OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 8 — Apr. 12, 2010
  • pp: 8261–8276

A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation

Zhongxi Zhang, Liang Chen, and Xiaoyi Bao  »View Author Affiliations

Optics Express, Vol. 18, Issue 8, pp. 8261-8276 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (389 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A fourth-order Runge-Kutta in the interaction picture (RK4IP) method is presented for solving the coupled nonlinear Schrödinger equation (CNLSE) that governs the light propagation in optical fibers with randomly varying birefringence. The computational error of RK4IP is caused by the fourth-order Runge-Kutta algorithm, better than the split-step approximation limited by the step size. As a result, the step size of RK4IP can have the same order of magnitude as the dispersion length and/or the nonlinear length of the fiber, provided the birefringence effect is small. For communication fibers with random birefringence, the step size of RK4IP can be orders of magnitude larger than the correlation length and the beating length of the fibers, depending on the interaction between linear and nonlinear effects. Our approach can be applied to the fibers having the general form of local birefringence and treat the Kerr nonlinearity without approximation. Our RK4IP results agree well with those obtained from Manakov-PMD approximation, provided the polarization state can be mixed enough on the Poincaré sphere.

© 2010 Optical Society of America

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(190.3270) Nonlinear optics : Kerr effect
(260.2030) Physical optics : Dispersion

ToC Category:
Nonlinear Optics

Original Manuscript: February 16, 2010
Revised Manuscript: March 26, 2010
Manuscript Accepted: March 28, 2010
Published: April 5, 2010

Zhongxi Zhang, Liang Chen, and Xiaoyi Bao, "A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schr¨odinger equation," Opt. Express 18, 8261-8276 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996). [CrossRef]
  2. D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997). [CrossRef]
  3. C. R. Menyuk and B. S. Marks, "Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems," J. Lightwave Technol. 24, 2806-2826 (2006). [CrossRef]
  4. C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002). [CrossRef]
  5. E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006). [CrossRef]
  6. C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. 23, 174-176 (1987). [CrossRef]
  7. M. Karlsson, "Modulational instability in lossy optical fibers," J. Opt. Soc. Am. B 12, 2071-2077 (1995).
  8. H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995). [CrossRef]
  9. A. V. T. Cartaxo, "Small-signal analysis for nonlinear and dispersive optical fibres, and its application to design of dispersion supported transmission systems with optical dispersion compensation," Proc. Inst. Elect. Eng. Optoelectron. 146(5), 213-222 (1999). [CrossRef]
  10. M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001). [CrossRef]
  11. A. Vannucci, P. Serena, and A. Bononi, "The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation," J. Lightwave Technol. 20, 1102-1112 (2002). [CrossRef]
  12. O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightwave Technol. 21, 61-68 (2003). [CrossRef]
  13. E. Ciaramella and E. Forestieri, "Analytical approximation of nonlinear distortions," IEEE Photon. Technol. Lett. 17, 91-93 (2005). [CrossRef]
  14. M. Secondini, E. Forestieri, and C. R. Menyuk, "A combined regular-logarithmic perturbation method for signalnoise interaction in amplified optical systems," J. Lightwave Technol. 27, 3358-3369 (2009). [CrossRef]
  15. J. Hult, "A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers," J. Lightwave Technol. 25, 3770-3775 (2007). [CrossRef]
  16. G. H. Weiss and A. A. Maradudin, "The Baker-Hausdorff formula and a problem in crystal physics," J. Math. Phys. 3, 771-777 (1962). [CrossRef]
  17. J. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, 2003). [CrossRef]
  18. J. N. Damask, Polarization Optics in Telecommunications (Springer 2005).
  19. A. Galtarossa and C. R. Menyuk, Polarization Mode Dispersion (Springer 2005). [CrossRef]
  20. K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008). [CrossRef]
  21. S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992). [CrossRef]
  22. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000). [CrossRef]
  23. J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004). [CrossRef]
  24. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, "Sideband instability induced by periodic power variation in long-distance fiber links," Opt. Lett. 18, 1499-1501 (1993). [CrossRef] [PubMed]
  25. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997). [CrossRef]
  26. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001). [CrossRef]
  27. J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, "Effects of DGE bandwidth on nonlinear ULH systems," in Proc. Optical Fiber Communications Conf., Anaheim, CA, 2005, paper OWA2.
  28. E. Ibragimov, C. Menyuk, andW. Kath, "PMD-induced reduction of nonlinear penalties in terrestrial optical fiber transmission," in Proc. OFC, 2000, pp. 195-197, Paper WL3.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited