Study on a New Split-Step Fourier Algorithm for Optical Fiber. Transmission Systems Simulations. T.T. Meirelles1, A.A. Rieznik1,2, and H.L. Fragnito1. 1.
Study on a New Split-Step Fourier Algorithm for Optical Fiber Transmission Systems Simulations T.T. Meirelles1, A.A. Rieznik1,2, and H.L. Fragnito1 1. Optics and Photonics Research Center, IFGW, Unicamp, Campinas, SP, Brazil. 2. PADTEC, Rodovia Campinas – Mogi-Mirim (SP 340) Km 118.5, Brazil. Abstract — Recently we presented a new Split-Step Fourier algorithm to the simulation of optical-fiber transmission systems. To check it, we show here its efficiency to model a two first-order soliton collision, which is a situation of practical interest because of its sensitivity to numerical errors. Index Terms — Optical Solitons , Split-Step Fourier Method (SSFM), Numerical Analysis.
I. INTRODUCTION In a recent publication [1] we introduced a new algorithm for the adaptive determination of the spatial step size to be used when implementing the Split-Step Fourier Method (SSFM) to the simulation of optical-fiber transmission systems. This new algorithm was called the Uncertainty Principle Method (UPM). A variety of spatial step-size selection algorithms have been proposed for optimizing the SSFM. One of the most used methods is the nonlinear phase-rotation method (NLPRM), in which the step size is chosen so that the phase change due to nonlinearity does not exceed a certain limit. This method is reasonably efficient only for soliton transmission systems [2] and also depends on the expertise of the user. The first user- and systemindependent algorithm, called Local Error Method (LEM), was proposed recently by Sinkin et al. [2]. In this method, the spatial step-size h is chosen so that the relative local error, a one-dimensional parameter defined in [2] and which gives a good idea on the point-to-point error at each step, is kept within a specified range. A new user- and system-independent algorithm, called UPM, was presented in [1]. The spatial step-size h is selected using the uncertainty relation between two noncommuting operators to calculate a maximum local error at each step. This method will be described in detail in Section 2. In this work we study the efficiency of the UPM to the simulation of a two first-order soliton collision, comparing its performance against those given by the LEM and the NLPRM. We find the regions of accuracy at
which each one of the methods works better and show that the longer the fiber length used in the simulations, the greater the region of accuracies in which the LEM is more efficient than the UPM. II. THEORY The Generalized Non-Linear Schrödinger Equation (GNLSE) [3] ∂A 1 ∂2A i ∂3A 2 = − β 2 2 − β3 3 + γ A A (1) ∂z 2 6 ∂t ∂t governs the wave propagation in a lossless optical fibers. In (1), A = A(z,t) is the slowly varying field amplitude, β2 and β3 are, respectively, the second and third order dispersion coefficients, γ is the nonlinear coefficient (optical Kerr effect), z is the position along the fiber, and t is the local time, i.e., in a reference frame that travels with the average group velocity of the wave. Equation (1) can be written as i
∂A( z, t ) = ( D + N [ A]) A( z , t ) , ∂z with D and N defined as ∂ ∂ 1 i D = β 2 22 + β 3 33 2 6 ∂t ∂t i
N = γ A( z , t )
(2)
(3)
2
where D is a differential operator that accounts for dispersion and N is a nonlinear operator that describes the nonlinearities of the fiber. The SSFM obtains an approximate solution by dividing the fiber in steps of length h (h can vary along the fiber) and then propagating the field at each step assuming first no dispersion (D = 0) and then no nonlinearity (N = 0). The error of the SSFM is given by the fact that D and N are non-commuting operators [3]. Moreover, it is easy to show that the dominant local error at each sample time point is given, in first order in h, by (1/2)h2[D,N]A(t) [13].
ε ≡ (1 / 2)h < [D, N ] > ≡ (1 / 2)h 2
∫ A ( z, t )[D, N ]A( z, t )dt ∫ A(t ) dt *
2
2
(4) where [D,N] = DN – ND is the commutator between D and N, and the mean value of [D,N] is taken in the “state” A(t) as given in QM. The uncertainty principle from QM gives an upper bound to the average of the commutator of several pairs of physical operators such as position and momentum. This principle arises from the Schwarz inequality [4] and thus applies not only in the context of quantum mechanical operators but to any pair of hermitian operators. In particular, it also applies the operators D and N of the NLSE (since they are hermitian) and takes the form < [D, N ] > ≤ 2∆D∆N ,
the global relative error, this last defined in [2, equation (12)] and in [1, equation (7)]. We do not enter here in the discussion on the validity of these parameters to measure the computational cost and accuracy of the simulations, but an extensive discussion can be found in [1]. The simulations results are shown in Figure 1 (a) and (b), respectively.
5
10
Number of FFTs
Since the error at each sample point is not interesting from a practical point of view, a new parameter was defined in [1] based on Quantum Mechanics(QM):
LEM NLPRM UPM
4
10
3
10
(a) 2
10 -7 10
(5)
10
-6
∫ A ( z, t )BA( z, t )dt = ∫ A(t ) dt *
1x10
-4
10
-3
ε (7) ∆D∆N in order to bound ε by any desired value chosen to replace it in Eq. (7). This equation states the Uncertainty Principle Method (UPM) [1]. h =
III. NUMERICAL EXPERIMENTS In this Section we show the simulation results for a collision of two first-order soliton of the form A(t) = η(β2/γ)1/2sech(ηt), with η = 0.44 ps-1, β2 = -0.1 ps2/km and γ = 2.2 W-1km-1. These values give a pulse duration of 4 ps and a peak power of 8.8 mW [2]. We used 3072 sample points (i.e., Fourier modes) and a simulation time windows of 400 ps. The central-frequency difference between both solitons is 800 Ghz and at the input they were separated in time by 100 ps. We simulated two propagating distances; first trough 400 km of fiber and then trough 200km. As a measure of the computational cost of each simulation we use the total number of FFTs performed and as a measure of accuracy
-2
5
LEM NLPRM UPM
2
(6) Combining equations (4) an (5) the step size h can be chosen according to
10
10
Number of FFTs
∆B = < ( B − < B >) > with B
-5
Global relative error
where ∆D and ∆N are standard deviation as defined in QM, calculated as 2
1x10
4
10
3
10
(b) 2
10 -7 10
-6
10
-5
1x10
-4
1x10
-3
10
-2
10
Global relative error
Fig. 1. Number of FFTs versus global relative error for a two first-order soliton collision propagating through (a) 200 km and (b) 400 km of fiber. Results from the UPM, LEM, and NPRM are shown.
IV. DISCUSSION Two results are relevant to be discussed. First, as observed in [1] the UPM works better at high global errors while the LEM is more efficient at lower values, when higher accuracies are required. The NLPRM is always the worst option. Second and more important in this work, introduced here as a novelty, the longer the fiber length used in the simulations, the greater the region of accuracies in which the LEM is more efficient than the
UPM, i.e., the point at which both curves (corresponding to the UPM and to the LEM simulations) cross each other goes to higher global errors –lower accuracies- as the fiber length increases. In Figure 1 is clearly seen that this -4 point is near the global error 10 for the propagation -3 distance of 200 km and near 10 for the distance of 400 km. In the following of this Section we discuss these two observations. That the LEM works better than the other methods at higher accuracies is a consequence that it is a higherorder scheme than the UPM and NLPRM. Thus, while the global error in the LEM is second-order in h, i.e., the 2 error is O(h ), the UPM and the NLPRM are O(h). When m we say that a method is O(h ) we mean that the global error accumulated at the end of the propagation is m bounded by Ch for some value C [2]. This fact was already pointed out in [1] and [2] and we confirm it here. The second relevant result, i.e., that the point at which both curves (corresponding to the UPM and to the LEM simulations) cross each other goes to higher global errors as the fiber length increases, deserves special attention and can be explained in the following simple way: The 2 global error of the LEM and UPM are given by, say, Kh and Ch, respectively. We stress that the value of h to be substituted here is an averaged value, since it changes adaptively during the propagation. If, after the propagation using the LEM and UPM, the averaged value of the spatial step size are h’ and h’’, both global errors 2 will be equal if and only if Kh’ = Ch’’. But the number of FFTs performed during the propagations is proportional to the number of steps, i.e., to L/h, were L is the fiber length. More strictly, the total number of FFTs is equal to nL/h, were n is an integer representing the number of FFTs performed at each step. Thus, the number of FFTs performed during the propagation using the LEM or UPM are equal if and only if mh’ = lh’’, were l is the number of FFTs performed per step using the LEM and m using the UPM ( m = 3, while l is larger, the exact number being dependent on, for instance, the chosen initial-step size). So, the point where both curves cross each other, i.e., where both the global error and the number of FFT are the same, is such that the averaged step-size is the same for both method an given by h = mC/lK. Therefore the cross-point is such that h respects this condition, independently of the target local accuracy. But, for a given value of h = mC/lK, the global error increases when the fiber length increases, because more spatial steps must be performed until the end of the propagation. Finalizing, the cross point between the UPM and LEM is such that the averaged h used in both
simulations is given by h = mC/lK, and, as the global error increases for a given h with the fiber length, both curves cross each other at higher global error values for higher fibers lengths L. This completes our discussion and gives a theoretical explanation of the behavior observed in Figure 1. IV. CONCLUSSION We showed that for the specific soliton collision studied here, the UPM method performs better at global -3 -4 errors longer than 10 and 10 , for the 400 and 200 km fiber propagation, respectively. For smaller global errors, higher accuracies, the LEM efficiency is greater. We gave a theoretical explanation with justifies the observed behavior and generalizes it. Thus, we demonstrate (1) that the LEM works better than the UPM at high accuracies and (2) that the longer the fiber length used in the simulations, the greater the region of accuracies in which the LEM is more efficient than the UPM. These two statements were theoretically discussed and proved to be true for any input parameters of the NLSE. Future improvements of this work include the development of algorithms capable of identifying the best method to be applied under different input parameters. Also more refined methods of even higher orders than those discussed here can be easily imaginable as a simple extension of these lasts. We are already working on these points and new results will be presented at the Conference. ACKNOWLEDGEMENT The authors wish to acknowledge CEPOF/FAPESP. REFERENCES [1] A.A. Rieznik, T.T. Meirelles, F.A Callegari and H.L. Fragnito , “Uncertainty principle and the optimization of optical-fiber communications systems simulations,” to be published in Optics Express. [2] Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” IEEE J. of Lightwave Technol. 21, 61-68 (2003). [3] G. P. Agrawal, Nonlinear Fiber Optics (London, U.K. Academic, 1995). [4] E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
