Abstract

We analyze the propagation of discrete solitons in a periodic system of weakly coupled nonlinear optical waveguides, i.e., a waveguide array. Soliton reflection, transmission, and trapping, as well as coherent and incoherent interaction with a linear guided wave [which can exist owing to defect (inhomogeneous) coupling between two neighboring waveguides], are demonstrated numerically and investigated analytically with a collective-coordinate approach. Some potential schemes of controllable and steerable soliton-based optical switching in nonlinear waveguide arrays are discussed. For the first scheme it is suggested that unstable soliton modes be used to achieve easily steerable propagation of discrete bright and dark solitons. This is to avoid mode trapping by the effective Peierls –Nabarro potential, which always appears because of the system discreteness. The other scheme is based on soliton control with the help of a linear guided wave that can be excited in an inhomogeneous array.

© 1996 Optical Society of America

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  1. See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  2. W. K. Burns, A. F. Milton, A. B. Lee, and E. J. West, “Optical modal evolution 3-dB coupler,” Appl. Opt. 15, 1053–1065 (1976).
    [CrossRef] [PubMed]
  3. D. B. Martimore and J. W. Arkwright, “Theory and fabrication of wavelength-flattened 1 × 7 single-mode coupler,” Appl. Opt. 29, 1814–1818 (1990); “Monolithic wavelength-flattened 1 × 7 single mode fused fiber couplers, theory, fabrication, and analysis,” Appl. Opt. 30, 650 (1991).
    [CrossRef]
  4. D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
    [CrossRef]
  5. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
    [CrossRef]
  6. N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities and chaos in a three-waveguide directional coupler,” Appl. Phys. Lett. 56, 2276–2279 (1990).
    [CrossRef]
  7. M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
    [CrossRef]
  8. C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
    [CrossRef]
  9. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
    [CrossRef] [PubMed]
  10. Yu. S. Kivshar, “Self-localization in array of defocusing waveguides,” Opt. Lett. 18, 7–9 (1993).
    [CrossRef] [PubMed]
  11. C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
    [CrossRef] [PubMed]
  12. W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, and M. Cronin-Golomb, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
    [CrossRef]
  13. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
    [CrossRef] [PubMed]
  14. A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, “Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays,” Opt. Lett. 19, 332–334 (1994).
    [CrossRef] [PubMed]
  15. A. V. Buryak and N. N. Akhmediev, “Stationary pulse propagation in N-core nonlinear fiber arrays,” IEEE J. Quantum Electron. 31, 682–688 (1995).
    [CrossRef]
  16. W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.
  17. R. Muschall, C. Schmidt-Hattenberger, and F. Lederer, “Spatially solitary waves in arrays of nonlinear waveguides,” Opt. Lett. 19, 323–325 (1994).
    [CrossRef] [PubMed]
  18. A. S. Davydov and N. I. Kislukha, “Solitary excitations in one-dimensional molecular chains,” Phys. Stat. Sol. B 59, 465–470 (1973); A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrech, 1985).
    [CrossRef]
  19. Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable equations.” Rev. Mod. Phys. 61, 763–815 (1989).
    [CrossRef]
  20. H. Feddersen, “Solitary wave solutions to the discrete Schrödinger equation,” in Nonlinear Coherent Structures in Physics and Biology, M. Remoissenent and M. Peyrard, eds., Vol. 393 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1991), p. 159.
    [CrossRef]
  21. K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
    [CrossRef]
  22. Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
    [CrossRef] [PubMed]
  23. Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
    [CrossRef]
  24. Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
    [CrossRef] [PubMed]
  25. Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro periodic barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
    [CrossRef]
  26. Yu. S. Kivshar, W. Królikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
    [CrossRef]
  27. See, e.g., F. R. N. Nabarro, Theory of Crystal Dislocations (Dover, New York, 1987), and references therein.
  28. Yu. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 28, 250–265 (1993).
    [CrossRef]

1995 (1)

A. V. Buryak and N. N. Akhmediev, “Stationary pulse propagation in N-core nonlinear fiber arrays,” IEEE J. Quantum Electron. 31, 682–688 (1995).
[CrossRef]

1994 (6)

1993 (4)

Yu. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 28, 250–265 (1993).
[CrossRef]

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro periodic barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[CrossRef]

M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
[CrossRef]

Yu. S. Kivshar, “Self-localization in array of defocusing waveguides,” Opt. Lett. 18, 7–9 (1993).
[CrossRef] [PubMed]

1992 (2)

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

1991 (3)

C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
[CrossRef] [PubMed]

C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
[CrossRef]

Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

1990 (3)

N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities and chaos in a three-waveguide directional coupler,” Appl. Phys. Lett. 56, 2276–2279 (1990).
[CrossRef]

D. B. Martimore and J. W. Arkwright, “Theory and fabrication of wavelength-flattened 1 × 7 single-mode coupler,” Appl. Opt. 29, 1814–1818 (1990); “Monolithic wavelength-flattened 1 × 7 single mode fused fiber couplers, theory, fabrication, and analysis,” Appl. Opt. 30, 650 (1991).
[CrossRef]

D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
[CrossRef]

1989 (1)

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable equations.” Rev. Mod. Phys. 61, 763–815 (1989).
[CrossRef]

1988 (2)

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

1976 (1)

1973 (1)

A. S. Davydov and N. I. Kislukha, “Solitary excitations in one-dimensional molecular chains,” Phys. Stat. Sol. B 59, 465–470 (1973); A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrech, 1985).
[CrossRef]

Aceves, A. B.

Akhmediev, N. N.

A. V. Buryak and N. N. Akhmediev, “Stationary pulse propagation in N-core nonlinear fiber arrays,” IEEE J. Quantum Electron. 31, 682–688 (1995).
[CrossRef]

Arkwright, J. W.

Burns, W. K.

Buryak, A. V.

A. V. Buryak and N. N. Akhmediev, “Stationary pulse propagation in N-core nonlinear fiber arrays,” IEEE J. Quantum Electron. 31, 682–688 (1995).
[CrossRef]

Campbell, D. K.

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro periodic barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Chen, Y.

D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
[CrossRef]

Christodoulides, D. N.

Chubykalo, O. A.

Yu. S. Kivshar, W. Królikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Cronin-Golomb, M.

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, and M. Cronin-Golomb, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
[CrossRef]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

Davydov, A. S.

A. S. Davydov and N. I. Kislukha, “Solitary excitations in one-dimensional molecular chains,” Phys. Stat. Sol. B 59, 465–470 (1973); A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrech, 1985).
[CrossRef]

De Angelis, C.

Deering, W. D.

M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
[CrossRef]

Feddersen, H.

H. Feddersen, “Solitary wave solutions to the discrete Schrödinger equation,” in Nonlinear Coherent Structures in Physics and Biology, M. Remoissenent and M. Peyrard, eds., Vol. 393 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1991), p. 159.
[CrossRef]

Fei, Zhang

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Finlayson, N.

N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities and chaos in a three-waveguide directional coupler,” Appl. Phys. Lett. 56, 2276–2279 (1990).
[CrossRef]

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Forinash, K.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

Joseph, R. I.

Kislukha, N. I.

A. S. Davydov and N. I. Kislukha, “Solitary excitations in one-dimensional molecular chains,” Phys. Stat. Sol. B 59, 465–470 (1973); A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrech, 1985).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar, W. Królikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Yu. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 28, 250–265 (1993).
[CrossRef]

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro periodic barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Yu. S. Kivshar, “Self-localization in array of defocusing waveguides,” Opt. Lett. 18, 7–9 (1993).
[CrossRef] [PubMed]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable equations.” Rev. Mod. Phys. 61, 763–815 (1989).
[CrossRef]

Królikowski, W.

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, and M. Cronin-Golomb, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
[CrossRef]

Yu. S. Kivshar, W. Królikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

Lederer, F.

Lee, A. B.

Malomed, B.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Malomed, B. A.

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable equations.” Rev. Mod. Phys. 61, 763–815 (1989).
[CrossRef]

Martimore, D. B.

Milton, A. F.

Mitchell, D. J.

D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
[CrossRef]

Molina, M. I.

M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
[CrossRef]

Muschall, R.

R. Muschall, C. Schmidt-Hattenberger, and F. Lederer, “Spatially solitary waves in arrays of nonlinear waveguides,” Opt. Lett. 19, 323–325 (1994).
[CrossRef] [PubMed]

C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
[CrossRef]

Nabarro, F. R. N.

See, e.g., F. R. N. Nabarro, Theory of Crystal Dislocations (Dover, New York, 1987), and references therein.

Peyrard, M.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Rubenchik, A. M.

Schmidt-Hattenberger, C.

R. Muschall, C. Schmidt-Hattenberger, and F. Lederer, “Spatially solitary waves in arrays of nonlinear waveguides,” Opt. Lett. 19, 323–325 (1994).
[CrossRef] [PubMed]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, and M. Cronin-Golomb, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
[CrossRef]

C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
[CrossRef] [PubMed]

C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
[CrossRef]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

Seaton, C. T.

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Snyder, A. W.

D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
[CrossRef]

Stegeman, G. I.

N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities and chaos in a three-waveguide directional coupler,” Appl. Phys. Lett. 56, 2276–2279 (1990).
[CrossRef]

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Trillo, S.

Trutschel, U.

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, and M. Cronin-Golomb, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
[CrossRef]

C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
[CrossRef]

C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
[CrossRef] [PubMed]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

Tsironis, G. P.

M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
[CrossRef]

Turitsyn, S. K.

Vázquez, L.

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Wabnitz, S.

West, E. J.

Wright, E. M.

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Yang, X.

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

Zanoni, R.

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities and chaos in a three-waveguide directional coupler,” Appl. Phys. Lett. 56, 2276–2279 (1990).
[CrossRef]

Electron. Lett. (1)

D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164 (1990).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, “Stationary pulse propagation in N-core nonlinear fiber arrays,” IEEE J. Quantum Electron. 31, 682–688 (1995).
[CrossRef]

Yu. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 28, 250–265 (1993).
[CrossRef]

J. Lightwave Technol. (1)

See, for instance, G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Opt. Commun. (1)

C. Schmidt-Hattenberger, U. Trutschel, R. Muschall, and F. Lederer, “Envelope description of an optical fiber array with circularly distributed multiple cores,” Opt. Commun. 82, 461–465 (1991).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (2)

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Zhang Fei, Yu. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interaction in the ϕ4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (3)

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro periodic barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Yu. S. Kivshar, W. Królikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

Yu. S. Kivshar, Zhang Fei, and L. Vázquez, “Resonant soliton-impurity interactions,” Phys. Rev. Lett. 67, 1177–1180 (1991).
[CrossRef] [PubMed]

Phys. Stat. Sol. B (1)

A. S. Davydov and N. I. Kislukha, “Solitary excitations in one-dimensional molecular chains,” Phys. Stat. Sol. B 59, 465–470 (1973); A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrech, 1985).
[CrossRef]

Physica D (1)

M. I. Molina, W. D. Deering, and G. P. Tsironis, “Optical switching in three-coupler configuration,” Physica D 66, 135 (1993).
[CrossRef]

Rev. Mod. Phys. (1)

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable equations.” Rev. Mod. Phys. 61, 763–815 (1989).
[CrossRef]

Other (3)

H. Feddersen, “Solitary wave solutions to the discrete Schrödinger equation,” in Nonlinear Coherent Structures in Physics and Biology, M. Remoissenent and M. Peyrard, eds., Vol. 393 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1991), p. 159.
[CrossRef]

W. Królikowski, U. Trutschel, C. Schmidt-Hattenberger, M. Cronin-Golomb, and X. Yang, in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 87.

See, e.g., F. R. N. Nabarro, Theory of Crystal Dislocations (Dover, New York, 1987), and references therein.

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Figures (9)

Fig. 1
Fig. 1

Schematic structure of an array of coupled waveguides.

Fig. 2
Fig. 2

Steering of a bright soliton in discrete array. Shown is the relative position of the soliton envelope center after propagation over the distance zk = 100 as a function of the initial phase variation. Filled squares correspond to a stable input pulse centered at a waveguide, whereas open circles correspond to an unstable input pulse initially centered between neighboring waveguides.

Fig. 3
Fig. 3

Same as in Fig. 2 but for a dark soliton in a waveguide array. The relative position of the minimum intensity of the dark soliton is shown at zk = 600 versus the soliton’s initial velocity, defined by the value of the parameter ξ. The curve shown with open circles corresponds to the B mode, and there exists no threshold for this mode to change direction. The oscillating dependence corresponds to the A mode. Above the threshold value of ξ the soliton dynamics is similar to that for the B mode, i.e., the soliton always moves in the direction selected by the initial conditions. However, below the threshold velocity ξth ≈ 0.045 the soliton may escape to the right or to the left, depending on values of either the initial soliton velocity or its position.

Fig. 4
Fig. 4

Amplitude profile of (a) the symmetric and (b) the antisymmetric linear guided modes that can exist at the defect waveguides.

Fig. 5
Fig. 5

Propagation of the envelope soliton at V0 = 0.5 through the region of locally decreased ( < 0) linear coupling. (a) Complete transmission at = −0.1; (b) total reflection at = −0.2; (c) partial transmission at = −0.13; (d) example of the weak trapping of the soliton by a repulsive coupling defect at = −0.0054.

Fig. 6
Fig. 6

Propagation of the discrete soliton at V0 = 0.2 through the region of locally increased linear coupling ( > 0). (a) Soliton reflection at = 0.5; (b) transmission at = 0.12; (c) and (d) soliton trapping by the defect for = 0.1 and = 0.063, respectively.

Fig. 7
Fig. 7

Result of the interaction of the discrete soliton with an excited symmetric linear guided mode as a function of the relative phase between the soliton and the linear mode. Initial amplitude of the linear mode a0 = 0.25; initial soliton velocity V0 = 0.2; coupling defect = 0.1.

Fig. 8
Fig. 8

Behavior of the discrete soliton interacting with an excited linear guided mode, corresponding to different values of the relative phase ϕ taken from Fig. 7. As the phase varies the soliton is either (a) reflected (ϕ = 2.2), (b) transmitted (ϕ = 1.5), (c) trapped (ϕ = 1.8), (d) again transmitted (ϕ = 0), or (e) trapped after an initial partial reflection (ϕ = 2.12).

Fig. 9
Fig. 9

Example of the resonant interaction of the discrete soliton with an excited linear guided mode; the soliton is initially trapped in the region of increased coupling but finally leaves the defect region almost completely, depleting the energy of the linear mode. In this case the soliton velocity V0 = 0.2, the coupling defect = 0.1, and the relative phase is ϕ = 1.6.

Tables (1)

Tables Icon

Table 1 Results of the Soliton Collision with the Waveguide Defecta

Equations (22)

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i d u n d z = u n 1 + u n + 1 + | u n | 2 u n .
i d u n d z = u n 1 + u n + 1 + | u n | 2 u n + ( δ n , m u m + 1 + δ n , m + 1 u m ) ,
= ( κ ˜ κ ) / κ .
i u z = 1 2 2 u x 2 + | u | 2 u + 2 δ ( x ) u ,
u n = { exp [ i k ( n n c ) ] cosh [ ( A / 2 ) ( n n c ) ] , n n c = 0 , ± 1 , ± 2 , 0 , otherwise
u n = tanh [ η ( n n c ) ] + i ξ ,
u s ( x , z ) = a exp [ i V x i Δ ( z ) ] cosh [ a ( x x 0 ( z ) ] ,
Δ ( z ) = ½ ( a 2 V 2 ) + Δ ( 0 ) , x 0 ( z ) = 2 V z + x 0 ( 0 ) ,
H = 1 2 + d x ( | u x | 2 | u | 4 4 δ ( x ) | u | 2 ) ,
H = ( V 2 a 1 3 a 3 ) 2 a 2 cosh 2 ( a x 0 ) .
p 1 = 2 V , q 1 = a x 0 , p 2 = 2 a , q 2 = Δ .
d a d z = 0 ,
d x 0 d z = V ,
d V d z = 2 a 2 tanh ( a x 0 ) cosh 2 ( a x 0 ) ,
d Δ d z = 1 2 ( a 2 V 2 ) + 2 a cosh 2 ( a x 0 ) .
d 2 ξ d z 2 = d U ( ξ ) d ξ ,
U ( ξ ) = a 3 cosh 2 ξ .
x 0 ( z ) = 1 a sinh 1 [ ( 2 a | | V 0 2 1 ) 1 / 2 cosh ( a V 0 z ) ] .
u n ( z ) = u ( 0 ) { 1 , n = m , m + 1 q | m n | , otherwise } exp ( i Λ + z ) ,
u n ( z ) = u ( 0 ) { ( 1 ) n , n = m , m + 1 ( q ) | m n | , otherwise } exp ( i Λ z ) ,
q = 1 / ( 1 + ) .
Λ ± = ± [ q + ( 1 / q ) ] ,

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