Abstract

For a relatively long optical pulse in a fiber with a dispersion distance z0 much larger than the loss distance, a soliton cannot exist in an ideal sense. However, with a proper choice of the initial amplitude and amplifier distance za, a nonlinear pulse (a guiding-center soliton) propagates like a soliton over a distance much larger than the dispersion distance when it is periodically amplified at distances much shorter than the dispersion distance. The guiding-center soliton is shown to satisfy the nonlinear Schrödinger equation with a correction of order (za/z0)2. Numerical examples supported by analytical results are presented for distortionless propagation of the guiding-center solitons with a pulse width of 40 psec in a dispersion-shifted fiber of D = 1 psec/(nm-km).

© 1990 Optical Society of America

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References

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  1. A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. Y. Kodama, A. Hasegawa, Opt. Lett. 7, 339 (1982).
    [CrossRef] [PubMed]
  3. L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
    [CrossRef]
  4. The term guiding center is taken from the guiding-center motion of a charged particle in a magnetic field, where the slow motion of the center of gyromotion of the particle is derived by averaging over the rapid gyrofrequency. The behavior of a guiding-center soliton is derived by using a similar averaging method; see, for examples, R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983) and M. S. Chance, Phys. Fluids 27, 2455 (1984).
    [CrossRef]
  5. The detailed structure of the solution for Q˜ may be calculated by the Lie transformation; details are in A. Hasegawa, Y. Kodama, “The guiding center solitons,” Phys. Rev. Lett. (to be published).
  6. L. F. Mollenauer, AT&T Bell Laboratories, Holmdel, N.J. (personal communication). However, this choice is unnecessary because for a choice of different value of Ao one can still obtain a one-soliton solution with a proper choice of the initial pulse width.
  7. K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
    [CrossRef]
  8. A. Hasegawa, Appl. Opt. 23, 3302 (1984).
    [CrossRef] [PubMed]
  9. E. Shiojiri, Y. Fujii, Appl. Opt. 24, 358 (1985).
    [CrossRef] [PubMed]
  10. H. Kubota, M. Nakazawa, IEEE J. Quantum Electron. 26, 692 (1990).
    [CrossRef]

1990 (1)

H. Kubota, M. Nakazawa, IEEE J. Quantum Electron. 26, 692 (1990).
[CrossRef]

1986 (1)

L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

1985 (1)

1984 (1)

1983 (1)

The term guiding center is taken from the guiding-center motion of a charged particle in a magnetic field, where the slow motion of the center of gyromotion of the particle is derived by averaging over the rapid gyrofrequency. The behavior of a guiding-center soliton is derived by using a similar averaging method; see, for examples, R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983) and M. S. Chance, Phys. Fluids 27, 2455 (1984).
[CrossRef]

1982 (2)

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Y. Kodama, A. Hasegawa, Opt. Lett. 7, 339 (1982).
[CrossRef] [PubMed]

1973 (1)

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

Blow, K. J.

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Doran, N. J.

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Fujii, Y.

Gordon, J. P.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Appl. Opt. 23, 3302 (1984).
[CrossRef] [PubMed]

Y. Kodama, A. Hasegawa, Opt. Lett. 7, 339 (1982).
[CrossRef] [PubMed]

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

The detailed structure of the solution for Q˜ may be calculated by the Lie transformation; details are in A. Hasegawa, Y. Kodama, “The guiding center solitons,” Phys. Rev. Lett. (to be published).

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

Kodama, Y.

Y. Kodama, A. Hasegawa, Opt. Lett. 7, 339 (1982).
[CrossRef] [PubMed]

The detailed structure of the solution for Q˜ may be calculated by the Lie transformation; details are in A. Hasegawa, Y. Kodama, “The guiding center solitons,” Phys. Rev. Lett. (to be published).

Kubota, H.

H. Kubota, M. Nakazawa, IEEE J. Quantum Electron. 26, 692 (1990).
[CrossRef]

Littlejohn, R. G.

The term guiding center is taken from the guiding-center motion of a charged particle in a magnetic field, where the slow motion of the center of gyromotion of the particle is derived by averaging over the rapid gyrofrequency. The behavior of a guiding-center soliton is derived by using a similar averaging method; see, for examples, R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983) and M. S. Chance, Phys. Fluids 27, 2455 (1984).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

L. F. Mollenauer, AT&T Bell Laboratories, Holmdel, N.J. (personal communication). However, this choice is unnecessary because for a choice of different value of Ao one can still obtain a one-soliton solution with a proper choice of the initial pulse width.

Nakazawa, M.

H. Kubota, M. Nakazawa, IEEE J. Quantum Electron. 26, 692 (1990).
[CrossRef]

Shiojiri, E.

Tappert, F. D.

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

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (2)

H. Kubota, M. Nakazawa, IEEE J. Quantum Electron. 26, 692 (1990).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, M. N. Islam, IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

J. Plasma Phys. (1)

The term guiding center is taken from the guiding-center motion of a charged particle in a magnetic field, where the slow motion of the center of gyromotion of the particle is derived by averaging over the rapid gyrofrequency. The behavior of a guiding-center soliton is derived by using a similar averaging method; see, for examples, R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983) and M. S. Chance, Phys. Fluids 27, 2455 (1984).
[CrossRef]

Opt. Commun. (1)

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Opt. Lett. (1)

Other (2)

The detailed structure of the solution for Q˜ may be calculated by the Lie transformation; details are in A. Hasegawa, Y. Kodama, “The guiding center solitons,” Phys. Rev. Lett. (to be published).

L. F. Mollenauer, AT&T Bell Laboratories, Holmdel, N.J. (personal communication). However, this choice is unnecessary because for a choice of different value of Ao one can still obtain a one-soliton solution with a proper choice of the initial pulse width.

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Figures (2)

Fig. 1
Fig. 1

Magnitude of q for a pair of guiding-center solitons shown at multiples of 500 km when they are amplified at distances of (a) 50 km and (b) 100 km. The enhanced level of |q| at 500 km in (b) compared with that at Z = 0 is considered to be the effect of the dispersive wave as well as that of Q ˜ in Eq. (26), although the numerical result is somewhat larger than the theoretical value of |q| in Eq. (26).

Fig. 2
Fig. 2

Magnitude of q for a pair of a0 = 1 solitons. The pair behaves like linear pulses.

Equations (26)

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i q Z + 1 2 2 q T 2 + | q | 2 q = i Γ q .
q = g λ E / ,
Z = 2 z / λ ,
T = ( t z / υ g ) / ( λ k ) 1 / 2 ,
k = λ 2 D / ( 2 πc ) .
Γ = γλ / 2 .
q = η sech η ( T + κZ ) exp [ iκT + i 2 ( η 2 κ 2 ) Z ] .
τ s P 0 = 2 . 9 λ 3 / 2 | D | S ,
= 1 . 76 ( λ k ) 1 / 2 / τ s ,
Γ = γ τ s 2 3 . 1 ( k ) = γ z 0 ,
z 0 = τ s 2 3 . 1 ( k ) .
i q Z + 1 2 2 q d T 2 + | q | 2 q = i Γ q + [ exp ( Γ Z a ) 1 ] × n = 1 N δ ( Z n Z a ) q ,
q ( Z , T ) = a ( Z ) Q ( Z , T ) ,
d a d Z = Γ a + [ exp ( Γ Z a ) 1 ] n = 1 N δ ( Z n Z a ) a .
i Q d Z + 1 2 2 Q T 2 + a ( Z ) 2 | Q | 2 Q = 0 .
Q = Q 0 + Q ˜ ,
Q ˜ 1 Z a 0 Z a Q ˜ ( Z ) d Z = 0 .
a 2 ( Z ) = A 0 + A ˜ ( Z ) , with A ˜ = 0 .
A 0 = a 2 ( Z ) = a 0 2 2 Γ Z a [ 1 exp ( 2 Γ Z a ) ] ,
i Q 0 Z + 1 2 2 Q 0 T 2 + A 0 | Q 0 | 2 Q 0 + nonlinear terms of Q ˜ = 0 .
Q ˜ = i A ˜ 1 ( Z ) | Q 0 | 2 Q 0 + O ( Z a 2 ) ,
d A ˜ 1 ( Z ) d Z = A ˜ ( Z ) , with A ˜ 1 = 0 ,
A ˜ 1 ( Z ) = [ 1 2 Γ + Z a 2 Z Z a exp ( 2 Γ Z ) 1 exp ( 2 Γ Z a ) ] A 0 ,
| A ˜ 1 ( Z ) | Z a A 0 / 2 .
a 0 = [ 2 Γ Z a 1 exp ( 2 Γ Z a ) ] 1 / 2 .
q ( Z , T ) = a 0 exp ( Γ Z ˆ ) ( Q 0 + Q ˜ ) for 0 < Z ˆ < Z a ,

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