Fiber parameters.

## 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

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

where

Usually, the field amplitudes can be written as

Assuming:

The amplitude

The small-signal theory implies that the frequency modulation or noise

The operation

The modulation or noise

So

And

When only intensity modulation is present and no phase modulation exists, the transfer function

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

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

Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case *dz* satisfies *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 *L* is separated into two parts 0 < *z* < *Lwo *and *Lwo * < *z* < *L*; *Lwo *is the walk-off length, *D* is the dispersion coefficient, and *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].

*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.

## 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

where

There are

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

Repeating the process of [3]

*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:

where

### 3.2 The solution by Green function

The solution has the form

Then, there is

Let

and taking the operator

Assuming

Its characteristic roots are

where

By the construction method, it is

At the point

Let

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 .

### 3.3 Estimation of the fourth-order dispersion coefficient β 4

The NLSE governing the wave’s transmission in fibers is

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

where

Constructing the iteration

where

The minimum value of

Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to

Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume

Firstly, we see what will be induced by the above items *is* *s* value to 0.0001 in (b), the split pulse’s symmetry is improved.

Is the pulse split in Figure 4(a) caused by

From the deviation between the red and black lines in Figure 5 , we can also detect the impact of *s =* 0 in Figure 5(a) ). This is similar to the self-phase modulation-broadening spectral and oscillation. The high nonlinear *s ≠* 0 ( Figure 5(b) ),

Generally, we do not take *γ*. But with the increase of *γ* ( Figure 5(c) ), the split pulse’s redshift is strengthened, so *z* = 9 *β* _{2} results in the impact of

So, we can utilize

a (dB/km) | γ (/km/W) | s | ^{2}/km) | ^{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 2 is the average of *is* ^{−26} ps^{4}/km quantity order for the typical SMF. Here, the impact of

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 |

## 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:

where *N* is the channel number. ASE noise generated in erbium-doped fiber amplifiers (EDFAs) is *h* is the Planck constant. *l*.

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

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

We now seek their traveling wave solution by taking [37]

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

(52) is differentiated to

Replacing

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

and

In the above calculation process, *B*, *c*, and *k* should be regarded as constants, and

### 4.2 Probability density function of ASE noise

Because

Here,

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

(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

## 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.