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 Optical Society of America
Self-similarity is a fundamental physical property that has been studied in many areas of physics and, in particular, in optics [1–5]. In addition, recent studies in nonlinear optics have revealed an important type of optical pulses (similaritons) with a parabolic profile in both the time and frequency domains, and having a linear chirp. These pulses propagate in nonlinear optical fibers with normal second-order group-velocity dispersion  and in optical fiber amplifiers with constant and distributed gain functions [7–9]. The propagating pulses in optical fiber amplifiers with normal dispersion are asymptotically self-similar and their asymptotic behavior depends only on the input energy. This remarkable property is connected with a global attractor  which directs the pulses with different initial conditions to the same self-similar structurally stable asymptotic form [10, 11].
Self-similar parabolic pulses are of fundamental interest because they represent a new class of solution to the nonlinear Schrdinger equation (NLSE) with gain, and have wide-ranging practical significance since their linear chirp leads to highly efficient pulse compression to the sub-100-fs domain . Moreover, fiber amplifiers and lasers which use self-similar propagating pulses in the normal dispersion regime have been demonstrated experimentally to achieve high-energy pulses [7, 13, 14].
We present in this paper a new analytical solution of NLSE describing the propagation of the parabolic similaritons in fiber amplifiers including the influence of saturated gain. This exact asymptotical solution is found by solving the second order differential equation which has been derived previously  for the propagating pulses in optical amplifiers with an arbitrary gain function. We use here the standard model equation for the saturation effect which follows from averaging the gain dynamics in the presence of the pulse train . As an example, such a saturation effect is important for the pulse evolution in normal fiber ring lasers .
We note that an approximate solution of the NLSE with the same model equation for the saturation effect has been proposed . In this parabolic solution, depending on three indeterminate parameters, the peak power is a constant asymptotically and the pulse duration increases linearly with distance in the saturation regime. In contrast to this solution, the peak power of the parabolic similaritons in our asymptotically exact solution is a decreasing function of the propagating distance and the pulse duration is not a linear function of the distance. Furthermore, there are no indeterminate parameters in the new solution presented in this paper. We have confirmed numerically that our analytical solution leads to an accurate description of parabolic pulses for long propagating distances when the dimensionless saturation energy ηs is greater than some critical parameter ηc ≃ 0.3 (see Section 5). In the cases when the condition ηs > ηc is not satisfied the parabolic similariton regimes do not exist.
We have also observed that fiber amplifiers support a new type of self-similar linearly chirped pulses when the condition ηs < ηc 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.
2. Parabolic similaritons in fiber amplifiers
In the presence of an arbitrary distributed gain function pulse propagation in fiber amplifiers and lasers in similariton propagation segment can be described by the generalized NLSE as15] which follows from averaging the gain dynamics in the presence of the pulse train: 16,18]. Thus the Eqs. (1), (2) with 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.
Using the standard definition of the real amplitude A(z,τ) and the phase Φ(z,τ) of the pulses ψ(z,τ) = A(z,τ) exp(iΦ(z,τ)) and the ansatz:Eq. (5) yields the system of equations for the real amplitude B(z,τ) and the phase Φ(z,τ): Eq. (4) one can also present this inequality in the form: 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) = B2(z,τ) and Eq. (8) we may reduce the system of Eqs. (6), (7) for the asymptotical solutions as 7–9] which means that the function (z,τ) and the phase Φ(z,τ) are quadratic functions of τ for |τ| < τp(z) and (z,τ) = 0 for |τ| ≥ τp(z). Here the function τp(z) is the effective width of the pulse which defines the region of τ (|τ| < τp(z)) where the function (z,τ) = B2(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 : Eqs. (9), (10). As an example, when g is a constant and E(z) = E0egz the solution of Eq. (14) is
We note that in the case when a global attractor exists, the asymptotical solution of Eq. (14) (for z → ∞) does not depend on the boundary conditions. Using the above parabolic solution we may present the condition given by Eq. (8) for |τ| < τp(z) asEq. (14) on z we may prove that τp(z) is an increasing function of z when z → ∞. Hence, the condition (15) with the pulse width τp(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 is small enough (ηs < ηc with ηc ≃ 0.3) then the pulses propagating in fiber amplifiers will evolve into the Hyper-Gaussian similariton regime.
3. Parabolic solution of NLSE with gain saturation
We present in this section a new asymptotically exact parabolic solution of NLSE with gain saturation. Using the standard model for the saturation effect  given by Eq. (2), the energy of the pulses can be found byEq. (16) we can find the energy of the pulse from the following equation: Eq. (17):
Similarly, we define the pulse width τp(z) of the similaritons in the form:Eq. (14) and Eq. (16) in the form:
This system of nonlinear differential equations for the functions T (ξ) and ε (ξ) yields the second order differential equation for the function W (ε) = T (ξ):Eq. (21) reduces to the equation:
Let us define a new function Y (x) by equation W (ε) = εY (x) with x = ln(ε/ε̄) where ε̄ is an integration constant. Hence, Eq. (22) can be written in terms of Y (x) asEq. (23) as follows: Eq. (24) and Eq. (25) together with above sequential algorithm yields the solution in the form: Eq. (24), however it is the required asymptotical solution when the gain is given as g (z) = g0(1 + E (z)/Es)−1.
We define the integration constant σ = −lnε̄, then x = ln(ε/ε̄) = σ + lnε and the function W (ε) = εU (σ + lnε)1/3 is
Finally, we obtain the analytical expression for the effective pulse width:Eq. (13):
It follows from Eq. (12) that the chirp of the pulses has the form Ω(z,τ) = −2C(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/(β2z).
The Eqs. (30), (31) demonstrate that the peak power 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 E0 of the pulses.
The asymptotical solution given by Eqs. (11)–(13) and Eqs. (29)–(31) becomes close to the exact solution when the condition 3lnε (ξ) ≫ |κ| is satisfied. Using Eq. (18) we can write this condition in an explicit form as
The integration constant κ defines the region (zb, +∞) where the asymptotical solution has a high accuracy. In principle, the asymptotical solution for large enough z does not depend on κ, but the bound zb depends on κ. To estimate the parameter κ which leads to the smallest bound zb for the asymptotical solution we define the value ε0 by the condition W (ε0) = ε0(κ + 3lnε0)1/3 = 0. Hence we set by definition κ = 3σ + 5 = −3lnε0. We note that z0 defined by relation ε0 = ε(z0) is a singular point for the analytical solution given by Eqs. (29)–(31) because P(z) ∼ (κ +3lnε (z))−1/3 and hence P(z0) = ∞ and τp(z0) = 0. We also require that W2(dW/dε) ≪ (1 +ε)3 for ε > ε0 since Eq. (22) follows from Eq. (21) when this condition is satisfied. Using the asymptotical solution W (ε) = ε (κ + 3lnε)1/3 we can write this condition as ε2(κ +3lnε)+ ε2 ≪ (1+ ε)3 for ε > ε0. This condition for κ = −3lnε0 and ε ≫ 1 has the form 1 + 3ln(ε/ε0) ≪ ε. It is satisfied for ε > ε0 with a minimal value for ε when ε ≃ ε0 ≃ 10, hence κ = −3lnε0 ≃ −7 and σ ≃ −4.
The parameter σ is defined as σ = −lnε̄ where ε̄ is some integration constant. We can choose this integration constant as ε̄ ≃ 55 which yields κ ≃ −7 and leads to a minimal value of the bound zb for the interval (zb, +∞) 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.
4. Numerical simulations of parabolic similaritons
It is useful for numerical simulations to define the dimensionless variables which do not depend on the parameters connected with initial condition for NLSE given by Eqs. (1) and (2). Thus, the natural dimensionless variables areEq. (2) are critically dependent on the value of the parameter ηs.
Using Eqs. (11)–(13) and Eqs. (29)–(31) we may rewrite the exact asymptotical solution of the NLSE with saturation effect in dimensionless form as χ (ξ,ζ) = u (ξ,ζ)exp(iϕ (ξ,ζ)) where the amplitude u (ξ,ζ) isEqs. (36)–(39) has a high accuracy. We choose in our simulations the Gaussian input pulse and the following parameters of the fiber amplifier: β2 = 0.02 ps2m−1, γ = 2·10−5 W−1m−1, g0 = 2m−1. The input energy E0 = 200 pJ and the saturation energy Es = 2 · 104 pJ lead to a dimensionless input energy η0 = 0.02 and dimensionless saturation energy ηs = 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 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.
In Fig. 3 and Fig. 4 we show the numerical solution (solid line) and the analytical solution (dotted line) for a large value of the saturation energy parameter ηs = 100 (with η0 = 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 ηs = 100 and ξ = 600. This is the result of an inevitable trade off between computational time and precision in the numerical simulations.
5. Hyper-Gaussian similaritons
We have found in the previous section that the dimensionless NLSE depends only on a single parameter which is the dimensionless saturation energy. Our numerical simulations have also shown that the propagation of the pulses in the fiber amplifiers with saturated gain are critically dependent on this parameter. It has been found that when the condition ηs > ηc (ηc ≃ 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 ηs < ηc 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 isEq. (2) we can represent the constant width parameter in the form μ = ρ(γβ2Esg0)1/3 where ρ is a dimensionless factor. Hence the width of HG pulse in this case is
From Eq. (31) and Eq. (43) it follows that asymptotically (at 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.
We have found analytically and confirmed numerically in our simulations that the higher order terms (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 σ2 = 1. Thus the power of the HG pulses in this case with a good accuracy is given byEq. (42) and Eq. (44) that the phase and the amplitude of HG similaritons for sufficient distances (z ≫ zc) depend on two parameters μ and σ which can be found numerically from Eq. (1).
Figure 6(a) demonstrates that the HG similaritons undergo only small spectral broadening with a very smooth shape which can be of particular interest for fiber based amplification systems. The linear chirp of HG similaritons (see Fig. 6(b)) is the most important feature of these pulses since it allows easy spectral manipulation and compression. We also emphasize that when the saturation effect is significant the peak power P(z,0) of HG similaritons given by Eq. (40) and Eq. (44) is asymptotically constant.
The asymptotic propagation of parabolic similaritons is connected with the global attractor of the NLSE when ηs > ηc. 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 . We have also observed that for some class of decreasing gain functions the pulses will evolve into the HG similariton regime. To study the existence of an attractor of the NLSE driving the pulses into the asymptotic HG similariton regime we have performed many simulations launching pulses with different temporal shapes. The result of these simulations is shown in Fig. 7. This figure demonstrates that all different input pulses evolve towards the HG regime when the condition ηs < ηc 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.
In conclusion, we have found a new asymptotically exact parabolic similariton solution of the generalized nonlinear Schrdinger equation for pulses propagating in fiber amplifiers and lasers with normal dispersion including the effect of saturation of the gain. Our numerical simulations have demonstrated that this analytical solution describing self-similar linearly chirped parabolic pulses is very accurate. We have also found numerically that for sufficiently small values of the dimensionless saturation energy parameter (ηs < ηc) 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 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 ηs > ηc and ηs < ηc (ηc ≃ 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.
References and links
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. 38, 1189–1202 (1991). [CrossRef]
3. V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992). [CrossRef]
4. T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. 23, 268–270 (1998). [CrossRef]
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]
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]
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]
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]
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 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003). [CrossRef] [PubMed]
15. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2001).
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]