Abstract

We study light propagation in arrays of weakly coupled nonlinear cavities driven by an inclined holding beam. We show analytically that both discreteness and inclination of the driving field can dramatically change the conditions for modulational instability in discrete nonlinear systems. We find numerically the families of resting and moving dissipative solitons for an arbitrary inclination angle of the driving field, both in the discrete and a quasi-continuous limits. We analyze a crossover between resting and moving cavity solitons, and also observe novel features in the soliton collision.

© 2007 Optical Society of America

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  1. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Waveguides to Photonic Crystals (Academic Press, San Diego, 2003), 540 pp.
  2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  3. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Berlin, 2005), pp. 450.
    [CrossRef]
  4. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
    [CrossRef]
  5. D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
    [CrossRef]
  6. U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
    [CrossRef]
  7. E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
    [CrossRef] [PubMed]
  8. E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
    [CrossRef] [PubMed]
  9. J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
    [CrossRef]
  10. K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
    [CrossRef]
  11. N. K. Efremidis and D. N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 26606 (2003).
    [CrossRef]
  12. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
    [CrossRef]
  13. K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
    [CrossRef]
  14. A. Mohamadou and T. C. Kofane, "Modulational instability and pattern formation in discrete dissipative systems," Phys. Rev. E 73, 46607 (2006).
    [CrossRef]
  15. U. Peschel, O. A. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909 (2004).
    [CrossRef] [PubMed]
  16. O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
    [CrossRef]
  17. O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
    [CrossRef]
  18. L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef] [PubMed]
  19. Yu. S. Kivshar and M. Peyrard, "Modulational instabilities in discrete lattices," Phys. Rev. A 46, 3198 (1992).
    [CrossRef] [PubMed]
  20. Yu. S. Kivshar and M. Salerno, "Modulational instabilities in the discrete deformable nonlinear Schr¨odinger equation," Phys. Rev. E 49, 3543 (1994).
    [CrossRef]
  21. S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
    [CrossRef]
  22. M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
    [CrossRef]
  23. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
    [CrossRef] [PubMed]
  24. R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, "Discrete modulational instability in periodically poled lithium niobate waveguide arrays," Opt. Express 13, 7794 (2005).
    [CrossRef] [PubMed]
  25. M. Stepic, C. Wirth, C. E. Rueter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31, 247 (2006).
    [CrossRef] [PubMed]
  26. H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, "Optical discrete solitons in waveguide arrays. I. Solitons formation," J. Opt. Soc. Am. B 19, 2938 (2002).
    [CrossRef]
  27. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. II. Dynamic properties," J. Opt. Soc. Am. B 19, 2637 (2002).
    [CrossRef]
  28. Yu. S. Kivshar, "Self-localization in arrays of defocusing waveguides," Opt. Lett. 18, 1147 (1993).
    [CrossRef] [PubMed]
  29. S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
    [CrossRef]
  30. M. Tlidi and A. G. Vladimirov, "Interaction and stability of periodic and localized structures in optical bistable systems," IEEE J. Quantum. Electron. 39, 216 (2003).
    [CrossRef]
  31. K. Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 53901 (2003).
    [CrossRef]

2006 (3)

A. Mohamadou and T. C. Kofane, "Modulational instability and pattern formation in discrete dissipative systems," Phys. Rev. E 73, 46607 (2006).
[CrossRef]

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

M. Stepic, C. Wirth, C. E. Rueter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31, 247 (2006).
[CrossRef] [PubMed]

2005 (4)

R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, "Discrete modulational instability in periodically poled lithium niobate waveguide arrays," Opt. Express 13, 7794 (2005).
[CrossRef] [PubMed]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
[CrossRef]

2004 (2)

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

U. Peschel, O. A. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909 (2004).
[CrossRef] [PubMed]

2003 (9)

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

M. Tlidi and A. G. Vladimirov, "Interaction and stability of periodic and localized structures in optical bistable systems," IEEE J. Quantum. Electron. 39, 216 (2003).
[CrossRef]

K. Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 53901 (2003).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 26606 (2003).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
[CrossRef] [PubMed]

U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
[CrossRef]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (1)

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

2000 (1)

J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
[CrossRef]

1997 (3)

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
[CrossRef]

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
[CrossRef]

1994 (1)

Yu. S. Kivshar and M. Salerno, "Modulational instabilities in the discrete deformable nonlinear Schr¨odinger equation," Phys. Rev. E 49, 3543 (1994).
[CrossRef]

1993 (1)

1992 (1)

Yu. S. Kivshar and M. Peyrard, "Modulational instabilities in discrete lattices," Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

1987 (1)

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Aitchison, J. S.

Akhmediev, N.

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
[CrossRef]

Ankiewicz, A.

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
[CrossRef]

Arnold, J. M.

Brambilla, M.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

Christodoulides, D. N.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
[CrossRef] [PubMed]

N. K. Efremidis and D. N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 26606 (2003).
[CrossRef]

Darmanyan, S.

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
[CrossRef]

Dizes, S.

J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
[CrossRef]

Efremidis, N. K.

N. K. Efremidis and D. N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 26606 (2003).
[CrossRef]

Egorov, O. A.

O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
[CrossRef]

U. Peschel, O. A. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909 (2004).
[CrossRef] [PubMed]

Eisenberg, H. S.

Etrich, C.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Fedorov, S.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Firth, W. J.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

Gal, P.

J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
[CrossRef]

Hadzievski, L.

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

Iwanow, R.

Kip, D.

M. Stepic, C. Wirth, C. E. Rueter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31, 247 (2006).
[CrossRef] [PubMed]

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and M. Salerno, "Modulational instabilities in the discrete deformable nonlinear Schr¨odinger equation," Phys. Rev. E 49, 3543 (1994).
[CrossRef]

Yu. S. Kivshar, "Self-localization in arrays of defocusing waveguides," Opt. Lett. 18, 1147 (1993).
[CrossRef] [PubMed]

Yu. S. Kivshar and M. Peyrard, "Modulational instabilities in discrete lattices," Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Kofane, T. C.

A. Mohamadou and T. C. Kofane, "Modulational instability and pattern formation in discrete dissipative systems," Phys. Rev. E 73, 46607 (2006).
[CrossRef]

Lange, C.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Lederer, F.

O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
[CrossRef]

U. Peschel, O. A. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909 (2004).
[CrossRef] [PubMed]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
[CrossRef] [PubMed]

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. II. Dynamic properties," J. Opt. Soc. Am. B 19, 2637 (2002).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
[CrossRef]

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
[CrossRef]

Lefever, R.

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Lugiato, L. A.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Maluckov, A.

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

Maruno, K.

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
[CrossRef]

Meier, J.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

Michaelis, D.

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
[CrossRef]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Min, Y.

Mohamadou, A.

A. Mohamadou and T. C. Kofane, "Modulational instability and pattern formation in discrete dissipative systems," Phys. Rev. E 73, 46607 (2006).
[CrossRef]

Morandotti, R.

Pennelli, G.

Pertsch, T.

Peschel, U.

O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
[CrossRef]

U. Peschel, O. A. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909 (2004).
[CrossRef] [PubMed]

U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. II. Dynamic properties," J. Opt. Soc. Am. B 19, 2637 (2002).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Peyrard, M.

Yu. S. Kivshar and M. Peyrard, "Modulational instabilities in discrete lattices," Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

Ravoux, J. F.

J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
[CrossRef]

Relke, I.

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
[CrossRef]

Rosanov, N.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Rueter, C. E.

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

M. Stepic, C. Wirth, C. E. Rueter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31, 247 (2006).
[CrossRef] [PubMed]

Salamo, G.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

Salerno, M.

Yu. S. Kivshar and M. Salerno, "Modulational instabilities in the discrete deformable nonlinear Schr¨odinger equation," Phys. Rev. E 49, 3543 (1994).
[CrossRef]

Schiek, R.

Silberberg, Y.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. II. Dynamic properties," J. Opt. Soc. Am. B 19, 2637 (2002).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, "Optical discrete solitons in waveguide arrays. I. Solitons formation," J. Opt. Soc. Am. B 19, 2938 (2002).
[CrossRef]

Skryabin, D. V.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Sohler, W.

Sorel, M.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
[CrossRef]

Spinelli, L.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

Staliunas, K.

K. Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 53901 (2003).
[CrossRef]

Stegeman, G. I.

R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, "Discrete modulational instability in periodically poled lithium niobate waveguide arrays," Opt. Express 13, 7794 (2005).
[CrossRef] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Stepic, M.

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

M. Stepic, C. Wirth, C. E. Rueter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31, 247 (2006).
[CrossRef] [PubMed]

Tlidi, M.

M. Tlidi and A. G. Vladimirov, "Interaction and stability of periodic and localized structures in optical bistable systems," IEEE J. Quantum. Electron. 39, 216 (2003).
[CrossRef]

Ultanir, E. A.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, "Stable spatial solitons in semiconductor optical amplifiers," Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

Vladimirov, A. G.

M. Tlidi and A. G. Vladimirov, "Interaction and stability of periodic and localized structures in optical bistable systems," IEEE J. Quantum. Electron. 39, 216 (2003).
[CrossRef]

Weiss, C. O.

U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
[CrossRef]

Wirth, C.

Yang, H.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

U. Peschel, D. Michaelis, and C. O. Weiss, "Spatial solitons in optical cavities," IEEE J. Quantum Electron. 39, 51 (2003).
[CrossRef]

IEEE J. Quantum. Electron. (1)

M. Tlidi and A. G. Vladimirov, "Interaction and stability of periodic and localized structures in optical bistable systems," IEEE J. Quantum. Electron. 39, 216 (2003).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Commun. (2)

M. Stepic, C. E. Rueter, D. Kip, A. Maluckov, and L. Hadzievski, "Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity," Opt. Commun. 267, 229 (2006).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equations," Opt. Commun. 221, 199 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (2)

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, "Motion and stability properties of solitons in discrete dissipative structures," Phys. Lett. A 314, 126 (2003).
[CrossRef]

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Dissipative solitons of the discrete complex cubic-quintic Ginzburg-Landau equation," Phys. Lett. A 347, 231 (2005).
[CrossRef]

Phys. Rev. A (2)

D. Michaelis, U. Peschel, and F. Lederer, "Multistable localized structures and superlattices in semiconductor optical resonators," Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Yu. S. Kivshar and M. Peyrard, "Modulational instabilities in discrete lattices," Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (8)

Yu. S. Kivshar and M. Salerno, "Modulational instabilities in the discrete deformable nonlinear Schr¨odinger equation," Phys. Rev. E 49, 3543 (1994).
[CrossRef]

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662 (1997).
[CrossRef]

J. F. Ravoux, S. Dizes, and P. Gal, "Stability analysis of plane wave solutions of the discrete Ginzburg-Landau equation," Phys. Rev. E 61, 390 (2000).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 26606 (2003).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 56612 (2005).
[CrossRef]

O. A. Egorov, U. Peschel, and F. Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 66603 (2005).
[CrossRef]

A. Mohamadou and T. C. Kofane, "Modulational instability and pattern formation in discrete dissipative systems," Phys. Rev. E 73, 46607 (2006).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Phys. Rev. Lett. (5)

K. Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 53901 (2003).
[CrossRef]

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, "Spatial soliton pixels in semiconductor devices," Phys. Rev. Lett. 79, 2042 (1997).
[CrossRef]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92, 163902 (2004).
[CrossRef] [PubMed]

Other (2)

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Waveguides to Photonic Crystals (Academic Press, San Diego, 2003), 540 pp.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Berlin, 2005), pp. 450.
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (2102 KB)     
» Media 2: AVI (2526 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Array of coupled waveguide cavities endowed with a Kerr nonlinearity.

Fig. 2.
Fig. 2.

(a) Modulus |b| of the plane wave amplitude vs. the holding beam amplitude E 0 (the dotted section indicates homogeneous instability - Q = 0); (b) modulationally unstable plane wave domains in |b| - q parameters space; shaded regions between the full and dashed lines depict MI domains for various Q values, whereas the dotted lines mark the boundaries of homogenous instability. Regions of MI in q - Q parameter space for (c) the upper (|b| = 2) and (d) the lower (|b| = 1.05) branch; parameters: Δ′ = -3, C = 1.

Fig. 3.
Fig. 3.

Families of discrete cavity solitons shown as the modulus of the field amplitude at the center |u| vs. the amplitude of the holding beam E 0 for different phase shifts (holding beam inclination) q: (a) q = 0, (b) q = π/2, and (c) q = π. Plane wave solutions are shown by black lines where the soliton branches emanate subcritically. The solid parts of these branches correspond to stable and the dashed ones to unstable solitons; parameters: Δ′ = -3 and C = 1.

Fig. 4.
Fig. 4.

(a) Existence domains of stable resting DCSs (red areas) in q - |b| (modulus of background amplitude) parameter space. Arrows point at the site where the respective DCS shown in the right panel exist. Profiles of stable resting DCSs: (b) Bright DCS for E 0 = 2, q = 0.12π, (c) dark DCS for E 0 = 9.4, q = 0; (d) dark DCS for E 0 = 3, q = π/2; and (e) dark DCS for E 0 = 1.8, q = π. Other parameters are: Δ′ = -3 and C = 1. The shaded area displays the MI domain and the dotted lines mark the limits of homogeneous instability.

Fig. 5.
Fig. 5.

(a) Existence domains of moving DCSs (blue areas) in q - |b| parameter space (dashed lines encircle domains of resting DCSs). Arrows point to the sites where DCSs shown on the right exist. (b,c) Profiles of bright (b) and dark moving DCSs (c) for the inclination parameter q = π/2, calculated both in the quasi-continuous (solid) and discrete models (diamonds). A movie (2.1MB) shows propagation of bright DCS (b) [Media 1]. Other parameters are: E 0 = 1.95, Δ′ = -3, and C = 1

Fig. 6.
Fig. 6.

Dependence of the velocity of bright DCS vs. the inclination q of the holding beam, calculated for the effective continuous model (5) (solid) and discrete model (1) (rhombuses). Parameters are: E 0 = 1.92, Δ′ = -3, and C = 1. Inserts show the soliton profiles.

Fig. 7.
Fig. 7.

A movie (2.5MB) shows collision between a resting and two moving DCSs and the formation of multi-soliton bound state. Parameters are: q = 0.204π, E 0 = 1.92, Δ′ = -3, and C = 1. [Media 2]

Equations (5)

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

i u n T + C ( u n + 1 + u n 1 2 u n ) + ( i + Δ ) u n + γ u n 2 u n = E 0 e iqn .
u n ( T ) = ( b + a e λT + iQn ) e iqn .
λ ( Q , q ) = 1 ± i ( Λ + γ b 2 ) ( Λ + 3 γ b 2 ) 2 iC sin Q sin q ,
b 1,2 2 = 1 3 γ ( 2 Λ ± Λ 2 3 ) .
i u T + i D ( 1 ) u x + D ( 2 ) 2 u x 2 + i D ( 3 ) 3 u x 3 + ( i + Δ ) u + γ u 2 u = E 0 ,

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