Fiber parameters.
Abstract
Firstly, based on the smallsignal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss is solved. It is also adapted to the NLSE with the highorder dispersion terms. Furthermore, a general theory on crossphase modulation (XPM) intensity fluctuation which adapted to all kinds of modulation formats (continuous wave, nonreturntozero wave, and returnzero 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 splitstep method in which the step size has to be carefully controlled. Additionally, the fourthorder dispersion coefficient of fibers can be estimated by the Green function solution of NLSE. The fourthorder 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 zerodispersion regime, the higherorder nonlinear effect (higher than selfsteepening) 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 FokkerPlanck equation including the dispersion effect is presented.
Keywords
 smallsignal analysis
 Green function
 traveling wave solution
 FokkerPlanck 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 dambreak approximation for the nonreturntozero 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 EulerLagrange 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 timeconsuming 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 pseudospectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step sizecontrolling method [12] and the predictorcorrector method [13] were proposed to speed up the implementation of splitstep Fourier method (SSFM). The cubic (or higher order) Bsplines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The RungeKutta 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 quasilinear firstorder hyperbolic system, the accurate simulations of the intensity and phase for the Schrödingertype pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higherorder NLSE in the presence of nonKerr quintic nonlinearities [18].
In this chapter, several methods to solve the NLSE will be presented: (1) The smallsignal analysis theory and splitstep 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 highorder 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 fourthorder dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its probability density function.
2. Smallsignal 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
where
Usually, the field amplitudes can be written as
Assuming:
The amplitude
The smallsignal 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 smallphase modulation index. Even for large modulation index
Obviously, the above process can be used to treat NLSE with higherorder dispersion (
The corresponding MI gain
Figure 2
shows a comparison of the gain spectra between Eq. (11) and [6] for the case
2.2 The general theory on crossphase modulation (XPM) intensity fluctuation
For the general case of two channels, the input optical powers are denoted by
This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from
The walkoff between copropagating 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
Repeating the process of [3]
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 fourthorder dispersion coefficient
β
4
The NLSE governing the wave’s transmission in fibers is
where
where
Constructing the iteration
where
The minimum value of
Next, we take the higherorder nonlinear effect into account. Constructing another iteration related to
Now, we can simulate the pulse shape affected by highorder dispersive and nonlinear effects. Assume
Firstly, we see what will be induced by the above items
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
Generally, we do not take
So, we can utilize







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





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 inphase and quadrature components of ASE noise
The field including the complex envelopes of signal and ASE noise is:
where
Substituting Eq. (47) into (1), we can get the equation that
So, the inphase 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,
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 FokkerPlanck equations of (60) and (61), the probabilities of ASE noise are
(66) and (67) are efficient in the models of Gaussian and correlated nonGaussian 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 smallsignal 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 highorder dispersion and nonlinear coefficients. For the conventional fibers, SMF, NZDSF, and DCF, the higherorder nonlinear effect contribution to
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.
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