## Coexistence of strong and weak pulses in a fiber laser with largely anomalous dispersion |

Optics Express, Vol. 19, Issue 7, pp. 5874-5887 (2011)

http://dx.doi.org/10.1364/OE.19.005874

Acrobat PDF (1608 KB)

### Abstract

The coexistence of weakly sech-shaped solitons and strongly dissipative solitons is experimentally observed in an ultra-large net-anomalous-dispersion mode-locked fiber laser for the first time to author’s best knowledge. Both sech-shaped and dissipative solitons appear to be the asymmetrically combined pulse state with one pulse component much smaller than the other. The energy of dissipative solitons is over three orders of magnitude larger than that of sech-shaped solitons. Two different types of pulse-shaping mechanisms coexist in the laser: one is the dissipative processes and the other is the balance between anomalous dispersion and nonlinear Kerr effect. Numerical simulations and analysis confirm the experimental observations.

© 2011 OSA

## 1. Introduction

1. H. Zhang, D. Y. Tang, L. M. Zhao, Q. Bao, and K. Loh, “Vector dissipative solitons in graphene mode locked fiber lasers,” Opt. Commun. **283**(17), 3334–3338 (2010). [CrossRef]

6. V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A **79**(4), 043829 (2009). [CrossRef]

8. K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. **28**(24), 2226–2228 (1992). [CrossRef]

11. A. Cabasse, G. Martel, and J. L. Oudar, “High power dissipative soliton in an Erbium-doped fiber laser mode-locked with a high modulation depth saturable absorber mirror,” Opt. Express **17**(12), 9537–9542 (2009). [CrossRef] [PubMed]

9. D. Y. Tang, L. M. Zhao, G. Q. Xie, and L. J. Qian, “Coexistence and competition between different solitons-shaping mechanisms in a laser,” Phys. Rev. A **75**(6), 063810 (2007). [CrossRef]

10. X. Liu, “Pulse evolution without wave breaking in a strongly dissipative-dispersive laser system,” Phys. Rev. A **81**(5), 053819 (2010). [CrossRef]

12. L. E. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B **65**(2), 277–294 (1997). [CrossRef]

12. L. E. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B **65**(2), 277–294 (1997). [CrossRef]

13. 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**(13), 1080 (1993). [CrossRef] [PubMed]

14. V. I. Kruglov, D. Méchin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A **81**(2), 023815 (2010). [CrossRef]

15. J. H. Im, S. Y. Choi, F. Rotermund, and D. I. Yeom, “All-fiber Er-doped dissipative soliton laser based on evanescent field interaction with carbon nanotube saturable absorber,” Opt. Express **18**(21), 22141–22146 (2010). [CrossRef] [PubMed]

18. X. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A **82**(6), 063834 (2010). [CrossRef]

19. B. Oktem, C. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics **4**(5), 307–311 (2010). [CrossRef]

20. F. W. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. **2**(1-2), 58–73 (2008). [CrossRef]

_{net}approaches zero, dispersion-managed solitons and sech-shaped solitons coexist in the same laser [9

9. D. Y. Tang, L. M. Zhao, G. Q. Xie, and L. J. Qian, “Coexistence and competition between different solitons-shaping mechanisms in a laser,” Phys. Rev. A **75**(6), 063810 (2007). [CrossRef]

_{net}, both stretched-pulse and self-similar operation can be observed in a laser [20

20. F. W. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. **2**(1-2), 58–73 (2008). [CrossRef]

_{net}is normal enough, the fiber laser either works on a status that is similar to an all-normal-dispersion laser or generates a type of pulses exhibited as the trapezoid-spectrum profile according to the pump power [21

21. X. Liu, “Dissipative soliton evolution in ultra-large normal-cavity-dispersion fiber lasers,” Opt. Express **17**(12), 9549–9557 (2009). [CrossRef] [PubMed]

22. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A **79**(3), 033840 (2009). [CrossRef]

25. S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express **17**(14), 11776–11781 (2009). [CrossRef] [PubMed]

## 2. Experimental setup and observations

### 2.1. Experimental setup

^{2}and 0.5 ps

^{2}, respectively. The proposed laser system has extremely large net-anomalous dispersion so that it may exploit two soliton-shaping mechanisms. One is the balance between the anomalous dispersion and the nonlinear Kerr effect and thus the conventional soliton-like pulses are generated. The other is the balance between gain and loss processes that induce the DS pulses.

### 2.2. Weakly sech-shaped soliton pulses

*u*and

*v*components of pulses at the pump power

*P*= 18 mW. Note that the y-axis of Fig. 3 is the linear scale, instead of the logarithmic scale in Fig. 2(b). It is found from Fig. 3 that the ratio

*R*of peak power of

_{P}*u*and

*v*components of pulses is about 11.5. According to the theory [27

27. X. M. Liu, “Mechanism of high-energy pulse generation without wave breaking in mode-locked fiber lasers,” Phys. Rev. A **82**(5), 053808 (2010). [CrossRef]

*u*and

*v*components also is 11.5.

12. L. E. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B **65**(2), 277–294 (1997). [CrossRef]

*λ*and wider pulse duration Δτ, e.g., Δ

*λ*≈1 nm and Δτ≈2.9 ps here instead of Δ

*λ*≈9 nm and Δτ≈0.45 ps in Ref [12

**65**(2), 277–294 (1997). [CrossRef]

### 2.3. Coexistence of two types of weak and strong pulses

*P*is beyond about 28 mW, a strange phenomenon occurs, i.e., the laser stably emits a type of pulses without spectral sidebands and with an order-of-magnitude increase in the pulse height. An experimental example for

*P*= 29 mW is shown in Fig. 4 . Figure 4(a) is the oscilloscope trace at the range from 0 to 100 μs, and Figs. 4(b) and 4(c) are the local view of Fig. 4(a) from 0 to 2.5 μs and from 3.2 to 3.4 μs, respectively. Figure 4(d) is the optical spectra, in which the ratio

*R*of peak power of

_{P}*u*and

*v*components is about 11.1.

*P*= 29 mW is about 40 nJ, increasing three orders of magnitude in comparison with the weak pulse. Comparing Fig. 4(d) to Figs. 2(b) and 2(c), we can see that the optical spectrum does not have spectral sidebands for

*P*>28 mW rather than have spectral sidebands at the pump power range from about 10 mW to about 25 mW. It originates from the fact that the strong pulses dominate the spectral property for

*P*>28 mW.

*P*the height of strong pulses hardly changes while their width broadens. Some experimental examples are shown in Fig. 5(a) , in which Δτ≈6.7 and 8.5 ns for

*P*= 40 and 50 mW, respectively. No internal fine structures are observed within the pulses. We note that the trailing and leading parts of the pulses are different [three examples are shown in Figs. 4(c) and 5(a)]. The difference is caused by the response time of our measurement systems. The optical spectra for

*P*= 50, 150, 300, 450 mW are shown in Fig. 5(b). One can see from Fig. 5(b) that the optical spectra approximately are the Gaussian profiles in the linear scale.

10. X. Liu, “Pulse evolution without wave breaking in a strongly dissipative-dispersive laser system,” Phys. Rev. A **81**(5), 053819 (2010). [CrossRef]

28. X. Wu, D. Y. Tang, H. Zhang, and L. M. Zhao, “Dissipative soliton resonance in an all-normal-dispersion erbium-doped fiber laser,” Opt. Express **17**(7), 5580–5584 (2009). [CrossRef] [PubMed]

29. D. Mao, X. Liu, L. Wang, H. Lu, and H. Feng, “Generation and amplification of high-energy nanosecond pulses in a compact all-fiber laser,” Opt. Express **18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

*R*of peak power of

_{P}*u*and

*v*components of pulses is more than 11. Then the ratio of pulse energy for

*u*and

*v*components of pulse is beyond 11 according to the theory [27

27. X. M. Liu, “Mechanism of high-energy pulse generation without wave breaking in mode-locked fiber lasers,” Phys. Rev. A **82**(5), 053808 (2010). [CrossRef]

## 3. Theoretical simulation and analysis

### 3.1. Coexistence of two types of weak and strong pulses

10. X. Liu, “Pulse evolution without wave breaking in a strongly dissipative-dispersive laser system,” Phys. Rev. A **81**(5), 053819 (2010). [CrossRef]

30. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A **81**(2), 023811 (2010). [CrossRef]

**81**(5), 053819 (2010). [CrossRef]

*u*and

*v*denote the envelopes of the optical pulses along the two orthogonal polarization axes of the fiber,

*α*is the loss coefficient of fiber,

*δ*is the group velocity difference between the two polarization modes,

*β*

_{2}represents the fiber dispersion,

*γ*refers to the cubic refractive nonlinearity of the medium, Ω

*is the bandwidth of the laser gain. The variable*

_{g}*T*and

*z*indicate the time and the propagation distance, respectively.

*g*is the net gain, which is nonzero only for the amplifier fiber. It describes the gain function of EDF and is expressed by [31

31. G. Agrawal, “Amplification of ultrashort solitons in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. **2**(12), 875–877 (1990). [CrossRef]

*g*

_{0}is the small-signal gain,

*E*is the gain saturation energy, and

_{s}*E*is the pulse energy. When the soliton propagates through the PAPM element, its intensity transmission,

_{p}*T*, is expressed aswhere

_{i}*ϕ*

_{1}is the phase delay caused by the polarization controllers and

*ϕ*

_{2}is the phase delay resulted from the fiber including both the linear phase delay and the nonlinear phase delay. The polarizer and analyzer have an orientation of angle

*θ*and

*φ*with respect to the fast axis of the fiber, respectively [30

30. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A **81**(2), 023811 (2010). [CrossRef]

32. X. Liu and B. Lee, “A fast method for nonlinear schrödinger equation,” IEEE Photon. Technol. Lett. **15**(11), 1549–1551 (2003). [CrossRef]

*α*= 0.2 dB/km,

*g*

_{0}= 2 m

^{−1},

*γ*= 4.5 W

^{−1}km

^{−1}for EDF and 1.3 W

^{−1}km

^{−1}for SMF, Ω

*= 30 nm, and*

_{g}*β*

_{2}= 53.5 × 10

^{−3}ps

^{2}/m for EDF and −21.7 × 10

^{−3}ps

^{2}/m for SMF. The simulation starts from an arbitrary signal and converges into a stable solution when the appropriate parameters are given.

*θ*= π/3.5,

*φ*= π/10,

*ϕ*

_{1}= 0.9 + π/2, and

*E*= 0.24 nJ, the weakly sech-shaped soliton solution is obtained, as shown in Fig. 6 . Figures 6(a) and 6(b) illustrate the temporal and spectral domain profiles of solitons. The autocorrelation trace of soliton is shown in Fig. 6(c). It is found from Fig. 6 that there are significant spectral sidebands [Fig. 6(b)] and the pedestal of solitons has the oscillating structure [Fig. 6(a)]. It is the oscillating structure of the pedestal that causes the complex structural spectrum. The numerical results show that the pulse duration and the spectral width are 3.1 ps and 0.9 nm, respectively, corresponding to the time–bandwidth product of about 0.34. The intensity peak of soliton is about 16 W and the soliton energy is about 0.048 nJ, as are in good agreement with the experimental results. Figures 6(a) and 6(b) show that the ratio

_{s}*R*of peak intensity of

_{P}*u*and

*v*components of pulses is about 11.2. The theoretical predictions [Figs. 6(b) and 6(c)] agree well with the experimental observations (Figs. 2(b), 2(d), and 3).

*ω*of the pulses is nearly zero at the time range from −3 ps to 3 ps and hence the pulse chirp is about zero across the pulse width. So this kind of solitons is stable chirp-free pulses with low energy and solitons are static, chirp-free solutions of the coupled complex NLSEs [i.e., Eq. (1)].

### 3.2. Theoretical modeling and results for strongly dissipative solitons

*v*component can be dealt with a small perturbation on the

*u*component. A pair of coupled complex NLSEs can be simplified to a single equation, e.g., Ginzburg-Landau equation.

^{15}= 32768. As a result, the numerical investigations of laser system are very time-consuming. When the pulse duration is more than 5 ns [Figs. 4(c) and 5(a)], the time interval should be more than 500 ns to meet the solution accuracy. So the numerical investigations for this case are hardly completed. As a result, we have to utilize the soliton amplitude and phase ansatzes to search the analytical stationary solution of Ginzburg-Landau equation modeling the dissipative processes.

*ε*, a real quintic saturable absorber term

*μ*, and a real quintic (reactive) nonlinearity term

*ν*, the complex cubic-quintic Ginzburg-Landau equation (CGLE) replaces the NLSE to describe the laser oscillators with more complicated structures [33

33. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. **85**(14), 2937–2940 (2000). [CrossRef] [PubMed]

37. W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of an external periodic potential on pairs of dissipative solitons,” Phys. Rev. A **80**(1), 013815 (2009). [CrossRef]

22. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A **79**(3), 033840 (2009). [CrossRef]

23. A. Komarov and F. Sanchez, “Structural dissipative solitons in passive mode-locked fiber lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **77**(6), 066201 (2008). [CrossRef] [PubMed]

*v*was ignored, Wise

*et al*. had predicted the flat-top solitons solutions existing in the normal-dispersion lasers by an analytic theory [20

20. F. W. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. **2**(1-2), 58–73 (2008). [CrossRef]

22. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A **79**(3), 033840 (2009). [CrossRef]

*ν*[22

**79**(3), 033840 (2009). [CrossRef]

38. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **53**(1), 1190–1201 (1996). [CrossRef] [PubMed]

41. S. Chen, “Theory of dissipative solitons in complex Ginzburg-Landau systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(2), 025601 (2008). [CrossRef] [PubMed]

*ψ*,

*t*, and

*z*are the normalized envelope of the field, the retarded time, and the propagation distance that the pulse travels, respectively.

*D*is the dispersion coefficient (it is positive (negative) in the anomalous (normal) dispersion regime [39

39. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A **78**(2), 023830 (2008). [CrossRef]

*σ*is the linear gain, and

*β*describes spectral filtering. In this report, we focus on the anomalous dispersion regime and the dispersion coefficient in Eq. (4) is given by

*D*= 1 [40

40. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. **79**(21), 4047–4051 (1997). [CrossRef]

38. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **53**(1), 1190–1201 (1996). [CrossRef] [PubMed]

41. S. Chen, “Theory of dissipative solitons in complex Ginzburg-Landau systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(2), 025601 (2008). [CrossRef] [PubMed]

*ω*is a real constant, and

*A*and Φ are real functions of

*t*. Inserting Eq. (5) into Eq. (4) and separating real and imaginary terms, we can achieve two coupled ordinary differential equations. We use a phase ansatzwhere

*d*is the chirp parameter and the initial phase is supposed to be equal to zero. By inserting the ansatz (6) into the intermediate differential equations and after some cumbersome transformations, one can obtain the reduced system of nonlinear equationswhere

*F*=

*A*

^{2}. Equation (7) is an elliptic-type differential equation, and its solution for our purposes is given by [38

38. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **53**(1), 1190–1201 (1996). [CrossRef] [PubMed]

*σ*satisfies the relationship of

*σ*

_{1}=

*σ*and take new

*n*is 1, 3, 5, 7, 9,…. Then we can achieve flat-top soliton solutions and their shapes are shown in Fig. 7 .

*n*), the pulse width broadens and the pulse height hardly varies. These theoretical predictions are in good agreement with the experimental observations. Therefore, both theoretical and experimental results prove that (1) with the increase of pumping strength the width of the quasi-rectangular DSs broadens with the fixed amplitude, (2) this type of DSs is a class of the wave-breaking-free pulses, (3) the ultra-large net-anomalous-dispersion PML fiber laser exploit a new way to generate high-energy nanosecond pulses, and (4) the dissipative processes based on the balance between the energy being supplied and lost in laser system governs the formation and evolution of DSs.

## 4. Discussions

29. D. Mao, X. Liu, L. Wang, H. Lu, and H. Feng, “Generation and amplification of high-energy nanosecond pulses in a compact all-fiber laser,” Opt. Express **18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

*φ*and

*θ*are about 0.1π and 0.28π, respectively. But the sech-shaped soliton solution is never achieved numerically when the parameters of laser oscillator of Ref [29

29. D. Mao, X. Liu, L. Wang, H. Lu, and H. Feng, “Generation and amplification of high-energy nanosecond pulses in a compact all-fiber laser,” Opt. Express **18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

^{2}in current work, instead of a 0.8-m length of EDF with normal dispersion of about 0.01 ps

^{2}in Ref [29

**18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

**18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

*φ*and

*θ*, but also the cavity parameters (e.g., the total normal and anomalous dispersions of cavity, the length of EDF and SMF, and even their location in cavity). The experimental results here show that, if the polarization controllers are not set to the appropriate orientation and pressure, the proposed laser does not deliver the weakly sech-shaped solitons and its operation is very similar to reports in Ref [29

**18**(22), 23024–23029 (2010). [CrossRef] [PubMed]

1. H. Zhang, D. Y. Tang, L. M. Zhao, Q. Bao, and K. Loh, “Vector dissipative solitons in graphene mode locked fiber lasers,” Opt. Commun. **283**(17), 3334–3338 (2010). [CrossRef]

*u*and

*v*components along the two orthogonal polarization axes of the fiber almost have the same spectral and temporal profiles (see Figs. 3, 4(d), 6(a), and 6(b) in this paper), except the intensity difference of

*u*and

*v*. For the vector solitons, however, the spectral and temporal profiles for the horizontal axis are completely different from those for the vertical axis (see Figs. 4(c), 5, and 6(b) in Ref [1

1. H. Zhang, D. Y. Tang, L. M. Zhao, Q. Bao, and K. Loh, “Vector dissipative solitons in graphene mode locked fiber lasers,” Opt. Commun. **283**(17), 3334–3338 (2010). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

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38. | N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

39. | W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A |

40. | N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. |

41. | S. Chen, “Theory of dissipative solitons in complex Ginzburg-Landau systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(140.3500) Lasers and laser optics : Lasers, erbium

(140.4050) Lasers and laser optics : Mode-locked lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: February 11, 2011

Manuscript Accepted: March 9, 2011

Published: March 15, 2011

**Citation**

Xueming Liu, "Coexistence of strong and weak pulses in a fiber laser with largely anomalous dispersion," Opt. Express **19**, 5874-5887 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5874

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### References

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