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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11776–11781
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Rains of solitons in a fiber laser

Souad Chouli and Philippe Grelu  »View Author Affiliations


Optics Express, Vol. 17, Issue 14, pp. 11776-11781 (2009)
http://dx.doi.org/10.1364/OE.17.011776


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Abstract

We report a novel and intriguing nonlinear dynamics observed in a fiber laser cavity, in which soliton pulses are created from an extended noisy background and drift until they reach a condensed phase comprising several tens of aggregated solitons. This soliton flow can be adjusted with manual cavity tuning, and can even be triggered by the injection of an external low-power cw laser.

© 2009 OSA

1. Introduction

Mode-locked fiber lasers are ideal tools for the exploration of new areas of soliton nonlinear dynamics. Even under moderate pumping power, significant interplays between dispersive and dissipative physical effects, such as Kerr nonlinearity, chromatic dispersion, linear and saturable losses, and bandwidth limited gain, can be experienced. In numerous situations, the concept of a dissipative soliton has been applied successfully to interpret dynamics that are otherwise unusual in the frame of soliton dynamics in conservative systems [1

1. Ph. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations,” in Dissipative solitons: from optics to biology and medicine, N. Akhmediev and A. Ankiewicz, eds. (Springer-Verlag, Berlin, 2008). [CrossRef]

]. For instance, long-period soliton pulsations [2

2. J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(6), 066612 (2004). [CrossRef]

], soliton collisions [3

3. Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12(14), 3184–3189 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184. [CrossRef]

,4

4. M. Olivier, V. Piché, M. Roy, and F. Babin, “Pulse collisions in the stretched-pulse fiber laser,” Opt. Lett. 29(13), 1461–1463 (2004). [CrossRef]

] and vibrations of soliton pairs [5

5. M. Grapinet and Ph. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006). [CrossRef]

,6

6. J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1), 016613 (2007). [CrossRef]

] were highlighted. An important feature for dissipative solitons is their trend to aggregate into stable complexes, or “soliton molecules” [1

1. Ph. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations,” in Dissipative solitons: from optics to biology and medicine, N. Akhmediev and A. Ankiewicz, eds. (Springer-Verlag, Berlin, 2008). [CrossRef]

]. The larger the number of interacting solitons, the less stable soliton molecules are in general [7

7. Ph. Grelu and J. M. Soto-Crespo, “Multisoliton states and pulse fragmentation in a passively mode-locked fibre laser,” J. Opt. B. 6, S271–S278 (2004). [CrossRef]

], although in specific cases, large and stable complexes akin to “one-dimensional soliton crystals” can be obtained [8

8. A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Analysis of soliton pattern formation in passively mode-locked fiber lasers,” Phys. Rev. A 78(4), 043806 (2008). [CrossRef]

].

With several tens to hundreds of solitons inside the laser cavity, complex collective behaviors can also manifest, such as the dynamics revealed in the present paper. We have found these dynamics in a range of cavity settings where soliton pulses coexist with a relatively large amount of quasi-cw background in the cavity. This situation corresponds to a weakly mode locked regime, i.e. a regime where low-intensity waves are not as efficiently filtered out as they are in usual mode locking. The existence of a cw component that mediates interactions between solitons can strongly affect the overall dynamics [9

9. S. Wabnitz, “Control of soliton train transmission, storage, and clock recovery by cw light injection,” J. Opt. Soc. Am. B 13(12), 2739–2749 (1996). [CrossRef]

12

12. A. Komarov, K. Komarov, H. Leblond, and F. Sanchez, “Spectral-selective management of dissipative solitons in passive mode-locked fibre lasers,” J. Opt. A, Pure Appl. Opt. 9(12), 1149–1156 (2007). [CrossRef]

]. In the present case, a large number of quasi-cw components produce a noisy background from which dissipative solitons can be formed continuously in the fiber laser cavity. As soon as they are formed, these solitons drift until they reach a condensed phase of bound solitons that also propagates in the cavity. The soliton flow can be adjusted with manual cavity tuning, and can even be triggered by the injection of an external low-power cw laser. Preliminary explanations are provided for this phenomenon.

2. Fiber laser experimental setup

Fig. 1. Fiber laser setup. EDF: erbium-doped fiber ; DCF: dispersion compensation fiber ; PC1, PC2: polarization controllers ; 980/1550: wavelength multiplexer ; DL: pumping diode laser.

Experiments are performed with the passively-mode-locked fiber laser sketched on Fig. 1. Mode locking is achieved through the ultrafast saturable absorber effect that results from the nonlinear polarization evolution that takes place in the fibers when followed by polarization-dependent transmission [13

13. V. Matsas, T. Newson, D. Richardson, and D. Payne, “Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation,” Electron. Lett. 28(15), 1391–1392 (1992). [CrossRef]

]. The all-fibered ring laser cavity features dual 980-nm pumping of a 2-meter-long erbium-doped fiber (EDF, normal dispersion D=-12.5 ps.nm-1.km-1). It also comprises successively a polarization insensitive optical isolator to ensure unidirectional laser emission at λ=1.5µm, a polarization controller made of small fiber-loops (PC1), a 3% output coupler, a polarization splitter essential to mode locking, a four-port 80/20 coupler that allows light injection, a second polarization controller (PC2), and a meter length of dispersion compensating fiber (D=-91 ps.nm-1.km-1). Except for a short length (0.15 m) of birefringent fiber at the output of the polarization splitter, fibered components are pigtailed with SMF-28. The overall cavity length is 13.5 meters; the roundtrip time being 66 ns. The cavity operates in an anomalous path-averaged chromatic dispersion (Daverage=+5 ps.nm-1.km-1).

The orientations of the paddles of the polarization controllers, which can be recorded, define the shape of the intensity transfer function that in turns controls most of the pulsed dynamics [14

14. G. Martel, C. Chédot, A. Hideur, and Ph. Grelu, “Numerical Maps for Fiber Lasers Mode Locked with Nonlinear Polarization Evolution: Comparison with Semi-Analytical Models,” Fib. Integr. Opt. 27(5), 320–340 (2008). [CrossRef]

]. The orientations of the paddles can be set to obtain stable mode locking with negligible background outside the wings of the pulses. The total injected pumping power can reach 800 mW, yielding in this case around one hundred coexisting soliton pulses inside the cavity. Real-time recordings of the output field intensity are performed using a 6-GHz oscilloscope Lecroy SDA6020, with a 5.6-GHz O/E converter.

3. Observation of soliton rains

Precise setting of the paddles allows for qualitative control of the formation of various multiple pulse configurations, from soliton molecules to harmonic mode locking for instance. In the following, we investigate unusual cavity settings for which cw components and soliton pulses coexist, which is somehow an intermediate setting between cw lasing and stable mode locking. A recording of the distribution of the field intensity in the cavity at a given time is given in Fig. 2(a).

Fig. 2. Recordings of the output field typical of “soliton rains”. (a): Oscilloscope recording, (b): vertical magnification of the oscilloscope trace, (c): Optical spectrum in logarithmic scale, (d): horizontal magnification of the condensed soliton phase, showing also one incoming soliton on the left part of the figure.

The large peak per roundtrip corresponds to an unresolved group of bound solitons, which we call the condensed soliton phase that spans over half of a nanosecond. Several isolated solitons are seen on the left of the large peak. Quantization of the soliton energy is quite obvious from this recording, and it is used to estimate the number of solitons that are comprised in the condensed phase. This number amounts here to 30, and can be adjusted between 20 and 50, typically, depending on the orientation of the polarization controllers and on the pumping power. Using vertical magnification (Fig. 2(b)), the relatively important noisy background is highlighted. Its total energy is here around twice the energy comprised in all solitons. This is consistent with the spectral recording of the output (Fig. 2(c)), which shows a large amount of quasi-cw components around 1557 nm. The other peaks, symmetrically located in the spectrum, are the well-known soliton sidebands that manifest in mode locked fiber lasers in anomalous dispersion regime, and correspond to dispersive waves phase-matched to soliton spectral components [15

15. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger equation,” J. Opt. Soc. Am. B 9(1), 91–97 (1992). [CrossRef]

]. Their amplitudes are much smaller, typically 20dB below, than the amplitude of the cw components around 1557 nm.

Fig. 3. Video recording of the real-time oscilloscope trace corresponding to vertically magnified Fig. 2(b), showing the dynamics of “soliton rains” (Media 1).

Once experimental settings for soliton rain dynamics are found, we can study how dynamics depend on cavity parameters such as the pumping power or the orientation of mode locking paddles. Figures 4(a) and 4(b) demonstrate how changing the orientation of one paddle of PC2, recorded as angle θ, is able to switch the laser operating regime. At θ=46°, the laser is stably mode locked without any significant background in the temporal trace, as reflected also in the plain optical spectrum. The large temporal peak comprises ~40 bound solitons. At θ=50°, approximately 50 irregularly spaced solitons fill the cavity, and the spectrum features soliton sidebands. At θ=53°, soliton bunching is obtained again but solitons coexist with a small cw background. The background increases gradually along with the increase of θ up to 59°. The soliton rain dynamics appears at θ~56°. Around θ=53°, an increase of the pumping power also increases the background, until the level of its fluctuations is sufficient to trigger the formation of soliton pulses, as displayed on Fig. 4(c). The number of solitons inside the condensed phase increases as well with the pumping power.

4. Triggering the rain

The soliton rain seems to appear above a certain threshold of the noisy cw background, although the required threshold also depends on the settings of the polarization controllers. When the polarization controllers are fixed, a fine control of the amount of the cw background should trigger or stop the soliton rain. We demonstrate this possibility by injecting inside the laser cavity an external cw laser from the available input port of the 80/20 coupler. The injected light is produced by a narrowband tunable laser Photonetics PRI, and its intensity is adjusted with an attenuator.

Fig. 4. (a): switching of dynamics allowed by simply turning the orientation of one paddle (from 46° to 59°) of the polarization controller (here PC2), (b): corresponding optical spectra, (c): influence of the pumping power (from 417 to 604 mW) on soliton rains.

The regime is first set close to the soliton rain threshold, with the injected laser “off” (Figs. 5(a) and 5(b)). Right after switching “on” the injected laser, the soliton rain starts and lasts as long as injection remains (Figs. 5(c) and 5(d)). We can see from the optical spectrum that the needed contribution of the injected laser is relatively small, the injected power being in the range of 10 µW. When injection is turned “off”, the soliton rain stops almost immediately, right after the soliton “droplets” created before finish their drifting to the condensed phase.

5. Discussion

The large number of cw spectral components produce, through their beats, large fluctuations that can grow and become stable solitons when they exceed a certain level. These cw spectral components can be fed by various mechanisms, such as amplified spontaneous emission and dispersive waves radiated from the solitons [10

10. J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003). [CrossRef]

,15

15. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger equation,” J. Opt. Soc. Am. B 9(1), 91–97 (1992). [CrossRef]

], however we see in the case illustrated above that the main cw components occupy a specific spectral portion (1556.5 to 1557 nm).

Fig. 5. Control of the soliton rain dynamics. (a): temporal optical intensity, injected laser “off”, (b): optical spectrum, injected laser “off”, (c): temporal optical intensity, injected laser “on”, (d): optical spectrum, injected laser “on”.

Although the coexistence of cw and soliton pulses looks contradictory in a mode-locked laser, it should be possible to understand it in the frame of high-order nonlinear propagation models, such as in the cubic-quintic Ginzburg-Landau equation (CGLE), which has been successfully used to model various dissipative soliton dynamics [1

1. Ph. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations,” in Dissipative solitons: from optics to biology and medicine, N. Akhmediev and A. Ankiewicz, eds. (Springer-Verlag, Berlin, 2008). [CrossRef]

7

7. Ph. Grelu and J. M. Soto-Crespo, “Multisoliton states and pulse fragmentation in a passively mode-locked fibre laser,” J. Opt. B. 6, S271–S278 (2004). [CrossRef]

]. In principle, cubic-quintic equations allow the coexistence of several cw and soliton solutions, although some of them can be unstable [16

16. J. M. Soto-Crespo, N. Akhmediev, and G. Town, “Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber,” J. Opt. Soc. Am. B 19(2), 234–242 (2002). [CrossRef]

]. In addition, the set of intensity transfer functions that can be obtained through nonlinear-polarization-evolution mode locking, when the settings of the polarization controllers are varied, is by far richer than the set obtained with a given saturable absorber element such as a SESAM for instance [14

14. G. Martel, C. Chédot, A. Hideur, and Ph. Grelu, “Numerical Maps for Fiber Lasers Mode Locked with Nonlinear Polarization Evolution: Comparison with Semi-Analytical Models,” Fib. Integr. Opt. 27(5), 320–340 (2008). [CrossRef]

]. So, it is possible that for subsets of the cavity settings, the corresponding CGLE parameters would provide simultaneously stable cw and soliton solutions, or that the presence of a low-energy unstable soliton solution determine a threshold above which fluctuations can grow and evolve into the stable soliton solution.

6. Conclusion

Mode-locked lasers that operate with a large number of pulses per roundtrip can reveal surprising collective behaviors, such as the one discovered here. The soliton rain dynamics presents the following features. A soliton cluster, or condensed phase, coexists with a noisy quasi-cw background. The energies of these two components are of the same order of magnitude. When the background level is sufficient, single solitons can arise spontaneously from its fluctuations. Then, they drift at a nearly constant relative velocity until they merge with the condensed phase. This process can go on forever in a quasi-stationary fashion. It is also possible to set the level of the background below the soliton rain threshold, so that the soliton rain can be triggered and stopped easily via the injection of a low-power external cw laser. The finding of the intriguing soliton rain dynamics that operates in a weakly mode locked regime, namely when quasi-continuous waves and noise are not strongly filtered out, is also interesting in the context of the renewed interest for mode-locking dynamics [17

17. A. Gordon, O. Gat, B. Fischer, and F. Kärtner, “Self-starting of passive mode locking,” Opt. Express 14(23), 11142–11154 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11142. [CrossRef]

].

Acknowledgements

We thank Profs. N. Akhmediev and J.M. Soto-Crespo for stimulating discussions and acknowledge support from the Agence Nationale de la Recherche (project ANR05-BLAN-0152-01). S.C. acknowledges financial support from Conseil Régional de Bourgogne and from Université de Bourgogne.

References and links

1.

Ph. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations,” in Dissipative solitons: from optics to biology and medicine, N. Akhmediev and A. Ankiewicz, eds. (Springer-Verlag, Berlin, 2008). [CrossRef]

2.

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(6), 066612 (2004). [CrossRef]

3.

Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12(14), 3184–3189 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184. [CrossRef]

4.

M. Olivier, V. Piché, M. Roy, and F. Babin, “Pulse collisions in the stretched-pulse fiber laser,” Opt. Lett. 29(13), 1461–1463 (2004). [CrossRef]

5.

M. Grapinet and Ph. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006). [CrossRef]

6.

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1), 016613 (2007). [CrossRef]

7.

Ph. Grelu and J. M. Soto-Crespo, “Multisoliton states and pulse fragmentation in a passively mode-locked fibre laser,” J. Opt. B. 6, S271–S278 (2004). [CrossRef]

8.

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Analysis of soliton pattern formation in passively mode-locked fiber lasers,” Phys. Rev. A 78(4), 043806 (2008). [CrossRef]

9.

S. Wabnitz, “Control of soliton train transmission, storage, and clock recovery by cw light injection,” J. Opt. Soc. Am. B 13(12), 2739–2749 (1996). [CrossRef]

10.

J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003). [CrossRef]

11.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005). [CrossRef]

12.

A. Komarov, K. Komarov, H. Leblond, and F. Sanchez, “Spectral-selective management of dissipative solitons in passive mode-locked fibre lasers,” J. Opt. A, Pure Appl. Opt. 9(12), 1149–1156 (2007). [CrossRef]

13.

V. Matsas, T. Newson, D. Richardson, and D. Payne, “Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation,” Electron. Lett. 28(15), 1391–1392 (1992). [CrossRef]

14.

G. Martel, C. Chédot, A. Hideur, and Ph. Grelu, “Numerical Maps for Fiber Lasers Mode Locked with Nonlinear Polarization Evolution: Comparison with Semi-Analytical Models,” Fib. Integr. Opt. 27(5), 320–340 (2008). [CrossRef]

15.

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger equation,” J. Opt. Soc. Am. B 9(1), 91–97 (1992). [CrossRef]

16.

J. M. Soto-Crespo, N. Akhmediev, and G. Town, “Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber,” J. Opt. Soc. Am. B 19(2), 234–242 (2002). [CrossRef]

17.

A. Gordon, O. Gat, B. Fischer, and F. Kärtner, “Self-starting of passive mode locking,” Opt. Express 14(23), 11142–11154 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11142. [CrossRef]

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(140.4050) Lasers and laser optics : Mode-locked lasers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 15, 2009
Revised Manuscript: May 12, 2009
Manuscript Accepted: May 12, 2009
Published: June 29, 2009

Citation
Souad Chouli and Philippe Grelu, "Rains of solitons in a fiber laser," Opt. Express 17, 11776-11781 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11776


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References

  1. Ph. Grelu, and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations,” in Dissipative solitons: from optics to biology and medicine, N. Akhmediev and A. Ankiewicz, eds. (Springer-Verlag, Berlin, 2008). [CrossRef]
  2. J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(6), 066612 (2004). [CrossRef]
  3. Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12(14), 3184–3189 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184 . [CrossRef]
  4. M. Olivier, V. Roy, M. Piché, and F. Babin, “Pulse collisions in the stretched-pulse fiber laser,” Opt. Lett. 29(13), 1461–1463 (2004). [CrossRef]
  5. M. Grapinet and Ph. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006). [CrossRef]
  6. J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1), 016613 (2007). [CrossRef]
  7. Ph. Grelu and J. M. Soto-Crespo, “Multisoliton states and pulse fragmentation in a passively mode-locked fibre laser,” J. Opt. B. 6, S271–S278 (2004). [CrossRef]
  8. A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Analysis of soliton pattern formation in passively mode-locked fiber lasers,” Phys. Rev. A 78(4), 043806 (2008). [CrossRef]
  9. S. Wabnitz, “Control of soliton train transmission, storage, and clock recovery by cw light injection,” J. Opt. Soc. Am. B 13(12), 2739–2749 (1996). [CrossRef]
  10. J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003). [CrossRef]
  11. D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005). [CrossRef]
  12. A. Komarov, K. Komarov, H. Leblond, and F. Sanchez, “Spectral-selective management of dissipative solitons in passive mode-locked fibre lasers,” J. Opt. A, Pure Appl. Opt. 9(12), 1149–1156 (2007). [CrossRef]
  13. V. Matsas, T. Newson, D. Richardson, and D. Payne, “Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation,” Electron. Lett. 28(15), 1391–1392 (1992). [CrossRef]
  14. G. Martel, C. Chédot, A. Hideur, and Ph. Grelu, “Numerical Maps for Fiber Lasers Mode Locked with Nonlinear Polarization Evolution: Comparison with Semi-Analytical Models,” Fib. Integr. Opt. 27(5), 320–340 (2008). [CrossRef]
  15. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger equation,” J. Opt. Soc. Am. B 9(1), 91–97 (1992). [CrossRef]
  16. J. M. Soto-Crespo, N. Akhmediev, and G. Town, “Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber,” J. Opt. Soc. Am. B 19(2), 234–242 (2002). [CrossRef]
  17. A. Gordon, O. Gat, B. Fischer, and F. Kärtner, “Self-starting of passive mode locking,” Opt. Express 14(23), 11142–11154 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11142 . [CrossRef]

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