Abstract

Collision-induced timing jitter is theoretically and numerically studied for two-channel wavelength-division-multiplexed optical soliton transmission systems with both weak and strong (practical) dispersion management. For a weakly managed case the timing jitter induced by the residual frequency shift strongly depends on bit rate per channel owing to the correlation in the relative time shift in neighboring pulses. We also discuss the effect of dispersion management for reducing the timing jitter. For a strongly managed case the timing jitter is mainly induced either by the accumulated position shift or by an incomplete-collision-induced residual frequency shift. It becomes significant for nearly zero path-averaged dispersion. We optimize the path-averaged dispersion of the line composed of 1.3-µm zero-dispersion single-mode fiber and its reversed dispersion fiber by minimizing the jitter caused by both collision and amplifier noise.

© 2001 Optical Society of America

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  1. D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.
  2. K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.
  3. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
    [CrossRef]
  4. A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21, 39–41 (1996).
    [CrossRef] [PubMed]
  5. S. Wabnitz, “Stabilization of sliding-filtered soliton wavelength division multiplexing transmissions by dispersion-compensating fibers,” Opt. Lett. 21, 638–640 (1996).
    [CrossRef] [PubMed]
  6. S. Kumar, Y. Kodama, and A. Hasegawa, “Optimal dispersion management schemes for WDM soliton systems,” Electron. Lett. 33, 459–461 (1997).
    [CrossRef]
  7. W. Forysiak, J. F. L. Devaney, N. J. Smith, and N. J. Doran, “Dispersion management for wavelength-division-multiplexed soliton transmission,” Opt. Lett. 22, 600–602 (1997).
    [CrossRef] [PubMed]
  8. H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton-WDM systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
    [CrossRef]
  9. J. F. L. Devaney, W. Forysiak, A. M. Niculae, and N. J. Doran, “Soliton collisions in dispersion-managed wavelength-division-multiplexed systems,” Opt. Lett. 22, 1695–1697 (1997).
    [CrossRef]
  10. Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
    [CrossRef]
  11. A. Mecozzi, “Timing jitter in wavelength-division-multiplexed filtered soliton transmission,” J. Opt. Soc. Am. B 15, 152–161 (1998).
    [CrossRef]
  12. H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
    [CrossRef]
  13. T. Hirooka and A. Hasegawa, “Chirped soliton interaction in strongly dispersion-managed wavelength-division-multiplexing systems,” Opt. Lett. 23, 768–770 (1998).
    [CrossRef]
  14. M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
    [CrossRef]
  15. A. M. Niculae, W. Forysiak, A. J. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
    [CrossRef]
  16. D. J. Kaup, B. A. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
    [CrossRef]
  17. H. Sugahara, A. Maruta, and Y. Kodama, “Optimal alloca-tion of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system,” Opt. Lett. 24, 145–147 (1999).
    [CrossRef]
  18. P. V. Mamyshev and L. F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. 24, 448–450 (1999).
    [CrossRef]
  19. H. Sugahara and A. Maruta, “Timing jitter of a strongly dispersion managed soliton in a wavelength-division-multiplexed system,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThA4.
  20. H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
    [CrossRef]
  21. D. J. Kaup, B. A. Malomed, and J. Yang, “Collision-induced pulse timing jitter in a wavelength-division-multiplexing system with strong dispersion management,” J. Opt. Soc. Am. B 16, 1628–1635 (1999).
    [CrossRef]
  22. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, Oxford, UK, 1995).
  23. L. F. Mollenauer, “Method for nulling nonrandom timing jitter in soliton transmission,” Opt. Lett. 21, 384–386 (1996).
    [CrossRef] [PubMed]
  24. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
    [CrossRef]
  25. Y. Kodama, S. Kumar, and A. Maruta, “Chirped nonlinear pulse propagation in a dispersion-compensated system,” Opt. Lett. 22, 1689–1691 (1997).
    [CrossRef]
  26. D. Anderson, “Variational approach to non-linear pulse-propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  27. S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
    [CrossRef]
  28. 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]

1999 (4)

1998 (7)

A. Mecozzi, “Timing jitter in wavelength-division-multiplexed filtered soliton transmission,” J. Opt. Soc. Am. B 15, 152–161 (1998).
[CrossRef]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

T. Hirooka and A. Hasegawa, “Chirped soliton interaction in strongly dispersion-managed wavelength-division-multiplexing systems,” Opt. Lett. 23, 768–770 (1998).
[CrossRef]

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

A. M. Niculae, W. Forysiak, A. J. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
[CrossRef]

D. J. Kaup, B. A. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (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]

1997 (6)

1996 (4)

1995 (1)

S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
[CrossRef]

1991 (1)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

1983 (1)

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

Ablowitz, M. J.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

Aceves, A. B.

S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
[CrossRef]

Anderson, D.

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

Bennion, I.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Biondini, G.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

Blow, K. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Burgo, S. D.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Chakravarty, S.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

Devaney, J. F. L.

Doran, N. J.

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

Favre, F.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Forysiak, W.

Fujii, K.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Fukuchi, K.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Georges, T.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Gloag, A. J.

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

Grot, D.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Guen, D. L.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Hasegawa, A.

T. Hirooka and A. Hasegawa, “Chirped soliton interaction in strongly dispersion-managed wavelength-division-multiplexing systems,” Opt. Lett. 23, 768–770 (1998).
[CrossRef]

S. Kumar, Y. Kodama, and A. Hasegawa, “Optimal dispersion management schemes for WDM soliton systems,” Electron. Lett. 33, 459–461 (1997).
[CrossRef]

A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21, 39–41 (1996).
[CrossRef] [PubMed]

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

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

Henry, M.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Hirooka, T.

Horne, R. L.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

Inada, Y.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Inoue, T.

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[CrossRef]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Ito, T.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Kakui, M.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Kato, H.

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton-WDM systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

Kaup, D. J.

Knox, F. M.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Kodama, Y.

H. Sugahara, A. Maruta, and Y. Kodama, “Optimal alloca-tion of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system,” Opt. Lett. 24, 145–147 (1999).
[CrossRef]

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[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]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton-WDM systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

S. Kumar, Y. Kodama, and A. Hasegawa, “Optimal dispersion management schemes for WDM soliton systems,” Electron. Lett. 33, 459–461 (1997).
[CrossRef]

Y. Kodama, S. Kumar, and A. Maruta, “Chirped nonlinear pulse propagation in a dispersion-compensated system,” Opt. Lett. 22, 1689–1691 (1997).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21, 39–41 (1996).
[CrossRef] [PubMed]

S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
[CrossRef]

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

Kumar, S.

Malomed, B. A.

Mamyshev, P. V.

Maruta, A.

H. Sugahara, A. Maruta, and Y. Kodama, “Optimal alloca-tion of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system,” Opt. Lett. 24, 145–147 (1999).
[CrossRef]

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[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]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Y. Kodama, S. Kumar, and A. Maruta, “Chirped nonlinear pulse propagation in a dispersion-compensated system,” Opt. Lett. 22, 1689–1691 (1997).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

H. Sugahara and A. Maruta, “Timing jitter of a strongly dispersion managed soliton in a wavelength-division-multiplexed system,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThA4.

Mecozzi, A.

Mollenauer, L. F.

Moulinard, M. L.

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

Niculae, A. M.

Nijhof, J. H. B.

Okamawari, T.

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]

Sasaki, A.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Shikii, S.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Shitomi, T.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Smith, N. J.

W. Forysiak, J. F. L. Devaney, N. J. Smith, and N. J. Doran, “Dispersion management for wavelength-division-multiplexed soliton transmission,” Opt. Lett. 22, 600–602 (1997).
[CrossRef] [PubMed]

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Sugahara, H.

H. Sugahara, A. Maruta, and Y. Kodama, “Optimal alloca-tion of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system,” Opt. Lett. 24, 145–147 (1999).
[CrossRef]

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[CrossRef]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton-WDM systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

H. Sugahara and A. Maruta, “Timing jitter of a strongly dispersion managed soliton in a wavelength-division-multiplexed system,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThA4.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Tsuzaki, T.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

Wabnitz, S.

S. Wabnitz, “Stabilization of sliding-filtered soliton wavelength division multiplexing transmissions by dispersion-compensating fibers,” Opt. Lett. 21, 638–640 (1996).
[CrossRef] [PubMed]

S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
[CrossRef]

Yang, J.

Electron. Lett. (4)

S. Kumar, Y. Kodama, and A. Hasegawa, “Optimal dispersion management schemes for WDM soliton systems,” Electron. Lett. 33, 459–461 (1997).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton-WDM systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

J. Lightwave Technol. (2)

H. Sugahara, H. Kato, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[CrossRef]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

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

Opt. Commun. (2)

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. Chakravarty, and R. L. Horne, “On timing jitter in wavelength-division multiplexed soliton systems,” Opt. Commun. 150, 305–318 (1998).
[CrossRef]

Opt. Fiber Technol.: Mater., Devices Syst. (1)

S. Wabnitz, Y. Kodama, and A. B. Aceves, “Control of optical soliton interactions,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 187–217 (1995).
[CrossRef]

Opt. Lett. (12)

L. F. Mollenauer, “Method for nulling nonrandom timing jitter in soliton transmission,” Opt. Lett. 21, 384–386 (1996).
[CrossRef] [PubMed]

Y. Kodama, S. Kumar, and A. Maruta, “Chirped nonlinear pulse propagation in a dispersion-compensated system,” Opt. Lett. 22, 1689–1691 (1997).
[CrossRef]

A. M. Niculae, W. Forysiak, A. J. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
[CrossRef]

D. J. Kaup, B. A. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
[CrossRef]

H. Sugahara, A. Maruta, and Y. Kodama, “Optimal alloca-tion of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system,” Opt. Lett. 24, 145–147 (1999).
[CrossRef]

P. V. Mamyshev and L. F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. 24, 448–450 (1999).
[CrossRef]

T. Hirooka and A. Hasegawa, “Chirped soliton interaction in strongly dispersion-managed wavelength-division-multiplexing systems,” Opt. Lett. 23, 768–770 (1998).
[CrossRef]

A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21, 39–41 (1996).
[CrossRef] [PubMed]

S. Wabnitz, “Stabilization of sliding-filtered soliton wavelength division multiplexing transmissions by dispersion-compensating fibers,” Opt. Lett. 21, 638–640 (1996).
[CrossRef] [PubMed]

J. F. L. Devaney, W. Forysiak, A. M. Niculae, and N. J. Doran, “Soliton collisions in dispersion-managed wavelength-division-multiplexed systems,” Opt. Lett. 22, 1695–1697 (1997).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

W. Forysiak, J. F. L. Devaney, N. J. Smith, and N. J. Doran, “Dispersion management for wavelength-division-multiplexed soliton transmission,” Opt. Lett. 22, 600–602 (1997).
[CrossRef] [PubMed]

Phys. Rev. A (1)

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

Other (4)

D. L. Guen, S. D. Burgo, M. L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier span,” in Optical Fiber Communications Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4.

K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, and A. Hasegawa, “1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely dispersion-managed transmission line employing 1.55/1.58-µm hybrid repeaters,” presented at the 25th European Conference on Optical Communication, Nice, France, September 26–30, 1999, paper PD2-10.

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

H. Sugahara and A. Maruta, “Timing jitter of a strongly dispersion managed soliton in a wavelength-division-multiplexed system,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThA4.

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Figures (11)

Fig. 1
Fig. 1

Standard deviations of absolute timing jitter after 3000-km propagation versus minimum pulse separation, which corresponds to the reciprocal of the bit rate per channel [constant dispersion case, map (a)]. The insets show the probability density function of arrival time and the bit-error-rate versus time acceptance window for 50-ps minimum pulse separation (20 Gbits/s/channel) and the tf=25 ps case.

Fig. 2
Fig. 2

Standard deviations of absolute timing jitter after 3000-km propagation versus minimum pulse separation, which corresponds to the reciprocal of the bit rate per channel [dispersion-managed cases, maps (b) and (c)].

Fig. 3
Fig. 3

Standard deviations of relative timing jitter after 3000-km propagation versus minimum pulse separation, which corresponds to the reciprocal of the bit rate per channel [constant dispersion case, map (a)]. The insets show the probability density function of arrival time for 40-ps and 50-ps minimum pulse separation (25 Gbits/s/channel and 20 Gbits/s/channel) and tf=tint/2 cases.

Fig. 4
Fig. 4

Standard deviations of relative timing jitter after 3000-km propagation versus minimum pulse separation, which corresponds to the reciprocal of the bit rate per channel [dispersion-managed cases, maps (b) and (c)].

Fig. 5
Fig. 5

Standard deviations of timing jitter versus transmission distance for constant dispersion line [map (a)] with tint=80 ps. The worst (best) tf conditions of tf=38.72 ps (46.64 ps) are chosen.

Fig. 6
Fig. 6

Standard deviations of timing jitter versus transmission distance for constant dispersion and dispersion-managed lines [maps (a) and (c)] with tint=70 ps and tf=35 ps.

Fig. 7
Fig. 7

Time shift after 9000-km propagation versus initial separation of colliding pulses.

Fig. 8
Fig. 8

Standard deviations of absolute timing jitter versus path-averaged dispersion after 3000-km, 6000-km, and 9000-km propagation. The insets show the probability density function of arrival time and the bit-error-rate versus the time acceptance window after 9000-km propagation for the Dav=0.1 ps/nm/km case.

Fig. 9
Fig. 9

Standard deviations of relative-timing jitter and Gordon–Haus jitter versus path-averaged dispersion after 3000-km and 9000-km propagation. The insets show the probability density function of arrival time after 9000-km propagation for the Dav=0.01 ps/nm/km and 0.1 ps/nm/km cases, respectively.

Fig. 10
Fig. 10

Standard deviations of timing jitter versus transmission distance for Dav=0.01 ps/nm/km and 0.1 ps/nm/km with tf=25 ps (10 Gbits/s/channel).

Fig. 11
Fig. 11

Eye patterns of the received soliton train after 3000-km propagation for Dav=0.01 ps/nm/km (left figures) and 0.1 ps/nm/km (right figures) with tf=tint/2=25 ps (20 Gbits/s/channel). The upper figures show the single-channel cases, and the lower figures show the two-channel WDM cases.

Tables (2)

Tables Icon

Table 1 System Model for Weakly Dispersion-Managed Line

Tables Icon

Table 2 System Model for Strongly Dispersion-Managed Line

Equations (73)

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iQz+k0(z) Qt-k0(z)2 2Qt2+ν(z)|Q|2Q
=-iγ(z)Q,
T=τt0,Z=zz0,q=QP0,
i qZ+d(Z)2 2qT2+s(Z)|q|2q=-iΓ(Z)q,
d(Z)=-k0(Z)z0t02,s(Z)=ν(Z)P0z0,
Γ(Z)=γ(z)z0.
zDz0=t02|k0|,
d(Z)=Dˆ(Z)[ps/nm/km]|Dˆ|[ps/nm/km],
s(Z)=π2250
×cˆ[×108m/s]nˆ2(Z)[×10-20m/W2]Pˆ0[W](tˆ0[ps])2(λˆ[µm])3Aˆeff (Z)[(µm)2]|Dˆ|[ps/nm/km],
Γ(Z)=π loge 10100 cˆ[×108m/s]αˆ[dB/km](tˆ0[ps])2(λˆ[µm])2|Dˆ|[ps/nm/km].
zNLz0=λAeff2πP0|n2|
d(Z)=250π2
×(λˆ[µm])3Dˆ(Z)[ps/nm/km]Aˆeff[(µm)2]cˆ[×108m/s]|nˆ2|[×10-20m/W2](tˆ0[ps])2Pˆ0[mW],
s(Z)=nˆ2(Z)[×10-20m/W2]|nˆ2|[×10-20m/W2] Aˆeff[(µm)2]Aˆeff (Z)[(µm)2],
Γ(Z)=5 loge 102π αˆ(Z)[dB/km]λˆ[µm]Aˆeff[(µm)2]|nˆ2|[×10-20m/W2]Pˆ0[mW].
a(Z)=a0 exp-nZdZΓ(ζ)dζ for0<Z-nZd<Zd,
i uZ+d(Z)2 2uT2+S(Z)|u|2u=0,
δTa=c=0N-1bcδT(Zc),
σa2=(δTa)2-(δTa)2.
PDF=12πσa exp-(T-δTa)22σa2.
BER10=erfc(αTint/2σa2),
δTr,k=c=0N-1δbc,kδT(Zc),
σr2=k=12-kσr,k2=k=12-k(δTr,k)2.
BER10=erfc(αTint/2σr2).
i uZ+12 2uT2+Sd(Z)|u|2u=0,
d(δκ)dZ=2Sd(Z)E0 -|u3-l|2 |ul|2T dT,
d(ΔTc)dZ=-[ΔB+2δκ(Z)],
ul=Al sech[Al(T-Tl)]exp[-iκl(T-Tl)+iθl],
δT=-ΔTc2-0Z ΔB2 dZ
δT(Zc)=(L-Zc)n=1an sin(ϕn,c)-n=1anπΔB coth(ϕn,c)-3Zd2πn×cos(ϕn,c)-4(ΔB)2,
an=16Zdπ3 xn4n sinh2 xn|αn|,
ϕn,c=2πn ZcZd+arg(αn),
xn=π2n/(ZdΔB),
αn=1Zd 0ZdS(Z)exp-i 2πnZd 0Zd(ζ)dζdZ.
δT(Zc)=(L-Zc)δκres(Zc),
δTa=c=0N-1bc(L-Zc)δκres(Zc)=c=0N-1bc[L-(Zf+cZs)]δκres(Zf+cZs),
δTa=12 c=0N-1[L-(Zf+cZs)]δκres(Zf+cZs)
(δTa)2=12 c=0N-1[L-(Zf+cZs)]2[δκres(Zf+cZs)]2+14 c1,c2=1c1c2N-1[L-(Zf+c1Zs)]×[L-(Zf+c2Zs)]×δκres(Zf+c1Zs)δκres(Zf+c2Zs).
σa2(Zf)=14 c=0N-1[L-(Zf+cZs)]2[δκres(Zf+cZs)]2.
σa2=L312Zs n=1an2ψ(n, c, n, c)+n1,n2=1n1n2an1an2ψ(n1, c, n2, c),
ψ(n1, c1, n2, c2)=1Zd 0Zd sin(ϕn1, c1)sin(ϕn2, c2)dZ.
σa2=L324Zs n=1an2.
δTr,k=c=0N-1δbc,k[L-(Zf+cZs)]δκres(Zf+cZs).
σr,k2(Zf)=(δTr,k)2=12 c=0N-1[L-(Zf+cZs)]2[δκres(Zf+cZs)]2-14 c=0N-1[L-(Zf+cZs)]×{L-[Zf+(c+k)Zs]}×δκres(Zf+cZs)δκres[Zf+(c+k)Zs].
σr2(Zf)=k=12-kσr,k2(Zf).
σr,k2=L36Zs n=1an2ψ(n, c, n, c)+n1,n2=1n1n2an1an2ψ(n1, c, n2, c)-L36Zs-kL24n=1an2ψ(n, c, n, c+k)+n1,n2=1n1n2an1an2ψ(n1, c, n2, c+k).
σr,k2=n=1an2L312Zs-L312Zs-kL28ψcs,
ψcs=1Zd 0Zd cos2πnZd ZcZc+kZsd(Z)dZdZc.
σr2=k=12-kn=1an2L312Zs-L312Zs-kL28ψcs.
c=0k-1bc[L-(Zf+cZs)]δκres(Zf+cZs)
-c=N-kN-1bc+k[L-(Zf+cZs)]δκres(Zf+cZs).
ul=Al exp-pl22(T-Tl)2expiCl2pl2(T-Tl)2-κl(T-Tl)+θl,
dpdZ=-p3Cd(Z),
dCdZ=p2(1+C2)d(z)-S(Z)E0p2π 1+2(1-x2)exp-x22,
d(δκ)dZ=2S(Z)E0p2x2π exp-x22,
d(ΔTc)dZ=-[ΔB+2δκ(Z)]d(Z),
δT(Tsep)=ΔTc(L)2-0L ΔB2d(Z)dZ,
δTa=n=-bnδT(Tf+nTint).
δTr,k=n=-(bn-bn+k)δT(Tf+nTint).
f(n, m, k)=1Zd 0Zd cos[Θn(2Zc)+Θm(2Zc+k)+θ0]dZ,
f(n, m, k)=cos{Θn(d1Z¯1)+Θm[d1(Z1+Z˜s)]+θ0}sin[Θn+m(d1Z¯1)]Θn+m(d1Zd)+cos{Θn[d1(Z¯1+Z1)]+Θm(2d1Z1+d2Z˜s)+θ0}sin[Θn(d1Z˜s)+Θm(d2Z˜s)]Θn(d1Zd)+Θm(d2Zd)+cos{Θn(2d1Z1+d2Z¯2)+Θm[2d1Z1+d2(Z2+Z˜s)]+θ0}sin[Θn+m(d2Z¯2)]Θn+m(d2Zd)+cos[Θn(2davZd-d2Z˜s)+Θm(2davZd+d1Z˜s)+θ0]sin[Θn(d2Z˜s)+Θm(d1Z˜s)]Θn(d2Zd)+Θm(d1Zd);
f(n, m, k)=cos{Θn(d1Z1)+Θm[2d1Z1+d2(Z˜s-Z¯1)]+θ0}sin[Θn(d1Z1)+Θm(d2Z1)]Θn(d1Zd)+Θm(d2Zd)+cos{Θn(2d1Z1+d2Z¯2)+Θm[2d1Z1+d2(Z2+Z˜s)]+θ0}sin[Θn+m(d2Z¯2)]Θn+m(d2Zd)+cos{Θn[2davZd+d2(Z¯1-Z˜s)]+Θm(2davZd+d1Z1)+θ0}sin[Θn(d2Z1)+Θm(d1Z1)]Θn(d2Zd)+Θm(d1Zd)-cos{Θn(2davZd+d2Z¯1)+Θm[4davZd-d2(Zd+Z¯2)]+θ0}sin[Θn+m(d2Z¯1)]Θn+m(d2Zd);
f(n, m, k)=cos{Θn(d1Z¯1)+Θm[d1(Z1+Z˜s)]+θ0}sin[Θn+m(d1Z¯1)]Θn+m(d1Zd)+cos{Θn[d1(2Z¯1+Z2)]+Θm(2d1Z1+d2Z2)+θ0}sin[Θn(d1Z2)+Θm(d2Z2)]Θn(d1Zd)+Θm(d2Zd)-cos{Θn[d1(2Z1+Z¯2)]+Θm(2davZd-d1Z¯2)+θ0}sin[Θn+m(d1Z¯2)]Θn+m(d1Zd)+cos{Θn(2davZd-d2Z2)+Θm[2davZd-d1(Z¯2-Z˜s)]+θ0}sin[Θn(d2Z2)+Θm(d1Z2)]Θn(d2Zd)+Θm(d1Zd);
f(n, m, k)=cos{Θn[d1(Z1+Z¯2)]+Θm[2davZd-d2(Z2+Z¯1)]+θ0}sin[Θn(d1Z¯d)+Θm(d2Z¯d)]Θn(d1Zd)+Θm(d2Zd)-cos{Θn[d1(2Z1+Z¯2)]+Θm(2davZd-d1Z¯2)+θ0}sin[Θn+m(d1Z¯2)]Θn+m(d1Zd)+cos{Θn[2davZd+d2(Z¯1-Z2)]+Θm[2davZd+d1(Z1-Z¯2)]+θ0}sin[Θn(d2Z¯d)+Θm(d1Z¯d)]Θn(d2Zd)+Θm(d1Zd)-cos{Θn(2davZd+d2Z¯1)+Θm[4davZd-d2(Zd+Z¯2)]+θ0}sin[Θn+m(d2Z¯1)]Θn+m(d2Zd).
f(n,-n, k)=Z¯1 cos[Θn(2d1Z˜s)]+Z¯2 cos[Θn(2d2Z˜s)]Zd+cos{Θn[(d1+d2)Z˜s]}sin[Θn(ΔDZ˜s)]Θn(ΔDZd/2);
f(n,-n, k)=Z¯2 cos[Θn(2d2Z˜s)]-Z˜1 cos{Θn[2(ΔDZ1+d2Z˜s)]}Zd+cos[Θn(ΔDZ1+2d2Z˜s)]sin[Θn(ΔDZ1)]Θn(ΔDZd/2);
f(n,-n, k)=Z¯1 cos[Θn(2d1Z˜s)]-Z¯2 cos{Θn[2(d1Z˜s-ΔDZ2)]}Zd+cos[Θn(ΔDZ2-2d1Z˜s)]sin[ΔDZ2]Θn(ΔDZd/2);
f(n,-n, k)=-Z¯1 cos{Θn[2(d2Z˜s+ΔDZ1)]}+Z¯2 cos{Θn[2(d1Z˜s-ΔDZ2)]}Zd+cos{Θn[ΔD(Z1-Z2)+(d1+d2)Z˜s]}sin[Θn(ΔDZ¯d)]Θn(ΔDZd/2).
ψ(n, c, n, c)=[1-f(n, n, 0)]/2,
ψ(n1, c, n2, c)=[f(n1,-n2, 0)-f(n1, n2, 0)]/2,
ψ(n, c, n, c+k)=[f(n,-n, k)-f(n, n, k)]/2,
ψ(n1, c, n2, c+k)=[f(n1,-n2, k)-f(n1, n2, k)]/2.

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