## Abstract

Dissipative soliton (DS) evolution in passively mode-locked fiber lasers with ultra-large net-normal-dispersion (as large as 1 ps^{2}) is investigated. The proposed DS laser operates on three statuses with respect to the pump power. The DS laser works on a status that is similar to an all-normal-dispersion laser when the pump power is low, whereas it creates a new type of pulses exhibited as the trapezoid-spectrum profile when the pump power is large. The laser cavity emits the unstable pulses between the above two statuses. The cubic–quintic Ginzburg–Landau equation can serve to qualitatively explain our experimental observations.

©2009 Optical Society of America

## 1. Introduction

Passively mode-locked fiber lasers have been extensively investigated in the past two decades both experimentally and theoretically because they are able to easily produce self-starting ultra-short pulses [1-5]. Ultra-short light pulses can help us directly observe some of the fastest processes in nature, along with studies of matter under extreme conditions. The utilization of saturable absorbers is the first method proposed for passive mode-locking. The nonlinear polarization rotation technique, one of the artificial saturable absorber methods, is a powerful tool for creating ultra-short pulses [6].

The group-velocity dispersion (GVD) of fiber laser cavity is a key parameter for achieving different ultra-short pulses [7]. When a fiber laser is made of fibers with purely anomalous GVD, the fiber dispersion (i.e., GVD) balances the fiber nonlinearity (i.e., self-phase modulation) to produce conventional soliton-like pulses that are nearly bandwidth limited. Most modern ultra-short pulse cavity has dispersion maps in which a fiber laser can operate either in the positive or negative cavity dispersion regime. As the net GVD of the cavity is anomalous enough, soliton-like pulses can be formed as a result of the balance between anomalous GVD and positive nonlinearity, similar to the case of purely anomalous GVD. When the net GVD approaches zero, the so-called stretched-pulse fiber laser is operating with the breathing solutions (i.e., dispersion-managed solitons) [8-11]. Recently, researchers have investigated that the mode-locked laser operations with the net normal GVD can achieve self-similar parabolic pulses [12-15], and its spectrum has an approximately parabolic top and steep sides. When the laser cavity contains components without anomalous GVD, this all-normal-dispersion laser exhibits a kind of solitons that have steep spectral edges [16-19].

Pulses that can exist at large net cavity GVD rely on the dissipative processes and, therefore, can be considered as dissipative solitons (DSs) [2]. The gain and loss coexist in the dissipative system and play an essential role in the formation of DSs [20-24]. As the energy of dissipative solutions has to be dissipated in the medium, they exist only if there is a continuous energy supply from an external source [23]. This attribute becomes one of their most prominent features. The balance between the energy being supplied and lost has to be exact in the system. Thus, the DSs must be self-organized and consequently the dynamics of DSs is different from that of conventional solitons. For instance, the physical parameters (e.g., amplitude, shape, width, etc.) of DSs drastically depend on the system parameters and are unique for the given external conditions [21, 23].

Usually the nonlinear Schrödinger (NLS) equation [25-28], which is derived from a high-frequency asymptotic expansion of Maxwell’s equations by using some approximations, can govern the underlying wave behavior in an optical fiber laser cavity. The master mode-locking model being a perturbation from the NLS is more general model and has the capacity of describing both the energy saturation and the pulse stabilization process. So the complex cubic Ginzburg-Landau equation (sometimes referred to as the “master equation” [2, 29]) is a powerful tool for the mode-locked laser systems [30-35]. Unfortunately, the master mode-locking model fails to capture the robust nature of the experimentally observed mode locking [31]. By introducing a quintic loss term, the quintic master mode-locking equation can overcome the above problems. The quintic loss term helps stabilize solutions to the governing equation by controlling the growth of the nonlinear gain term [31].

The passively mode-locked fiber lasers with the net normal GVD of less than 0.2 ps^{2} are investigated extensively. When the net cavity GVD in dispersion maps is as large as 1 ps^{2}, what happens in the laser system? The current work answers this question. Three different operation statuses are observed in the proposed laser. A kink of trapezoid-spectrum pulses is reported for the first time to the author’s best knowledge.

## 2. Design of dissipative-soliton laser

It is generally useful to classify mode-locking regimes according to dispersion maps, with segments of normal and anomalous GVD. Pulse evolutions in conventional-soliton, stretched-pulse, self-similar, and all-normal-dispersion lasers are qualitatively distinct from each other. They are controlled by the distinctive types of soliton-shaping mechanisms in mode-locked fiber lasers. However, when the net cavity GVD βnet approaches zero [Fig. 1(a)], dispersion-managed soliton and conventional soliton coexist in the same mode-locked fiber laser [25]. With fixed and moderate normal GVD (e.g., β_{net}≈0.01 ps^{2}) [Fig. 1(b)], both stretched-pulse and self-similar operation can be observed [2]. When the total positive dispersion is extremely larger than the total negative dispersion, the net cavity GVD β_{net} in the dispersion map is ultra-large (e.g., β_{net}≈1 ps^{2}) [Fig. 1(c)]. So the designed laser cavity as shown in Fig. 1(c) can be approximately similar to the cavity configuration of all-normal-dispersion laser. On the other hand, the proposed laser cavity (Fig. 1(c)) has the dispersion maps and its net cavity GVD is positive, roughly similar to the stretched-pulse and self-similar lasers. The cavity configuration of the proposed laser is then expected to produce two distinctive types of pulses at the same cavity. The experimental configuration and results are shown in next section in detail.

## 3. Experimental results

The fiber laser oscillator for DSs is shown schematically in Fig. 2. It consists of a 20-m-long erbium-doped fiber with absorption 6 dB/m at 980 nm, a polarization-sensitive isolator (PS-ISO), two sets of polarization controllers (PCs), a fused coupler with 10% output, and a wavelength-division-multiplexed (WDM) coupler. The polarization-sensitive isolator provides unidirectional operation and polarization selectivity in a ring-cavity configuration. The erbium-doped fiber has a dispersion parameter of about -42 ps/nm/km (corresponding to GVD *β*
_{2}=0.0535 ps^{2}/m) at 1550 nm and other fibers in cavity are the standard single-mode fiber (SMF), which have a dispersion parameter of about 17 ps/nm/km (corresponding to *β*
_{2}=-0.0217 ps^{2}/m). The total length of cavity is 23.8 m, including the SMF of 3.8 m. The net cavity GVD β_{net} is then ~1 ps^{2}. The 977-nm laser diode (LD) can provide the pump power of up to 500 mW. An autocorrelator, an optical spectrum analyzer (ANDO AQ-6315A), and an 11-GHz oscilloscope (LeCroy SDA) together with a 12-GHz photodetector are used to simultaneously monitor the laser output.

Obviously the proposed laser cavity has very large net-normal GVD, whose value is several times larger than that of the typical all-normal-dispersion laser cavity [16, 19] and is about two orders of magnitude larger than that of the self-similar pulse laser cavity [12]. On the other hand, like the configuration of the self-similar pulse system, our cavity has dispersion maps with the total anomalous GVD of -0.082 ps^{2}. The proposed laser cavity has particular dispersion configuration and therefore creates the unusual results. The experimental results are shown as follows.

The nonlinear polarization rotation (NPR) technique is used to achieve self-started mode locking of DS laser. By appropriately adjusting two polarization controllers, self-started mode locking of the laser is achieved at the threshold pump power of ~70 mW. After mode locking, the laser cavity emits stable pulses with the fundamental cavity repetition rate. Figures 3 and 4 show an example for the pump power *P _{p}*=71 mW. The optical spectrum of the pulses, the corresponding oscilloscope and autocorrelation traces are shown in Figs. 3(a), 3(b) and 4, respectively. Figure 3(b) shows that the fundamental cavity repetition rate of the proposed laser is 8.72 MHz. Figure 3(a) has the typical spectrum characteristics of the all-normal-dispersion pulses, i.e., the steep spectral edges and the sharp peaks on the edges of the spectrum [2, 16]. The 3-dB spectral bandwidth of solitons is about 16.8 nm and the autocorrelation trace has a full width at half maximum (FWHM) of about 19.5 ps. If a Gaussian pulse profile is assumed, the pulse width is about 13.8 ps, which gives a time–bandwidth product of ~28.8. Therefore the pulse is strongly chirped.

When the pump power *P _{p}* is increased from 71 mW to 121 mW, the power spectra of laser approximately keep the same profile except that the spectrum bandwidth and the pulse duration broaden. These results are very different from the typical feature of the solitons in the purely normal dispersion laser [16], in which the top of the spectral profile evolves from a rectangular shape to a parabolic shape when the pump power increases. In the case of

*P*=120 mW, the optical spectrum and the corresponding autocorrelation trace are shown in Fig. 5(a) and Fig. 4, respectively. The 3-dB spectral bandwidth and the pulse duration are about 20.6 nm and 14.9 ps, respectively, and the corresponding time–bandwidth product is ~38. This value is much larger than the time–bandwidth product of the self-similar pulse laser [12]. Because of weaker dispersion map, the proposed laser cavity can generate highly chirped pulses and the large frequency chirp is enhanced with the increase of

_{p}*P*. Moreover the pedestal of power spectrum at the wavelength range of

_{p}*λ*>1566 nm and

*λ*<1548.2 nm in Fig. 3(a) and

*λ*>1568 nm and

*λ*<1546.1 nm in Fig. 5(a) originates from the amplified spontaneous emission (ASE) of erbium-doped fiber, respectively.

When *P _{p}* is slightly increased from 121 mW (e.g., 121.5 mW), the pulse and spectrum become very unstable. Figures 4 and 5(b) demonstrate the typical results of the autocorrelation trace and optical spectrum at

*P*=122 mW. The experiments show that the top and edges of the spectrum have unstable variation. Especially, the edges of the spectrum have strong fluctuations (e.g., Fig. 5(b)). In this stage, the oscilloscope and autocorrelation traces are extremely unstable.

_{p}While the pump power *P _{p}* is enhanced to ~123 mW, the spectral profile and pulse traces are quite stable again. An experimental example for

*P*=130 mW is shown in Figs. 4 and 5(c). These results exhibit that the pulse width broadens to 19.7 ps although the 3-dB spectrum bandwidth of pulses extends a little. The corresponding time–bandwidth product is increased to 50 and hence the pulse chirp is enhanced further. We can observe from Fig. 5(c) that (1) there are no sharp peaks on the spectrum edges (this result is different from Figs. 3(a) and 5(a)), (2) the top of the spectral profile is similar to a parabolic shape (this result is roughly the same as the optical spectrum of the parabolic pulse laser), (3) the spectrum edges significantly broaden at the pedestal of power spectrum, and (4) the spectral profile is similar to the trapezoid-spectrum shape rather than the rectangle-spectrum shape. Although our laser cavity operates in the net-normal GVD regime and the spectra exhibit some similarities to those of the parabolic pulse lasers, however, our laser cavity produces a new type of pulses.

_{p}With further increasing the pump power, the proposed fiber laser stably emits the pulses and the optical spectra are similar to Fig. 5(c). At the same time, the pulse duration and strength are increased. In our experiments, the proposed laser can stably evolve until *P _{p}*≈300 mW. Then our laser is unstable for

*P*>300 mW.

_{p}## 4. Theoretical analysis

By using the Ginzburg-Landau equation, Tang *et al*. numerically and experimentally proved that the two different types of solitons (i.e., dispersion-managed soliton and conventional soliton) coexist in a mode-locked fiber laser when the net cavity GVD approaches zero [25]. Kalashnikov *et al*. proved that the solutions with three types of pulse spectra (parabolic-like, Π- and M-like) are covered in the distributed complex cubic–quintic Ginzburg–Landau equation [34]. To describe optical pulse evolution in the dissipative systems, the quintic master mode-locking equation with cubic and quintic saturable absorber terms is given by [31]

where *u* denotes the electric field envelope normalized by the peak field power, *D* represents the average laser-cavity dispersion, the variable *t* and *z* indicate the normalized time and distance respectively, *δ* is the fiber loss, the parameter *τ* controls the spectral gain bandwidth of the mode-locking process whose value varies from 0.08 to 0.32, *β* characterizes the strength of the nonlinear (cubic) loss–gain, *γ* refers to the cubic refractive nonlinearity of the medium, *µ* measures the strength of the quintic attenuation, and g describes the saturated gain behavior.

Soto-Crespo *et al*. proved that the pulselike solutions of complex Ginzburg-Landau equation are in general unstable [36]. For the cubic–quintic Ginzburg–Landau equation, they found a region in the space of the system parameters where the pulselike solutions are stable [36, 37]. By using a variational method to solve Eq. (1) [31], Bale *et al*. found a stable node that is the attracting state for the laser. The achieved physical parameters are *D*=-0.4, *γ*=1, *τ*=0.2, *δ*=1, *g*
_{0}=1.5, *β*=0.7, and *µ*=0.2. Their results showed the agreement between the pulse parameters from the full numerical simulation of Eq. (1) with those obtained from the variational method. By using an ansatz, Eq. (1) admits exact solutions of the form [31]

Here *A, B, η, ω* and *ϕ* are real constants, and *B* is essential in capturing the observed spectral profiles. Bale *et al*. proved that the different spectral structures can be generated as a function of the ansatz parameter *B* [31]. With the ansatz parameter *B*>1, Renninger *et al*. proved that stable solutions to the cubic–quintic Ginzburg–Landau equation exist [4]. Their experimental observation confirmed the theoretical predictions.

According to the reports on [36, 37], without previous knowledge of at least some of the regions of soliton existence, the search could have taken years of numerical simulations. In this paper, therefore, we use the theoretical predictions obtained from Refs. [4, 31] to explain our experimental observations. According to the theoretical results obtained by Bale and Renninger, our experimental results as shown in Fig. 5 are in quantitative agreement with the theoretical predictions (e.g., Fig. 7 in Ref. [31] and Fig. 1 in Ref. [4]).

## 5. Discussion

The net cavity GVD β_{net} in the proposed laser, the self-similar pulse laser [12], the all-normal-dispersion laser [19], and the dissipative soliton laser [16] is about 1 ps^{2}, 0.007 ps^{2}, 0.17 ps^{2}, and 0.2 ps^{2}, respectively. So, β_{net} of our laser cavity is about 143, 6, and 5 times larger than that of the later three laser cavities, respectively. The experimental configuration and parameters of our cavity are greatly different from those of the self-similar and all-normal-dispersion laser cavity. As a consequence, the proposed net-normal-dispersion fiber laser constitutes a new type of pulse shaping in mode-locked lasers, and it emits the pulses that are completely different from the other net-normal-dispersion lasers (e.g., parabolic pulses) or purely normal dispersion lasers. For instance, the time-bandwidth product in our laser can be more than 50, which is far larger than that in the other net- or all-normal-dispersion lasers [12, 16]. The proposed laser with extremely large positive-GVD and slight negative-GVD would presumably have to exploit dissipative processes to a greater degree in the steady-state pulse-shaping. Actually, experimental observations of DSs are rare [2]. Our laser provides an experimental setting for the study of this class of solitary wave, which currently attracts much attention (albeit mostly theoretical) in the nonlinear-waves community [21].

## 6. Conclusions

Passively mode-locked fiber lasers with ultra-large net-normal GVD (as large as 1 ps^{2}) are investigated both theoretically and experimentally. The net cavity GVD in our laser is two orders of magnitude larger than that in the self-similar laser cavity although both cavities are constituted by dispersion maps. Our fiber laser emits DSs and works on three statuses as a function of the pump power *P _{p}*. The DS laser cavity operates on a status, similar to the all-normal-dispersion lasers, when

*P*is low (

_{p}*P*≈70-121 mW). However when

_{p}*P*is large (

_{p}*P*≈123-300 mW), the cavity creates a new type of pulse exhibited as the trapezoid-spectrum profile, which to the best of our knowledge has not been observed in any experimental setting. The spectral characteristics of this new kind of pulse are qualitatively distinct from those of the conventional-soliton pulse, stretched-pulse, self-similar pulse, and all-normal-dispersion pulse. Between the above-mentioned two statuses, the cavity emits the unstable pulses for

_{p}*P*≈121-123 mW. The theoretical predictions (e.g., Refs. [4, 31]) from the cubic–quintic Ginzburg–Landau equation can qualitatively explain our experimental observations.

_{p}## Acknowledgments

The author would especially like to thank Leiran Wang, Yongkang Gong, and Xiaohong Hu for help with the experiments. The author would like to thank Dr. Asifullah Khan for editing and improving the English of this paper. This work was supported by the “Hundreds of Talents Programs” of the Chinese Academy of Sciences and by the National Natural Science Foundation of China under Grants 10874239 and 10604066.

## References and links

**1. **M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A **77**, 033828 (2008).
[CrossRef]

**2. **F. W. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. **2**, 58–73 (2008).
[CrossRef]

**3. **X. Liu, T. Wang, C. Shu, L. Wang, A. Lin, K. Lu, T. Zhang, and W. Zhao, “Passively harmonic mode-locked erbium-doped fiber soliton laser with a nonlinear polarization rotation,” Laser Phys. **18**, 1357–1361 (2008).
[CrossRef]

**4. **W. H. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A **77**, 023814 (2008).
[CrossRef]

**5. **R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, “Statistical light-mode dynamics of multipulse passive mode locking,” Phys. Rev. E **76**, 031112 (2007).
[CrossRef]

**6. **V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self- starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. **28**, 1391–1393 (1992).
[CrossRef]

**7. **Y. Logvin and H. Anis, “Similariton pulse instability in mode-locked Yb-doped fiber laser in the vicinity of zero cavity dispersion,” Opt. Express **15**, 13607–13612 (2007)
[CrossRef] [PubMed]

**8. **K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. **18**, 1080–1082 (1993).
[CrossRef] [PubMed]

**9. **M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express **17**, 405–418 (2009).
[CrossRef] [PubMed]

**10. **F. Haxsen, A. Ruehl, M. Engelbrecht, D. Wandt, U. Morgner, and D. Kracht, “Stretched-pulse operation of a thulium-doped fiber laser,” Opt. Express **16**, 20471–20476 (2008).
[CrossRef] [PubMed]

**11. **O. Y. Schwartz and S. K. Turitsyn, “Multiple-period dispersion-managed solitons,” Phys. Rev. A **76**, 043819 (2007).
[CrossRef]

**12. **F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. **92**, 213902 (2004).
[CrossRef] [PubMed]

**13. **B. Ortaç, A. Hideur, M. Brunel, C. Chédot, J. Limpert, A. Tünnermann, and F. Ö. Ilday, “Generation of parabolic bound pulses from a Yb-fiber laser,” Opt. Express **14**, 6075–6083 (2006).
[CrossRef] [PubMed]

**14. **C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express **15**, 15824–15835 (2007).
[CrossRef] [PubMed]

**15. **D. Ruehl, U. Wandt, D. Morgner, and Kracht, “Normal dispersive ultrafast fiber oscillators,” IEEE J. Sel. Top. Quantum Electro. **15**, 170–181 (2009).
[CrossRef]

**16. **L. M. Zhao, D. Y. Tang, and J. Wu, “Gain-guided soliton in a positive group-dispersion fiber laser,” Opt. Lett. **31**, 1778–1790 (2006).
[CrossRef]

**17. **M. Schultz, H. Karow, O. Prochnow, D. Wandt, U. Morgner, and D. Kracht, “All-fiber ytterbium femtosecond laser without dispersion compensation,” Opt. Express **16**, 19562–19567 (2008)
[CrossRef] [PubMed]

**18. **C. Lecaplain, A. Chédot, B. Hideur, J. Ortaç, and Limpert, “High-power all-normal-dispersion femtosecond pulse generation from an Yb-doped large-mode-area microstructure fiber laser,” Opt.Lett. **32**, 2738–2740 (2007).
[CrossRef] [PubMed]

**19. **W. H. Chong, F. W. Renninger, and Wise, “Environmentally stable all-normal-dispersion femtosecond fiber laser,” Opt. Lett. **33**, 1071–1073 (2008).
[CrossRef] [PubMed]

**20. **F. Komarov and Sanchez, “Structural dissipative solitons in passive mode-locked fiber lasers,” Phys. Rev. E **77**, 066201 (2008).
[CrossRef]

**21. **N. Akhmediev and A. Ankiewics, Eds. *Dissipative Solitons* (Springer, Berlin, 2005).
[CrossRef]

**22. **B. Cabasse, G. Ortaç, A. Martel, J. Hideur, and Limpert, “Dissipative solitons in a passively mode-locked Er-doped fiber with strong normal dispersion,” Opt. Express **16**, 19322–19329 (2008).
[CrossRef]

**23. **W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A **78**, 023830 (2008).
[CrossRef]

**24. **D. Mihalache, F. Mazilu, H. Lederer, B. A. Leblond, and Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A **75**, 033811 (2007).
[CrossRef]

**25. **Y. Tang, L. M. Zhao, G. Q. Xie, and L. J. Qian, “Coexistence and competition between different soliton-shaping mechanisms in a laser,” Phys. Rev. A **75**, 063810 (2007).
[CrossRef]

**26. **G. P. Agrawal, *Nonlinear Fiber Optics*, 4th ed. (Burlington, MA, 2006).

**27. **J. N. Kutz, “Mode-Locked Soliton Lasers,” SIAM Rev. **48**, 629–678 (2006).
[CrossRef]

**28. **M. E. Fermann, V. I. Kruglov, B.C. Thomsen, J.M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. **84**, 6010 (2000).
[CrossRef] [PubMed]

**29. **H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B **8**, 2068–2076 (1991).
[CrossRef]

**30. **J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E **70**, 066612 (2004).
[CrossRef]

**31. **G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B **25**, 1193–1202 (2008).
[CrossRef]

**32. **V. L. Kalashnikov, A. Fernández, and A. Apolonski, “High-order dispersion in chirped-pulse oscillators,” Opt. Express **16**, 4206–4216 (2008).
[CrossRef] [PubMed]

**33. **P. Bélanger, “Stable operation of mode-locked fiber lasers: similariton regime,” Opt. Express **15**, 11033–11041 (2007).
[CrossRef] [PubMed]

**34. **V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. **7**, 217 (2005).
[CrossRef]

**35. **W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg- Landau equations,” Physica D **56**, 303–367 (1992).
[CrossRef]

**36. **J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E **55**, 4783–4796 (1997).
[CrossRef]

**37. **N. Akhmediev, J.M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A **372**, 3124–3128 (2008).
[CrossRef]