Nonlinear Schrödinger Equation

Abstract

Firstly, based on the small-signal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss is solved. It is also adapted to the NLSE with the high-order dispersion terms. Furthermore, a general theory on cross-phase modulation (XPM) intensity fluctuation which adapted to all kinds of modulation formats (continuous wave, non-return-to-zero wave, and return-zero pulse wave) is presented. Secondly, by the Green function method, the NLSE is directly solved in the time domain. It does not bring any spurious effect compared with the split-step method in which the step size has to be carefully controlled. Additionally, the fourth-order dispersion coefficient of fibers can be estimated by the Green function solution of NLSE. The fourth-order dispersion coefficient varies with distance slightly and is about 0.002 ps4/km, 0.003 ps4/nm, and 0.00032 ps4/nm for SMF, NZDSF, and DCF, respectively. In the zero-dispersion regime, the higher-order nonlinear effect (higher than self-steepening) has a strong impact on the short pulse shape, but this effect degrades rapidly with the increase of β 2. Finally, based on the traveling wave solution of NLSE for ASE noise, the probability density function of ASE by solving the Fokker-Planck equation including the dispersion effect is presented.

1. Introduction

The numerical simulation and analytical models of nonlinear Schrödinger equation (NLSE) play important roles in the design optimization of optical communication systems. They help to understand the underlying physics phenomena of the ultrashort pulses in the nonlinear and dispersion medium.

The inverse scattering [1], variation, and perturbation methods [2] could obtain the analytical solutions under some special conditions. These included the inverse scattering method for classical solitons [3], the dam-break approximation for the non-return-to-zero pulses with the extremely small chromatic dispersion [4], and the perturbation theory for the multidimensional NLSE in the field of molecular physics [5]. When a large nonlinear phase was accumulated, the Volterra series approach was adopted [6]. With the assumption of the perturbations, the NLSE with varying dispersion, nonlinearity, and gain or absorption parameters was solved in [7]. In [8], the generalized Kantorovitch method was introduced in the extended NLSE. By introducing Rayleigh’s dissipation function in Euler-Lagrange equation, the algebraic modification projected the extended NLSE as a frictional problem and successfully solved the soliton transmission problems [9].

Since the numerical computation of solving NLSE is a huge time-consuming process, the fast algorithms and efficient implementations, focusing on (i) an accurate numerical integration scheme and (ii) an intelligent control of the longitudinal spatial step size, are required.

The finite differential method [10] and the pseudo-spectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step size-controlling method [12] and the predictor-corrector method [13] were proposed to speed up the implementation of split-step Fourier method (SSFM). The cubic (or higher order) B-splines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The Runge-Kutta method in the interaction picture was applied to calculate the effective refractive index, effective area, dispersion, and nonlinear coefficients [15].

Recently, the generalized NLSE, taking into account the dispersion of the transverse field distribution, is derived [16]. By an inhomogeneous quasi-linear first-order hyperbolic system, the accurate simulations of the intensity and phase for the Schrödinger-type pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higher-order NLSE in the presence of non-Kerr quintic nonlinearities [18].

In this chapter, several methods to solve the NLSE will be presented: (1) The small-signal analysis theory and split-step Fourier method to solve the coupled NLSE problem, the MI intensity fluctuation caused by SPM and XPM, can be derived. Furthermore, this procedure is also adapted to NLSE with high-order dispersion terms. The impacts of fiber loss on MI gain spectrum can be discussed. The initial stage of MI can be described, and then the whole evolution of MI can also be discussed in this way; (2) the Green function to solve NLSE in the time domain. By this solution, the second-, third-, and fourth-order dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its probability density function.

2. Small-signal analysis solution of NLSE for MI generation

2.1 Theory for continuous wave

The NLSE governing the field in nonlinear and dispersion medium is

$$\frac{\partial u}{\partial z}+{\beta }_1\frac{\partial u}{\partial t}+\frac{i}{2}{\beta }_2\frac{{\partial }^{2}u}{\partial t^2}+\frac{a}{2}u=\mathit{i\gamma }\left[{\left|u\right|}^2+2{\left|u^{\prime }\right|}^2\right]u$$ E1

where β ₁ and β ₂ are the dispersions, γ is the nonlinear coefficient, and α is the fiber loss. In the frequency domain, the solution is

$$u\left(z+\mathit{dz},\omega \right)=exp\left(\mathit{dz}\widehat{D}\right)exp\left(\mathit{dz}\widehat{N}\right)u\left(z,\omega \right)$$ E2

where $\widehat{D}=\frac{i}{2}{\omega }^2{\beta }_2+i{\mathit{\omega \beta }}_1-\frac{a}{2}$ and $\widehat{N}=\mathit{i\gamma }\left[{\left|u\right|}^2+i2{\left|u^{\prime }\right|}^2\right]$ [19] ( Figure 1 ).

Figure 1.

Schematic illustration of medium. u(z, t) and u(z + dz, t) correspond to the field amplitudes at z and z + dz, respectively.

Usually, the field amplitudes can be written as

$$u\left(z,\omega \right)=\sqrt{P\left(z,\omega \right)}exp\left[\mathit{i\varphi }\left(z,\omega \right)\right]$$ E3

$\varphi \left(z,\omega \right)$ is caused by the nonlinear effect, and $\varphi \left(z,\omega \right)={\int }_0^z\gamma \left[P\left(z,\omega \right)+2P^{\prime }\left(z,\omega \right)\right]\mathit{dz}$ [3].

$u\left(z+\mathit{dz},\omega \right)$ (is)

$$\begin{array}{ll}u\left(z+\mathit{dz},\omega \right)& =exp\left(\mathit{dz}\widehat{D}\right)\sqrt{P\left(z,\omega \right)}exp\left\{\mathit{i\phi }\left(z,\omega \right)+\mathit{i\gamma }\left[P+2P^{\prime }\right]\mathit{dz}\right\}\\ & =e^{-\mathit{adz}/2}exp\left({\beta }_1\mathit{\omega dz}\right)exp\left({\beta }_2/2{\omega }^2\mathit{dz}\right)\sqrt{P\left(z,\omega \right)}{e}^{\mathit{i\phi }\left(z+\mathit{dz},\omega \right)}\\ & =\sqrt{P\left(z+\mathit{dz},\omega \right)}exp\left[\mathit{i\phi }\left(z+\mathit{dz}.\omega \right)\right]\end{array}$$ E4

Assuming: $P\left(z,\omega \right)=⟨P\left(z\right)⟩+\mathrm{\Delta }P\left(z,\omega \right)$

$⟨P\left(z\right)⟩$ is the average signal intensity. $\mathrm{\Delta }P\left(z,\omega \right)$ is the noise or modulation term. There is [20] $⟨P\left(z\right)⟩\mathrm{\Delta }P\left(z,\omega \right)$

The amplitude $\sqrt{P\left(z,\omega \right)}$ can be regarded as

$$\sqrt{P\left(z,\omega \right)}\approx \sqrt{⟨P\left(z\right)⟩}\left(1+\frac{\mathrm{\Delta }P\left(z,\omega \right)}{2P\left(z\right)}\right)$$ E5

The small-signal theory implies that the frequency modulation or noise $\dot{\phi }\left(z+\mathit{dz},\omega \right)=\frac{d\dot{\phi }\left(z+\mathit{dz},\omega \right)}{\mathit{dt}}$ is small enough. Finally ([21])

$$\begin{array}{ll}P\left(z+\mathit{dz},\omega \right)& =⟨P\left(z\right)⟩+2e^{-\mathit{adz}/2}\times \\ & Re\left\{⟨P\left(z\right)⟩exp\left(i{\mathit{\omega \beta }}_1\mathit{dz}+i{\omega }^2{\beta }_2\mathit{dz}\right)\left[\frac{\mathrm{\Delta }P\left(z,\omega \right)}{2⟨P\left(z\right)⟩}+\mathit{i\phi }\left(z+\mathit{dz},\omega \right)\right]\right\}\end{array}$$ E6

The operation $exp\left(i{\mathit{\omega \beta }}_1\mathit{dz}+i{\omega }^2{\beta }_2\mathit{dz}\right)$ can be split into its real and imaginary parts:

$$exp\left(i{\mathit{\omega \beta }}_1\mathit{dz}+i{\omega }^2{\beta }_2\mathit{dz}\right)=cos\left({\mathit{\omega \beta }}_1\mathit{dz}+{\omega }^2{\beta }_2\mathit{dz}\right)+isin\left({\mathit{\omega \beta }}_1\mathit{dz}+{\omega }^2{\beta }_2\mathit{dz}\right)$$ E7

The modulation or noise $\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)$ is $\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)\approx P\left(z+\mathit{dz},\omega \right)-⟨P\left(z\right)⟩$

So

$$\begin{array}{c}P\left(z+\mathit{dz},\omega \right)=e^{-\mathit{adz}/2-i{\mathit{\omega \beta }}_1\mathit{dz}}\\ \left[cos\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)\mathrm{\Delta }P\left(z,\omega \right)+sin\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)2⟨P\left(z\right)⟩\phi (z+\mathit{dz},\omega )\right]\end{array}$$ E8

And

$$\begin{array}{c}\left(\begin{array}{c}\frac{\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)}{2⟨P\left(z\right)⟩}\\ \phi \left(z+\mathit{dz},\omega \right)\end{array}\right)=e^{-\mathit{adz}/2-i{\mathit{\omega \beta }}_1\mathit{dz}}{e}^{\mathit{i\gamma }\left[⟨P\left(z\right)⟩+2⟨P^{\prime }\left(z\right)⟩\right]\mathit{dz}}\\ \left(\begin{array}{cc}cos\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)& -sin\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)\\ sin\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)& cos\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)\end{array}\right)\left(\begin{array}{c}\frac{\mathrm{\Delta }P\left(z,\omega \right)}{2⟨P\left(z\right)⟩}\\ \phi \left(z,\omega \right)\end{array}\right)\end{array}$$ E9

When only intensity modulation is present and no phase modulation exists, the transfer function $cos\left(\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}\right)$ is obtained. The 3 dB cutoff frequency corresponds to $\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}=\pi /4$ in [22, 23]. This treatment is also adaptable to the case that only the nonlinear phase (frequency) modulation is present; then, the intensity modulation $\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)$ due to FM-IM conversion is given as

$$\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)=2⟨P\left(z\right)⟩e^{-\mathit{adz}/2-i{\mathit{\omega \beta }}_1\mathit{dz}}sin\left(\frac{1}{2}{\beta }_2\mathit{dz}{\omega }^2\right)\phi \left(z+\mathit{dz},\omega \right)$$ E10

This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index $\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}=\pi /2$ , the difference is within 0.5 dB. Eq. (10) does not include a Bessel function, so it is simpler than that in [24].

Obviously, the above process can be used to treat NLSE with higher-order dispersion (β ₃, β ₄) [25]. Similarly, the result in Eq. (10) will include ω ³ and ω ⁴.

The corresponding MI gain g_MI in the side bands of ω ₀ (the frequency of signal) is given by

$$\begin{array}{ll}{g}_{\mathit{MI}}\left(z,\omega \right)& =\frac{\left|\mathrm{\Delta }P\left(z+\mathit{dz},\omega \right)-\mathrm{\Delta }P(z,\omega )\right|}{⟨P\left(z\right)⟩\mathit{dz}}\\ & =2e^{-\mathit{adz}/2-i{\mathit{\omega \beta }}_1\mathit{dz}}sin\left(\frac{1}{2}{\beta }_2\mathit{dz}{\omega }^2\right)\left\{\gamma {\int }_z^{z+\mathit{dz}}\left[P\left(z,\omega \right)+2P^{\prime }\left(z,\omega \right)\right]\mathit{dz}\right\}/\mathit{dz}\end{array}$$ E11

Figure 2.

MI gain spectra. +++ result of small-signal analysis. –––– result of perturbation approach. The parameters are P₀ = 10 dBm, β₂ = 15 ps²/km, λ = 1550 nm, a = 0.21 dB/km, γ = 0.015W⁻¹/m, and z = 0 m.

Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case $⟨P\left(z\right)⟩/⟨P^{\prime }\left(z\right)⟩=1$ . The maximum frequency modulation index caused by dispersion corresponds to $\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}=\pi$ [22, 23], and the maximum value of the sideband is ${\omega }_c=\sqrt{4\gamma ⟨P\left(z\right)⟩/\left|{\beta }_2\right|}$ , so the choice of dz satisfies $\frac{1}{2}{\beta }_2{\omega }^2\mathit{dz}=\pi$ , which makes Eq. (11) have the same frequency regime as [26]. In Figure 2 , the curves are different but have the same maximum value of g_MI . In practice, researchers generally utilize the maximum value of g_MI to estimate the amplified noises and SNR [3]. The result of small-signal analysis in Figure 2 has a phase delay of around ω ₀. Compared with the experiment result of [27], the reason is taking the fiber loss into account, the gain spectrum exhibits a phase delay close to ω ₀, and the curve descends a little [27]. Fiber loss results in the difference of g_MI between the small-signal analysis method and the perturbation approach.

2.2 The general theory on cross-phase modulation (XPM) intensity fluctuation

For the general case of two channels, the input optical powers are denoted by $P\left(t\right),P^{\prime }\left(t\right)$ , respectively [28]. Only in the first walk-off length, the nonlinear interaction (XPM) is taken into account; in the remaining fibers, signals are propagated linearly along the fibers, and dispersion acts on the phase-modulated signal resulting in intensity fluctuation. According to [4], the whole length L is separated into two parts 0 < z < L_wo and L_wo  < z < L; L_wo is the walk-off length, $L_{\mathit{wo}}=\mathrm{\Delta }t/\left(D\mathrm{\Delta }\lambda \right)$ . $\mathrm{\Delta }t$ is the edge duration of the carrier wave, D is the dispersion coefficient, and $\mathrm{\Delta }\lambda$ is the wavelength spacing between the channels. By the small-signal analysis, the phase modulation in channel 1 originating in dz at z can be expressed as

$$d{\varphi }_{\mathit{XPM}}\left(z,t\right)=\gamma 2P^{\prime }\left(z,t-z{\beta }_1^{\prime }\right)e^{-\mathit{az}}\mathit{dz}$$ E12

This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from z to the receiver. So, at the fiber output, the intensity fluctuation originating in dz in the frequency domain is given by [29].

$$\begin{array}{ll}{\mathit{dP}}_{\mathit{XPM}}\left(z,\omega \right)& =2\left[e^{\mathit{i\omega z}{\beta }_1}P\left(z,\omega \right)\right]\otimes \left\{e^{-a\left(L-z\right)}\cdot e^{i{\mathit{\omega \beta }}_1\left(L-z\right)}sin\left[b\left(L-z\right)\right]d{\phi }_{\mathit{XPM}}\left(z,\omega \right)\right\}\\ & =4\gamma \left[e^{\mathit{i\omega z}{\beta }_1}P\left(z,\omega \right)\right]\otimes \left\{e^{-a\left(L-z\right)}\cdot e^{-\mathit{az}}\cdot e^{\mathit{i\omega }{\beta }_1^{\prime }z}\cdot e^{i{\mathit{\omega \beta }}_1\left(L-z\right)}{P}^{\prime }\left(z,\omega \right)sin\left[b\left(L-z\right)\right]\right\}\mathit{dz}\end{array}$$ E13

$\otimes$ representing the convolution operation $b={\omega }^{2}D{\lambda }^2/\left(4\mathit{\pi c}\right)$ , where c is the speed of light. At the fiber output, the XPM-induced intensity fluctuation is the integral of Eq. (13) with z ranging from 0 to L:

$$\begin{array}{ll}{P}_{\mathit{XPM}}& ={\int }_0^L{\mathit{dP}}_{\mathit{XPM}}\left(z,\omega \right)\mathit{dz}\\ & ={\int }_0^{L}4\gamma \left[e^{\mathit{i\omega z}{\beta }_1}P\left(z,\omega \right)\right]\otimes \left\{e^{-a\left(L-z\right)}\cdot e^{-\mathit{az}}\cdot e^{\mathit{i\omega }{\beta }_1^{\prime }z}\cdot e^{i{\mathit{\omega \beta }}_1\left(L-z\right)}{P}^{\prime }\left(z,\omega \right)sin\left[b\left(L-z\right)\right]\right\}\mathit{dz}\end{array}$$ E14

The walk-off between co-propagating waves is regulated by the convolution operation.

3. Green function method for the time domain solution of NLSE

3.1 NLSE including the resonant and nonresonant cubic susceptibility tensors

From Maxwell’s equation, the field in fibers satisfies

$${\nabla }^2\overrightarrow{E}-\frac{1}{c^2}\frac{{\partial }^2\overrightarrow{E}}{\partial t^2}=-u_0\frac{{\partial }^2{\overrightarrow{P}}_L}{\partial t^2}-u_0\frac{{\partial }^2{\overrightarrow{P}}_{\mathit{NL}}}{\partial t^2}$$ E15

$$\begin{array}{ll}{\overrightarrow{P}}_L\left(\overrightarrow{r},t\right)& ={\epsilon }_0{\int }_{-\infty }^{+\infty }{\chi }^{\left(1\right)}\left(t-t^{\prime }\right)\overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)d{t}^{\prime }\\ & ={\epsilon }_0{\int }_{-\infty }^{+\infty }{\chi }^{\left(1\right)}\left(\omega \right)\overrightarrow{E}\left(\overrightarrow{r},\omega \right)exp\left(\mathit{i\omega t}\right)\mathit{d\omega }\end{array}$$ E16

$${\chi }^{\left(1\right)}\left(\omega \right)={\int }_{-\infty }^{+\infty }d{\mathit{\tau \chi }}^{\left(1\right)}\left(\tau \right)exp\left(-\mathit{j\omega \tau }\right)$$ E17

where $\overrightarrow{E}$ is the vector field and ${\chi }^{\left(1\right)}$ is the linear susceptibility. ${\overrightarrow{P}}_L$ and ${\overrightarrow{P}}_{\mathit{NL}}$ represent the linear and nonlinear induced fields, respectively [30]. The cubic susceptibility tensor including the resonant and nonresonant terms is

$${\chi }^{\left(3\right)}\left(\omega \right)={\chi }_{\mathit{NR}}^{\left(3\right)}+{\chi }_R^{\left(3\right)}\left(\omega \right)$$ E18

There are

$$\begin{array}{c}{\overrightarrow{P}}_{\mathit{NL},\mathit{NR}}\left(\overrightarrow{r},t\right)={\epsilon }_0\underset{\infty }{\iiint }{\mathit{dt}}_1{\mathit{dt}}_2{\mathit{dt}}_3{\chi }_{\mathit{NR}}^{\left(3\right)}\left(t_1,t_2,t_3\right)⋮\overrightarrow{E}\left(\overrightarrow{r},t-t_1\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t-t_2\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t-t_3\right)\\ ={\epsilon }_0\underset{\infty }{\iiint }d{\omega }_{1}d{\omega }_{2}d{\omega }_3{\chi }_{\mathit{NR}}^{\left(3\right)}\left(-{\omega }_1-{\omega }_2-{\omega }_3;{\omega }_1+{\omega }_2+{\omega }_3\right)\\ \overrightarrow{E}\left(\overrightarrow{r},t_1\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t_2\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t_3\right)exp\left(\mathit{j\omega t}\right)\delta \left(\omega -{\omega }_1-{\omega }_2-{\omega }_3\right)\end{array}$$ E19

$$\begin{array}{c}{\chi }_{\mathit{NR}}^{\left(3\right)}\left(-{\omega }_1-{\omega }_2-{\omega }_3;{\omega }_1+{\omega }_2+{\omega }_3\right)=\underset{\infty }{\iiint }{\mathit{dt}}_1{\mathit{dt}}_2{\mathit{dt}}_3{\chi }_{\mathit{NR}}^{\left(3\right)}\left(t_1,t_2,t_3\right)\\ exp\left(-j{\omega }_1{t}_1-j{\omega }_2{t}_2-j{\omega }_3{t}_3\right)\end{array}$$ E20

$$\begin{array}{c}{\overrightarrow{P}}_{\mathit{NL},R}\left(\overrightarrow{r},t\right)={\epsilon }_0\underset{\infty }{\iiint }{\mathit{dt}}_1{\mathit{dt}}_2{\mathit{dt}}_3{\chi }_R^{\left(3\right)}\left(t,t_1,t_2,t_3\right)⋮\overrightarrow{E}\left(\overrightarrow{r},t-t_1\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t-t_2\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t-t_3\right)\\ ={\epsilon }_0\underset{\infty }{\iiint }d{\omega }_{1}d{\omega }_{2}d{\omega }_3{\chi }_R^{\left(3\right)}\left(t,-{\omega }_1-{\omega }_2-{\omega }_3;{\omega }_1+{\omega }_2+{\omega }_3\right)\\ \overrightarrow{E}\left(\overrightarrow{r},t_1\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t_2\right)\cdot \overrightarrow{E}\left(\overrightarrow{r},t_3\right)exp\left(\mathit{j\omega t}\right)\delta \left(\omega -{\omega }_1-{\omega }_2-{\omega }_3\right)\end{array}$$ E21

$$\begin{array}{c}{\chi }_R^{\left(3\right)}\left(t\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{+\infty }\frac{a}{\omega -\left({\omega }_1+{\omega }_2+{\omega }_3\right)+i\mathrm{\Gamma }}{e}^{-\mathit{i\omega t}}\mathit{d\omega }\\ =-\sqrt{\frac{\pi }{2}}a\left(1+\frac{\mathrm{\Gamma }}{\left|\mathrm{\Gamma }\right|}\right)e^{-\left|\mathrm{\Gamma }\right|t+i\left({\omega }_1+{\omega }_2+{\omega }_3\right)t-i\frac{\pi }{2}}\end{array}$$ E22

$\mathrm{\Gamma }$ and a are the attenuation and absorption coefficients, respectively [31].

Repeating the process of [3]

$E=F\left(x,y\right)A\left(z,t\right)exp\left(\mathit{i\beta z}\right)$ , there is

$$\begin{array}{c}\frac{\partial A}{\partial z}+\frac{i}{2}{\beta }_2\frac{{\partial }^{2}A}{\partial t^2}-\frac{1}{6}{\beta }_3\frac{{\partial }^{3}A}{\partial t^3}=-\frac{a}{2}A+i\frac{3k_0}{8{\mathit{nA}}_{\mathit{eff}}}{\chi }_{\mathit{NR}}^{\left(3\right)}{\left|A\right|}^{2}A+\frac{{\mathit{ik}}_{0}g\left({\omega }_0\right)\left[1-\mathit{\text{if}}\left({\omega }_0\right)\right]}{2{\mathit{nA}}_{\mathit{eff}}}\\ A{\int }_{-\infty }^t{\chi }_R^{\left(3\right)}\left(t-\tau \right){\left|A\left(\tau \right)\right|}^2\mathit{d\tau }\end{array}$$ E23

$k_0={\omega }_0/c$ , where ${\omega }_0$ is the center frequency. A_eff is the effective core area. n is the refractive index. The last term is responsible for the Raman scattering, self-frequency shift, and self-steepening originating from the delayed response:

$$f\left({\omega }_1+{\omega }_2+{\omega }_3\right)=\frac{2\left({\omega }_1+{\omega }_2+{\omega }_3\right)\left(1-\left|\mathrm{\Gamma }\right|\right)}{-2{\left({\omega }_1+{\omega }_2+{\omega }_3\right)}^2-2\left|\mathrm{\Gamma }\right|+{\left|\mathrm{\Gamma }\right|}^2}$$ E24

$$g\left({\omega }_1+{\omega }_2+{\omega }_3\right)=\left[-2{\left({\omega }_1+{\omega }_2+{\omega }_3\right)}^2-2\left|\mathrm{\Gamma }\right|+{\left|\mathrm{\Gamma }\right|}^2\right]$$ E25

where $g\left({\omega }_1+{\omega }_2+{\omega }_3\right)$ is the Raman gain and $f\left({\omega }_1+{\omega }_2+{\omega }_3\right)$ is the Raman non-gain coefficients.

3.2 The solution by Green function

The solution has the form

$$A\left(z,t\right)=\phi \left(t\right)e^{-\mathit{iEz}}$$ E26

Then, there is

$$\frac{1}{2}{\beta }_2\frac{{\partial }^2\varphi }{\partial t^2}+\frac{i}{6}{\beta }_3\frac{{\partial }^3\varphi }{\partial t^3}-\frac{3k_0}{8{\mathit{nA}}_{\mathit{eff}}}{\chi }_{\mathit{NR}}^{\left(3\right)}{\left|\varphi \right|}^2\varphi -\frac{k_{0}g\left({\omega }_s\right)\left[1-\mathit{\text{if}}\left({\omega }_s\right)\right]}{2{\mathit{nA}}_{\mathit{eff}}}\varphi {\int }_{-\infty }^{+\infty }{\chi }_R^{\left(3\right)}\left(t-\tau \right){\left|\varphi \left(\tau \right)\right|}^2\mathit{d\tau }=\mathit{E\varphi }$$ E27

Let

$${\widehat{H}}_0\left(t\right)=\frac{1}{2}{\beta }_2\frac{{\partial }^2}{\partial t^2}+\frac{i}{6}{\beta }_3\frac{{\partial }^3}{\partial t^3}$$ E28

$$\widehat{V}\left(t\right)=\frac{-3k_0}{8{\mathit{nA}}_{\mathit{eff}}}{\chi }_{\mathit{NR}}^{\left(3\right)}\left|\varphi \right|-\frac{k_{0}g\left({\omega }_s\right)\left[1-\mathit{\text{if}}\left({\omega }_s\right)\right]}{2{\mathit{nA}}_{\mathit{eff}}}{\int }_{-\infty }^{+\infty }{\chi }_R^{\left(3\right)}\left(t-\tau \right){\left|\varphi \left(\tau \right)\right|}^2\mathit{d\tau }$$ E29

and taking the operator $\widehat{V}\left(t\right)$ as a perturbation item, we first solve the eigen equation $-\sum _{n=2}^k\frac{i^n}{n!}{\beta }_n\frac{{\partial }^n\phi }{\partial T^n}=\mathit{E\phi }$ .

$$\frac{1}{2}{\beta }_2\frac{{\partial }^2\varphi }{\partial T^2}+\frac{i}{6}{\beta }_3\frac{{\partial }^3\varphi }{\partial T^3}=\mathit{E\varphi }$$ E30

Assuming $E=1$ , we get the corresponding characteristic equation:

$$-\frac{1}{2}{\beta }_2{r}^2+\frac{{\beta }_3}{6}{r}^3=E$$ E31

Its characteristic roots are $r_1,r_2,r_3$ . The solution can be represented as

$$\varphi =c_1{\varphi }_1+c_2{\varphi }_2+c_3{\varphi }_3$$ E32

where ${\varphi }_m=exp\left({\mathit{ir}}_{m}t\right),m=1,2,3$ and $c_1,c_2,c_3$ are determined by the initial pulse. The Green function of (30) is

$$\left(E-{\widehat{H}}_0\left(t\right)\right)G_0\left(t,t^{\prime }\right)=\delta \left(t-t^{\prime }\right)$$ E33

By the construction method, it is

$$G_0\left(t,t^{\prime }\right)=\left\{\begin{array}{c}{a}_1{\phi }_1+a_2{\phi }_2+a_3{\phi }_3,t>t^{\prime }\\ b_1{\phi }_1+b_2{\phi }_2+b_3{\phi }_3,t<t^{\prime }\end{array}\right.$$ E34

At the point $t=t^{\prime }$ , there are

$$a_1{\varphi }_1\left(t^{\prime }\right)+a_2{\varphi }_2\left(t^{\prime }\right)+a_3{\varphi }_3\left(t^{\prime }\right)=b_1{\varphi }_1\left(t^{\prime }\right)+b_2{\varphi }_2\left(t^{\prime }\right)+b_3{\varphi }_3\left(t^{\prime }\right)$$ E35

$$a_1{\varphi }_1^{\prime }\left(t^{\prime }\right)+a_2{\varphi }_2^{\prime }\left(t^{\prime }\right)+a_3{\varphi }_3^{\prime }\left(t^{\prime }\right)=b_1{\varphi }_1^{\prime }\left(t^{\prime }\right)+b_2{\varphi }_2^{\prime }\left(t^{\prime }\right)+b_3{\varphi }_3^{\prime }\left(t^{\prime }\right)$$ E36

$$a_1{\varphi }_1^{″}\left(t^{\prime }\right)+a_2{\varphi }_2^{″}\left(t^{\prime }\right)+a_3{\varphi }_3^{″}\left(t^{\prime }\right)-b_1{\varphi }_1^{″}\left(t^{\prime }\right)-b_2{\varphi }_2^{″}\left(t^{\prime }\right)-b_3{\varphi }_3^{″}\left(t^{\prime }\right)=-6i/{\beta }_3$$ E37

Let $b_1=b_2=b_3=0$ , then

$$a_1=\frac{{\phi }_2{\dot{\phi }}_3-{\dot{\phi }}_2{\phi }_3}{W\left(t^{\prime }\right)},a_2=\frac{{\phi }_3{\dot{\phi }}_1-{\dot{\phi }}_3{\phi }_1}{W\left(t^{\prime }\right)},a_3=\frac{{\phi }_1{\dot{\phi }}_2-{\dot{\phi }}_1{\phi }_2}{W\left(t^{\prime }\right)}$$ E38

$$W\left(t^{\prime }\right)=\left|\begin{array}{ccc}{\varphi }_1& {\varphi }_2& {\varphi }_3\\ {\varphi }_1^{\left(1\right)}& {\varphi }_2^{\left(1\right)}& {\varphi }_3^{\left(1\right)}\\ {\varphi }_1^{\left(2\right)}& {\varphi }_2^{\left(2\right)}& {\varphi }_3^{\left(2\right)}\end{array}\right|$$ E39

Finally, the solution of (27) can be written with the eigen function and Green function:

$$\begin{array}{ll}\phi \left(t\right)& =\varphi \left(t\right)+\int G_0\left(t,t^{\prime }\right)V\left(t^{\prime }\right)\phi \left(t^{\prime }\right)d{t}^{\prime }\\ & =\varphi \left(t\right)+\int G_0\left(t,t^{\prime },E\right)V\left(t^{\prime }\right)\varphi \left(t^{\prime }\right)d{t}^{\prime }+\int d{t}^{\prime }{G}_0\left(t,t^{\prime },E\right)V\left(t^{\prime }\right)\int G_0\left(t^{\prime },t^{″},E\right)V\left(t^{″}\right)\phi \left(t^{″}\right)d{t}^{″}\\ & =\varphi \left(t\right)+\int G_0\left(t,t^{\prime },E\right)V\left(t^{\prime }\right)\varphi \left(t^{\prime }\right)d{t}^{\prime }+\int d{t}^{\prime }{G}_0\left(t,t^{\prime },E\right)V\left(t^{\prime }\right)\int G_0\left(t^{\prime },t^{″},E\right)V\left(t^{″}\right)\varphi \left(t^{″}\right)d{t}^{″}+\cdots \\ & +\underset{\text{times}l}{\underbrace{\int d{t}^{\prime }{G}_0\left(t,t^{\prime }\right)V\left(t^{\prime }\right)\int G_0\left(t^{\prime },t^{″}\right)V\left(t^{″}\right)d{t}^{″}\cdots }}\int G_0\left(t^l,t^{l+1}\right)V\left(t^{l+1}\right)\phi \left(t^{l+1}\right){\mathit{dt}}^{l+1}\end{array}$$ E40

The accuracy can be estimated by the last item of (40). The algorithm is plotted in Figure 3 .

Figure 3.

The Green algorithm for solving NLSE.

3.3 Estimation of the fourth-order dispersion coefficient ${\mathit{\beta }}_{\mathbf{4}}$

The NLSE governing the wave’s transmission in fibers is

$$\frac{\partial u}{\partial z}+\frac{i}{2}{\beta }_2\frac{{\partial }^{2}u}{\partial t^2}-\frac{1}{6}{\beta }_3\frac{{\partial }^{3}u}{\partial t^3}-\mathit{i\gamma }exp\left(-2\mathit{\alpha z}\right)\left[{\left|u\right|}^{2}u+\mathit{\text{is}}\frac{\partial {\left|u\right|}^2}{\partial t}u+\mathit{\text{is}}{\left|u\right|}^2\frac{\partial u}{\partial t}\right]=0$$ E41

where s is the self-steepening parameter. In the frequency domain, its solution is

$$u\left(z+\mathit{dz},\omega \right)=exp\left(\mathit{dz}\widehat{D}\right)exp\left(\mathit{dz}\widehat{N}\right)u\left(z,\omega \right)$$ E42

where $\widehat{D}=\frac{i}{2}{\omega }^2{\beta }_2-\frac{i}{6}{\omega }^3{\beta }_3$ , $\widehat{N}=\mathrm{\Gamma }\left\{\mathit{i\gamma }exp\left(-2\mathit{\alpha z}\right)\left[{\left|u\right|}^2+\mathit{\text{is}}\frac{\partial {\left|u\right|}^2}{\partial t}+\mathit{\text{is}}{\left|u\right|}^2\frac{\partial }{\partial t}\right]\right\}$ , and $\mathrm{\Gamma }$ represents the Fourier transform [32]. Let $\widehat{L}=\frac{\partial }{\partial z}-\widehat{D}-\widehat{N}$ and $\widehat{L}G\left(z,z^{\text{'}},\omega \right)=\delta \left(z-z^{\text{'}}\right)$ ; we obtain the Green function

$$G\left(z,z^{\text{'}},\omega \right)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }\frac{exp\left[-\mathit{ik}\left(z-z^{\text{'}}\right)\right]}{\mathit{ik}-\widehat{D}-\widehat{N}}\mathit{dk}$$ E43

Constructing the iteration ${\beta }_3={\beta }_3^0+{\mathit{\delta \beta }}_3$ , $u\left(z,\omega \right)=u^0\left(z,\omega \right)+\mathit{\delta u}\left(z,\omega \right)$ , then there is

$$\mathit{\delta u}\left(z,\omega \right)=\int G\left(z,z^{\text{'}},\omega \right)Z\left(z^{\text{'}},\omega ,{\mathit{\delta \beta }}_3\left(z^{\text{'}}\right),u^0\left(z^{\text{'}},\omega \right)\right){\mathit{dz}}^{\text{'}}$$ E44

where $Z\left(z^{\text{'}},\omega ,{\mathit{\delta \beta }}_3\left(z^{\text{'}}\right),u^0\left(z^{\text{'}},\omega \right)\right)=-\frac{i}{6}{\mathit{\delta \beta }}_3\left(z^{\text{'}}\right){\omega }^3{u}^0\left(z^{\text{'}},\omega \right)$ and $u^0\left(z^{\text{'}},\omega ,{\beta }_3^0\right)$ is determined by (42).

The minimum value of $\mathit{\delta u}\left(z,\omega \right)$ satisfies $\partial \mathit{\delta u}\left(z,\omega \right)/\partial \omega =0,R\left[{\partial }^2\mathit{\delta u}\left(z,\omega \right)/\partial {\omega }^2\right]>0$ , so

$${\mathit{\delta \beta }}_3=exp\left[{\int }_{-\infty }^{+\infty }\left(-\frac{1}{G}\frac{\partial G}{\partial \omega }-\frac{3}{\omega }-\frac{1}{u^0}\frac{\partial u^0}{\partial \omega }\right)\mathit{d\omega }\right]$$ E45

Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to $\mathit{\delta \gamma }:$ $\gamma ={\gamma }^0+\mathit{\delta \gamma }$ , $u\left(z,\omega \right)=u^0\left(z,\omega \right)+\mathit{\delta u}\left(z,\omega \right)$ and repeating the above process, we get

$$\mathit{\delta \gamma }\approx exp\left[{\int }_{-\infty }^{+\infty }\left(-\frac{1}{G}\frac{\partial G}{\partial \omega }-\frac{3\mathit{\text{is}}}{1-3\mathit{is\omega }}-\frac{1}{u^0}\frac{\partial u^0}{\partial \omega }\right)\mathit{d\omega }\right]$$ E46

Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume $L_D=t_0^2/\mid {\beta }_2\mid$ and $u\left(0,t\right)={\int }_{-\infty }^{+\infty }u\left(0,\omega \right)exp\left(-\mathit{i\omega t}\right)\mathit{d\omega }=u_{0}exp\left(-t^2/t_0^2/2\right)$ .

Firstly, we see what will be induced by the above items ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ . To extrude their impact, we choose the other parameters to be small values in Figures 4 and 5 . The deviation between the red and the black lines in Figure 4(a) indicates the impact of ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ ; that is, they induce the pulse’s symmetrical split. This split does not belong to the SPM-induced broadening oscillation spectral or ${\beta }_3$ -induced oscillation in the tailing edge of the pulse, because here $\gamma$ is very small and ${\beta }_3=0$ [3]. The self-steepening effect attributing to is $\partial \left({\left|u\right|}^{2}u\right)/\partial t$ is also shown explicitly in the black line. When we reduce the s value to 0.0001 in (b), the split pulse’s symmetry is improved.

Figure 4.

The pulse shapes with and without ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ . The red line: without ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ ; the black line: with ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ . $\nu =\omega /2/\pi$ , ${\beta }_3^0=0\left({\mathrm{ps}}^3/\mathrm{km}\right)$ , $\gamma =1.3\times {10}^{-2}\left(/\mathrm{km}/\mathrm{W}\right)$ , $t_0=80\left(\mathrm{fs}\right)$ , $z=3.7\times t_0^2/\left|{\beta }_2\right|$ , ${\beta }_2=-21.7/150\left({\mathrm{ps}}^2/\mathrm{km}\right)$ , $u_0=\left|{\beta }_2\right|/\gamma /t_0^2$ . (a) s = 0.01 and (b) s = 0.0001.

Figure 5.

The evolutions of pulse. The red line: without $\mathit{\delta \gamma }$ ; the black line: with ${\mathit{\delta \beta }}_3$ and $\mathit{\delta \gamma }$ . (a) $s=0$ , $\gamma =1.3\times {10}^{-4}\left(/\mathrm{km}/\mathrm{W}\right)$ ; (b) $s=0.01$ , $\gamma =1.3\times {10}^{-4}\left(/\mathrm{km}/\mathrm{W}\right)$ ; (c) $s=0.01$ , $\gamma =1.3\left(/\mathrm{km}/\mathrm{W}\right)$ . Other parameters are the same as Figure 4 .

Is the pulse split in Figure 4(a) caused by ${\mathit{\delta \beta }}_3$ or $\mathit{\delta \gamma }$ ? The red lines in Figure 5 describe the evolution of pulse affected by the very small second-order dispersion and nonlinear (including self-steepening) coefficients. Here, ${\mathit{\delta \beta }}_3$ induces the pulse’s symmetrical split, and the maximum peaks of split pulse alter and vary from the spectral central to the edge and to the central again. Therefore, its effect is equal to that of the fourth-order dispersion ${\beta }_4$ [33, 34, 3].

From the deviation between the red and black lines in Figure 5 , we can also detect the impact of $\mathit{\delta \gamma }$ . It only accelerates the pulse’s split when the self-steepening effect is ignored (s = 0 in Figure 5(a) ). This is similar to the self-phase modulation-broadening spectral and oscillation. The high nonlinear $\gamma$ accelerating pulse’s split is validated in [35, 36]. If s ≠ 0 ( Figure 5(b) ), $\mathit{\delta \gamma }$ simultaneously leads to the split pulse’s redshift.

Generally, we do not take $\mathit{\delta \gamma }$ into account, so we should clarify in which case it creates impact. Compared (c) with (b) in Figure 5 , the red lines change little means that ${\mathit{\delta \beta }}_3$ has a tiny relationship with γ. But with the increase of γ ( Figure 5(c) ), the split pulse’s redshift is strengthened, so $\mathit{\delta \gamma }$ has a relationship with $\gamma$ . In Figure 6 , the pulse is not split until z = 9 $L_D$ , and the black line with $\mathit{\delta \gamma }$ is completely overlapped by the red line without $\mathit{\delta \gamma }$ , so the high second-order dispersion β ₂ results in the impact of $\mathit{\delta \gamma }$ covered and the impact of ${\mathit{\delta \beta }}_3$ weakened. Therefore, only in the zero-dispersion regime, $\mathit{\delta \gamma }$ should be taken into account in the simulation of pulse shape.

Figure 6.

The pulse shapes with and without $\mathit{\delta \gamma }$ . ${\beta }_2=-21.7\left({\mathrm{ps}}^2/\mathrm{km}\right)$ , $s=0.01$ , $\gamma =1.3\left(/\mathrm{km}/\mathrm{W}\right)$ . Other parameters are the same as Figure 5 .

So, we can utilize ${\mathit{\delta \beta }}_3$ to determine the fourth-order dispersion coefficient ${\beta }_4$ . Fiber parameters are listed in Table 1 . The process is shown in Figure 7 , and the dispersion operator including ${\beta }_4$ is $\widehat{D}=\frac{i}{2}{\omega }^2{\beta }_2-\frac{i}{6}{\omega }^3{\beta }_3+\frac{i}{24}{\omega }^4{\beta }_4$ .

	a (dB/km)	γ (/km/W)	s	${\mathit{\beta }}_{\mathit{2}}$ (ps^2/km)	${\mathit{\beta }}_{\mathit{3}}$ (ps^3/km)
DCF	0.59	5.5	0.01	110	0.1381
NZDSF	0.21	2.2	0.01	−5.6	0.115
SMF	0.21	1.3	0.01	−21.7	−0.5

Table 1.

Fiber parameters.

Figure 7.

The process of calculating ${\beta }_4$ .

Table 2 is the average of ${\beta }_4$ . They are different from those determined by FWM or MI where ${\beta }_4$ is related to power and broadening frequency [35, 36]. By our method, the fourth-order dispersion is also a function of distance, and every type of fibers has its special average ${\beta }_4$ which reveals the characteristic of fibers. These values are similar to those experiment results in highly nonlinear fibers [35, 36]. Although we take the higher-order nonlinear effect $\mathit{\delta \gamma }$ into account which upgrades the pulse’s symmetrical split and redshift, the items is $\mathit{\text{δγ}}\partial \left({\left|u\right|}^{2}u\right)/\partial t$ and $\mathit{i\delta \gamma }exp\left(-2\mathit{\alpha z}\right){\left|u\right|}^{2}u$ have a very tiny contribution to ${\beta }_4$ , only 10⁻²⁶ ps⁴/km quantity order for the typical SMF. Here, the impact of $\mathit{\delta \gamma }$ is hidden by the relative strong ${\beta }_2$ .

	Z = 1.5LD	Z = 5LD	Z = 50LD
DCF	0.0003	0.00035	0.00032
NZDSF	0.0022	0.003	0.0032
SMF	0.0012	0.002	0.0025

Table 2.

The average.

Units (ps⁴/km).

4. Traveling wave solution of NLSE for ASE noise

4.1 The in-phase and quadrature components of ASE noise

The field including the complex envelopes of signal and ASE noise is:

$$U\left(z,t\right)=\sum _{l=1}^N\left[u_l\left(z,t\right)+A_l\left(z,t\right)\right]exp\left(-i{\omega }_{l}t\right)$$ E47

where $u_l\left(z,t\right)$ and $A_l\left(z,t\right)$ are the complex envelopes of signal and ASE noise, respectively [37, 38]. N is the channel number. ASE noise generated in erbium-doped fiber amplifiers (EDFAs) is $A_l\left(0,t\right)=A_{\mathit{lR}}\left(0,t\right)+{\mathit{iA}}_{\mathit{lI}}\left(0,t\right)$ , $A_{\mathit{lR}}\left(0,t\right)$ and $A_{\mathit{lI}}\left(0,t\right)$ are statistically real independent stationary white Gaussian processes, and $⟨A_{\mathit{lR}}\left(0,t+\tau \right)A_{\mathit{lR}}^{\ast }\left(0,t\right)⟩=⟨A_{\mathit{lI}}\left(0,t+\tau \right)A_{\mathit{lI}}^{\ast }\left(0,t\right)⟩=n_{\mathit{sp}}{\mathit{hv}}_l\left(G_l-1\right)\mathrm{\Delta }{v}_l\delta \left(\tau \right)$ . In the complete inversion case, $n_{\mathit{sp}}=1$ . h is the Planck constant. $G_l$ is the gain for channel l.

Substituting Eq. (47) into (1), we can get the equation that $A_l\left(z,t\right)$ satisfies:

$$\begin{array}{c}i\frac{\partial A_l\left(z,t\right)}{\partial z}=\frac{{\beta }_2}{2}\left(-{\omega }_l^2+\frac{{\partial }^2}{\partial t^2}-i2{\omega }_l\frac{\partial }{\partial t}\right)A_l\left(z,t\right)-\\ \gamma \left(z\right)exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2{A}_l\left(z,t\right)\end{array}$$ E48

So, the in-phase and quadrature components of ASE noise obey:

$$\begin{array}{c}\frac{\partial A_{\mathit{lR}}\left(z,t\right)}{\partial z}=-{\beta }_2{\omega }_l\frac{\partial A_{\mathit{lR}}\left(z,t\right)}{\partial t}+\frac{1}{2}{\beta }_2\frac{{\partial }^2{A}_{\mathit{lI}}\left(z,t\right)}{\partial t^2}-\\ \frac{1}{2}{\beta }_2{\omega }_l^2{A}_{\mathit{lI}}-\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2{A}_{\mathit{lI}}\end{array}$$ E49

$$\begin{array}{c}\frac{\partial A_{\mathit{lI}}\left(z,t\right)}{\partial z}=-{\beta }_2{\omega }_l\frac{\partial A_{\mathit{lI}}\left(z,t\right)}{\partial t}-\frac{1}{2}{\beta }_2\frac{{\partial }^2{A}_{\mathit{lR}}\left(z,t\right)}{\partial t^2}+\frac{1}{2}{\beta }_2{\omega }_l^2{A}_{\mathit{lR}}\left(z,t\right)+\\ \gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2{A}_{\mathit{lR}}\end{array}$$ E50

We now seek their traveling wave solution by taking [37] $A_{\mathit{lR}}=\varphi \left(\xi \right),$ $A_{\mathit{lI}}=\phi \left(\xi \right)$ , and $\xi =t-\mathit{cz}$ .

Then, (49) and (50) are converted into

$$\varphi \text{'}\left({\beta }_2{\omega }_l-c\right)=-\left[\frac{1}{2}{\beta }_2{\omega }_l^2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]\phi +\frac{1}{2}{\beta }_2\phi \text{'}\text{'}$$ E51

$$\phi \text{'}\left({\beta }_2{\omega }_l-c\right)=\left[\frac{1}{2}{\beta }_2{\omega }_l^2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]\varphi -\frac{1}{2}{\beta }_2\varphi \text{'}\text{'}$$ E52

(52) is differentiated to $\xi$

$$\phi \text{'}\text{'}\left({\beta }_2{\omega }_l-c\right)=\left[\frac{1}{2}{\beta }_2{\omega }_l^2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]{\varphi }^{\prime }-\frac{1}{2}{\beta }_2\varphi \text{'}\text{'}\text{'}$$ E53

Replacing ${\varphi }^{\prime }$ and $\varphi \text{'}\text{'}\text{'}$ in (53) with (51) and the differential of (51), there are

$$\begin{array}{c}\varphi \text{'}\text{'}{\left({\beta }_2{\omega }_l-c\right)}^2=-{\left[\frac{1}{2}{\beta }_2{\omega }_l^2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]}^2\varphi +\\ {\beta }_2\left[\frac{1}{2}{\beta }_2{\omega }_l^2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]\varphi ″+\frac{1}{4}{\beta }_2^2{\varphi }^{\left(4\right)}\end{array}$$ E54

From (51) and (54), we can easily obtain

$$\phi =B\left\{\left[{\beta }_2{\omega }_l^2/2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]cos\mathit{k\xi }+{\beta }_2{k}^2/2\cdot cos\mathit{k\xi }\right\}/\left({\beta }_2{\omega }_l-c\right)/k$$ E55

$$\phi =Bsin\mathit{k\xi }$$ E56

and

$$\begin{array}{c}B=A_{\mathit{lR}}\left(0,t\right)\left({\beta }_2{\omega }_l-c\right)k/\\ \left\{\left[{\beta }_2{\omega }_l^2/2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2\right]cos\mathit{kt}+{\beta }_2{k}^2/2\cdot cos\mathit{kt}\right\}\end{array}$$ E57

$$\begin{array}{c}c=\pm \{{\beta }_2^2{k}^2/4+\left[{\beta }_2{\omega }_l^2/2+\gamma \exp {\left(-2\alpha z\right)}^2{\left|{\displaystyle \sum }_{j=1}^N{u}_j\left(z\text{,}t\right)+A_j(z\text{,}t)\right|}^2\right]/k^2+{\beta }_2^2{\omega }_l^2/2+\\ {\gamma {\beta }_2\exp \left(-2\alpha z\right){\left|{\displaystyle \sum }_{j=1}^N{u}_j\left(z\text{,}t\right)+A_j(z\text{,}t)\right|}^2\}}^{1/2}+{\beta }_2{\omega }_l\end{array}$$ E58

$$k=arcsin\left(A_{\mathit{lI}}\left(0,t\right)/B\right)/t$$ E59

In the above calculation process, B, c, and k should be regarded as constants, and $A_{\mathit{lR}},A_{\mathit{lI}}$ are the functions of the solo variable $\xi$ , respectively.

4.2 Probability density function of ASE noise

Because $A_{\mathit{lR}}$ and $A_{\mathit{lI}}$ have been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around $A_{\mathit{lR}}$ and $A_{\mathit{lI}}$ are

$$\frac{\partial A_{\mathit{lR}}\left(z,t\right)}{\partial z}=f\left(A_{\mathit{lR}}\left(z,t\right)\right)+g\left(A_{\mathit{lR}}\left(z,t\right)\right)A_{\mathit{lR},z=0}$$ E60

$$\frac{\partial A_{\mathit{lI}}\left(z,t\right)}{\partial z}=f^{\text{'}}\left(A_{\mathit{lI}}\left(z,t\right)\right)+g^{\text{'}}\left(A_{\mathit{lI}}\left(z,t\right)\right)A_{\mathit{lI},z=0}$$ E61

Here,

$$\begin{array}{c}f\left(A_{\mathit{lR}}\left(z,t\right)\right)={\beta }_{2}k{\omega }_l\\ \sqrt{{\left[B\frac{{\beta }_2{\omega }_l^2/2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2+{\beta }_2{k}^2/2}{\left({\beta }_2{\omega }_l-c\right)k}\right]}^2-A_{\mathit{lR}}^2\left(z,t\right)}\end{array}$$ E62

$$\begin{array}{c}g\left(A_{\mathit{lR}}\left(z,t\right)\right)=-\frac{\left({\beta }_2{\omega }_l-c\right)k}{A_{\mathit{lR},z=0}}\\ \sqrt{{\left[B\frac{{\beta }_2{\omega }_l^2/2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2+{\beta }_2{k}^2/2}{\left({\beta }_2{\omega }_l-c\right)k}\right]}^2-A_{\mathit{lR}}^2\left(z,t\right)}\end{array}$$ E63

$$f^{\text{'}}\left(A_{\mathit{lI}}\left(z,t\right)\right)=-{\beta }_{2}k{\omega }_l\sqrt{B^2-A_{\mathit{lI}}^2\left(z,t\right)}$$ E64

$$\begin{array}{c}{g}^{\prime }\left(A_{\mathit{lI}}\left(z,t\right)\right)={\left[B\frac{{\beta }_2{\omega }_l^2/2+\gamma exp\left(-2\mathit{\alpha z}\right){\left|\sum _{j=1}^N{u}_j\left(z,t\right)+A_j(z,t)\right|}^2+{\beta }_2{k}^2/2}{\left({\beta }_2{\omega }_l-c\right)k}\right]}^2\\ \frac{\left({\beta }_2{\omega }_l-c\right)k}{{\mathit{BA}}_{\mathit{lI},z=0}}\sqrt{B^2-A_{\mathit{lI}}^2\left(z,t\right)}\end{array}$$ E65

Now, they can be regarded as the stationary equations, and we can gain their probabilities according to Sections (7.3) and (7.4) in [39]. By solving the corresponding Fokker-Planck equations of (60) and (61), the probabilities of ASE noise are

$$p_{\mathit{lR}}=\frac{C}{{\left[g\left(A_{\mathit{lR}}\right)\right]}^2}exp\left[2{\int }_{-\infty }^{A_{\mathit{lR}}}\frac{f\left(s\right)}{{\left[g\left(s\right)\right]}^2}\mathit{ds}\right]$$ E66

$$p_{\mathit{lI}}=\frac{C^{\prime }}{{\left[g^{\prime }\left(A_{\mathit{lI}}\right)\right]}^2}exp\left[2{\int }_{-\infty }^{A_{\mathit{lI}}}\frac{f^{\prime }\left(s\right)}{{\left[g^{\prime }\left(s\right)\right]}^2}\mathit{ds}\right]$$ E67

$C,C^{\prime }$ are determined by ${\int }_{-\infty }^{+\infty }\mathit{pdp}=1$ . Compared with [40], these probabilities of ASE noise take dispersion effect into account. This is the first time that the p.d.f. of ASE noise simultaneously including dispersion and nonlinear effects is presented.

(66) and (67) are efficient in the models of Gaussian and correlated non-Gaussian processes as our (49) and (50). Obviously, the Gaussian distribution has been distorted. They are no longer symmetrical distributions, and both have phase shifts consistent with [40], and as its authors have expected that “if the dispersion effect was taken into account, the asymmetric modulation side bands occur.” The reasons are that item $-i{\beta }_2{\omega }_l\frac{\partial }{\partial t}{A}_l\left(z,t\right)$ in (48) brings the phase shift and item $\frac{{\beta }_2}{2}\frac{{\partial }^2}{\partial t^2}{A}_l\left(z,t\right)$ brings the expansion and induces the side bands, the self-phase modulation effects, and the cross-phase modulation effects. Their synthesis impact is amplified by (66) and (67) and results in the complete non-Gaussian distributions.

5. Conclusion

NLSE is solved with small-signal analyses for the analyses of MI, and it can be broadened to all signal formats. The equation can be solved by introducing the Green function in the time domain, and it is used as the tool for the estimations of high-order dispersion and nonlinear coefficients. For the conventional fibers, SMF, NZDSF, and DCF, the higher-order nonlinear effect contribution to β ₄ can be neglected. This can be deduced that each effect has less impact for another coefficient’s estimation. The Green function can also be used for the solving of 3 + 1 dimension NLSE.

By the traveling wave methods, the p.d.f. of ASE noise can be obtained, and it provides a method for the calculation of ASE noise in WDM systems. So, the properties of MI, pulse fission, coefficient value, and ASE noise’s probability density function are also discussed for demonstrations of the theories.