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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 12174–12182
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On the profile of pulses generated by fiber lasers:the highly-chirped positive dispersion regime (similariton)

Pierre-André Bélanger  »View Author Affiliations


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

2. The distributed laser model

The typical fiber laser cavity consists of two concatenated fiber segments with normal dispersion, gain and anomalous dispersion followed by a mode-locking mechanism. In order to study numerically this complex nonlinear system, a master differential equation [6

6. N. Akhmediev and A. Ankiewitz, Solitons, nonlinear pulses and beams (Chapman and Hall, London, 1997).

] has been derived and is commonly known as the extended nonlinear Schroedinger equation (ENLS) where the different physical effects are uniformly distributed along the propagation axis x. Here, in order to study the propagation of very short pulses, I have included the third-order dispersion (TOD) term (β 3), SS ( γω0 ) and the IRS (TR ) effects. The second order disperion (SOD) term β 2 is complex (β 2=β¯2igT02), where β¯ is the group velocity dispersion (GVD), 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+ 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:

iVx+β22Vττi(gl)νγV2V+iβ36Vτττiγ0ω0(V2V)τ+γ0TR(V2)τV=0
(1)

As it is the case for the GL differential equation, the EGLE supports a chirped solitary wave solution [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]

] and is given by:

V(τ,x)=V0{sech[α(τ+bx)]}1iβexp[i(aτΓx)]
(2)

TRω0=β4(β2+9)(β21)
(3)

According to Eq. (3), three chirp parameters β can be calculated from this third-order algebric relation. Here, in the framework of the laser system under study, the IRS term TRω 0 must be positive and the chirp parameter must also be positive [see Eq. (A8)]. Hence, as shown in Fig. (1), when TRω 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 (TRw 0≥4). Assuming that the Raman term TR =3fs and 5fs respectively, the Raman parameter TRω 0 are epproximately equal to 3.64 and 6 for a fiber laser operating at 1550nm whereas for a laser operating at a wavelength of 1030nm, the Raman parameter is 5 and 9 respectively. When TRω 0<4, two operating modes of the laser can be achieved in the same positive average dispersion regine ( β¯ 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 TRω 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 TRω 0=5.

Fig. 1. Chirp parameter β as a function of the IRS effect TRω 0. The different operating regimes can be seen for TRω 0=5. The solitonic regime corresponding to the case where β=0 is also included in the figure.

3. The highly-chirped (similariton) regime

For a realistic IRS parameter (TRω 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

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

]) for different values of the chirp parameter 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]

] in a fiber laser where a clear and steepest decay is observed near ν=νf2 . We also depict in Fig. (2) the phase profile [(b) and (d)] for each value of β.

Fig. 2. Spectral [(a) and (c)] (given by Eq. (2) of Ref. [2]) and phase distribution ((b) and (d)) for different values of β ranging from 0–60. The amplitude of the various depicted spectra, while being arbitrary, ensures that all the shown spectra have the same energy.

The spectral full-width-half-maximum (FWHM) of the pulse is given by [2

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

]:

νf=2απ2arcsinh[cosh(π2β)]
(4)

For large values of the chirp parameter β, Eq. (4) can be approximated by:

νfαβπ
(5)

In a real fiber system [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]

], the pulse will propagate through the mode-locking device before being ejected from the resonator. In the present work, I have modelled the saturable absorber with a small nonlinear parameter ε 0. In appendix B, it is shown that the passage of a rectangular spectral profile 0 (ν) through the absorber will be transformed to:

V̂(ν)=V̂0exp(1.386ν2νf2)forννf2
(6a)
V̂(ν)=0forν>νf2
(6b)

The spectral amplitude is shown in Fig. 3(a) and is typical of the one calculated and measured in the laser system of Refs. [5

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

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]

]. Now, assuming that the pulse is chirp-free outside the resonator, the temporal profile is depicted in Fig. 3(b).

Fig. 3. Spectral (a) profile given by Eqs. (6a) and (6b) and corresponding temporal profile (b) calculated from the inverse Fourier transform of Eq. (6a).

For Eqs. (6a) and (6b), the calculated time-bandwidth product (TBP) is:

τfνf=0.9309
(7)

The so-called 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]

] and was inspired at first from the idea of using the parabolic pulse asymptotic solution of the nonlinear Schroedinger equation (NLS) which was first reported by 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]

]. However, they were well aware that this asymptotic solution cannot be compatible with the periodic boundary condition of a laser resonator. According to the present model, the average pulse profile inside the laser is given by:

V(τ)=V0sech(ατ)exp{iβln[sech(ατ)]}
(8)

where the small frequency shift effect a has been neglected. In order to show that this profile is more realistic that a parabolic one in such a highly-chirped regime, I shall compare it with the results reported recently by Ruehl et al. [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]

]. In this paper, the authors model the laser system using the usual split-step Fourier algorithm and they use a two-level-model to calculate the parameters of the gain fiber. The calculated temporal phase and amplitude profile corresponds to Fig. 1(a) of their article. The pulse width of 6.5ps corresponds to a root-mean-square (RMS) width of 2.76ps for a Gaussian profile and to 3.34ps for the chirped pulse distribution given by Eq. (8) which is to be compared to their calculated RMS width of 3.1ps. For large values of t, their phase profile is linear with a slope close to αβ=19 whereas for the pulse distribution given by Eq. (8), the predicted slope will be given by αβ and when combined with the result of Eq. (5), for large β, gives:

αβ=πνf=22
(9)

(γ0L)V02=(ΓL)
(10)

where the total phase shift for a laser of length L is given by:

(ΓL)=π2νf22(β¯L)
(11)

assuming that for a parabolic gain profile, the gain parameter (2gT02) can be approximated by:

2gLT020.05νg2
(12)

where νg is the frequency gain bandwidth. Using Eqs. (A11) and (11), it can be shown that:

νf2νg212(ΓL)β
(13)

Refering again to the laser system of Ref. [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]

], with β¯ L=0.013, Eq. (13) yields a phase shift (ΓL=π). With the estimated chirp parameter β=80, we can predict via the use of Eq. (13) that the gain frequency width should be 10THz which is close to the gain baindwidth of 12.7THz used in Ref. [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]

]. In a recent paper by Zhao et al. [12

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. 31, 2957–2959 (2006). [CrossRef] [PubMed]

] on the propagation of gain-guided solitons in a DM laser operating with a large positive dispersion, the authors approximately report the experimental observation of the same typical spectrum. Finally, following the numerical integration of the GL equation, they have obtained a similar type of spectrum [13

13. 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]

] as the one described analytically in the present paper.

4. Conclusion

After including the SS and IRS terms into the GL differential equation, I have shown how to derive a characteristic equation for the chirp parameter 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

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

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

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]

]. The parabolic pulse is an asymptotic solution of a pulse propagating into an amplifier and is not compatible with the periodic conditions imposed by a laser resonator. The parabolic pulse propagating into an amplifier of gain (g-l) will have a RMS chirp parameter C given by:

C=0.66(gl)Lβ¯L
(14)

according to Ref. [10

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]

]. For the chirped hyperbolic-secant distribution, the RMS chirp parameter, for large β, is given by:

C=1.22(gl)Lβ¯L
(15)

This result shows that the laser resonator 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]

] should continue to be used to define this high energy laser regime.

A. Appendix A

It has already been observed [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]

] that the chirped hyperbolic secant ansatz given by Eq. (2) is a solitary wave solution to the EGLE given by Eq. (1) under specific conditions. After direct substitution of ansatz (2) in Eq. (1), the resulting equation can be seperated into a time-symmetrical part and a time-antisymmetrical part. Ansatz (2) is a solution of the symmetrical differential equation if:

α2(1iβ)(iβ2)=2V02(γγ0aω0)(β2β3a)
(A1)

and

α2(1iβ)2=2[Γa22(β2β3a3)i(gl)(β2β3a)]
(A2)

In order to force the antisymmetrical part to satisfy ansatz (2), the two following additional complex conditions need to be satisfied:

α2(1iβ)(iβ2)=6γ0V02β3ω0(β2+9)[(β2+9)2TRω0β+6iTRω0]
(A3)
α2(1iβ)2=6β3(β2aβ3a22+b)
(A4)

These four complex relations result into eight real equations for the six characteristic parameters (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]

]) and as a consequence of this assumption, Eq. (A3) can be solved and allows one to fix the chirp parameter β relative to the Raman term namely TR such as:

β4(β2+9)(β21)=TRω0
(A5)

Next, after imposing the compatibility relation between Eqs. (A1) and (A2), one finds that the phase shift parameter a must satisfy the following relation:

a2ω02aω0[1ε0(β22)3β]=α2(β2+1)(β2+4)6(β21)
(A6)

The phase shift parameter 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 a2ω02 . Therefore, within this approximation, the rest of the external parameters are given by:

Γ=(gl)(β2+2)β
(A7)
γ0V02=3(gl)(β2+4)[3βε0(β22)]
(A8)
α2=3(gl)gT0(β2+1)
(A9)
b=β¯a(β2+1)[βε0(β2+2)]2β[(β22)+3βε0]
(A10)

Finally, the two residual compatibility relations read as:

gT02=β¯2[3βε0(β22)][(β22)+3βε0]
(A11)
β3ω0=3β¯(β2+1)(β2+4)2(β21)[(β22)+3βε0]
(A12)

Notice that if the the self-steepening is not included into the model (ω 0→∞), Eq. [A3] fixes the chirp parameter to β≡1, and as discussed in Ref. [2

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

], this chirp parameter (β≡1) is typical of the negative DM regime.

B. Appendix B

The propagation of a pulse V (τ,x) propagating through a saturable absorber is modelled by the differential equation:

Vx=ε0γ0V2V
(B1)

FT[V2V]=[(1iβ)2+(2πνα)2](2iβ)(1iβ)V̂
(B2)

V̂=V̂iexp[ε0γ0x0V02(2πνα)2(2iβ)(1iβ)]
(B3)

For very large values of β this relation reads as

V̂=V̂iexp[ε0γ0x0V02(4ν2νf2)]
(B4)

where the spectral width for large values of β has been used accordingly with Eq. (5). When β is large, the incoming spectrum i is rectangular and has been normalised to 1. Assuming that νf is still the FWHM of the output spectrum, it is straightforward to show that:

V̂(ν)=V̂0exp(1.386ν2νf2)forννf2
(B5a)
V̂(ν)=0forν>νf2
(B5b)

References and links

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]

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]

6.

N. Akhmediev and A. Ankiewitz, Solitons, nonlinear pulses and beams (Chapman and Hall, London, 1997).

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]

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]

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]

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]

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]

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. 31, 2957–2959 (2006). [CrossRef] [PubMed]

13.

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]

14.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products. (Academic press, New York, 2000).

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

  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]
  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. 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]
  6. N. Akhmediev, and A. Ankiewitz, Solitons, nonlinear pulses and beams (Chapman and Hall, London, 1997).
  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]
  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]
  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]
  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]
  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]
  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. 31,2957-2959 (2006). [CrossRef] [PubMed]
  13. 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]
  14. I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integrals, Series and Products. (Academic press, New York, 2000).

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