Abstract

Four-wave mixing in wavelength-division-multiplexed soliton systems with damping and amplification is studied. An analytical model is introduced that explains the dramatic growth of the four-wave terms. The model yields a resonance condition relating the soliton frequency and the amplifier distance. It correctly predicts all essential features regarding the resonant growth of the four-wave contributions.

© 1996 Optical Society of America

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References

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  1. M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).
  2. P. V. Mamyshev, L. F. Mollenauer, Opt. Lett. 21, 396 (1996).
    [CrossRef] [PubMed]
  3. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
    [CrossRef]
  4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  5. A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
    [CrossRef] [PubMed]
  6. L. F. Mollenauer, S. G. Evangelides, H. A. Haus, J. Lightwave Technol. 9, 194 (1991).
    [CrossRef]
  7. S. Chakravarty, M. J. Ablowitz, J. R. Sauer, R. B. Jenkins, Opt. Lett. 20, 136 (1995).
    [CrossRef] [PubMed]

1996

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

P. V. Mamyshev, L. F. Mollenauer, Opt. Lett. 21, 396 (1996).
[CrossRef] [PubMed]

1995

1991

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

1990

Ablowitz, M. J.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

S. Chakravarty, M. J. Ablowitz, J. R. Sauer, R. B. Jenkins, Opt. Lett. 20, 136 (1995).
[CrossRef] [PubMed]

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

Biondini, G.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

Chakravarty, S.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

S. Chakravarty, M. J. Ablowitz, J. R. Sauer, R. B. Jenkins, Opt. Lett. 20, 136 (1995).
[CrossRef] [PubMed]

Evangelides, S. G.

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

Hasegawa, A.

Haus, H. A.

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

Jenkins, R. B.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

S. Chakravarty, M. J. Ablowitz, J. R. Sauer, R. B. Jenkins, Opt. Lett. 20, 136 (1995).
[CrossRef] [PubMed]

Kodama, Y.

Mamyshev, P. V.

Mollenauer, L. F.

P. V. Mamyshev, L. F. Mollenauer, Opt. Lett. 21, 396 (1996).
[CrossRef] [PubMed]

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

Sauer, J. R.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

S. Chakravarty, M. J. Ablowitz, J. R. Sauer, R. B. Jenkins, Opt. Lett. 20, 136 (1995).
[CrossRef] [PubMed]

Segur, H.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

J. Lightwave Technol.

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

Opt. Lett.

Prog. Appl. Math. Rep.

M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, J. R. Sauer, Prog. Appl. Math. Rep. 283, 1 (1996).

Other

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

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Figures (3)

Fig. 1
Fig. 1

Two-soliton collision in the presence of damping and periodic amplification. Γ = 10, za = 0.2, Ω1 = − Ω2 = − 3, A1 = A2 = 1, and T1 = − T2 = 5.

Fig. 2
Fig. 2

Fourier spectrum relative to the collision shown in Fig. 1.

Fig. 3
Fig. 3

Amplitude and frequency of the secondary maxima of | q ˆ | for Ω = 12 as a function of the amplifier distance. Solid curves, numerical results from Eq. (1); dashed curves, theoretical predictions from expression (10).

Equations (10)

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iq z + 1 2 q tt + | q | 2 q = i Γ q + i [ exp ( Γ z a ) 1 ] n = 1 N δ ( z nz a ) q ,
A z + Γ A [ exp ( Γ z a ) 1 ] n = 1 N δ ( z nz a ) A = 0 ,
iu z + 1 2 u tt + g ( z ) | u | 2 u = 0 ,
iu z + 1 2 u tt + g ( z ) u 2 2 u 1 * = 0 ,
iF z + 1 2 F tt + i ω 4 F t ( 1 2 ω 4 2 k 4 ) F + g ( z ) × u 2 2 u 1 * exp [ i ( k 4 z ω 4 t ) ] = 0 .
iF z ( 1 2 ω 4 2 k 4 ) F = g ( z ) u 2 2 u 1 * exp [ i ( k 4 z ω 4 t ) ] .
z a = 4 / ( 2 Ω 2 + 1 ) .
i F ˆ z [ 1 2 ( ω + ω 4 ) 2 k 4 ] F ˆ + g ( z ) exp ( ik 4 z ) ω [ u 2 2 ( z , t ) u 1 * ( z , t ) exp ( i ω 4 t ) ] = 0 .
F ˆ ( z , ω ) = i π Ω sech ( πω / 2 ) exp [ 1 2 ( ω 2 + 3 Ω + 2 Ω 2 ) z ] × n = + g n Ω z d ζ exp [ i γ n ( ω ) ζ ] I ( ζ , ω ) ,
F ˆ ( z , ω ) ~ z i π 2 Ω exp [ 1 / 2 ( ω 2 + 3 Ω ω + 2 Ω 2 ) z ] × n = + n g n [ γ n ( ω ) + ω ] sinh { 1 / 2 π [ γ n ( ω ) + ω ] } cosh [ 1 / 2 π γ n ( ω ) ] .

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