Abstract

We study the nonlinear evolution in optical fibers of a modulationally unstable continuous wave. We find that a good description of the propagation can be analytically obtained by a simple truncated three-wave model. The dynamic viewpoint may give an important physical insight in different processes, such as frequency conversion of modulated signals, all-optical generation of temporal codes, and the onset of spatiotemporal chaos.

© 1991 Optical Society of America

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References

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  1. T. J. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
    [CrossRef]
  2. N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).
  3. K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  4. K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
    [CrossRef]
  5. M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
    [CrossRef] [PubMed]
  6. A. Hasegawa, K. Tai, Opt. Lett. 14, 512 (1989).
    [CrossRef] [PubMed]
  7. N. N. Akhmediev, V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
    [CrossRef]
  8. M. J. Ablowitz, B. M. Herbst, SIAM J. Appl. Math. 50, 339 (1990).
    [CrossRef]
  9. N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).
  10. H. T. Moon, Phys. Rev. Lett. 64, 412 (1990).
    [CrossRef] [PubMed]
  11. E. Infeld, Phys. Rev. Lett. 47, 717 (1981).
    [CrossRef]
  12. G. Cappellini, S. Trillo, J. Opt. Soc. Am. B 8, 824 (1991).
    [CrossRef]
  13. A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
    [CrossRef]

1991 (1)

1990 (3)

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

H. T. Moon, Phys. Rev. Lett. 64, 412 (1990).
[CrossRef] [PubMed]

M. J. Ablowitz, B. M. Herbst, SIAM J. Appl. Math. 50, 339 (1990).
[CrossRef]

1989 (1)

1988 (2)

N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
[CrossRef] [PubMed]

1986 (4)

N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

N. N. Akhmediev, V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
[CrossRef]

1981 (1)

E. Infeld, Phys. Rev. Lett. 47, 717 (1981).
[CrossRef]

1967 (1)

T. J. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, B. M. Herbst, SIAM J. Appl. Math. 50, 339 (1990).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

N. N. Akhmediev, V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
[CrossRef]

Benjamin, T. J.

T. J. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Bishop, A. R.

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

Cappellini, G.

Ercolani, N.

N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).

Feir, J. E.

T. J. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Forest, M. G.

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).

Hasegawa, A.

A. Hasegawa, K. Tai, Opt. Lett. 14, 512 (1989).
[CrossRef] [PubMed]

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Haus, H. A.

M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
[CrossRef] [PubMed]

Herbst, B. M.

M. J. Ablowitz, B. M. Herbst, SIAM J. Appl. Math. 50, 339 (1990).
[CrossRef]

Infeld, E.

E. Infeld, Phys. Rev. Lett. 47, 717 (1981).
[CrossRef]

Jewell, J. L.

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

N. N. Akhmediev, V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
[CrossRef]

McLaughlin, D. W.

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).

Mitskevich, N. V

N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Moon, H. T.

H. T. Moon, Phys. Rev. Lett. 64, 412 (1990).
[CrossRef] [PubMed]

Nakazawa, M.

M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
[CrossRef] [PubMed]

Overman, A. E.

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
[CrossRef] [PubMed]

Tai, K.

A. Hasegawa, K. Tai, Opt. Lett. 14, 512 (1989).
[CrossRef] [PubMed]

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Tomita, A.

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

Trillo, S.

Appl. Phys. Lett. (1)

K. Tai, A. Tomita, J. L. Jewell, A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

J. Fluid Mech. (1)

T. J. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Lett. A (1)

A. R. Bishop, M. G. Forest, D. W. McLaughlin, A. E. Overman, Phys. Lett. A 144, 17 (1990).
[CrossRef]

Phys. Rev. A (1)

M. Nakazawa, K. Suzuki, H. A. Haus, Phys. Rev. A 38, 5193 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

H. T. Moon, Phys. Rev. Lett. 64, 412 (1990).
[CrossRef] [PubMed]

E. Infeld, Phys. Rev. Lett. 47, 717 (1981).
[CrossRef]

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Physica (1)

N. Ercolani, M. G. Forest, D. W. McLaughlin, Physica 18D, 472 (1986); Geometry of the Modulational Instability. Part I: Local Analysis (University of Arizona Preprint, Tucson, Ariz., 1986).

SIAM J. Appl. Math. (1)

M. J. Ablowitz, B. M. Herbst, SIAM J. Appl. Math. 50, 339 (1990).
[CrossRef]

Sov. Phys. JETP (1)

N. N. Akhmediev, V. I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Theor. Math. Phys. (1)

N. N. Akhmediev, V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Evolution of a modulated wave: (a) = 10−2, ϕ0 = 0 (amplitude modulation); (b) = 10−2, ϕ0 = π (frequency modulation); (c) = 10−4, ϕ0 = −π/2 (separatrix).

Fig. 2
Fig. 2

Phase-space portraits for κ = −2: (a) three-wave model, (b) projection from the exact solution of the NLS equation.

Fig. 3
Fig. 3

Normalized intensities in the pump (solid curve), in the n = 1 (dotted–dashed curve), and n = 2 (dotted curve) modes, where κ = −1.

Equations (7)

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i ( u / ξ ) ± ½ ( 2 u / t 2 ) + | u | 2 u = 0 ,
u ( ξ = 0 , t ) = 1 + exp ( i ϕ 0 / 2 ) cos ( Ω t ) ,
u ( ξ , t ) = A 0 ( ξ ) + A 1 ( ξ ) exp ( i Ω t ) + A 1 ( ξ ) exp ( i Ω t )
i ( d A 0 / d ξ ) = ( | A 0 | 2 + 2 | A 1 | 2 + 2 | A 1 | 2 ) A 0 + 2 A 1 A 1 A 0 * , i ( d A 1 / d ξ ) ± ½ Ω 2 A 1 = ( | A 1 | 2 + 2 | A 0 | 2 + 2 | A 1 | 2 ) A 1 + A 1 * A 0 2 , i ( d A 1 / d ξ ) ± ½ Ω 2 A 1 = ( | A 1 | 2 + 2 | A 0 | 2 + 2 | A 1 | 2 ) A 1 + A 1 * A 0 2 .
H = ± 1 2 Ω 2 j j 2 | A j | 2 1 2 j = 1 1 | A j | 4 2 i < j | A i | 2 | A j | 2 ( A 0 * 2 A 1 A 1 + A 0 2 A 1 * A 1 * ) .
η ˙ = H ˆ / ϕ , ϕ ˙ = H ˆ / η ,
H ˆ ( η , ϕ ) = 2 η ( 1 η ) cos ( ϕ ) ( κ 1 ) η 3 2 η 2 .

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