Abstract

The use of guided-wave Mach–Zehnder interferometers to generate dark solitons with constant background is further examined. Under optimal conditions, both the input optical power and the driving voltage can be reduced by 30% as compared with the case of complete modulation. Dark solitons are also found to experience compression through amplification. When the gain coefficient is small, adiabatic amplification is possible. Besides being used for loss compensation, Raman amplification can be used as the gain mechanism for adiabatic amplification. The frequency and time shifts caused by intrapulse stimulated Raman scattering are both found to be smaller than those for bright solitons by a factor of 2. Finally, the propagation properties of even dark pulses are described quantitatively.

© 1992 Optical Society of America

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  1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 5, 364–372 (1972).
  2. V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).
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    [CrossRef]
  4. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
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  5. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 172 (1973).
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    [CrossRef]
  7. P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
    [CrossRef]
  8. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse,” Sov. Phys. Usp. 29, 642–677 (1986).
    [CrossRef]
  9. L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London Ser. A 315, 437–450 (1985).
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  10. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  11. K. J. Blow, N. J. Doran, and D. Wood, “Generation and stabilization of short soliton pulses in amplified nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 5, 381–391 (1988).
    [CrossRef]
  12. A. Hasegawa, “Amplification and reshaping of optical solitons in a glass fiber—IV: Use of the stimulated Raman process,” Opt. Lett. 8, 650–652 (1983).
    [CrossRef] [PubMed]
  13. N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
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  14. A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,” Appl. Opt. 23, 3302–3309 (1984).
    [CrossRef] [PubMed]
  15. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
    [CrossRef]
  16. L. F. Mollenauer and K. Smith, “Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain,” Opt. Lett. 13, 675–677 (1988).
    [CrossRef] [PubMed]
  17. L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
    [CrossRef]
  18. D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
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    [CrossRef]
  29. L. F. Mollenauer, R. H. Stolen, and M. N. Islam, “Experimental demonstration of soliton propagation in long fibers: loss compensated by Raman gain,” Opt. Lett. 10, 229–231 (1985).
    [CrossRef] [PubMed]
  30. S. A. Gredeskul and Y. S. Kivshar, “Dark soliton generation in optical fibers,” Opt. Lett. 14, 1281–1283 (1989).
    [CrossRef] [PubMed]
  31. W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. QE-24, 365–372 (1988).
    [CrossRef]
  32. F. M. Mitschke and L. F. Mollenauer, “Discovery of soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  33. J. P. Gordon, “Theory of soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  34. A. W. Weiner, R. N. Thurston, W. J. Tomlinson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, “Temporal and spectral self-shifts of dark optical solitons,” Opt. Lett. 14, 868–870 (1989).
    [CrossRef] [PubMed]
  35. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  36. Y. S. Kivshar, “Perturbation-induced dynamics of small-amplitude dark optical solitons,” Opt. Lett. 15, 1273–1275 (1990);Y. S. Kivshar and V. V. Afanasjev, “Decay of dark solitons due to the stimulated Raman effect,” Opt. Lett. 16, 285–287 (1991).
    [CrossRef] [PubMed]
  37. K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
    [CrossRef]
  38. K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
    [CrossRef] [PubMed]
  39. W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, and R. N. Thurston, “Dark optical solitons with finite-width background pulses,” J. Opt. Soc. Am. B 6, 329–334 (1989).
    [CrossRef]
  40. K. J. Blow and N. J. Doran, “Multiple dark soliton solutions of the nonlinear Schrodinger equation,” Phys. Lett. A 107, 55–58 (1985).
    [CrossRef]

1990 (4)

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

W. Zhao and E. Bourkoff, “Generation of dark solitons under cw background using waveguide EO modulators,” Opt. Lett. 15, 405–407 (1990).
[CrossRef] [PubMed]

J. A. Giannini and R. I. Joseph, “The propagation of bright and dark solitons in lossy optical fibers,” IEEE J. Quantum Electron. QE-26, 2109–2114 (1990).
[CrossRef]

Y. S. Kivshar, “Perturbation-induced dynamics of small-amplitude dark optical solitons,” Opt. Lett. 15, 1273–1275 (1990);Y. S. Kivshar and V. V. Afanasjev, “Decay of dark solitons due to the stimulated Raman effect,” Opt. Lett. 16, 285–287 (1991).
[CrossRef] [PubMed]

1989 (8)

1988 (8)

1987 (1)

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

1986 (4)

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse,” Sov. Phys. Usp. 29, 642–677 (1986).
[CrossRef]

F. M. Mitschke and L. F. Mollenauer, “Discovery of soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
[CrossRef] [PubMed]

J. P. Gordon, “Theory of soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

1985 (3)

L. F. Mollenauer, R. H. Stolen, and M. N. Islam, “Experimental demonstration of soliton propagation in long fibers: loss compensated by Raman gain,” Opt. Lett. 10, 229–231 (1985).
[CrossRef] [PubMed]

L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London Ser. A 315, 437–450 (1985).
[CrossRef]

K. J. Blow and N. J. Doran, “Multiple dark soliton solutions of the nonlinear Schrodinger equation,” Phys. Lett. A 107, 55–58 (1985).
[CrossRef]

1984 (1)

1983 (3)

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1974 (1)

J. Satruma and N. Yajima, “Initial value problems of one-dimensional self-phase modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–305 (1974).
[CrossRef]

1973 (3)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 172 (1973).

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 5, 364–372 (1972).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 5.

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse,” Sov. Phys. Usp. 29, 642–677 (1986).
[CrossRef]

Barthelemy, A.

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Bekki, N.

Blow, K. J.

K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
[CrossRef]

K. J. Blow, N. J. Doran, and D. Wood, “Generation and stabilization of short soliton pulses in amplified nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 5, 381–391 (1988).
[CrossRef]

K. J. Blow and N. J. Doran, “Multiple dark soliton solutions of the nonlinear Schrodinger equation,” Phys. Lett. A 107, 55–58 (1985).
[CrossRef]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[CrossRef]

Bourkoff, E.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse,” Sov. Phys. Usp. 29, 642–677 (1986).
[CrossRef]

Cohen, L. G.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

Doran, N. J.

K. J. Blow, N. J. Doran, and D. Wood, “Generation and stabilization of short soliton pulses in amplified nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 5, 381–391 (1988).
[CrossRef]

K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
[CrossRef]

K. J. Blow and N. J. Doran, “Multiple dark soliton solutions of the nonlinear Schrodinger equation,” Phys. Lett. A 107, 55–58 (1985).
[CrossRef]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[CrossRef]

Emplit, P.

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Evangelides, S. G.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

Froehly, C.

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Giannini, J. A.

J. A. Giannini and R. I. Joseph, “The propagation of bright and dark solitons in lossy optical fibers,” IEEE J. Quantum Electron. QE-26, 2109–2114 (1990).
[CrossRef]

Giuliani, G.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Gordon, J. P.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef]

J. P. Gordon, “Theory of soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Gredeskul, S. A.

Grischkowsky, D.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Halas, N. J.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Hamaide, J. P.

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
[CrossRef] [PubMed]

A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,” Appl. Opt. 23, 3302–3309 (1984).
[CrossRef] [PubMed]

A. Hasegawa, “Amplification and reshaping of optical solitons in a glass fiber—IV: Use of the stimulated Raman process,” Opt. Lett. 8, 650–652 (1983).
[CrossRef] [PubMed]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 172 (1973).

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Haus, H. A.

Hawkins, R. J.

Heritage, J. P.

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and M. N. Islam, “Experimental demonstration of soliton propagation in long fibers: loss compensated by Raman gain,” Opt. Lett. 10, 229–231 (1985).
[CrossRef] [PubMed]

Joseph, R. I.

J. A. Giannini and R. I. Joseph, “The propagation of bright and dark solitons in lossy optical fibers,” IEEE J. Quantum Electron. QE-26, 2109–2114 (1990).
[CrossRef]

Kirschner, E. M.

Kivshar, Y. S.

Krökel, D.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Leaird, D. E.

Mitschke, F. M.

Mollenauer, L. F.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

L. F. Mollenauer and K. Smith, “Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain,” Opt. Lett. 13, 675–677 (1988).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

F. M. Mitschke and L. F. Mollenauer, “Discovery of soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and M. N. Islam, “Experimental demonstration of soliton propagation in long fibers: loss compensated by Raman gain,” Opt. Lett. 10, 229–231 (1985).
[CrossRef] [PubMed]

L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London Ser. A 315, 437–450 (1985).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

Reynaud, R.

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Satruma, J.

J. Satruma and N. Yajima, “Initial value problems of one-dimensional self-phase modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–305 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 5, 364–372 (1972).

Simpson, J. R.

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 16, 1203–1205 (1990).
[CrossRef]

Smith, K.

Stolen, R. H.

Tai, K.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 172 (1973).

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Thurston, R. N.

Tomlinson, W. J.

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse,” Sov. Phys. Usp. 29, 642–677 (1986).
[CrossRef]

Weiner, A. M.

Weiner, A. W.

Wood, D.

Yajima, N.

J. Satruma and N. Yajima, “Initial value problems of one-dimensional self-phase modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–305 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 5, 364–372 (1972).

Zhao, W.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 172 (1973).

IEEE J. Quantum Electron. (4)

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[CrossRef]

W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. QE-24, 365–372 (1988).
[CrossRef]

J. A. Giannini and R. I. Joseph, “The propagation of bright and dark solitons in lossy optical fibers,” IEEE J. Quantum Electron. QE-26, 2109–2114 (1990).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

P. Emplit, J. P. Hamaide, R. Reynaud, C. Froehly, and A. Barthelemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun. 62, 374–379 (1987).
[CrossRef]

Opt. Lett. (16)

A. Hasegawa, “Amplification and reshaping of optical solitons in a glass fiber—IV: Use of the stimulated Raman process,” Opt. Lett. 8, 650–652 (1983).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

W. Zhao and E. Bourkoff, “Propagation properties of dark solitons,” Opt. Lett. 14, 703–705 (1989).
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[CrossRef] [PubMed]

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Phys. Lett. A (1)

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Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 5.

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Figures (9)

Fig. 1
Fig. 1

Dark solitons generated by the waveguide Mach–Zehnder interferometer. The amplitude of the input cw light is chosen to be a = π/2 for (a)–(c). The parameter δ is (a) 0.8, (b) 0.5, and (c) 0.2. (d) Is the case of optimal operation for which a = 1.33 and δ = 0.7. In all cases, the output pulse shapes are plotted as solid curves, and the dashed curves are input pulse shapes. The pulses shown here are at a propagation distance of z = 4.

Fig. 2
Fig. 2

Dark solitons under constant gain: output pulse shapes (solid curves) when (a) Γ = 0.05 and (b) Γ = 1, after propagation distance Γz = 1.6, compared with input pulse shapes (dashed curves); (c) pulse duration relative to its input as a function of Γz at various values of Γ. The solid curve is the adiabatic approximation obtained by the perturbation method. Three values of Γ are used: Γ = 0.05 (dotted curve), Γ = 0.2 (dashed–dotted curve), and Γ = 1 (dashed curve). Negative Γz depicts loss.

Fig. 3
Fig. 3

Pulse shapes of amplified dark solitons: (a) δ = 0.5, β = 2 In 1.05, ΓpL = 2, after 8 amplifying cycles (solid curve); (b) δ = 0.5, β = 2 In 1.02, ΓpL = 2, after 16 amplifying cycles (solid curve); (c) δ = 0.5, β = 2 In 1.02, ΓpL = 0.5, after 16 amplifying cycles (solid curve); (d) input pulse is the same as in Fig. 1(c), β = 2 In 1.05, after 8 amplifying cycles (solid curve). The input pulse shapes are plotted as dashed curves.

Fig. 4
Fig. 4

Shape of a fundamental dark soliton after a propagation distance of 40 (solid curve). Normalized time delay τd = 0.01. The dashed curve is the input pulse shape. (b) Trace of the soliton (solid curve) as a function of propagation distance for the situation described by (a). The dotted curve represents the case for a fundamental bright soliton under similar conditions.

Fig. 5
Fig. 5

Shape of a higher-order dark soliton, 2 tanh(t), after a propagation distance of 12 for τd = 0.01 (solid curve). The dotted curve is the pulse if τd = 0, i.e., without ISRS.

Fig. 6
Fig. 6

Shape of an adiabatically amplified fundamental dark soliton (solid curve); Γ = 0.05, z = 16, and τd = 0.01. The dotted curve corresponds to the pulse shape without ISRS. (b) The trace of the soliton (solid curve) for the case of (a). The dotted curve is a fit as described by Eq. (11) in the text.

Fig. 7
Fig. 7

Even dark pulses when the input pulse (dashed curve) is κ0|tanh t|: (a) κ0 = 1.56 and z = 8 (solid curve), (b) κ0 = 4 and z = 3.75 (solid curve). In (c) three different input pulses are assumed: 8|tanh t| (solid curve), 8[1 − exp(−t2/τg2)]1/2 (dotted curve), and 8[1 − sech(t/τs)] (dashed curve). The propagation distance is z = 8.

Fig. 8
Fig. 8

Even dark pulses generated from MZI. The pulse after MZI is 2 cos(π/2 sech2t) (dashed curve), and the shape of secondary dark solitons is shown by the solid curve for z = 4.

Fig. 9
Fig. 9

Loss-compensated even dark pulses. The input pulse is 2 cos(π/2 sech2t) (dotted curve). The secondary solitons with fiber losses compensated for by stimulated Raman scattering are shown by the solid curve. The pulse shape without fiber losses is shown by the dashed curve (same as Fig. 8) for comparison. The propagation distance is z = 4.

Tables (1)

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Table 1 Amplitudes of Secondary Even Dark Pulses

Equations (18)

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u = ( 2 π n 2 τ 0 2 λ A eff | β 2 | ) 1 / 2 A ,
L D = τ 0 2 β 2 .
u ( 0 , t ) = a sin [ δ π / 2 tanh ( t ) ] ,
i u z 1 / 2 u tt + | u | 2 u = i Γ u ,
t = t e Γ z ,
z = e 2 Γ z 1 2 Γ ,
u = ν e Γ z .
i ν z 1 2 υ t t + | υ | 2 υ = Γ t 2 Γ z + 1 υ t .
u ( z , t ) = exp ( i e 2 Γ z 1 2 Γ ) e Γ z tanh ( t e Γ z ) .
Γ = g ( e 2 Γ p z + e 2 Γ p ( L z ) ) Γ s ,
g = Γ p ( Γ s + β ) L sinh ( Γ p L ) e Γ p L ,
κ ( z ) = κ 0 exp ( β z ) ,
i u z 1 2 u t t + | u | 2 u = τ d | u | 2 t u ,
d ω d z = 4 τ d 15 κ 4 ,
d θ d z = ω ,
θ = τ d 60 Γ 2 ( e 4 Γ z 1 4 Γ z ) .
κ n = κ 0 Δ n ,
u n ( z , t ) = κ 0 ( λ n i ν n ) 2 + ν n exp [ 2 ν n ( t t n 0 λ n z ) ] 1 + ν n exp [ 2 ν n ( t t n 0 λ n z ) ] e iz ,

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