ICTON-MW'07 Fr4B.4 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ This research was carried out in the framework of COST Action 291 Towards Digital Optical Networks, and it received a support from national science funds as a research grant COST/51/2006 in 2006-2008. 978-1-4244-1639-4/07/$25.00 ©2007 IEEE 1 Pre-Simulated Local-Error Method for Modelling of Light Propagation in Wavelength-Division-Multiplexed Links Marek Jaworski, Member, IEEE, Marian Marciniak, Senior Member, IEEE National Institute of Telecommunications, Department of Transmission and Fiber Optics 1 Szachowa Str., 04-894 Warsaw, Poland Phone: +48 22 512 82 60, E-mail: M.Jaworski@itl.waw.pl ABSTRACT Different approaches to Split-Step-Fourier-Method (SSFM) applied to simulation of signal propagation in wavelength-division-multiplexed (WDM) links are considered in this paper. We propose novel simulation method, which contains two stages: step size optimization is carried out in the initial stage, using two recently proposed methods: local-error method and pre-simulation with signal spectrum averaging; in the second stage conventional SSFM is used, applying optimal steps obtained in the initial stage. Overall time savings reach 50%, depending of simulated system scenario. We called this novel procedure Pre-simulated Local Error S-SSFM (PsLE S-SSFM). Keywords: Split-Step-Fourier-Method, Local Error Method, simulation, DWDM systems. 1. INTRODUCTION Split-step-Fourier-method (SSFM) is commonly used for simulating of light propagation in optical fibre, described by nonlinear Schrödinger equation (NLSE) [1], due to its high numerical efficiency. In many publications optimisation of the simulation time and accuracy is considered [2-11]. Higher order numerical methods (i.e. explicit Adams–Bashforth and implicit Adams–Moulton, etc.) or predictor-corrector methods [2] are used. In higher order methods signal from several preceding points are taken into account to find subsequent step solution. Comparing to conventional symmetrical SSFM, the numerical effectiveness of higher order methods increases with higher desired accuracy. The higher order methods are especially useful for simulations of soliton propagation, where linear (L) and nonlinear (N) operators in SSFM are self-balanced. In WDM transmission, higher dispersion and lower nonlinearity occur, comparing to soliton transmission. As a consequence, special tailored methods should be applied for simulation of signal propagation in WDM links. Additionally, relatively low desired accuracy (of the order of 10 -2 – 10 -3 ) caused that symmetrized SSFM (S-SSFM), which has order of O(h 2 ), is preferred for WDM signal simulations. Besides common used S-SSFM, another methods are used in special cases, e.g. split-step wavelet collocation is faster then S-SSFM in very wideband simulations [3], but is applicable only for zero dispersion slope ( 3 0 β = ). The optimal step size in S-SSFM is basic factor to improve numerical efficiency. Lately, methods known in quantum mechanics was used to step size calculation [4]. The optimal step size optimal h can be estimated analytically for desired global error δ G . This procedure is fast in the case of lossless fiber. In more realistic case, with lossy fiber, the optimal step can be estimated as well, but with additional computational effort [4]. We propose novel simulation method, which contains two stages: step size optimization () optimal h z is carried out in the initial stage, using two recently invented methods: local error method [5] and pre-simulation with signal spectrum averaging [6]; in the second stage conventional S-SSFM is used, applying optimal steps obtained in the initial stage. Overall time savings reach 50%, depending on the simulated system scenario. We called this novel procedure Pre-simulated Local Error S-SSFM (PsLE S-SSFM). 2. SIMULATIONS OF SIGNAL PROPAGATION IN WDM LINKS We may write the equation describing the signal propagation in optical fibre as: 2 2 2 2 2 u u i i u u u z t β α γ ′′ ∂ ∂ - + - = ∂ ∂ N L    , (1) where: - u envelope of the signal wave; - z and t distance and retarded time; - β ′′ group velocity dispersion; - γ loss coefficient. For WDM systems with well-separated channels, we may also write [7]: