## Numerical Simulation of Nonuniformly Time-Sampled Pulse Propagation in Nonlinear Fiber

Journal of Lightwave Technology, Vol. 23, Issue 8, pp. 2434- (2005)

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### Abstract

Numerical simulation of pulse propagation in nonlinear optical fiber based on nonlinear Schrodinger equation plays a significant role in the design, analysis, and optimization of optical communication systems. Unconditionally stable operator-splitting techniques such as the split-step Fourier method or the split-step wavelet method have been successfully used for numerical simulation of uniformly time-sampled pulses along nonlinear optical fibers. Even though uniform time sampling is widely used in optical communication systems simulation, nonuniform time sampling is better or even desired for certain applications. For example, a sampling strategy that uses denser sampling points in regions where the signal changes rapidly and sparse sampling in regions where the signal change is gradual would result in a better replica of the signal. In this paper, we report a novel method that extends the standard operator-splitting techniques to handle nonuniformly sampled optical pulse profiles in the time domain. The proposed method relies on using cubic (or higher order) B-splines as a basis set for representing optical pulses in the time domain. We show that resulting operator matrices are banded and sparse due to the compact support of B-splines. Moreover, we use an algorithm based on Krylov subspace to exploit the sparsity of matrices for calculating matrix exponential operators. We present a comprehensive set of analytical and numerical simulation results to demonstrate the validity and accuracy of the proposed method.

© 2005 IEEE

**Citation**

Malin Premaratne, "Numerical Simulation of Nonuniformly Time-Sampled Pulse Propagation in Nonlinear Fiber," J. Lightwave Technol. **23**, 2434- (2005)

http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-23-8-2434

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### References

- G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. New York: Academic, 2001.
- A. Hasegawa, M. Matsumoto and P. I. Kattan, Optical Solitons in Fibers, 3rd ed. New York: Springer-Verlag, 2000.
- M. Brandt-Pearce, I. Jacobs and J. K. Shaw, "Optimal input Gaussian pulse width for transmission in dispersive nonlinear fiber", J. Opt. Soc. Amer. B, vol. 16, no. 8, pp. 1189-1196, 1999.
- V. Sinkin, R. Holzlohner, J. Zweck and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communication systems", J. Lightw. Technol., vol. 21, no. 1, pp. 61-68, Jan. 2003.
- Q. Chang, E. Jia and W. Suny, "Difference schemes for solving the generalized nonlinear Schrodinger equation", J. Comput. Phys., vol. 148, no. 2, pp. 397-415, 1999.
- L. R. Watkins, "Modeling propagating in optical fibers using wavelets", J. Lightw. Technol., vol. 12, no. 9, pp. 1536-1542, Dec. 1994.
- G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini and S. Bendedetto, "Suppression 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.
- X. Liu and B. Lee, "A fast method for nonlinear Schrodinger equation", IEEE Photon. Technol. Lett., vol. 15, no. 11, pp. 1549-1551, Nov. 2003.
- B. Fornberg and T. A. Driscoll, "A fast spectral algorithm for nonlinear wave equations with linear dispersion", J. Comput. Phys., vol. 155, no. 2, pp. 456-467, 1999.
- C. De Boor, A Practical Guide to Splines, New York: Springer-Verlag, 2001.
- Y. Saad, "Analysis of some Krylov approximation to the matrix exponential operator", SIAM J. Numer. Anal., vol. 29, no. 1, pp. 209-228, 1992.
- C. Moler and C. Van Loan, "Nineteen dubious ways to compute the exponential of a matrix, twenty five years later", SIAM Rev., vol. 45, no. 1, pp. 3-49, 2003.
- T. Blu, P. Thevenaz and M. Unser, "MOMS: Maximal-order interpolation of minimal support", IEEE Trans. Image Process., vol. 10, no. 7, pp. 1069-1080, Jul. 2001.
- W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, 2nd ed. London: U.K.: Cambridge Univ. Press, 2002.
- I. J. Schoenberg, Cardinal Spline Interpolation, Philadelphia, PA: SIAM, 1973,vol. 12.
- A. Sommerfeld, "Eine besonders anschauliche Ableitung des Gaussischen Fehlergesetzes," in Festschrift Ludwig Boltzmann Gewidmet Zum 60. Geburtstage, Leipzig: Germany: Barth, Feb. 1904, pp. 848-859.
- G. Polya, "Berechnung eines bestimmten integrals", Math. Ann., vol. 74, no. 20, pp. 204-212, 1913.
- R. Bartels, J. Beatty and B. Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, San Mateo, CA: Morgan Kaufmann, 1987.
- E. Cohen, R. F. Riesenfeld and G. Elber, Geometric Modeling with Splines: An Introduction, Natick, MA: A.K. Peters Ltd., 2001.
- M. Unser, A. Aldroubi and M. Eden, "B-spline signal processing-Part I: Theory", IEEE Trans. Signal Process., vol. 41, no. 2, pp. 821-833, Feb. 1993.
- M. Unser, "Splines: A perfect fit for signal and image processing", IEEE Signal Process. Mag., vol. 16, no. 6, pp. 22-38, Nov. 1999.
- M. Premaratne, "Split step spline method for modelling optical fibre communications systems", IEEE Photon. Technol. Lett., vol. 16, no. 5, pp. 1304-1306, May 2004.
- C. De Boor, "On calculating with B-splines", J. Approx. Theory, vol. 6, no. 1, pp. 50-62, 1972.
- C. De Boor, "On uniform approximation by splines", J. Approx. Theory, vol. 1, no. 1, pp. 219-235, 1968.
- I. J. Schoenberg, "Contribution to the problem of approximation of equidistant data by analytic functions", Quart. Appl. Math., vol. 4, no. 1, pp. 45-99, 1946.
- G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. New York: The Johns Hopkins Univ. Press, 1996.
- R. B. Sidje, "EXPOKIT: Software package for computing matrix exponentials", ACM Trans. Math. Softw., vol. 24, no. 1, pp. 130-158, 1998.
- L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Philadelphia, PA: SIAM, 1997.
- T. Lyche and K. Morken, "A discrete approach to knot removal and degree reduction for splines," in Algorithms for Approximations, J. C. Mason, and M. G. Cox, Eds. Oxford: U.K.: Clarendon, 1987, pp. 67-82.
- H. Schwetlick and T. Schutze, "Least square approximation by splines with free knots", Nordisk Tidskrift for InformationsBehandling, vol. 35, no. 3, pp. 361-384, 1995.

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