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

Vladimir I. Kruglov; Claude Aguergaray; John D. Harvey

doi:10.1364/OE.20.008741

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_c ≃ 0.3 (see Section 5). In the cases when the condition η_s > η_c 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 η_s < η_c 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 as

i ψ_{z} = \frac{β_{2}}{2} ψ_{τ τ} - γ {| ψ |}^{2} ψ + i \frac{g (z)}{2} ψ + i \frac{g (z)}{2} σ ψ_{τ τ},

where ψ(z,τ) is the slowly varying pulse envelope in a comoving frame, β₂ and γ are respectively the second-order dispersion parameter and the nonlinearity coefficient and g(z) is the distributed gain along the fibre. Here

σ = Ω_{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:

g (z) = g_{0} {(1 + \frac{1}{E_{s}} \int_{- \infty}^{+ \infty} {| ψ (z, τ) |}^{2} d τ)}^{- 1} .

We note that this approach neglects the dependence of the gain on the frequency of the signal. It is well-known however, that the gain bandwidth of the amplifier provides a limitation to the propagation of similaritons [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:

\begin{matrix} ψ (z, τ) = exp (\frac{1}{2} G (z)) \tilde{ψ} (z, τ), & G (z) = \int_{0}^{z} g (z^{'}) d z^{'}, \end{matrix}

we define a new wave function of the pulses in the form ψ̃(z,τ) = B(z,τ) exp(iΦ(z,τ)). The real amplitudes A(z,τ) and B(z,τ) are connected by the relation:

\begin{matrix} A (z, τ) = \sqrt{\frac{E (z)}{E_{0}}} B (z, τ), & E (z) = E_{0} exp (G (z)), \end{matrix}

where

E (z) = \int_{- \infty}^{+ \infty} {| ψ (z, τ) |}^{2} d τ

is the energy of the pulses and E₀ = E(0) is the input energy. The above definitions allow us to transform the generalized NLSE to the NLSE without gain:

\begin{matrix} i {\tilde{ψ}}_{z} = \frac{β_{2}}{2} {\tilde{ψ}}_{τ τ} - Γ (z) {| \tilde{ψ} |}^{2} \tilde{ψ}, & Γ (z) = γ exp (G (z)) . \end{matrix}

The Eq. (5) yields the system of equations for the real amplitude B(z,τ) and the phase Φ(z,τ):

2 B B_{z} = β_{2} Φ_{τ τ} B^{2} + 2 β_{2} Φ_{τ} B B_{τ}

Φ_{z} = Γ B^{2} + \frac{β_{2}}{2} {(Φ_{τ})}^{2} - \frac{β_{2}}{2} (\frac{B_{τ τ}}{B}) .

We will show below that in the cases when E(z) → ∞ with z → ∞ the condition ΓB² ≫ (β₂/2)|B_ττ/B| is satisfied for sufficient propagation distances when β₂ > 0. Using Eq. (4) one can also present this inequality in the form:

2 γ A^{2} ≫ β_{2} | \frac{A_{τ τ}}{A} |,

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) = B²(z,τ) and Eq. (8) we may reduce the system of Eqs. (6), (7) for the asymptotical solutions as

𝒡_{z} = β_{2} Φ_{τ τ} 𝒡 + β_{2} Φ_{τ} 𝒡_{τ}

Φ_{z} = Γ 𝒡 + \frac{β_{2}}{2} {(Φ_{τ})}^{2} .

This system of equations has the parabolic solution [7–9] which means that the function 𝒡(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) = 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 [8]:

A (z, τ) = P {(z)}^{1 / 2} {(1 - \frac{τ^{2}}{τ_{p} {(z)}^{2}})}^{1 / 2} θ (τ_{p} (z) - | τ |),

Φ (z, τ) = ϕ_{0} + \frac{3 γ}{4} \int_{0}^{z} \frac{E (z^{'})}{τ_{p} (z^{'})} d z^{'} + C (z) τ^{2},

where 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

\begin{matrix} P (z) = \frac{3 E (z)}{4 τ_{p} (z)}, & C (z) = - \frac{1}{2 β_{2} τ_{p} (z)} \frac{d τ_{p} (z)}{d z} \end{matrix} .

The effective width of the pulse τ_p(z) can be found by solving the equation:

\frac{d^{2} τ_{p} (z)}{d z^{2}} = (\frac{3 γ β_{2}}{2}) \frac{E (z)}{τ_{p} {(z)}^{2}},

which follows from Eqs. (9), (10). As an example, when g is a constant and E(z) = E₀e^gz the solution of Eq. (14) is

τ_{p} (z) = 3 {(\frac{γ β_{2} E_{0}}{2 g^{2}})}^{1 / 3} e^{g z / 3} .

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

(\frac{3 γ}{2 β_{2}}) E (z) τ_{p} (z) ≫ {(1 - \frac{τ^{2}}{τ_{p} {(z)}^{2}})}^{- 3} .

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} = γ E_{s} / \sqrt{g_{0} β_{2}}

is small enough (η_s < η_c with η_c ≃ 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 [15] given by Eq. (2), the energy of the pulses can be found by

\frac{d E (z)}{d z} = \frac{g_{0} E (z)}{1 + E (z) / E_{s}} .

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

E (z) = E_{0} + E_{s} g_{0} z - E_{s} ln \frac{E (z)}{E_{0}},

with E₀ = E(0). Introducing the dimensionless energy ε (ξ) = E(z)/E_s and the distance ξ = g₀z, we can find the dimensionless energy with a good accuracy using a first iteration of Eq. (17):

\begin{matrix} ε (ξ) = α^{- 1} + ξ - ln (α ξ), & α = E_{s} / E_{0}, \end{matrix}

when the condition ξ ≫ |α⁻¹ − ln(αξ)| is satisfied.

Similarly, we define the pulse width τ_p(z) of the similaritons in the form:

τ_{p} (z) = {(\frac{3 γ β_{2} E_{s}}{2 g_{0}^{2}})}^{1 / 3} T (ξ),

where the distance dependent function T (ξ) is dimensionless. Using this expression we can rewrite Eq. (14) and Eq. (16) in the form:

\begin{matrix} \frac{d^{2} T}{d ξ^{2}} = \frac{ε}{T^{2}}, & \frac{d ε}{d ξ} = \frac{ε}{1 + ε} . \end{matrix}

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):

\frac{d^{2} W}{d ε^{2}} + \frac{1}{ε (1 + ε)} \frac{d W}{d ε} = \frac{{(1 + ε)}^{2}}{ε W^{2}} .

In the case when ε ≫ 1 we can prove that the condition W²(dW/dε) ≪ (1 +ε)³ is satisfied and hence Eq. (21) reduces to the equation:

\frac{d^{2} W}{d ε^{2}} = \frac{ε}{W^{2}} .

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/ε̄) where ε̄ is an integration constant. Hence, Eq. (22) can be written in terms of Y (x) as

\frac{d^{2} Y}{d x^{2}} + \frac{d Y}{d x} = \frac{1}{Y^{2}} .

Finally, applying the ultimate transformation Y (x) = U (x)^1/3 one can write Eq. (23) as follows:

U \frac{d^{2} U}{d x^{2}} - \frac{2}{3} {(\frac{d U}{d x})}^{2} + U \frac{d U}{d x} - 3 U = 0.

We have found the particular solution of this nonlinear differential equation in the form of a series:

U (x) = \sum_{n} \sum_{m} B_{n m} \frac{{(ln x)}^{m}}{x^{n}},

where n = −1, 0, 1, 2,… and m = 0, 1, 2,… . The important characteristic of this series is that the coefficients (matrix B_nm) can be found using a sequential algorithm: first we find all non zero values of B_nm for n = −1; secondly we find all non zero values of B_nm for n = 0; … finally we find all non zero values of B_nm for 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:

\begin{array}{l} U (x) & = & 3 x + 5 + 2 ln x + \frac{4 ln x}{3 x} - \frac{31}{9 x^{2}} + \frac{8 ln x}{9 x^{2}} - \frac{4 {(ln x)}^{2}}{9 x^{2}} \\ - \frac{766}{81 x^{3}} + \frac{140 ln x}{27 x^{3}} - \frac{8 {(ln x)}^{2}}{9 x^{3}} + \frac{16 {(ln x)}^{3}}{81 x^{3}} + \dots . \end{array}

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) = g₀(1 + E (z)/E_s)⁻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

\begin{array}{l} W (ε) = ε [3 (σ + ln ε) + 5 + 2 ln (σ + ln ε) + \frac{4 ln (σ + ln ε)}{3 (σ + ln ε)} \\ {- \frac{31}{9 {(σ + ln ε)}^{2}} + \frac{8 ln (σ + ln ε)}{9 {(σ + ln ε)}^{2}} - \frac{4 {(ln (σ + ln ε))}^{2}}{9 {(σ + ln ε)}^{2}} + \dots]}^{1 / 3} . \end{array}

Selecting only the main asymptotical terms in this series we find the asymptotical solution in the form W (ε) = ε (κ +3lnε)^1/3 where κ = 3σ + 5. Therefore, since T (ξ) = W (α⁻¹ +ξ − ln(αξ)) we can write the asymptotical solution for the dimensionless width as

T (ξ) = (α^{- 1} + ξ - ln (α ξ)) {[κ + 3 ln (α^{- 1} + ξ - ln (α ξ))]}^{1 / 3} .

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

τ_{p} (z) = {(\frac{3 γ β_{2} E_{s}}{2 g_{0}^{2}})}^{1 / 3} (α^{- 1} + g_{0} z - ln (α g_{0} z)) {[κ + 3 ln (α^{- 1} + g_{0} z - ln (α g_{0} z))]}^{1 / 3} .

Hence, for sufficient propagation distances (3lnε (ξ) ≫ |κ|), the effective pulse width is proportional to ε (ξ)[lnε (ξ)]^1/3. The peak power P(z) and the phase function C(z) of the pulses can be found using Eq. (13):

P (z) = \frac{3}{4} {(\frac{2 g_{0}^{2} E_{s}^{2}}{3 β_{2} γ})}^{1 / 3} {[κ + 3 ln (α^{- 1} + g_{0} z - ln (α g_{0} z))]}^{- 1 / 3},

C (z) = - \frac{(g_{0} z - 1)}{2 β_{2} z (α^{- 1} + g_{0} z - ln (α g_{0} z))} {1 + \frac{1}{κ + 3 ln (α^{- 1} + g_{0} z - ln (α g_{0} z))}} .

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

3 ln (α^{- 1} + g_{0} z - ln (α g_{0} z)) ≫ | κ | .

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_b depends on κ. To estimate the parameter κ which leads to the smallest bound z_b 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 τ_p(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 z_b for the interval (z_b, +∞) 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 are

\begin{matrix} χ (ξ, ζ) = \sqrt{\frac{γ}{g_{0}}} ψ (z, τ), & ξ = g_{0} z, & ζ = \sqrt{\frac{g_{0}}{β_{2}}} τ . \end{matrix}

In this case the NLSE, describing the model saturation effect with the gain g(z) = g₀(1 + E(z)/E_s)⁻¹ and σ = 0 has the form:

i χ_{ξ} = \frac{1}{2} χ_{ζ ζ} - {| χ |}^{2} χ + \frac{i}{2} {(1 + η_{s}^{- 1} \int_{- \infty}^{+ \infty} {| χ (ξ, ζ) |}^{2} d ζ)}^{- 1} χ .

The dimensionless energy of the pulses

η (ξ) = \int_{- \infty}^{+ \infty} {| χ (ξ, ζ) |}^{2} d ζ

can be written as

η (ξ) = γ E (z) / \sqrt{g_{0} β_{2}}

, and the dimensionless saturation energy η_s and input energy η₀ of the pulses are given by

\begin{matrix} η_{s} = \frac{γ E_{s}}{\sqrt{g_{0} β_{2}}}, & η_{0} = \frac{γ E_{0}}{\sqrt{g_{0} β_{2}}}, \end{matrix}

with η₀ = η(0) and E₀ = E(0). Hence, the dimensionless NLSE depends only on a single parameter η_s. 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 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

u (ξ, ζ) = p {(ξ)}^{1 / 2} {(1 - \frac{ζ^{2}}{τ {(ξ)}^{2}})}^{1 / 2} θ (τ (ξ) - | ζ |),

τ (ξ) = {(\frac{3 η_{s}}{2})}^{1 / 3} (α^{- 1} + ξ - ln (α ξ)) {[κ + 3 ln (α^{- 1} + ξ - ln (α ξ))]}^{1 / 3},

p (ξ) = \frac{3}{4} {(\frac{2 η_{s}^{2}}{3})}^{1 / 3} {[κ + 3 ln (α^{- 1} + ξ - ln (α ξ))]}^{- 1 / 3} .

The dimensionless chirp ω (ξ,ζ) = −ϕ_ζ (ξ,ζ) of the similaritons is given by

ω (ξ, ζ) = \frac{(ξ - 1)}{ξ (α^{- 1} + ξ - ln (α ξ))} {1 + \frac{1}{κ + 3 ln (α^{- 1} + ξ - ln (α ξ))}} ζ .

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_b of the interval (ξ_b, +∞) of distances where the asymptotical solution given by Eqs. (36)–(39) has a high accuracy. We choose in our simulations the Gaussian input pulse

χ (0, ζ) = π^{- 1 / 4} \sqrt{η_{0}} exp (- ζ^{2} / 2)

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 E_s = 2 · 10⁴ pJ lead to a dimensionless input energy η₀ = 0.02 and dimensionless saturation energy η_s = 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.

Fig. 1 Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) solutions for distance parameter ξ = 400 with saturation energy parameter η_s = 2 and input energy parameter η₀ = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).

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Fig. 2 Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) solutions for distance parameter ξ = 4000 with saturation energy parameter η_s = 2 and input energy parameter η₀ = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).

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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 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 η_s = 100 and ξ = 600. This is the result of an inevitable trade off between computational time and precision in the numerical simulations.

Fig. 3 Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) pulse for distance parameter ξ = 100 with saturation energy parameter η_s = 100 and input energy parameter η₀ = 10⁻³. The dimensionless energy η (ξ) is illustrated in the inset diagram (a).

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Fig. 4 Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) pulse for distance parameter ξ = 600 with saturation energy parameter η_s = 100 and input energy parametyer η₀ = 10⁻³. The dimensionless pulse energy η (ξ) is illustrated in the inset diagram (a).

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5. Hyper-Gaussian similaritons

We have found in the previous section that the dimensionless NLSE depends only on a single parameter $η_{s} = γ E_{s} / \sqrt{g_{0} β_{2}}$ 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_c ≃ 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 η_s < η_c 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

P (z, τ) = \frac{Λ E (z)}{w (z)} exp [- \sum_{n = 1}^{\infty} σ_{n} {(\frac{τ}{w (z)})}^{n}],

where w(z) = μ(z_c + z) is the width of HG similariton which increases linearly with distance. Here μ, z_c and σ_n are constant parameters depending on the input wave function ψ₀(τ) (ψ|_z₌₀ = ψ₀(τ)). The relation

E (z) = \int_{- \infty}^{+ \infty} P (z, τ) d τ

yields the constant parameter

Λ^{- 1} = \int_{- \infty}^{+ \infty} exp [- \sum_{k = 1}^{N} σ_{k} x^{k}] d x

. When the gain is given by Eq. (2) we can represent the constant width parameter in the form μ = ρ(γβ₂E_sg₀)^1/3 where ρ is a dimensionless factor. Hence the width of HG pulse in this case is

w (z) = ρ {(γ β_{2} E_{s} g_{0})}^{1 / 3} (z_{c} + z) .

The phase of HG similaritons has the form:

Φ (z, τ) = ϕ_{0} + Λ γ \int_{0}^{z} \frac{E (z^{'})}{μ (z_{c} + z^{'})} d z^{'} - \frac{τ^{2}}{2 β_{2} (z_{c} + z)} .

This equation yields the chirp of HG similaritons at z → ∞ as

Ω (z, τ) = - Φ_{τ} (z, τ) = \frac{τ}{β_{2} z} .

Fig. 5 Pulse power of numerical HG pulse (solid line), analytical parabolic solution (blue dotted line) and analytical HG pulse power profile (green dotted line) for distance parameters ξ = 8000 (a) and ξ = 14000 (b) with saturation energy parameter η_s = 0.1 and input energy parameter η₀ = 10⁻⁴. The Hyper-Gaussian pulse power profile is given by Eq. (44).

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Fig. 6 (a) Pulse spectrum of input pulse (dotted line) and numerical output HG pulse (solid line) for distance parameter ξ = 14000. (b) Chirp of numerical output HG pulse (red curve) and analytical HG pulse (blue line) with distance parameter ξ = 14000. Here the saturation energy parameter and input energy parameter are η_s = 0.1 and η₀ = 10⁻⁴.

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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 by

P (z, τ) = \frac{Λ E (z)}{w (z)} exp [- {(\frac{τ}{w (z)})}^{2} - σ {(\frac{τ}{w (z)})}^{4}] .

where w(z) = μ(z_c + z) and

Λ^{- 1} = \int_{- \infty}^{+ \infty} exp [- x^{2} - σ x^{4}] d x

. 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_c) 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 η_s > η_c. 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 η_s < η_c 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.

Fig. 7 Power profiles for numerical HG (solid line) and analytical HG (dotted line) pulses for distance parameter ξ = 8000 with four different input pulses and η_s = 0.1, η₀ = 10⁻⁴. The parabolic analytical solution is also shown (dashed line) which differ substantially from numerical and analytical HG similaritons.

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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_c) 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 η_s > η_c and η_s < η_c (η_c ≃ 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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Parabolic and hyper-Gaussian similaritons in fiber amplifiers and lasers with gain saturation

Abstract

1. Introduction

2. Parabolic similaritons in fiber amplifiers

3. Parabolic solution of NLSE with gain saturation

4. Numerical simulations of parabolic similaritons

5. Hyper-Gaussian similaritons

6. Conclusion

References and links

Cited By

Figures (7)

Equations (45)

Optics Express