Abstract

It is shown that discrete self-focusing may be realized in an array of coupled defocusing nonlinear waveguides. Strongly localized self-focusing patterns are found, and it is pointed out that stable beam propagation corresponds to the case in which the electric field is equivalently distributed between two neighboring waveguides with relative phase difference π.

© 1993 Optical Society of America

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References

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  1. A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, Y. Chen, J. Opt. Soc. Am. B 8, 2102 (1991).
    [CrossRef]
  2. D. J. Mitchell, A. W. Snyder, Y. Chen, Electron. Lett. 26, 1164 (1990).
    [CrossRef]
  3. N. Finlayson, G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
    [CrossRef]
  4. J. M. Soto-Crespo, E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
    [CrossRef]
  5. S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
    [CrossRef]
  6. D. N. Christodoulides, R. T. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  7. C. Schmidt-Hattenberger, U. Truschel, F. Lederer, Opt. Lett. 16, 295 (1991).
    [CrossRef]
  8. S. Trillo, S. Wabnitz, E. M. Wright, G. I. Stegeman, Opt. Lett. 13, 672 (1988).
    [CrossRef] [PubMed]
  9. Yu. S. Kivshar, Opt. Lett. 18, 7 (1993).
    [CrossRef] [PubMed]
  10. K. Tai, A. Tomita, A. Hasegawa, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  11. V. I. Karpman, JETP Lett. 6, 227 (1967).
  12. M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011 (1976).
    [CrossRef]
  13. Yu. S. Kivshar, B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  14. Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
    [CrossRef]
  15. M. Peyrard, M. Kruskal, Physica D 13, 88 (1984).
    [CrossRef]
  16. A. J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988).
    [CrossRef] [PubMed]

1993 (2)

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Yu. S. Kivshar, Opt. Lett. 18, 7 (1993).
[CrossRef] [PubMed]

1991 (3)

C. Schmidt-Hattenberger, U. Truschel, F. Lederer, Opt. Lett. 16, 295 (1991).
[CrossRef]

A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, Y. Chen, J. Opt. Soc. Am. B 8, 2102 (1991).
[CrossRef]

J. M. Soto-Crespo, E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[CrossRef]

1990 (2)

D. J. Mitchell, A. W. Snyder, Y. Chen, Electron. Lett. 26, 1164 (1990).
[CrossRef]

N. Finlayson, G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[CrossRef]

1989 (1)

Yu. S. Kivshar, B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

1988 (3)

1986 (1)

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

1984 (1)

M. Peyrard, M. Kruskal, Physica D 13, 88 (1984).
[CrossRef]

1982 (1)

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

1976 (1)

M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011 (1976).
[CrossRef]

1967 (1)

V. I. Karpman, JETP Lett. 6, 227 (1967).

Ablowitz, M. J.

M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011 (1976).
[CrossRef]

Chen, Y.

Christodoulides, D. N.

Claude, Ch.

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Finlayson, N.

N. Finlayson, G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[CrossRef]

Hasegawa, A.

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

Jensen, S. M.

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

Joseph, R. T.

Karpman, V. I.

V. I. Karpman, JETP Lett. 6, 227 (1967).

Kivshar, Yu. S.

Yu. S. Kivshar, Opt. Lett. 18, 7 (1993).
[CrossRef] [PubMed]

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Yu. S. Kivshar, B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Kluth, O.

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Kruskal, M.

M. Peyrard, M. Kruskal, Physica D 13, 88 (1984).
[CrossRef]

Ladik, J. F.

M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011 (1976).
[CrossRef]

Lederer, F.

C. Schmidt-Hattenberger, U. Truschel, F. Lederer, Opt. Lett. 16, 295 (1991).
[CrossRef]

Malomed, B. A.

Yu. S. Kivshar, B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Mitchell, D. J.

Peyrard, M.

M. Peyrard, M. Kruskal, Physica D 13, 88 (1984).
[CrossRef]

Poladian, L.

Rowland, D. R.

Schmidt-Hattenberger, C.

C. Schmidt-Hattenberger, U. Truschel, F. Lederer, Opt. Lett. 16, 295 (1991).
[CrossRef]

Sievers, A. J.

A. J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988).
[CrossRef] [PubMed]

Snyder, A. W.

Soto-Crespo, J. M.

J. M. Soto-Crespo, E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[CrossRef]

Spatschek, K. H.

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Stegeman, G. I.

Tai, K.

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

Takeno, S.

A. J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988).
[CrossRef] [PubMed]

Tomita, A.

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

Trillo, S.

Truschel, U.

C. Schmidt-Hattenberger, U. Truschel, F. Lederer, Opt. Lett. 16, 295 (1991).
[CrossRef]

Wabnitz, S.

Wright, E. M.

Appl. Phys. Lett. (1)

N. Finlayson, G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[CrossRef]

Electron. Lett. (1)

D. J. Mitchell, A. W. Snyder, Y. Chen, Electron. Lett. 26, 1164 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

J. Appl. Phys. (1)

J. M. Soto-Crespo, E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[CrossRef]

J. Math. Phys. (1)

M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011 (1976).
[CrossRef]

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

JETP Lett. (1)

V. I. Karpman, JETP Lett. 6, 227 (1967).

Opt. Lett. (4)

Phys. Rev. B (1)

Ch. Claude, Yu. S. Kivshar, O. Kluth, K. H. Spatschek, Phys. Rev. B 47, 14228 (1993).
[CrossRef]

Phys. Rev. Lett. (2)

A. J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988).
[CrossRef] [PubMed]

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

Physica D (1)

M. Peyrard, M. Kruskal, Physica D 13, 88 (1984).
[CrossRef]

Rev. Mod. Phys. (1)

Yu. S. Kivshar, B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

(a) Symmetric (unstable) and (b) antisymmetric (stable) self-focusing patterns in an array of defocusing nonlinear waveguides. Shown is the function fn defined through Eqs. (6) and (7). The filled circles represent the waveguide locations. The dashed curves are used to show the field envelope Ψn = |fn|, which, in the continuum limit approximation, is described by Eq. (4).

Equations (8)

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

i d E n d z + β E n + c ( E n + 1 + E n - 1 ) - λ E n 2 E n - μ ( E n + 1 2 + E n - 1 2 ) E n = 0 ,
q = β + 2 c cos ( k D ) - ( λ + 2 μ ) E 0 2 ,
[ Q + 2 c sin ( k D ) sin ( K D ) ] 2 = 8 c cos ( k D ) sin 2 ( K D 2 ) [ 2 c cos ( k D ) sin 2 ( K D 2 ) + λ E 0 2 + 2 μ E 0 2 cos ( K D ) ] ,
i Ψ z = c D 2 2 Ψ x 2 + ( λ + 2 μ ) Ψ 2 Ψ ,
E 1 , 2 2 = 1 2 P [ 1 ± 1 - ( P cr / P ) 2 ] .
E n = A exp ( i K z ) ( , 0 , - η , 1 , - η , 0 , ) ,
E n = B exp ( i Ω z ) ( , 0 , - ξ , 1 - 1 , ξ , 0 , ) ,
H = - c n ( E n E n + 1 * + E N * E n - 1 ) + 1 2 λ n E n 4 ,

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