Abstract

The effects of filtering on the Gordon–Haus timing jitter of chirped optical pulses in dispersion-managed communication systems with lumped and distributed amplification are studied in detail. Evolution equations for the average frequency and time delay are derived and solved analytically. Distributed amplification produces less timing jitter than lumped amplification. For short distances the timing jitter depends explicitly on the chirp, whereas for long distances it does not. The implicit chirp dependence that remains is inherited from the soliton energy. Simple formulas are derived for the asymptotic rates at which timing jitter grows. Numerical examples are described for 10-Gb/s systems and their relevance to 40-Gb/s systems is discussed. Although filtering limits the timing jitter in dispersion-managed systems, it precludes the reduction of timing jitter by dispersion postcompensation.

© 2002 Optical Society of America

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  1. 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]
  2. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
    [CrossRef]
  3. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).
  4. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997).
  5. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998).
  6. R. J. Essiambre and G. P. Agrawal, “Timing jitter of ultrashort solitons in high-speed communication systems: General formalism and application to dispersion-decreasing fibers,” J. Opt. Soc. Am. B 14, 314–322 (1997).
    [CrossRef]
  7. S. Kumar and F. Lederer, “Gordon–Haus effect in dispersion-managed soliton systems,” Opt. Lett. 22, 1870–1872 (1997).
    [CrossRef]
  8. T. Okamawari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
    [CrossRef]
  9. T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion-compensated optical transmission system,” Opt. Lett. 23, 694–696 (1998).
    [CrossRef]
  10. 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]
  11. V. S. Grigoryan, C. R. Menyuk, and R. M. Mu, “Calculation of amplitude and timing jitter in dispersion-managed optical fiber communications using linearization,” J. Lightwave Technol. 17, 1347–1356 (1999).
    [CrossRef]
  12. J. Santhanam, C. J. McKinstrie, T. I. Lakoba, and G. P. Agrawal, “Effects of precompensation and postcompensation on timing jitter in dispersion-managed systems,” Opt. Lett. 26, 1131–1133 (2001).
    [CrossRef]
  13. C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon–Haus timing jitter in dispersion-managed systems with lumped amplification: analytical approach,” J. Opt. Soc. Am. B 19, 640–649 (2002).
    [CrossRef]
  14. C. J. McKinstrie, “Gordon–Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001).
    [CrossRef]
  15. C. J. McKinstrie and L. F. Mollenauer, “Dependence of Gordon–Haus timing jitter on the ratio of the forward and backward pump powers,” Opt. Lett. 26, 1663–1665 (2001).
    [CrossRef]
  16. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
    [CrossRef] [PubMed]
  17. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  18. 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]
  19. L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
    [CrossRef]
  20. L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
    [CrossRef]
  21. L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gb/s and achievement of 375 Gb/s-Mm, error free, at 12.5 and 15 Gb/s,” Opt. Lett. 19, 704–706 (1994).
    [CrossRef] [PubMed]
  22. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
    [CrossRef]
  23. D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  24. N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
    [CrossRef]
  25. G. M. Carter, J. M. Jacob, C. R. Menyuk, E. A. Golovchenko, and A. N. Pilipetskii, “Timing-jitter reduction for a dispersion-managed soliton system: experimental evidence,” Opt. Lett. 22, 513–515 (1997).
    [CrossRef] [PubMed]
  26. M. Matsumoto and H. A. Haus, “Stretched-pulse optical fiber communications,” IEEE Photon. Technol. Lett. 9, 785–787 (1997).
    [CrossRef]
  27. G. M. Carter and J. M. Jacob, “Dynamics of solitons in filtered dispersion-managed systems,” IEEE Photon. Technol. Lett. 10, 546–548 (1998).
    [CrossRef]
  28. This effect was discovered during a collaboration with L. F. Mollenauer on a different project.
  29. L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
    [CrossRef]
  30. W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
    [CrossRef]
  31. M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
    [CrossRef]

2002 (1)

2001 (4)

C. J. McKinstrie, “Gordon–Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001).
[CrossRef]

C. J. McKinstrie and L. F. Mollenauer, “Dependence of Gordon–Haus timing jitter on the ratio of the forward and backward pump powers,” Opt. Lett. 26, 1663–1665 (2001).
[CrossRef]

J. Santhanam, C. J. McKinstrie, T. I. Lakoba, and G. P. Agrawal, “Effects of precompensation and postcompensation on timing jitter in dispersion-managed systems,” Opt. Lett. 26, 1131–1133 (2001).
[CrossRef]

L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
[CrossRef]

1999 (2)

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

V. S. Grigoryan, C. R. Menyuk, and R. M. Mu, “Calculation of amplitude and timing jitter in dispersion-managed optical fiber communications using linearization,” J. Lightwave Technol. 17, 1347–1356 (1999).
[CrossRef]

1998 (4)

T. Okamawari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion-compensated optical transmission system,” Opt. Lett. 23, 694–696 (1998).
[CrossRef]

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]

G. M. Carter and J. M. Jacob, “Dynamics of solitons in filtered dispersion-managed systems,” IEEE Photon. Technol. Lett. 10, 546–548 (1998).
[CrossRef]

1997 (4)

1996 (1)

N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
[CrossRef]

1994 (1)

1993 (2)

W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

1992 (3)

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

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]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

1991 (2)

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[CrossRef] [PubMed]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

1986 (1)

1983 (1)

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

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Agrawal, G. P.

Anderson, D.

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

Biondini, G.

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Blow, K. J.

W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
[CrossRef]

Carter, G. M.

Charavarty, S.

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Doran, N. J.

N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
[CrossRef]

W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
[CrossRef]

Essiambre, R. J.

Evangelides, S. G.

Forysiak, W.

L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
[CrossRef]

N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
[CrossRef]

W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
[CrossRef]

Golovchenko, E. A.

Gordon, J. P.

Grigoryan, V. S.

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Horne, R.

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Jacob, J. M.

Kodama, Y.

Kumar, S.

Lai, Y.

Lakoba, T. I.

Lederer, F.

Lichtman, E.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Mamyshev, P. V.

Maruta, A.

T. Okamawari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion-compensated optical transmission system,” Opt. Lett. 23, 694–696 (1998).
[CrossRef]

Matsumoto, M.

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

McKinstrie, C. J.

Mecozzi, A.

Menyuk, C. R.

Mollenauer, L. F.

C. J. McKinstrie and L. F. Mollenauer, “Dependence of Gordon–Haus timing jitter on the ratio of the forward and backward pump powers,” Opt. Lett. 26, 1663–1665 (2001).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gb/s and achievement of 375 Gb/s-Mm, error free, at 12.5 and 15 Gb/s,” Opt. Lett. 19, 704–706 (1994).
[CrossRef] [PubMed]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

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]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

Moores, J. D.

Mu, R. M.

Neubelt, M. J.

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gb/s and achievement of 375 Gb/s-Mm, error free, at 12.5 and 15 Gb/s,” Opt. Lett. 19, 704–706 (1994).
[CrossRef] [PubMed]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Nijhof, J. H. B.

L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
[CrossRef]

Nyman, B. M.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Okamawari, T.

T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion-compensated optical transmission system,” Opt. Lett. 23, 694–696 (1998).
[CrossRef]

T. Okamawari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Pilipetskii, A. N.

Richardson, L. J.

L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
[CrossRef]

Santhanam, J.

Smith, N. J.

N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
[CrossRef]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Electron. Lett. (4)

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 5 Gb/s, single-channel, and over more than 11000 km at 10 Gb/s in two-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gb/s, single channel, and over more than 13 Mm at 20 Gb/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

N. J. Smith, W. Forysiak, and N. J. Doran, “Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems,” Electron. Lett. 32, 2085–2086 (1996).
[CrossRef]

W. Forysiak, K. J. Blow, and N. J. Doran, “Reduction of Gordon–Haus jitter by posttransmission dispersion compensation,” Electron. Lett. 29, 1225–1226 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

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

G. M. Carter and J. M. Jacob, “Dynamics of solitons in filtered dispersion-managed systems,” IEEE Photon. Technol. Lett. 10, 546–548 (1998).
[CrossRef]

J. Lightwave Technol. (2)

V. S. Grigoryan, C. R. Menyuk, and R. M. Mu, “Calculation of amplitude and timing jitter in dispersion-managed optical fiber communications using linearization,” J. Lightwave Technol. 17, 1347–1356 (1999).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

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

Opt. Commun. (4)

C. J. McKinstrie, “Gordon–Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001).
[CrossRef]

T. Okamawari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

M. J. Ablowitz, G. Biondini, S. Charavarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

L. J. Richardson, J. H. B. Nijhof, and W. Forysiak, “An interpretation of the energy variations of dispersion-managed solitons in terms of effective average dispersion,” Opt. Commun. 189, 63–67 (2001).
[CrossRef]

Opt. Lett. (11)

J. Santhanam, C. J. McKinstrie, T. I. Lakoba, and G. P. Agrawal, “Effects of precompensation and postcompensation on timing jitter in dispersion-managed systems,” Opt. Lett. 26, 1131–1133 (2001).
[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]

G. M. Carter, J. M. Jacob, C. R. Menyuk, E. A. Golovchenko, and A. N. Pilipetskii, “Timing-jitter reduction for a dispersion-managed soliton system: experimental evidence,” Opt. Lett. 22, 513–515 (1997).
[CrossRef] [PubMed]

T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion-compensated optical transmission system,” Opt. Lett. 23, 694–696 (1998).
[CrossRef]

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]

S. Kumar and F. Lederer, “Gordon–Haus effect in dispersion-managed soliton systems,” Opt. Lett. 22, 1870–1872 (1997).
[CrossRef]

C. J. McKinstrie and L. F. Mollenauer, “Dependence of Gordon–Haus timing jitter on the ratio of the forward and backward pump powers,” Opt. Lett. 26, 1663–1665 (2001).
[CrossRef]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[CrossRef] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

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]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gb/s and achievement of 375 Gb/s-Mm, error free, at 12.5 and 15 Gb/s,” Opt. Lett. 19, 704–706 (1994).
[CrossRef] [PubMed]

Phys. Rev. A (1)

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

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Other (4)

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997).

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

This effect was discovered during a collaboration with L. F. Mollenauer on a different project.

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Figures (8)

Fig. 1
Fig. 1

Pulse chirp, width, and bandwidth plotted as functions of distance for a DM system with lumped amplification.

Fig. 2
Fig. 2

Standard deviations of the frequency and time shifts plotted as functions of distance for systems with lumped amplification. The solid curves represent an ensemble of DM solitons for which the kick strengths were calculated exactly; the dotted-dashed curves represent an ensemble for which the kick strengths were calculated approximately; the dashed curves represent the associated ensemble of CD solitons.

Fig. 3
Fig. 3

Pulse chirp and width plotted as functions of distance for a DM system with distributed amplification. The ratio of the forward and backward pump powers is 0.5.

Fig. 4
Fig. 4

Pulse bandwidth and combination of chirp and residual dispersion plotted as functions of distance for a DM system with distributed amplification. The ratio of the forward and backward pump powers is 0.5.

Fig. 5
Fig. 5

Standard deviations of the frequency and time shifts plotted as functions of distance for systems with distributed amplification. The ratio of the forward and backward pump powers is 0.5. The solid curves represent an ensemble of DM solitons for which the kick strengths were calculated exactly; the dotted–dashed curves represent an ensemble for which the kick strengths were calculated approximately; the dashed curves represent the associated ensemble of CD solitons.

Fig. 6
Fig. 6

Pulse chirp and width plotted as functions of distance for a DM system with distributed amplification. The ratio of the forward and backward pump powers is 0.08.

Fig. 7
Fig. 7

Pulse bandwidth and combination of chirp and residual dispersion plotted as functions of distance for a DM system with distributed amplification. The ratio of the forward and backward pump powers is 0.08.

Fig. 8
Fig. 8

Standard deviations of the frequency and time shifts plotted as functions of distance for systems with distributed amplification. The ratio of the forward and backward pump powers is 0.08. The solid curves represent an ensemble of DM solitons for which the kick strengths were calculated exactly; the dotted–dashed curves represent an ensemble for which the kick strengths were calculated approximately; the dashed curves represent the associated ensemble of solitons.

Equations (98)

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E=-|A|2dt,
T=1E-t|A|2dt,
Ω=i2E-(A*At-AAt*)dt,
dΩ/dz=0,
dT/dz=βΩ,
Ω2i=Ω2i-1+δ Ω2i,
ΩTi=ΩTi-1+dΩ2i-1+δ ΩδTi,
T2i=T2i-1+2dΩTi-1+d2Ω2i-1+δT2i,
A(t, z)=a exp[-iΩt-(1+ic)(t-T)2/2τ2],
δ Ω2=(S/E)[(1+c2)/τ2],
δ ΩδT=(S/E)c,
δT2=(S/E)τ2.
v=τ2/(1+c2),
w=cτ2/(1+c2),
A(t)=a exp[-iΩt-(t-T)2/2(v-iw)].
c=w/v,
τ2=v+w2/v.
F(ω)=exp(-uω2/2),
a+=v-iwu+v-iw1/2 exp-Ω2uv2(u+v)a-,
Ω+=vu+vΩ-,
T+=T--uwu+vΩ-.
q=vu+v1/2 exp-Ω2uvu+v.
q=στ(1+c2+σ2τ2)1/2exp-Ω2τ21+c2+σ2τ2,
r=σ2τ21+c2+σ2τ2,
s=-cτ21+c2+σ2τ2.
Ω2+=r2Ω2-,
ΩT+=r(ΩT-+sΩ2-),
T2+=T2-+2sΩT-+s2Ω2-.
Ω2i-=r2Ω2i-1-+δ Ω2i,
ΩTi-=r[ΩTi-1-+(s+rd)Ω2i-1-]+δ ΩδTi,
T2i-=T2i-1-+2(s+rd)ΩTi-1-+(s+rd)2Ω2i-1-+δT2i.
Ω2n=δ Ω21-r2n1-r2,
ΩTn=δ ΩδT1-rn1-r+δ Ω2d(1-rn)(r-rn)(1-r)(1-r2),
T2n=δT2n+δ ΩδT2d[n(1-r)-(1-rn)](1-r)2+δ Ω2×d2[n(1-r2)-2(1+r)(1-rn)+(1-r2n)](1-r)2(1-r2).
Ω2nδ Ω211-r2,
ΩTnδ ΩδT11-r+δ Ω2dr(1-r)(1-r2),
T2nδT2+δ ΩδT2d1-r+δ Ω2d2(1-r)2n.
δ Ω2S/E,
δ ΩδT(S/E)c,
δT2(S/E)(1+c2).
T2n(S/E){1+[c+(s+rd)/(1-r)]2}n,
T2n(S/E)[1+r2d2/(1-r)2]n.
Ω2i=r2Ω2i-1+δ Ω2i,
ΩTi=r[ΩTi-1+(d-+s+rd+)Ω2i-1]+δ ΩδTi,
T2i=T2i-1+2(d-+s+rd+)ΩTi-1+(d-+s+rd+)2Ω2i-1+δT2i.
s(z)-s(0)+(1-r)d-(z).
df=-ΩTn/Ω2n
T2f=T2n-ΩTn2/Ω2n.
Ω2i=Ω2i-1+Ai,
ΩTi=ΩTi-1+dΩ2i-1+Bi,
T2i=T2i-1+2dΩTi-1+d2Ω2i-1+Ci,
Ai=zi-1ziσa(z)dz,
Bi=zi-1ziσb(z)dz+zi-1ziσa(z)ρ(z, zi)dz,
Ci=zi-1ziσc(z)dz+2zi-1ziσb(z)ρ(z, zi)dz+zi-1ziσa(z)ρ2(z, zi)dz
ρ(z, zi)=zziβ(z)dz.
σa=(Sz/E)[(1+c2)/τ2],
σb=(Sz/E)c,
σc=(Sz/E)τ2,
Ω2i=r2Ω2i-1+r2A-+A+,
ΩTi=r[ΩTi-1+(d-+s+rd+)Ω2i-1]+r[B-+(s+rd+)A-]+B+,
T2i=T2i-1+2(d-+s+rd+)ΩTi-1+(d-+s+rd+)2Ω2i-1+[C-+2(s+rd+)B-+(s+rd+)2A-]+C+.
δ Ω2i=r2zi-1zfσa(z)dz+zfziσa(z)dz,
δ ΩδTi=rzi-1zfσb(z)dz+zfziσb(z)dz+rzi-1zfσa(z)[ρ(z, zf)+s+rρ(zf, zi)]dz+zfziσa(z)ρ(z, zi)dz,
δT2i=zi-1ziσc(z)dz+2zi-1zfσb(z)[ρ(z, zf)+s+rρ(zf, zi)]dz+2zfziσb(z)ρ(z, zi)dz+zi-1zfσa(z)[ρ(z, zf)+s+rρ(zf, zi)]2dz+zfziσa(z)ρ2(z, zi)dz.
δΩ2i=zi-1ziσ(z)dz,
δ ΩδTi=zi-1ziσ(z)[c(z)+ρ(z, zi)]dz,
δT2i=zi-1ziσ(z){1+[c(z)+ρ(z, zi)]2}dz,
T2nzi-1ziσ(z){1+[c(z)+ρ(z, zi)+(s+rd)/(1-r)]2}dzn.
δ Ω2i=zi-1ziσ(z)dz,
δ ΩδTi=zi-1ziσ(z)dzc(zi),
δT2i=zi-1ziσ(z)dz[1+c2(zi)].
T2nzi-1ziσ(z)dz[1+r2d2/(1-r)2]n.
dzΩ=-κΩ,
dzT=βΩ,
Ω2i=Ω2i-1 exp(-2κl)+δ Ω2i,
ΩTi=[ΩTi-1+ρ(zi-1, zi)Ω2i-1]×exp(-κl)+δ ΩTi,
T2i=T2i-1+2ρ(zi-1, zi)ΩTi-1+ρ2(zi-1, zi)Ω2i-1+δT2i,
ρ(zi-1, zi)=zi-1ziβ(z)exp[-κ(z-zi-1)]dz
δ Ω2i=zi-1ziσa(z)exp[-2κ(zi-z)]dz,
δ ΩδTi=zi-1zi[σb(z)+σa(z)ρ(z, zi)]×exp[-κ(zi-z)]dz,
δT2i=zi-1zi[σc(z)+2σb(z)ρ(z, zi)+σa(z)ρ2(z, zi)]dz.
ρ(zi-1, zi)-c(zi)τm2[1-exp(-κl)]+d exp(-κl),
Ω2(zi, zj)=σa(zi)dzi exp[-2κ(zj-zi)],
ΩT(zi, zj)=[σb(zi)dzi+σa(zi)dziρ(zi, zj)]×exp[-κ(zj-zi)],
T2(zi, zj)=σc(zi)dzi+2σb(zi)dziρ(zi, zj)+σa(zi)dziρ2(zi, zj),
ρ(zi, zj)=zizjβ(z)exp[-κ(z-zi)]dz.
Ω2n=0znσa(z)exp[-2κ(zn-z)]dz,
ΩTn=0zn[σb(z)+σa(z)ρ(z, zn)]×exp[-κ(zn-z)]dz,
T2n=0zn[σc(z)+2σb(z)ρ(z, zn)+σa(z)ρ2(z, zn)]dz.
dzΩ2=-2κΩ2+σa,
dzΩT=-κΩT+βΩ2+σb,
dzT2=2βΩT+σc.
dζU=σu,
dζV=U+σv,
dζW=2V+σw.
U(ζn)=0ζnσu(ζ)dζ,
V(ζn)=0ζn[σv(ζ)+σu(ζn-ζ)]dζ,
W(ζn)=0ζn[σw(ζ)+2σv(ζn-ζ)+σu(ζ)(ζn-ζ)2]dζ.

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