Abstract

We show that a nonlinear array of coupled waveguides can exhibit discrete self-focusing that in the continuum approximation obeys the so-called nonlinear Schrödinger equation. This process has much in common with the biophysical model of Davydov.

© 1988 Optical Society of America

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References

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  1. M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Berlin, 1981).
    [Crossref]
  2. E. Fermi, J. Pasta, S. Ulam, “Studies of nonlinear problems,” (Los Alamos Scientific Laboratory, Los Alamos, N. M., 1955), Tech. Rep. LA-1940.
  3. U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
    [Crossref]
  4. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  5. S. M. Jensen, IEEE J. Quantum Electron, QE-18, 1580 (1982).
    [Crossref]
  6. S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
    [Crossref]
  7. W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
    [Crossref] [PubMed]
  8. J. E. Sipe, H. G. Winful, Opt. Lett. 13, 132 (1988).
    [Crossref] [PubMed]
  9. A. S. Davydov, J. Theor. Biol. 38, 559 (1973); Biology and Quantum Mechanics (Pergamon, Oxford, 1982).
    [Crossref]
  10. A. C. Scott, Philos. Trans. R. Soc. London Ser. A 315, 423 (1985); P. Lomdahl, S. Layne, I. Bigio, Los Alamos Sci. (Spring1984), p. 3.
    [Crossref]
  11. A. C. Scott, L. Macneil, Phys. Lett. A 98, 87 (1983). In this paper their quantities an and (γ/J) correspond to our fn and p, respectively.
    [Crossref]
  12. N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976).
  13. K. Tai, A. Tomita, A. Hasegawa, Phys. Rev. Lett. 56, 135 (1986).
    [Crossref] [PubMed]
  14. S. Kawakami, H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 (1986).
    [Crossref]
  15. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
    [Crossref]
  16. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [Crossref]
  17. M. J. Ablowitz, J. F. Ladik, J. Math. Phys. 17, 1011(1976).
    [Crossref]
  18. J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
    [Crossref]

1988 (1)

1987 (3)

U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
[Crossref]

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

1986 (2)

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

S. Kawakami, H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 (1986).
[Crossref]

1985 (2)

A. C. Scott, Philos. Trans. R. Soc. London Ser. A 315, 423 (1985); P. Lomdahl, S. Layne, I. Bigio, Los Alamos Sci. (Spring1984), p. 3.
[Crossref]

J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
[Crossref]

1983 (1)

A. C. Scott, L. Macneil, Phys. Lett. A 98, 87 (1983). In this paper their quantities an and (γ/J) correspond to our fn and p, respectively.
[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]

1973 (1)

A. S. Davydov, J. Theor. Biol. 38, 559 (1973); Biology and Quantum Mechanics (Pergamon, Oxford, 1982).
[Crossref]

1972 (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[Crossref]

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

Ablowitz, M. J.

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

Andrejco, M. J.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976).

Bhattacharya, P.

U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
[Crossref]

Chen, W.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Chen, Y.

U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
[Crossref]

Das, U.

U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
[Crossref]

Davydov, A. S.

A. S. Davydov, J. Theor. Biol. 38, 559 (1973); Biology and Quantum Mechanics (Pergamon, Oxford, 1982).
[Crossref]

Eilbeck, J. C.

J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
[Crossref]

Fermi, E.

E. Fermi, J. Pasta, S. Ulam, “Studies of nonlinear problems,” (Los Alamos Scientific Laboratory, Los Alamos, N. M., 1955), Tech. Rep. LA-1940.

Friberg, S. R.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Hasegawa, A.

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

Haus, H. A.

S. Kawakami, H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 (1986).
[Crossref]

Jensen, S. M.

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

Kawakami, S.

S. Kawakami, H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 (1986).
[Crossref]

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

Ladik, J. F.

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

Lomdahl, P. S.

J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
[Crossref]

Macneil, L.

A. C. Scott, L. Macneil, Phys. Lett. A 98, 87 (1983). In this paper their quantities an and (γ/J) correspond to our fn and p, respectively.
[Crossref]

Mermin, N. D.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976).

Mills, D. L.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Oliver, M.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Pasta, J.

E. Fermi, J. Pasta, S. Ulam, “Studies of nonlinear problems,” (Los Alamos Scientific Laboratory, Los Alamos, N. M., 1955), Tech. Rep. LA-1940.

Saifi, M. A.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Scott, A. C.

A. C. Scott, Philos. Trans. R. Soc. London Ser. A 315, 423 (1985); P. Lomdahl, S. Layne, I. Bigio, Los Alamos Sci. (Spring1984), p. 3.
[Crossref]

J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
[Crossref]

A. C. Scott, L. Macneil, Phys. Lett. A 98, 87 (1983). In this paper their quantities an and (γ/J) correspond to our fn and p, respectively.
[Crossref]

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[Crossref]

Silberberg, Y.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Sipe, J. E.

Smith, P. W.

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

Tai, K.

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

Toda, M.

M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Berlin, 1981).
[Crossref]

Tomita, A.

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

Ulam, S.

E. Fermi, J. Pasta, S. Ulam, “Studies of nonlinear problems,” (Los Alamos Scientific Laboratory, Los Alamos, N. M., 1955), Tech. Rep. LA-1940.

Winful, H. G.

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[Crossref]

Appl. Phys. Lett. (2)

U. Das, Y. Chen, P. Bhattacharya, Appl. Phys. Lett. 51, 1679 (1987); P. Li Kam Wa, J. Sitch, N. Mason, J. Roberts, P. Robson, Electron. Lett. 21, 26 (1985).
[Crossref]

S. R. Friberg, Y. Silberberg, M. Oliver, M. J. Andrejco, M. A. Saifi, P. W. Smith, Appl. Phys. Lett. 51, 1135 (1987).
[Crossref]

IEEE J. Lightwave Technol. (1)

S. Kawakami, H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 (1986).
[Crossref]

IEEE J. Quantum Electron, (1)

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

J. Math. Phys. (1)

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

J. Theor. Biol. (1)

A. S. Davydov, J. Theor. Biol. 38, 559 (1973); Biology and Quantum Mechanics (Pergamon, Oxford, 1982).
[Crossref]

Opt. Lett. (1)

Philos. Trans. R. Soc. London Ser. A (1)

A. C. Scott, Philos. Trans. R. Soc. London Ser. A 315, 423 (1985); P. Lomdahl, S. Layne, I. Bigio, Los Alamos Sci. (Spring1984), p. 3.
[Crossref]

Phys. Lett. A (1)

A. C. Scott, L. Macneil, Phys. Lett. A 98, 87 (1983). In this paper their quantities an and (γ/J) correspond to our fn and p, respectively.
[Crossref]

Phys. Rev. Lett. (3)

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

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Physica D (1)

J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
[Crossref]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); J. Satsuma, N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[Crossref]

Other (4)

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976).

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Berlin, 1981).
[Crossref]

E. Fermi, J. Pasta, S. Ulam, “Studies of nonlinear problems,” (Los Alamos Scientific Laboratory, Los Alamos, N. M., 1955), Tech. Rep. LA-1940.

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Figures (2)

Fig. 1
Fig. 1

A nonlinear array of coupled waveguides.

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Fig. 2
Fig. 2

Intensity patterns of the self-focusing formation. The solid bars represent the quantities |fn|2(discrete model), and the dashed curves represent the soliton envelope sech2(x/x0) as obtained from the continuum approximation for the same value of p. (a) Self-focusing formation when p ≃ 1.92; (b) self-focusing formation for p ≃ 6.135. All figures are normalized to a maximum value of unity.

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Equations (11)

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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 ,
k z = β + 2 c   cos ( k x D ) + λ ∣ A ∣ 2 .
i d Φ n d z + c ( Φ n + 1 + Φ n - 1 - 2 Φ n ) + λ ∣ Φ n ∣ 2 Φ n = 0.
Φ n ( z ) = ( p c / λ ) 1 / 2 f n ( z ) ,
i d f n d z + c ( f n + 1 + f n - 1 - 2 f n ) + p c ∣ f n ∣ 2 f n = 0.
δ n ( z ) = ∊ 1   exp [ i ( K z z - K x x n ) ] + ∊ 2   exp [ - i ( K z z - K x x n ) ] ,
K z 2 = 8 c   sin 2 ( D K x / 2 ) [ 2 c   sin 2 ( D K x / 2 ) - λ q 0 2 ] .
Φ n ± 1 = Φ ± D ∂ Φ ∂ x + D 2 2 ∂ 2 Φ ∂ x 2 + … .
i ∂ Φ ∂ z + c D 2 ∂ 2 Φ ∂ x 2 + λ ∣ Φ ∣ 2 Φ = 0 ,
Φ = Φ ^ 0   sech ( x / x 0 ) exp ( i σ z ) ,
K z 2 = c D 2 K x 2 ( c D 2 K x 2 - 2 λ q 0 2 ) .

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