Abstract
Extensions of the split-step Fourier method (SSFM) for Schrödinger-type pulse propagation equations for simulating femto-second pulses in single- and two-mode optical communication fibers are developed and tested for Gaussian pulses. The core idea of the proposed numerical methods is to adopt an operator splitting approach, in which the nonlinear sub-operator, consisting of Kerr nonlinearity, the self-steepening and stimulated Raman scattering terms, is reformulated using Madelung transformation into a quasilinear first-order system of signal intensity and phase. A second-order accurate upwind numerical method is derived rigorously for the resulting system in the single-mode case; a straightforward extension of this method is used to approximate the four-dimensional system resulting from the nonlinearities of the chosen two-mode model. Benchmark SSFM computations of prototypical ultra-fast communication pulses in idealized single- and two-mode fibers with homogeneous and alternating dispersion parameters and also high nonlinearity demonstrate the reliable convergence behavior and robustness of the proposed approach.
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References
Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. Academic Press (2007)
Amorim, A.A., Tognetti, M.V., Oliveira, P., Silva, J.L., Bernardo, L.M., Kärtner, F.X., Crespo, H.M.: Sub-two-cycle pulses by soliton self-compression in highly-nonlinear photonic crystal fibers. Opt. Lett. 34, 3851 (2009)
Atre, R., Panigrahi, P.: Controlling pulse propagation in optical fibers through nonlinearity and dispersion management. Phys. Rev. A 76, 043,838 (2007)
Blow, K.J., Wood, D.: Theoretical description of transient stimulated Raman scattering in optical fibers. IEEE J. Quantum Electronics 25 (12), 2665–2673 (1989)
Deiterding, R., Glowinski, R., Oliver, H., Poole, S.: A reliable split-step Fourier method for the propagation equation of ultra-fast pulses in single-mode optical fibers. J. Lightwave Technology 31, 2008–2017 (2013)
Glowinski, R.: Finite element methods for incompressible viscous flows. In: P.G. Ciarlet, J.L. Lions (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176, North-Holland, Amsterdam (2003)
Gnauck, A.H., Charlet, G., Tran, P., Winzer, P.J., Doerr, C.R., Centanni, J.C., Burrows, E.C., Kawanishi, T., Sakamoto, T., Higuma, K.: 25.6 Tb/s WDM transmission of polarization-multiplexed RZ-DQPSK signals. J. Lightwave Technology 26, 79 (2008)
Guo, S., Huang, Z.: Densely dispersion-managed fiber transmission system with both decreasing average dispersion and decreasing local dispersion. Optical Engineering 43, 1227 (2004)
Hager, W.: Applied Numerical Linear Algebra. Prentice Hall, Englewood Cliffs, NJ (1988)
Hohage, T., Schmidt, F.: On the numerical solution of nonlinear Schrödinger type equations in fiber optics. Tech. Rep. ZIB-Report 02–04, Konrad-Zuse-Zentrum für Informationstechnik Berlin (2002)
Kalithasan, B., Nakkeeran, K., Porsezian, K., Tchofo Dinda, P., Mariyappa, N.: Ultra-short pulse propagation in birefringent fibers – the projection operator method. J. Opt. A: Pure Appl. Opt. 10, 085,102 (2008)
Ketcheson, D.I., LeVeque, R.J.: WENOClaw: a higher order wave propagation method. In: Hyperbolic Problems: Theory, Numerics, Applications, pp. 609–616. Springer, Berlin (2008)
Lax, P.D.: Gibbs phenomena. J. Scientific Comput. 28 (2/3), 445–449 (2006)
van Leer, B.: Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, New York (2002)
Long, V.C., Viet, H.N., Trippenback, M., Xuan, K.D.: Propagation technique for ultrashort pulses II: Numerical methods to solve the pulse propagation equation. Comp. Meth. Science Techn. 14 (1), 13–19 (2008)
Madelung, E.: Quantentheorie in hydrodynamischer Form. Zeitschrift für Physik 40 (3–4), 322–326 (1927)
Malomed, B.A.: Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach. Opt. Comm. 136, 313–319 (1997)
Muslu, G.M., Erbay, H.A.: A split-step Fourier method for the complex modified Korteweg-de Vries equation. Computers and Mathematics with Applications 45, 503–514 (2003)
Richardson, L.J., Forsyiak W. Blow, K.J.: Single channel 320Gbit/s short period dispersion managed transmission over 6000km. Optics Letters 36, 2029 (2000)
Sinkin, O.V., Holzlöhner, R., Zweck, J., Menyuk, C.R.: Optimization of the split-step Fourier method in modeling optical-fiber communication systems. J. Lightwave Technology 21 (1), 61–68 (2003)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1982)
Spiegel, E.A.: Fluid dynamical form of the linear and nonlinear Schrödinger equations. Physica D: Nonlinear Phenomena 1 (2), 236–240 (1980)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Num. Anal. 5, 506–517 (1968)
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This work was supported by the Department of Defense and used resources of the Extreme Scale Systems Center at Oak Ridge National Laboratory.
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Deiterding, R., Poole, S.W. (2016). Robust Split-Step Fourier Methods for Simulating the Propagation of Ultra-Short Pulses in Single- and Two-Mode Optical Communication Fibers. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_18
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DOI: https://doi.org/10.1007/978-3-319-41589-5_18
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