## Propagation dynamics of weakly localized cnoidal waves in dispersion-managed fiber: from stability to chaos

Optics Express, Vol. 11, Issue 26, pp. 3574-3582 (2003)

http://dx.doi.org/10.1364/OE.11.003574

Acrobat PDF (357 KB)

### Abstract

We study the dynamics of propagation of the pulse train modeled by truncated cnoidal-type wave in a nonlinear dispersion-managed (DM) fiber. Computer simulations permit to select fiber parameters and waveform to ensure self-repeating of wave after the dispersion map period. It is shown that the long-period maps lead to the complicated chaotic behavior of cnoidal type wave, namely the Kolmogorov-Arnold-Moser (KAM) chaos.

© 2003 Optical Society of America

## 1. Introduction

1. A. Hasegawa, “Soliton-Based Optical Communications: An Overview,” IEEE J. Sel. Top. Quantum Electron. **6**, 1161 (2000). [CrossRef]

2. M. Nakazawa, A. Sahara, and H. Kubota, “Propagation of a solitonlike nonlinear pulse in average normal group-velocity dispersion and its unsuitability for high-speed, long-distance optical transmission,” J. Opt. Soc. Am. B **18**, 409 (2001). [CrossRef]

3. H. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. **6**, 1173 (2000). [CrossRef]

4. R.-M. Mu, V.S. Grigoryan, and C.R. Menyuk, “Comparison of Theory and Experiment for Dispersion-Managed Solitons in a Recirculating Fiber Loop,” IEEE J. Sel. Top. Quantum Electron. **6**, 248 (2000). [CrossRef]

5. L. Berge, V.K. Mezentzev, J.J. Rasmussen, P.L. Christiansen, and Yu.B. Gaididei, “Self-guiding light in layered nonlinear media,” Opt. Lett. **25**, 1037 (2000). [CrossRef]

6. I. Towers and B. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr non-linearity”, J. Opt. Soc. Am. B **19**, 537 (2002). [CrossRef]

1. A. Hasegawa, “Soliton-Based Optical Communications: An Overview,” IEEE J. Sel. Top. Quantum Electron. **6**, 1161 (2000). [CrossRef]

7. V. Cautaerts, A. Maruto, and Y. Kodama, “On the dispersion managed soliton,” Chaos **10**, 515 (2000). [CrossRef]

8. Y. Chen, “Dark solitons in dispersion compensated fiber transmission systems,” Opt. Commun. **161**, 267 (1999). [CrossRef]

9. C.P are and P.-A. Belanger, “Antisymmetric soliton in a dispersion-managed system,” Opt. Commun. **168**, 103 (1999); [CrossRef]

10. M.J. Ablowitz and Z.H. Musslimani, “Dark and gray strong dispersion-managed solitons,” Phys. Rev. E **67**, 025601(R) (2003). [CrossRef]

11. P. V. Mamyshev and L.F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. **24**, 448 (1999) [CrossRef]

12. C. Xu, C. Xie, and L. Mollenauer, “Analysis of soliton collisions in a wavelength-division-multiplexed dispersion-managed soliton transmission system,” Opt. Lett. **27**, 1303 (2002). [CrossRef]

13. Ya.V. Kartashov, V.A. Vysloukh, E. Marti-Panameño, D. Artigas, and L. Torner, “Dispersion-managed cnoidal pulse trains”, Phys. Rev. E **68**, 026613 (2003). [CrossRef]

14. E. Infeld, “Quantitative theory of the Fermi-Pasta-Ulam Resonance in the nonlinear Schrödinger equation,” Phys. Rev. Lett. **47**, 717 (1981). [CrossRef]

15. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. **16**, 986 (1991). [CrossRef] [PubMed]

17. N. Korneev, “Polarization chaos in nonlinear birefringent resonators,” Opt. Commun. **211**, 153 (2002). [CrossRef]

## 2. Mathematical model and system parameters

*d*(

*ξ*):

*q*(

*η,ξ*)=(

*L*

_{dis}/

*L*

_{spm})

^{1/2}

*A*(

*η,ξ*) is the normalized complex amplitude;

*A*(

*η,ξ*) is the slowly varying envelope;

*I*

_{0}is the peak input intensity;

*η*=(

*t*-

*z*/

*u*

_{gr})/

*τ*

_{0}is the running time;

*τ*

_{0}is the characteristic time scale;

*k*

_{0}=

*k*(

*ω*

_{0}) is the wave number;

*ω*

_{0}is the carrying frequency;

*ξ*=

*z*/

*L*

_{dis}is the normalized propagation distance;

*L*

_{dis}=

*β*

_{2}| is the dispersion length; coefficient

*L*

_{spm}=2

*c*/(

*ω*

_{0}

*n*

_{2}

*I*

_{0}) is the self-phase modulation length;

*n*

_{2}is the coefficient of nonlinearity.

*L*

_{0}=(2

*a*+

*b*)

*L*is the period of the dispersion map,

*d*

_{0}>0 is one half of the dispersion difference,

*a,b*are positive parameters; the average dispersion is given by

*d*

_{av}=(2

*a*-

*b*)/(2

*a*+

*b*)

*d*

_{0}, and L is the characteristic length. Note, that the first segment with anomalous group velocity dispersion (focusing) is followed by a segment with a normal GVD (defocussing) and terminated with the segment with the anomalous GVD.

*d*(

*ξ*)=

*d*

_{av}>0 in a form of elliptic dn- and cn- waves.

*d*(

*ξ*)=

*d*

_{av}<0:

*cn*(

*η,ξ*),

*sn*(

*η,ξ*),

*dn*(

*η,ξ*) are Jacobi elliptic functions; 0≤

*m*≤1 is the modulus of the elliptic function that describes the degree of localization of the wave field energy;

*κ*>0 is the arbitrary form-factor;

*ψ*

_{0}is the constant phase. The transverse period of the dn-wave equals

*l*

_{dn}=2

*K*(

*m*)/

*κ*, where

*K*(

*m*) is the elliptic integral of the first kind, whereas transverse periods of cn- and sn- waves are equal to to

*l*

_{cn}=

*l*

_{sn}=4

*K*(

*m*)/

*κ*. It is worth mentioning that the analytical solutions given by Eqs. (3–5) are a good initial guess for calculation of the profile of the true breathing elliptic wave by the numeric averaging method [4

4. R.-M. Mu, V.S. Grigoryan, and C.R. Menyuk, “Comparison of Theory and Experiment for Dispersion-Managed Solitons in a Recirculating Fiber Loop,” IEEE J. Sel. Top. Quantum Electron. **6**, 248 (2000). [CrossRef]

7. V. Cautaerts, A. Maruto, and Y. Kodama, “On the dispersion managed soliton,” Chaos **10**, 515 (2000). [CrossRef]

*m*→0,

*K*(

*m*)→

*π*/2,

*S*

_{n}(

*ξ*) are the amplitudes of harmonics, the sum is taken over appropriate frequencies given by Eqs. (6). For our normalization, the smallest nonzero Ω

_{n}is of an order of unity. By substitution of Eq. (7) into Eq. (1), it is possible to obtain the system of ordinary differential equations in the Hamiltonian form:

*l*takes any possible value of

*n*.

*I*=∑

*S*

_{n}

*S**

_{n}=

*const*. If we restrict the minimization procedure to real amplitudes, the resulting values coincide quite well with the Fourier coefficients of the cnoidal wave expansion Eq. (6) with the same period and the same intensity, if the number of harmonics is sufficient to represent well the cnoidal wave with these parameters.We will call these solutions truncated elliptic waves. Dn-type truncated elliptic waves are obtained if the number of harmonics is odd, cn- and sn- waves, if the number of harmonics is even, depending on the symmetry/antisymmetry of coefficients. Truncated cnoidal waves are neutrally stable, small variations of initial amplitudes produce a small jitter around the trajectory of the cnoidal wave in the phase space. This is the consequence of the complete integrability of the nonlinear Schrödinger equation - the solution moves to the nearby invariant torus.

## 3. Truncated elliptic waves in a dispersion-managed fiber

*m*≪1). Note, that both dn- and cn- type solutions can be approximated, if we take into account the symmetry of coefficients

*S*

_{n}=

*S*

_{-n}. Sn-type solutions are obtained with an even number of harmonics and antisymmetric amplitudes.

*A*(

*ξ*),

*B*(

*ξ*),

*C*(

*ξ*) lie on a sphere. Their shape can be determined by rewriting Hamiltonian in terms of Stokes parameters, the trajectories are intersections of a sphere with parabolic cylinders for the case of dn-wave, and with elliptic cylinders in a case of cnwaves (see [18

18. N. Korneev, “Analytical solutions for three and four diffraction orders interaction in Kerr media,” Opt. Express **7**, 299 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-299. [CrossRef] [PubMed]

*d*

_{0}=1,

*a*=0.5,

*b*=0.2. The length

*L*from Eq. (2) served as a variable parameter (the period of the map is then

*L*

_{0}=1.2

*L*). For the completely integrable case (uniform fiber with positive dispersion, no DM) the mapping points form the closed lines corresponding to the solution trajectories.

*L*is small in comparison with a typical longitudinal period of the cnoidal waves (which is physically of an order of dispersion length in our case), the trajectory pattern is very similar to that one with a dispersion equal the average

*d*

_{av}=2/3 (see Fig. 1-a). When the map period grows, chaotic trajectories appear close to the unstable periodic points of the map. To trace the chaos development one could change the nonlinearity strength or the temporal period instead. The length changes were chosen because the transition there is clearly seen.

*L*=0.75 (Fig. 1(b)), the chaotic region is quite small, but it increases rapidly with

*L*, (Figs. 1(c) and (d)). The process is characterized by a formation of a typical homoclinic tangle near the unstable periodic points. The point in the phase space corresponding the breathing dn-type cnoidal wave is quite close to this chaotic region. While the nearest vicinity of cnoidal wave remains stable with growing dispersion map period

*L*, the region of the neutral stability diminishes, and finally even the trajectories very close to the cnoidal wave point become chaotic. For moderate dispersion management there is also a big number of chains with alternating stable and unstable periodic points which is typical for KAM chaos (Figs. 1(c) and (d)).

*B*=

*C*=0 on the sphere. One stable and one unstable manifold start in the vicinity of this stationary point. They are formed by points which either approximate the stationary point with any iteration moving forward or backwards in time. The manifolds produce the eight-figure, the point which starts on the unstable manifold in the close vicinity of the stationary point first moves away and then approximates it on the stable manifold.

*L*the chains of periodic points of the map appear, and then the chaos develops close to unstable periodic points of these chains. Thus, for the cn-wave for the same period and intensity as the dn-wave, the chaotic region develops for bigger dispersion management length, and starts quite far away from the the region of truncated cnoidal wave in the phase space. Consequently, cn-waves are more stable with respect to a long-scale dispersion management. Note, that for the weakly localized sn-wave patterns on the sphere are qualitatively similar to that for cn-wave, so we will not go into details.

*L*the solution retains its structure. The threshold value of

*L*producing developed chaotic behavior diminishes if additional harmonics are involved. If the perturbation is not symmetric, the cnoidal wave can start to move, but generally retains its structure (Fig. 3(e)).

## 4. Concluding remarks

## Acknowledgments

## References and links

1. | A. Hasegawa, “Soliton-Based Optical Communications: An Overview,” IEEE J. Sel. Top. Quantum Electron. |

2. | M. Nakazawa, A. Sahara, and H. Kubota, “Propagation of a solitonlike nonlinear pulse in average normal group-velocity dispersion and its unsuitability for high-speed, long-distance optical transmission,” J. Opt. Soc. Am. B |

3. | H. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. |

4. | R.-M. Mu, V.S. Grigoryan, and C.R. Menyuk, “Comparison of Theory and Experiment for Dispersion-Managed Solitons in a Recirculating Fiber Loop,” IEEE J. Sel. Top. Quantum Electron. |

5. | L. Berge, V.K. Mezentzev, J.J. Rasmussen, P.L. Christiansen, and Yu.B. Gaididei, “Self-guiding light in layered nonlinear media,” Opt. Lett. |

6. | I. Towers and B. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr non-linearity”, J. Opt. Soc. Am. B |

7. | V. Cautaerts, A. Maruto, and Y. Kodama, “On the dispersion managed soliton,” Chaos |

8. | Y. Chen, “Dark solitons in dispersion compensated fiber transmission systems,” Opt. Commun. |

9. | C.P are and P.-A. Belanger, “Antisymmetric soliton in a dispersion-managed system,” Opt. Commun. |

10. | M.J. Ablowitz and Z.H. Musslimani, “Dark and gray strong dispersion-managed solitons,” Phys. Rev. E |

11. | P. V. Mamyshev and L.F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. |

12. | C. Xu, C. Xie, and L. Mollenauer, “Analysis of soliton collisions in a wavelength-division-multiplexed dispersion-managed soliton transmission system,” Opt. Lett. |

13. | Ya.V. Kartashov, V.A. Vysloukh, E. Marti-Panameño, D. Artigas, and L. Torner, “Dispersion-managed cnoidal pulse trains”, Phys. Rev. E |

14. | E. Infeld, “Quantitative theory of the Fermi-Pasta-Ulam Resonance in the nonlinear Schrödinger equation,” Phys. Rev. Lett. |

15. | S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. |

16. | D.K. Arrowsmith and C.M. Place |

17. | N. Korneev, “Polarization chaos in nonlinear birefringent resonators,” Opt. Commun. |

18. | N. Korneev, “Analytical solutions for three and four diffraction orders interaction in Kerr media,” Opt. Express |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 12, 2003

Revised Manuscript: December 11, 2003

Published: December 29, 2003

**Citation**

N. Korneev, V. Vysloukh, and E. Rodriguez, "Propagation dynamics of weakly localized cnoidal waves in dispersion-managed fiber: from stability to chaos," Opt. Express **11**, 3574-3582 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3574

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### References

- A. Hasegawa, �??Soliton-Based Optical Communications: An Overview,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000). [CrossRef]
- M. Nakazawa, A. Sahara, H. Kubota, �??Propagation of a solitonlike nonlinear pulse in average normal group-velocity dispersion and its unsuitability for high-speed, long-distance optical transmission,�?? J. Opt. Soc. Am.B 18, 409 (2001). [CrossRef]
- H. Haus, �??Mode-Locking of Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000). [CrossRef]
- R.-M. Mu, V.S. Grigoryan, C.R. Menyuk, �??Comparison of Theory and Experiment for Dispersion-Managed Solitons in a Recirculating Fiber Loop,�?? IEEE J. Sel. Top. Quantum Electron. 6, 248 (2000). [CrossRef]
- L. Berge, V.K. Mezentzev, J.J. Rasmussen, P.L. Christiansen, Yu.B. Gaididei, �??Self-guiding light in layered nonlinear media,�?? Opt. Lett. 25, 1037 (2000). [CrossRef]
- I. Towers, B. Malomed, �??Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity�?? J. Opt. Soc. Am. B 19, 537 (2002). [CrossRef]
- V. Cautaerts, A. Maruto, Y. Kodama, �??On the dispersion managed soliton,�?? Chaos 10, 515 (2000). [CrossRef]
- Y. Chen, �??Dark solitons in dispersion compensated fiber transmission systems,�?? Opt. Commun. 161, 267 (1999). [CrossRef]
- C. Pare, P.-A. Belanger, �??Antisymmetric soliton in a dispersion-managed system,�?? Opt. Commun. 168, 103 (1999); [CrossRef]
- M.J. Ablowitz, Z.H. Musslimani, �??Dark and gray strong dispersion-managed solitons,�?? Phys. Rev. E 67, 025601(R) (2003). [CrossRef]
- P. V. Mamyshev, L.F. Mollenauer, �??Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,�?? Opt. Lett. 24, 448 (1999) [CrossRef]
- C. Xu, C. Xie, L. Mollenauer, �??Analysis of soliton collisions in a wavelength-division-multiplexed dispersionmanaged soliton transmission system,�?? Opt. Lett. 27, 1303 (2002). [CrossRef]
- Ya.V. Kartashov, V.A. Vysloukh, E.Marti-Panameño, D.Artigas, and L.Torner, �??Dispersion-managed cnoidal pulse trains�??, Phys. Rev. E 68, 026613 (2003). [CrossRef]
- E. Infeld, �??Quantitative theory of the Fermi-Pasta-Ulam Resonance in the nonlinear Schrödinger equation,�?? Phys.Rev. Lett. 47, 717 (1981). [CrossRef]
- S. Trillo, S. Wabnitz, �??Dynamics of the nonlinear modulational instability in optical fibers,�?? Opt. Lett. 16, 986(1991). [CrossRef] [PubMed]
- D.K. Arrowsmith, C.M. Place An introduction to Dynamical Systems, (Cambrige University Press, N.Y., 1990)
- N. Korneev, �??Polarization chaos in nonlinear birefringent resonators,�?? Opt. Commun. 211, 153 (2002). [CrossRef]
- 18. N. Korneev, �??Analytical solutions for three and four diffraction orders interaction in Kerr media,�?? Opt. Express7, 299 (2000), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-299">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-299</a> [CrossRef] [PubMed]

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