Abstract

The system of coupled discrete equations describing a two-component superlattice with interlaced linear and nonlinear constituents is studied as a basis for investigating binary waveguide arrays, such as ribbed AlGaAs structures, among others. Compared to the single nonlinear lattice, the interlaced system exhibits an extra band-gap controlled by the, suitably chosen by design, relative detuning. In more general physics settings, this system represents a discretization scheme for the single-equation-based continuous models in media with transversely modulated linear and nonlinear properties. Continuous wave solutions and the associated modulational instability are fully analytically investigated and numerically tested for focusing and defocusing nonlinearity. The propagation dynamics and the stability of periodic modes are also analytically investigated for the case of zero Bloch momentum. In the band-gaps a variety of stable discrete solitary modes, dipole or otherwise, in-phase or of staggered type are found and discussed.

© 2008 Optical Society of America

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  1. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  2. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
    [CrossRef]
  3. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freeman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
    [CrossRef] [PubMed]
  4. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap Solitons in Waveguide Arrays," Phys. Rev. Lett.  92, 093,904-1-4 (2004).
    [CrossRef]
  5. D.N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001).
    [CrossRef] [PubMed]
  6. D. N. Christodouldes and E. D. Eugenieva, "Minimizing bending losses in two-dimensional discrete soliton networks," Opt. Lett. 23,1876, (2001).
    [CrossRef]
  7. E. D. Eugenieva, N. Efremidis, and D. N. Christodoulides "Design of switching junctions for two-dimensional discrete soliton network," Opt. Lett. 26, 1978 (2001).
    [CrossRef]
  8. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Controlled switching of discrete solitons in waveguide arrays," Opt. Lett. 28, 1942 (2003).
    [CrossRef] [PubMed]
  9. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefrigent wave guide arrays," Opt. Lett. 29, 2905 (2004).
    [CrossRef]
  10. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, "Parametric amplification of soliton steering in optical lattices," Opt. Lett. 29, 1102 (2004).
    [CrossRef] [PubMed]
  11. F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
    [CrossRef]
  12. G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
    [CrossRef]
  13. V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," "Phys. Rev. A 65, 021602(R) (2002).
  14. N. K. Efremidis and D. N. Christodoulides, "Lattice solitons in Bose-Einstein condensates," Phys. Rev. A 67, 063,608-1-9 (2003).
    [CrossRef]
  15. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Matter-wave dark solitons in optical lattices," J. Opt. B: Quantum Semiclassical Opt. 6, S309 (2004).
    [CrossRef]
  16. Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
    [CrossRef]
  17. M. Johansson and Y. S. Kivshar, "Discreteness-induced oscillatory instabilities in dark solitons," Phys. Rev. Lett. 82, 85 (1999).
    [CrossRef]
  18. V. V. Konotop and S. Takeno, "Stationary dark localized modes: Discrete nonlinear Schrödinger equations," Phys. Rev. E 60, 1001 (1999).
    [CrossRef]
  19. B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation," Phys. Rev. E 71, 036627 (2005).
    [CrossRef]
  20. H. Susanto and M. Johansson, "Discrete dark solitons with multiple holes," Phys. Rev. E 72, 016605 (2005).
    [CrossRef]
  21. Y. Kominis and K. Hizanidis, "Power dependent soliton location and stability in complex photonic structures," Opt. Express 16, 12124 (2008).
    [CrossRef] [PubMed]
  22. S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
    [CrossRef]
  23. A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
    [CrossRef] [PubMed]
  24. A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals," Phys. Rev. E 65, 036609 (2002).
    [CrossRef]
  25. Y. Kominis, "Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures," Phys. Rev. E 73, 066619 (2006).
    [CrossRef]
  26. Y. Kominis and K. Hizanidis, "Lattice solitons in self-defocusing optical media: Analytical solutions," Opt. Lett. 31, 2888 (2006).
    [CrossRef] [PubMed]
  27. Y. Kominis, A. Papadopoulos and K. Hizanidis, "Surface solitons in waveguide arrays: Analytical solutions," Opt. Express 15, 10041 (2007).
    [CrossRef] [PubMed]
  28. A. A. Sukhorukov and Y. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112 (2002).
    [CrossRef]
  29. R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890 (2004).
    [CrossRef]
  30. A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
    [CrossRef]

2008 (1)

2007 (1)

2006 (2)

Y. Kominis and K. Hizanidis, "Lattice solitons in self-defocusing optical media: Analytical solutions," Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

Y. Kominis, "Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures," Phys. Rev. E 73, 066619 (2006).
[CrossRef]

2005 (4)

A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
[CrossRef]

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation," Phys. Rev. E 71, 036627 (2005).
[CrossRef]

H. Susanto and M. Johansson, "Discrete dark solitons with multiple holes," Phys. Rev. E 72, 016605 (2005).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freeman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

2004 (4)

2003 (2)

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Controlled switching of discrete solitons in waveguide arrays," Opt. Lett. 28, 1942 (2003).
[CrossRef] [PubMed]

2002 (3)

A. A. Sukhorukov and Y. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112 (2002).
[CrossRef]

G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals," Phys. Rev. E 65, 036609 (2002).
[CrossRef]

2001 (5)

E. D. Eugenieva, N. Efremidis, and D. N. Christodoulides "Design of switching junctions for two-dimensional discrete soliton network," Opt. Lett. 26, 1978 (2001).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

D.N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

D. N. Christodouldes and E. D. Eugenieva, "Minimizing bending losses in two-dimensional discrete soliton networks," Opt. Lett. 23,1876, (2001).
[CrossRef]

1999 (2)

M. Johansson and Y. S. Kivshar, "Discreteness-induced oscillatory instabilities in dark solitons," Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

V. V. Konotop and S. Takeno, "Stationary dark localized modes: Discrete nonlinear Schrödinger equations," Phys. Rev. E 60, 1001 (1999).
[CrossRef]

1997 (1)

S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
[CrossRef]

1994 (1)

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
[CrossRef]

1988 (1)

Abdullaev, F. Kh.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Aitchison, J. S.

Alfimov, G. L.

G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
[CrossRef]

Baizakov, B. B.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Bartal, G.

Buljan, H.

Christodouldes, D. N.

D. N. Christodouldes and E. D. Eugenieva, "Minimizing bending losses in two-dimensional discrete soliton networks," Opt. Lett. 23,1876, (2001).
[CrossRef]

Christodoulides, D. N.

Christodoulides, D.N.

D.N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

Chubykalo, O. A.

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Cohen, O.

Darmanyan, S. A.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Efremidis, N.

Efremidis, N. K.

Eisenberg, H. S.

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Eugenieva, E. D.

D.N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

D. N. Christodouldes and E. D. Eugenieva, "Minimizing bending losses in two-dimensional discrete soliton networks," Opt. Lett. 23,1876, (2001).
[CrossRef]

E. D. Eugenieva, N. Efremidis, and D. N. Christodoulides "Design of switching junctions for two-dimensional discrete soliton network," Opt. Lett. 26, 1978 (2001).
[CrossRef]

Fleischer, J. W.

Freeman, B.

Hizanidis, K.

Johansson, M.

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation," Phys. Rev. E 71, 036627 (2005).
[CrossRef]

H. Susanto and M. Johansson, "Discrete dark solitons with multiple holes," Phys. Rev. E 72, 016605 (2005).
[CrossRef]

M. Johansson and Y. S. Kivshar, "Discreteness-induced oscillatory instabilities in dark solitons," Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

Joseph, R. I.

Kartashov, Y. V.

Kivshar, Y. S.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefrigent wave guide arrays," Opt. Lett. 29, 2905 (2004).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890 (2004).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Matter-wave dark solitons in optical lattices," J. Opt. B: Quantum Semiclassical Opt. 6, S309 (2004).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Controlled switching of discrete solitons in waveguide arrays," Opt. Lett. 28, 1942 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals," Phys. Rev. E 65, 036609 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

M. Johansson and Y. S. Kivshar, "Discreteness-induced oscillatory instabilities in dark solitons," Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Kominis, Y.

Konotop, V. V.

G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
[CrossRef]

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

V. V. Konotop and S. Takeno, "Stationary dark localized modes: Discrete nonlinear Schrödinger equations," Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Krolikowsi, W.

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Leontidis, E.

S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
[CrossRef]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Matter-wave dark solitons in optical lattices," J. Opt. B: Quantum Semiclassical Opt. 6, S309 (2004).
[CrossRef]

Malomed, B. A.

A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
[CrossRef]

Mandelik, D.

Manela, O.

Merhasin, I. M.

A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
[CrossRef]

Molina, M. I.

Morandotti, R.

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Matter-wave dark solitons in optical lattices," J. Opt. B: Quantum Semiclassical Opt. 6, S309 (2004).
[CrossRef]

Papadopoulos, A.

Salerno, M.

G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
[CrossRef]

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Sanchez-Rey, B.

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation," Phys. Rev. E 71, 036627 (2005).
[CrossRef]

Schwartz, T.

Segev, M.

Silberberg, Y.

Sorel, M.

Sukhorukov, A. A.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890 (2004).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals," Phys. Rev. E 65, 036609 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Susanto, H.

H. Susanto and M. Johansson, "Discrete dark solitons with multiple holes," Phys. Rev. E 72, 016605 (2005).
[CrossRef]

Takeno, S.

V. V. Konotop and S. Takeno, "Stationary dark localized modes: Discrete nonlinear Schrödinger equations," Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Theodorakis, S.

S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
[CrossRef]

Torner, L.

Vicencio, R. A.

Vysloukh, V. A.

Zafrany, A.

A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
[CrossRef]

Chaos (1)

A. Zafrany, B. A. Malomed, and I. M. Merhasin, "Solitons in a linearly coupled system with separated dispersion and nonlinearity," Chaos 15, 037108 (2005).
[CrossRef]

Europhys. Lett. (1)

G. L. Alfimov, V. V. Konotop, and M. Salerno, "Matter solitons in Bose-Einstein condensates with optical lattices," Europhys. Lett. 58, 7 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Matter-wave dark solitons in optical lattices," J. Opt. B: Quantum Semiclassical Opt. 6, S309 (2004).
[CrossRef]

J. Phys. A (1)

S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (9)

D. N. Christodouldes and E. D. Eugenieva, "Minimizing bending losses in two-dimensional discrete soliton networks," Opt. Lett. 23,1876, (2001).
[CrossRef]

Y. Kominis and K. Hizanidis, "Lattice solitons in self-defocusing optical media: Analytical solutions," Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

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

E. D. Eugenieva, N. Efremidis, and D. N. Christodoulides "Design of switching junctions for two-dimensional discrete soliton network," Opt. Lett. 26, 1978 (2001).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112 (2002).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Controlled switching of discrete solitons in waveguide arrays," Opt. Lett. 28, 1942 (2003).
[CrossRef] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, "Parametric amplification of soliton steering in optical lattices," Opt. Lett. 29, 1102 (2004).
[CrossRef] [PubMed]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890 (2004).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefrigent wave guide arrays," Opt. Lett. 29, 2905 (2004).
[CrossRef]

Phys. Rev. A (1)

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, "Nonlinear excitations in arrays in Bose-Einstein condensates," Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Phys. Rev. E (6)

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, "Dark solitons in discrete lattices," Phys. Rev. E 50, 5020 (1994).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals," Phys. Rev. E 65, 036609 (2002).
[CrossRef]

Y. Kominis, "Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures," Phys. Rev. E 73, 066619 (2006).
[CrossRef]

V. V. Konotop and S. Takeno, "Stationary dark localized modes: Discrete nonlinear Schrödinger equations," Phys. Rev. E 60, 1001 (1999).
[CrossRef]

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation," Phys. Rev. E 71, 036627 (2005).
[CrossRef]

H. Susanto and M. Johansson, "Discrete dark solitons with multiple holes," Phys. Rev. E 72, 016605 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
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[CrossRef]

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[CrossRef] [PubMed]

Other (3)

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[CrossRef]

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[CrossRef]

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Figures (15)

Fig. 1.
Fig. 1.

A binary ribbed AlGaAs structure (W and S are Aluminum mole fraction respectively in the ribbed and the substrate regions). The shaded areas sketch coupled beams of light.

Fig. 2.
Fig. 2.

Diffraction condition for the existence of propagating modes DD(shaded areas) for selffocusing (first row) and self-de focusing (second row) nonlinearity, for interlaced lattices (a, b, d, e) and single nonlinear (c, f).

Fig. 3.
Fig. 3.

Gray scale diagram of the MI growth rate for the upper edge of the Brillouin zone (q=π) as function of the effective propagation constant of the original modulationally unstable CW, κ, and the spatial frequency of the modulation Q for self-focusing nonlinearity (σ=1) and Δε=-1.

Fig. 4.
Fig. 4.

(a), (c): Growth rates for κ=0.5, 2 (respectively within the secondary and the main lobes shown in Fig. 3). (b), (d): The respective propagation constants; the unstable modes shown in gray.

Fig. 5.
Fig. 5.

Propagation of two perturbed modes with initial (ζ=0) amplitudes satisfying Eq.(7). The perturbations on the amplitude and phase at ζ=0 are random within ±5%.: (a) corresponding to Fig. 4c-d with κ=2 (within the instability region); (b) stable propagation of a perturbed mode with κ=4 (beyond the instability region shown in Fig. 3).

Fig. 6.
Fig. 6.

Gray scale diagram of the MI growth rate for the lower edge of the Brillouin zone (q=0) as function of the effective propagation constant of the original modulationally unstable CW, κ, and the spatial frequency of the modulation Q for self-focusing nonlinearity (σ=1) and Δε=-1.

Fig. 7.
Fig. 7.

(a), (c): Growth rates for κ=1.7, 3.5 (respectively near the edge and in the bulk of the instability region shown in Fig. 6). (b), (d): The respective propagation constants; the unstable modes shown in gray.

Fig. 8.
Fig. 8.

Propagation of three perturbed modes [the perturbations on the amplitude and phase at the launching point (ζ=0) are random within ±5%.]: (a), (b): unstable, corresponding to Fig. 7; (c) stable propagation of a perturbed mode with κ=0.4 (that is, beyond the instability region shown in Fig. 6).

Fig. 9.
Fig. 9.

Gray scale diagram of the MI growth rate within the Brillouin zone [near the upper edge of (q=2π/3)] as function of the effective propagation constant of the original modulationally unstable CW, κ, and the spatial frequency of the modulation Q for Δε=-1 and: (a) σ=1, (b) σ=- 1.

Fig. 10.
Fig. 10.

D-S0 diagram of the fixed points loci for various values of Δε. The red line corresponds to Δε=0. (a): for σcos[Δφ(ζ=0)]>0 [that is, for focusing (defocusing) nonlinearity and phase difference 0 (π)] there exist three fixed points beyond a particular value of the constant of propagation, S0 ; (b): for σcos[Δφ(ζ=0)]<0 [that is, for focusing (defocusing) nonlinearity and phase difference π (0)] D(S0) there is always a single fixed point.

Fig. 11.
Fig. 11.

Δε-S0 diagram for the nature and the number of fixed points. (a), (b): Δφ=0, σ=1 (both elliptic and hyperbolic fixed points exist); (c) Δφ=0, σ=-1 (only elliptic fixed points exist); (d) Δφ=π, σ=1 (only elliptic fixed points exist); (e), (f): Δφ=π, σ=-1 (only hyperbolic point exist).

Fig. 12.
Fig. 12.

A case of three fixed points for σ=1, Δφ(ζ=0)=0, Δε=-2.7, two elliptic (O, red and green) and one hyperbolic (X). The constant of propagation is set to S0 =5.76, while the other constant of propagation, namely the Hamiltonian, H0 , acquires the values indicated. The red contour is the separatrix.

Fig. 13.
Fig. 13.

Localized modes with propagation constant above the acoustic branch (κ=0.85) for Δε=1.15 and self-focusing nonlinearity. (a,b): intensity profile upon propagation, (c,d): initial (at ζ=0) profiles; The latter are perturbed randomly in amplitude and phase within 10%.

Fig. 14.
Fig. 14.

Localized modes with propagation constant below the optical branch (κ=-3) for Δε=1.15 and self-defocusing nonlinearity. (a,b): intensity profile upon propagation, (c,d): initial (at ζ=0) profiles; The latter are perturbed randomly in amplitude and phase within 5%.

Fig. 15.
Fig. 15.

Localized modes with propagation constant inside the bounded band-gap (κ=0.46) for Δε=-1.15 and self-focusing nonlinearity. (a,b): intensity profile upon propagation, (c,d): initial (at ζ=0) profiles; The latter are perturbed randomly in amplitude and phase within 5%.

Equations (35)

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i E z + 1 2 β 0 2 E x 2 + k 0 Δ n ( x ) E + k 0 n 2 ( x ) E 2 E = 0 ,
i d A 2 m dz + C even ( A 2 m 1 + A 2 m + 1 ) + V even A 2 m + G even A 2 m 2 A 2 m = 0
i d A 2 m ± 1 dz + C odd ( A 2 m + A 2 m ± 2 ) + V odd A 2 m ± 1 + G odd A 2 m ± 1 2 A 2 m ± 1 = 0
i d ψ 2 m d ζ + 1 2 ( ψ 2 m 1 + ψ 2 m + 1 2 ψ 2 m ) + V even 2 C ψ 2 m + ψ 2 m 2 ψ 2 m = 0
i d ψ 2 m ± 1 d ζ + 1 2 ( ψ 2 m + ψ 2 m ± 2 2 ψ 2 m ± 1 ) + V odd 2 C ψ 2 m ± 1 + G odd G even ψ 2 m ± 1 2 ψ 2 m ± 1 = 0
i d ψ 2 m d ζ + 1 2 ( ψ 2 m 1 + ψ 2 m + 1 2 ψ 2 m ) + ε even ψ 2 m + σ ψ 2 m ψ 2 m 2 = 0
i d ψ 2 m ± 1 d ζ + 1 2 ( ψ 2 m ± 2 + ψ 2 m 2 ψ 2 m ± 1 ) + ε odd ψ 2 m ± 1 = 0
i u ζ + ( ε even 1 ) u + v + σ u u 2 = 0
i v ζ + ( ε odd 1 ) v + u + ε 4 2 u χ 2 = 0
( k z + 1 ε even ) v 2 m + 1 2 ( v 2 m 1 + v 2 m + 1 ) + σ v 2 m v 2 m 2 = 0
( k z + 1 ε odd ) v 2 m ± 1 + 1 2 ( v 2 m ± 2 + v 2 m ) = 0
[ k z + 1 ε even 1 2 ( k z + 1 ε odd ) ] v 2 m + v 2 m + 2 + v 2 m 2 4 ( k z + 1 ε odd ) + σ v 2 m v 2 m 2 = 0
σ 2 ( 2 κ 1 + cos q κ + Δ ε ) = A 2 0
σ ( κ cos q ) = A 2 0
ψ ˜ 2 m = [ A + u e i ( Γ ζ 2 m Q ) + w * e i ( Γ * ζ 2 m Q ) ] e i ( k z ζ m q ) ,
ψ ˜ 2 m ± 1 = [ A 1 + e iq κ + Δ ε + u 1 + e i ( q + 2 Q ) κ + Δ ε + Γ e i ( Γ ζ 2 mQ ) + w * 1 + e i ( q 2 Q ) κ + Δ ε Γ * e ( Γ * ζ 2 mQ ) ] e i ( k z ζ mq ) 2
0 = { [ κ Γ + 1 + cos ( q + 2 Q ) 2 ( κ + Δ ε + Γ ) + 2 σ A 2 ] u + σ A 2 w } e i ( Γ ζ 2 mQ )
+ { [ κ + Γ * + 1 + cos ( q 2 Q ) 2 ( κ + Δ ε Γ * ) + 2 σ A 2 ] w * + σ A 2 u * } e i ( Γ * ζ 2 mQ )
[ κ Γ + 1 + cos ( q + 2 Q ) 2 ( κ + Δ ε + Γ ) + 2 σ A 2 ] [ κ + Γ + 1 + cos ( q 2 Q ) 2 ( κ + Δ ε Γ ) + 2 σ A 2 ] = A 4 .
ψ 2 m + 1 = ψ odd ( ζ ) = ρ odd ( ζ ) exp [ i ϕ odd ( ζ ) ] , ψ 2 m = ψ even ( ζ ) = ρ even ( ζ ) exp [ i ϕ even ( ζ ) ] .
d ρ even d ζ = ρ odd sin Δ ϕ , d ρ odd d ζ = ρ even sin Δ ϕ
d Δ ϕ d ζ = ( ρ odd ρ even ρ even ρ odd ) cos Δ ϕ + Δ ε + σ ρ even 2
Δ ϕ ϕ even ϕ odd , Δ ε ε even ε odd , S ρ even 2 + ρ odd 2 = constant S 0 > 0
d Δ ϕ d ζ = D ρ even ρ odd cos Δ ϕ + Δ ε + σ ρ even 2
d D d ζ = 4 ρ even ρ odd sin Δ ϕ
ρ even 2 = S + D 2 , ρ odd 2 = S D 2
d Δ ϕ d ζ = 2 D S 0 2 D 2 cos Δ ϕ + Δ ε + σ 2 ( S 0 + D ) , d D d ζ = 2 S 0 2 D 2 sin Δ ϕ
d ϕ d ζ = ε + σ ρ 2 , ρ 2 = S 0 2 = constant , ψ ( ζ ) = ρ exp [ i ( ε + σ ρ 2 ) ζ ]
Δ ε = 2 D S 0 2 D cos Δ ϕ 0 σ 2 ( S 0 + D )
( D , Δ ϕ ) = 2 S 0 2 D 2 cos Δ ϕ + ( Δ ε + σ 2 S 0 ) D + σ 4 D 2 = constant = H 0
d Δ ϕ d ζ = D , dD ζ = Δ ϕ
ζ 4 = ± D ( ζ = 0 ) D ( ζ ) dw a 0 + a 1 w a 2 w 2 a 3 w 3 w 4
a 0 = 16 ( 4 S 0 2 H 0 2 ) , a 1 = 16 H 0 σ ( S 0 + 2 σ Δ ε )
a 2 = 4 S 0 ( S 0 + 4 σ Δ ε ) + 8 ( 8 + 2 Δ ε 2 σ H 0 ) , a 3 = 4 ( S 0 + 2 σ Δ ε )
Δ = 2 D 2 2 Δ ϕ 2 ( 2 D Δ ϕ ) 2

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