Abstract

We experimentally studied the transition dynamics between consecutive multi-pulsing states, specifically the build-up and annihilation of soliton pulses between a double pulsing and a three-pulse state, utilizing the dispersive Fourier transform technique. The birth of an additional pulse in a mode-locked soliton fiber laser in a multi-pulsing regime arises from a dispersive wave-induced narrow-band pulse that experiences strong intensity fluctuations, while the other soliton pulses maintain their shapes. During the decaying process to a double pulsing state, it is observed that all the pulses undergo a unique breathing behavior before settling into a steady state.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Passively mode-locked lasers have been widely used, either at the fundamental repetition rate or in multi-pulsing states. However, the generation of multi-pulsing states relies on an intricate interplay of different mechanisms. In a soliton laser, additional pulses can be generated above a certain pumping threshold due to energy quantization, including harmonic mode locking (HML) [1], multi-pulsing [2,3] and soliton bunching [3], depending on various phenomena, e.g., soliton interaction, gain depletion and recovery dynamics [4], and acoustic effects [5]. While the main mechanisms behind multi-soliton formation in mode-locked lasers [6,7] can be attributed to pulse splitting and interaction with non-solitonic components, transition dynamics from ${\rm N}$ to ${\rm N} + {1}$ pulses can also undergo a Hopf bifurcation leading to chaotic states [810].

The dispersive Fourier transform (DFT) [11] offers a high temporal resolution based on a time-wavelength mapping induced by dispersion. Thus, measurements based on the DFT technique can capture the transient spectral-temporal dynamics of ultrafast phenomena and have offered new insights into the pulse-to-pulse spectral dynamics in various ultrafast laser operating regimes, including rogue waves [12], soliton explosions [13], soliton molecules [14,15], and noise-like pulses [16]. Moreover, the build-up dynamics of dissipative solitons [17,18], soliton molecules [19], and stretched-pulse lasers has been investigated [20]. The soliton formation process in a Ti:sapphire Kerr-lens mode-locked laser was first resolved experimentally, unveiling that solitons developed from background picosecond fluctuations [21]. The creation of solitons in a fiber laser was characterized by ${\rm Q}$-switched mode locking, spectral beating, and transient bound states before stable mode locking was reached [22]. Similar phenomena of ${\rm Q}$-switching and self-phase modulation (SPM) were observed in the build-up process of a multi-pulsing state [23]. Studies captured the transition directly from noise to a single pulse that split into five pulses and was stabilized to a 5th HML state by an acoustic wave [24]. During the decaying process from a double pulsing state, the two pulses either disappeared simultaneously or one pulse was weakened mainly due to gain competition [25]. While these findings revealed important insights into selective transitions, the evolution is associated with specific laser systems and pulsing regimes. As the birth of new pulses and multi-pulsing generations are complex processes that can be governed by different transition dynamics, the most commonly induced evolution between consecutive multi-pulsing states and their real-time dynamics is studied here for the first time, to the best of our knowledge.

In this Letter, we characterize the pulse-by-pulse transition dynamics between consecutive multi-pulsing states in an erbium (Er)-doped fiber ring laser utilizing DFT. The soliton formation via pulse shaping of a dispersive wave (DW) and modulation of a narrow-band pulse from a 2nd HML to a three-pulse state are presented in detail. A unique spectral breathing is reported that occurs simultaneously among all oscillating pulses during the decaying process from a three-pulse to a two-pulse state.

An Er-doped soliton mode-locked fiber ring laser is used to study multi-pulsing transitions, as illustrated in Fig. 1(a). The laser consists of an 85 cm long Er gain fiber (Liekki Er80-8/125) and a single-walled carbon nanotube saturable absorber. The total length of the laser cavity of 4.6 m corresponds to a fundamental repetition rate of 44.1 MHz. The laser gain fiber is backward pumped through a wavelength division multiplexer (WDM) with a 980 nm laser diode (LD). Three polarization controllers (PCs) are incorporated into the cavity. The pulses are externally amplified in a home-built Er-doped fiber amplifier.

 figure: Fig. 1.

Fig. 1. (a) Er-doped soliton mode-locked fiber ring laser and DFT setup. (b) Optical spectrum recorded with an OSA (left) and retrieved from the DFT measurement (right) for a 2nd HML state.

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 figure: Fig. 2.

Fig. 2. Real-time recording of the transition dynamics from a 2nd HML to a three-pulse state. (a) Optical spectrum from an OSA. (b) DFT measurement over 880,000 cavity roundtrips. Pump power (66 mW to 78 mW, brown) and pulse dynamics (blue). (c) Close-up of the transition region of $\sim{1100}$ cavity roundtrips featuring strong amplitude fluctuations. (d) Retrieved energy of individual pulses. (e) 2D contour plot over 15,000 cavity roundtrips. (f) Close-up of new pulse evolution P2 from the small dotted area in (e). (g) 3D shot-to-shot spatial-temporal evolution from the large dashed box in (e). (h) Temporal-spectral cross sections for selective roundtrips during the new pulse formation process.

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To obtain insights into the real-time transition dynamics in the multi-pulsing regime, the spectral-temporal evolution is studied. The pulses are stretched in a $\sim{5.4}\;{\rm km}$ long SMF-28 passive fiber spool to induce a time-wavelength mapping based on the DFT. The fiber spool provides a total group-delay dispersion of $ - {177}\;{{\rm ps}^2}$, yielding a spectral resolution of 0.4 nm. The stretched pulses are coupled into a high-speed photodetector PD2 (EOT 3600, 22 GHz) and digitized with a real-time fast oscilloscope (Tektronix); see Fig. 1(a). A small fraction of the pump light is split off and coupled to PD1 (Thorlabs PDA36A), which triggers the real-time data acquisition based on the pump power ramp-up process.

For the described laser system, the real-time transition dynamics between a double pulsing state and a state with three pulses are studied. The multi-pulsing state is generated by launching $\sim{90}\;{\rm mW}$ of pump power into the WDM and properly adjusting the three PCs. For pump powers down to 62 mW, double pulsing is obtained. By further reducing the pump power, a transition can be induced from the double pulsing state to a 2nd HML state, where the pulses are equally redistributed in time due to the passive modulation of the net gain at lower pump powers [26]. This 2nd HML state is supported for pump power values up to 70 mW and marks the starting point for the studies. The left plot of Fig. 1(b) shows the optical spectrum of a 2nd HML state measured by an optical spectrum analyzer (OSA) for a coupled pump power of 65 mW. The pulses are centered at 1566.5 nm with a full-width at half-maximum (FWHM) spectral bandwidth (BW) of 6.7 nm. To validate the DFT technique, the corresponding spectrum averaged over 5000 cavity roundtrips for the same 2nd HML state [see Fig. 1(b)] agrees well with the OSA steady-state spectrum and its characteristic sidebands.

The real-time transition dynamics from a 2nd HML state (red dashed) to a three-pulse state (blue solid) are investigated; see optical spectra in Fig. 2(a). The spectrum is centered at 1567 nm with a FWHM of 7 nm for the 2nd HML state and 6.6 nm for the three-pulse state, corresponding to a transform-limited pulse duration of 368 fs and 391 fs, respectively. To capture the intricate details of the transition, a 20 ms continuous temporal recording captures more than 880,000 consecutive cavity roundtrips; see Fig. 2(b). The pump power increases linearly at an estimated rate of 1.03 mW/ms or 23 nW/cavity roundtrip, from 66 mW to 78 mW. As a result, the output power of the laser cavity increases from 1.1 mW to 1.5 mW, corresponding to a change of intracavity pulse energy from 123 pJ for the 2nd HML state to 113 pJ for the three-pulse state. A noticeable spike in the intensity is recorded when the pump power reaches a value around $\sim{69.7}\;{\rm mW}$. The zoom-in into these intensity fluctuations reveals in Fig. 2(c) that the intensity increases beyond the normalized steady-state intensity of 1 to 1.8 over a time span of $\sim{17.5}\;{\unicode{x00B5} {\rm s}}$ or $\sim{770}$ cavity roundtrips. To further investigate the transition dynamics, the continuous temporal trace is segmented into periods corresponding to the fundamental roundtrip time and stacked back to back. Thus, the individual pulses during each cavity roundtrip are visualized, and their energies are retrieved; see Fig. 2(d). P1 and P3 refer to the two originally oscillating pulses, while the new pulse P2 is generated during the increase in pump power, see Fig. 2(e). Around the roundtrip number of 350,000, the energy of the new pulse P2 rapidly increases, while the other two pulses maintain energy values close to their steady-state values. Before reaching a steady state, the energy of the three pulses is damped slightly, since the effective gain for each soliton decreases. In Fig. 2(e), an interval around the transition spanning over 340 µs (15,000 cavity roundtrips) is shown. A close-up is translated into the spectral domain in Fig. 2(f), demonstrating the birth of the new pulse in more detail. The new pulse is generated from a DW contribution that gets modulated into a narrow-band pulse, which is visible as a narrow line before P2 is formed and which does not split off from the existing oscillating pulses in the cavity. With three pulses oscillating in the cavity, the temporal spacing between P1 and P2 and P2 and P3 is 6.68 ns and 4.63 ns, respectively, while the temporal spacing between P1 and P3 is 11.31 ns and remains unchanged before and after the transition. As shown in the close-up of the new pulse formation in Fig. 2(f), P2 is spectrally broadened dramatically from a narrow pulse while undergoing a redshift of its center wavelength. In Fig. 2(g), the 3D shot-to-shot spatial-temporal evolution reveals a beating between P1 and a background pulse evolved from a DW. While this leads to a slight increase in peak intensity for P1, it does not disturb P1, and the weak pulse dies out eventually, potentially due to gain competition [7]. Depending on the transition dynamics and the evolution of additional weak background pulses in the cavity, these do not necessarily interfere with the oscillating solitons. In Fig. 2(h), five selected cross sections emphasize the evolution of the new pulse formation process in more detail. The increase of the effective gain amplifies the narrow pulse instead of the existing solitons due to peak power limiting effects [6], leading to a continuous increase of the peak intensity, while the spectral BW remains narrow. The combination of the data from Fig. 2 provides detailed insights into the transition process: due to soliton energy quantization, a background pulse is initiated from a DW that slowly grows in strength for a given pump power that undergoes a slight oscillatory behavior leading up to the transition. With more pump power available in the cavity, the gain dynamics can promote intensity fluctuations for the new pulse that can exceed the normalized intensity of the steady-state pulses. Once a certain intensity threshold is reached with a high peak intensity, a soliton starts evolving through nonlinear pulse shaping and spectral broadening by SPM. Until the soliton is completely shaped, a spike remains, as shown in the spectrum at the roundtrip number of 349,497 in Fig. 2(h). During the pulse broadening process, a redshift of the center pulse is observed, similar to Refs. [21,23] during the initiation of the mode-locking process. Balanced by the net-cavity gain and loss dynamics combined with saturable absorption, the spike disappears, and a steady-state solution is reached. This transition process spans over $\sim{562}\;{\unicode{x00B5} {\rm s}}$ or 24,800 roundtrips. While spectral modulations, including Hopf bifurcations, can be observed during some multi-pulsing transitions [6,7,9,10], for the presented mode-locked laser system no additional modulation of the existing pulses or any other chaotic phenomena associated with transition dynamics or bound solitons are observed. Depending on the initial pump power and the PC settings in the cavity and different polarization conditions, the extent of the amplitude fluctuations of the spike can vary slightly; however, the observed transitions are governed by the same phenomena and underlying mechanisms.

 figure: Fig. 3.

Fig. 3. Real-time recording of the annihilation dynamics from a 3rd HML to a two-pulse state. (a) Optical spectrum from an OSA. (b) DFT measurement: pump power (71 mW to 61 mW, brown) and respective pulse dynamics (blue) over 880,000 cavity roundtrips. (c) 2D contour plot over 12,000 cavity roundtrips during the transition. (d) 3D shot-to-shot spatial-temporal evolution. (e) Total energy evolution during the pulse decaying process. (f) Retrieved spectral FWHM of the three individual pulses show a strong breathing behavior before one pulse disappears.

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After studying the build-up of consecutive multi-pulsing states, the reverse decay process is examined. Figure 3(a) displays the corresponding optical spectrum in a steady state from the OSA, where the FWHM changes from 5.9 nm for the 3rd HML state (blue dashed) to 6.2 nm for a double pulsing state (red solid). The pump power is decreased linearly at a rate of $\sim{0.91}\;{\rm mW/ms}$ or 21 nW/cavity roundtrip, from 71 mW to 61 mW. A 20 ms long oscilloscope trace, including the shot-to-shot pulses and pump dynamics, is recorded, as shown in Fig. 3(b). During this time window, the output power of the laser decreases from 1.3 mW to 0.9 mW, corresponding to an intracavity pulse energy from 96 pJ for the 3rd HML state to 101 pJ for the double pulsing state. When the pump power reaches a value around 61.2 mW, there is not enough energy to support three pulses, leading to the decay of one pulse. Compared to the pumping threshold of $\sim{70}\;{\rm mW}$ for generating an extra soliton, a pump power hysteresis [8] is noted. In Fig. 3(c), a contour map illustrates the temporal dynamics during the pulse decaying process between roundtrips 722,000 and 734,000, corresponding to a duration of 272 µs. All three pulses undergo a similar breathing behavior until one of the pulses disappears. After annihilation of one of the pulses, the remaining two pulses settle into a steady state. While the three pulses in the 3rd HML state feature an equal spacing of $\,\sim{7.56}\;{\rm ns}$ in time, the remaining two pulses are separated by 7.54 and 15.13 ns and do not rearrange themselves for the given pump power. In Fig. 3(d), the $\sim{68}\;\unicode{x00B5} {\rm s}$ long ($\sim{3000}$ roundtrips) 3D shot-to-shot spatial-temporal evolution is shown. One of the pulses decays gradually over a period of $\sim{4.5}\;\unicode{x00B5} {\rm s}$ ($\sim{200}$ roundtrips), while the other two pulses remain relatively stable, as shown in the inset of Fig. 3(d). During this transition period, it is observed that all three pulses undergo a unique breathing behavior, as shown in the evolution of the total energy in Fig. 3(e) and the FWHM of the individual pulses; see Fig. 3(f). The total energy for a cavity roundtrip exhibits increased oscillations over 227 µs (more than 10,000 cavity roundtrips) with a maximum total energy fluctuation of 16.7% compared to the three-pulse steady state. As revealed in Fig. 3(f), the FWHM exhibits similar simultaneous oscillating features that match the breathing effect observed in Fig. 3(c). Each of the three pulses is characterized by a FWHM of 4.5 nm in the 3rd HML state before the transition, and the FWHM varies between 3.4 nm and 5.4 nm. After one of the pulses decays, the spectral FWHM of the remaining two pulses increases to 6.1 nm in a steady state. The energy fluctuations capture changes in peak intensity, and the varying strengths of nonlinear contributions leading to spectral changes, and are directly correlated with the oscillations in the optical spectrum. We believe that this phenomenon can be associated with instabilities at a bifurcation, as well as gain and recovery dynamics. Close to the soliton formation threshold energy, gain variations and relaxation oscillations can lead to transient higher peak intensities which are coupled with stronger nonlinear contributions and an increase in spectral BW. Additional theoretical studies can potentially shed more insights into this evolution. The chosen triggering scheme and the recording of pumping dynamics for both transitions allows a direct mapping of the pump strength with the pulse evolution dynamics. This can enable a close comparison with theoretical results, since any changes in gain and population inversion can trigger these different dynamics during the transition.

In summary, the real-time dynamics of soliton formation and annihilation between two and three pulse states have been studied. The described results and underlying transition mechanisms are representative of other consecutive multi-pulsing transitions, e.g., between the fundamentally mode-locked state and a double pulsing state, and hold true for even and odd numbered multi-pulsing operation. In the presented mode-locked laser system, an additional soliton is formed through shaping of a narrow-band pulse arising from a DW, thus differing from pulse splitting processes that originated from a single pulse [24]. Strong intensity fluctuations occur only in the new pulse, rather than in all oscillating pulses in the cavity, which is observed and characterized for the first time, to the best of our knowledge. During the annihilation process, the pulses in the cavity undergo a unique simultaneous breathing process, which is reflected in energy fluctuations, as well as oscillations in the spectral BW. These findings offer important insights into the multi-pulsing soliton dynamics that are coupled with the observation of new phenomena during the transition, which are crucial for laser design and optimization. This research can further advance the modeling and design of the associated laser stability for the various operating regimes.

Funding

National Science Foundation (ECCS-1710849).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. A. B. Grudinin and S. Gray, J. Opt. Soc. Am. B 14, 144 (1997). [CrossRef]  

2. J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B 16, 674 (1999). [CrossRef]  

3. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005). [CrossRef]  

4. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998). [CrossRef]  

5. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, Opt. Lett. 20, 907 (1995). [CrossRef]  

6. D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005). [CrossRef]  

7. W. S. Man, H. Y. Tam, M. S. Demokan, and D. Y. Tang, Opt. Quantum Electron. 33, 1139 (2001). [CrossRef]  

8. A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 71, 053809 (2005). [CrossRef]  

9. J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008). [CrossRef]  

10. B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, Opt. Express 17, 23137 (2009). [CrossRef]  

11. K. Goda and B. Jalali, Nat. Photonics 7, 102 (2013). [CrossRef]  

12. M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016). [CrossRef]  

13. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, Optica 2, 36 (2015). [CrossRef]  

14. Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019). [CrossRef]  

15. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, Science 356, 50 (2017). [CrossRef]  

16. O. S. Torres-Muñoz, O. Pottiez, Y. Bracamontes-Rodriguez, J. P. Lauterio-Cruz, H. E. Ibarra-Villalon, J. C. Hernandez-Garcia, M. Bello-Jimenez, and E. A. Kuzin, Opt. Express 27, 17521 (2019). [CrossRef]  

17. J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and H. Zeng, Commun. Phys. 1, 20 (2018). [CrossRef]  

18. H.-J. Chen, M. Liu, J. Yao, S. Hu, J.-B. He, A.-P. Luo, W.-C. Xu, and Z.-C. Luo, Opt. Express 26, 2972 (2018). [CrossRef]  

19. X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018). [CrossRef]  

20. D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019). [CrossRef]  

21. G. Herink, B. Jalali, C. Ropers, and D. R. Solli, Nat. Photonics 10, 321 (2016). [CrossRef]  

22. X. Liu and Y. Cui, Adv. Photonics 1, 1 (2019). [CrossRef]  

23. Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017). [CrossRef]  

24. X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019). [CrossRef]  

25. G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018). [CrossRef]  

26. A. Komarov, H. Leblond, and F. Sanchez, Opt. Commun. 267, 162 (2006). [CrossRef]  

References

  • View by:

  1. A. B. Grudinin and S. Gray, J. Opt. Soc. Am. B 14, 144 (1997).
    [Crossref]
  2. J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B 16, 674 (1999).
    [Crossref]
  3. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
    [Crossref]
  4. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
    [Crossref]
  5. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, Opt. Lett. 20, 907 (1995).
    [Crossref]
  6. D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
    [Crossref]
  7. W. S. Man, H. Y. Tam, M. S. Demokan, and D. Y. Tang, Opt. Quantum Electron. 33, 1139 (2001).
    [Crossref]
  8. A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 71, 053809 (2005).
    [Crossref]
  9. J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008).
    [Crossref]
  10. B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, Opt. Express 17, 23137 (2009).
    [Crossref]
  11. K. Goda and B. Jalali, Nat. Photonics 7, 102 (2013).
    [Crossref]
  12. M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
    [Crossref]
  13. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, Optica 2, 36 (2015).
    [Crossref]
  14. Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
    [Crossref]
  15. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, Science 356, 50 (2017).
    [Crossref]
  16. O. S. Torres-Muñoz, O. Pottiez, Y. Bracamontes-Rodriguez, J. P. Lauterio-Cruz, H. E. Ibarra-Villalon, J. C. Hernandez-Garcia, M. Bello-Jimenez, and E. A. Kuzin, Opt. Express 27, 17521 (2019).
    [Crossref]
  17. J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and H. Zeng, Commun. Phys. 1, 20 (2018).
    [Crossref]
  18. H.-J. Chen, M. Liu, J. Yao, S. Hu, J.-B. He, A.-P. Luo, W.-C. Xu, and Z.-C. Luo, Opt. Express 26, 2972 (2018).
    [Crossref]
  19. X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).
    [Crossref]
  20. D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
    [Crossref]
  21. G. Herink, B. Jalali, C. Ropers, and D. R. Solli, Nat. Photonics 10, 321 (2016).
    [Crossref]
  22. X. Liu and Y. Cui, Adv. Photonics 1, 1 (2019).
    [Crossref]
  23. Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
    [Crossref]
  24. X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019).
    [Crossref]
  25. G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
    [Crossref]
  26. A. Komarov, H. Leblond, and F. Sanchez, Opt. Commun. 267, 162 (2006).
    [Crossref]

2019 (5)

O. S. Torres-Muñoz, O. Pottiez, Y. Bracamontes-Rodriguez, J. P. Lauterio-Cruz, H. E. Ibarra-Villalon, J. C. Hernandez-Garcia, M. Bello-Jimenez, and E. A. Kuzin, Opt. Express 27, 17521 (2019).
[Crossref]

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
[Crossref]

D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
[Crossref]

X. Liu and Y. Cui, Adv. Photonics 1, 1 (2019).
[Crossref]

X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019).
[Crossref]

2018 (4)

G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
[Crossref]

J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and H. Zeng, Commun. Phys. 1, 20 (2018).
[Crossref]

H.-J. Chen, M. Liu, J. Yao, S. Hu, J.-B. He, A.-P. Luo, W.-C. Xu, and Z.-C. Luo, Opt. Express 26, 2972 (2018).
[Crossref]

X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).
[Crossref]

2017 (2)

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
[Crossref]

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, Science 356, 50 (2017).
[Crossref]

2016 (2)

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, Nat. Photonics 10, 321 (2016).
[Crossref]

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
[Crossref]

2015 (1)

2013 (1)

K. Goda and B. Jalali, Nat. Photonics 7, 102 (2013).
[Crossref]

2009 (1)

2008 (1)

2006 (1)

A. Komarov, H. Leblond, and F. Sanchez, Opt. Commun. 267, 162 (2006).
[Crossref]

2005 (3)

A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 71, 053809 (2005).
[Crossref]

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
[Crossref]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

2001 (1)

W. S. Man, H. Y. Tam, M. S. Demokan, and D. Y. Tang, Opt. Quantum Electron. 33, 1139 (2001).
[Crossref]

1999 (1)

1998 (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

1997 (1)

1995 (1)

Akhmediev, N. N.

Bale, B. G.

Bello-Jimenez, M.

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

Billet, C.

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D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
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G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
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D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
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Liu, X.

X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019).
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X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).
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Luo, Z.-C.

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W. S. Man, H. Y. Tam, M. S. Demokan, and D. Y. Tang, Opt. Quantum Electron. 33, 1139 (2001).
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Merolla, J.-M.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
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X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019).
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D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
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M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
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D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
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Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
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M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
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Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
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Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
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M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
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D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
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Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
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G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
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Yao, X.

X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).
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Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
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G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
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J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and H. Zeng, Commun. Phys. 1, 20 (2018).
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D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
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D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
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D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
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D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
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Adv. Photonics (1)

X. Liu and Y. Cui, Adv. Photonics 1, 1 (2019).
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Appl. Phys. Lett. (1)

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, Appl. Phys. Lett. 110, 201107 (2017).
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Commun. Phys. (1)

J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and H. Zeng, Commun. Phys. 1, 20 (2018).
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IEEE J. Quantum Electron. (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
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Infrared Phys. Technol. (1)

D. Han, Z. Hui, J. Xie, K. Ren, J. Gong, F. Zhao, J. Dong, D. Li, and X. Xin, Infrared Phys. Technol. 102, 102984 (2019).
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J. Opt. Soc. Am. B (2)

Laser Photonics Rev. (1)

X. Liu and M. Pang, Laser Photonics Rev. 13, 1800333 (2019).
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Nat. Commun. (2)

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, Nat. Commun. 7, 13675 (2016).
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Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, Nat. Commun. 10, 830 (2019).
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Nat. Photonics (2)

K. Goda and B. Jalali, Nat. Photonics 7, 102 (2013).
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G. Herink, B. Jalali, C. Ropers, and D. R. Solli, Nat. Photonics 10, 321 (2016).
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Opt. Commun. (1)

A. Komarov, H. Leblond, and F. Sanchez, Opt. Commun. 267, 162 (2006).
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Opt. Express (4)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

W. S. Man, H. Y. Tam, M. S. Demokan, and D. Y. Tang, Opt. Quantum Electron. 33, 1139 (2001).
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Optica (1)

Photonics Res. (1)

G. Wang, G. Chen, W. Li, C. Zeng, and H. Yang, Photonics Res. 6, 825 (2018).
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Phys. Rev. A (2)

A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 71, 053809 (2005).
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D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, Phys. Rev. A 72, 043816 (2005).
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Phys. Rev. E (1)

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

Phys. Rev. Lett. (1)

X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).
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Science (1)

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, Science 356, 50 (2017).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. (a) Er-doped soliton mode-locked fiber ring laser and DFT setup. (b) Optical spectrum recorded with an OSA (left) and retrieved from the DFT measurement (right) for a 2nd HML state.
Fig. 2.
Fig. 2. Real-time recording of the transition dynamics from a 2nd HML to a three-pulse state. (a) Optical spectrum from an OSA. (b) DFT measurement over 880,000 cavity roundtrips. Pump power (66 mW to 78 mW, brown) and pulse dynamics (blue). (c) Close-up of the transition region of $\sim{1100}$ cavity roundtrips featuring strong amplitude fluctuations. (d) Retrieved energy of individual pulses. (e) 2D contour plot over 15,000 cavity roundtrips. (f) Close-up of new pulse evolution P2 from the small dotted area in (e). (g) 3D shot-to-shot spatial-temporal evolution from the large dashed box in (e). (h) Temporal-spectral cross sections for selective roundtrips during the new pulse formation process.
Fig. 3.
Fig. 3. Real-time recording of the annihilation dynamics from a 3rd HML to a two-pulse state. (a) Optical spectrum from an OSA. (b) DFT measurement: pump power (71 mW to 61 mW, brown) and respective pulse dynamics (blue) over 880,000 cavity roundtrips. (c) 2D contour plot over 12,000 cavity roundtrips during the transition. (d) 3D shot-to-shot spatial-temporal evolution. (e) Total energy evolution during the pulse decaying process. (f) Retrieved spectral FWHM of the three individual pulses show a strong breathing behavior before one pulse disappears.

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