Effect of Fiber Self Phase Modulation on the Splitting Error using the Strang Formulas

The generalized nonlinear Schrodinger equation describes the different physical phenomena encountered when ultrashort pulses propagate through dispersive and nonlinear fibers. If the pulse duration is of picoseconds order, the nonlinear Schrodinger equation can be simp lified. However the analyt ical solution remains inaccessible except for some special cases like soliton. The symmetric split-step Fourier method (S-SSFM) which is derived fro m the St rang formulas, subdivides the global propagation distance into small steps of length h to calculate the numerical solution of this equation. By using only the fact that the dispersive and nonlinear operators do not commute the Baker-Campbell-Hausdorff formu la shows that the global relative error of this method is O(h 2 ). Our numerical simulation results show that this error depends also on the self phase modulation nonlinear term. For this purpose, we emp loy in this work an exp licit representation of the nonlinear operator and we present four imp lementations: the S-SSFM 1, S-SSFM 2, T-SM1 and T-SM 2 obtained respectively fro m


Introduction
Ultrashort pulse propagation in dispersive and nonlinear optical fibers is described by the generalized nonlinear Schrödinger equation (G-NLSE) [1,2]: , is the amplitude of the variable field, β 2 and β 3 are respectively, the second and the third order d ispersion coefficients, α is the attenuation coefficient and γ is the nonlinear coefficient of the self phase modulat ion due to optical Kerr effect. R T is the slope o f th e Raman gain (stimu lated Raman scattering (SRS)) and 0 ω is the center angular frequency. The term proportional to in the second member represents the self-steepening effect. The quantity is the retarded time where z is the position along the fiber, ' t is the physical t ime and g v is the group velocity at the center wavelength. We are not considering here the effect of higher order dispersion (coefficients β 4 , β 5 …).
The generalized nonlinear Schrödinger equation is known to be applicable for femtosecond pulses; it can be simplified when the pulse width is of p icosecond order (not less than 5ps). The Raman effect and the self-steepening effect terms can be neglected compared to the self phase modulation term [3]. For such pulses the contribution of the third order dispersion is also quite small co mpared to the second order dispersion term unless operation is near the zero of the group velocity dispersion (where β 2 is null). The simp lified NLSE is then given by the follo wing equation: , ( This equation can be written as with a linear operator D  and a nonlinear operator N  defined as Equation (2) cannot be solved analytically except for some special cases like soliton where the d ispersion effects are compensated by the optical Kerr effect [4][5][6].
Several methods has been developed to determine the numerical solution of this equation, the most important are called operator splitting methods and the widely used are the The nonlinear term is solved in t ime domain, whereas the dispersion term is solved in the frequency domain and requires some fast Fourier transform (FFT) routines. Fro m the Lie formu las derive the split-step Fourier method (SSFM), and the Strang formulas lead to the symmetric split step Fourier method (S-SSFM).
In the split-step Fourier method (SSFM ), the global propagation distance is subdivides into steps of length h, sufficiently s mall, and the approximate solution is obtained by supposing that along each step the effects of dispersion and nonlinearity are assumed to be independent. The error of this method derives fro m the fact that the operators of dispersion and nonlinearity do not commute. To imp rove the precision, we must consider another procedure for the pulse propagation where the effects of nonlinearity (or d ispersion) are inserted at the middle of each step of the fiber. Th is is known as the symmetric split step Fourier method (S-SSFM) and its description are given in paragraph 2. In the paragraph 3, we study the effect of the nonlinear term 2 ) , ( t z A on the S-SSFM global relat ive error. We present for this purpose four implementations: the S-SSFM 1, S-SSFM2, T-SM 1 and T-SM 2 and we will show numerically that this error depends on the nonlinear term and is O(h) or O(h 2 ). Ho wever, the S-SSFM algorith ms are more accurate than the T-SM. We present the numerical results in paragraph 4 and a conclusion of this work is given in paragraph 5.

The Symmetric Split-Step Fourier Method
If the step h is sufficiently s mall, we can neglect the variations of the nonlinear operator N  in the interval[z, z+h] and the formal solution of the variable field amplitude A(z+h, t) in terms of A(z, t) is given by the following equation: The two operators D  and N  do not commute thus the calculation of A(z, t) is difficult to realize. Equation (5) can be approached by the following equation: (6) or by the equation: An argument based on the Baker-Campbell-Hausdorff formula [9] shows that the local error, which is the error along one step of length h of the symmetric split-step scheme is O(h 3 ). Since the total nu mber of the steps in a fiber is inversely proportional to the spatial step length h, the global relative error accu mulated along the whole fiber is O(h 2 ) [10,11].

Evaluation of the Nonlinear Operator
The nonlinear operator can be appro ximated by using the trapezoidal rule: cannot be known at z+h/2, then we propose two implementations.
We explo it an explicit representation of the nonlinear operator N  where the attenuation factor α is neglected and by using equations (2) and (7), we co mpute numerically the output pulse along a step fro m an input pulse A(0,t) at the entry of the fiber as fo llo wing: Step by step, we compute recursively the output pulse at the end of the fiber. The choice of the coefficients c 0 and c 1 is crucial for the co mputation errors of the method. In order to evaluate the performance of these imp lementations, we use the global relative error δ g iven by the following equation [12]: a a n A A A − = δ (10) Where A a is the fine numerical solution at the end of the fiber co mputed for a very small and constant value of the step size h; the norm A is defined by: In this work, we present a numerical estimation of the global relat ive error (GRE) of the S-SSFM for two imp lementations: (c 0 , c 1 ) = (0, 1) and (c 0 , c 1 ) = (1, 0) wh ich we design respectively by S-SSFM1 and S-SSFM2. We show that the GRE is respectively O(h) and O(h 2 ).
Then, we consider the second formula of the Strang splitting and we make a permutation in the position of the two operators D  and N  in equation (7), the solution will be numerically co mputed from the input pulse A(0,t) as following: We can see that the nonlinear operator will be applied in two steps. First, we must introduce the intensity A pondered respectively by three coefficients c 0 , c 1 and c 2 to compute the nonlinear term of A 3 . Th is approach is also called the three-split method and designed by T-SM [13].

Numerical Results
We consider in our simulat ion a silica single mode fiber and we study in the dispersive and nonlinear regime, the propagation of a Gaussian pulse for which the amplitude of the incident field can be written as: The corresponding parameters are as following: β 2 = -20 ps 2 /km in an optical teleco mmun ication window around λ 0 = 1,55 µm, γ = 2 W -1 km -1 , P 0 = 5 mW and t 0 = 40 ps. We compute the numerical output solutions A n , using the four implementations, by considering the transmission distance z = 200 km, with a spatial step size h = 1 km. The correspondent curves are plotted in Figure 1a.
We note a temporal broadening of the output pulses, due to dispersion and a diminution of the maximal intensity. It seems that the numerical results are identical, but if we zoom the framed part in Figure 1a, we obtain the Figure 1b in which we can see the differences between the numerical values.  We have measured the variation of the global relative error for the four implementations: S-SSFM1, S-SSFM2, T-SM 1 and T-SM2 against the spatial step size h. We note that the fine numerical solution A a is computed for h = 7.8 m. The corresponding curves are plotted in Figure 2. We prove by using a graphic log-log that the global relative error can be estimated by formulas of type δ = Ch α . More precisely we have verified by a linear regression in the graphic, that the order of h is pract ically equal to 1 for the S-SSFM 1 and T-SM 1 and 2 for the S-SSFM2 and T-SM2. The detailed results are presented in table 1. We can also show fro m the curves of Figure 2 that the S-SSFM imp lementations are more accurate than the T-SM.  The estimated GRE for the S-SSFM 2 and T-SM2 is O(h 2 ) which corresponds substantially to the commutator error. The nonlinear term has practically no influence on these implementations rather than the S-SSFM 1 and the T-SM1 for wh ich the GRE is O(h), decreased by one order and depends on the nonlinear term.
In the Figure 3, we plot the GRE variation curves for the four imp lementations for h = 1 km versus the propagation length z. These curves will be an indicator of accuracy and allo w decid ing the step size to be taken in a wide range of situations of practical interest.

Conclusions
The propagation of picoseconds pulses in optical fibers is described by the simp lified Schrödinger equation. The symmetric split-step Fourier method is often used to calculate the numerical solution of this equation. If we consider the only fact that the dispersive and nonlinear operators do not commute the Baker-Campbell-Hausdorff formu la shows that the global relat ive error of this method is O(h 2 ). We have analysed in this paper, the effect of the nonlinear self phase modulation, wh ich depends on the intensity term 2 ) , ( t z A , on the global relat ive error. By using the two Strang splitting formulas and an exp licit representation of the nonlinear operator, we have presented four implementations: the S-SSFM 1, S-SSFM 2, T-SM 1 and T-SM 2 obtained from different forms of the appro ximate nonlinear term and respectively for the weighting coefficients (c 0 , c 1 ) = (0, 1), (c 0 , c 1 ) = (1, 0), (c 0 ,c 1 ,c 2 ) = (-1,1,1) and (c 0 ,c 1 ,c 2 ) = (1,-1,1). Our numerical results prove that this error is O(h) or O(h 2 ). We conclude that the nonlinear term has an influence on the splitting error, the S-SSFM is more accurate than the T-SM and in order to obtain an indicator of accuracies, we present the variations of the global relative errors for some values of the propagation length of the fiber.