Abstract

We numerically study the initial-value problem of the nonlinear Schrödinger equation in the normal-dispersion regime of an optical fiber. A nonchirped hyperbolic tangent input pulse having arbitrary amplitude is found to evolve into a primary dark soliton having a constant amplitude and speed. The effect of the input amplitude is to alter the pulse width of the primary dark soliton. In addition, a set of secondary dark solitons of smaller amplitude moving away from the primary pulse is also generated. It is also shown that nonlinear dark pulses in optical fibers are more stable than bright pulses with respect to loss and noise.

© 1989 Optical Society of America

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  1. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973) ,Appl. Phys. Lett. 23, 171 (1973) .
    [CrossRef]
  2. J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974);V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
    [CrossRef]
  3. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).
  4. S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
    [CrossRef]
  5. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.
  6. L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
    [CrossRef]
  7. K. J. Blow, N. J. Doran, Phys. Lett. 107A, 55 (1985).
  8. A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1146 (1981).
    [CrossRef]
  9. W. Zhao, E. Bourkoff, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper WF44.

1986 (1)

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

1985 (1)

K. J. Blow, N. J. Doran, Phys. Lett. 107A, 55 (1985).

1981 (1)

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1146 (1981).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
[CrossRef]

1974 (1)

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974);V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
[CrossRef]

1973 (2)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973) ,Appl. Phys. Lett. 23, 171 (1973) .
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Blow, K. J.

K. J. Blow, N. J. Doran, Phys. Lett. 107A, 55 (1985).

Bourkoff, E.

W. Zhao, E. Bourkoff, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper WF44.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Doran, N. J.

K. J. Blow, N. J. Doran, Phys. Lett. 107A, 55 (1985).

Gordon, J. P.

L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1146 (1981).
[CrossRef]

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973) ,Appl. Phys. Lett. 23, 171 (1973) .
[CrossRef]

Hawkins, R. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Kirchner, E. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Kodama, Y.

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1146 (1981).
[CrossRef]

Leaird, D. E.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Mollenauer, L. F.

L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
[CrossRef]

Satsuma, J.

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974);V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
[CrossRef]

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Stolen, R. L.

L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973) ,Appl. Phys. Lett. 23, 171 (1973) .
[CrossRef]

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Tomlinson, W. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Weiner, A. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

Yajima, N.

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974);V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Zhao, W.

W. Zhao, E. Bourkoff, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper WF44.

Appl. Phys. Lett. (1)

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973) ,Appl. Phys. Lett. 23, 171 (1973) .
[CrossRef]

Phys. Lett. (2)

L. F. Mollenauer, R. L. Stolen, J. P. Gordon, Phys. Lett. 45, 1045 (1980).
[CrossRef]

K. J. Blow, N. J. Doran, Phys. Lett. 107A, 55 (1985).

Proc. IEEE (1)

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1146 (1981).
[CrossRef]

Progr. Theor. Phys. Suppl. (1)

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974);V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Sov. Phys. Usp. (1)

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Other (2)

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, W. J. Tomlinson, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD24.

W. Zhao, E. Bourkoff, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper WF44.

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Figures (3)

Fig. 1
Fig. 1

Dark-pulse propagation for different initial amplitudes u0. (a) N = 4, α = 0.3; (b) N = 3, α = 0; (c) N = 2, α = 0.5. The pulse shapes at increasing propagation distance are shown.

Fig. 2
Fig. 2

Effect of the fiber loss on the pulse width. The ratio of FWHM at z to that at the input for the dark pulse (solid curve) increases more slowly than that for the bright pulse (dotted curve) when γ = 0.05, κ = 1.

Fig. 3
Fig. 3

Effect of both loss and noise. When both fiber losses (γ = 0.2) and background noise (2% rms) are included, the input pulses (dotted curves) and output pulses (solid curves) at z = 1 for both bright and dark pulses are plotted as a function of normalized time.

Equations (11)

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i ( u z + γ u ) = ( D / 2 ) u t t | u | 2 u ,
u b ( z , t ) = κ sech ( κ t ) exp ( i κ 2 z / 2 ) , D = 1 ,
u d ( z , t ) = κ tanh ( κ t ) exp ( i κ 2 z ) , D = 1 ,
u b ( 0 , t ) = u 0 sech ( t ) ,
u d ( 0 , t ) = u 0 tanh ( t ) ,
κ n = ± ( N n α ) for n = 1 , 2 ,…, N 1 ,
u d n ( z , t ) = e i z ( λ n i ν n ) 2 + ν n exp [ 2 ν ( t t n 0 λ n z ) ] 1 + ν n exp [ 2 ν ( t t n 0 λ n z ) ]
| u d n ( z , t ) | 2 = 1 ν n 2 sech 2 [ ν n ( t t n 0 λ n z + 1 2 ν n ln ν n ) ] ,
E d ( z ) = + ( | u ( z , t ) | 2 u 0 2 ) d t .
E d ( ) = + ( N α ) 2 sech 2 ( N α ) d t 2 n = 1 N 1 + κ n 2 sech 2 κ n t d t = 2 ( N α ) 2 2 α ( 1 α ) .
E ( z ) = E 0 e 2 γ z .

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