Chapter 3 Simulation Methods
3.2 Split-step Fourier method
One of the pseudospecral methods that have been used extensively to solve the pulse-propagation problem in nonlinear dispersive media is the split-step Fourier method. The main reason for the faster speed of the split-step method compared with the most finite-difference schemes is the use of the FFT.
To understand the philosophy behind the split-step Fourier method, it is useful to rewrite Eq. 2.2.3 in the form [21]
where is a differential operator that accounts for the dispersion and absorption in a media and is a nonlinear operator that presents the effect of fiber nonlinearities on pulse propagation. These operators are given by [21]
Dˆ Nˆ
n
In general, the dispersion and nonlinearity effects act together along the length of the fiber. The split-step Fourier method assumes that the dispersive and nonlinear effects can be pretended to act independently inside a small distance h and therefore obtains an approximation solution. More specifically, propagation along the fiber from the position z to z + h is carried out in two steps. In the first step, the nonlinearity acts alone, and =0 in Eq. 3.2.1. In the second step, dispersion acts alone, and =0. Mathematically [21],
Dˆ The first step can be evaluated in the time domain while the second step should be calculated in the frequency domain. The process is shown as the following prescription
(
z h,T)
F(
exp[ hDˆ (iω)] F [exp(hNˆ)A(z,T)])
A + ≈ T−1 ∗ T (3.2.5)
where , , are the FFT operation, the inverse FFT operation and the Fourier transform of in Eq. 3.2.5. Notice that the differential operator
FT FT−1 Dˆ(iω) dispersion effect should be calculated in the frequency space. After finishing the dispersion effect on , we should change the calculation from the frequency domain to the time domain by the inverse FFT. The use of the FFT makes numerical evaluation of Eq. 3.2.5 relatively fast. It is for this reason that the split-step Fourier method is faster up to two orders of magnitude compared with most of the finite-difference schemes.
) , (z T A
To estimate the accuracy of the split-step Fourier method, we note that a formally exact solution of Eq. 3.2.1 is given by
) if is assumed to be z independent. At this point, it is useful to recall the Baker -Hausdorff formula [35] for two noncommutating operatorsa
Nˆ split-step Fourier method ignores the noncommutating feature of the operators and . By using Eq. 3.2.7 with and , the dominant error term is found resulting from the single commutator
a method is accurate to the second order in the step size h.
The accuracy of the split-step method can be improved by adopting a different procedure to propagate the optical pulse over one segment from z to z+h. In this procedure Eq. 3.2.4 is replaced by
)
The procedure divides into 3 parts [See Fig. 3.1]. At first, the dispersion effect acts alone in the first half of distance h. Then the effect of nonlinearity acts alone in the middle of segment. Finally the dispersion effect acts again in the rest of length h/2.
Similar to Eq. 3.2.5, the dispersion effects at the both sides of the segment is calculated in the frequency domain by the FFT whereas the nonlinear effect at the middle part is calculated in time domain.
Dispersion Only
Z=0 h
A(z,T) Nonlinear effect Only
Fig. 3.1 Schematic illustration of the symmetrized split-step Fourier method. Fiber length is divided into a large number of segments of width h. Within the segment, the effect of nonlinearity acts at the midplane shown by a dashed line.
Because of the symmetric form of Eq. 3.2.8, this scheme is known as the symmetrized split-step Fourier method [36]. The integral in the middle exponential considers the z dependent of the nonlinear operator . If the step size h is small enough, the integral can be approximated by . The most important advantage of using the symmetrized form of Eq. 3.2.8 is that the leading error term comes from the double commutator in Eq. 3.2.7 is of the third order in the step size h.
This can be proved by applying Eq. 3.2.8 twice in Eq. 3.2.7.
Nˆ ˆ) exp(hN
The accuracy of the symmetrized split-step Fourier method can be further improved by evaluating the integral in Eq. 3.2.8 more accurately than approximating it by . A simple approach is to employ the trapezoidal rule and approximate the integral by [37]
)
However, the implementation of Eq. 3.2.9 is not simple because is unknown at the mid-segment located at . It is necessary to use an iterative procedure that is initiated by replacing by . Equation is than used to estimate
which in turn is used to calculate the new value of . Although the iteration procedure is time-consuming, it can still reduce the overall computing time if the step size h can be increased because of the improved accuracy of the numerical algorithm. Two iterations are generally enough in practice.
)
Let us see Eq. 3.2.3 again, the nonlinear operator contains an integral part and the differential part which corresponds to the Raman effect and the SS. It is more complicated to deal with them. We rewrite by using Eq. 2.2.4 and Eq. 3.2.3 as the
The integral part can be solved by the convolution theory by inversely FFT the product of hR(Ω) (the FFT ofhR(T) ) and the FFT of A( Tz, )2 [38]. The differential part can be solved in the frequency domain by replacing∂ /∂Twithiω. Therefore,Nˆcan be written as
⎟⎠ pute the nonlinear operator in use of the convolution theory and the FFT algorithm.
In many papers, it is usually solved the SS term using the Runge-Kutta method with treating the differential term as a perturbation [33][38][39]. However, we use the
Nˆ
FFT algorithm, which is simpler and more straightforward than the Runge-Kutta method. We therefore simulate the evolution of the pulse spectrum using the split-step Fourier method. We also combine the plug-in program Matfor 4.0 with Compaq Visual Fortran 6.6. Matfor is a very powerful tool which can draft the evolution of spectrum synchronously [Fig. 3.2].
Fig. 3.2 The plug-in program Matfor 4.0