## 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 Optical Society of America

## 1. Introduction

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 [6] 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 [10] 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 [12]. 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 [8] 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 [15]. As an example, such a saturation effect is important for the pulse evolution in normal fiber ring lasers [16].

We note that an approximate solution of the NLSE with the same model equation for the saturation effect has been proposed [17]. 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 *η _{s}* is greater than some critical parameter

*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}We have also observed that fiber amplifiers support a new type of self-similar linearly chirped pulses when the condition *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

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 as

*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 $\sigma ={\mathrm{\Omega}}_{g}^{-2}$ is the parameter of the bandwidth-limited gain in the fiber. We also use here a standard model equation for the saturation effect [15] which follows from averaging the gain dynamics in the presence of the pulse train:

*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(*i*Φ(*z,τ*)) and the ansatz:

*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:

*E*

_{0}=

*E*(0) is the input energy. The above definitions allow us to transform the generalized NLSE to the NLSE without gain:

*B*(

*z,τ*) and the phase Φ(

*z,τ*):

*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

**(

*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]:

*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

*τ*(

_{p}*z*) can be found by solving the equation:

*g*is a constant and

*E*(

*z*) =

*E*

_{0}

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

^{gz}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| <* *τ _{p}*(

*z*) as

*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 ${\eta}_{s}=\gamma {E}_{s}/\sqrt{{g}_{0}{\beta}_{2}}$ is small enough (

*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

We present in this section a new asymptotically exact parabolic solution of NLSE with gain saturation. Using the standard model for the saturation effect [15] given by Eq. (2), the energy of the pulses can be found by

Solving Eq. (16) we can find the energy of the pulse from the following equation: with*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):

*ξ*≫ |

*α*

^{−1}− ln(

*αξ*)| is satisfied.

Similarly, we define the pulse width *τ _{p}*(

*z*) of the similaritons in the form:

*T*(

*ξ*) is dimensionless. Using this expression we can rewrite 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*):

*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:

It follows from Eq. (18) that *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.

Let us define a new function *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

*Y*(

*x*) =

*U*(

*x*)

^{1/3}one can write Eq. (23) as follows:

*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:

*g*(

*z*) =

*g*

_{0}(1 +

*E*(

*z*)

*/E*)

_{s}^{−1}.

We define the integration constant *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

Finally, we obtain the analytical expression for the effective pulse width:

*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):

It follows from Eq. (12) that the chirp of the pulses has the form Ω(*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*).

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 *E*_{0} of the pulses.

The asymptotical solution given by Eqs. (11)–(13) and Eqs. (29)–(31) becomes close to the exact solution when the condition 3ln*ε* (*ξ*) ≫ |*κ*| is satisfied. Using Eq. (18) we can write this condition in an explicit form as

The integration constant *κ* defines the region (*z _{b}*, +∞) 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

*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.

The parameter *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 _{b}* for the interval (

*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

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 are

*g*(

*z*) =

*g*

_{0}(1 +

*E*(

*z*)/

*E*)

_{s}^{−1}and

*σ*= 0 has the form:

*η*and input energy

_{s}*η*

_{0}of the pulses are given by

*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}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*) is

*ω*(

*ξ,ζ*) = −

*ϕ*(

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

*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 $\chi (0,\zeta )={\pi}^{-1/4}\sqrt{{\eta}_{0}}\text{exp}\hspace{0.17em}(-{\zeta}^{2}/2)$ and the following parameters of the fiber amplifier:

_{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.

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 *η _{s}* = 100 (with

*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

We have found in the previous section that the dimensionless NLSE depends only on a single parameter
${\eta}_{s}=\gamma {E}_{s}/\sqrt{{g}_{0}{\beta}_{2}}$ 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 *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

_{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 $E(z)={\int}_{-\infty}^{+\infty}P(z,\tau )d\tau $ yields the constant parameter ${\mathrm{\Lambda}}^{-1}={\int}_{-\infty}^{+\infty}\text{exp}\hspace{0.17em}\left[-{\sum}_{k=1}^{N}{\sigma}_{k}{x}^{k}\right]\hspace{0.17em}dx$. When the gain is given by Eq. (2) we can represent the constant width parameter in the form

*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:

*z*→ ∞ as

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

*w*(

*z*) =

*μ*(

*z*+

_{c}*z*) and ${\mathrm{\Lambda}}^{-1}={\int}_{-\infty}^{+\infty}\text{exp}\hspace{0.17em}\left[-{x}^{2}-\sigma {x}^{4}\right]\hspace{0.17em}dx$. We name these pulses Hyper-Gaussian similaritons because the shape of such self-similar pulses is a product of Gaussian and Super-Gaussian distributions. It also follows from Eq. (42) and Eq. (44) that the phase and the amplitude of HG similaritons for sufficient distances (

*z*≫

*z*) depend on two parameters

_{c}*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 *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]. 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

_{c}*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

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 (*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.

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