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 [15]. 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 [79]. 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 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 as

iψz=β22ψττγ|ψ|2ψ+ig(z)2ψ+ig(z)2σψττ,
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=Ωg2 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)=g0(1+1Es+|ψ(z,τ)|2dτ)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:

ψ(z,τ)=exp(12G(z))ψ˜(z,τ),G(z)=0zg(z)dz,
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:
A(z,τ)=E(z)E0B(z,τ),E(z)=E0exp(G(z)),
where E(z)=+|ψ(z,τ)|2dτ 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:
iψ˜z=β22ψ˜ττΓ(z)|ψ˜|2ψ˜,Γ(z)=γexp(G(z)).
The Eq. (5) yields the system of equations for the real amplitude B(z,τ) and the phase Φ(z,τ):
2BBz=β2ΦττB2+2β2ΦτBBτ
Φz=ΓB2+β22(Φτ)2β22(BττB).
We will show below that in the cases when E(z) → ∞ with z → ∞ the condition ΓB2 ≫ (β2/2)|Bττ/B| is satisfied for sufficient propagation distances when β2 > 0. Using Eq. (4) one can also present this inequality in the form:
2γA2β2|AττA|,
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=β2Φττ𝒡+β2Φτ𝒡τ
Φz=Γ𝒡+β22(Φτ)2.
This system of equations has the parabolic solution [79] 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 [8]:
A(z,τ)=P(z)1/2(1τ2τp(z)2)1/2θ(τp(z)|τ|),
Φ(z,τ)=ϕ0+3γ40zE(z)τp(z)dz+C(z)τ2,
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
P(z)=3E(z)4τp(z),C(z)=12β2τp(z)dτp(z)dz.
The effective width of the pulse τp(z) can be found by solving the equation:
d2τp(z)dz2=(3γβ22)E(z)τp(z)2,
which follows from Eqs. (9), (10). As an example, when g is a constant and E(z) = E0egz the solution of Eq. (14) is
τp(z)=3(γβ2E02g2)1/3egz/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| < τp(z) as

(3γ2β2)E(z)τp(z)(1τ2τp(z)2)3.
Integrating the left and right sides of Eq. (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 ηs=γEs/g0β2 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 [15] given by Eq. (2), the energy of the pulses can be found by

dE(z)dz=g0E(z)1+E(z)/Es.
Solving Eq. (16) we can find the energy of the pulse from the following equation:
E(z)=E0+Esg0zEslnE(z)E0,
with E0 = E(0). Introducing the dimensionless energy ε (ξ) = E(z)/Es and the distance ξ = g0z, we can find the dimensionless energy with a good accuracy using a first iteration of Eq. (17):
ε(ξ)=α1+ξln(αξ),α=Es/E0,
when the condition ξ ≫ |α−1 − ln(αξ)| is satisfied.

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

τp(z)=(3γβ2Es2g02)1/3T(ξ),
where the distance dependent function T (ξ) is dimensionless. Using this expression we can rewrite Eq. (14) and Eq. (16) in the form:
d2Tdξ2=εT2,dεdξ=ε1+ε.

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):

d2Wdε2+1ε(1+ε)dWdε=(1+ε)2εW2.
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:
d2Wdε2=εW2.

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

d2Ydx2+dYdx=1Y2.
Finally, applying the ultimate transformation Y (x) = U (x)1/3 one can write Eq. (23) as follows:
Ud2Udx223(dUdx)2+UdUdx3U=0.
We have found the particular solution of this nonlinear differential equation in the form of a series:
U(x)=nmBnm(lnx)mxn,
where n = −1, 0, 1, 2,… and m = 0, 1, 2,… . The important characteristic of this series is that the coefficients (matrix Bnm) can be found using a sequential algorithm: first we find all non zero values of Bnm for n = −1; secondly we find all non zero values of Bnm for n = 0; … finally we find all non zero values of Bnm 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:
U(x)=3x+5+2lnx+4lnx3x319x2+8lnx9x24(lnx)29x276681x3+140lnx27x38(lnx)29x3+16(lnx)381x3+.
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)/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

W(ε)=ε[3(σ+lnε)+5+2ln(σ+lnε)+4ln(σ+lnε)3(σ+lnε)319(σ+lnε)2+8ln(σ+lnε)9(σ+lnε)24(ln(σ+lnε))29(σ+lnε)2+]1/3.
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)=(α1+ξln(αξ))[κ+3ln(α1+ξln(αξ))]1/3.

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

τp(z)=(3γβ2Es2g02)1/3(α1+g0zln(αg0z))[κ+3ln(α1+g0zln(αg0z))]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)=34(2g02Es23β2γ)1/3[κ+3ln(α1+g0zln(αg0z))]1/3,
C(z)=(g0z1)2β2z(α1+g0zln(αg0z)){1+1κ+3ln(α1+g0zln(αg0z))}.

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

3ln(α1+g0zln(αg0z))|κ|.

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 εε0 ≃ 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 are

χ(ξ,ζ)=γg0ψ(z,τ),ξ=g0z,ζ=g0β2τ.
In this case the NLSE, describing the model saturation effect with the gain g(z) = g0(1 + E(z)/Es)−1 and σ = 0 has the form:
iχξ=12χζζ|χ|2χ+i2(1+ηs1+|χ(ξ,ζ)|2dζ)1χ.
The dimensionless energy of the pulses η(ξ)=+|χ(ξ,ζ)|2dζ can be written as η(ξ)=γE(z)/g0β2, and the dimensionless saturation energy ηs and input energy η0 of the pulses are given by
ηs=γEsg0β2,η0=γE0g0β2,
with η0 = η(0) and E0 = E(0). Hence, the dimensionless NLSE depends only on a single parameter ηs. 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 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( (ξ,ζ)) where the amplitude u (ξ,ζ) is

u(ξ,ζ)=p(ξ)1/2(1ζ2τ(ξ)2)1/2θ(τ(ξ)|ζ|),
τ(ξ)=(3ηs2)1/3(α1+ξln(αξ))[κ+3ln(α1+ξln(αξ))]1/3,
p(ξ)=34(2ηs23)1/3[κ+3ln(α1+ξln(αξ))]1/3.
The dimensionless chirp ω (ξ,ζ) = −ϕζ (ξ,ζ) of the similaritons is given by
ω(ξ,ζ)=(ξ1)ξ(α1+ξln(αξ)){1+1κ+3ln(α1+ξln(αξ))}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 ξb of the interval (ξb, +∞) 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)=π1/4η0exp(ζ2/2) 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 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.

 figure: Fig. 1

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 ηs = 2 and input energy parameter η0 = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).

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 figure: Fig. 2

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 ηs = 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 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.

 figure: Fig. 3

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 ηs = 100 and input energy parameter η0 = 10−3. The dimensionless energy η (ξ) is illustrated in the inset diagram (a).

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 figure: Fig. 4

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 ηs = 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 ηs=γEs/g0β2 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 is

P(z,τ)=ΛE(z)w(z)exp[n=1σn(τw(z))n],
where w(z) = μ(zc + z) is the width of HG similariton which increases linearly with distance. Here μ, zc and σn are constant parameters depending on the input wave function ψ0(τ) (ψ|z=0 = ψ0(τ)). The relation E(z)=+P(z,τ)dτ yields the constant parameter Λ1=+exp[k=1Nσkxk]dx. When the gain is given by Eq. (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
w(z)=ρ(γβ2Esg0)1/3(zc+z).
The phase of HG similaritons has the form:
Φ(z,τ)=ϕ0+Λγ0zE(z)μ(zc+z)dzτ22β2(zc+z).
This equation yields the chirp of HG similaritons at z → ∞ as
Ω(z,τ)=Φτ(z,τ)=τβ2z.

 figure: Fig. 5

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 ηs = 0.1 and input energy parameter η0 = 10−4. The Hyper-Gaussian pulse power profile is given by Eq. (44).

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 figure: Fig. 6

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 ηs = 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)=ΛE(z)w(z)exp[(τw(z))2σ(τw(z))4].
where w(z) = μ(zc + z) and Λ1=+exp[x2σx4]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 (zzc) 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 [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 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.

 figure: Fig. 7

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 ηs = 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 (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.

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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]  

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]  

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]  

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]  

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]  

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]  

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]  

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]  

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]  

References

  • View by:

  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]

2010 (4)

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]

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]

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

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

2007 (1)

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

2006 (1)

2004 (1)

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]

2003 (1)

2002 (1)

2000 (3)

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]

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]

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]

1998 (1)

1993 (1)

1992 (2)

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]

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

1991 (1)

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]

Afanas’ev, A. A.

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]

Agrawal, G. P.

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2001).

Anderson, D.

Bale, B. G.

Bergman, K.

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]

Buckley, J. R.

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]

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]

Chong, A.

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]

Clark, W. G.

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]

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]

de Sterke, C. M.

Desaix, M.

Dudley, J. M.

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

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]

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]

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]

Fermann, M. E.

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]

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]

Finot, C.

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

Hartl, I.

Harvey, J. D.

Ilday, F. O.

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]

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]

Jakyte, R.

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]

Karlsson, M.

Kruglov, V. I.

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

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]

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]

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]

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]

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]

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]

Krylov, D.

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]

Levi, D.

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

Lim, H.

Lisak, M.

Logvin, Yu. A.

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]

Marcinkevicius, A.

Mechin, D.

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

Menyuk, C. R.

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

Millar, P. D.

Millot, G.

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

Monro, T. M.

Peacock, A. C.

Poladian, L.

Quiroga-Teixeiro, M. L.

Renninger, W. H.

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]

Richardson, D. J.

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

Ruehl, A.

Samson, B. A.

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]

Sears, S.

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]

Segev, M.

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]

Soljacic, M.

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]

Thomson, B. C.

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]

Volkov, V. M.

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]

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]

Wabnitz, S.

Winternitz, P.

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

Wise, F. W.

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]

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]

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]

J. Mod. Opt. (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]

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]

J. Opt. Soc. Am. B (3)

Nat. Phys. (1)

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

Opt. Lett. (5)

Phys. Rev. A. (2)

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

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]

Phys. Rev. Lett. (4)

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]

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]

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

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]

Other (1)

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2001).

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Figures (7)

Fig. 1
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 ηs = 2 and input energy parameter η0 = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).
Fig. 2
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 ηs = 2 and input energy parameter η0 = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).
Fig. 3
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 ηs = 100 and input energy parameter η0 = 10−3. The dimensionless energy η (ξ) is illustrated in the inset diagram (a).
Fig. 4
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 ηs = 100 and input energy parametyer η0 = 10−3. The dimensionless pulse energy η (ξ) is illustrated in the inset diagram (a).
Fig. 5
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 ηs = 0.1 and input energy parameter η0 = 10−4. The Hyper-Gaussian pulse power profile is given by Eq. (44).
Fig. 6
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 ηs = 0.1 and η0 = 10−4.
Fig. 7
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 ηs = 0.1, η0 = 10−4. The parabolic analytical solution is also shown (dashed line) which differ substantially from numerical and analytical HG similaritons.

Equations (45)

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i ψ z = β 2 2 ψ τ τ γ | ψ | 2 ψ + i g ( z ) 2 ψ + i g ( z ) 2 σ ψ τ τ ,
g ( z ) = g 0 ( 1 + 1 E s + | ψ ( z , τ ) | 2 d τ ) 1 .
ψ ( z , τ ) = exp ( 1 2 G ( z ) ) ψ ˜ ( z , τ ) , G ( z ) = 0 z g ( z ) d z ,
A ( z , τ ) = E ( z ) E 0 B ( z , τ ) , E ( z ) = E 0 exp ( G ( z ) ) ,
i ψ ˜ z = β 2 2 ψ ˜ τ τ Γ ( z ) | ψ ˜ | 2 ψ ˜ , Γ ( z ) = γ exp ( G ( z ) ) .
2 B B z = β 2 Φ τ τ B 2 + 2 β 2 Φ τ B B τ
Φ z = Γ B 2 + β 2 2 ( Φ τ ) 2 β 2 2 ( B τ τ B ) .
2 γ A 2 β 2 | A τ τ A | ,
𝒡 z = β 2 Φ τ τ 𝒡 + β 2 Φ τ 𝒡 τ
Φ z = Γ 𝒡 + β 2 2 ( Φ τ ) 2 .
A ( z , τ ) = P ( z ) 1 / 2 ( 1 τ 2 τ p ( z ) 2 ) 1 / 2 θ ( τ p ( z ) | τ | ) ,
Φ ( z , τ ) = ϕ 0 + 3 γ 4 0 z E ( z ) τ p ( z ) d z + C ( z ) τ 2 ,
P ( z ) = 3 E ( z ) 4 τ p ( z ) , C ( z ) = 1 2 β 2 τ p ( z ) d τ p ( z ) d z .
d 2 τ p ( z ) d z 2 = ( 3 γ β 2 2 ) E ( z ) τ p ( z ) 2 ,
τ p ( z ) = 3 ( γ β 2 E 0 2 g 2 ) 1 / 3 e g z / 3 .
( 3 γ 2 β 2 ) E ( z ) τ p ( z ) ( 1 τ 2 τ p ( z ) 2 ) 3 .
d E ( z ) d z = g 0 E ( z ) 1 + E ( z ) / E s .
E ( z ) = E 0 + E s g 0 z E s ln E ( z ) E 0 ,
ε ( ξ ) = α 1 + ξ ln ( α ξ ) , α = E s / E 0 ,
τ p ( z ) = ( 3 γ β 2 E s 2 g 0 2 ) 1 / 3 T ( ξ ) ,
d 2 T d ξ 2 = ε T 2 , d ε d ξ = ε 1 + ε .
d 2 W d ε 2 + 1 ε ( 1 + ε ) d W d ε = ( 1 + ε ) 2 ε W 2 .
d 2 W d ε 2 = ε W 2 .
d 2 Y d x 2 + d Y d x = 1 Y 2 .
U d 2 U d x 2 2 3 ( d U d x ) 2 + U d U d x 3 U = 0.
U ( x ) = n m B n m ( ln x ) m x n ,
U ( x ) = 3 x + 5 + 2 ln x + 4 ln x 3 x 31 9 x 2 + 8 ln x 9 x 2 4 ( ln x ) 2 9 x 2 766 81 x 3 + 140 ln x 27 x 3 8 ( ln x ) 2 9 x 3 + 16 ( ln x ) 3 81 x 3 + .
W ( ε ) = ε [ 3 ( σ + ln ε ) + 5 + 2 ln ( σ + ln ε ) + 4 ln ( σ + ln ε ) 3 ( σ + ln ε ) 31 9 ( σ + ln ε ) 2 + 8 ln ( σ + ln ε ) 9 ( σ + ln ε ) 2 4 ( ln ( σ + ln ε ) ) 2 9 ( σ + ln ε ) 2 + ] 1 / 3 .
T ( ξ ) = ( α 1 + ξ ln ( α ξ ) ) [ κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) ] 1 / 3 .
τ p ( z ) = ( 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 .
P ( z ) = 3 4 ( 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 ) = ( g 0 z 1 ) 2 β 2 z ( α 1 + g 0 z ln ( α g 0 z ) ) { 1 + 1 κ + 3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) } .
3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) | κ | .
χ ( ξ , ζ ) = γ g 0 ψ ( z , τ ) , ξ = g 0 z , ζ = g 0 β 2 τ .
i χ ξ = 1 2 χ ζ ζ | χ | 2 χ + i 2 ( 1 + η s 1 + | χ ( ξ , ζ ) | 2 d ζ ) 1 χ .
η s = γ E s g 0 β 2 , η 0 = γ E 0 g 0 β 2 ,
u ( ξ , ζ ) = p ( ξ ) 1 / 2 ( 1 ζ 2 τ ( ξ ) 2 ) 1 / 2 θ ( τ ( ξ ) | ζ | ) ,
τ ( ξ ) = ( 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 ) = 3 4 ( 2 η s 2 3 ) 1 / 3 [ κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) ] 1 / 3 .
ω ( ξ , ζ ) = ( ξ 1 ) ξ ( α 1 + ξ ln ( α ξ ) ) { 1 + 1 κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) } ζ .
P ( z , τ ) = Λ E ( z ) w ( z ) exp [ n = 1 σ n ( τ w ( z ) ) n ] ,
w ( z ) = ρ ( γ β 2 E s g 0 ) 1 / 3 ( z c + z ) .
Φ ( z , τ ) = ϕ 0 + Λ γ 0 z E ( z ) μ ( z c + z ) d z τ 2 2 β 2 ( z c + z ) .
Ω ( z , τ ) = Φ τ ( z , τ ) = τ β 2 z .
P ( z , τ ) = Λ E ( z ) w ( z ) exp [ ( τ w ( z ) ) 2 σ ( τ w ( z ) ) 4 ] .

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