## Formation of discrete solitons in light-induced photonic lattices

Optics Express, Vol. 13, Issue 6, pp. 1816-1826 (2005)

http://dx.doi.org/10.1364/OPEX.13.001816

Acrobat PDF (286 KB)

### Abstract

We present both experimental and theoretical results on discrete solitons in two-dimensional optically-induced photonic lattices in a variety of settings, including fundamental discrete solitons, vector-like discrete solitons, discrete dipole solitons, and discrete soliton trains. In each case, a clear transition from two-dimensional discrete diffraction to discrete trapping is demonstrated with a waveguide lattice induced by partially coherent light in a bulk photorefractive crystal. Our experimental results are in good agreement with the theoretical analysis of these effects.

© 2005 Optical Society of America

## 1. Introduction

1. D.N. Christodoulides, F. Lederer, and Y. Siberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817 (2003). [CrossRef] [PubMed]

2. D. Campbell, S. Flach, and Y.S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today , **57**, 43 (2004). [CrossRef]

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

4. H.S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Observation of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. **81**, 3383 (1998). [CrossRef]

5. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-Focusing and Defocusing in Waveguide Arrays,” Phys. Rev. Lett. **86**, 3296 (2001). [CrossRef] [PubMed]

6. N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

7. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,” Phys. Rev. Lett. **90**, 023902 (2003). [CrossRef] [PubMed]

9. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710 (2003). [CrossRef] [PubMed]

10. H. Martin, E.D. Eugenieva, Z. Chen, and D.N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004); [CrossRef] [PubMed]

11. Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. **92**, 143902 (2004). [CrossRef] [PubMed]

12. B.A. Malomed and P.G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E **64**, 026601 (2001). [CrossRef]

13. J. Yang and Z.H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**, 2094 (2003). [CrossRef] [PubMed]

14. D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically-induced photonic lattices,” Phys. Rev. Lett. **92**, 123903 (2004). [CrossRef] [PubMed]

15. J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D.N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices.” Phys. Rev. Lett. **92**, 123904 (2004). [CrossRef] [PubMed]

*coherent*multi-beam interference [7

7. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,” Phys. Rev. Lett. **90**, 023902 (2003). [CrossRef] [PubMed]

9. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710 (2003). [CrossRef] [PubMed]

*incoherent*photonic lattice. This in turn enables stable lattice formation due to suppression of incoherent modulation instability [16

16. M. Soljacic, M Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. **84**, 467 (2000). [CrossRef] [PubMed]

10. H. Martin, E.D. Eugenieva, Z. Chen, and D.N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004); [CrossRef] [PubMed]

11. Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. **92**, 143902 (2004). [CrossRef] [PubMed]

^{3}, r

_{33}=280 pm/V and r

_{13}=24 pm/V) illuminated by an argon laser beam (λ=488 nm) passing through a rotating diffuser and an amplitude mask as shown in Fig. 1. The biased crystal provides a self-focusing noninstantaneous nonlinearity. The amplitude mask provides spatial modulation after the diffuser on the otherwise uniform beam, which exhibits a pixel-like intensity pattern at the input face of the crystal [17

17. Z. Chen and K. MaCarthy, “Spatial soliton pixles from partially coherent light,” Opt. Lett. **27**, 2019 (2002). [CrossRef]

18. Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. , **29**, 1656 (2004). [CrossRef] [PubMed]

19. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole solitons in optically-induced two-dimensional photonic lattices,” Opt. Lett. , **29**, 1662 (2004). [CrossRef] [PubMed]

*n*

_{e}=[

_{33}

*E*

_{0}/2](1+I)

^{-1}and Δ

*n*

_{o}=[

*r*

_{13}

*E*

_{0}/2](1+I)

^{-1}for e-polarized and o-polarized beams, respectively [6

6. N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

11. Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. **92**, 143902 (2004). [CrossRef] [PubMed]

*E*

_{0}is the applied electric field along the crystalline c-axis (x-direction), and

*I*is the intensity of the beam normalized to the background illumination. Due to the difference between the nonlinear electro-optic coefficient

*r*

_{33}and

*r*

_{13}, Δ

*n*

_{e}is more than 10 times larger than Δ

*n*

_{o}under the same experimental conditions. Thus, the o-polarized lattice beam experiences only weak nonlinear index changes as compared with the e-polarized probe beam, so the lattice can be considered as

*linear*during propagation. In Fig. 1, the insert shows a typical example of a 2D lattice pattern created in experiment. The square lattice has its principal axes orientated in the 450 directions relative to the x- and y-axis, with a spatial period of 20 µm. Indeed, as the bias field is increased to more than 3 kV/cm, the lattice structure remains nearly invariant, except a slight change in its contrast at high bias due to nonzero

*r*

_{13}.

## 2. Fundamental discrete solitons

8. J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147 (2003). [CrossRef] [PubMed]

13. J. Yang and Z.H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**, 2094 (2003). [CrossRef] [PubMed]

20. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. **29**, 268 (2004). [CrossRef] [PubMed]

21. D. N. Christodoulides, T. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. **78**, 646 (1997). [CrossRef]

10. H. Martin, E.D. Eugenieva, Z. Chen, and D.N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004); [CrossRef] [PubMed]

## 3. Discrete dipole and vector solitons

22. M.J. Ablowitz and Z.H. Musslimani, “Discrete vector spatial solitons in a nonlinear waveguide array,” Phys. Rev. E **65**, 056618 (2002). [CrossRef]

23. J. Hudock, P. G. Kevrekidis, B. A. Malomed, and D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: Solutions, stability, and dynamics,” Phys. Rev. E **67**, 056618 (2003). [CrossRef]

24. J. Meier, J. Hudock, D.N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays ”. Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

18. Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. , **29**, 1656 (2004). [CrossRef] [PubMed]

18. Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. , **29**, 1656 (2004). [CrossRef] [PubMed]

19. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole solitons in optically-induced two-dimensional photonic lattices,” Opt. Lett. , **29**, 1662 (2004). [CrossRef] [PubMed]

29. S.R. Singh and D.N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. **118**, 569 (1995). [CrossRef]

25. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically-induced Two-dimensional Photonic Lattices: Theory and Experiment,” *Stud. Appl. Math*. **113**, 389 (2004). [CrossRef]

## 4. Discrete soliton trains

*linear*regime, we observe that the stripe beam breaks up into 2D filaments, and then it evolves into a train of 2D discrete solitons as the level of the nonlinearity for the stripe is gradually increased.

**92**, 143902 (2004). [CrossRef] [PubMed]

32. Z Chen, H. Martin, A. Bezryadina, D.N. Neshev, Y.S. Kivshar, and D.N. Christodoulides, “Experiments on Gaussian beams and vortices in optically-induced photonic lattices,” J. Opt. Soc. Am. B, to appear 2005. [CrossRef]

## 5. Summary

## Acknowledgments

## References and Links:

1. | D.N. Christodoulides, F. Lederer, and Y. Siberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | D. Campbell, S. Flach, and Y.S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today , |

3. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

4. | H.S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Observation of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. |

5. | R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-Focusing and Defocusing in Waveguide Arrays,” Phys. Rev. Lett. |

6. | N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E |

7. | J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,” Phys. Rev. Lett. |

8. | J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

9. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

10. | H. Martin, E.D. Eugenieva, Z. Chen, and D.N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices,” Phys. Rev. Lett. |

11. | Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. |

12. | B.A. Malomed and P.G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E |

13. | J. Yang and Z.H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. |

14. | D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically-induced photonic lattices,” Phys. Rev. Lett. |

15. | J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D.N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices.” Phys. Rev. Lett. |

16. | M. Soljacic, M Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. |

17. | Z. Chen and K. MaCarthy, “Spatial soliton pixles from partially coherent light,” Opt. Lett. |

18. | Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. , |

19. | J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole solitons in optically-induced two-dimensional photonic lattices,” Opt. Lett. , |

20. | J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. |

21. | D. N. Christodoulides, T. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. |

22. | M.J. Ablowitz and Z.H. Musslimani, “Discrete vector spatial solitons in a nonlinear waveguide array,” Phys. Rev. E |

23. | J. Hudock, P. G. Kevrekidis, B. A. Malomed, and D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: Solutions, stability, and dynamics,” Phys. Rev. E |

24. | J. Meier, J. Hudock, D.N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays ”. Phys. Rev. Lett. |

25. | J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically-induced Two-dimensional Photonic Lattices: Theory and Experiment,” |

26. | P. G. Kevrekidis, A.R. Bishop, and K. Rasmussen, “Twisted localized modes,” Phys. Rev. E |

27. | A.A. Sukhorukov and Y.S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E |

28. | P.G. Kevrekidis, B.A. Malomed, and A.R. Bishop, “Bound states of two-dimensional solitons in the discrete nonlinear Schrodinger equation.” J. Phys. A |

29. | S.R. Singh and D.N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. |

30. | J. Yang, I. Makasyuk, H. Martin, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and Z. Chen, “Necklace-like solitons in optically induced photonic lattices,” Phys. Rev. Lett, submitted. |

31. | A. Smerzi, A. Trombettoni, P. G. Kevrekidis, and A. R. Bishop, Dynamical superfluid-insulator transition in a chain of weakly coupled Bose-Einstein condensates,” Phys. Rev. Lett. |

32. | Z Chen, H. Martin, A. Bezryadina, D.N. Neshev, Y.S. Kivshar, and D.N. Christodoulides, “Experiments on Gaussian beams and vortices in optically-induced photonic lattices,” J. Opt. Soc. Am. B, to appear 2005. [CrossRef] |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(270.5530) Quantum optics : Pulse propagation and temporal solitons

**ToC Category:**

Focus Issue: Discrete solitons in nonlinear optics

**History**

Original Manuscript: December 8, 2004

Revised Manuscript: January 6, 2005

Published: March 21, 2005

**Citation**

Zhigang Chen, Hector Martin, Eugenia Eugenieva, Jingjun Xu, and Jianke Yang, "Formation of discrete solitons in light-induced photonic lattices," Opt. Express **13**, 1816-1826 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1816

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### References

- D.N. Christodoulides, F. Lederer and Y. Siberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices,�?? Nature 424, 817 (2003). [CrossRef] [PubMed]
- D. Campbell, S. Flach and Y.S. Kivshar, 'Localizing energy through nonlinearity and discreteness,�?? Phys. Today, 57, 43 (2004). [CrossRef]
- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]
- H.S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Observation of discrete solitons in optical waveguide arrays,�?? Phys. Rev. Lett. 81, 3383 (1998). [CrossRef]
- R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, "Self-Focusing and Defocusing in Waveguide Arrays,�?? Phys. Rev. Lett. 86, 3296 (2001). [CrossRef] [PubMed]
- N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer and M. Segev, �??Discrete solitons in photorefractive optically induced photonic lattices,�?? Phys. Rev. E 66, 046602 (2002). [CrossRef]
- J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, "Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,�?? Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
- J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,�?? Nature 422, 147 (2003). [CrossRef] [PubMed]
- Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings,�?? Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
- H. Martin, E.D. Eugenieva, Z. Chen and D.N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices,�?? Phys. Rev. Lett. 92, 123902 (2004); [CrossRef] [PubMed]
- Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, and A. Bezryadina, "Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,�?? Phys. Rev. Lett. 92, 143902 (2004). [CrossRef] [PubMed]
- B.A. Malomed and P.G. Kevrekidis, "Discrete vortex solitons,�?? Phys. Rev. E 64, 026601 (2001). [CrossRef]
- J. Yang and Z.H. Musslimani, "Fundamental and vortex solitons in a two-dimensional optical lattice,�?? Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]
- D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, �??Observation of discrete vortex solitons in optically-induced photonic lattices,�?? Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed]
- J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, �??Observation of vortex-ring discrete solitons in 2D photonic lattices.�?? Phys. Rev. Lett. 92, 123904 (2004). [CrossRef] [PubMed]
- M. Soljacic, M, Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media,�?? Phys. Rev. Lett. 84, 467 (2000). [CrossRef] [PubMed]
- Z. Chen and K. MaCarthy, "Spatial soliton pixles from partially coherent light,�?? Opt. Lett. 27, 2019 (2002). [CrossRef]
- Z. Chen, I. Makasyuk, A. Bezryadina, and J. Yang, "Observation of two-dimensional lattice vector solitons,�?? Opt. Lett., 29, 1656 (2004). [CrossRef] [PubMed]
- J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, "Dipole solitons in optically-induced two-dimensional photonic lattices,�?? Opt. Lett., 29, 1662 (2004). [CrossRef] [PubMed]
- J. Hudock, N. K. Efremidis, and D. N. Christodoulides, "Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,�?? Opt. Lett. 29, 268 (2004). [CrossRef] [PubMed]
- D. N. Christodoulides, T. Coskun, M. Mitchell and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media,�?? Phys. Rev. Lett. 78, 646 (1997). [CrossRef]
- M.J. Ablowitz and Z.H. Musslimani, "Discrete vector spatial solitons in a nonlinear waveguide array,�?? Phys. Rev. E 65, 056618 (2002). [CrossRef]
- J. Hudock, P. G. Kevrekidis, B. A. Malomed, and D. N. Christodoulides, "Discrete vector solitons in two-dimensional nonlinear waveguide arrays: Solutions, stability, and dynamics,�?? Phys. Rev. E 67, 056618 (2003). [CrossRef]
- J. Meier, J. Hudock, D.N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays ". Phys. Rev. Lett. 91, 143907 (2003). [CrossRef] [PubMed]
- J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, "Dipole and Quadrupole Solitons in Optically-induced Two-dimensional Photonic Lattices: Theory and Experiment,�?? Stud. Appl. Math. 113, 389 (2004). [CrossRef]
- P. G. Kevrekidis, A.R. Bishop, and K. Rasmussen, "Twisted localized modes,�?? Phys. Rev. E 63, 036603 (2001). [CrossRef]
- A.A. Sukhorukov and Y.S. Kivshar, "Spatial optical solitons in nonlinear photonic crystals,�?? Phys. Rev. E 65, 036609 (2002). [CrossRef]
- P.G. Kevrekidis, B.A. Malomed, A.R. Bishop, �??Bound states of two-dimensional solitons in the discrete nonlinear Schrodinger equation.�?? J. Phys. A 34, 9615 (2001). [CrossRef]
- S.R. Singh and D.N. Christodoulides, �??Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,�?? Opt. Commun. 118, 569 (1995). [CrossRef]
- J. Yang, I. Makasyuk, H. Martin, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices,�?? Phys. Rev. Lett, submitted.
- A. Smerzi, A. Trombettoni, P. G. Kevrekidis and A. R. Bishop, Dynamical superfluid-insulator transition in a chain of weakly coupled Bose-Einstein condensates,�?? Phys. Rev. Lett. 89, 170402 (2002). [CrossRef] [PubMed]
- Z, Chen, H. Martin, A. Bezryadina, D.N. Neshev, Y.S. Kivshar, and D.N. Christodoulides, "Experiments on Gaussian beams and vortices in optically-induced photonic lattices,�?? J. Opt. Soc. Am. B, to appear 2005. [CrossRef]

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