Abstract

We introduce a model in which completely stable propagation of dispersion-managed (DM) solitons is provided by periodic coupling of the system fiber’s core to short segments of a parallel lossy core. Considering the model in the distributed approximation, we identified a vast stability domain in its parameter space. We also obtained a set of DM-strength–power characteristics for stable solitons at fixed values of the DM period. We found that, as for the filtered single-core DM model, stable transmission of the solitons is possible at virtually all the values of the DM strength, whereas the normalized peak power of the pulse must exceed a certain minimum value. Close to this minimum, we found a stable transmission regime with moderate values of the DM strength, S3, and a DM period of approximately 1.5–2 soliton periods (defined as for the homogeneous fiber), which may be promising for reducing soliton–soliton interactions. We also found cases in which the soliton transmission remained stable in the corresponding strongly discrete (lumped) model as well as at normal average dispersion.

© 2000 Optical Society of America

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References

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  1. A. Hasegawa, ed., New Trends in Optical Soliton Transmission Systems (Kluwer Academic, Dordrecht, The Netherlands, 1998).
  2. Feature on fundamental challenges in ultrahigh-capacity optical fiber communication systems, IEEE J. Quantum Electron 34, 2053–2108 (1998).
  3. S. Kumar and F. Lederer, “Gordon–Haus effect in dispersion-managed soliton systems,” Opt. Lett. 22, 1870–1872 (1997); A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997); T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion compensated optical transmission system,” Opt. Lett. OPLEDP 23, 694–696 (1998); B. A. Malomed, “Suppression of soliton jitter and interactions by means of dispersion management,” Opt. Commun. OPCOB8 147, 157–162 (1998).
    [CrossRef]
  4. B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
    [CrossRef]
  5. H. A. Haus, Y. Lai, A. Mecozzi, and J. D. Moores, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991); 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]
  6. M. Matsumoto, “Analysis of filter control of dispersion-managed soliton transmission,” J. Opt. Soc. Am. B 15, 2831–2837 (1998); “Instability of dispersion-managed solitons in a system with filtering,” Opt. Lett. 23, 1901–1903 (1998).
    [CrossRef]
  7. L. F. Mollenauer, P. V. Mamyshev, and J. P. Gordon, “Effect of guiding filters on the behavior of dispersion-managed solitons,” Opt. Lett. 24, 220–222 (1999).
    [CrossRef]
  8. A. Berntson and B. A. Malomed, “Dispersion-management with filtering,” Opt. Lett. 24, 507–509 (1999).
    [CrossRef]
  9. B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
    [CrossRef]
  10. J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component systems,” Phys. Rev. E 54, 4371–4374 (1996).
    [CrossRef]
  11. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
    [CrossRef]
  12. A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
    [CrossRef]
  13. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, 1995).
  14. P. L. Chu, G. D. Peng, B. A. Malomed, H. Hatami-Hanza, and I. Skinner, “Time domain soliton filter based on a semidissipative dual-core coupler,” Opt. Lett. 20, 1092–1094 (1995).
    [CrossRef] [PubMed]
  15. M. C. Cross and P. C. Hohenberg, “Pattern formation out of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
    [CrossRef]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1995).
  17. J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
    [CrossRef]
  18. 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, 900–903 (1998).
    [CrossRef]
  19. T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
    [CrossRef]
  20. S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed solitons in optical amplifier transmission systems with zero average dispersion,” Opt. Lett. 23, 682–684 (1998); J. N. Kutz and S. G. Evangelides, “Dispersion-managed breathers with average normal dispersion,” Opt. Lett. 23, 685–687 (1998).
    [CrossRef]
  21. 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]
  22. I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–330 (1996); B. A. Malomed, “Pulse propagation in a nonlinear-optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997); J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B JOBPDE 15, 87–96 (1998); S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical propagation in dispersion compensated transmission systems,” Opt. Commun. OPCOB8 151, 117–135 (1998).
    [CrossRef]
  23. T. Yu, E. A. Golovchenko, A. N. Pipipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett. 22, 793–795 (1997); T. Georges, “Soliton interaction in dispersion-managed links,” J. Opt. Soc. Am. B 15, 1553–1560 (1998); S. Kumar, M. Wald, F. Lederer, and A. Hasegawa, “Soliton interaction in strongly dispersion managed optical fibers,” Opt. Lett. OPLEDP 23, 1019–1021 (1998); M. Matsumoto, “Analysis of interaction between stretched pulses propagating in dispersion-managed fibers,” IEEE Photon. Technol. Lett. IPTLEL 10, 373–375 (1998).
    [CrossRef] [PubMed]

1999 (2)

1998 (4)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
[CrossRef]

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, 900–903 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

1997 (2)

B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
[CrossRef]

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

1996 (2)

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

1995 (2)

P. L. Chu, G. D. Peng, B. A. Malomed, H. Hatami-Hanza, and I. Skinner, “Time domain soliton filter based on a semidissipative dual-core coupler,” Opt. Lett. 20, 1092–1094 (1995).
[CrossRef] [PubMed]

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]

1993 (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation out of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Berntson, A.

Chu, P. L.

A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
[CrossRef]

P. L. Chu, G. D. Peng, B. A. Malomed, H. Hatami-Hanza, and I. Skinner, “Time domain soliton filter based on a semidissipative dual-core coupler,” Opt. Lett. 20, 1092–1094 (1995).
[CrossRef] [PubMed]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation out of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

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, 900–903 (1998).
[CrossRef]

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

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, 900–903 (1998).
[CrossRef]

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

Gordon, J. P.

Hatami-Hanza, H.

Haus, H. A.

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]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation out of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

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]

Kaup, D. K.

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Knox, F. M.

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

Lakoba, T. I.

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Malomed, B. A.

A. Berntson and B. A. Malomed, “Dispersion-management with filtering,” Opt. Lett. 24, 507–509 (1999).
[CrossRef]

A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
[CrossRef]

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
[CrossRef]

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

P. L. Chu, G. D. Peng, B. A. Malomed, H. Hatami-Hanza, and I. Skinner, “Time domain soliton filter based on a semidissipative dual-core coupler,” Opt. Lett. 20, 1092–1094 (1995).
[CrossRef] [PubMed]

Mamyshev, P. V.

Matera, F.

B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
[CrossRef]

Mollenauer, L. F.

Mostofi, A.

A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
[CrossRef]

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, 900–903 (1998).
[CrossRef]

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

Peng, G. D.

Settembre, M.

B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
[CrossRef]

Skinner, I.

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]

Winful, H. G.

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

Yang, J.

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Electron. Lett. (1)

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion-managed systems 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]

Opt. Commun. (3)

T. I. Lakoba, J. Yang, D. K. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

A. Mostofi, B. A. Malomed, and P. L. Chu, “Effect of coupling strength fluctuations on soliton switching in nonlinear couplers,” Opt. Commun. 145, 274–280 (1998).
[CrossRef]

B. A. Malomed, F. Matera, and M. Settembre, “Reduction of the jitter for return-to-zero signals,” Opt. Commun. 143, 193–198 (1997).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. A (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

Phys. Rev. E (2)

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation out of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Other (10)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1995).

Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, 1995).

S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed solitons in optical amplifier transmission systems with zero average dispersion,” Opt. Lett. 23, 682–684 (1998); J. N. Kutz and S. G. Evangelides, “Dispersion-managed breathers with average normal dispersion,” Opt. Lett. 23, 685–687 (1998).
[CrossRef]

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

M. Matsumoto, “Analysis of filter control of dispersion-managed soliton transmission,” J. Opt. Soc. Am. B 15, 2831–2837 (1998); “Instability of dispersion-managed solitons in a system with filtering,” Opt. Lett. 23, 1901–1903 (1998).
[CrossRef]

A. Hasegawa, ed., New Trends in Optical Soliton Transmission Systems (Kluwer Academic, Dordrecht, The Netherlands, 1998).

Feature on fundamental challenges in ultrahigh-capacity optical fiber communication systems, IEEE J. Quantum Electron 34, 2053–2108 (1998).

S. Kumar and F. Lederer, “Gordon–Haus effect in dispersion-managed soliton systems,” Opt. Lett. 22, 1870–1872 (1997); A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997); T. Okamawari, A. Maruta, and Y. Kodama, “Analysis of Gordon–Haus jitter in a dispersion compensated optical transmission system,” Opt. Lett. OPLEDP 23, 694–696 (1998); B. A. Malomed, “Suppression of soliton jitter and interactions by means of dispersion management,” Opt. Commun. OPCOB8 147, 157–162 (1998).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–330 (1996); B. A. Malomed, “Pulse propagation in a nonlinear-optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997); J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B JOBPDE 15, 87–96 (1998); S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical propagation in dispersion compensated transmission systems,” Opt. Commun. OPCOB8 151, 117–135 (1998).
[CrossRef]

T. Yu, E. A. Golovchenko, A. N. Pipipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett. 22, 793–795 (1997); T. Georges, “Soliton interaction in dispersion-managed links,” J. Opt. Soc. Am. B 15, 1553–1560 (1998); S. Kumar, M. Wald, F. Lederer, and A. Hasegawa, “Soliton interaction in strongly dispersion managed optical fibers,” Opt. Lett. OPLEDP 23, 1019–1021 (1998); M. Matsumoto, “Analysis of interaction between stretched pulses propagating in dispersion-managed fibers,” IEEE Photon. Technol. Lett. IPTLEL 10, 373–375 (1998).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Region of stable transmission of the DM soliton in the dual-core model with zero average dispersion and equal lengths of anomalous- and normal-dispersion segments, L1=L2L. The values of the filtering coefficient in the main core and of the loss coefficient in the additional core are fixed, γ1=0.30 and Γ=1.35, respectively.

Fig. 2
Fig. 2

Set of DM-strength–normalized-power characteristics for the solitons in the dual-core model. The solid curve corresponds to the right border (vertical line) in Fig. 1. The other curves correspond to various constant values of DM half-period L inside the stability domain shown in Fig. 1.

Fig. 3
Fig. 3

Shape of the stable soliton at a position in which it is narrowest. The parameters are γ0=0.65, S=2.62, and P=3.35. For comparison, the shapes of the Gaussian and sech best fits to the soliton’s core are also shown.

Fig. 4
Fig. 4

Example of stable propagation of the soliton in the lumped version of the model; the amplifier and the filter are placed at the midpoint of the anomalous-dispersion segment. In this case the initial pulse is a Gaussian; the waveforms are plotted at every 40th DM period.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

iuz+[(1/2)D(z)-iγ1]uττ+|u|2u-iγ0u+κv=0,
ivz=iΓv+κu=0,
D(z)=D+>0,0<z<L+D-<0,L+<z<L++L-,
S(D+L++|D-|L-)/T2,PP0T2(1.76)2γ1,
LD+=L|D-|1.
γ0<Γ,Γγ0<1.

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