## Parabolic and hyper-Gaussian similaritons in fiber amplifiers and lasers with gain saturation |

Optics Express, Vol. 20, Issue 8, pp. 8741-8754 (2012)

http://dx.doi.org/10.1364/OE.20.008741

Acrobat PDF (1444 KB)

### Abstract

We present a new asymptotically exact analytical similariton solution of the generalized nonlinear Schrdinger equation for pulses propagating in fiber amplifiers and lasers with normal dispersion including the effect of gain saturation. Numerical simulations are in excellent agreement with this analytical solution describing self-similar linearly chirped parabolic pulses. We have also found that for small enough values of the dimensionless saturation energy parameter the fiber amplifiers and lasers can generate a new type of linearly chirped self-similar pulses, which we call Hyper-Gaussian similaritons. The analytical Hyper-Gaussian similariton solution of the generalized nonlinear Schrdinger equation is also in a good agreement with numerical simulations.

© 2012 OSA

## 1. Introduction

1. C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. **69**, 3048–3051 (1992). [CrossRef] [PubMed]

5. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. **84**, 1902–1905 (2000). [CrossRef] [PubMed]

6. 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]

7. M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

9. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B **19**, 461–469 (2002). [CrossRef]

10. V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B **23**, 2541–2550 (2006). [CrossRef]

10. V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B **23**, 2541–2550 (2006). [CrossRef]

11. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. **3**, 597–603 (2007). [CrossRef]

12. A. Ruehl, A. Marcinkevicius, M. E. Fermann, and I. Hartl, “80 W, 120 fs Yb-fiber frequency comb,” Opt. Lett. **35**, 3015–3017 (2010). [CrossRef] [PubMed]

7. M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

13. F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 *μ*m from a wave-breaking-free fiber laser,” Opt. Lett. **28**, 1365–1367 (2003). [CrossRef] [PubMed]

14. F. O. 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]

8. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

16. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. **82**, 021805 (2010). [CrossRef]

17. B. G. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. **35**, 2466–2468 (2010). [CrossRef] [PubMed]

*η*is greater than some critical parameter

_{s}*η*≃ 0.3 (see Section 5). In the cases when the condition

_{c}*η*>

_{s}*η*is not satisfied the parabolic similariton regimes do not exist.

_{c}*η*<

_{s}*η*is satisfied. The shape of such pulses differs significantly from the parabolic profile, but the self-similarity of the pulses has been confirmed numerically with a high accuracy. We also show with a high accuracy that the shape of these similaritons is a product of Gaussian and super-Gaussian pulses and we call such pulses Hyper-Gaussian (HG) similaritons. The theory for HG similaritons developed here is in a good agreement with numerical simulations. We anticipate that this new type of HG similaritons may find applications in chirped pulse amplification systems where gain saturation is important since their linear chirp and smooth spectral density facilitates pulse compression.

_{c}## 2. Parabolic similaritons in fiber amplifiers

*ψ*(

*z,τ*) is the slowly varying pulse envelope in a comoving frame,

*β*

_{2}and

*γ*are respectively the second-order dispersion parameter and the nonlinearity coefficient and

*g*(

*z*) is the distributed gain along the fibre. Here

16. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. **82**, 021805 (2010). [CrossRef]

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

*σ*= 0 can be used for describing the propagation of similaritons for all distances such that the pulse spectral width is less than the gain bandwidth. This condition is satisfied for all numerical simulations presented below and this limitation is also assumed in our analytical solutions.

*A*(

*z,τ*) and the phase Φ(

*z,τ*) of the pulses

*ψ*(

*z,τ*) =

*A*(

*z,τ*) exp(

*i*Φ(

*z,τ*)) and the ansatz: we define a new wave function of the pulses in the form

*ψ*̃(

*z,τ*) =

*B*(

*z,τ*) exp(

*i*Φ(

*z,τ*)). The real amplitudes

*A*(

*z,τ*) and

*B*(

*z,τ*) are connected by the relation: where

*E*

_{0}=

*E*(0) is the input energy. The above definitions allow us to transform the generalized NLSE to the NLSE without gain: The Eq. (5) yields the system of equations for the real amplitude

*B*(

*z,τ*) and the phase Φ(

*z,τ*): We will show below that in the cases when

*E*(

*z*) → ∞ with

*z*→ ∞ the condition Γ

*B*

^{2}≫ (

*β*

_{2}/2)|

*B*/

_{ττ}*B*| is satisfied for sufficient propagation distances when

*β*

_{2}> 0. Using Eq. (4) one can also present this inequality in the form: for asymptotical solutions of Eqs. (6), (7). We may neglect the last term on the right-hand side of Eq. (7) for the asymptotical solutions when the Eq. (8) is satisfied. Thus, using the definition

**(

*z,τ*) =

*B*

^{2}(

*z,τ*) and Eq. (8) we may reduce the system of Eqs. (6), (7) for the asymptotical solutions as This system of equations has the parabolic solution [7

7. M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

9. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B **19**, 461–469 (2002). [CrossRef]

**(

*z,τ*) and the phase Φ(

*z,τ*) are quadratic functions of

*τ*for |

*τ*| <

*τ*(

_{p}*z*) and

**(

*z,τ*) = 0 for |

*τ*| ≥

*τ*(

_{p}*z*). Here the function

*τ*(

_{p}*z*) is the effective width of the pulse which defines the region of

*τ*(|

*τ*| <

*τ*(

_{p}*z*)) where the function

**(

*z,τ*) =

*B*

^{2}(

*z,τ*) is positive. Using Eq. (4) we may present the solution of the Eqs. (9), (10) with varying gain function

*g*(

*z*) in the form [8

8. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

*θ*(

*x*) is the step function:

*θ*(

*x*) = 1 for

*x*≥ 0 and

*θ*(

*x*) = 0 otherwise. Here the distance dependent peak power

*P*(

*z*) and the function

*C*(

*z*) are The effective width of the pulse

*τ*(

_{p}*z*) can be found by solving the equation: which follows from Eqs. (9), (10). As an example, when

*g*is a constant and

*E*(

*z*) =

*E*

_{0}

*e*the solution of Eq. (14) is

^{gz}*z*→ ∞) does not depend on the boundary conditions. Using the above parabolic solution we may present the condition given by Eq. (8) for |

*τ| <*

*τ*(

_{p}*z*) as Integrating the left and right sides of Eq. (14) on

*z*we may prove that

*τ*(

_{p}*z*) is an increasing function of

*z*when

*z*→ ∞. Hence, the condition (15) with the pulse width

*τ*(

_{p}*z*) given by Eq. (14) can be satisfied with any accuracy for sufficient distances. However, the conditions (8) or (15) are necessary but not sufficient for the existence of the parabolic solutions. We demonstrate this point in the following section for fiber amplifiers with gain saturation. It is shown that if the dimensionless saturation energy parameter

*η*<

_{s}*η*with

_{c}*η*≃ 0.3) then the pulses propagating in fiber amplifiers will evolve into the Hyper-Gaussian similariton regime.

_{c}## 3. Parabolic solution of NLSE with gain saturation

*E*

_{0}=

*E*(0). Introducing the dimensionless energy

*ε*(

*ξ*) =

*E*(

*z*)/

*E*and the distance

_{s}*ξ*=

*g*

_{0}

*z*, we can find the dimensionless energy with a good accuracy using a first iteration of Eq. (17): when the condition

*ξ*≫ |

*α*

^{−1}− ln(

*αξ*)| is satisfied.

*τ*(

_{p}*z*) of the similaritons in the form: where the distance dependent function

*T*(

*ξ*) is dimensionless. Using this expression we can rewrite Eq. (14) and Eq. (16) in the form:

*T*(

*ξ*) and

*ε*(

*ξ*) yields the second order differential equation for the function

*W*(

*ε*) =

*T*(

*ξ*): In the case when

*ε*≫ 1 we can prove that the condition

*W*

^{2}(

*dW/dε*) ≪ (1 +

*ε*)

^{3}is satisfied and hence Eq. (21) reduces to the equation:

*ε*(

*ξ*) =

*ξ*asymptotically when

*ξ*→ ∞. Setting

*T*(

*ξ*) =

*W*(

*ε*) and

*ξ*=

*ε*we can also derive Eq. (22) from the first equation of the system (20) for

*ε*≫ 1.

*Y*(

*x*) by equation

*W*(

*ε*) =

*εY*(

*x*) with

*x*= ln(

*ε/ε*̄) where

*ε*̄ is an integration constant. Hence, Eq. (22) can be written in terms of

*Y*(

*x*) as Finally, applying the ultimate transformation

*Y*(

*x*) =

*U*(

*x*)

^{1/3}one can write Eq. (23) as follows: We have found the particular solution of this nonlinear differential equation in the form of a series: where

*n*= −1, 0, 1, 2,… and

*m*= 0, 1, 2,… . The important characteristic of this series is that the coefficients (matrix

*B*) can be found using a sequential algorithm: first we find all non zero values of

_{nm}*B*for

_{nm}*n*= −1; secondly we find all non zero values of

*B*for

_{nm}*n*= 0; … finally we find all non zero values of

*B*for

_{nm}*n*=

*k*where

*k*is an arbitrary positive integer number. Thus, the Eq. (24) and Eq. (25) together with above sequential algorithm yields the solution in the form: This solution is not a unique solution for Eq. (24), however it is the required asymptotical solution when the gain is given as

*g*(

*z*) =

*g*

_{0}(1 +

*E*(

*z*)

*/E*)

_{s}^{−1}.

*σ*= −ln

*ε*̄, then

*x*= ln(

*ε/ε*̄) =

*σ*+ ln

*ε*and the function

*W*(

*ε*) =

*εU*(

*σ*+ ln

*ε*)

^{1/3}is

*W*(

*ε*) =

*ε*(

*κ*+3ln

*ε*)

^{1/3}where

*κ*= 3

*σ*+ 5. Therefore, since

*T*(

*ξ*) =

*W*(

*α*

^{−1}+

*ξ*− ln(

*αξ*)) we can write the asymptotical solution for the dimensionless width as

*ε*(

*ξ*) ≫ |

*κ*|), the effective pulse width is proportional to

*ε*(

*ξ*)[ln

*ε*(

*ξ*)]

^{1/3}. The peak power

*P*(

*z*) and the phase function

*C*(

*z*) of the pulses can be found using Eq. (13):

*z,τ*) = −2

*C*(

*z*)

*τ*where the phase function

*C*(

*z*) is given by Eq. (31). This yields, in the asymptotic regime, the chirp function Ω(

*z,τ*) =

*τ*/(

*β*

_{2}

*z*).

*P*(

*z*) and the chirp Ω(

*z,τ*) of the parabolic similaritons in a fiber amplifiers under the influence of gain saturation are asymptotically decreasing functions of the propagating distance

*z*. Furthermore, for long propagating distances the asymptotical solution is independent of

*α*and hence it is independent of the input energy

*E*

_{0}of the pulses.

*ε*(

*ξ*) ≫ |

*κ*| is satisfied. Using Eq. (18) we can write this condition in an explicit form as

*κ*defines the region (

*z*, +∞) where the asymptotical solution has a high accuracy. In principle, the asymptotical solution for large enough

_{b}*z*does not depend on

*κ*, but the bound

*z*depends on

_{b}*κ*. To estimate the parameter

*κ*which leads to the smallest bound

*z*for the asymptotical solution we define the value

_{b}*ε*

_{0}by the condition

*W*(

*ε*

_{0}) =

*ε*

_{0}(

*κ*+ 3ln

*ε*

_{0})

^{1/3}= 0. Hence we set by definition

*κ*= 3

*σ*+ 5 = −3ln

*ε*

_{0}. We note that

*z*

_{0}defined by relation

*ε*

_{0}=

*ε*(

*z*

_{0}) is a singular point for the analytical solution given by Eqs. (29)–(31) because

*P*(

*z*) ∼ (

*κ*+3ln

*ε*(

*z*))

^{−1/3}and hence

*P*(

*z*

_{0}) = ∞ and

*τ*(

_{p}*z*

_{0}) = 0. We also require that

*W*

^{2}(

*dW/dε*) ≪ (1 +

*ε*)

^{3}for

*ε*>

*ε*

_{0}since Eq. (22) follows from Eq. (21) when this condition is satisfied. Using the asymptotical solution

*W*(

*ε*) =

*ε*(

*κ*+ 3ln

*ε*)

^{1/3}we can write this condition as

*ε*

^{2}(

*κ*+3ln

*ε*)+

*ε*

^{2}≪ (1+

*ε*)

^{3}for

*ε*>

*ε*

_{0}. This condition for

*κ*= −3ln

*ε*

_{0}and

*ε*≫ 1 has the form 1 + 3ln(

*ε/ε*

_{0}) ≪

*ε*. It is satisfied for

*ε*>

*ε*

_{0}with a minimal value for

*ε*when

*ε*≃

*ε*

_{0}≃ 10, hence

*κ*= −3ln

*ε*

_{0}≃ −7 and

*σ*≃ −4.

*σ*is defined as

*σ*= −ln

*ε*̄ where

*ε*̄ is some integration constant. We can choose this integration constant as

*ε*̄ ≃ 55 which yields

*κ*≃ −7 and leads to a minimal value of the bound

*z*for the interval (

_{b}*z*, +∞) of distances where the asymptotical solution has a high accuracy. This result is also confirmed by our numerical simulations which we present in the next section.

_{b}## 4. Numerical simulations of parabolic similaritons

*g*(

*z*) =

*g*

_{0}(1 +

*E*(

*z*)/

*E*)

_{s}^{−1}and

*σ*= 0 has the form: The dimensionless energy of the pulses

*η*and input energy

_{s}*η*

_{0}of the pulses are given by with

*η*

_{0}=

*η*(0) and

*E*

_{0}=

*E*(0). Hence, the dimensionless NLSE depends only on a single parameter

*η*. It is shown below that the propagation of the pulses in the fiber amplifiers with the gain given by Eq. (2) are critically dependent on the value of the parameter

_{s}*η*.

_{s}*χ*(

*ξ,ζ*) =

*u*(

*ξ,ζ*)exp(

*iϕ*(

*ξ,ζ*)) where the amplitude

*u*(

*ξ,ζ*) is The dimensionless chirp

*ω*(

*ξ,ζ*) = −

*ϕ*(

_{ζ}*ξ,ζ*) of the similaritons is given by Hence for

*ξ*≫ 1 the chirp has the form

*ω*(

*ξ,ζ*) =

*ζ/ξ*. We can use in these equations an arbitrary integration constant

*κ*for sufficiently large distances

*ξ*, but the value

*κ*= −7 yields a minimal value for the bound

*ξ*of the interval (

_{b}*ξ*, +∞) of distances where the asymptotical solution given by Eqs. (36)–(39) has a high accuracy. We choose in our simulations the Gaussian input pulse

_{b}*β*

_{2}= 0.02

*ps*

^{2}

*m*

^{−1},

*γ*= 2·10

^{−5}

*W*

^{−1}

*m*

^{−1},

*g*

_{0}= 2

*m*

^{−1}. The input energy

*E*

_{0}= 200

*pJ*and the saturation energy

*E*= 2 · 10

_{s}^{4}

*pJ*lead to a dimensionless input energy

*η*

_{0}= 0.02 and dimensionless saturation energy

*η*= 2. Using these parameters the numerical solution (solid line) and the analytical solution (dotted line), given by Eqs. (36)–(39), are plotted in Fig. 1 and Fig. 2 for two different dimensionless distances

_{s}*ξ*, respectively 400 and 4000. The agreement between the numerical and the analytical temporal profile and chirp of the pulses is good in both cases. Therefore we can conclude that the pulse in Fig. 1 with

*ξ*= 400 has already reached the self-similar regime.

*η*= 100 (with

_{s}*η*

_{0}= 10

^{−3}) and for dimensionless distances

*ξ*= 100 and

*ξ*= 600 respectively. We observe again a good match between our analytical solution and the numerical simulations. Small differences between the numerical and the analytical power profiles in Fig. 4 are connected with the numerical error for the split-step Fourier method with the large parameters

*η*= 100 and

_{s}*ξ*= 600. This is the result of an inevitable trade off between computational time and precision in the numerical simulations.

## 5. Hyper-Gaussian similaritons

*η*>

_{s}*η*(

_{c}*η*≃ 0.3) is satisfied, the input pulses will evolve into a similariton regime with a parabolic shape and linear chirp as described in the above sections. In contrast, when the condition

_{c}*η*<

_{s}*η*is satisfied the input pulses evolve into a different similariton regime with a linear chirp. This new type of HG (Hyper-Gaussian) similariton regime is demonstrated in Fig. 5. In this figure the numerical solutions (red solid lines) show the new similariton which differs from the parabolic analytic solutions (blue dot lines), but the chirp of both pulses is the same (see Fig. 6(b)). However, the numerical simulations are in good agreement with the HG similariton solution of Eq. (1) which is also presented in this figure (green dotted lines). The analytical HG similariton solution of Eq. (1) for the pulse power is where

_{c}*w*(

*z*) =

*μ*(

*z*+

_{c}*z*) is the width of HG similariton which increases linearly with distance. Here

*μ*,

*z*and

_{c}*σ*are constant parameters depending on the input wave function

_{n}*ψ*

_{0}(

*τ*) (

*ψ*|

_{z}_{=0}=

*ψ*

_{0}(

*τ*)). The relation

*μ*=

*ρ*(

*γβ*

_{2}

*E*

_{s}g_{0})

^{1/3}where

*ρ*is a dimensionless factor. Hence the width of HG pulse in this case is The phase of HG similaritons has the form: This equation yields the chirp of HG similaritons at

*z*→ ∞ as

*z*→ ∞) the chirp for HG and parabolic similaritons is the same. The derivation of the HG similariton solution given by Eqs. (40)–(42) for a particular class of gain functions and in particular for distributed gain as in Eq. (2) will be presented elsewhere.

*n*> 4) in the expansion of Eq. (40) can be neglected with good accuracy. Moreover, if the shape of the input pulse is symmetric then the expansion in Eq. (40) has only terms with even values of

*n*. We note that without loss of generality one can also choose in Eq. (40) the parameter

*σ*

_{2}= 1. Thus the power of the HG pulses in this case with a good accuracy is given by where

*w*(

*z*) =

*μ*(

*z*+

_{c}*z*) and

*z*≫

*z*) depend on two parameters

_{c}*μ*and

*σ*which can be found numerically from Eq. (1).

*P*(

*z,*0) of HG similaritons given by Eq. (40) and Eq. (44) is asymptotically constant.

*η*>

_{s}*η*. For some class of gain functions this attractor will force any input pulse, regardless of its shape, to evolve into the similariton regime with parabolic power profile [10

_{c}10. V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B **23**, 2541–2550 (2006). [CrossRef]

*η*<

_{s}*η*is satisfied. A good fit is obtained for all of them, confirming the existence of an attractor of the NLSE driving different input pulses into an asymptotic HG similariton regime.

_{c}## 6. Conclusion

*η*<

_{s}*η*) the fiber amplifiers and lasers can form a new type of self-similar linearly chirped pulses, the Hyper-Gaussian similaritons, with a smooth spectral density. We have also found the analytical Hyper-Gaussian similariton solution of the generalized nonlinear Schrdinger equation which is in a good agreement with numerical simulations. The analytical solution and numerical simulations have shown that asymptotically (at

_{c}*z*→ ∞) the chirp of HG and parabolic similaritons is the same. Our numerical simulations have also demonstrated the existence of two different attractors of the NLSE (with saturation effect in the gain), for the conditions

*η*>

_{s}*η*and

_{c}*η*<

_{s}*η*(

_{c}*η*≃ 0.3), evolving different input pulses asymptotically into parabolic and HG similariton regime respectively. These newly discovered linearly chirped HG similaritons can find applications in the systems which use pulses with smooth spectral density since they are suitable for further amplification and compression.

_{c}## References and links

1. | C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. |

2. | A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. |

3. | V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. |

4. | T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. |

5. | S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. |

6. | 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 |

7. | M. E. Fermann, V. I. Kruglov, B. C. Thomson, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. |

8. | V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in fiber amplifiers,” Opt. Lett. |

9. | V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B |

10. | V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B |

11. | J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. |

12. | A. Ruehl, A. Marcinkevicius, M. E. Fermann, and I. Hartl, “80 W, 120 fs Yb-fiber frequency comb,” Opt. Lett. |

13. | F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 |

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

15. | G. P. Agrawal, |

16. | W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. |

17. | B. G. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. |

18. | V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. |

**OCIS Codes**

(140.3280) Lasers and laser optics : Laser amplifiers

(140.3510) Lasers and laser optics : Lasers, fiber

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 17, 2012

Manuscript Accepted: March 14, 2012

Published: March 30, 2012

**Citation**

Vladimir I. Kruglov, Claude Aguergaray, and John D. Harvey, "Parabolic and hyper-Gaussian similaritons in fiber amplifiers and lasers with gain saturation," Opt. Express **20**, 8741-8754 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8741

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### References

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