DWDM transmission optimization in nonlinear optical fibres with a fast split-step wavelet collocation method T. Krempa and W. Freudeb a Institut
f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen University of Technology, Templergraben 55, 52056 Aachen, Germany; b High-Frequency and Quantum Electronics Laboratory, University of Karlsruhe, Kaiserstr. 12, 76128 Karlsruhe, Germany. ABSTRACT
To meet rapidly increasing bandwidth requirements, extensive numerical simulations are an important optimization step for optical networks. Using a basis of cardinal functions with compact support, we developed a new split-step wavelet collocation method (SSWCM) as a solver for the generalized nonlinear Schr¨odinger equation describing pulse propagation in nonlinear optical fibers. With N as the number of discretization points, this technique has the optimum complexity O(N ) for any fixed accuracy, which is superior to the complexity O(N log 2 N ) of the standard split-step Fourier method (SSFM). For an accurate simulation considering third order dispersion, self-steepening and the Raman effect in a large 40 Gbit/s dense wavelength division multiplexing (DWDM) system with 64 channels, the SSWCM requires less than 40 % of computation time compared to the SSFM. This improvement allows investigations of the bit error rate as a function of the WDM system parameters. As an example, we determine the optimum launch power for both on-off keying and differential phase shift keying. The maximum spectral efficiency is compared with predictions from recently published analytical methods. Keywords: WDM systems, differential phase shift keying, nonlinear Schr¨odinger equation, initial boundary value problem, collocation method, Deslaurier-Dubuc interpolating wavelets.
1. INTRODUCTION With the growth of internet traffic, the bandwidth demands on wavelength division multiplexing (WDM) systems are subject to a continuous rise. For an optimum design of such systems, numerical tools for solving the nonlinear Schr¨odinger equation (NLSE) are employed. The standard numerical method used by most professional simulators is the split-step Fourier method (SSFM). Using a fast Fourier transform (FFT) in each propagation step, the SSFM has the complexity O(N log 2 N ) for N discretization points. Due to the long transmission distances and the extremely large signal bandwidths, the computation time can amount to several days. Hence, simplified models [1–3], analytical approximations [4–6] or engineering rules of thumb are often employed, but these do not offer the necessary flexibility and accuracy to optimize general real-world WDM systems. To achieve a speed-up for accurate propagation simulations, several wavelet techniques have been presented in the literature [7–9]. However, for accurate simulations of large WDM systems, to the best of our knowledge, a significant reduction of the computation time is not yet reported for any total field propagation method. This is due to the fact that even wavelets with a large number of vanishing moments do not allow a significant compression to obtain a sparse representation of dense WDM (DWDM) signals which are sampled close to the Nyquist-Shannon limit. However, a substantial speed-up is possible if a sparse and sufficiently accurate representation of the propagation operator of the NLSE is found. This is achieved using a basis of cardinal functions with compact support, leading to the split-step wavelet collocation method (SSWCM) [10–13], which is closely related to the implicit wavelet split-step method proposed in [14, Sect. 9.3.1] using orthogonal Daubechies’ wavelets. In this paper, we use the SSWCM for the first time for simulating a differential phase shift keying (DPSK) [15] DWDM system. We compare the bit error probability (BER) with a standard on-off keying (OOK) scheme and evaluate the optimum launch power. The maximum spectral efficiency is estimated numerically and compared to recently published analytical methods [4–6]. Further author information: T. Kremp: E-mail:
[email protected], Telephone: +49 241 809 7107 W. Freude: E-mail:
[email protected], Telephone: +49 721 608 2492
¨ 2. NONLINEAR SCHRODINGER EQUATION In scalar and slowly varying envelope approximation, pulse propagation in a nonlinear optical medium is described by the generalized nonlinear Schr¨odinger equation (GNLSE). With the frequency-dependent propagation (n) constant β, we define β0 := (dnβ/dω n )(ω0 ), n = 0, 1, 2, . . ., at the reference angular frequency ω0 . Thus, the (1) retarded time T := t − β0 z is measured in a reference frame moving in the z-direction with the group velocity (1) 1/β0 . With the power attenuation constant α, the nonlinearity parameter γ and the Raman time constant T R , (0) the GNLSE for the envelope A(T, z) of the electric field E ∼ A(T, z) e j(ω0 t−β0 z) reads [16, (2.3.41)] ¤ ∂A(T, z) £ = L + N (A) A(T, z), ∂z (3) (2) d2 β0 d3 β α + , L := − + j 0 3 2 · 2 dT 2 6 dT µ ¶ ½ ¾¸ j j ∗ ∂A 2 ∗ ∂A N (A) := − j γ |A| − A − + TR 2< A . ω0 ∂T ω0 ∂T
(1) (2) (3)
Here, A∗ is the complex conjugate of A, and the symbol “ 0, y := < 0. x := √ N1 N1
The optimum threshold value uth is found by the requirement that the BER is minimum: µ ¶ µ √ ¶ 1 1 √ p N 2 S1 √ S1 ! ∂ BER (19) 2 √1 0 = 0= − − ln uth + − uth + ∂uth N1 N0 N1 N1 p0 2 πN1 S1 | {z } | {z } | {z } =:a
⇒ uth
" Ã !#2 r b b2 1 − − . = − ac a 2 4
=:b
(19) (20)
=:c
(21)
Using the SSWCM Eq. (9), we propagate the total field envelope A for NRZ-OOK modulation in the DWDM system described in Sect. 6. After each span, we add Gaussian distributed white ASE noise with spectral power density Ghf0 and measure the noise powers N0 and N1 for logical zeros and ones numerically (see Eqs. (17) and (18)). In Fig. 4(c), the BER from Eqs. (20–21) in the center channel k = M/2 is depicted as a function of the launch power P (0) per channel, for M = 16 WDM channels and ns = 1, 2, 4 spans. As illustrated by the asymptotes for vanishing nonlinear noise (dotted lines), ASE noise dominates for small powers, whereas nonlinearity-induced noise dominates for large powers. For ns = 4 spans, the optimum launch power per channel is P (0) = −2.2 dBm = 0.61 mW, leading to BER = 8.2 · 10−26 (lowest point of uppermost curve in Fig. 4).
0
100
10
8
ns = 4
50
p P0
PSfrag replacements 25 0 0
6
PSfrag replacements
4
p P1
BER
75
PDF (a. u.)
replacements
PDF (a. u.)
−10
10
ns = 2
−20
10
ns = 1
−30
10
2
−40
10
0
0.01 0.02 0.03 0.8 0.9 1.0 −20 −10 0 10 optical power (mW) optical power (mW) (a) (c) optical launch power P (0) (dBm) (b) Figure 4. NRZ-OOK modulation with M = 16 WDM channels. PDF of logical zeros (a) and ones (b) corresponding to eye diagrams Fig. 3(a/b) after ns = 4 spans for P (0) = 0.45 mW/channel. Histograms (—) and analytical PDF (Eqs. (17) and (18)) (−−). (c) BER (Eqs. (20–21)) (—) as a function of the launch power P (0) per channel for n s = 1, 2, 4 spans (from bottom to top). Asymptotes for vanishing nonlinear noise (· · · ).
7.2. Differential phase shift keying (DPSK) with balanced detection In the case of differential phase shift keying (DPSK) [15], the information is coded in the phase difference ∆φ between adjacent bits. For conventional DPSK, ∆φ = 0 corresponds to a logical one, and ∆φ = π to a logical zero. In the case of balanced detection (see, e. g., [15]), two photodetectors and a Mach-Zehnder interferometer are employed to mix the envelope a(T, L) in channel k after the propagation length L with the copy a(T − τ, L), time-shifted by the inverse bit rate τ = 1/ft . The electrical differential signal u of the two photodetectors is proportional to cos(∆φ): u ∼ |a(T, L) + a(T − τ, L)|2 − |a(T, L) − a(T − τ, L)|2 = 4
