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

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References

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  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]
  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]
  16. D.K. Arrowsmith and C.M. PlaceAn 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. 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]

2003 (2)

M.J. Ablowitz and 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]

2002 (3)

2001 (1)

2000 (6)

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

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]

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]

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

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

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]

1999 (3)

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

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

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

1991 (1)

1981 (1)

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

Ablowitz, M.J.

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

are, C.P

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

Arrowsmith, D.K.

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

Artigas, D.

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]

Belanger, P.-A.

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

Berge, L.

Cautaerts, V.

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

Chen, Y.

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

Christiansen, P.L.

Gaididei, Yu.B.

Grigoryan, V.S.

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]

Hasegawa, A.

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

Haus, H.

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

Infeld, E.

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

Kartashov, Ya.V.

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]

Kodama, Y.

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

Korneev, N.

Kubota, H.

Malomed, B.

Mamyshev, P. V.

Marti-Panameño, E.

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]

Maruto, A.

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

Menyuk, C.R.

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]

Mezentzev, V.K.

Mollenauer, L.

Mollenauer, L.F.

Mu, R.-M.

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]

Musslimani, Z.H.

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

Nakazawa, M.

Place, C.M.

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

Rasmussen, J.J.

Sahara, A.

Torner, L.

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]

Towers, I.

Trillo, S.

Vysloukh, V.A.

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]

Wabnitz, S.

Xie, C.

Xu, C.

Chaos (1)

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

IEEE J. Sel. Top. Quantum Electron. (3)

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, 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]

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

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

C.P are and 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 (1)

Opt. Lett. (4)

Phys. Rev. E (2)

M.J. Ablowitz and 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. (1)

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

Other (1)

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

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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.

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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.

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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

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Equations (16)

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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 ,

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