Abstract

We analyze pulse propagation in a nonlinear optical fiber in which linear loss in the fiber is balanced by a chain of periodically spaced, phase-sensitive, degenerate parametric amplifiers. Our analysis shows that no pulse evolution occurs over a soliton period owing to attenuation in the quadrature orthogonal to the amplified quadrature. Evidence is presented that indicates that stable pulse solutions exist on length scales much longer than the soliton period. These pulses are governed by a nonlinear fourth-order evolution equation, which describes the exponential decay of arbitrary initial pulses (within the stability regime) onto stable, steady-state, solitonlike pulses.

© 1993 Optical Society of America

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References

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  1. A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990); Phys. Rev. Lett. 66, 161 (1991).
    [CrossRef] [PubMed]
  2. L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
    [CrossRef]
  3. Y. Kodama, A. Hasegawa, Opt. Lett. 17, 31 (1992); L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, Electron. Lett. 28, 792 (1992).
    [CrossRef] [PubMed]
  4. J. P. Gordon, H. A. Haus, Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  5. H. Yuen, Opt. Lett. 17, 73 (1992).
    [CrossRef] [PubMed]
  6. If the carrier frequency of the evolving signal pulses is ν0, then the pump pulses are at 2ν0.
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  8. J. Kevorklan, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981), Chap. 3.
  9. M. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
    [CrossRef]

1992 (2)

1991 (1)

L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
[CrossRef]

1990 (1)

1986 (1)

1985 (1)

M. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

Cole, J. D.

J. Kevorklan, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981), Chap. 3.

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
[CrossRef]

Gordon, J. P.

Hasegawa, A.

Haus, H. A.

L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
[CrossRef]

J. P. Gordon, H. A. Haus, Opt. Lett. 11, 665 (1986).
[CrossRef] [PubMed]

Kevorklan, J.

J. Kevorklan, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981), Chap. 3.

Kodama, Y.

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
[CrossRef]

Weinstein, M.

M. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

Yuen, H.

J. Lightwave Technol. (1)

L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
[CrossRef]

Opt. Lett. (4)

SIAM J. Math. Anal. (1)

M. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

Other (3)

If the carrier frequency of the evolving signal pulses is ν0, then the pump pulses are at 2ν0.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

J. Kevorklan, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981), Chap. 3.

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Figures (3)

Fig. 1
Fig. 1

Schematic of a nonlinear optical fiber transmission line in which loss is balanced by a chain of periodically spaced, phase-matched, degenerate optical parametric amplifiers (DOPA’s).

Fig. 2
Fig. 2

Example of stable pulse propagation for Γ l = 1 (corresponding to a parametric gain of 2.7), l = 1.5, κ = 1, Γ2 = −0.1, and an initial pulse R(T, 0) = 1.6 sech(T).

Fig. 3
Fig. 3

Physically realizable degenerate optical parametric amplification (DOPA) scheme for use in long-distance fiber-optic transmission.

Equations (9)

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q Z = i 2 2 q 2 T + i q 2 q + 1 h ( Z ) q + 1 f ( Z ) q * .
h ( ζ ) q = - Γ q + [ cosh ( B P z a ) - 1 ] × n = 1 N δ ( ζ - n l ) q ( n l - , T ) ,
f ( ζ ) q * = exp ( i ϕ ) sinh ( B P z a ) n = 1 N δ ( ζ - n l ) q * ( n l - , T ) .
q = q 0 ( ζ , Z , ξ , T ) + q 1 ( ζ , Z , ξ , T ) + .
q 0 ζ = h ( ζ ) q 0 + f ( ζ ) q 0 * .
R Z = 0 ,
q 1 = i exp [ i ϕ ( Z ) / 2 ] 4 Γ sinh Γ l { [ 1 - exp ( - 2 Γ l ) ] R 3 + Γ l ( 2 R T 2 - d ϕ d Z R ) } .
2 l tanh ( Γ l ) R ξ + 1 4 4 R T 4 - κ 2 2 R T 2 + ( κ 2 4 + Γ 2 ) R - β 1 κ R 3 + β 2 R 5 + β 3 R ( R T ) 2 + β 4 R 2 2 R T 2 = 0 ,
L R ˜ = - 2 l tanh ( Γ l ) R ˜ ξ ,

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