Abstract

The modulation instability of an extended nonlinear Schrödinger equation was investigated. It was found that the odd-order higher dispersion [β3] does not contribute to the modulation-instability frequency. The time derivative of the nonlinearity was included and shown to affect the instability. Finally, nonlinear retardation effects are shown to alter the results significantly.

© 1987 Optical Society of America

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  1. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  3. N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981);M. Jain, N. Tzoar, Opt. Lett. 3, 202 (1978).
    [CrossRef] [PubMed]
  4. D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
    [CrossRef]
  5. D. Anderson, Opt. Commun. 48, 107 (1983).
    [CrossRef]
  6. K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
    [CrossRef]
  7. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).
  8. D. N. Christodoulides, R. I. Joseph, Appl. Phys. Lett. 47, 76 (1985).
    [CrossRef]
  9. D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
    [CrossRef]
  10. G. P. Agrawal, M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
    [CrossRef] [PubMed]
  11. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, H. H. Chen, Opt. Lett. 11, 464 (1986).
    [CrossRef] [PubMed]
  12. V. A. Vyshoukh, Sov. J. Quantum Electron. 13, 1113 (1983).
    [CrossRef]
  13. M. J. Potasek, G. P. Agrawal, S. C. Pinault, J. Opt. Soc. Am. B 3, 205 (1986).
    [CrossRef]
  14. T. Taniuti, Progr. Theor. Phys. Suppl. No. 55, 1 (1974);Y. Kodama, J. Stat. Phys. 39, 597 (1985);Y. Kodama, J. Phys. Soc. Jpn. 45, 311 (1978);Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
    [CrossRef]
  15. B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
    [CrossRef]
  16. A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
    [CrossRef]
  17. K. Tai, A. Tomita, A. Hasegawa, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  18. N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).
  19. D. Anderson, M. Lisak, Opt. Lett. 9, 468 (1984).
    [CrossRef] [PubMed]
  20. B. Hermansson, D. Yevick, Opt. Commun. 52, 99 (1984).
    [CrossRef]
  21. P. K. Shukla, J. J. Rasmussen, Opt. Lett. 11, 171 (1986).
    [CrossRef] [PubMed]
  22. D. J. Kaup, A. C. Newell, J. Math. Phys. 19, 798 (1978);E. Molhus, J. Plasma Phys. 19, 437 (1978).
    [CrossRef]
  23. F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
    [CrossRef] [PubMed]
  24. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]
  25. A. Hasegawa, Opt. Lett. 9, 288 (1984).
    [CrossRef] [PubMed]

1986 (7)

1985 (3)

N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

D. N. Christodoulides, R. I. Joseph, Appl. Phys. Lett. 47, 76 (1985).
[CrossRef]

1984 (4)

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

D. Anderson, M. Lisak, Opt. Lett. 9, 468 (1984).
[CrossRef] [PubMed]

B. Hermansson, D. Yevick, Opt. Commun. 52, 99 (1984).
[CrossRef]

A. Hasegawa, Opt. Lett. 9, 288 (1984).
[CrossRef] [PubMed]

1983 (4)

V. A. Vyshoukh, Sov. J. Quantum Electron. 13, 1113 (1983).
[CrossRef]

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

D. Anderson, Opt. Commun. 48, 107 (1983).
[CrossRef]

K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
[CrossRef]

1981 (1)

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981);M. Jain, N. Tzoar, Opt. Lett. 3, 202 (1978).
[CrossRef] [PubMed]

1980 (2)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

1979 (1)

B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
[CrossRef]

1978 (1)

D. J. Kaup, A. C. Newell, J. Math. Phys. 19, 798 (1978);E. Molhus, J. Plasma Phys. 19, 437 (1978).
[CrossRef]

1974 (1)

T. Taniuti, Progr. Theor. Phys. Suppl. No. 55, 1 (1974);Y. Kodama, J. Stat. Phys. 39, 597 (1985);Y. Kodama, J. Phys. Soc. Jpn. 45, 311 (1978);Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

1973 (1)

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

Agrawal, G. P.

Akhmedieva, N. N.

N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).

Anderson, D.

D. Anderson, M. Lisak, Opt. Lett. 9, 468 (1984).
[CrossRef] [PubMed]

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

D. Anderson, Opt. Commun. 48, 107 (1983).
[CrossRef]

Blow, K. J.

K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
[CrossRef]

Brinkman, W. F.

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Chen, H. H.

Christodoulides, D. N.

D. N. Christodoulides, R. I. Joseph, Appl. Phys. Lett. 47, 76 (1985).
[CrossRef]

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

Cummins, E.

K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
[CrossRef]

Dasgupta, B.

B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
[CrossRef]

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

Doran, N. J.

K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
[CrossRef]

Eleonskii, V. M.

N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

Gordon, J. P.

J. P. Gordon, Opt. Lett. 11, 662 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Gupta, M. R.

B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
[CrossRef]

Hasegawa, A.

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

A. Hasegawa, Opt. Lett. 9, 288 (1984).
[CrossRef] [PubMed]

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

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

Hermansson, B.

B. Hermansson, D. Yevick, Opt. Commun. 52, 99 (1984).
[CrossRef]

Jain, M.

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981);M. Jain, N. Tzoar, Opt. Lett. 3, 202 (1978).
[CrossRef] [PubMed]

Joseph, R. I.

D. N. Christodoulides, R. I. Joseph, Appl. Phys. Lett. 47, 76 (1985).
[CrossRef]

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

Kaup, D. J.

D. J. Kaup, A. C. Newell, J. Math. Phys. 19, 798 (1978);E. Molhus, J. Plasma Phys. 19, 437 (1978).
[CrossRef]

Kulagin, N. E.

N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).

Lee, Y. C.

Lisak, M.

D. Anderson, M. Lisak, Opt. Lett. 9, 468 (1984).
[CrossRef] [PubMed]

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Newell, A. C.

D. J. Kaup, A. C. Newell, J. Math. Phys. 19, 798 (1978);E. Molhus, J. Plasma Phys. 19, 437 (1978).
[CrossRef]

Pinault, S. C.

Potasek, M. J.

Prokhorov, A. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

Rasmussen, J. J.

Shukla, P. K.

Som, B. K.

B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
[CrossRef]

Sorkin, V. N.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Tai, K.

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

Taniuti, T.

T. Taniuti, Progr. Theor. Phys. Suppl. No. 55, 1 (1974);Y. Kodama, J. Stat. Phys. 39, 597 (1985);Y. Kodama, J. Phys. Soc. Jpn. 45, 311 (1978);Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

Tappert, F.

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

Tomita, A.

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

Tzoar, N.

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981);M. Jain, N. Tzoar, Opt. Lett. 3, 202 (1978).
[CrossRef] [PubMed]

Vyshoukh, V. A.

V. A. Vyshoukh, Sov. J. Quantum Electron. 13, 1113 (1983).
[CrossRef]

Wai, P. K. A.

Yevick, D.

B. Hermansson, D. Yevick, Opt. Commun. 52, 99 (1984).
[CrossRef]

Appl. Phys. Lett. (2)

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

D. N. Christodoulides, R. I. Joseph, Appl. Phys. Lett. 47, 76 (1985).
[CrossRef]

Electron. Lett. (1)

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

J. Math. Phys. (1)

D. J. Kaup, A. C. Newell, J. Math. Phys. 19, 798 (1978);E. Molhus, J. Plasma Phys. 19, 437 (1978).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

B. K. Som, M. R. Gupta, B. Dasgupta, J. Phys. Soc. Jpn. 47, 1296 (1979).
[CrossRef]

JETP Lett. (1)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Sorkin, JETP Lett. 28, 876 (1985).

Opt. Commun. (3)

D. Anderson, Opt. Commun. 48, 107 (1983).
[CrossRef]

K. J. Blow, N. J. Doran, E. Cummins, Opt. Commun. 48, 181 (1983).
[CrossRef]

B. Hermansson, D. Yevick, Opt. Commun. 52, 99 (1984).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (3)

G. P. Agrawal, M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
[CrossRef] [PubMed]

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981);M. Jain, N. Tzoar, Opt. Lett. 3, 202 (1978).
[CrossRef] [PubMed]

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

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

Progr. Theor. Phys. (1)

T. Taniuti, Progr. Theor. Phys. Suppl. No. 55, 1 (1974);Y. Kodama, J. Stat. Phys. 39, 597 (1985);Y. Kodama, J. Phys. Soc. Jpn. 45, 311 (1978);Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. A. Vyshoukh, Sov. J. Quantum Electron. 13, 1113 (1983).
[CrossRef]

Sov. Phys. JETP (1)

N. N. Akhmedieva, V. M. Eleonskii, N. E. Kulagin, Sov. Phys. JETP 62, 894 (1985).

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Figures (3)

Fig. 1
Fig. 1

ωMI as a function of power for three values of . Solid line, = 0; dashed line, = 2; dashed–dotted line, = 5.

Fig. 2
Fig. 2

γimag as a function of ω for q02 = 0.15. Solid line, = 0; dashed line, = 2; dashed–dotted line, = 2.38.

Fig. 3
Fig. 3

γimag as a function of ω for q02 = 0.15. Solid line, = 0.133,η = 0.066; dashed line, α = 0, η = 0.066.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

i ϕ z 1 2 β ( 2 ) 2 ϕ s 2 + n 2 ω 0 c | ϕ | 2 ϕ i β ( 3 ) 6 3 ϕ s 3 + i a 1 s ( | ϕ | 2 ϕ ) + i ( a 2 + i a 3 ) ϕ s | ϕ | 2 = 0 ,
a 1 = 1 β ( 1 ) c 2 A eff ω [ ω 2 n n 2 | ψ ( r , θ , ω ) | 4 d S ] ,
a 2 = ω 2 β ( 1 ) c 2 A eff n n 2 ω | ψ ( r , θ , ω ) | 4 d S ,
a 3 ϕ ( t ) t d l f ( l s ) | ϕ ( l ) | 2 ,
ξ = z / λ , τ = s / λ β ( 2 ) , q = 2 π n 2 ϕ ,
i q ξ 1 2 2 q τ 2 + | q | 2 q i μ 3 q τ 3 + i ν τ ( | q | 2 q ) + i ( χ + i η ) q τ | q | 2 = 0.
μ = β ( 3 ) 6 λ [ β ( 2 ) ] 3 , ν = K a 1 , χ = K a 2 , η = K a 3 , K = [ λ / 4 β ( 2 ) π 2 n 2 2 ] 1 / 2 .
q ( ξ , t ) = q 0 exp ( i q 0 2 ξ ) .
i a ξ 1 2 2 a τ 2 + q 0 2 exp ( 2 i q 0 2 ξ ) a * + 2 q 0 2 a i μ 3 a τ 3 + i ν [ q 0 2 exp ( 2 i q 0 2 ξ ) a * τ + 2 q 0 2 a τ ] + i ( χ + i η ) [ q 0 2 exp ( 2 i q 0 2 ξ ) a * τ + q 0 2 a τ ] = 0.
( U V ) = ( U 0 V 0 ) exp [ i ( γ ξ ω τ ) ]
| i γ i μ ω 3 i ω q 0 2 ( 3 ν + 2 χ ) ω 2 / 2 ω 2 / 2 + 2 q 0 2 + i 2 η q 0 2 ω i γ + i ω q 0 2 ν + i μ ω 3 | = 0. .
γ 2 γ [ 2 μ ω 3 + ω q 0 2 ( 4 ν + 2 χ ) ] + μ 2 ω 6 + q 0 4 ν ( 3 ν + 2 χ ) ω 2 + μ q 0 2 ( 4 ν + 2 χ ) ω 4 ω 4 4 ω 2 q 0 2 i η q 0 2 ω 3 = 0.
γ = 2 μ ω 3 + q 0 2 ( 4 ν + 2 χ ) ω 2 ± i 2 α + i δ ,
α = 4 q 0 2 ( 1 q 0 2 2 ) ω 2 ω 4 , δ = 4 q 0 2 η ω 3 ,
γ imag = 1 2 [ ( α 2 + δ 2 ) 1 / 2 + α 2 ] 1 / 2 .
γ imag = 1 2 [ 4 q 0 2 ( 1 q 0 2 2 ) ω 2 ω 4 ] 1 / 2 ,
ω MI = ± ( 2 q 0 2 2 q 0 4 2 ) 1 / 2 .
γ imag = ± 1 2 ( 4 q 0 2 ω 2 ω 4 ) 1 / 2 ,
ω MI = ± 2 q 0 2 ,
γ imag = 1 2 [ 4 q 0 2 ( 1 4 q 0 2 ω 0 2 ) ω 2 ω 4 ] 1 / 2
ω MI = ± ( 8 q 0 4 ω 0 2 2 q 0 2 ) 1 / 2 ,

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