Abstract

We investigate some of the unique properties encountered by pulse solutions in a dispersion-managed optical fiber in the regime of net positive dispersion. Two pulse solutions of the same pulse width and different energies exist in this regime. Collisions of pulses of the same pulse width and different energies can lead to the destruction of the less-energetic pulse. Sideband generation can occur without a periodic phase-matching condition. The amplified spontaneous-emission-induced timing jitter follows a simple law that is expressible analytically.

© 1999 Optical Society of America

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References

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  1. N. S. Bergano, “Undersea amplified lightwave systems design,” in Optical Fiber Telecommunications IIA, I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, 1997).
  2. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  3. L. Boivin, M. C. Nuss, J. Shah, D. A. B. Miller, and H. A. Haus, “Optical receiver sensitivity improvement by impulsive coding,” in Ultrafast Electronics and Optoelectronics M. Nuss and J. Bowers, eds., Vol. 13 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 663–667.
  4. M. Suzuki, N. Edagawa, I. Morita, S. Yamamoto, and S. Akiba, “Soliton-based return-to-zero transmission over transoceanic distances by periodic dispersion compensation,” J. Opt. Soc. Am. B 14, 2953–2959 (1997).
    [CrossRef]
  5. J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
    [CrossRef]
  6. V. S. Grigoryan and C. R. Menyuk, “Dispersion-managed solitons at normal average dispersion,” Opt. Lett. 23, 609–611 (1998).
    [CrossRef]
  7. A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
    [CrossRef]
  8. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
    [CrossRef]
  9. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  10. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
    [CrossRef]
  11. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  12. H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
    [CrossRef]
  13. R.-M. Mu, V. S. Grigoryan, C. R. Menyuk, E. A. Golovchenko, and A. N. Pilipetskii, “Timing-jitter reduction in a dispersion-managed soliton system,” Opt. Lett. 23, 930–932 (1998).
    [CrossRef]
  14. M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photonics Technol. Lett. 9, 785–787 (1997).
    [CrossRef]

1998 (3)

1997 (3)

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photonics Technol. Lett. 9, 785–787 (1997).
[CrossRef]

M. Suzuki, N. Edagawa, I. Morita, S. Yamamoto, and S. Akiba, “Soliton-based return-to-zero transmission over transoceanic distances by periodic dispersion compensation,” J. Opt. Soc. Am. B 14, 2953–2959 (1997).
[CrossRef]

1996 (1)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

1995 (1)

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

1992 (2)

1986 (1)

1983 (1)

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

Akiba, S.

Anderson, D.

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

Berntson, A.

Doran, N. J.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Edagawa, N.

Evangelides, S. G.

Forysiak, W.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Golovchenko, E. A.

Gordon, J. P.

Grigoryan, V. S.

Haus, H. A.

M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photonics Technol. Lett. 9, 785–787 (1997).
[CrossRef]

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

Ippen, E. P.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Kelly, S. M. J.

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

Knox, F. M.

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Matsumoto, M.

M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photonics Technol. Lett. 9, 785–787 (1997).
[CrossRef]

Menyuk, C. R.

Mollenauer, L. F.

Morita, I.

Mu, R.-M.

Nelson, L. E.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Nijhof, J. H. B.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Pilipetskii, A. N.

Suzuki, M.

Tamura, K.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Yamamoto, S.

Electron. Lett. (2)

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed system with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photonics Technol. Lett. 9, 785–787 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (5)

Phys. Rev. A (1)

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

Rev. Mod. Phys. (1)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Other (2)

N. S. Bergano, “Undersea amplified lightwave systems design,” in Optical Fiber Telecommunications IIA, I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, 1997).

L. Boivin, M. C. Nuss, J. Shah, D. A. B. Miller, and H. A. Haus, “Optical receiver sensitivity improvement by impulsive coding,” in Ultrafast Electronics and Optoelectronics M. Nuss and J. Bowers, eds., Vol. 13 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 663–667.

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Figures (13)

Fig. 1
Fig. 1

Variation of dispersion from half-cell to half-cell.

Fig. 2
Fig. 2

Numerical solutions in (a) the net positive and (b) the net negative dispersion regimes, where FWHM=20 ps, L=100 km, and (a) Δk=0.1 ps2/km and k=17 ps2/km, and (b) Δk=-0.1 ps2/km and k=8.25 ps2/km. Pulse-amplitude profiles at reference planes are sampled in every tenth cell, and (a) and (b) have the same energy. Plotted in the figure is the normalized amplitude A=τ0a(κ/|Δk|)1/2 with κ the nonlinear coefficient and T=t/τ0.

Fig. 3
Fig. 3

Steady-state pulse amplitude and phase profiles in individual half-cells of Fig. 2, where (a) is for net positive, and (b) is for net negative dispersion.

Fig. 4
Fig. 4

Effective dispersion keff versus Δk for different stretch ratios, where τFWHM=1.66τ0.

Fig. 5
Fig. 5

Two branches of solutions in the net positive average dispersion regime from variational principle. Two numerically computed points are shown with circles.

Fig. 6
Fig. 6

Numerical solutions in the net positive dispersion regime for a high-energy pulse with the parameters of Fig. 2(a) (showing the low-energy pulse).

Fig. 7
Fig. 7

Numerical solutions of steady-state pulse propagation: (a) no Kerr nonlinearity in a negatively dispersive half-cell; (b) no Kerr nonlinearity in a positively dispersive half-cell. All the parameters are as in Fig. 2(a).

Fig. 8
Fig. 8

Propagation in one cell of the steady-state solutions displayed in Fig. 7: (a) and (c) are the amplitude and the phase for the case of Fig. 7(a); (b) and (d) correspond to Fig. 7(b).

Fig. 9
Fig. 9

Phase-matching diagram of solitons and stretched pulses in the regime of (a) Δk<0 and (b) Δk>0.

Fig. 10
Fig. 10

Computer simulations displaying sideband generation: (a) Frequency spectrum of the case in Fig. 2(a); sidebands are generated by Gaussian initial excitation, which differs slightly from the steady-state profile. (b) Initial excitation with higher energy than that of the steady state. (c) Filters of spectral width ten times the pulse spectral width placed at the planes of minimum pulse width; gain of 0.07 dB/cell included for filter-loss compensation.

Fig. 11
Fig. 11

Numerical simulations of the Gordon–Haus effect and comparison with analytic formula (11).

Fig. 12
Fig. 12

Collision of two pulses of (a) equal energy and (b) unequal energies with the parameters of Figs. 2(a) and 6, respectively.

Fig. 13
Fig. 13

Pulse evolution of the cases shown in Fig. 12 within the fifth cell (before the pulses meet at the reference plane).

Equations (26)

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ζa(ζ, t)=jΔk22t2a(ζ, t)+j12keff2t2a(ζ, t)-jδ|A0|21-μt2τ02.
a=A exp-t22τ2(1+jβ)-jψz
12z0z0+2L1ak(z)2at2dz
=12z0z0+2Lk(z)1-β2τ4dzt2-k(z)τ2dz.
jaz=-k22at2+κ|a|2a
ddzτ=2kβ2τ2τ,
ddzβ2τ2=2k14τ4-β2τ22+κE22τ3.
δ=κτ01τ,
()12Lz0z0+2Ldz().
μ=τ021τ31τ.
|keff|-Δk2τ04=1τ3κE.
Δt2=1.76ω0κnsp|Δk|(G-1)z39τFWHMLaEsolE,
d2τdz2=κkE2τ2+k2τ3,
12dτdz2=-κkE2τ-k22τ2+c,
c±=k±2τ±k±2τ±+κE.
±z-L2=f(c+, k+, τ)-f(c+, k+, τ+),
0<z<L,
±z-3L2=f(c-, k-, τ)-f(c-, k-, τ-),
L<z<2L,
f(c±, k±, τ)=1c±c±τ2-2k±κEτ-k±2+κk±E2c±3/2ln[2c±τ-2k±κE+2c±(c±τ2-2k±κEτ-k±2)].
L2=f(c+, k+, τb)-f(c+, k+, τ+),
L2=f(c-, k-, τb)-f(c-, k-, τ-).
z0z0+2Lκ|a|2dz=κE1τexp-t2τ2dzκE1τdz-1τ3dzt2,
12Lz0z0+2Ldzk(z)1-β2τ4keff+Δkτ04,
δ=κτ01τ,
μ=τ021τ31τ.

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