Abstract

We study the evolution of solitary waves in inhomogeneous nonlinear arrays of waveguides. We use both a continuous and a discrete approach. For the continuous approximation we find a Lagrangean that permits us to change to a set of ordinary differential equations. The evolution of the solitary waves is studied for various inhomogeneous arrays.

© 1994 Optical Society of America

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References

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  1. T. Holstein, Ann. Phys. (Leipzig) 8, 325 (1959).
    [Crossref]
  2. A. S. Davydov, N. I. Kishlukha, Phys. Status Solidi B 59, 465 (1973).
    [Crossref]
  3. J. C. Eilbeck, P. S. Lomdahl, A. C. Scott, Physica D 16, 318 (1985).
    [Crossref]
  4. J. C. Eilbeck, Phys. Lett. A 155, 407 (1991).
    [Crossref]
  5. A. C. Scott, L. MacNeil, Phys. Lett. A 98, 87 (1983).
    [Crossref]
  6. A. R. Bishop, D. K. Campbell, S. Pnevmatikos, eds., Disorder and Nonlinearity, Vol. 39 of Springer Series in Physics (Springer-Verlag, Berlin, 1989).
  7. R. Scharf, A. R. Bishop, Phys. Rev. E 47, 1375 (1993).
    [Crossref]
  8. H. A. Haus, L. Molter-Orr, IEEE J. Quantum Electron. QE-19, 840 (1983).
    [Crossref]
  9. H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
    [Crossref]
  10. M. Kuznetsov, IEEE J. Quantum Electron. QE-21, 1893 (1985).
    [Crossref]
  11. E. Kapon, J. Katz, A. Yariv, Opt. Lett. 9, 125 (1984).
    [Crossref] [PubMed]
  12. R. R. A. Syms, IEEE J. Quantum Electron. QE-23, 525 (1987).
    [Crossref]
  13. D. N. Christodoulides, R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [Crossref] [PubMed]
  14. F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
    [Crossref]

1993 (2)

R. Scharf, A. R. Bishop, Phys. Rev. E 47, 1375 (1993).
[Crossref]

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

1991 (1)

J. C. Eilbeck, Phys. Lett. A 155, 407 (1991).
[Crossref]

1988 (1)

1987 (1)

R. R. A. Syms, IEEE J. Quantum Electron. QE-23, 525 (1987).
[Crossref]

1985 (2)

M. Kuznetsov, IEEE J. Quantum Electron. QE-21, 1893 (1985).
[Crossref]

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

1984 (2)

H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
[Crossref]

E. Kapon, J. Katz, A. Yariv, Opt. Lett. 9, 125 (1984).
[Crossref] [PubMed]

1983 (2)

H. A. Haus, L. Molter-Orr, IEEE J. Quantum Electron. QE-19, 840 (1983).
[Crossref]

A. C. Scott, L. MacNeil, Phys. Lett. A 98, 87 (1983).
[Crossref]

1973 (1)

A. S. Davydov, N. I. Kishlukha, Phys. Status Solidi B 59, 465 (1973).
[Crossref]

1959 (1)

T. Holstein, Ann. Phys. (Leipzig) 8, 325 (1959).
[Crossref]

Bishop, A. R.

R. Scharf, A. R. Bishop, Phys. Rev. E 47, 1375 (1993).
[Crossref]

Boardman, A. D.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Christodoulides, D. N.

Davydov, A. S.

A. S. Davydov, N. I. Kishlukha, Phys. Status Solidi B 59, 465 (1973).
[Crossref]

Eilbeck, J. C.

J. C. Eilbeck, Phys. Lett. A 155, 407 (1991).
[Crossref]

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

Haus, H. A.

H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
[Crossref]

H. A. Haus, L. Molter-Orr, IEEE J. Quantum Electron. QE-19, 840 (1983).
[Crossref]

Holstein, T.

T. Holstein, Ann. Phys. (Leipzig) 8, 325 (1959).
[Crossref]

Joseph, R. I.

Kapon, E.

Katz, J.

Kishlukha, N. I.

A. S. Davydov, N. I. Kishlukha, Phys. Status Solidi B 59, 465 (1973).
[Crossref]

Kuznetsov, M.

M. Kuznetsov, IEEE J. Quantum Electron. QE-21, 1893 (1985).
[Crossref]

Lederer, F.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Leine, L.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Leonberger, F. J.

H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
[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).
[Crossref]

Molter-Orr, L.

H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
[Crossref]

H. A. Haus, L. Molter-Orr, IEEE J. Quantum Electron. QE-19, 840 (1983).
[Crossref]

Muschall, R.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Scharf, R.

R. Scharf, A. R. Bishop, Phys. Rev. E 47, 1375 (1993).
[Crossref]

Schmidt-Hattenberger, C.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Scott, A. C.

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).
[Crossref]

Syms, R. R. A.

R. R. A. Syms, IEEE J. Quantum Electron. QE-23, 525 (1987).
[Crossref]

Trutschel, U.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Wachter, C.

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Yariv, A.

Ann. Phys. (1)

T. Holstein, Ann. Phys. (Leipzig) 8, 325 (1959).
[Crossref]

Appl. Phys. Lett. (1)

H. A. Haus, L. Molter-Orr, F. J. Leonberger, Appl. Phys. Lett. 45, 19 (1984).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Kuznetsov, IEEE J. Quantum Electron. QE-21, 1893 (1985).
[Crossref]

H. A. Haus, L. Molter-Orr, IEEE J. Quantum Electron. QE-19, 840 (1983).
[Crossref]

R. R. A. Syms, IEEE J. Quantum Electron. QE-23, 525 (1987).
[Crossref]

Opt. Commun. (1)

F. Lederer, L. Leine, R. Muschall, C. Schmidt-Hattenberger, U. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (2)

J. C. Eilbeck, Phys. Lett. A 155, 407 (1991).
[Crossref]

A. C. Scott, L. MacNeil, Phys. Lett. A 98, 87 (1983).
[Crossref]

Phys. Rev. E (1)

R. Scharf, A. R. Bishop, Phys. Rev. E 47, 1375 (1993).
[Crossref]

Phys. Status Solidi B (1)

A. S. Davydov, N. I. Kishlukha, Phys. Status Solidi B 59, 465 (1973).
[Crossref]

Physica D (1)

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

Other (1)

A. R. Bishop, D. K. Campbell, S. Pnevmatikos, eds., Disorder and Nonlinearity, Vol. 39 of Springer Series in Physics (Springer-Verlag, Berlin, 1989).

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Figures (3)

Fig. 1
Fig. 1

Evolution of the field envelope |u|2 along the direction z in arrays with different effective potentials (continuous approach): (a) constant potential (C0 = 1, C1 = C2 = 0), (b) linear potential (C0 = 1.0034, C1 = −0.332, C2 = 0), (c) parabolic potential C0 = 1, C1 = 0, C2 = − 12.5.

Fig. 2
Fig. 2

Comparison between the continuous and the discrete models for (a) linear and (b) parabolic potential. The solid curves describe the evolution of the center of mass ξ of the sech solution of Eq. (5), and the filled diamonds denote the evolution of the corresponding discrete entity n along the propagation direction.

Fig. 3
Fig. 3

Influence of the particular phase difference δϕ between adjacent channels on the evolution of the field envelope |u|2 (the discrete approach) for (a) δ ϕ = 0 and (b) δ ϕ = π/15.

Equations (11)

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- i z u n = C n , n + 1 u n + 1 + C n , n - 1 u n - 1 + u n 2 u n ,
C n , n ± 1 = C ( x ± h 2 ) = C ( x ) ± h 2 x C ( x ) + h 2 8 x x C ( x ) ,
u n ± 1 = u ( x ± h ) = u ( x ) ± h x u ( x ) + h 2 2 x x u ( x ) .
L = i 2 ( u * z u - z u * u ) + 1 2 u 4 + 2 C u 2 + h 2 4 x C ( u * x u + x u * u ) + h 2 4 C ( u * x x u + x x u * u ) .
u ( z , x ) = A ( z ) η ( z ) sech { η ( z ) [ x - ξ ( z ) ] } × exp { i [ Φ ( z ) + v ( z ) x + w ( z ) x 2 ] } ,
z A = 0 ,
z η = - h 2 [ 7 5 η π 2 C 2 w + 4 η w ( C 0 + 2 C 1 ξ + 3 C 2 ξ 2 ) + 2 η v ( C 1 + 2 ξ C 2 ) ] ,
z ξ = h 2 [ 2 ( v + 2 ξ w ) ( C 0 + C 1 ξ + C 2 ξ 2 ) + π 2 6 η 2 ( C 2 v + 2 C 1 w + 6 C 2 w ξ ) ] ,
z Φ = 2 C 0 + 7 π 4 h 2 C 2 w 2 60 η 4 + 4 π 2 ξ 2 η 4 h 2 × ( C 0 + C 1 ξ + C 2 ξ 2 ) - 4 h 2 ξ 3 w ( C 2 v + 3 C 2 ξ w + C 1 w ) - h 2 η 2 ( C 1 ξ + 2 C 0 ) 3 + h 2 C 2 ( π 2 - 6 ) 36 - 2 A 2 ξ 2 η 3 π 2 + 5 A 2 η 6 - h 2 C 0 v 2 + π 2 h 2 w ξ ( C 2 v + C 1 w + 6 5 C 2 ξ w ) η 2 ,
z v = 2 C 1 + 4 π 2 A 2 ξ η 3 - 8 π 2 ξ η 4 h 2 × ( C 0 + C 1 ξ + C 2 ξ 2 ) + 4 h 2 ξ 2 w ( 3 C 2 v + 3 C 1 w + 8 C 2 ξ w ) - 1 3 h 2 η 2 ( C 1 + 2 ξ C 2 ) - h 2 v ( C 1 v + 4 C 0 w ) + π 2 h 2 w η 2 ( - w C 1 - v C 2 + 8 / 5 w ξ C 2 ) ,
z w = 2 C 2 - 14 π 2 h 2 C 2 w 2 5 η 2 - 2 η 3 A 2 π 2 + 4 η 4 h 2 π 2 × ( C 0 + C 1 ξ + C 2 ξ 2 ) - 4 h 2 w 2 ( C 0 + 3 C 1 ξ + 6 C 2 ξ 2 ) - h 2 v ( 4 C 1 w + 12 C 2 ξ w + C 2 v ) .

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