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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 13128–13139
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Steady and oscillating multiple dissipative solitons in normal-dispersion mode-locked Yb-doped fiber laser

Mohamed A. Abdelalim, Yury Logvin, Diaa A. Khalil, and Hanan Anis  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 13128-13139 (2009)
http://dx.doi.org/10.1364/OE.17.013128


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Abstract

Multiple dissipative solitons are numerically studied in the normal-cavity-dispersion Yb-doped fiber laser. Soliton pairs and triplets of different types are found in parameter domain where single soliton mode-locking is unstable. Similar scenario is found for soliton pair and triplet: an additional weak soliton supplements already existing solitons, then with increasing gain the solitons start oscillating and at higher gain the soliton complex becomes stable. Spectral profiles of the solitons dynamically change taking either parabolic-like or Π-like or M-like shape similar to those described in the literature for individual dissipative soliton at different pump level.

© 2009 Optical Society of America

1. Introduction

Generation of a single ultra-short energetic pulse is hindered by the harmonic mode-locking when multiple pulses are formed inside the laser cavity with increasing external pump level. Interaction between multiple solitons has been extensively studied both theoretically and experimentally. It has been shown that due to the cavity pulse peak clamping effect, number of pulses increases with gain level with a hysteresis transition from one multiple soliton state to another [9

9. A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005). [CrossRef]

,10

10. 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, 043816 (2005). [CrossRef]

]. Another study has demonstrated that solitons may form pairs with predetermined separation and phase difference between two pulses [11

11. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo “Multisoliton solution of the complex Ginzburg-Landau equation,” Phy. Rev. Lett. 79, 4047–4051 (1997). [CrossRef]

, 12

12. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am B 15, 2, 515–523 (1998). [CrossRef]

]. Most recent work show that both gain-guided solitons [13

13. L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, and C. Lu, “Generation of Multiple Gain-Guided Solitons in a Fiber Laser,” Opt. Lett. 32(11), 1581 (2007). [CrossRef]

] and dissipative solitons [14

14. J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007). [CrossRef]

] may demonstrate even more sophisticated interaction dynamics including vibrating, shaking and mixed soliton pairs.

2. Laser cavity model

The laser cavity model used in the current work, illustrated in Fig. 1 is the same as that introduced before in [15

15. M. A. Abdelalim, Y. Logvin, D. Khalil, and H. Anis, “Properties and stability limits of an optimized Yb-doped femtosecond fiber laser,” Opt. Express 17(4), 2264–2279 (2009). [CrossRef]

]. Here, we briefly describe the model: the Yb-doped fiber with a length of 60 cm is considered as the main element in the laser cavity. The model includes also few centimeters of single mode fiber (SMF) as a part of WDM coupler to provide external pumping but the presence of the SMF is not essential for the obtained results. Polarization beam splitter (PBS) is assumed to output light from the cavity as well as to initiate mode-locking through nonlinear polarization evolution (NPE). A wave plate (WP) is used for controlling the polarization state of the propagating light, and the isolator is used to provide unidirectional propagation.

Fig. 1. Schematic view of the laser cavity: WDM – wavelength division multiplexer coupler, PBS – polarization beam splitter, WP – wave plate, SMF – single mode fiber.

The pulse evolution in the Yb-doped fiber is governed by the Ginzburg-Landau equation for the complex amplitude A:

Az=α2Ajβ222At2+jγA2A+(g01+EpulseEsat.)(1+1Ωg22t2)A,
(1)

where z is the propagation coordinate, t is the retarding time, A(z, t) is the complex field amplitude, α is the linear loss taken as 0.04 m-1, β2 is the GVD parameter taken as 24000 fs2/m that corresponds to the normal cavity dispersion, γ is the nonlinear parameter taken as 0.005 W-1m-1, go is the small signal gain parameter with parabolic frequency dependence and a bandwidth Ωg taken as 35 nm, Epulse is the pulse energy and Esat is the gain saturation energy (5 nJ). Light transmission through the polarizer has been modeled by a transfer function T=1-lo/[1+ Ppulse/Psat], where lo=0.5 is the unsaturated loss, Ppulse is the pulse power and Psat=12 kW is the saturation power. The total loss of the cavity has been taken as 10 dB corresponding to losses at all interfaces as well as the useful loss due to light output from the cavity. The numerical method used is based on the split-step Fourier transform algorithm. As an initial condition, white noise has been used. The accuracy of the numerical results has been verified by reproducing them at different spatial and temporal grid parameters.

3. Results and discussion

3.1. Soliton pairs: characteristics and classification.

Fig. 2. The light intensity profiles of the OSP for go=47 dB/m in linear (a) and logarithmic (b) scales at three moments of time corresponding to different roundtrips (RT) number: blue at RT=9840, green at RT=9930, and red at RT=10010.
Fig. 3. The normalized pulse profiles of the OSP for go=47 dB/m at different parts of the cavity within the same roundtrip: (a) RT=9840, and (b) RT=10010. Blue represents SMF output, green represents Yb output, and red represents SA output.

When analyzing the spectrum of the soliton pair in Fig. 4(b) and comparing it with the spectra of Fig. 4(a), it becomes clear that the spectral width is determined by the more energetic narrower soliton of the pair while the weaker soliton creates modulation on the spectrum pedestal.

It should be noted that in our case there is no direct interaction between solitons through their tails which is different from the situation of the vibrating solitons described by Soto-Crespo et al [10

10. 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, 043816 (2005). [CrossRef]

]. We find some resemblance between our oscillating solitons and the nonstationary behavior of the multiple gain-guided solitons reported by Zhao et al [13

13. L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, and C. Lu, “Generation of Multiple Gain-Guided Solitons in a Fiber Laser,” Opt. Lett. 32(11), 1581 (2007). [CrossRef]

] in the study of normal-dispersion erbium doped fiber laser. We do not find any correlations between the phases of the oscillating solitons what follows from the fact that solitons do not interact through their tails. We see that phase difference between soliton peaks changes quite chaotically in time which can be explained by the fact that electric field decays significantly (in other words, goes through numerical zero) between the pulses as it is shown in Fig. 2(b). We assume that random change of the phase difference is associated with numerical noise at the points of field singularities.

Fig. 4. Oscillating soliton pair (OSP) spectrum at go=47 dB/m: (a) spectra of the separated pulses where the blue curve represents the M-like spectrum at maximum pulse energy (L or R) while the green one is the parabolic-type spectrum at minimum energy (L or R), (b) the spectrum of the OSP when one pulse is at maximum (L or R) and the other is at minimum (R or L).

With increasing pump level we see that the amplitude of oscillations decreases. Eventually, beyond gain parameter go value of 49dB/m we observe formation of the stable soliton pair (SSP) with characteristics presented in Figs. 5 and 6. It is seen that the solitons, L and R in the SSP are identical and the spectrum has Π-like shape similar to what has been reported for individual dissipative solitons in our system [15

15. M. A. Abdelalim, Y. Logvin, D. Khalil, and H. Anis, “Properties and stability limits of an optimized Yb-doped femtosecond fiber laser,” Opt. Express 17(4), 2264–2279 (2009). [CrossRef]

].

Fig. 5. The stable soliton pair (SSP) at go=50 dB/m in linear (a) and logarithmic (b) scales.
Fig. 6. Stable soliton pair (SSP) spectra at go=50 dB/m: (a) individual pulse spectrum and (b) spectrum of the SSP.

Fig. 7. Evolution of the total normalized light energy ‘Q’ in the cavity for different values of the gain parameter go: energy oscillations for the case of the OSP from go=45 dB/m to go=48 dB/m and steady-state final value of Q for SSP from go=49 dB/m to go=54 dB/m.

3.3. Soliton triplets: characteristics and classification.

When the pump parameter go is taken above 55 dB/m, as a final state we observe formation of the oscillating soliton triplet (OST) whose characteristics is presented in Figs. 810. In Fig. 8 temporal intensity profiles of the OST are shown in linear (a) and logarithmic scale (b) at go=60 dB/m. Two snap-shots correspond to the moments of time separated by 75 roundtrips when either the single soliton on the left (L) or the other two nearly identical solitons on center (C) and on right (R) reach maximum power. We observe that the soliton triplet consists of a soliton pair and a separate soliton: The solitons in the pair are nearly identical and oscillate in phase while the individual soliton oscillates in the opposite phase. Although the solitons are oscillating, the peaks positions do not change; hence the distances among the solitons are fixed. Moreover, the triplet profiles at the different positions of the cavity are almost the same, like in the case of the soliton pair Again, from the analysis of the profiles in the logarithmic scale we see that the solitons do not interact directly through their tails. Another interesting feature is that while the shape of the C and R pulses in the pair does not change much with the number of roundtrip, the width of the separate soliton L decreases when its energy grows.

The spectra of the individual pulses as well as that of the whole OST are presented in Fig. 9 and 10. Three profiles in Fig. 9(a) are the spectra of the separated pulses of Fig 8 when the individual soliton L reaches its minimum energy. The spectrum of this individual soliton has M-like shape and is different from the Π-like profiles of the other two solitons C and R. The Π-like spectra are wider and they determine the spectral width of the whole OST in Fig. 9(b) while the individual soliton L contribution is the modulation in the spectrum center.

As time (the roundtrip number) progresses, the energy of the individual soliton L grows and its M-like spectrum widens. Fig. 10(a) presents spectra at the instant of time when the separate pulse L reaches maximum energy (the pair, C and R pulses is at minimum energy). It is seen that the M-type spectrum is now wider than the Π-type spectrum of the pulses of the pair. The spectrum width of the whole triplet in Fig. 10(b) is determined now by the individual soliton and the signature of the pair is the modulation.

Fig.8. Eemporal intensity profiles of the OST in linear (a) and logarithmic scale (b) at go=60 dB/m. Two curves correspond to the moments of time separated by 75 roundtrips when either the single soliton on the left L or the other two nearly identical solitons, C and R have maximum power.
Fig. 9. Oscillating soliton triplet (OTS) spectrum at go=60 dB/m at RT=14000: (a) spectra of the individual pulses where blue curve is the spectrum of the L pulse at minimum energy while the green (red) curve is the spectrum of the C (R) pulse at maximum, (b) the spectrum of the whole OTS.
Fig. 10. Oscillating soliton triplet (OTS) spectrum at go=60 dB/m at RT=14075: (a) spectra of the individual pulses where blue curve is the spectrum of the L pulse at maximum energy while the green (red) curve is the spectrum of the C (R) pulse at minimum, (b) the spectrum of the whole OTS.

When simulations are performed at higher levels of the gain parameter go we observe that oscillations decay at approximately go=60 dB/m and stable soliton triplet (SST) is formed in the cavity. In Fig. 11, the intensity profiles of the pulses are shown for go=61 dB/m. The pulses are almost identical, i.e. they have nearly the same width and peak power. The spectra of the solitons look identical. An example of the individual soliton spectrum is presented in Fig. 12(a), it is seen that the spectrum profile seems to be intermediate between the M-type and the Π-type shapes. Fig. 12(b) illustrates the spectrum of the whole triplet.

Fig. 11. Temporal intensity profiles of the stable soliton triplet (SST) in linear (a) and logarithmic scale (b) at go=61 dB/m..

The transient dynamics of the normalized energy inside the cavity for the case of soliton triplet is presented in Fig. 13. It is seen that after transient process, periodic oscillations settle down. The period of the oscillations decreases with growing gain while the oscillation amplitude reaches maximum at g0=57 dB/m and then decreases. In the oscillation shape one can note two scales: a slow change from one maximum to another and then a sudden drop after which the process is repeated. When we follow individual pulse dynamics we see that slow scale is associated with the change of the energy of the soliton pair while sharp drop is due to sudden change of the separate soliton energy. For gain parameter above g0=60 dB/m we observe decay of the oscillations and the steady soliton triplet previously described in Fig. 12 establishes. One can see that dynamics presented for soliton triplets in Fig. 13 is similar to the case of the soliton pair in Fig. 7. With increasing gain we see that oscillation behavior vanishes and the stable multiple solitons appear. It should be noted that stable soliton triplets as well as collision between soliton pair and the single soliton have been found as solutions of the cubic-quintic complex Ginzburg–Landau equation [12

12. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am B 15, 2, 515–523 (1998). [CrossRef]

, 18]. In our case, interaction between solitons leading to specific type of oscillatory behavior is different from that described earlier in the literature.

Fig.12. Stationary soliton triplet (SST) spectrum at go=61dB/m: (a) spectrum of the individual pulse and (b) spectrum of whole SST.
Fig. 13. Total normalized energy ‘Q’ versus roundtrip number for different values of the gain parameter go : from go=56 dB/m to go=60 dB/m the final state is the oscillating soliton triplet while at go=61 dB/m – the stable soliton triplet.

3.3. Asymmetric border states.

At some values of the gain parameter we observe extremely long transient dynamics which corresponds to the boundaries between different attractors of our infinite-dimensional dynamical system. Examples of such dynamics are presented in Fig. 14. It is seen that the transient time to reach stable attractor here is more than two times longer as compared to Figs. 7 and 13. As a result of extremely long transient we observe formation of stable asymmetric soliton pairs and triplets presented in Figs. 15 and 16 respectively. Stable asymmetric soliton pair in Fig. 14 obtained at the gain parameter go=44 dB/m is a border state which corresponds to emergence of the soliton pair. Below go=42 dB/m we observe formation of the single dissipative soliton in the laser cavity.

Fig.14. The total normalized energy ‘Q’ for the soliton pair and triplet border states.
Fig. 15. Stable asymmetric soliton pair at go=44 dB/m : (a) intensity profile and (b) the corresponding spectrum.

Formation of the soliton pair shown in Fig. 15(a), with one soliton being much weaker than another, can be explained by the cavity pulse peak clamping mechanism. This is generally considered as a cause of harmonic mode-locking. With increasing gain, the weaker soliton grows and gets closer to the more powerful one. Later we observe oscillatory behavior described above of the OSP. The spectrum of the asymmetric soliton pair is presented in Fig. 15(b). The total spectrum width is determined by more energetic soliton while the weaker one creates modulation in the spectrum center.

Fig.16. Stable asymmetric soliton triplet at go=55 dB/m: (a) intensity profile and (b) the corresponding spectrum.

Stable asymmetric soliton triplet obtained at go=55 dB/m is presented in Fig. 16. It is seen from Fig. 16(a) that third pulse has quite low amplitude compared to other two. Two powerful solitons determine spectrum envelope in Fig. 16(b), whereas the weakest one is responsible for narrow modulation in the spectrum center.

4. Conclusion

In the present paper, multiple dissipative soliton states are found in the model of the mode-locked normal-cavity-dispersion Yb-doped fiber laser. Soliton pairs and triplets arise in the parameter domain where single dissipative soliton is unstable. The following sequence has been observed. At the border of single soliton stability, formation of the asymmetric soliton pair is found with one soliton being much weaker than another. When gain parameter is increased the solitons form symmetric oscillating soliton pair, and for higher gain the oscillations decay and symmetric soliton pair becomes stable. It should be noted that solitons do not interact directly through their tails so that no correlation between phases of the solitons has been observed.

Similar scenario has been found for soliton triplets for higher value of the gain parameter. Beyond stability border of the soliton pair, third weak soliton is added to the pair so that the soliton triplet emerges. As the gain increases further, the triplet starts oscillating in such a way that the two identical solitons oscillate in the same phase while the third one oscillates in the opposite phase. Finally at higher gain values, all three solitons become stable. Spectral profiles of the solitons dynamically change taking either parabolic-like or Π-like or M-like shape similar to those observed for individual solitons at different pump level.

References and links

1.

J. Limpert, F. Roser, T. Schreiber, and A. Tunnermann, “High-power ultrafast fiber laser systems,” IEEE J. Sel. Topics in Quantum Electron. 12, 233–244 (2006). [CrossRef]

2.

D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993). [CrossRef]

3.

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

4.

L. M. Zhao, D. Y. Tang, and J. Wu, “Gain-guided soliton in a positive group-dispersion fiber laser,” Opt. Lett. 31, 1788–1790 (2006). [CrossRef] [PubMed]

5.

L. M. Zhao, D. Y. Tang, H. Zhang, T. H. Cheng, H. Y. Tam, and C. Lu, “Dynamics of gain-guided solitons in an all-normal-dispersion fiber laser,” Opt. Lett. 32, 1806–1808 (2007). [CrossRef] [PubMed]

6.

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of Normal-Dispersion Femtosecond Fiber Lasers,” J. Opt. Soc. Am B 25(2), 140–148 (2008). [CrossRef]

7.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative Solitons in Normal-Dispersion Fiber Lasers,” Phys. Rev. A 77, 023814 (2008). [CrossRef]

8.

K. Kieu, W. H. Renninger, A. Chong, and F. W. Wise, “Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser,” Opt. Lett. 34, 593–595 (2009). [CrossRef] [PubMed]

9.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005). [CrossRef]

10.

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, 043816 (2005). [CrossRef]

11.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo “Multisoliton solution of the complex Ginzburg-Landau equation,” Phy. Rev. Lett. 79, 4047–4051 (1997). [CrossRef]

12.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am B 15, 2, 515–523 (1998). [CrossRef]

13.

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, and C. Lu, “Generation of Multiple Gain-Guided Solitons in a Fiber Laser,” Opt. Lett. 32(11), 1581 (2007). [CrossRef]

14.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007). [CrossRef]

15.

M. A. Abdelalim, Y. Logvin, D. Khalil, and H. Anis, “Properties and stability limits of an optimized Yb-doped femtosecond fiber laser,” Opt. Express 17(4), 2264–2279 (2009). [CrossRef]

16.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-Pulse Oscillators: Theory and Experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

17.

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

18.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu,” Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005). [CrossRef]

OCIS Codes
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(140.4050) Lasers and laser optics : Mode-locked lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: May 27, 2009
Revised Manuscript: July 9, 2009
Manuscript Accepted: July 9, 2009
Published: July 16, 2009

Citation
Mohamed A. Abdelalim, Yury Logvin, Diaa A. Khalil, and Hanan Anis, "Steady and oscillating multiple dissipative solitons in normal-dispersion mode-locked Yb-doped fiber laser," Opt. Express 17, 13128-13139 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13128


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References

  1. J. Limpert, F. Roser, T. Schreiber, and A. Tunnermann, "High-power ultrafast fiber laser systems,"IEEE J. Sel. Top. Quantum Electron. 12, 233 - 244 (2006). [CrossRef]
  2. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Am. B 10, 1185-1190 (1993). [CrossRef]
  3. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. E. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]
  4. L. M. Zhao, D. Y. Tang, and J. Wu, "Gain-guided soliton in a positive group-dispersion fiber laser," Opt. Lett. 31, 1788-1790 (2006). [CrossRef] [PubMed]
  5. L. M. Zhao, D. Y. Tang, H. Zhang, T. H. Cheng, H. Y. Tam, and C. Lu, "Dynamics of gain-guided solitons in an all-normal-dispersion fiber laser," Opt. Lett. 32, 1806-1808 (2007). [CrossRef] [PubMed]
  6. A. Chong, W. H. Renninger, and F. W. Wise, "Properties of Normal-Dispersion Femtosecond Fiber Lasers," J. Opt. Soc. Am B 25(2), 140-148 (2008). [CrossRef]
  7. W.H. Renninger, A. Chong, and F. W. Wise, "Dissipative Solitons in Normal-Dispersion Fiber Lasers," Phys. Rev. A 77, 023814 (2008). [CrossRef]
  8. K. Kieu, W. H. Renninger, A. Chong, and F. W. Wise, "Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser," Opt. Lett. 34, 593-595 (2009). [CrossRef] [PubMed]
  9. A. Komarov, H. Leblond, and F. Sanchez, "Multistability and hysteresis phenomena in passively mode-locked fiber lasers," Phys. Rev. A 71, 053809 (2005). [CrossRef]
  10. 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, 043816 (2005). [CrossRef]
  11. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo "Multisoliton solution of the complex Ginzburg-Landau equation," Phy. Rev. Lett. 79, 4047-4051 (1997). [CrossRef]
  12. N. Akhmediev, A. Ankiewicz, and J.M. Soto-Crespo "Stable soliton pairs in optical transmission lines and fiber lasers," J. Opt. Soc. Am B 15, 2, 515-523 (1998). [CrossRef]
  13. L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, and C. Lu, "Generation of Multiple Gain-Guided Solitons in a Fiber Laser," Opt. Lett. 32(11), 1581 (2007). [CrossRef]
  14. J. M. Soto-Crespo, P. Grelu, N. Akhmediev and N. Devine, "Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs," Phys. Rev. E 75, 016613 (2007). [CrossRef]
  15. M. A. Abdelalim, Y. Logvin, D. Khalil, and H. Anis, "Properties and stability limits of an optimized Yb-doped femtosecond fiber laser," Opt. Express 17(4), 2264 - 2279 (2009). [CrossRef]
  16. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, "Chirped-Pulse Oscillators: Theory and Experiment," Appl. Phys. B 83, 503 - 510 (2006). [CrossRef]
  17. N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu,"Roadmap to ultra-short record high-energy pulses out of laser oscillators," Phy. Lett. A 372, 3124-3128 (2008). [CrossRef]
  18. N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu," Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005). [CrossRef]

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