta _{1}\in [5,5.2]$ and $\Delta _{1}\in [5.8,6.3]$, while one peak is observed when $\Delta _{1}\in [5.3,5.7]$ and $\Delta _{1}\in [6.4,6.5]$. Different polarization angle of the pump and interval between two orthogonal resonances will change the abovementioned detuning range. Vector solitonic pulse can be interpreted as a pair of stationary and stably interlocked switching waves, also called domain walls or fronts, connecting the upper and lower homogeneous steady-state solutions of the bi-stability curve in Kerr microresonators [31,59]. The two-solitonic pulses are due to the bifurcation from the vicinity of saddle node in bifurcation diagram and undergo collapsed defect-mediated snaking [60]. The number of generated vector pulses is determined by the combination of detuning, interval of two orthogonal resonances and FC lifetime. One can obviously see the distortion of the temporal profile of vector pulses, arising from nonlinear coupling between two modes (see Figs. 2(b) and 2(h)). Such distortion usually appears at the high power part of the wider dark pulse component. Only the temporal profile of the TM field is distorted because there is no mismatch between the two components ($\epsilon$ = 0). It is hard to observe distortion on the top of the TE field. Pulse width and repetition rate are also partly determined by detuning and deviation of two orthogonal resonances.
For the third case (see Fig. 4), one can observe anharmonic oscillations of two modes from their energy traces in Figs. 4(a) and 4(e). These oscillations of the intracavity power are regular and continuous as they evolve. Their oscillation cycle and temporal profiles (Figs. 4(d) and 4(h)) are also related to simulation parameters. Oscillating structures result from the Hopf bifurcation in bifurcation diagram which leads to stable temporal breathing pulses resembling bright breathers in the anomalous dispersion regime [60,61].
The detailed parameter spaces for the generation of vector pulse are illustrated in Fig. 5. It was revealed that vector pulse can be directly generated (blue region), enter oscillating (green region) or C.W. states (pink region) when turn-key detuning changes. In Figs. 5(a)–5(b), two vector pulse generation domains are separated by an oscillation domain (green region). The height of blue region almost linearly changes with the growth of the interval between pumped modes $\delta$ (see Fig. 5(a)). The upper and lower boundary values increase monotonically as the deviation increases, exhibiting broad range for vector pulse excitation. As $\delta$ increases, new generation domains separated by C.W. region emerge at the lower detuning region. The normalized peak power of TM field is hundreds of times smaller than that of the TE field when $\delta$ exceeds 15. In this case, it can be considered that the vector pulse is not generated, where the weak power originates from the XPM effect. Figures 5(c)–5(d) show that pulse excitation range changes with the increase of polarization angle $\theta$ using two different $\delta$. The transformation of the pulse generated in the first mode ($\theta$ = 0, dashed lines) to vector pulse (blue region), then to the pulse in the second mode ($\theta$ = 0.5$\pi$, dashed lines) is observed with the increase of polarization angle. The region of the detuning for the vector pulse generation changes nonlinearly with the growth of the pump polarization angle (blue region). Similarly, two vector pulse generation domains are also separated by an oscillation domain (green region). Figures 5(e)–5(f) also depict the pulse excitation range with varying polarization angle $\theta$ using two different $F$. A stronger pump tends to result in a larger detuning region for vector and oscillating pulse excitation. At the lower branch of Fig. 5(f), vector pulse can even be generated in the blue detuning side. These results using the turn-key approach pave the way for the study of complex solitonic structure and microcomb source generation, avoiding additional mode coupling and much faster than thermal tuning method [62].
Fig. 5. Parameter spaces for vector pulse excitation. Stable vector pulse (blue region), oscillation (green region) and C.W. (pink region) domains versus different deviation of two orthogonal resonances (a)-(b) and polarization angle (c)-(f). (a) $\theta$ = 0.25$\pi$, (b) $\theta$ = 0.2$\pi$. Other parameters: $\epsilon$ = 0, $\tau _{c}$=1.2, $F$ = 6. (c) $\delta$=2, (d) $\delta$=5. Other parameters: $\epsilon$ = 0, $\tau _{c}$=1.2, $F$ = 6. (e) $F$ = 4, (f) $F$ = 8. Other parameters: $\delta$=0, $\epsilon$ = 0, $\tau _{c}$=1.2. The dashed lines in (c)-(f) mark the scalar (one-component) pulse generation domains.
4. Pump frequency scanning
Alternatively, the traditional pump frequency scanning method can still be used to generate vector pulses. Several possible regimes have been revealed under different deviation between pumped modes. Figures 6(a)-(c) show evolution of the intracavity energy for both polarization modes with $\delta$=1, 3 and 6. Simultaneous excitation and decay of nonlinearly coupled solitonic pulses are revealed at both polarizations (see Fig. 6(a)). Independent excitation of vector solitonic pulses, that is, two pairs of pulses emerge one after another during laser sweeping (see Fig. 6(c)). Figures 6(d)–6(f) give the linear detuning, temporal evolution, effective detuning and nonlinear loss corresponding to 6(b), respectively. Although the laser frequency is linearly tunned in the whole process, the FCD induced cavity detuning is changed over roundtrips.
Fig. 6. Vector pulse excitation using pump frequency scanning method. (a)-(c) Evolution of the intracavity energy for both polarization modes with different deviation between pumped modes. $\theta$ = 0.25$\pi$, $\epsilon$ = 0, $\tau _{c}$=0.7, $F$ = 6. (d) linear detuning, (e) temporal evolution and (f) effective detuning and loss corresponding to (b).
The pump polarization angle $\theta$ and FC lifetime $\tau _{c}$ provide additional degrees of freedom for adjusting the properties of generated vector pulses. Figure 7 shows vector pulse with different polarization angle. One can observe the pulse width, peak power and nonlinear coupling depth vary with the increasing of $\theta$. The nonlinear coupling is more pronounced at high power parts of the dark pulses. Similar phenomena can be accessed in Fig. 8 where a shorter $\tau _{c}$ is used. Interestingly, the number of vector pulse is associated with $\theta$ and $\tau _{c}$. In detail, FC not only assists vector pulse generation, but also imposes influences on the dynamics of pulse evolution. A longer $\tau _{c}$ induces a larger nonlinear detuning thus a narrower pulse width. Different polarization angle means different power ratio of two pump modes and a larger $\theta$ tends to result in larger pulse width.
Fig. 7. Vector pulse with different polarization angle $\theta$ when $\delta$ = 4 and $\tau _{c}$ = 0.7. (a)-(d) Temporal profile of TE (blue curve) and TM (red curve) modes with $\theta$ = 0.15$\pi$, 0.2$\pi$, 0.26$\pi$ and 0.32$\pi$, respectively.
Fig. 8. Vector pulse with different polarization angle $\theta$ when $\delta$ = 3 and $\tau _{c}$ = 0.4. (a)-(d) Temporal profile of TE (blue curve) and TM (red curve) modes with $\theta$ = 0.15$\pi$, 0.2$\pi$, 0.26$\pi$ and 0.32$\pi$, respectively. In all cases, $F$ = 6.
Furthermore, the evolution process of pulse depends largely on FC lifetime. In Fig. 9(a), one can observe the number of pulses decreases from four to zero (marked with black arrows) with the increasing of detuning, while the number of pulses decreases from two to zero with a larger FC coefficient in Fig. 9(b). As continue increase of the FC lifetime, the vector pulses enter oscillating states, as shown in Figs. 9(c) and 9(d). Longer FC lifetime results in less pulses on the one hand, and causes breathing structures on the other. In contrast, the blue and green regions in Fig. 9 correspond to vector (blue) and oscillating (green) states via the turn-key approach with the same parameters but different detuning.
Fig. 9. Vector pulse excitation using pump frequency scanning method under different FC lifetime. (a)-(d) Evolution of the intracavity energy for both polarization modes with $\tau _{c}$=0.7, 1.5, 1.7 and 2, respectively. In all cases $\delta$=2, $\theta$ = 0.25$\pi$, $\epsilon$ = 0, $F$ = 6. The blue and green regions in these figures correspond to vector and oscillating states via the turn-key approach with the same parameters, respectively.
5. Influences of group velocity mismatch and thermal effects
The influences of group velocity mismatch between two polarization modes have also been studied. Figure 10 shows the results of turn-key scheme with different detuning and mismatch parameters. The drift of vector pulses (see Figs. 10(a), 10(d) and 10(g)) is due to the group velocity mismatch at a chosen detuning. The temporal profile and spectra become asymmetric. Moreover, intracavity pattern may enter the oscillation state when a larger mismatch is introduced (compare Figs. 10(g) and 10(j)). Comparatively, Fig. 11 depicts the results of frequency scanning method with mismatch parameters $\epsilon$ = 0, 0.2 and 0.5. With the increase of the mismatch value, the drift speed of vector pulses becomes larger (compare Figs. 11(a)-(b), 11(d)-(e) and 11(g)-(h)) and the distortion parts of the high power region becomes asymmetric. One can also observe that the emergence and decay locations of the pulses are altered by different mismatch. A large group velocity mismatch is undesirable for vector pulse excitation. When $\epsilon$ exceeds a certain value (e.g. $\epsilon$ = 5 in Fig. 11), only one soliton component can be generated. The threshold value is also changed with varied pump power and resonance interval. Given a group velocity mismatch between two polarization states, one can clearly see that the TM field also has an influence on the TE field. The TE pulse pedestal is asymmetric as shown by the zoomed-in part of Fig. 11(h). In Figs. 11(f) and 11(i), this interaction causes a slight change in the spectral envelope of both pulses, which leads to asymmetric spectra around the pump (also see Figs. 10(c), 10(f) and 10(i)).
Fig. 10. Vector pulse excitation via turn-key approach under group velocity mismatch. (a), (d), (g), (j) Temporal evolution for both polarization modes, (b), (e), (h), (k) temporal profiles and (c), (f), (i), (l) spectra at the locations marked with white dashed line with $\epsilon$ = 2, 5, 1, 2 respectively. In all cases $\tau _{c}$=1.2, $\delta$=2, $\theta$ = 0.25$\pi$, $F$ = 6.
Fig. 11. Vector pulse excitation via turn-key approach under group velocity mismatch. (a), (d), (g), (j) Temporal evolution for both polarization modes, (b), (e), (h), (k) temporal profiles and (c), (f), (i), (l) spectra at the locations marked with white dashed line with $\epsilon$ = 2, 5, 1, 2 respectively. In all cases $\tau _{c}$=1.2, $\delta$=2, $\theta$ = 0.25$\pi$, $F$ = 6.
Thermal instability will cause the pump to fall out of resonance due to blueshift of the cavity resonance associated with thermal-optical effect, which is crucial for soliton microcomb generation in practical microresonators. Thermal control [62], backward tuning [63], auxiliary laser heating [64] or slow frequency tuning [65] technics have been proposed to mitigate this obstacle for single soliton generation. Dissipative heating has also been studied extensively in simulations [66,67]. We incorporate thermal effects in Eqn. (4) and perform numerous simulations to uncover the effect of thermal coefficients. For weak thermal effects (e.g. $\kappa$=-0.005), vector pulses tend to be generated after a delay in all 40 simulations with different pump power using turn-key or laser sweeping schemes. In contrast, vector pulse can not be observed with strong thermal effects (e.g. $\kappa$=-0.05) in all 40 simulations. Figure 12 depicts the representative results of vector pulses generation when thermal coefficient changes. Delayed spontaneous vector pulse generation is observed with relatively small thermal coefficient as shown in Fig. 12(a) and 12(b). However, parametric process is inhibited in both Fig. 12(c) and 12(d) when a large thermal coefficient is considered. Therefore, accessing vector solitonic states with strong thermal effects could be challenging and require improved resonator performances. Such as reducing the optical absorption rate or engineering the device dimension to increase the microresonator thermal conductance [65].
Fig. 12. Vector pulse excitation with different thermal coefficient. (a)-(c) Turn-key scheme with $\kappa$=-0.01 and-0.02, respectively. In all cases $\tau _{c}$=1.2, $\delta$=2, $\theta$ = 0.25$\pi$, $F$ = 6. (b)-(d) Frequency scanning scheme with $\kappa$=-0.02 and-0.05, respectively. In all cases $\tau _{c}$=0.7, $\delta$=2, $\theta$ = 0.25$\pi$, $F$ = 6.
6. Conclusion
In summary, we have presented the excitation of vector solitonic pulses comprising orthogonally polarized components via free-carrier effects in normal GVD microresonators numerically. In the turn-key excitation regime, single- and dual-peak vector pulses as well as oscillation states are feasible with a single CW pump, circumventing traditional technics like mode interaction. The detuning range for pulse generation versus varying interval between pumped modes and polarization angle have been unveiled. In the pump scanning scheme, simultaneous and independent excitation regimes are identified due to different deviation of pumped modes. The distortion of the wider vector pulses is due to the nonlinear coupling between two modes. The pulse width, number of the vector solitonic microcombs can be controlled by the pump polarization angle and the FC life time. Breathing dynamics is also observed with proper combinations of parameters. Finally, we observe delayed spontaneous solitonic pulse generation or pulse annihilation with a lower or larger thermal-optical coefficient, mainly attributed to the blueshift of resonance associated with intracavity power drop. These results will give deep insight into the FC associated complex nonlinear process in microresonators and facilitate development of vector solitonic microcomb sources.
Funding
National Natural Science Foundation of China (12004312, 12204380, 12204381, 52002331); Natural Science Basic Research Program of Shaanxi Province (2021JQ-130, 2022JQ-066, 2022JQ-688, 2452022377, 2452022379).
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.
References
1. C. Xiang, J. Liu, J. Guo, L. Chang, R. N. Wang, W. Weng, J. Peters, W. Xie, Z. Zhang, J. Riemensberger, J. Selvidge, T. J. Kippenberg, and J. E. Bowers, “Laser soliton microcombs heterogeneously integrated on silicon,” Science 373(6550), 99–103 (2021). [CrossRef]
2. B. Shen, L. Chang, J. Liu, H. Wang, Q.-F. Yang, C. Xiang, R. N. Wang, J. He, T. Liu, W. Xie, J. Guo, D. Kinghorn, L. Wu, Q.-X. Ji, T. J. Kippenberg, K. Vahala, and J. E. Bowers, “Integrated turnkey soliton microcombs,” Nature 582(7812), 365–369 (2020). [CrossRef]
3. P. J. Marchand, J. Riemensberger, J. C. Skehan, J.-J. Ho, M. H. Pfeiffer, J. Liu, C. Hauger, T. Lasser, and T. J. Kippenberg, “Soliton microcomb based spectral domain optical coherence tomography,” Nat. Commun. 12(1), 427 (2021). [CrossRef]
4. G. Moille, E. F. Perez, J. R. Stone, A. Rao, X. Lu, T. S. Rahman, Y. K. Chembo, and K. Srinivasan, “Ultra-broadband kerr microcomb through soliton spectral translation,” Nat. Commun. 12(1), 7275 (2021). [CrossRef]
5. S. Fujii, K. Wada, R. Sugano, H. Kumazaki, S. Kogure, Y. K. Kato, and T. Tanabe, “Versatile tuning of kerr soliton microcombs in crystalline microresonators,” Commun. Phys. 6(1), 1 (2023). [CrossRef]
6. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014). [CrossRef]
7. G. Lihachev, W. Weng, J. Liu, L. Chang, J. Guo, J. He, R. N. Wang, M. H. Anderson, Y. Liu, J. E. Bowers, and T. J. Kippenberg, “Platicon microcomb generation using laser self-injection locking,” Nat. Commun. 13(1), 1771 (2022). [CrossRef]
8. X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9(9), 594–600 (2015). [CrossRef]
9. X. Xue, P.-H. Wang, Y. Xuan, M. Qi, and A. M. Weiner, “Microresonator Kerr frequency combs with high conversion efficiency,” Laser Photonics Rev. 11(1), 1600276 (2017). [CrossRef]
10. B. Y. Kim, J. K. Jang, Y. Okawachi, X. Ji, M. Lipson, and A. L. Gaeta, “Synchronization of nonsolitonic Kerr combs,” Sci. Adv. 7(43), eabi4362 (2021). [CrossRef]
11. H. Wang, B. Shen, Y. Yu, Z. Yuan, C. Bao, W. Jin, L. Chang, M. A. Leal, A. Feshali, M. Paniccia, J. Bowers, and K. Vahala, “Self-regulating soliton switching waves in microresonators,” Phys. Rev. A 106(5), 053508 (2022). [CrossRef]
12. G. N. Campbell, S. Zhang, L. Del Bino, P. Del’Haye, and G.-L. Oppo, “Counterpropagating light in ring resonators: Switching fronts, plateaus, and oscillations,” Phys. Rev. A 106(4), 043507 (2022). [CrossRef]
13. M.-G. Suh, X. Yi, Y.-H. Lai, S. Leifer, I. S. Grudinin, G. Vasisht, E. C. Martin, M. P. Fitzgerald, G. Doppmann, J. Wang, D. Mawet, S. B. Papp, S. A. Diddams, C. Beichman, and K. Vahala, “Searching for exoplanets using a microresonator astrocomb,” Nat. Photonics 13(1), 25–30 (2019). [CrossRef]
14. E. Obrzud, M. Rainer, A. Harutyunyan, M. H. Anderson, J. Liu, M. Geiselmann, B. Chazelas, S. Kundermann, S. Lecomte, M. Cecconi, A. Ghedina, E. Molinari, F. Pepe, F. Wildi, F. Bouchy, T. J. Kippenberg, and T. Herr, “A microphotonic astrocomb,” Nat. Photonics 13(1), 31–35 (2019). [CrossRef]
15. M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354(6312), 600–603 (2016). [CrossRef]
16. M. Yu, Y. Okawachi, A. G. Griffith, N. Picqué, M. Lipson, and A. L. Gaeta, “Silicon-chip-based mid-infrared dual-comb spectroscopy,” Nat. Commun. 9(1), 1869 (2018). [CrossRef]
17. Q.-F. Yang, B. Shen, H. Wang, M. Tran, Z. Zhang, K. Y. Yang, L. Wu, C. Bao, J. Bowers, A. Yariv, and K. Vahala, “Vernier spectrometer using counterpropagating soliton microcombs,” Science 363(6430), 965–968 (2019). [CrossRef]
18. R. Niu, S. Wan, J. Li, R.-C. Zhao, C.-L. Zou, G.-C. Guo, and C.-H. Dong, “Fast spectroscopy based on a modulated soliton microcomb,” IEEE Photonics J. 13(4), 1–4 (2021). [CrossRef]
19. Y. Wang, Z. Wang, X. Wang, W. Shao, L. Huang, B. Liang, B. E. Little, S. T. Chu, W. Zhao, W. Wang, and W. Zhang, “Scanning dual-microcomb spectroscopy,” Sci. China Phys. Mech. Astron. 65(9), 294211 (2022). [CrossRef]
20. J. Riemensberger, A. Lukashchuk, M. Karpov, W. Weng, E. Lucas, J. Liu, and T. J. Kippenberg, “Massively parallel coherent laser ranging using a soliton microcomb,” Nature 581(7807), 164–170 (2020). [CrossRef]
21. P. Trocha, M. Karpov, D. Ganin, M. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018). [CrossRef]
22. J. Wang, Z. Lu, W. Wang, F. Zhang, J. Chen, Y. Wang, J. Zheng, S. T. Chu, W. Zhao, B. E. Little, X. Qu, and W. Zhang, “Long-distance ranging with high precision using a soliton microcomb,” Photonics Res. 8(12), 1964–1972 (2020). [CrossRef]
23. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546(7657), 274–279 (2017). [CrossRef]
24. Y. Geng, X. Huang, W. Cui, Y. Ling, B. Xu, J. Zhang, X. Yi, B. Wu, S.-W. Huang, K. Qiu, C. W. Wong, and H. Zhou, “Terabit optical OFDM superchannel transmission via coherent carriers of a hybrid chip-scale soliton frequency comb,” Opt. Lett. 43(10), 2406–2409 (2018). [CrossRef]
25. Y. Geng, H. Zhou, X. Han, W. Cui, Q. Zhang, B. Liu, G. Deng, Q. Zhou, and K. Qiu, “Coherent optical communications using coherence-cloned Kerr soliton microcombs,” Nat. Commun. 13(1), 1070 (2022). [CrossRef]
26. Z. L. Newman, V. Maurice, T. Drake, et al., “Architecture for the photonic integration of an optical atomic clock,” Optica 6(5), 680–685 (2019). [CrossRef]
27. S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1(1), 10 (2014). [CrossRef]
28. T. E. Drake, T. C. Briles, J. R. Stone, D. T. Spencer, D. R. Carlson, D. D. Hickstein, Q. Li, D. Westly, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Terahertz-rate Kerr-microresonator optical clockwork,” Phys. Rev. X 9, 031023 (2019). [CrossRef]
29. Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Counter-propagating solitons in microresonators,” Nat. Photonics 11(9), 560–564 (2017). [CrossRef]
30. S. Zhang, T. Bi, G. N. Ghalanos, N. P. Moroney, L. Del Bino, and P. Del’Haye, “Dark-bright soliton bound states in a microresonator,” Phys. Rev. Lett. 128(3), 033901 (2022). [CrossRef]
31. C. Bao, P. Liao, A. Kordts, L. Zhang, A. Matsko, M. Karpov, M. H. P. Pfeiffer, G. Xie, Y. Cao, A. Almaiman, M. Tur, T. J. Kippenberg, and A. E. Willner, “Orthogonally polarized frequency comb generation from a Kerr comb via cross-phase modulation,” Opt. Lett. 44(6), 1472–1475 (2019). [CrossRef]
32. A. U. Nielsen, B. Garbin, S. Coen, S. G. Murdoch, and M. Erkintalo, “Coexistence and interactions between nonlinear states with different polarizations in a monochromatically driven passive Kerr resonator,” Phys. Rev. Lett. 123(1), 013902 (2019). [CrossRef]
33. R. Suzuki, S. Fujii, A. Hori, and T. Tanabe, “Theoretical study on dual-comb generation and soliton trapping in a single microresonator with orthogonally polarized dual pumping,” IEEE Photonics J. 11(1), 1–11 (2018). [CrossRef]
34. L. Guo, L. Wang, Q. Sun, M. Liu, G. Wang, W. Wang, P. Xie, W. Fan, and W. Zhao, “Mid-infrared dual-comb generation via the cross-phase modulation effect in a normal-dispersion microcavity,” Appl. Opt. 59(7), 2101–2107 (2020). [CrossRef]
35. B. Kostet, Y. Soupart, K. Panajotov, and M. Tlidi, “Coexistence of dark vector soliton Kerr combs in normal dispersion resonators,” Phys. Rev. A 104(5), 053530 (2021). [CrossRef]
36. V. E. Lobanov, A. E. Shitikov, R. R. Galiev, K. N. Min’kov, O. V. Borovkova, and N. M. Kondratiev, “Generation of vector flat-top solitons and hybrid bright–flat-top soliton complexes in optical microresonators via modulated pump,” Phys. Rev. A 104(6), 063511 (2021). [CrossRef]
37. B. Y. Kim, Y. Okawachi, J. K. Jang, M. Yu, X. Ji, Y. Zhao, C. Joshi, M. Lipson, and A. L. Gaeta, “Turn-key, high-efficiency Kerr comb source,” Opt. Lett. 44(18), 4475–4478 (2019). [CrossRef]
38. V. E. Lobanov, G. Lihachev, T. J. Kippenberg, and M. L. Gorodetsky, “Frequency combs and platicons in optical microresonators with normal GVD,” Opt. Express 23(6), 7713–7721 (2015). [CrossRef]
39. J. K. Jang, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson, and A. L. Gaeta, “Dynamics of mode-coupling-induced microresonator frequency combs in normal dispersion,” Opt. Express 24(25), 28794–28803 (2016). [CrossRef]
40. N. M. Kondratiev and V. E. Lobanov, “Modulational instability and frequency combs in whispering-gallery-mode microresonators with backscattering,” Phys. Rev. A 101(1), 013816 (2020). [CrossRef]
41. N. M. Kondratiev, V. E. Lobanov, A. E. Shitikov, R. R. Galiev, D. A. Chermoshentsev, N. Y. Dmitriev, A. N. Danilin, E. A. Lonshakov, K. N. Min’kov, D. M. Sokol, S. J. Cordette, Y.-H. Luo, W. Liang, J. Liu, and I. A. Bilenko, “Recent advances in laser self-injection locking to high-Q microresonators,” Front. Phys. 18(2), 21305 (2023). [CrossRef]
42. W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A. Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, “Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high- Q microresonators,” Nat. Photonics 15, 1–8 (2021). [CrossRef]
43. N. M. Kondratiev, V. E. Lobanov, E. A. Lonshakov, N. Y. Dmitriev, A. S. Voloshin, and I. A. Bilenko, “Numerical study of solitonic pulse generation in the self-injection locking regime at normal and anomalous group velocity dispersion,” Opt. Express 28(26), 38892–38906 (2020). [CrossRef]
44. V. E. Lobanov, N. M. Kondratiev, and I. A. Bilenko, “Thermally induced generation of platicons in optical microresonators,” Opt. Lett. 46(10), 2380–2383 (2021). [CrossRef]
45. L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear group IV photonics based on silicon and germanium: from near-infrared to mid-infrared,” Nanophotonics 3(4-5), 247–268 (2014). [CrossRef]
46. M. Sinobad, C. Monat, B. Luther-davies, P. Ma, S. Madden, D. J. Moss, A. Mitchell, D. Allioux, R. Orobtchouk, S. Boutami, J.-M. Hartmann, J.-M. Fedeli, and C. Grillet, “Mid-infrared octave spanning supercontinuum generation to 8.5 μm in silicon-germanium waveguides,” Optica 5(4), 360–366 (2018). [CrossRef]
47. M. Montesinos-Ballester, C. Lafforgue, J. Frigerio, A. Ballabio, V. Vakarin, Q. Liu, J. M. Ramirez, X. Le Roux, D. Bouville, A. Barzaghi, C. Alonso-Ramos, L. Vivien, G. Isella, and D. Marris-Moroni, “On-chip mid-infrared supercontinuum generation from 3 to 13 μm wavelength,” ACS Photonics 7(12), 3423–3429 (2020). [CrossRef]
48. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010). [CrossRef]
49. J. Yuan, Z. Kang, F. Li, X. Zhang, X. Sang, Q. Wu, B. Yan, K. Wang, X. Zhou, K. Zhong, G. Zhou, C. Yu, C. Lu, H. Y. Tam, and P. K. A. Wai, “Mid-infrared octave-spanning supercontinuum and frequency comb generation in a suspended germanium-membrane ridge waveguide,” J. Lightwave Technol. 35(14), 2994–3002 (2017). [CrossRef]
50. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
51. M. Yu, Y. Okawachi, A. G. Griffith, M. Lipson, and A. L. Gaeta, “Mode-locked mid-infrared frequency combs in a silicon microresonator,” Optica 3(8), 854–860 (2016). [CrossRef]
52. R. Haldar, A. Roy, P. Mondal, V. Mishra, and S. K. Varshney, “Free-carrier-driven Kerr frequency comb in optical microcavities: Steady state, bistability, self-pulsation, and modulation instability,” Phys. Rev. A 99(3), 033848 (2019). [CrossRef]
53. M. Liu, H. Huang, Z. Lu, Y. Wang, Y. Cai, and W. Zhao, “Dynamics of dark breathers and Raman-Kerr frequency combs influenced by high-order dispersion,” Opt. Express 29(12), 18095–18107 (2021). [CrossRef]
54. L. Hill, G.-L. Oppo, M. T. Woodley, and P. Del’Haye, “Effects of self-and cross-phase modulation on the spontaneous symmetry breaking of light in ring resonators,” Phys. Rev. A 101(1), 013823 (2020). [CrossRef]
55. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18(4), 3582–3591 (2010). [CrossRef]
56. F. Bessin, A. M. Perego, K. Staliunas, S. K. Turitsyn, A. Kudlinski, M. Conforti, and A. Mussot, “Gain-through-filtering enables tuneable frequency comb generation in passive optical resonators,” Nat. Commun. 10(1), 4489 (2019). [CrossRef]
57. A. M. Perego, S. K. Turitsyn, and K. Staliunas, “Gain through losses in nonlinear optics,” Light: Sci. Appl. 7(1), 43 (2018). [CrossRef]
58. M. Rowley, P.-H. Hanzard, A. Cutrona, H. Bao, S. T. Chu, B. E. Little, R. Morandotti, D. J. Moss, G.-L. Oppo, J. S. Totero Gongora, M. Peccianti, and A. Pasquazi, “Self-emergence of robust solitons in a microcavity,” Nature 608(7922), 303–309 (2022). [CrossRef]
59. E. Nazemosadat, A. Fülöap, Ó. B. Helgason, P.-H. Wang, Y. Xuan, D. E. Leaird, M. Qi, E. Silvestre, A. M. Weiner, and V. Torres-Company, “Switching dynamics of dark-pulse Kerr frequency comb states in optical microresonators,” Phys. Rev. A 103(1), 013513 (2021). [CrossRef]
60. P. Parra-Rivas, E. Knobloch, D. Gomila, and L. Gelens, “Dark solitons in the lugiato-lefever equation with normal dispersion,” Phys. Rev. A 93(6), 063839 (2016). [CrossRef]
61. P. Parra-Rivas, D. Gomila, E. Knobloch, S. Coen, and L. Gelens, “Origin and stability of dark pulse Kerr combs in normal dispersion resonators,” Opt. Lett. 41(11), 2402–2405 (2016). [CrossRef]
62. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton modelocking in microresonators,” Opt. Lett. 41(11), 2565–2568 (2016). [CrossRef]
63. H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13(1), 94–102 (2017). [CrossRef]
64. H. Zhou, Y. Geng, W. Cui, S.-W. Huang, Q. Zhou, K. Qiu, and C. W. Wong, “Soliton bursts and deterministic dissipative Kerr soliton generation in auxiliary-assisted microcavities,” Light: Sci. Appl. 8(1), 50 (2019). [CrossRef]
65. Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4(2), 193–203 (2017). [CrossRef]
66. Z. Kang, F. Li, J. Yuan, K. Nakkeeran, J. N. Kutz, Q. Wu, C. Yu, and P. Wai, “Deterministic generation of single soliton Kerr frequency comb in microresonators by a single shot pulsed trigger,” Opt. Express 26(14), 18563–18577 (2018). [CrossRef]
67. A. Leshem, Z. Qi, T. F. Carruthers, C. R. Menyuk, and O. Gat, “Thermal instabilities, frequency-comb formation, and temporal oscillations in Kerr microresonators,” Phys. Rev. A 103(1), 013512 (2021). [CrossRef]
