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Abstract
Cavity solitons are shape-preserving waveforms infinitely revolving around a cavity, which have numerous applications from spectroscopy to telecommunications. Although the cavity solitons have been widely studied for their special time-frequency characteristics over the past decade, the spectral flatness will be a large limitation in some applications without any extra filtering process. In this paper, we report on the generation of a distinct cavity soliton in a cyclic polarization permutation fiber resonator. With the simultaneous excitation of two orthogonal polarization modes with equally opposite dispersion, vectorial cavity solitons possessing broader and flatter spectra can be generated. In order to verify the concept, a numerical model of the polarization-maintaining fiber is proposed and the soliton with a flattened spectrum can be formed. The results enrich the soliton dynamics in the vectorial dissipation system.
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1. Introduction
Soliton is a kind of highly self-localized pattern that exists in many nonlinear dynamic systems including Bose-Einstein condensates [1], polaritons [2], hydrodynamics [3], and mode-locked lasers [4,5]. Coherently-driven Kerr resonator, as a typical nonlinear dissipating system, has been demonstrated to sustain the generation of solitons which are usually called by temporal cavity solitons (CSs). First experimentally observed in the macroscopic fiber ring resonator [6], and soon thereafter in the integrated on-chip microresonator [7], CSs have attracted great attention due to their remarkable contributions to fundamental physics [8–15] in the past decade. Relying on the composite balance between anomalous dispersion and Kerr nonlinearity, as well as parametric gain and cavity losses, the generated CSs can retain their temporal waveform (bright pulse) and indefinitely revolve around the cavity. Owing to their highly temporal robustness, the CSs have been demonstrated to be ideal support for optical signal processing by using the all-optical trapping and manipulation technique [16–18]. Because the periodic pulse trains correspond to a series of discrete spectral lines in the frequency domain, the CS acts as an equidistant multi-wavelength optical source named microcomb in numerous applications, such as spectroscopy [19,20], microwave generation [21,22], optical frequency synthesis [23], astro-comb calibration [24,25], ultrafast optical ranging [26,27], and telecommunication [28], etc.
Although the traditional CSs mentioned above have many prominent time-frequency characteristics, the sech-shaped envelope extremely limits the spectral flatness and bandwidth in this regime. The formation process of the conventional soliton optical comb depends on the mode-locking guided by the interaction between anomalous dispersion and nonlinearity, and the spectrum formed is hyperbolic secant with limited flatness and bandwidth. Stephane Coen et al. proposed a method to estimate the comb bandwidth for a given pump, which points out that the constant coefficients related to the sech-shaped pulse limit the bandwidth [29]. This kind of limitation requires more complicated filtering processes to achieve spectral shaping, which can degrade the total signal-to-noise ratio during the data transmission [28,32]. The methods directly tailoring the spectrum inside the cavity can be of great potential to reduce the extra-cavity filtering complexity. In 2015, the stretched CSs with a Gaussian profile have been theoretically demonstrated in the dispersion-managed Kerr resonators by Bao et al. [30] and subsequently observed in the experiments by Dong et al. [31], where the longitudinal dispersion variation makes the CSs experience periodic compression and expansion. Besides the bright CSs in the anomalous dispersion, their counterpart commonly called ‘dark soliton’ in the normal dispersion regime is observed to have a distinct spectrum [33,34]. Despite a higher conversion efficiency of the pump power into the comb, the modulation envelope in its spectrum still hinders the improvement of the flatness [35,36]. Maitland et al. have recently reported on a CS generated in a Möbius resonator, in which the Möbius CS can be sustained by the cavity with opposite dispersion [37]. Although the resonator needs two rings connecting with each other, it provides new guidance to form CSs in the dispersion-alternating condition. The specific waveform could enlarge the spectral width to some extent, whilst a bulge at the spectrum center remains to be eliminated.
The above investigations have hitherto been predominantly considered single transverse mode families sharing the same spatial distribution or polarization property. To break down the barriers brought by the only excitation of a single mode, the simultaneous excitation of multi modes could be an effective method by providing another controllable dimension. Actually, the interactions of different spatial modes and orthogonal polarization modes have been proven to sustain the generation of non-identical CSs displayed in the repetition rate [38,39], number [39], and amplitude [40,41]. These results give insight into the features of the vectorial Kerr resonators.
In this paper, the different soliton dynamics are revealed in the resonator with cyclic polarization permutation. By simultaneously pumping two orthogonal polarization modes with equally opposite dispersion, the optical field permutation and cross-phase modulation between the two modes produce specific vector CSs with broader and flatter spectra than the traditional CSs. To satisfy the generated condition, we then optimize the dispersion of two polarization modes in a photonic crystal hybrid polarization maintaining fiber. The numerical simulations based on the designed parameters are consistent with the normalized results. Our proposed cyclic polarization permutation fiber resonator not only provides a potential way to broaden and flatten the frequency comb spectrum but also gives more insights into the CSs dynamics in the multi-mode system.
2. Theoretical model and method
To better understand the mechanism of mutual circulation between the two polarization states, the schematic illustration of the resonator is shown in Fig. 1. The polarization beam splitter (PBS) divides the light field into two paths representing two orthogonal polarization states. Polarization beam combiner (PBC) combines the light fields of two polarization states. Before the PBS, the fast and slow axis of the fiber is exchanged which can be achieved by rotation splicing. In this condition, the optical fields will experience polarization permutation before outputting each optical coupler (OC), which means the two polarizations share the same optical routes and possess equal frequency detuning.
Fig. 1. Schematic of the cyclic polarization permutation Kerr resonator. CW: continuous wave, PBS: polarization beam splitter, PBC: polarization beam combiner, OC: optical coupler, LP01, x and LP01, y denote two orthogonal polarization states, respectively.
To obtain the evolving polarization modes of the optical field within the cavity, we assume a birefringent fiber-based Kerr resonator that supports two orthogonal polarization mode families. The resonator is driven with a monochromatic pump laser that splits into two orthogonal polarizations to excite both modes simultaneously. The evolution of the light field in the cavity corresponds to both polarization modes. In dimensionless form, the nonlinear Schrödinger (NLS) equation can be described in the following way:
In this paper, the excitation of CSs is achieved by increasing the intensity perturbation to the continuous wave. Firstly, we recover the typical state behavior of the ring resonator steady state, such as modulation instability, chaos, oscillation, and stable cavity solitons with β2(1) = -1, β2(2) = -1. When the absolute dispersions of the two polarization states are equal but with opposite signs, β2(1) = -1, β2(2) = 1, instead of the usual Kerr cavity solitons, a new observed state called cyclic polarization permutation cavity solitons (CPP CSs) shows stable waveforms in both polarization states, as shown in Fig. 2. The CPP CSs appear in a small detuning range δ∈(0.28, 0.32), and their structures change slightly in this range. Out of the detuning range, there is no apparent modulation instability state nor chaotic steady state during the evolution. When the detuning amount is relatively small (0.28-0.29), the soliton possesses an oscillation background, with larger detuning such background disappears. It can be found that the pulses of the two polarization states share the same optical path but show different widths. This is because the pulses experience stretching and compressing during traveling the cavity, while polarization 1 and polarization 2 undergo anomalous dispersion and normal dispersion, respectively before outputting the OCs.
Fig. 2. Steady state power distribution in polarization state 1 (a) and polarization state 2 (b) over time. Ein = 0.14, θ = 2/15, α = 1/200, β2(1) = -1, β2(2) = 1, β3(1) =β3(2) = 0. The detuning amounts of the two polarization states are consistent δ1 =δ2.
In the cyclic polarization permutation Kerr resonator, the cyclic evolution of the light field between two polarization states is that the output of one polarization state is fed back to the input of the other polarization state. When the light field experiences periodic second-order dispersion, two loops of periodic behavior will appear, as shown in Fig. 3. The steady-state time-domain waveforms in two of the polarization states are plotted in Fig. 3(a). The spectral range of the CPP CSs is much wider than that of standard cavity solitons, as shown in Fig. 3(b). The standard Kerr cavity soliton is drawn by the black line, which is formed in an equivalent resonator with the same dispersion, same detuning, same pump power, and initial conditions. On the premise that the GVD of one polarization state is positive and the other is negative, the spectra under several different conditions are illustrated in Fig. 3(c). It can be concluded that the smaller the absolute value of the group velocity, the broader the spectrum. Therefore, this kind of CPP CS is a promising approach for generating broadband frequency combs.
Fig. 3. Example of intracavity steady-state power and spectrum of the cyclic polarization permutation resonators consisting of two orthogonal polarization states with opposite and equal GVD β2(1) = -1, β2(2) = 1. Other parameters: Ein = 0.14, θ = 2/15, α = 1/200, δ1 =δ2 = 0.3, β3(1) =β3(2) = 0. (a) Intracavity power profiles from polarization 1 and polarization 2 after 800 round trips. (b) The spectrum of polarization 1, polarization 2, and standard CSs. Comparison of the CPP CSs spectrum with that of the standard Kerr CSs. (c) Spectral comparison of CPP CSs with different dispersions. Blue line: β2(1) = -1, β2(2) = 1; black line: β2(1) = -2, β2(2) = 2; red line: β2(1) = -3, β2(2) = 3.
Next, we consider the TOD on the CPP CS. Including the TOD in both polarization states, β3(1) = β3(2) = 2, the waveform in the time-domain becomes asymmetric and a finite group velocity is obtained. It is stable to a certain detuning range but unlike the previous results with β3 = 0, the time domain waveforms of the two fields become more similar, as shown in Fig. 4(a). The CPP CSs are characterized by the high oscillation structure in the time domain and narrower spectra. The resonance radiation generated by the TOD is obvious, as shown in Fig. 4(b). Since the Ikeda boundary conditions result in a range of possible phase-matching wavenumbers, there are multiple peaks according to the phase-matching condition [42,43]:
Fig. 4. (a) Power profiles from polarization 1 and polarization 2 given δ1 =δ2 = 0.28, β2(1) = -1, β2(2) = 1, β3(1) =β3(2) = 2. (b) The corresponding spectrum in the case of adding TOD.
3. Numerical model
To verify the concept in the previous section, a polarization maintaining fiber with opposite and equal GVD is designed as shown in Fig. 5. The fiber is of a hybrid structure of step refractive index and photonic crystal structure with two stress rods and an elliptical core (SIPCPMF) [44]. The dispersion distribution of the fiber is influenced by the combination of a core and a series of rods made of a high refractive index material GeO2. For the holes around the fiber core, the diameter of the first layer is d1, and the diameters of the air holes in the second to third layers are d2. The distance between any two adjacent pores is denoted by P. The fiber core is Ge-doped with long and short half axes b/2 and a/2, respectively. The material dispersion of SIPCPMF is given by the Sellmeier equation [45]. The Ge-doped concentration of the core is expressed as MC1, and the Ge-doped concentration of the surrounding pores is MC2. The distance between the stress rod and the core is 38 µm and its B-doped concentration is 15% so the refractive index is calculated to be 1.426. In addition, a set of conventional process parameters can be used in the preparation of the fiber [46]. Doping the stress rod with B2O3 increases the thermal expansion coefficient by an order of magnitude, causing stress in the fiber.
By optimizing the fiber parameters, the dispersion profile can be adjusted to obtain two orthogonal polarization states with equal and opposite signs in a certain wavelength. At the same time, the horizontal dispersion is kept as low and flat as possible in order to obtain a wider spectrum according to the discussion in the previous section. The fiber properties are obtained after optimization with a = 3 µm, b = 2.1 µm, MC1 = 10.39%, MC2 = 9.4%, d1 = 2 µm, and d2 = 2.9 µm, P = 4.9 µm. The dispersion curves of the two orthogonal polarization states are plotted by the red and black lines, as shown in Fig. 6(a), where the insets are the electric field intensity distributions for x-polarization and y-polarization at 1550 nm wavelength, respectively. The two intersection points where the blue dotted line intersects the two dispersion curves indicate that the GVD of the two polarization states at 1553 nm are equal and opposite in sign, and the β2 are -1.8 × 10−28 s2/m and 1.8 × 10−28 s2/m, respectively, as shown in Fig. 6(b).
Fig. 6. (a) Dispersion profiles of two orthogonal polarization states. The two insets represent the electric field distributions for the two orthogonal polarization states, respectively. (b) The detailed zoom of dispersion in the wavelength range of 1540 nm-1580 nm. The dashed light green line marks the zero-dispersion level.
4. Result and discussion
In order to further quantitatively study the spectral quality of the CPP CS, the designed fiber parameters are applied to the resonator. The parameters of the resonator are shown in Table 1.
Table 1. Relative physical parameters
With the absence of higher-order dispersion, the characteristics of CPP CSs are shown in Fig. 7. The peak power of the two polarization states is 3.5 W and 1.8 W respectively, Fig. 7(a). Flat CPP CSs with the 3-dB bandwidth of 84 nm can be obtained as shown in Fig. 7(b). Figure 7(c) and (d) plot the dynamics of the time-domain optical field evolution for two polarization states, respectively. In addition, when the TOD is not zero, the balance of the mutual cancellation of the GVD is broken. The introduction of TOD increases the peak power of the two pulses but narrows the spectral bandwidth, as shown in Fig. 8. The time-domain waveform spectra are in good agreement with the normalized results above. Finally, CPP CSs with the 3-dB bandwidth of 13 nm are obtained. Assuming that the TOD can also be mutually compensated e. g. β3(1) = 5.15 × 10−41 s3/m, β3(2) = -5.15 × 10−41 s3/m, the 3-dB bandwidth and 10-dB bandwidth of the CPP CS can achieve 31 nm and 52 nm, respectively.
Fig. 7. Characterizations of pulses in the cyclic polarization permutation Kerr resonator with the absence of higher-order dispersion. (a) Power profiles for two polarization states. (b) Spectrum. (c), (d) Time-domain evolution diagram of CPP CSs.
Fig. 8. Characterizations of pulses in the cyclic polarization permutation Kerr resonator with GVD and TOD. (a) The temporal profiles and (b) spectra at the 300th roundtrip.
The spectral bandwidth also has a strong dependence on the power of the injecting power, as shown in Fig. 9. The spectral bandwidth varies linearly with the driving power, where the slope is about 40 nm/W. To analyze the superiority of the CPP CS spectrum, it is compared with the conventional Kerr CSs, as shown in Fig. 10. The black and red lines represent the temporal profiles and spectra of bright soliton with β2= -1.8 × 10−28 s2/m and dark soliton with β2= 1.8 × 10−28 s2/m, respectively. The green line represents the temporal profile and the spectrum generated in a dispersion-managed fiber resonator (the half length of the fiber resonator is normal dispersion, and the other half length is abnormal dispersion.) Comparisons of the spectral properties of CPP CS and conventional Kerr CSs are shown in Table 2. CS+ in the resonator composed of the normal dispersion fiber has a relatively small bandwidth and low sensitivity to the TOD. In addition, the bright soliton in the anomalous dispersion resonator has a large bandwidth only when the TOD is small. The bandwidth of the dispersion-managed CS0 gradually increases as the TOD decreases. For CPP CS, it can be found that with large TOD, the bandwidth of the CPP CS and CS0 is nearly the same. However, once the TOD is further decreased, the CPP CS experiences a more severe bandwidth extension compared to the CS0. Therefore, the dispersion-flattened fiber allows the CPP CS to offer a greater advantage.
Fig. 9. The effect of driving power on the spectrum. (a) Measured spectrum as a function of drive power. (b) 10-dB bandwidth under several different powers.
Fig. 10. (a) Temporal profiles of CSs without TOD in traditional Kerr resonator. (b) Comparison of the CPP CS spectrum with other Kerr CSs spectra under the same pumping power.
Table 2. Comparison of the bandwidth of different CSs
For a wider and flatter soliton comb generation, the key factor is to obtain a lower and flatter dispersion curve in the wide wavelength range. For example, when the dispersion of two polarization states is β2(1) = -1 × 10−28 s2/m, β2(2) = 1 × 10−28 s2/m, the 3-dB bandwidth and 10-dB bandwidth of the CS further reach 112 nm and 166 nm without considering TOD, respectively. At the same time, optimizing the third-order dispersion on the basis of ensuring the low dispersion level is also crucial to expand the bandwidth. The structure of the cyclic polarization permutation is also expected to be combined with other approaches, such as dispersion-managed fiber rings [31], dispersion-shifted fiber rings [47], coupled fiber rings [48], etc.
5. Conclusion
In summary, we reveal a new kind of soliton dynamics namely CPP CS in a cyclic polarization permutation Kerr resonator. It is found that such CPP CS shows different time/frequency features compared with the traditional CSs generated in an anomalous dispersion or normal dispersion cavity. Particularly, it can be used to produce the flat optical frequency comb. Based on the analysis of the normalized model, we further designed the SIPCPMF that satisfies the GVD of the CPP CS. The numerical results show that the 3-dB bandwidth and 10-dB bandwidth of the CPP CS can be broadened to 84 nm and 125 nm, respectively, which is much wider than the ones that only consider anomalous or normal dispersion. The CPP CS is of great potential in applications that need wide and flat frequency comb lines.
Funding
National Natural Science Foundation of China (62275097); Wuhan National Laboratory for Optoelectronics (2019WNLOKF005); Natural Science Foundation of Guangdong Province; Fundamental Research Funds for the Central Universities (CUGDCJJ202204).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. C. Becker, S. Stellmer, P. Soltan-Panahi, S. Drscher, M. Baumert, E.-M. Richter, J. Kronjger, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4(6), 496–501 (2008). [CrossRef]
2. A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemnager, R. Houdr, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332(6034), 1167–1170 (2011). [CrossRef]
3. A. Chabchoub, M. Onorato, and N. Akhmediev, Rogue and Shock Waves in Nonlinear Dispersive Media, Lecture Notes in Physics, M. Onorato, S. Resitori, and F. Baronio, eds., (Springer International Publishing, Cham, 2016) pp. 55–87.
4. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]
5. W. Liu, L. Pang, H. Han, W. Tian, H. Chen, M. Lei, P. Yan, and Z. Wei, “Generation of dark solitons in erbium-doped fiber lasers based Sb2Te3 saturable absorbers,” Opt. Express 23(20), 26023–26031 (2015). [CrossRef]
6. F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4(7), 471–476 (2010). [CrossRef]
7. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014). [CrossRef]
8. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21(7), 9180–9191 (2013). [CrossRef]
9. M. Anderson, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Observations of spatiotemporal instabilities of temporal cavity solitons,” Optica 3(10), 1071–1074 (2016). [CrossRef]
10. M. Anderson, Y. Wang, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Coexistence of multiple nonlinear states in a tristable passive Kerr resonator,” Phys. Rev. X 7(3), 031031 (2017). [CrossRef]
11. E. Lucas, M. Karpov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Breathing dissipative solitons in optical microresonators,” Nat. Commun. 8(1), 736 (2017). [CrossRef]
12. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018). [CrossRef]
13. J. Pan, Z. Cheng, T. Huang, C. Song, P. P. Shum, and G. Brambilla, “Fundamental and third harmonic mode coupling induced single soliton generation in Kerr microresonators,” J. Lightwave Technol. 37(21), 5531–5536 (2019). [CrossRef]
14. J. K. Jang, A. Klenner, X. Ji, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Synchronization of coupled optical microresonators,” Nat. Photonics 12(11), 688–693 (2018). [CrossRef]
15. T. Huang, J. Pan, Z. Cheng, G. Xu, Z. Wu, T. Du, S. Zeng, and P. P. Shum, “Nonlinear-mode-coupling-induced soliton crystal dynamics in optical microresonators,” Phys. Rev. A 103(2), 023502 (2021). [CrossRef]
16. J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6(1), 7370 (2015). [CrossRef]
17. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Writing and erasing of temporal cavity solitons by direct phase modulation of the cavity driving field,” Opt. Lett. 40(20), 4755–4758 (2015). [CrossRef]
18. Y. Wang, B. Garbin, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Addressing temporal Kerr cavity solitons with a single pulse of intensity modulation,” Opt. Lett. 43(13), 3192–3195 (2018). [CrossRef]
19. 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]
20. A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4(3), e1701858 (2018). [CrossRef]
21. S. Diallo and Y. K. Chembo, “Optimization of primary Kerr optical frequency combs for tunable microwave generation,” Opt. Lett. 42(18), 3522–3525 (2017). [CrossRef]
22. X. Xue, Y. Xuan, H.-J. Kim, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Programmable single-bandpass photonic RF filter based on Kerr comb from a microring,” J. Lightwave Technol. 32(20), 3557–3565 (2014). [CrossRef]
23. D. T. Spencer, T. Drake, T. C. Briles, et al., “An optical-frequency synthesizer using integrated photonics,” Nature 557(7703), 81–85 (2018). [CrossRef]
24. E. Obrzud, M. Rainer, A. Harutyunyan, M. H. Anderson, J. Q. Liu, M. Geiselmann, B. Chazelas, S. Kundermann, S. Lecomte, and M. Cecconi, “A microphotonic astrocomb,” Nat. Photonics 13(1), 31–35 (2019). [CrossRef]
25. M. G. Suh, X. Yi, Y. H. Lai, S. Leifer, I. S. Grudinin, G. Vasisht, E. C. Martin, M. P. Fitzgerald, G. Doppmann, and J. Wang, “Searching for exoplanets using a microresonator astrocomb,” Nat. Photonics 13(1), 25–30 (2019). [CrossRef]
26. P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359(6378), 887–891 (2018). [CrossRef]
27. M.-G. Suh and K. J. Vahala, “Soliton microcomb range measurement,” Science 359(6378), 884–887 (2018). [CrossRef]
28. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, and M. H. Anderson, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546(7657), 274–279 (2017). [CrossRef]
29. S. Coen and M. Erkintalo, “Universal Scaling Laws of Kerr Frequency Combs,” Opt. Lett. 38(11), 1790–1792 (2013). [CrossRef]
30. C. Bao and C. Yang, “Stretched cavity soliton in dispersion-managed Kerr resonators,” Phys. Rev. A 92(2), 023802 (2015). [CrossRef]
31. X. Dong, Q. Yang, C. Spiess, V. G. Bucklew, and W. H. Renninger, “Stretched-pulse soliton Kerr resonators,” Phys. Rev. Lett. 125(3), 033902 (2020). [CrossRef]
32. B. Corcoran, M. Tan, X. Xu, A. Boes, J. Wu, T. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Ultra-dense optical data transmission over standard fiber with a single chip source,” Nat. Commun. 11(1), 2568 (2020). [CrossRef]
33. 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]
34. X. Xue, F. Leo, Y. Xuan, J. A. Jaramillo-Villegas, P. H. Wang, D. E. Leaird, M. Erkintalo, M. Qi, and A. M. Weiner, “Second-harmonic-assisted four-wave mixing in chip-based microresonator frequency comb generation,” Light: Sci. Appl. 6(4), e16253 (2017). [CrossRef]
35. A. Fülöp, M. Mazur, A. Lorences-Riesgo, Ó. B. Helgason, P. H. Wang, Y. Xuan, D. E. Leaird, M. Qi, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “High-order coherent communications using mode-locked dark-pulse Kerr combs from microresonators,” Nat. Commun. 9(1), 1598 (2018). [CrossRef]
36. J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, and R. Schmogrow, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014). [CrossRef]
37. C. Maitland, M. Conforti, A. Mussot, and F. Biancalana, “Stationary states and instabilities of a Möbius fiber resonator,” Phys. Rev. Res. 2(4), 043195 (2020). [CrossRef]
38. J. Pan, Z. Cheng, T. Huang, M. Zhu, Z. Wu, and P. P. Shum, “Numerical investigation of all-optical manipulation for polarization-multiplexed cavity solitons,” J. Lightwave Technol. 39(2), 582–591 (2021). [CrossRef]
39. E. Lucas, G. Lihachev, R. Bouchand, N. G. Pavlov, A. S. Raja, M. Karpov, M. L. Gorodetsky, and T. J. Kippenberg, “Spatial multiplexing of soliton microcombs,” Nat. Photonics 12(11), 699–705 (2018). [CrossRef]
40. E. Averlant, M. Tlidi, K. Panajotov, and L. Weicker, “Coexistence of cavity solitons with different polarization states and different power peaks in all-fiber resonators,” Opt. Lett. 42(14), 2750–2753 (2017). [CrossRef]
41. 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]
42. J. Jang, M. Erkintalo, S. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39(19), 5503–5506 (2014). [CrossRef]
43. K. Luo, Y. Xu, M. Erkintalo, and S. G. Murdoch, “Resonant radiation in synchronously pumped passive Kerr cavities,” Opt. Lett. 40(3), 427–430 (2015). [CrossRef]
44. T. Huang, S. Feng, X. Zeng, G. Xu, J. Pan, F. Xiao, Z. Wu, J. Zhang, L. Han, and P. P. Shum, “Polarization-decoupled cavity solitons generation in Kerr resonators with flattened near-zero dispersion,” Opt. Express 30(12), 20767–20782 (2022). [CrossRef]
45. D. Michalik, T. Stefaniuk, and R. Buczyński, “Dispersion management in hybrid optical fibers,” J. Lightwave Technol. 38(6), 1427–1434 (2020). [CrossRef]
46. Y. Yang, J. Gao, S. Fu, M. Tang, X. Zhang, and D. Liu, “PANDA type four-core fiber with the efficient use of stress rods,” IEEE Photonics J. 11(6), 1–9 (2019). [CrossRef]
47. Z. Li, Y. Xu, S. Coen, S. G. Murdoch, and M. Erkintalo, “Experimental observations of bright dissipative cavity solitons and their collapsed snaking in a Kerr resonator with normal dispersion driving,” Optica 7(9), 1195–1203 (2020). [CrossRef]
48. X. Xue, X. Zheng, and B. Zhou, “Super-efficient temporal solitons in mutually coupled optical cavities,” Nat. Photonics 13(9), 616–622 (2019). [CrossRef]
