Abstract

In wavelength-division-multiplexed communication systems phase jitter is driven by amplifier noise and mediated by cross-phase modulation. The variational method is used to derive formulas for the absolute phase variance of an ensemble of solitons and the relative phase variance of ensembles of neighboring solitons. The predictions of these formulas are consistent with the results of numerical simulations.

© 2003 Optical Society of America

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References

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  1. J. P. Gordon and L. F. Mollenauer, Opt. Lett. 23, 1351 (1990).
    [CrossRef]
  2. D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
    [CrossRef] [PubMed]
  3. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  4. W. J. Firth, Opt. Commun. 22, 226 (1977).
    [CrossRef]
  5. D. Anderson, Phys. Rev. A 27, 3135 (1983).
    [CrossRef]
  6. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.
  7. M. Hanna, H. Porte, J. P. Goedgebuer, and W. T. Rhodes, Opt. Lett. 24, 732 (1999).
    [CrossRef]
  8. C. J. McKinstrie and C. Xie, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
    [CrossRef]
  9. The variational equations for pulse–pulse interactions have been known since the 1980s. For a convenient derivation of these equations, see C. J. McKinstrie, Opt. Commun. 205, 123 (2002).
    [CrossRef]
  10. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

2002 (2)

C. J. McKinstrie and C. Xie, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[CrossRef]

The variational equations for pulse–pulse interactions have been known since the 1980s. For a convenient derivation of these equations, see C. J. McKinstrie, Opt. Commun. 205, 123 (2002).
[CrossRef]

1999 (1)

1990 (3)

1983 (1)

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

1977 (1)

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Anderson, D.

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

Firth, W. J.

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

Goedgebuer, J. P.

Gordon, J. P.

Hanna, M.

Haus, H. A.

Iannone, E.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Lai, Y.

Matera, F.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.

McKinstrie, C. J.

C. J. McKinstrie and C. Xie, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[CrossRef]

The variational equations for pulse–pulse interactions have been known since the 1980s. For a convenient derivation of these equations, see C. J. McKinstrie, Opt. Commun. 205, 123 (2002).
[CrossRef]

Mecozzi, A.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.

Mollenauer, L. F.

Porte, H.

Rhodes, W. T.

Settembre, M.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Xie, C.

C. J. McKinstrie and C. Xie, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

IEEE J. Sel. Top. Quantum Electron. (1)

C. J. McKinstrie and C. Xie, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[CrossRef]

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

Opt. Commun. (2)

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

The variational equations for pulse–pulse interactions have been known since the 1980s. For a convenient derivation of these equations, see C. J. McKinstrie, Opt. Commun. 205, 123 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Other (1)

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998), pp. 158 and 163.

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Figures (4)

Fig. 1
Fig. 1

Standard deviation of the (normalized) noise-induced energy shifts plotted as a function of distance.

Fig. 2
Fig. 2

Absolute phase variances plotted as functions of distance for systems with 3, 5, and 9 channels, separated by multiples of 100 GHz. Dashed curves denote the predictions of Eq. (12); solid curves denote the simulation results. For clarity, the five- and nine-channel curves are displaced upward by 1 and 2, respectively.

Fig. 3
Fig. 3

Phase correlation coefficients plotted as functions of distance for systems with 3, 5, and 9 channels, separated by multiples of 100 GHz. Dotted, dashed, and solid curves denote 3-, 5-, and 9-channel correlations, respectively.

Fig. 4
Fig. 4

Relative phase variances plotted as functions of distance for systems with 3, 5, and 9 channels, separated by multiples of 100 GHz. Dashed curves denote the predictions of Eq. (14); solid curves denote the simulation results. For clarity, the 5- and 9-channel curves are displaced upward by 1 and 2, respectively.

Equations (14)

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zA=g-αA/2-iβttA/2+iγA2A+R,
Aiz,t=ai expiϕi-iωit-ti-1-icit-ti2/2τi2,
γEi/2π1/2τi=β/τi2
ϕiz=3γEiz/42π1/2τi.
dzϕi=δϕz+β/2τi2+5γEi/42π1/2τi.
δϕi2z=Sz/Eic1z+c33γEi/42π1/2τi2z3,
dϕidz=γEjπτi2+τj21/23τi2+2τj2τi2+τj2-2τi2ti-tj2τi2+τj22exp-ti-tj2τi2+τj2.
δϕi2γEj/βωij
δϕiz=4γEz/T.
δEj2zc=δEj20+2SzEzc.
δϕi2=4γ2δEj2/βωij2.
δϕi2z=8γ2SzEz2/βωijT.
δϕi1-δϕi2k=1nijδEjkk-1/2zij-k=2nij+1δEjkk-3/2zij.
δϕi2z=32γ2SzEz/βωij2.

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