Nonlinear Schrödinger Equation

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 ps/km, 0.003 ps/nm, and 0.00032 ps/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 β2. 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.


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,

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 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 whereD ¼ i 2 ω 2 β 2 þ iωβ 1 À a 2 andN ¼ iγ u j j 2 þ i2 u 0 j j 2 h i [19] (Figure 1).
Usually, the field amplitudes can be written as ϕ z; ω ð Þis caused by the nonlinear effect, and ϕ z; iis the average signal intensity. ΔP z; ω ð Þ is the noise or modulation term. There is [20] can be regarded as The small-signal theory implies that the frequency modulation or noise The operation exp iωβ 1 dz þ iω 2 β 2 dz ð Þ can be split into its real and imaginary parts: The modulation or noise ΔP z þ dz; And 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.
When only intensity modulation is present and no phase modulation exists, the transfer function cos 1 2 β 2 ω 2 dz À Á is obtained. The 3 dB cutoff frequency corresponds to 1 2 β 2 ω 2 dz ¼ π=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 ΔP z þ dz; ω ð Þdue to FM-IM conversion is given as This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index 1 2 β 2 ω 2 dz ¼ π=2, the difference is within 0.5 dB. Eq. (10) does not include a Bessel function, so it is simpler than that in [24].
The corresponding MI gain g MI in the side bands of ω 0 (the frequency of signal) is given by  Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case P z ð Þ h i= P 0 z ð Þ h i¼ 1. The maximum frequency modulation index caused by dispersion corresponds to 1 2 β 2 ω 2 dz ¼ π [22,23], and the maximum value of the sideband is ω c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , so the choice of dz satisfies 1 2 β 2 ω 2 dz ¼ π, 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.

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 t ð Þ, P 0 t ð Þ, 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, Δt is the edge duration of the carrier wave, D is the dispersion coefficient, and Δλ is the wavelength spacing between the channels. By the smallsignal analysis, the phase modulation in channel 1 originating in dz at z can be expressed as 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].
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: The walk-off between co-propagating waves is regulated by the convolution operation.

NLSE including the resonant and nonresonant cubic susceptibility tensors
From Maxwell's equation, the field in fibers satisfies where E ! is the vector field and χ 1 ð Þ is the linear susceptibility. P L ! and P NL ! represent the linear and nonlinear induced fields, respectively [30]. The cubic susceptibility tensor including the resonant and nonresonant terms is There are Γ and a are the attenuation and absorption coefficients, respectively [31].
Repeating the process of [3] where ω 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, selffrequency shift, and self-steepening originating from the delayed response: where g ω 1 þ ω 2 þ ω 3 ð Þ is the Raman gain and f ω 1 þ ω 2 þ ω 3 ð Þ is the Raman nongain coefficients.

The solution by Green function
The solution has the form Then, there is and taking the operatorV t ð Þ as a perturbation item, we first solve the eigen Assuming E ¼ 1, we get the corresponding characteristic equation: Its characteristic roots are r 1 , r 2 , r 3 . The solution can be represented as where ϕ m ¼ exp ir m t ð Þ, m ¼ 1, 2, 3 and c 1 , c 2 , c 3 are determined by the initial pulse. The Green function of (30) is By the construction method, it is At the point t ¼ t 0 , there are Finally, the solution of (27) can be written with the eigen function and Green function: The accuracy can be estimated by the last item of (40). The algorithm is plotted in Figure 3.
Firstly, we see what will be induced by the above items δβ 3 and δγ. 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 δβ 3 and δγ; that is, they induce the pulse's symmetrical split. This split does not belong to the SPM-induced broadening oscillation spectral or β 3 -induced oscillation in the tailing edge of the pulse, because here γ is very small and β 3 ¼ 0 [3]. The self-steepening effect attributing to is ∂ u j j 2 u =∂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. Is the pulse split in Figure 4(a) caused by δβ 3 or δγ? 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, δβ 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 β 4 [33,34,3].
From the deviation between the red and black lines in Figure 5, we can also detect the impact of δγ. 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 modulationbroadening spectral and oscillation. The high nonlinear γ accelerating pulse's split is validated in [35,36]. If s 6 ¼ 0 ( Figure 5(b)), δγ simultaneously leads to the split pulse's redshift.
Generally, we do not take δγ 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 δβ 3 has a tiny relationship with γ. But with the increase of γ ( Figure 5(c)), the split pulse's redshift is strengthened, so δγ has a relationship with γ. In Figure 6, the pulse is not split until z = 9 L D , and the black line with δγ is completely overlapped by the red line without δγ, so the high second-order dispersion β 2 results in the impact of δγ covered and the impact of δβ 3 weakened. Therefore, only in the zerodispersion regime, δγ should be taken into account in the simulation of pulse shape. So, we can utilize δβ 3 to determine the fourth-order dispersion coefficient β 4 . Fiber parameters are listed in Table 1. The process is shown in Figure 7, and the dispersion operator including β 4 isD ¼ i 2 ω 2 β 2 À i 6 ω 3 β 3 þ i 24 ω 4 β 4 . Table 2 is the average of β 4 . They are different from those determined by FWM or MI where β 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 Other parameters are the same as Figure 4.    fibers has its special average β 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 δγ into account which upgrades the pulse's symmetrical split and redshift, the items is ∂ u j j 2 u =∂t and iδγ exp À2αz ð Þu j j 2 u have a very tiny contribution to β 4 , only 10 À26 ps 4 /km quantity order for the typical SMF. Here, the impact of δγ is hidden by the relative strong β 2 .

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: where u l z; t ð Þ and A l z; t ð Þ 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 0; t ð Þ ¼ A lR 0; t ð ÞþiA lI 0; t ð Þ, A lR 0; t ð Þ and A lI 0; t ð Þ are statistically real independent stationary white Gaussian processes, and In the complete inversion case, n sp ¼ 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 z; t ð Þ satisfies: So, the in-phase and quadrature components of ASE noise obey: We now seek their traveling wave solution by taking [37] A lR ¼ ϕ ξ ð Þ, A lI ¼ φ ξ ð Þ, and ξ ¼ t À cz. Then, (49) and (50) are converted into Replacing ϕ 0 and ϕ 000 in (53) with (51) and the differential of (51), there are From (51) and (54), we can easily obtain and u j z; t ð Þ þA j ðz; tÞ (57) In the above calculation process, B, c, and k should be regarded as constants, and A lR , A lI are the functions of the solo variable ξ, respectively.

Probability density function of ASE noise
Because A lR and A lI have been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around A lR and A lI are effect was taken into account, the asymmetric modulation side bands occur." The reasons are that item Àiβ 2 ω l ∂ ∂t A l z; t ð Þ in (48) brings the phase shift and item β 2 2 ∂ 2 ∂t 2 A l z; t ð Þ 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.

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.

Author details
Jing Huang Physics Department, South China University of Technology, Guangzhou, China *Address all correspondence to: jhuang@scut.edu.cn © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.