Abstract

We present a nonperturbative analysis of certain dynamical aspects of breathers (dispersion-managed solitons) including the effects of third-order dispersion. The analysis highlights the similarities to and differences from the well-known analogous procedures for second-order dispersion. We discuss in detail the phase-space evolution of breathers in dispersion-managed systems in the presence of third-order dispersion.

© 2001 Optical Society of America

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References

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  1. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  2. T. I. Lakoba and G. P. Agrawal, “Effects of third-order dispersion on dispersion-managed solitons,” J. Opt. Soc. Am. B 16, 1332–1343 (1999).
    [CrossRef]
  3. 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]
  4. T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A 260, 68–77 (1999).
    [CrossRef]
  5. N. Akhmediev and M. Karlsson, “Cherenkov radiation emited by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
    [CrossRef] [PubMed]
  6. I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
    [CrossRef]
  7. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  9. T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
    [CrossRef]
  10. F. A. Scheck, Mechanics: From Newton’s Laws to Deterministic Chaos, 3rd. ed. (Springer, New York, 1999).
  11. P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
    [CrossRef]
  12. J. Kurnasako, M. Matsumoto, and S. Waiyapot, “Linear stability analysis of dispersion-managed solitons controlled by filters,” J. Lightwave Technol. 18, 1064–1068 (2000).
    [CrossRef]
  13. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, New York, 2000).

2000 (2)

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

J. Kurnasako, M. Matsumoto, and S. Waiyapot, “Linear stability analysis of dispersion-managed solitons controlled by filters,” J. Lightwave Technol. 18, 1064–1068 (2000).
[CrossRef]

1999 (3)

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

T. I. Lakoba and G. P. Agrawal, “Effects of third-order dispersion on dispersion-managed solitons,” J. Opt. Soc. Am. B 16, 1332–1343 (1999).
[CrossRef]

T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A 260, 68–77 (1999).
[CrossRef]

1998 (1)

1995 (3)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emited by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
[CrossRef]

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]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Agrawal, G. P.

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emited by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Evangelides, S. G.

Ghafouri-Shiraz, H.

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]

Gölles, M.

I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
[CrossRef]

Gordon, J. P.

Holmes, P.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

Inoue, T.

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emited by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

Kodama, Y.

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

Kurnasako, J.

Kutz, J. N.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

Lakoba, T. I.

T. I. Lakoba and G. P. Agrawal, “Effects of third-order dispersion on dispersion-managed solitons,” J. Opt. Soc. Am. B 16, 1332–1343 (1999).
[CrossRef]

T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A 260, 68–77 (1999).
[CrossRef]

Lederer, F.

I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
[CrossRef]

Maruta, A.

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

Matsumoto, M.

Nagata, M.

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]

Shum, P.

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]

Suguhara, H.

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

Uzunov, I. M.

I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
[CrossRef]

Waiyapot, S.

IEEE J. Quantum Electron. (1)

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]

IEEE Photonics Technol. Lett. (1)

T. Inoue, H. Suguhara, A. Maruta, and Y. Kodama, “Interactions between dispersion-managed solitons in optical time-division-multiplexed system,” IEEE Photonics Technol. Lett. 12, 299–301 (2000).
[CrossRef]

J. Lightwave Technol. (1)

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

Phys. Lett. A (1)

T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A 260, 68–77 (1999).
[CrossRef]

Phys. Rev. A (2)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emited by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Phys. Rev. E (1)

I. M. Uzunov, M. Gölles, and F. Lederer, “Soliton interaction near the zero-dispersion wavelength,” Phys. Rev. E 52, 1059–1071 (1995).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

Other (3)

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, New York, 2000).

F. A. Scheck, Mechanics: From Newton’s Laws to Deterministic Chaos, 3rd. ed. (Springer, New York, 1999).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

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Figures (5)

Fig. 1
Fig. 1

Position of pulse center c(t), normalized to the input pulsewidth, versus propagation distance z, normalized to the single-soliton period. The data points are from a split-step Fourier simulation, and the solid curve is from numerical integration of Eq. (12.1), which was derived from theoretical considerations.

Fig. 2
Fig. 2

Phase portraits for normal dispersion: level sets of the Hamiltonian in the (η, β) plane with σ=-1. Note that all trajectories start and end at the origin.

Fig. 3
Fig. 3

Phase portraits for anomalous dispersion: level sets of the Hamiltonian in the (η, β) plane with σ=1. Trajectories within the separatrix, shown by the dotted curve, are periodic.

Fig. 4
Fig. 4

Construction of dispersion maps based on Figs. 2 and 3. Transitions between the two types of fiber are indicated with circles. Two possible trajectories traversed by a breather are indicated by arrows.

Fig. 5
Fig. 5

Numerical simulation of pulse propagation over several periods along a dispersion map constructed obtained by Fig. 4.

Equations (30)

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i uz+σ(z)2 2ut2+|u|2u-iγ 3ut3=0,
δLδuLu-t L(u/t)-z L(u/z)+2t2 L(2u/t2)=0,
L(u, u*)=iu u*z-u* uz+σ(z)ut2-|u|4-iγ2ut2 u*t-2u*t2 ut,
u(z, t)=Aη sech[η(t-c)]×expiω(t-c)+β(t-c)2+ϕ2,
-u(z, t)u(z, t)*dt=2A2.
iu u*z-u* uz
2[(t-c)2β+(t-c)×(ω-2βc)+(ϕ-ωc)]uu*, 
σut2σ{4(t-c)2β2+η2 tanh[η(t-c)]2+4(t-c)βω+ω2}uu*,
-|u|4-A2η sech[η(t-c)]2uu*,
-iγ2ut2 u*t-2u*t2 ut
γ16(t-c)3β3+24(t-c)2β2ω+2ω(η2+ω2)-2β{η tanh[η(t-c)]-(t-c)(η2+3ω2)}uu*.
L=A2π23η2 dβdz+4dϕdz-ω dcdz-4A2 η3+2σω2+13 π2β2η2+η2+4γωπ2β2η2+ω2+η2.
pβ=βL=π2A23η2,
pc=cL=-4A2ω
L=A2γpcpc216A6+π23pβ+3pββ2A4+σ2π2A29pβ+pc28A4+2pββ2A2-4πA333pβ+4 dϕdz+pββ+pcc.
H(β, c;η, ω;z)=pββ+pcc-L=2A2-2γω+σ3π2β2η2+η2+ω2+23 A2η-ω2σ-2 dϕdz,
dβdz=-2π2A2η3-2(σ+6γω)β2-η4π2,
dηdz=-2βη(σ+6γω),
dcdz=σω+γπ2β2η2+η2+3ω2,
dωdz=0,
β=±ηπ 23 A2ησ/3+2γω-η21/2.
δβ=2π3 A22π(σ/3+2γω)2,
β2(z)β1(z)=-5.5+1,L2L1+L2=0.20.8+0.2.
 (I)β=0,η=0,
(II)β=0,η=A2σ+6γΩη0.
Δη=-2A2Δβ,
Δβ=2π2A2η02Δη,
λ±=±i 2πη02(σ+6γω)=±i 2π A4σ+6γω
v±=±i πη+0, 1.
·(dη/dz, dβ/dz)=-6β(σ±+6γω)

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