On the efficiency of different numerical methods for the calculation of intrapulse Raman scattering of optical solitons

  • Ivan M. Uzunov
  • Todor N. ArabadzhievEmail author


In this paper we compare the performance of different numerical methods for the calculation of the asymptotic evolution of soliton self-frequency shift in the presence of intrapulse Raman scattering (IRS) in optical fibers. First we have calculated the order of global accuracy for the fundamental soliton and the second-order bound state of the unperturbed nonlinear Schrödinger equation for the following numerical methods: the simple split step (SS) method, the full SS method, the reduced SS method, the reduced SS method with fourth order Runge–Kutta (RK4), the Blow–Wood method, the fourth order Runge–Kutta in the interaction picture (RK4IP) method and the Agrawal SS method with one and two iterations. We have shown that the asymptotic evolution of soliton self-frequency shift in the presence of IRS in optical fiber can be best described by the Agrawal SS method (compared to the Blow–Wood method and the RK4IP method). The obtained numerical results for the soliton position and frequency are in agreement with the predictions for these parameters according to the perturbation theory. We have shown that in the presence of IRS the fundamental soliton quickly develops an oscillating tail on its left. The generated tail hardly influences the soliton propagation at large distances. At large distances the form of the soliton gradually becomes asymmetric. The increase of the frequency resolution leads to a notable increase of the maximum propagation distance of the numerical soliton under the influence of IRS.


Nonlinear fiber optics Intrapulse Raman scattering Optical solitons Methods for the numerical solution of the nonlinear Schrödinger equation (NLSE) and a perturbed NLSE 



  1. Agrawal, G.P.: Nonlinear Fiber Optics, 3rd edn. Academic Press, San Diego (2001a)zbMATHGoogle Scholar
  2. Agrawal, G.P.: Applications of Nonlinear Fibre Optics. Academic Press, Cambridge (2001b)Google Scholar
  3. Akhmediev, N.N., Krolikowski, W., Lowery, A.J.: Influence of the Raman-effect on solitons in optical fibers. Opt. Commun. 131, 260–266 (1996)ADSCrossRefGoogle Scholar
  4. Balac, S., Fenandez, A.: Mathematical analysis of adaptive step-size techniques when solving the nonlinear Schrödinger equation for simulating light-wave propagation in optical fibers. Opt. Commun. 329, 1–9 (2014)ADSCrossRefGoogle Scholar
  5. Besse, C., Bidegaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40 (2002)MathSciNetCrossRefGoogle Scholar
  6. Blow, K.J., Wood, D.: Theoretical description of transient stimulated Raman scattering in optical fibers. IEEE J. Quantum Electron. 25, 2665–2673 (1989)ADSCrossRefGoogle Scholar
  7. Caradoc-Davies, B.M.: Ph.D. Dissertation, Univ. Utago, Dunedin (2000)Google Scholar
  8. Dianov, E.M., Grudinin, A.B., Prokhorov, A.M., Serkin, V.N., Taylor, J.R. (eds.): Optical Solitons-Theory and Experiment, p. 197. Cambridge University Press, Cambridge (1992). (Chap. 7) Google Scholar
  9. Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1184 (2006)ADSCrossRefGoogle Scholar
  10. Erkinalo, M., Genty, G., Wetzel, B., Dudley, J.M.: Limitations of the linear Raman gain approximation in modeling broadband nonlinear propagation in optical fibers. Opt. Express 18, 25449–25460 (2010)ADSCrossRefGoogle Scholar
  11. Facao, M., Carvalho, M.I., Parker, D.F.: Soliton self-frequency shift: self-similar solutions and their stability. Phys. Rev. E 81, 046604 (2010)ADSMathSciNetCrossRefGoogle Scholar
  12. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)CrossRefGoogle Scholar
  13. Fleck, J.A., Morris, J.R., Feit, M.D.: Time-dependent propagation of high energy laser beams through the atmosphere. Appl. Phys. 10, 129–160 (1976)ADSCrossRefGoogle Scholar
  14. Gagnon, L., Belanger, P.A.: Soliton self-frequency shift versus Galilean-like symmetry. Opt. Lett. 15, 466–468 (1990)ADSCrossRefGoogle Scholar
  15. Golovchenko, E., Pilipetskii, A.N.: Unified analysis of four-photon mixing, modulational instability, and stimulated Raman scattering under various polarization conditions in fibers. JOSA B 11, 92–101 (1994)ADSCrossRefGoogle Scholar
  16. Gordon, J.P.: Theory of the soliton self-frequency shift. Opt. Lett. 11, 662–664 (1986)ADSCrossRefGoogle Scholar
  17. Hardin, R.H., Tappert, F.D.: SIAM Rev. Chronicle 15, 423 (1973)Google Scholar
  18. Hasegawa, A., Kodama, Y.: Solitons in Optical Communications. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  19. Heidt, A.M.: Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers. J. Lightwave Technol. 27, 3984–3991 (2009)ADSCrossRefGoogle Scholar
  20. Hohage, T., Schmidt, F.: Konrad-Zuse-Zentrum fur informationstechnik. Technical Report ZIB-Report 02–04, Berlin (2002)Google Scholar
  21. Hult, J.: A fourth-order Runge–Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers. J. Lightwave Technol. 25, 3770–3775 (2007)ADSCrossRefGoogle Scholar
  22. Ivanov, S.K., Kamchatnov, A.M.: arXiv:904.05784v1 [niln.PS] 11 April 2019
  23. Kaup, D.J., Malomed, B.A.: Tails and decay of a Raman-driven pulse in a nonlinear-optical fiber. JOSA B 12, 1656–1662 (1995)ADSCrossRefGoogle Scholar
  24. Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. 23, 510–524 (1987)ADSCrossRefGoogle Scholar
  25. Mamyshev, P.V., Chernikov, S.V.: Ultrashort-pulse propagation in optical fibers. Opt. Lett. 15, 1076–1078 (1990)ADSCrossRefGoogle Scholar
  26. Menyuk, C.R.: Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes. J. Opt. Soc. Am. B 5(2), 392–402 (1988)ADSCrossRefGoogle Scholar
  27. Mitschke, F.M., Mollenauer, L.F.: Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661 (1986)ADSCrossRefGoogle Scholar
  28. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1986)CrossRefGoogle Scholar
  29. Potasek, M.J., Agrawal, G.P., Pinault, S.C.: Analytic and numerical study of pulse broadening in nonlinear dispersive optical fibers. J. Opt. Soc. Am. B 3, 205–211 (1986)ADSCrossRefGoogle Scholar
  30. Serkin, V.N.: ‘Colored’ envelope solitons in fiber-optic waveguides, Sov. Tech. Phys. Lett. 13, 320–323 (1987)Google Scholar
  31. Sinkin, O.V., Holzlohner, R., Zweck, J., Menyuk, C.R.: Optimization of the split-step Fourier method in modeling optical-fiber communications systems. J. Lightwave Technol. 21, 61–68 (2003)ADSCrossRefGoogle Scholar
  32. Stolen, R.H., Gordon, J.P., Tomlinson, W.J., Haus, H.A.: Raman response function of silica-core fibers. JOSA B 6, 1159–1166 (1989)ADSCrossRefGoogle Scholar
  33. Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55, 203–230 (1984)ADSMathSciNetCrossRefGoogle Scholar
  34. Uzunov, I.M., Gerdjikov, V.S.: Self-frequency shift of dark solitons in optical fibers. Phys. Rev. A 47, 1582–1585 (1993)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied PhysicsTechnical University SofiaSofiaBulgaria

Personalised recommendations