## On the profile of pulses generated by fiber lasers:the highly-chirped positive dispersion regime (similariton)

Optics Express, Vol. 14, Issue 25, pp. 12174-12182 (2006)

http://dx.doi.org/10.1364/OE.14.012174

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### Abstract

I model the nonlinear fiber laser using an expanded Ginzburg-Landau equation (GLE) which includes the self-steepening (SS) and intrapulse Raman scattering (IRS) effects. I show that above a minimum value of the Raman effect, it is possible to find two chirped solitary pulses for the laser system. The smaller chirped solitary wave corresponds to the dispersion-managed (DM) regime whereas the larger chirped solitary wave corresponds to the so-called *similariton* regime.

© 2006 Optical Society of America

## 1. Introduction

1. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. **30**, 200–208 (1994). [CrossRef]

2. P. -A. Bélanger, “On the profile of pulses generated by fiber lasers,” Opt. Express **13**, 8089–8096 (2005). [CrossRef] [PubMed]

2. P. -A. Bélanger, “On the profile of pulses generated by fiber lasers,” Opt. Express **13**, 8089–8096 (2005). [CrossRef] [PubMed]

*L*=2

*π*,

*β*=1) also implies that all these systems will generate pulses with the same peak power for any average negative dispersion. For the positive average dispersion regime, no specific operating point emerges from this model. Hence, in this paper, after extending the GL distributed model by including higher-order nonlinear terms such as the SS and IRS effects, I shall show that the resulting solution of this equation leads to two specific values for the chirp parameter

*β*. The solitary pulse having the largest chirp parameter

*β*has a spectral profile typical of the profile observed experimentally and numerically in the so-called

*similariton*regime [3–5

3. F. Ö. 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]

## 2. The distributed laser model

*β*

_{3}), SS (

*T*

_{R}) effects. The second order disperion (SOD) term

*β*

_{2}is complex (

*β*

_{2}=

*T*

_{0}is the inverse gain bandwidth, g stands for the gain and

*l*for the loss. The nonlinear parameter

*γ*is complex (

*γ*=

*γ*

_{0}(1+

*iε*

_{0})) where

*γ*

_{0}is the Kerr nonlinearity and where the small saturable absorber parameter

*ε*

_{0}stands for an approximation of the mode-locking mechanism. Following the presence of gain in this ENLS, this master equation can be called an extended Ginzburg-Landau equation (EGLE) and is given by:

7. Z. Li, L. Li, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse formwith self-frequancy shift,” Phys. Rev. Lett. **89**, 263901 (2002). [CrossRef] [PubMed]

2. P. -A. Bélanger, “On the profile of pulses generated by fiber lasers,” Opt. Express **13**, 8089–8096 (2005). [CrossRef] [PubMed]

*β*≅1 to explain most of the experimental observations when the average dispersion is negative (

*<0). For positive average dispersion (*β ¯
L

*>0), no fixed operation point for the chirp parameter*β ¯
L

*β*could be found neither from the model nor from the experiments. Here this extended model yields a direct access to a unique chirp parameter for the different operating regimes. This result comes from the analysis of Eq. (A3) after seperating the real and imaginary parts and assuming that

*β*

_{3}is real. When this is carried out, the following characteristic equation can be deduced for the chirp parameter

*β*:

*β*can be calculated from this third-order algebric relation. Here, in the framework of the laser system under study, the IRS term

*T*

_{R}

*ω*

_{0}must be positive and the chirp parameter must also be positive [see Eq. (A8)]. Hence, as shown in Fig. (1), when

*T*

_{R}

*ω*

_{0}>1.65, two positive chirp parameters

*β*are found for a given value of the IRS effect. Assuming that the saturable absorber parameter

*ε*

_{0}is very small, we can deduce from Eq. (A11) that for

*β*<√2, the average dispersion must be negative. Then, according to Eq. (3), DM operation in the negative average dispersion regime is always possible if th IRS term is larger than 4 (

*T*

_{R}

*w*

_{0}≥4). Assuming that the Raman term

*T*

_{R}=3

*fs*and 5

*fs*respectively, the Raman parameter

*T*

_{R}

*ω*

_{0}are epproximately equal to 3.64 and 6 for a fiber laser operating at 1550

*nm*whereas for a laser operating at a wavelength of 1030

*nm*, the Raman parameter is 5 and 9 respectively. When

*T*

_{R}

*ω*

_{0}<4, two operating modes of the laser can be achieved in the same positive average dispersion regine (

*L*>0):one with a low chirp parameter

*β*and the other corresponding to a large value of the chirp parameter

*β*. For the case corresponding to

*T*

_{R}

*ω*

_{0}>4, the operation of the laser is in the DM negative average dispersion regime or in a highly-chirped positive dispersion regime. The case corresponding to the situation where

*β*=0 is simply the pure solitonic regime, which occurs in the anolamous dispersion region. Figure (1) summarises well what can be deduced from Eq. (3) and more specifically, we show the operating regimes corresponding to the case where

*T*

_{R}

*ω*

_{0}=5.

## 3. The highly-chirped (similariton) regime

*T*

_{R}

*ω*

_{0}>5), Eq. (3) always yields one large possible value of the chirp parameter

*β*. The chirped solitary pulse will be obviously highly-chirped for large values of

*β*. As from now, the discussion will be restricted to this situation and I will show that this solitary wave solution has most of the features of the so-called

*similariton*operating regime which occurs for large positive average dispersion in a fiber laser. Figure (2) shows the amplitude of the spectrum [(a) and (c)] of the solitary pulse (Eq. (2) of Ref. [2

**13**, 8089–8096 (2005). [CrossRef] [PubMed]

*β*. It is to be noted that the amplitude of the various depicted spectra, while being arbitrary, ensures that all the shown spectra have the

*same*energy. The spectrum tends to a nearly rectangular shape as the value of chirp parameter

*β*increases. This profile appears to be close to the shape of the

*similariton*previously measured and calculated [9

9. A. Ruehl, O. Prochnow, D. Wandt, D Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in a ultrafast fiber laser,” Opt. Lett. **31**, 2734–2736 (2006). [CrossRef] [PubMed]

*β*.

**13**, 8089–8096 (2005). [CrossRef] [PubMed]

*β*, Eq. (4) can be approximated by:

9. A. Ruehl, O. Prochnow, D. Wandt, D Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in a ultrafast fiber laser,” Opt. Lett. **31**, 2734–2736 (2006). [CrossRef] [PubMed]

*ε*

_{0}. In appendix B, it is shown that the passage of a rectangular spectral profile

*V̂*

_{0}(

*ν*) through the absorber will be transformed to:

5. 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]

9. A. Ruehl, O. Prochnow, D. Wandt, D Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in a ultrafast fiber laser,” Opt. Lett. **31**, 2734–2736 (2006). [CrossRef] [PubMed]

*similariton*regime was first introduced by

*Ilday et al.*[3

3. F. Ö. 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]

*Anderson et al.*[8

8. 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]

*ν*

_{f}=7 for their calculated spectral width. The measured spectrum appears to be shifted as inferred by the present model through the presence of the TOD. From the result of Eq. (9) and from the knowledge of the pulse width, the value of

*β*can be estimated to be around 80 which implies an IRS term of

*T*

_{R}

*ω*

_{0}≈20. It is important here to understand that the distributed model gives only an

*averaged*information of the pulse that propagates in the laser and no accurate information on the local temporal and phase profiles is available. However, as pointed out and discussed in a previous paper [2

**13**, 8089–8096 (2005). [CrossRef] [PubMed]

*β*and assuming that the saturable absorber term

*ε*

_{0}and the frequency chirp to be very small, the main characteristics of the chirped averaged similariton pulse can be obtained approximately from the equations of appendix A:

*L*is given by:

## 4. Conclusion

*β*. This main result defines clearly three different operating regimes for a fiber laser system. The first regime is the solitonic regime which appears as a unique point in Fig. (1). Two other values for the chirp parameter

*β*can also be found to satisfy the phase of the chirped pulse solution. The smaller one defines the DM regime laser operation either in anomalous or normal average dispersion regime. The anomalous dispersion regime is always possible for a realistic Raman parameter and the chirp parameter will be close to 1. The normal average dispersion regime seems to be possible if the Raman parameter is very small and if the chirp parameter is larger than √2. However, for a laser operating in a positive dispersion regime, the chirp parameter is always large and the corresponding highly-chirped pulse appears to have the characteristics features of the so-called

*similariton*regime recently introduced as a modification of what is commonly known as the temporal parabolic pulse regime [3

3. F. Ö. 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]

4. F. Ilday, F Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express **12**, 2731–2738 (2004). [CrossRef] [PubMed]

5. 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]

**31**, 2734–2736 (2006). [CrossRef] [PubMed]

*g*-

*l*) will have a RMS chirp parameter

*C*given by:

10. 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**, 6010–6013 (2000). [CrossRef] [PubMed]

*β*, is given by:

*doubles*the optimal chirp that the amplifier could have achieved. The present model has thus demonstrated that it is

*not*necessary to introduce the self-similar temporal parabolic pulse in order to explain the

*similariton*regime. However, the chirped solitary pulse used here is without doubt a self-similar pulse and hence the term

*similariton*[11

11. C. Finot, G. Millot, C. Billet, and J. M. Dudley, “Experimental generation of parabolic pulses via Raman amplification in optical fiber,” Opt. Express **11**, 1547–1552 (2003). [CrossRef] [PubMed]

## A. Appendix A

7. Z. Li, L. Li, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse formwith self-frequancy shift,” Phys. Rev. Lett. **89**, 263901 (2002). [CrossRef] [PubMed]

*V*

_{0},

*a*,

*b*,Γ,

*α*,

*β*) of ansatz (2). However, Eqs. (A1) and (A3) as well as Eqs. (A2) and (A4) impose certain relations among some of the internal and external parameters of the laser system. Here, I will assume that

*β*

_{3}is real (it is to be pointed out that this is

*not*the case in Ref. [7

7. Z. Li, L. Li, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse formwith self-frequancy shift,” Phys. Rev. Lett. **89**, 263901 (2002). [CrossRef] [PubMed]

*β*relative to the Raman term namely

*T*

_{R}such as:

*a*must satisfy the following relation:

*a*is of course assumed to be much smaller than the central frequency

*ω*

_{0}and from here I shall neglect all the contribution in

*ω*

_{0}→∞), Eq. [A3] fixes the chirp parameter to

*β*≡1, and as discussed in Ref. [2

**13**, 8089–8096 (2005). [CrossRef] [PubMed]

*β*≡1) is typical of the negative DM regime.

## B. Appendix B

*V*(

*τ*,

*x*) propagating through a saturable absorber is modelled by the differential equation:

*β*this relation reads as

*β*has been used accordingly with Eq. (5). When

*β*is large, the incoming spectrum

*V̂*

_{i}is rectangular and has been normalised to 1. Assuming that

*ν*

_{f}is still the FWHM of the output spectrum, it is straightforward to show that:

## References and links

1. | H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. |

2. | P. -A. Bélanger, “On the profile of pulses generated by fiber lasers,” Opt. Express |

3. | F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. |

4. | F. Ilday, F Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express |

5. | 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 |

6. | N. Akhmediev and A. Ankiewitz, |

7. | Z. Li, L. Li, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse formwith self-frequancy shift,” Phys. Rev. Lett. |

8. | 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 |

9. | A. Ruehl, O. Prochnow, D. Wandt, D Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in a ultrafast fiber laser,” Opt. Lett. |

10. | 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. |

11. | C. Finot, G. Millot, C. Billet, and J. M. Dudley, “Experimental generation of parabolic pulses via Raman amplification in optical fiber,” Opt. Express |

12. | L. M. Zhao, D. Y. Tang, T. H. Cheng, and C. Lu, “Gain-guided solitons in dispersion-managed fiber lasers with large net cavity dispersion,” Opt. Lett. |

13. | L. M. Zhao, D. Y. Tang, and C. Lu, “Gain-guided solitons in a positive group-dispersion fiber laser,” Opt. Lett. |

14. | I. S. Gradshteyn and I. M. Ryzhik, |

**OCIS Codes**

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

(140.3510) Lasers and laser optics : Lasers, fiber

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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 27, 2006

Manuscript Accepted: November 28, 2006

Published: December 11, 2006

**Citation**

Pierre-André Bélanger, "On the profile of pulses generated by fiber lasers:the highly-chirped positive dispersion regime (similariton)," Opt. Express **14**, 12174-12182 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12174

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### References

- H. A. Haus, E. P. Ippen, and K. Tamura, "Additive-pulse mode-locking in fiber lasers," IEEE J. Quantum Electron. 30,200-208 (1994). [CrossRef]
- P. -A. Bélanger, "On the profile of pulses generated by fiber lasers," Opt. Express 13,8089-8096 (2005). [CrossRef] [PubMed]
- F. Ö. 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]
- F. Ilday, F. Wise, and F. Kaertner, "Possibility of self-similar pulse evolution in a Ti:sapphire laser," Opt. Express 12,2731-2738 (2004). [CrossRef] [PubMed]
- B. Ortac¸, 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]
- N. Akhmediev, and A. Ankiewitz, Solitons, nonlinear pulses and beams (Chapman and Hall, London, 1997).
- Z. Li, L. Li, G. Zhou, and K. H. Spatschek, "Chirped femtosecond solitonlike laser pulse formwith self-frequancy shift," Phys. Rev. Lett. 89,263901 (2002). [CrossRef] [PubMed]
- 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]
- A. Ruehl, O. Prochnow, D. Wandt, D, Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of parabolic pulses in a ultrafast fiber laser," Opt. Lett. 31,2734-2736 (2006). [CrossRef] [PubMed]
- 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. 84,6010-6013 (2000). [CrossRef] [PubMed]
- C. Finot, G. Millot, C. Billet, and J. M. Dudley, "Experimental generation of parabolic pulses via Raman amplification in optical fiber," Opt. Express 11,1547-1552 (2003). [CrossRef] [PubMed]
- L. M. Zhao, D. Y. Tang, T. H. Cheng, and C. Lu, "Gain-guided solitons in dispersion-managed fiber lasers with large net cavity dispersion," Opt. Lett. 31,2957-2959 (2006). [CrossRef] [PubMed]
- L. M. Zhao, D. Y. Tang, and C. Lu, "Gain-guided solitons in a positive group-dispersion fiber laser," Opt. Lett. 31,1788-1790 (2006). [CrossRef] [PubMed]
- I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integrals, Series and Products. (Academic press, New York, 2000).

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