Open access peer-reviewed chapter

Nonlinear Schrödinger Equation

By Jing Huang

Submitted: May 9th 2018Reviewed: August 23rd 2018Published: December 10th 2018

DOI: 10.5772/intechopen.81093

Downloaded: 419

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.

Keywords

  • small-signal analysis
  • Green function
  • traveling wave solution
  • Fokker-Planck equation
  • nonlinear Schrodinger equation

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

uz+β1ut+i2β22ut2+a2u=u2+2u2uE1

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

uz+dzω=expdzD̂expdzN̂uzωE2

where D̂=i2ω2β2+iωβ1a2and N̂=u2+i2u2[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

uzω=PzωexpzωE3

ϕzωis caused by the nonlinear effect, and ϕzω=0zγPzω+2Pzωdz[3].

uz+dzω(is)

uz+dzω=expdzD̂Pzωexpzω+P+2Pdz=eadz/2expβ1ωdzexpβ2/2ω2dzPzωez+dzω=Pz+dzωexpz+dz.ωE4

Assuming: Pzω=Pz+ΔPzω

Pzis the average signal intensity. ΔPzωis the noise or modulation term. There is [20] PzΔPzω

The amplitude Pzωcan be regarded as

PzωPz1+ΔPzω2PzE5

The small-signal theory implies that the frequency modulation or noise φ̇z+dzω=dφ̇z+dzωdtis small enough. Finally ([21])

Pz+dzω=Pz+2eadz/2×RePzexpiωβ1dz+iω2β2dzΔPzω2Pz+z+dzωE6

The operation expiωβ1dz+iω2β2dzcan be split into its real and imaginary parts:

expiωβ1dz+iω2β2dz=cosωβ1dz+ω2β2dz+isinωβ1dz+ω2β2dzE7

The modulation or noise ΔPz+dzωis ΔPz+dzωPz+dzωPz

So

Pz+dzω=eadz/2iωβ1dzcos12β2ω2dzΔPzω+sin12β2ω2dz2Pzφ(z+dzω)E8

And

ΔPz+dzω2Pzφz+dzω=eadz/2iωβ1dzePz+2Pzdzcos12β2ω2dzsin12β2ω2dzsin12β2ω2dzcos12β2ω2dzΔPzω2PzφzωE9

When only intensity modulation is present and no phase modulation exists, the transfer function cos12β2ω2dzis obtained. The 3 dB cutoff frequency corresponds to 12β2ω2dz=π/4in [22, 23]. This treatment is also adaptable to the case that only the nonlinear phase (frequency) modulation is present; then, the intensity modulation ΔPz+dzωdue to FM-IM conversion is given as

ΔPz+dzω=2Pzeadz/2iωβ1dzsin12β2dzω2φz+dzωE10

This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index 12β2ω2dz=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].

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 gMI in the side bands of ω 0 (the frequency of signal) is given by

gMIzω=ΔPz+dzωΔP(zω)Pzdz=2eadz/2iωβ1dzsin12β2dzω2γzz+dzPzω+2Pzωdz/dzE11

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 Pz/Pz=1. The maximum frequency modulation index caused by dispersion corresponds to 12β2ω2dz=π[22, 23], and the maximum value of the sideband is ωc=4γPz/β2, so the choice of dz satisfies 12β2ω2dz=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 gMI . In practice, researchers generally utilize the maximum value of gMI 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 gMI 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 Pt,Pt, 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 < Lwo and Lwo  < z < L; Lwo is the walk-off length, Lwo=Δt/DΔλ. Δtis the edge duration of the carrier wave, D is the dispersion coefficient, and Δλis 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

dϕXPMzt=γ2Pztzβ1eazdzE12

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].

dPXPMzω=2eiωzβ1PzωeaLzeiωβ1LzsinbLzdφXPMzω=4γeiωzβ1PzωeaLzeazeβ1zeiωβ1LzPzωsinbLzdzE13

representing the convolution operation b=ω2Dλ2/4πc, 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:

PXPM=0LdPXPMzωdz=0L4γeiωzβ1PzωeaLzeazeβ1zeiωβ1LzPzωsinbLzdzE14

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

2E1c22Et2=u02PLt2u02PNLt2E15
PLrt=ε0+χ1ttErtdt=ε0+χ1ωErωexpiωtE16
χ1ω=+dτχ1τexpjωτE17

where Eis the vector field and χ1is the linear susceptibility. PLand PNLrepresent the linear and nonlinear induced fields, respectively [30]. The cubic susceptibility tensor including the resonant and nonresonant terms is

χ3ω=χNR3+χR3ωE18

There are

PNL,NRrt=ε0dt1dt2dt3χNR3t1t2t3Ertt1Ertt2Ertt3=ε0dω1dω2dω3χNR3ω1ω2ω3ω1+ω2+ω3Ert1Ert2Ert3expjωtδωω1ω2ω3E19
χNR3ω1ω2ω3ω1+ω2+ω3=dt1dt2dt3χNR3t1t2t3expjω1t1jω2t2jω3t3E20
PNL,Rrt=ε0dt1dt2dt3χR3tt1t2t3Ertt1Ertt2Ertt3=ε0dω1dω2dω3χR3tω1ω2ω3ω1+ω2+ω3Ert1Ert2Ert3expjωtδωω1ω2ω3E21
χR3t=12π+aωω1+ω2+ω3+iΓeiωt=π2a1+ΓΓeΓt+iω1+ω2+ω3tiπ2E22

Γand a are the attenuation and absorption coefficients, respectively [31].

Repeating the process of [3]

E=FxyAztexpiβz, there is

Az+i2β22At216β33At3=a2A+i3k08nAeffχNR3A2A+ik0gω01ifω02nAeffAtχR3tτAτ2E23

k0=ω0/c, where ω0is the center frequency. Aeff 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:

fω1+ω2+ω3=2ω1+ω2+ω31Γ2ω1+ω2+ω322Γ+Γ2E24
gω1+ω2+ω3=2ω1+ω2+ω322Γ+Γ2E25

where gω1+ω2+ω3is the Raman gain and fω1+ω2+ω3is the Raman non-gain coefficients.

3.2 The solution by Green function

The solution has the form

Azt=φteiEzE26

Then, there is

12β22ϕt2+i6β33ϕt33k08nAeffχNR3ϕ2ϕk0gωs1ifωs2nAeffϕ+χR3tτϕτ2=E27

Let

Ĥ0t=12β22t2+i6β33t3E28
V̂t=3k08nAeffχNR3ϕk0gωs1ifωs2nAeff+χR3tτϕτ2E29

and taking the operator V̂tas a perturbation item, we first solve the eigen equation n=2kinn!βnnφTn=.

12β22ϕT2+i6β33ϕT3=E30

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

12β2r2+β36r3=EE31

Its characteristic roots are r1,r2,r3. The solution can be represented as

ϕ=c1ϕ1+c2ϕ2+c3ϕ3E32

where ϕm=expirmt,m=1,2,3and c1,c2,c3are determined by the initial pulse. The Green function of (30) is

EĤ0tG0tt=δttE33

By the construction method, it is

G0tt=a1φ1+a2φ2+a3φ3,t>tb1φ1+b2φ2+b3φ3,t<tE34

At the point t=t, there are

a1ϕ1t+a2ϕ2t+a3ϕ3t=b1ϕ1t+b2ϕ2t+b3ϕ3tE35
a1ϕ1t+a2ϕ2t+a3ϕ3t=b1ϕ1t+b2ϕ2t+b3ϕ3tE36
a1ϕ1t+a2ϕ2t+a3ϕ3tb1ϕ1tb2ϕ2tb3ϕ3t=6i/β3E37

Let b1=b2=b3=0, then

a1=φ2φ̇3φ̇2φ3Wt,a2=φ3φ̇1φ̇3φ1Wt,a3=φ1φ̇2φ̇1φ2WtE38
Wt=ϕ1ϕ2ϕ3ϕ11ϕ21ϕ31ϕ12ϕ22ϕ32E39

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

φt=ϕt+G0ttVtφtdt=ϕt+G0ttEVtϕtdt+dtG0ttEVtG0ttEVtφtdt=ϕt+G0ttEVtϕtdt+dtG0ttEVtG0ttEVtϕtdt++dtG0ttVtG0ttVtdttimeslG0tltl+1Vtl+1φtl+1dtl+1E40

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 β4

The NLSE governing the wave’s transmission in fibers is

uz+i2β22ut216β33ut3exp2αzu2u+isu2tu+isu2ut=0E41

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

uz+dzω=expdzD̂expdzN̂uzωE42

where D̂=i2ω2β2i6ω3β3, N̂=Γexp2αzu2+isu2t+isu2t, and Γrepresents the Fourier transform [32]. Let L̂=zD̂N̂and L̂Gzz'ω=δzz'; we obtain the Green function

Gzz'ω=12π+expikzz'ikD̂N̂dkE43

Constructing the iteration β3=β30+δβ3, uzω=u0zω+δuzω, then there is

δuzω=Gzz'ωZz'ωδβ3z'u0z'ωdz'E44

where Zz'ωδβ3z'u0z'ω=i6δβ3z'ω3u0z'ωand u0z'ωβ30is determined by (42).

The minimum value of δuzωsatisfies δuzω/ω=0,R2δuzω/ω2>0, so

δβ3=exp+1GGω3ω1u0u0ωE45

Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to δγ:γ=γ0+δγ, uzω=u0zω+δuzωand repeating the above process, we get

δγexp+1GGω3is13isω1u0u0ωE46

Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume LD=t02/β2and u0t=+u0ωexpiωt=u0expt2/t02/2.

Firstly, we see what will be induced by the above items δβ3and 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 δβ3and δγ; that is, they induce the pulse’s symmetrical split. This split does not belong to the SPM-induced broadening oscillation spectral or β3-induced oscillation in the tailing edge of the pulse, because here γis very small and β3=0[3]. The self-steepening effect attributing to is u2u/tis 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 δβ 3 and δγ . The red line: without δβ 3 and δγ ; the black line: with δβ 3 and δγ . ν = ω / 2 / π , β 3 0 = 0 ps 3 / km , γ = 1.3 × 10 − 2 / km / W , t 0 = 80 fs , z = 3.7 × t 0 2 / β 2 , β 2 = − 21.7 / 150 ps 2 / km , u 0 = β 2 / γ / t 0 2 . (a) s = 0.01 and (b) s = 0.0001.

Figure 5.

The evolutions of pulse. The red line: without δγ ; the black line: with δβ 3 and δγ . (a) s = 0 , γ = 1.3 × 10 − 4 / km / W ; (b) s = 0.01 , γ = 1.3 × 10 − 4 / km / W ; (c) s = 0.01 , γ = 1.3 / km / W . Other parameters are the same as Figure 4 .

Is the pulse split in Figure 4(a) caused by δβ3or 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, δβ3induces 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 β4[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 modulation-broadening spectral and oscillation. The high nonlinear γaccelerating pulse’s split is validated in [35, 36]. If s ≠ 0 ( Figure 5(b) ), δγsimultaneously leads to the split pulse’s redshift.

Generally, we do not take δγinto 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 δβ3has a tiny relationship with γ. But with the increase of γ ( Figure 5(c) ), the split pulse’s redshift is strengthened, so δγhas a relationship with γ. In Figure 6 , the pulse is not split until z = 9 LD, and the black line with δγis completely overlapped by the red line without δγ, so the high second-order dispersion β 2 results in the impact of δγcovered and the impact of δβ3weakened. Therefore, only in the zero-dispersion regime, δγshould be taken into account in the simulation of pulse shape.

Figure 6.

The pulse shapes with and without δγ . β 2 = − 21.7 ps 2 / km , s = 0.01 , γ = 1.3 / km / W . Other parameters are the same as Figure 5 .

So, we can utilize δβ3to determine the fourth-order dispersion coefficient β4. Fiber parameters are listed in Table 1 . The process is shown in Figure 7 , and the dispersion operator including β4is D̂=i2ω2β2i6ω3β3+i24ω4β4.

a (dB/km)γ (/km/W)sβ2(ps2/km)β3(ps3/km)
DCF0.595.50.011100.1381
NZDSF0.212.20.01−5.60.115
SMF0.211.30.01−21.7−0.5

Table 1.

Fiber parameters.

Figure 7.

The process of calculating β 4 .

Table 2 is the average of β4. They are different from those determined by FWM or MI where β4is 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 β4which 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 δγinto account which upgrades the pulse’s symmetrical split and redshift, the items is δγu2u/tand iδγexp2αzu2uhave a very tiny contribution to β4, only 10−26 ps4/km quantity order for the typical SMF. Here, the impact of δγis hidden by the relative strong β2.

Z = 1.5LDZ = 5LDZ = 50LD
DCF0.00030.000350.00032
NZDSF0.00220.0030.0032
SMF0.00120.0020.0025

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:

Uzt=l=1Nulzt+AlztexpiωltE47

where ulztand Alztare 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 Al0t=AlR0t+iAlI0t, AlR0tand AlI0tare statistically real independent stationary white Gaussian processes, and AlR0t+τAlR0t=AlI0t+τAlI0t=nsphvlGl1Δvlδτ. In the complete inversion case, nsp=1. h is the Planck constant. Glis the gain for channel l.

Substituting Eq. (47) into (1), we can get the equation that Alztsatisfies:

iAlztz=β22ωl2+2t2i2ωltAlztγzexp2αzj=1Nujzt+Aj(zt)2AlztE48

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

AlRztz=β2ωlAlRztt+12β22AlIztt212β2ωl2AlIγexp2αzj=1Nujzt+Aj(zt)2AlIE49
AlIztz=β2ωlAlIztt12β22AlRztt2+12β2ωl2AlRzt+γexp2αzj=1Nujzt+Aj(zt)2AlRE50

We now seek their traveling wave solution by taking [37] AlR=ϕξ,AlI=φξ, and ξ=tcz.

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

ϕ'β2ωlc=12β2ωl2+γexp2αzj=1Nujzt+Aj(zt)2φ+12β2φ''E51
φ'β2ωlc=12β2ωl2+γexp2αzj=1Nujzt+Aj(zt)2ϕ12β2ϕ''E52

(52) is differentiated to ξ

φ''β2ωlc=12β2ωl2+γexp2αzj=1Nujzt+Aj(zt)2ϕ12β2ϕ'''E53

Replacing ϕand ϕ'''in (53) with (51) and the differential of (51), there are

ϕ''β2ωlc2=12β2ωl2+γexp2αzj=1Nujzt+Aj(zt)22ϕ+β212β2ωl2+γexp2αzj=1Nujzt+Aj(zt)2ϕ+14β22ϕ4E54

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

φ=Bβ2ωl2/2+γexp2αzj=1Nujzt+Aj(zt)2cos+β2k2/2cos/β2ωlc/kE55
φ=BsinE56

and

B=AlR0tβ2ωlck/β2ωl2/2+γexp2αzj=1Nujzt+Aj(zt)2coskt+β2k2/2cosktE57
c=±{β22k2/4+[β2ωl2/2+γexp(2αz)2|j=1Nuj(z,t)+Aj(z,t)|2]/k2+β22ωl2/2+γβ2exp(2αz)|j=1Nuj(z,t)+Aj(z,t)|2}1/2+β2ωlE58
k=arcsinAlI0t/B/tE59

In the above calculation process, B, c, and k should be regarded as constants, and AlR,AlIare the functions of the solo variable ξ, respectively.

4.2 Probability density function of ASE noise

Because AlRand AlIhave been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around AlRand AlIare

AlRztz=fAlRzt+gAlRztAlR,z=0E60
AlIztz=f'AlIzt+g'AlIztAlI,z=0E61

Here,

fAlRzt=β2kωlBβ2ωl2/2+γexp2αzj=1Nujzt+Aj(zt)2+β2k2/2β2ωlck2AlR2ztE62
gAlRzt=β2ωlckAlR,z=0Bβ2ωl2/2+γexp2αzj=1Nujzt+Aj(zt)2+β2k2/2β2ωlck2AlR2ztE63
f'AlIzt=β2kωlB2AlI2ztE64
gAlIzt=Bβ2ωl2/2+γexp2αzj=1Nujzt+Aj(zt)2+β2k2/2β2ωlck2β2ωlckBAlI,z=0B2AlI2ztE65

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

plR=CgAlR2exp2AlRfsgs2dsE66
plI=CgAlI2exp2AlIfsgs2dsE67

C,Care determined by +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 iβ2ωltAlztin (48) brings the phase shift and item β222t2Alztbrings 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.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Jing Huang (December 10th 2018). Nonlinear Schrödinger Equation, Nonlinear Optics - Novel Results in Theory and Applications, Boris I. Lembrikov, IntechOpen, DOI: 10.5772/intechopen.81093. Available from:

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