Abstract

We address basic properties and stability of two-dimensional solitons in photonic lattices induced by the nondiffracting Mathieu beams. Such lattices allow for smooth topological transformation of radially symmetric Bessel lattices into quasi-one-dimensional periodic ones. The transformation of lattice topology drastically affects the properties of ground-state and dipole-mode solitons, including their shape, stability, and transverse mobility.

© 2006 Optical Society of America

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  1. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  2. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
    [CrossRef] [PubMed]
  3. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
    [CrossRef]
  4. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
    [CrossRef] [PubMed]
  5. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
    [CrossRef] [PubMed]
  6. J. Yang and Z. Musslimani, Opt. Lett. 28, 2094 (2003).
    [CrossRef] [PubMed]
  7. Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, Opt. Lett. 29, 1918 (2004).
    [CrossRef] [PubMed]
  8. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
    [CrossRef] [PubMed]
  9. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
    [CrossRef] [PubMed]
  10. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
    [CrossRef] [PubMed]
  11. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
    [CrossRef] [PubMed]
  12. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
    [CrossRef]
  13. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 30, 637 (2005).
    [CrossRef] [PubMed]
  14. Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Express 13, 1774 (2005).
    [CrossRef] [PubMed]
  15. Z. Xu, Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 30, 1180 (2005).
    [CrossRef] [PubMed]
  16. J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, Opt. Lett. 25, 1493 (2000).
    [CrossRef]
  17. J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
    [CrossRef]
  18. S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
    [CrossRef]

2005 (3)

2004 (7)

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, Opt. Lett. 29, 1918 (2004).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
[CrossRef] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

2003 (4)

J. Yang and Z. Musslimani, Opt. Lett. 28, 2094 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
[CrossRef] [PubMed]

2002 (2)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

2001 (1)

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

2000 (1)

Alexander, T. J.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Allison, I.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Bartal, G.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Bezryadina, A.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Chavez-Cerda, S.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, Opt. Lett. 25, 1493 (2000).
[CrossRef]

Chen, Z.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Christodoulides, D. N.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, Opt. Lett. 29, 1918 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Cohen, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Courtial, J.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Egorov, A. A.

Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, Opt. Lett. 29, 1918 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Fleischer, J. W.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Gutierrez-Vega, J. C.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, Opt. Lett. 25, 1493 (2000).
[CrossRef]

Hudock, J.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, Opt. Lett. 25, 1493 (2000).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y.

Kivshar, Y. S.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Krolikowski, W.

MacVicar, I.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Makasyuk, I.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
[CrossRef] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Manela, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Martin, H.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Musslimani, Z.

Neshev, D.

Neshev, D. N.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

New, G. H. C.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

O'Neil, A. T.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Ostrovskaya, E.

Ostrovskaya, E. A.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Padgett, M. J.

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Ramirez, G. A.

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Rodriguez-Dagnino, R. M.

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Tepichin, E.

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Torner, L.

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 30, 637 (2005).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 30, 1180 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

Xu, Z.

Yang, J.

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

S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, J. Opt. B: Quantum Semiclassical Opt. 4, S52 (2002).
[CrossRef]

Nature (London) (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003).
[CrossRef]

Opt. Commun. (1)

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez-Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. E (2)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Phys. Rev. Lett. (5)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Lowest-order even (top) and odd (bottom) Mathieu lattices with different values of interfocal parameter a.

Fig. 2
Fig. 2

Top row, ground-state soliton in an even lattice at b = 2.9 , a = 500 , p = 5 (left) and dipole-mode soliton in an odd lattice at b = 2.84 , a = 30 , p = 5 (right). Focusing medium. Bottom row, simplest gap soliton in an even lattice at b = 3.2 , a = 12 , p = 15 (left) and combination of in-phase gap solitons in an odd lattice at b = 3.2 , a = 12 , p = 15 (right). Defocusing medium.

Fig. 3
Fig. 3

Properties of ground-state solitons in even Mathieu lattices. (a) Energy flow versus propagation constant. (b) Propagation constant cutoff versus interfocal parameter. (c) Boundaries of the instability domain versus the interfocal parameter. (d) Real part of the perturbation growth rate versus the propagation constant. In all cases p = 1 . Focusing medium.

Fig. 4
Fig. 4

Properties of dipole-mode solitons in odd Mathieu lattices. (a) Energy flow versus propagation constant. (b) Propagation constant cutoff versus the interfocal parameter. (c) Real part of the perturbation growth rate versus the propagation constant. (d) Trajectory of the center of a ground-state soliton with b = 4 and input tilt α η = 1 launched into even lattices corresponding to different a. In all cases p = 5 . Focusing medium.

Equations (2)

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i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) + σ q q 2 p R ( η , ζ ) q .
q M ( η , ζ , ξ ) = exp ( i b lin ξ ) A ( ϕ ) exp [ i ( 2 b lin ) 1 2 ( η cos ϕ + ζ sin ϕ ) ] d ϕ ,

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