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

Full Article  |  PDF Article

References

  • View by:
  • |

  1. A. Hasegawa, �??Soliton-Based Optical Communications: An Overview,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
    [CrossRef]
  2. 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]
  3. H. Haus, �??Mode-Locking of Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. V. Cautaerts, A. Maruto, 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. Pare, P.-A. Belanger, �??Antisymmetric soliton in a dispersion-managed system,�?? Opt. Commun. 168, 103 (1999);
    [CrossRef]
  10. M.J. Ablowitz, Z.H. Musslimani, �??Dark and gray strong dispersion-managed solitons,�?? Phys. Rev. E 67, 025601(R) (2003).
    [CrossRef]
  11. P. V. Mamyshev, L.F. Mollenauer, �??Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,�?? Opt. Lett. 24, 448 (1999)
    [CrossRef]
  12. 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]
  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, S. Wabnitz, �??Dynamics of the nonlinear modulational instability in optical fibers,�?? Opt. Lett. 16, 986(1991).
    [CrossRef] [PubMed]
  16. D.K. Arrowsmith, C.M. Place An introduction to Dynamical Systems, (Cambrige University Press, N.Y., 1990)
  17. N. Korneev, �??Polarization chaos in nonlinear birefringent resonators,�?? Opt. Commun. 211, 153 (2002).
    [CrossRef]
  18. 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]

Chaos

V. Cautaerts, A. Maruto, Y. Kodama, �??On the dispersion managed soliton,�?? Chaos 10, 515 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Hasegawa, �??Soliton-Based Optical Communications: An Overview,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[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]

J. Opt. Soc. Am. B

J. Opt. Soc. Am.B

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]

Opt. Commun.

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]

N. Korneev, �??Polarization chaos in nonlinear birefringent resonators,�?? Opt. Commun. 211, 153 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

M.J. Ablowitz, Z.H. Musslimani, �??Dark and gray strong dispersion-managed solitons,�?? Phys. Rev. E 67, 025601(R) (2003).
[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]

Phys.Rev. Lett.

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

Other

D.K. Arrowsmith, C.M. Place An introduction to Dynamical Systems, (Cambrige University Press, N.Y., 1990)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Phase portraits of the dn-type wave on the Poincaré sphere for 3 harmonics taken into account. B=23/2 Im(S Ω S*0), C=23/2 Re(S Ω S*0), Ω=2.The dispersion management period is L 0=1.2L. a - L=0.1. The center of the pattern B=C=0,A=1 corresponds to a constant intensity. It is unstable stationary point of the map. Two stable periodic points in upper and lower parts correspond to truncated cnoidal waves, the upper is marked with an arrow. b - L=0.75, c - L=0.95, d - L=1.25.

Fig. 2.
Fig. 2.

The same, as in Fig. 1, but for the cn-type wave with 4 harmonics taken into account. A=2(S 1/2Ω S*1/2Ω-S 3/2Ω S*3/2Ω), C=2Re(S 3/2Ω S*1/2Ω), Ω=2. The dispersion management periods are L=0.1, 1.4, 1.8, and 2.8 for pictures a–d.

Fig. 3.
Fig. 3.

Influence of higher harmonics on the cnoidal wave. The intensity distributions over 200 periods in a wave which is close to the cn-wave and has initial parameters S 1/2Ω=cos(0.29)/√2, S 3/2Ω=sin(0.29)/√2 are shown. a - 4 harmonics, L=0.1, b - 10 harmonics, L=0.1 c - 4 harmonics, L=3, d - 8 harmonics, L=2.4, e - same as b, but initial conditions are not symmetric S -3/2Ω=(1+0.2i)S 3/2Ω, S -1/2Ω=(1+0.2i)S1/2Ω

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

i q ξ + d ( ξ ) 2 2 q η 2 + q 2 q = 0 ,
d ( ξ ) = d 0 , 0 < ξ < a L
d ( ξ ) = d 0 , a L < ξ < ( a + b ) L
d ( ξ ) = d 0 , ( a + b ) L < ξ < ( 2 a + b ) L
q dn ( η , ξ ) = κ d av 1 2 d n ( κ η ; m ) exp [ i κ 2 ( 1 m 2 2 ) ξ + i ψ 0 ]
q cn ( η , ξ ) = m κ d av 1 2 c n ( κ η ; m ) exp [ i κ 2 ( m 2 1 2 ) ξ + i ψ 0 ] ,
q sn ( η , ξ ) = m κ d av 1 2 sn ( κ η ; m ) exp [ i κ 2 ( 1 + m 2 ) ξ 2 + i ψ 0 ] .
dn ( η ; m ) = π l dn 1 + 4 π l dn 1 n = 1 ρ n ( 1 + ρ 2 n ) 1 cos [ 2 π n η l dn ] ,
cn ( η ; m ) = 8 π l cn 1 n = 1 ρ n 1 2 ( 1 + ρ 2 n 1 ) 1 cos [ 2 π ( 2 n 1 ) η l cn ] ,
q el ( η , ξ ) = n = N n = N S n ( ξ ) exp ( i Ω n η ) ,
i S n ξ = H S n * .
H ( S , S * , ξ ) = d ( ξ ) 2 n Ω n 2 S n S n * 1 2 l 1 + l 2 = l 3 + l 4 S l 1 S l 2 S l 3 * S l 4 * ,
A = I 1 I 2 = S 1 S 1 * S 2 S 2 *
B = i ( S 1 S 2 * S 1 * S 2 )
C = S 1 S 2 * + S 1 * S 2 .
A 2 + B 2 + C 2 = const ,

Metrics