DWDM transmission optimization in nonlinear optical fibres with a fast split-step wavelet collocation method

Transcription

1 DWDM transmission optimization in nonlinear optical fibres with a fast split-step wavelet collocation method T. Kremp a and W. Freude b a Institut für Geometrie und Praktische Mathematik, RWTH Aachen University of Technology, Templergraben 55, 556 Aachen, Germany; b High-Frequency and Quantum Electronics Laboratory, University of Karlsruhe, Kaiserstr. 1, 7618 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 Schrödinger 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 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 4 Gbit/s dense wavelength division multiplexing (DWDM system with 64 channels, the SSWCM requires less than 4 % 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 Schrödinger 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 Schrödinger 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 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 [1 13], which is closely related to the implicit wavelet split-step method proposed in [14, Sect ] 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: kremp@igpm.rwth-aachen.de, Telephone: W. Freude: w.freude@etec.uni-karlsruhe.de, Telephone:

2 . NONLINEAR SCHRÖDINGER EQUATION In scalar and slowly varying envelope approximation, pulse propagation in a nonlinear optical medium is described by the generalized nonlinear Schrödinger equation (GNLSE. With the frequency-dependent propagation constant β, we define β (n := (d n β/dω n (ω, n =, 1,,..., at the reference angular frequency ω. Thus, the retarded time T := t β (1 1/β (1 z is measured in a reference frame moving in the z-direction with the group velocity. With the power attenuation constant α, the nonlinearity parameter γ and the Raman time constant T R, the GNLSE for the envelope A(T, z of the electric field E A(T, z e j(ωt β( z reads [16, (.3.41] A(T, z z = [ L + N (A ] A(T, z, (1 L := α + j β( N (A := j γ d [ A j ω A A T d 3 dt + β(3 6 dt 3, ( ( { j + T R R A A }]. (3 ω T Here, A is the complex conjugate of A, and the symbol R denotes the real part. The linear operator L represents loss and dispersion, and the nonlinear function N (A describes the optical Kerr effect, self-steepening and Raman scattering. 3. DISCRETIZATION OF THE TIME VARIABLE T : COLLOCATION METHOD To obtain a discrete representation, we expand the envelope A(T, z in a series of N known basis functions φ m (T and evaluate this series at N collocation points T n : A n (z := A(T n, z = N m=1 c m (zφ m (T n, }{{} n = 1,..., N. (4 =:φ nm Introducing the N 1 column vectors A(z and c(z with elements A n (z := A(T n, z and c n (z, and the constant N N matrix Φ with elements φ nm := φ m (T n, we write Eq. (4 for all collocation points T n in matrix form: A(z = Φc(z c(z = Φ 1 A(z. (5 Defining the N N matrix LΦ with elements (Lφ m (T n and using Eqs. (4 and (5, the N 1 column vector LA(z with elements (LA(T n, z is given by LA(z (4 = (LΦc(z = (LΦΦ 1 A(z = LA(z. (6 }{{} =:L Introducing the N N diagonal matrix N(z with elements N nn (z = N (A(T n, z, the N 1 column vector NA with elements ( N (AA (T n, z = ( N (A (T n, z A(T n, z = N nn (za n (z is given by NA(z = N(zA(z. (7 With Eqs. (6 and (7, we can write Eq. (1 for all collocation points T n in a single collocation equation da(z dz = [ L + N(z ] A(z. (8 Hence, the collocation technique transforms the partial differential equation (1 into the system of N ordinary differential equations (8.

3 4. INTEGRATION ALONG SPATIAL VARIABLE z: EXPONENTIAL SPLIT-STEP Dividing the total propagation length into sufficiently short segments, we employ the well-known exponential split-step integration method (see, e. g., [16, Eq. (.4.4] to solve Eq. (8. Therefore, introducing the N N matrix e Lh, which is constant as long as the step size h remains unchanged, and the N N diagonal matrix e N(zh with elements e Nnn(zh, we obtain the general split-step collocation method (SSCM [17] A(z + h = e Lh e N(zh A(z. (9 Since the matrices L (in the lossless case α = and assuming a symmetric discretization and N (trivially are skew-hermitian, the matrices e Lh and e Nh are unitary. Hence, this fast explicit integration method (9 is energy-conserving and therefore unconditionally stable. If Eq. (9 is implemented symmetrically, i. e., with half-steps e Lh/ at the beginning and the end of the total propagation distance, Eq. (9 is accurate to the order O(h 3. The matrix e Lh is in general fully filled, leading to a complexity C SSCM = O(N. However, using cardinal ( interpolating basis functions φ m with compact support, we can obtain an asymptotically optimum linear complexity for any fixed accuracy, as shown in the following Sect COMPACTLY SUPPORTED CARDINAL BASIS: SPLIT-STEP WAVELET COLLOCATION METHOD (SSWCM Using equidistant collocation points T n := n T, we select the basis functions φ n as translated copies of the compactly supported Deslaurier-Dubuc interpolating function φ [18], Fig. 1, φ n (T := φ(t T n, T n := n T. (1 The circles in Fig. 1 illustrate the cardinal (interpolating property φ(n T = δ n (Kronecker symbol. Hence, using this basis of equidistantly translated Deslaurier-Dubuc functions in the series Eq. (4, the matrix Φ in Eq. (5 equals the identity matrix I, leading to an identity of the function values A n (z and the coefficients c n (z: φ nm = δ nm, Φ = I, A(z = c(z. (11 This is an advantage over other wavelet methods [7 9] and over the SSFM [16]. 1.5 φ(t PSfrag replacements T/ T Figure 1. Interpolating scaling function φ(t for SSWCM, support s = 5. There exists no closed form definition of the function φ. Instead, it is defined recursively by a symmetric equidistant interpolation procedure using polynomials of degree s (see, e. g., [11, Sect ]. Hence, φ is uniquely determined by the odd number s. Each basis function φ n is compactly supported, i. e., φ n vanishes identically outside its support interval (T n s T ; T n + s T : φ(t T s T, s odd. (1 Thus, at any T, only s 1 N basis functions φ n (T contribute a nonzero function value in the sum Eq. (4. Hence, the matrix L := (LΦΦ 1 in Eq. (6 is banded and has the bandwidth s 1. The function φ (Fig. 1 is called scaling function because it fulfills the scaling equation [11, Eq. (5.15], φ(t = n Z φ(n T/φ(T n T. (13

4 Due to Eqs. (1 and (13, the elements of the matrix L are the solution of a homogeneous linear system of equations of dimension s [13], and φ generates a compactly supported interpolating wavelet. In contrast to methods [8] with a basis not fulfilling such a scaling equation (13, the SSWCM allows an adaptive multiresolution algorithm employing a fast wavelet transform with complexity O(N [1]. If we impose periodic boundary conditions with respect to the time coordinate T, the matrix L is a circulant matrix, defined uniquely by its first column l. Thus, the required storage space is drastically reduced from O(N to O(N. Since circulant matrices have harmonic eigenvectors [19, (.5.4 and (3..4], they are diagonal in the Fourier basis. Hence, any function f of a circulant matrix can be calculated very efficiently using a fast Fourier transform (FFT, see, e. g., [11, Sect ]. In the case of the matrix exponential function e Lh in Eq. (9, the first column f, which defines the circulant matrix e Lh, is given by f = IFFT { exp ( FFT { l } h }. (14 With the variable threshold parameter p 1, the resulting e Lh is reduced to a banded matrix with bandwidth S N [13]. Thus, the storage space for e Lh is further reduced to S 1... storage cells. The bandwidth S of the matrix e Lh is constant with respect to N, but, as we found empirically for a wide range of parameters [11, Sect ], it depends logarithmically on p and almost linearly on the support s and the step size h. Hence, the complexity of the SSWCM is C SSWCM = O(SN N, S sh log p. (15 Thus, for any fixed accuracy, which is a priori controlled by the choice of the threshold parameter p and the support s, the SSWCM has the optimum complexity O(N. For large N, this is superior to the SSFM with complexity O(N log N. Hence, the SSWCM is especially well suited for large problems. 6. APPLICATION: WDM SIMULATIONS To compare SSFM and SSWCM, we simulate a DWDM system with bit rate f t = 4 Gbit/s, channel spacing f = 1 GHz, central wavelength λ = 1.55 µm, and a 5.75 km span, consisting of km conventional single-mode fiber (α =.484 km 1, β ( =.4 ps 1 km, γ = W m followed by 7. km of dispersioncompensating fiber (α =.138 km 1, β ( = 17.5 ps 1 km, γ = W m, both with T R = 3. fs. The launched signal has the average power P (z = =.45 mw per channel and a pseudo-random word length of 14 bit/channel. We simulate both non-return-to-zero (NRZ on-off-keying (OOK with direct detection as well as return-to-zero (RZ differential phase-shift keying (DPSK with balanced detection. For a minimum meansquare bandwidth, we launch a chirp-free signal [], i. e., we assume a Mach-Zehnder interferometer in push-pull configuration with an infinite extinction ratio [1]. We employ compiled C code in a Matlab environment on a 51 MB RAM, 1. GHz AMD Athlon processor. For the SSFM, a radix- FFT algorithm [] is used. In Fig. (a, the computation time per span is displayed as a function of the total number of WDM channels for both SSFM and SSWCM. Depending on the support s and on the threshold parameter p, the SSWCM requires only between 5 %... 5 % of the computation time of the SSFM for M 8 channels. The accuracy of the SSWCM increases with increasing s and decreasing p, and due to Eq. (15, this corresponds also to an increasing complexity and computation time (Fig. (a. However, even for a very high accuracy with a relative error per span [13, Sect. V-A] of less than 1 1 (s = 39, p = 1 9, dashed line in Fig. (b, the SSWCM requires less than 4 % of computation time compared to the SSFM (Fig. (a. In this case, the eye diagrams of both methods in Fig. 3 are indistinguishable for both modulation formats. Due to its superior performance (Fig., we employ solely the SSWCM for the propagation simulations in the following sections, using the parameters s = 39, p = BIT ERROR PROBABILITY (BER The nonlinear function N (A of the GNLSE (1 causes nonlinear interactions of the WDM channels, perturbing the original signal. Furthermore, after each span, the signal is optically amplified by a power amplification factor

5 PSfrag replacements computation time (h SSFM PSfrag replacements SSWCM (a total number M of WDM channels relative error (b SSFM SSWCM total number M of WDM channels Figure. Computation time (a and relative error (b after n s = 1 span of length L = 5.75 km as a function of the total number M of WDM channels. SSWCM: s = 59, p = 1 1 ( ; s = 39, p = 1 9 ( ; s = 3, p = 1 7 (. SSFM (. Figure 3. Eye diagram after n s = 4 spans (3 km for M = 16 channels. Launch power P ( =.45 mw/channel. First row: OOK, second row: DPSK. First column : SSFM, second column: SSWCM (s = 39, p = 1 9. G 1 for compensating the span loss. Hence, amplified spontaneous emission (ASE noise is added to the signal envelope A after each span. With the channel envelopes a k (T, z and the offset frequencies ω k = ω k ω with respect to the reference angular frequency ω, the total envelope for M channels is A(T, z = M k=1 a k(t, z e j ωkt. The channel envelopes a k can be obtained from the total field envelope A via suitable optical filtering. With an optical amplifier noise figure F, an optical center frequency f, and Planck s constant h, the ASE noise power per degree of freedom after n s spans in channel k having the optical channel bandwidth B is given by N ASE = n s GF hf B = n sgf hf B. (16 As all the considerations in the following Sections 7 and 8 refer to this fixed channel numbered k, we omit the index k in the notation for convenience. In the following simulations, we assume an optimum optical amplifier noise figure F =.

6 7.1. On-off keying (OOK with direct detection In the case of amplitude-shift keying (ASK, the information is coded in the modulus of the transmitted signal and can therefore be restored using simple direct detection. With a(t, z being the optical field envelope of channel k, the electrical signal u after direct detection is proportional to the instantaneous optical power a. At sampling time, the power a has a noncentral chi-squared probability density function (PDF, see, e. g., [3, ( ] [13, Eq. (54]. In this section, we assume conventional on-off-keying (OOK, i. e., binary ASK, and raised cosine impulses. Hence, in absence of noise, the optical signal powers at sampling time for logical ones (subscript 1 and logical zeros (subscript are S 1 = P (double the average channel power P and S =, respectively. In presence of noise, with the noise powers N 1 and N averaged over many sampling instances, the optical power is modified. The powers P 1 and P both represent the sum of signal and noise powers at sampling time. Again, P 1 is the average of P 1 over many sampling instances, and σ P represents the standard deviation of P. The PDF p P1, of the powers P 1, for logical ones and zeros at sampling time are approximately Gaussian and exponential, respectively, see Fig. 4(a/b [13, Sect. V-B]: ( ( 1 p P1 (P 1 P1 S 1 exp πn 1 S 1 N 1 p P (P 1 ( exp P N N, N 1 = P 1 S 1, (17, N = σ P. (18 If the detected signal u is larger than a pre-defined electrical threshold value u th, this symbol is interpreted as a logical one, otherwise as a logical zero. The bit error probability (BER is the probability that a transmitted symbol ( or 1 is wrongly interpreted as the complementary binary symbol (1 or. With the probabilities p and p 1 = 1 p for the transmission of a logical zero and a logical one, respectively, the BER for OOK is given by (erf(x = x π dt is the error function e t p 1 BER = πn 1 S 1 = p 1 x := uth ( exp ( u S1 N 1 [ 1 πx (e x e y + erf(y + erf(x S1 N1 >, y := uth S 1 N1 <. du + p exp ( un du (19 N u ] ( th + p exp u th, ( N The optimum threshold value u th is found by the requirement that the BER is minimum: =! BER ( (19 1 = 1 ( uth + S 1 uth + S 1 p 1 N ln u th N 1 N }{{ N }}{{ 1 N } 1 p πn 1 S }{{ 1 } =:a =:b =:c [ ( 1 u th = b ] b a 4 ac. (1 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 Ghf and measure the noise powers N and N 1 for logical zeros and ones numerically (see Eqs. (17 and (18. In Fig. 4(c, the BER from Eqs. ( 1 in the center channel k = M/ is depicted as a function of the launch power P ( per channel, for M = 16 WDM channels and n s = 1,, 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 n s = 4 spans, the optimum launch power per channel is P ( =. dbm =.61 mw, leading to BER = (lowest point of uppermost curve in Fig. 4.

7 replacements PDF (a. u. (a p P PSfrag replacements.1..3 optical power (mw PDF (a. u PSfrag replacements p P (b optical power (mw BER (c n s =4 n s = n s =1 1 1 optical launch power P ( (dbm 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 n s = 4 spans for P ( =.45 mw/channel. Histograms ( and analytical PDF (Eqs. (17 and (18 (. (c BER (Eqs. ( 1 ( as a function of the launch power P ( per channel for n s = 1,, 4 spans (from bottom to top. Asymptotes for vanishing nonlinear noise (. 7.. 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 = 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/f t. The electrical differential signal u of the two photodetectors is proportional to cos( φ: u a(t, L + a(t τ, L a(t, L a(t τ, L = 4R{a(T, La (T τ, L} cos( φ(t, L. In the following, we assume equal probabilities p = p 1 = 1/ for the transmission of a logical zero and a logical one, respectively, as well as a threshold value u th =, centered between the points cos(π = 1 and cos( = 1. With the optical power P and the ASE noise power N ASE (see Eq. (16 in channel k, the linear optical signal-to-noise ratio (OSNR is ρ := P/N ASE. Hence, in absence of nonlinear effects, the BER is given by [3, Eq. (4..117] [15, Eq. (1] BER lin = e ρ, ρ := P. ( N ASE The nonlinear function N (A of the GNLSE (1 leads to a nonlinear differential phase shift φ NL having the variance σ φ NL. If we assume that the total differential phase shift φ = φ ASE + φ NL has a Gaussian PDF with variance σ φ, the BER after balanced detection is given by [15, Eqs. (18/19] [4, Eq. (4] ( π BER Gauss = erfc. (3 σ φ In the case of Gaussian PDFs, the variances of the differential phases equal double the variances of the phases, e. g., σ φ ASE = σφ ASE = 1/ρ [5, Eq. (4], and the variances can simply be added: σ φ = σ φ ASE + σ φ NL = 1/ρ + σ φ NL. This assumption Eq. (3 of a Gaussian PDF for the differential phase shift φ is satisfactorily accurate at sufficiently high power levels with dominating nonlinear effects [4, Fig. ], see Fig. 5(a. However, since the PDF of the ASE-noise induced differential phase shift φ ASE is not exactly Gaussian, a more precise model must be applied for low powers where ASE noise is not negligible [4, Eq. (5] [6] (modified Bessel function I n : BER = 1 ρ e ρ n= ( 1 n [ ( ρ ( ρ ] [ ] I n + I n+1 exp (n + 1 σ φ NL. (4 n + 1 Using the SSWCM Eq. (9, we propagate the total field envelope A for RZ-DPSK modulation in the DWDM system described in Sect. 6. Without adding ASE noise, we measure the variance σ φ NL of the resulting nonlinearityinduced differential phase shift φ NL numerically after each span. In Fig. 5(b, the BER from Eqs. (3 and (4

8 in the center channel k = M/ is depicted as a function of the launch power P ( per channel, for M = 16 WDM channels and n s = 1,, 4 spans. As expected, for high powers, the simple Gaussian model Eq. (3 leads to the same BER as the more involved series model Eq. (4. However, Eq. (3 is too optimistic for low powers with dominating ASE noise, indicated by the asymptotes (dotted lines for vanishing nonlinear noise (Eq. (. For n s = 4 spans, the optimum launch power per channel is P ( =.1 dbm =.6 mw, leading to BER = (lowest point of uppermost curve in Fig. 5. Hence, DPSK leads to substantially smaller BER than OOK (Fig. 4(c, and is less sensitive to an increasing number n s of spans. In the linear regime for small powers, DPSK requires only half the power of OOK for the same BER, because in the complex phasor plane, the distance between the (DPSK-points cos( = 1 and cos(π = 1 equals double the distance between the (OOK-points 1 and. We note that due to inevitable rounding errors, the smallest BER which can be calculated numerically with the series representation Eq. (4 is approximately given by half the relative floating point accuracy ε/, because of the leading term 1/. Hence, with standard double precision accuracy ε =. 1 16, it is impossible to compute the BER curves in Fig. 5(b reaching values of the order Instead, it is necessary to employ a high precision arithmetic package [7]. This is also necessary in the case of the more involved models in [4] which are obtained if the Gaussian assumption for φ NL is released, and if the stochastic dependence between φ NL and φ ASE is considered. However, the difference between the BER calculated from these models and the BER obtained from Eq. (4 is even smaller than the difference between Eqs. (3 and (4 in Fig. 5(b [4, Fig. ]. Hence, the accuracy of the model Eq. (4 is sufficient for all practical purposes, and the simple Gaussian model Eq. (3 can be used if ASE noise is not dominant n s =4 replacements l power (mw PDF (a. u. (a 1 p φ PSfrag replacements differential phase φ (rad BER (b n s = n s = optical launch power P ( (dbm Figure 5. RZ-DPSK modulation with M = 16 WDM channels. (a PDF p φ ( φ ( and p φπ ( φ + π ( of differential phases φ and φ π for logical ones and zeros, respectively, corresponding to eye diagrams Fig. 3(c/d after n s = 4 spans for P ( =.45 mw/channel. Gaussian fits (thin lines. (b BER Eqs. (4 ( and (3 ( as a function of the launch power P ( per channel for n s = 1,, 4 spans (from bottom to top. Asymptotes for vanishing nonlinear noise (. 8. MAXIMUM SPECTRAL EFFICIENCY FOR COHERENT (HOMODYNE DETECTION For a linear Gaussian channel propagating the signal power P and additive white Gaussian noise (AWGN with an average power N, the maximum spectral efficiency C lin (in units bit/(s Hz per degree of freedom for errorfree coherent (homodyne detection is given by C lin = log (1 + P/N, see [8]. For a nonlinear channel described by the GNLSE (1, the noise power N depends on the signal power P.

9 8.1. Analytical estimates Recently, several analytical estimates for performance evaluations of DWDM systems where published [4 6], employing different approximations. In the following, we give the resulting expressions for the maximum spectral efficiency C Neglecting dispersion In the case β ( =, i. e., in absence of group delay dispersion, the GNLSE (1 can be solved analytically. If nonlinear effects from noise can be neglected, and for flat channel spectra with an optical bandwidth B = f each, the maximum spectral efficiency C reads for the center channel k = M/ [4, Eqs. (6-7] P ( 1+Q C = log (1 +, N ASE + SP ( Q := (n s MγP ( 1 e αl Q n, S := a n (n + 1 πα (1 + Q n+, a n := n + 1 n+1 n= Neglecting four-wave mixing (FWM n m= ( 1 m (n + 1 m n m!(n + 1 m! Neglecting four-wave mixing (FWM, the nonlinear channel is related to a linear channel with multiplicative noise in [5]. This approximation is valid for large channel spacings f and small optical channel bandwidths B. The resulting maximum spectral efficiency C reads [5, Eqs. (-3] C = log ( 1 + P ( N NL N ASE + N NL, N NL := [ P ( 1 e ( Q ] P (, Q := (5 πα ( β B f (. (6 n s γ ln M Small nonlinearity Assuming that the changes in the signal spectrum are negligible compared to the original DWDM spectrum, and for rectangular channel spectra with an optical bandwidth B f each, the maximum spectral efficiency C reads [6, Eqs. (7 1] C = log ( 1 + P ( N NL N ASE + N NL 8.. Simulation and discussion, N NL := n sγ P ( 3 ( πα β ( f ln ns π ( β α M f. (7 Using the SSWCM Eq. (9, we propagate the total field envelope A numerically for RZ-DPSK modulation in the DWDM system described in Sect. 6. Without adding ASE noise, we measure the nonlinear noise powers N NL, and N NL,1 for logical zeros and ones numerically after each span. Defining the nonlinear noise power N NL as the arithmetic mean value of N NL, and N NL,1, we obtain in analogy to Eqs. (6 and (7 as a numerical estimate for the maximum spectral efficiency C = log ( 1 + P ( N NL N ASE + N NL, N NL := N NL, + N NL,1. (8 In Fig. 6, the maximum spectral efficiency C in the center channel k = M/ is plotted as a function of the launch power P ( per WDM channel for n s = 1,, 4 spans and M = 16 channels. As dispersion leads to destructive interference of FWM products, the author of [4] expects that his model (5 (dash-dotted lines in Fig. 6, originally derived for the case of a vanishing dispersion β ( =, underestimates C for β (. Hence, in presence of dispersion (β (, this model Eq. (5 [4] is not suitable for a quantitative parameter optimization. In contrast, the models Eqs. (6 [5] and (7 [6] (dashed and dotted lines are in good agreement with the

10 simulation using Eq. (8 (solid lines. For n s = 4 spans, the simulation with Eq. (8 estimates that error-free coherent detection is possible with a maximum spectral efficiency C = 9.65 bit/(s Hz (peak of the lowest solid curve in Fig. 6, at an optimum launch power P ( =.67 dbm = 1.17 mw per channel. Similarly, the models Eqs. (6 and (7 predict a maximum spectral efficiency for error-free coherent detection of C = 9.1 bit/(s Hz and C = 8.7 bit/(s Hz, respectively (peaks of lowest dashed and dotted curves in Fig. 6. The corresponding optimum launch powers P ( are 1.1 dbm =.78 mw and.3 dbm =.59 mw, respectively. PSfrag replacements maximum spectral efficiency C (bit/(s Hz n s = optical launch power P ( (dbm Figure 6. Maximum spectral efficiency C for coherent detection as a function of the launch power P ( per channel. Eq. (8 using SSWCM simulation (. Analytical models Eqs. (6 [5] (, (5 [4] (, and (7 [6] (. M = 16 WDM channels, n s = 1,, 4 spans (from top to bottom. We note that these very high spectral efficiencies can only be reached with an optimum modulation format. In this case, the resulting nonlinear noise power may be different from the value N NL obtained from Eq. (8 for the simulated RZ-DPSK scheme. Hence, based on this simulation, Eq. (8 gives only an approximation for the maximum possible spectral efficiency. However, due to the close agreement with the models Eqs. (6 and (7 (Fig. 6, we can assume that N NL is a suitable approximation for the nonlinear noise power of an optimum modulation scheme. Conversely, it is possible to use the models Eqs. (6 and (7 for an approximate optimization of DWDM systems fulfilling the assumptions made in [5] and [6]. Comparing with the optimum launch powers P ( for n s = 4 spans with NRZ-OOK and RZ-DPSK modulation,. dbm =.61 mw and.1 dbm =.6 mw (Sects. 7.1 and 7., respectively, we see that for the present DWDM system (see Sect. 6, especially the model Eq. (7 is well suited for an approximative first guess of useful values of system parameters like the optimum launch power P (. Nevertheless, for a verification of such analytical predictions and an accurate quantitative optimization of WDM system parameters, a numerical solution of the GNLSE (1 with an efficient propagation method like the SSWCM is indispensable. 9. CONCLUSION Using a basis of cardinal functions with compact support, we developed a new split-step wavelet collocation method (SSWCM as a general solver for the nonlinear Schrödinger equation. With N as the number of discretization points, this technique has the optimum complexity O(N for any fixed accuracy, making the method especially well suited for large problems with large time windows. Apart from a variable threshold criterion for storing the linear propagation operator e Lh with only a constant number S N of memory cells, the SSWCM uses the same basic assumptions and approximations as the standard split-step Fourier method (SSFM with complexity O(N log N. For the simulation of a 4 Gbit/s NRZ DWDM systems with 64 channels, the SSWCM requires less than 4 % of computation time compared to the SSFM. This improvement allows systematic investigations of the bit error rate as a function of the WDM system parameters. As an example, we determine the

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