## Observation of light localization in modulated Bessel optical lattices

Optics Express, Vol. 14, Issue 7, pp. 2825-2830 (2006)

http://dx.doi.org/10.1364/OE.14.002825

Acrobat PDF (465 KB)

### Abstract

We generate higher-order azimuthally modulated Bessel optical lattices in photorefractive crystals by employing a phase-imprinting technique. We report on the experimental observation of self-trapping and nonlinear localization of light in such segmented lattices in the form of ring-shaped and single-site states. The experimental results agree well with numerical simulations accounting for an anisotropic and spatially nonlocal nonlinear response of photorefractive crystals.

© 2006 Optical Society of America

## 1. Introduction

2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

3. 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**, 046602-5 (2002). [CrossRef]

4. 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**, 147–150 (2003). [CrossRef] [PubMed]

5. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

6. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902-4 (2004). [CrossRef] [PubMed]

7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. **93**, 093904-4 (2004). [CrossRef] [PubMed]

8. X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices”, Phys. Rev. Lett. **96**, 083904-4 (2005). [CrossRef]

9. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

*Bessel beams*. In spite of the fact that the Bessel beams appear as eigenmode solutions of Maxwell’s equations of infinite transverse extent and energy, they have been demonstrated in experiments with finite beams, where the characteristics of stationary propagation are sustained over long distances [10

10. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

11. Y. Lin, W. Seka, J. H. Eberly, H. Huang, and D. L. Brown, “Experimental investigation of bessel beam characteristics,” Appl. Opt. **31**, 2708–2713 (1992). [CrossRef] [PubMed]

12. T. Wulle and S. Herminghaus, “Nonlinear optics of bessel beams,” Phys. Rev. Lett. **70**, 1401–1404 (1993); Erratum: Phys. Rev. Lett. **71**, 209 (1993). [CrossRef] [PubMed]

13. C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, “Production of high-order Bessel beams with a Mach-Zehnder interferometer,” Appl. Optics **43**, 5060–5063 (2004). [CrossRef]

14. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill III, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. **28**, 2183–2185 (2003). [CrossRef] [PubMed]

15. S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett. **28**, 1867–1869 (2003). [CrossRef] [PubMed]

16. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

17. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express **11**, 158–166 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-158. [CrossRef] [PubMed]

18. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices,” Phys. Rev. E **70**, 065602-4 (2004). [CrossRef]

19. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A **57**, 522–534 (1998). [CrossRef]

20. A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kar-tashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. **30**, 869–871 (2005). [CrossRef] [PubMed]

## 2. Generation of nondiffracting Bessel lattices

*z*, and it can be presented as a superposition of an infinite number of plane waves. In past years, a number of practical ways to generate Bessel beams have been suggested including the use of passive optical systems such as ring apertures, Fabry-Perot etalons, and axicons [21

21. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A **6**, 150–152 (1989). [CrossRef]

22. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A **6**, 1748–1754 (1989). [CrossRef] [PubMed]

23. W. X. Cong, N. X. Chen, and B. Y. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A **15**, 2362–2364 (1998). [CrossRef]

24. J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. **190**, 117–122 (2001). [CrossRef]

14. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill III, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. **28**, 2183–2185 (2003). [CrossRef] [PubMed]

15. S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett. **28**, 1867–1869 (2003). [CrossRef] [PubMed]

_{4}laser (532nm) is split into two orthogonally polarized beams by a polarizing beam splitter (PBS), where the splitting ratio is set by a half-wave plate. The beam transmitted through the PBS is expanded approximately 10 times and illuminates a phase modulator (Hamamatsu X8267). The modulator is programmed to reproduce the exact phase profile of a modulated Bessel beam given by

*ρ*and

*φ*are the transverse polar coordinates,

*w*is the spatial scale, and

*J*

_{n}is the

*n*-th order Bessel function. Such a field distribution represents a well-known nondiffracting beams, and it is ideal for the application of the optical induction technique in a similar way as realized earlier for square lattices induced by four interfering beams [4

4. 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**, 147–150 (2003). [CrossRef] [PubMed]

25. R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Yu. S. Kivshar, “Reduced-symmetry two-dimensional solitons in square photonic lattices,” Phys. Rev. Lett. **96**, 023905(4) (2006). [CrossRef] [PubMed]

*w*= 7.6

*μ*m is of the order of 2cm. In fact, the phase modulator only reproduces the phase structure of the Bessel beam (1) and to obtain a real nondiffracting beam we employ Fourier filtering in the focal plane of the telescope which images the active plane of the modulator onto the front face of a 20mm long SBN photorefractive crystal. The crystal is externally biased by a DC electric field (3500V/cm) applied horizontally along the crystal c-axis, allowing the study of self-action effects.

## 3. Light localization in a Bessel lattice

3. 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**, 046602-5 (2002). [CrossRef]

4. 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**, 147–150 (2003). [CrossRef] [PubMed]

5. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

6. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902-4 (2004). [CrossRef] [PubMed]

*μ*m onto the front face of the crystal. The input and output crystal faces can be imaged by a lens and recorded onto a CCD camera for analyzing the transverse intensity distribution. Additionally, a small portion of the probe beam is used as an interferometric reference beam to monitor the phase profile of the probe beam at the output.

*μ*m at the crystal output, as shown in Fig. 2(b), which corresponds to 12 diffractions lengths of propagation. Then we switch on the PPM, which is programmed to reproduce the phase distribution of the Bessel beam [Eq. (1)]. The resulting intensity distribution of this lattice forming beam (after Fourier filtering) has the form shown in Fig. 2(a). Under the action of the bias electric field this beam induces a refractive index change following its intensity profile. It is important to note that we have to select the proper orientation of the lattice pattern with respect to the crystalline c-axis, in order to minimize the distortion of the refractive index modulation due to the anisotropy of the photorefractive crystal as previously discussed for square lattices [20

20. A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kar-tashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. **30**, 869–871 (2005). [CrossRef] [PubMed]

*n*= 1,2,3, and 4. For the low-order lattices, the two (

*n*= 1) or four (

*n*= 2) sites in the inner ring dominate the structure of induced refractive index and represent almost decoupled waveguides for the probe beam. For the higher-order Bessel beams (

*n*= 4) the effect of the crystal anisotropy leads to strong vertical merging of the closely located lattice sites. Therefore, below we present results for the representative example of the third-order modulated Bessel beams (

*n*= 3).

2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

## 4. Numerical results for the anisotropic model

19. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A **57**, 522–534 (1998). [CrossRef]

*φ*of the optically-induced space-charge field that satisfies the relation [19

19. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A **57**, 522–534 (1998). [CrossRef]

^{2}=

*x͂*,

*y͂*, and

*z͂*correspond to their dimensionless counterparts as (

*x͂*,

*y͂*) = (

*wx*,

*wy*) and

*z͂*= 2

*κw*

^{2}

*z*, here

*w*is the transverse scale factor [Eq. (1)] and

*κ*= 2

*πn*

_{0}/

*λ*is the carrier wave vector with the linear refractive index

*n*

_{0}. Parameter

*γ*=

*w*

^{2}

*κ*

^{2}

*r*

_{eff}ℰ is defined through the effective electro-optic coefficient

*r*

_{eff}and externally applied bias DC field ℰ . The total intensity is given by the sum

*r*

^{2}=

*x*

^{2}+

*y*

^{2},

*φ*= tan

^{-1}

*y*/

*x*, and I is measured in units of the background illumination intensity, necessary for the formation of spatial solitons in such a medium. We solve Eqs. (3), (4) and find that the modulated Bessel beam induces the refractive index modulation which has a symmetry lower than the symmetry of the intensity pattern [cf. Fig. 3(a) and (b)]. Stationary solutions of the system (2), (3) are sought in the standard form

*E*(

*x*,

*y*,

*z*) =

*U*(

*x*,

*y*)exp(

*iβz*), where the real envelope

*U*satisfies the equation

**57**, 522–534 (1998). [CrossRef]

*β*exceeds some threshold, which corresponds to the threshold for soliton power and peak intensity, our relaxation procedure converges to the on-site single soliton, shown in Fig. 3(d). We conclude that two families of solutions may be linked through a bifurcation, which may correspond to the onset of symmetry-breaking instability for the effectively two-lobe Ȝin-phase” soliton shown in Fig. 3(c).

## 5. Conclusions

## Acknowledgments

## References and links

1. | Yu. S. Kivshar and G. P. Agrawal, |

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

3. | 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 |

4. | 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 |

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

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

7. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. |

8. | X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices”, Phys. Rev. Lett. |

9. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

10. | J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

11. | Y. Lin, W. Seka, J. H. Eberly, H. Huang, and D. L. Brown, “Experimental investigation of bessel beam characteristics,” Appl. Opt. |

12. | T. Wulle and S. Herminghaus, “Nonlinear optics of bessel beams,” Phys. Rev. Lett. |

13. | C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, “Production of high-order Bessel beams with a Mach-Zehnder interferometer,” Appl. Optics |

14. | N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill III, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. |

15. | S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett. |

16. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

17. | D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express |

18. | Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices,” Phys. Rev. E |

19. | A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A |

20. | A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kar-tashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. |

21. | G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A |

22. | A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A |

23. | W. X. Cong, N. X. Chen, and B. Y. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A |

24. | J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. |

25. | R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Yu. S. Kivshar, “Reduced-symmetry two-dimensional solitons in square photonic lattices,” Phys. Rev. Lett. |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 23, 2006

Revised Manuscript: March 17, 2006

Manuscript Accepted: March 18, 2006

Published: April 3, 2006

**Citation**

Robert Fischer, Dragomir N. Neshev, Servando Lopez-Aguayo, Anton S. Desyatnikov, Andrey A. Sukhorukov, Wieslaw Krolikowski, and Yuri S. Kivshar, "Observation of light localization in modulated Bessel optical lattices," Opt. Express **14**, 2825-2830 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2825

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### References

- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003). [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 66,046602-5 (2002).Q1 [CrossRef]
- 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 422,147-150 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28,710-712 (2003). [CrossRef] [PubMed]
- H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2 [CrossRef] [PubMed]
- X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3 [CrossRef]
- J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58,1499-1501 (1987). [CrossRef] [PubMed]
- Y. Lin, W. Seka, J. H. Eberly, H. Huang, and D. L. Brown, "Experimental investigation of bessel beam characteristics," Appl. Opt. 31,2708-2713 (1992). [CrossRef] [PubMed]
- 12. T. Wulle and S. Herminghaus, "Nonlinear optics of bessel beams," Phys. Rev. Lett. 70,1401-1404 (1993);Erratum: Phys. Rev. Lett. 71,209 (1993). Q4Q5 [CrossRef] [PubMed]
- C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004). [CrossRef]
- N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, III, and R. Roy, "Generation of nondiffracting Bessel beams by use of a spatial light modulator," Opt. Lett. 28,2183-2185 (2003). [CrossRef] [PubMed]
- S. H. Tao, W. M. Lee, and X. C. Yuan, "Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator," Opt. Lett. 28,1867-1869 (2003). [CrossRef] [PubMed]
- D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46,15-28 (2005). [CrossRef]
- D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11,158-166 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-158. [CrossRef] [PubMed]
- Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6 [CrossRef]
- A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998). [CrossRef]
- A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005). [CrossRef] [PubMed]
- G. Indebetouw, "Nondiffracting optical fields: some remarks on their analysis and synthesis," J. Opt. Soc. Am. A 6,150-152 (1989). [CrossRef]
- A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6,1748-1754 (1989). [CrossRef] [PubMed]
- W. X. Cong, N. X. Chen, and B. Y. Gu, "Generation of nondiffracting beams by diffractive phase elements," J. Opt. Soc. Am. A 15,2362-2364 (1998). [CrossRef]
- J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001). [CrossRef]
- R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006). [CrossRef] [PubMed]

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