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, a new split-step wavelet collocation method (SSWCM) was developed as a general solver for the nonlinear Schrödinger equation describing pulse propagation in nonlinear optical fibers. With N as the number of discretization points, this technique has the optimum complexity cal(N) for a fixed accuracy, which is superior to the complexity cal(N log2 N) of the standard split-step Fourier method (SSFM). For the simulation of a large 40-Gb/s dense-wavelength-division-multiplexing (DWDM) system with 64 channels, the SSWCM requires less than 40% of computation time compared with the SSFM. This improvement allows a systematic optimization of wavelength-division-multiplexing (WDM) system parameters to achieve a minimum bit-error rate.

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Other (35)

T. Yu, W. M. Reimer, V. S. Grigoryan and C. R. Menyuk, "A mean field approach for simulating wavelength-division multiplexed systems", IEEE Photon. Technol. Lett., vol. 12, no. 4, pp. 443-445, Apr. 2000.

M. Plura, J. Kissing, M. Gunkel, J. Lenge, J.-P. Elbers, C. Glingener, D. Schulz and E. Voges, "Improved split-step method for efficient fiber simulations", Electron. Lett., vol. 37, no. 5, pp. 286-287, Mar. 2001.

J. Leibrich and W. Rosenkranz, "Efficient numerical simulation of multichannel WDM transmission systems limited by XPM", IEEE Photon. Technol. Lett., vol. 15, no. 3, pp. 395-397, Mar. 2003.

P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fiber communications", Nature, vol. 411, pp. 1027-1030, Jun. 28, 2001.

J. Tang, "The multispan effects of Kerr nonlinearity and amplifier noises on Shannon channel capacity of a dispersion-free nonlinear optical fiber", J. Lightw. Technol., vol. 19, no. 8, pp. 1110-1115, Aug. 2001.

L. Gagnon and J. M. Lina, "Symmetric Daubechies' wavelets and numerical solution of NLS equations", J. Phys. A, Math. Gen., vol. 27, pp. 8207-8230, 1994.

L. R. Watkins and Y. R. Zhou, "Modeling propagation in optical fibers using wavelets", J. Lightw. Technol., vol. 12, no. 9, pp. 1536-1542, Sep. 1994.

I. Pierce, P. Rees and K. A. Shore, "Wavelet operators for nonlinear optical pulse propagation", J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 17, no. 12, pp. 2431-2439, Dec. 2000.

T. Kremp, A. Killi, A. Rieder and W. Freude, "Split-step wavelet collocation method for nonlinear optical pulse propagation", IEICE Trans. Electron. (Special Issue on Signals, Systems, and Electronics Technology), vol. E85-C, no. 3, pp. 534-543, Mar. 2002.

T. Kremp, "Split-step wavelet collocation methods for linear and nonlinear optical wave propagation", Ph.D. dissertation, High-Frequency and Quantum Electronics Laboratory, University of Karlsruhe, Cuvillier Verlag Göttingen, Feb. 2002.

T. Kremp, A. Killi, A. Rieder and W. Freude, "Adaptive multiresolution split-step wavelet collocation method for nonlinear optical pulse propagation", presented at the Conf. Lasers Electro-Optics (CLEO 2002), Long Beach, CA, May 2002,Paper CThO40.

O. M. Nielsen, "Wavelets in scientific computing", Ph.D. dissertation, Department of Mathematical Modeling, Technical University of Denmark, Lyngby, Denmark, 1998.

T. Kremp and W. Freude, "Fast wavelet collocation method for WDM system parameter optimization", presented at the Optical Fiber Communication Conf. (OFC 2003), Atlanta, GA, Mar. 2003,Paper MF1.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. San Diego, CA: Academic, 2001.

B. A. Finlayson, The Method of Weighted Residuals, New York: Academic, 1972.

H.-J. Reinhardt, Analysis of Approximation Methods for Differential and Integral Equations , New York: Springer-Verlag, 1985.

K. Hayata, A. Misawa and M. Koshiba, "Split-step finite-element method applied to nonlinear integrated optics", J. Opt. Soc. Amer. B, Opt. Phys., vol. 7, no. 9, pp. 1772-1784, Sep. 1990.

B. Hermansson, D. Yevick, W. Bardyszewski and M. Glasner, "The unitarity of split-operator finite difference and finite-element methods: Applications to longitudinally varying semiconductor rib waveguides", J. Lightw. Technol., vol. 8, no. 12, pp. 1866-1873, Dec. 1990.

F. Di Pasquale and H. E. Hernandez-Figueroa, "Improved all-optical switching in a three-slab nonlinear directional coupler with gain", IEEE J. Quantum Electron., vol. 30, no. 5, pp. 1254-1258, May 1994.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices", IEEE J. Select. Topics Quantum Electron. , vol. 6, no. 1, pp. 150-162, Jan.-Feb. 2000.

A. Sharma and A. Taneja, "Unconditionally stable procedure to propagate beams through optical waveguides using the collocation method", Opt. Lett. , vol. 16, no. 15, pp. 1162-1164, Aug. 1991.

G. Deslaurier and S. Dubuc, "Symmetric iterative interpolation processes", Constr. Approx., vol. 5, pp. 49-68, 1989.

S. Goedecker, Wavelets and Their Application, Lausanne: Switzerland: Presses Polytechniques et Universitaires Romandes, 1998.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. San Diego, CA: Academic, 1999.

M. Fujii and W. J. R. Hoefer, "A wavelet formulation of the finite-difference method: Full-vector analysis of optical waveguide junctions", IEEE J. Quantum Electron., vol. 37, no. 8, pp. 1015-1029, Aug. 2001.

G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", SIAM. J. Numer. Anal., vol. 6, no. 6, pp. 1716-1740, Dec. 1992.

P. J. Davis, Circulant Matrices, New York: Wiley, 1979.

A. Iserles, "How large is the exponential of a banded matrix?", New Zealand J. Math., vol. 29, pp. 177-192, 2000.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini and S. Benedetto, "Supression of spurious tones induced by the split-step method in fiber systems simulation", IEEE Photon. Technol. Lett., vol. 12, no. 5, pp. 489-491, May 2000.

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, 2nd ed. , Cambridge: U.K.: Cambridge Univ. Press, 1992.

C. Francia, "Constant step-size analysis in numerical simulation for correct four-wave-mixing power evaluation in optical fiber transmission systems", IEEE Photon. Technol. Lett., vol. 11, no. 1, pp. 69-71, Jan. 1999.

C. J. Rasmussen, "Simple and fast method for step size determination in computations of signal propagation through nonlinear fibers", presented at the Optical Fiber Communication Conf. (OFC 2001), Anaheim, CA, Mar. 2001,Paper WDD29-1.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. : Singapore: McGraw-Hill, 1984.

J. G. Proakis, Digital Communications, 1st ed. New York: McGraw-Hill, 1983.

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers", J. Lightw. Technol., vol. 9, no. 11, pp. 1576-1582, Nov. 1991.

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