## Two-dimensional self-trapped nonlinear photonic lattices

Optics Express, Vol. 14, Issue 7, pp. 2851-2863 (2006)

http://dx.doi.org/10.1364/OE.14.002851

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### Abstract

We predict theoretically and generate experimentally in pho-torefractive crystal two-dimensional self-trapped periodic waves of different symmetries, including vortex lattices – patterns of phase dislocations with internal energy flows. We demonstrate that these nonlinear waves exist with nonlocal nonlinearity even when the optically-induced periodic refractive index becomes highly anisotropic, and it depends on the orientation of the two-dimensional lattice relative to the crystallographic c-axis.

© 2006 Optical Society of America

## 1. Introduction

2. S. F. Mingaleev, Yu. S. Kivshar, and R. Sammut, in: *Soliton-driven Photonics*, A. D. Boardman and A. P. Sukho-rukov, Eds. (Kluwer, Dordrecht, Netherlands, 2001), pp. 487–504. [CrossRef]

3. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photore-fractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (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 (2003). [CrossRef] [PubMed]

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

6. A. S. Desyatnikov, E. A. Ostrovskaya, Yu. S. Kivshar, and C. Denz, “Composite band-gap solitons in nonlinear optically induced lattices,” Phys. Rev. Lett. **91**, 153902 (2003). [CrossRef] [PubMed]

7. D. Neshev, Yu. S. Kivshar, H. Martin, and Z. G. Chen, “Soliton stripes in two-dimensional nonlinear photonic lattices,” Opt. Lett. **29**, 486 (2004). [CrossRef] [PubMed]

8. 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]

*pixel-like*spatial solitons have been demonstrated experimentally in parametric systems [9

9. S. Minardi, S. Sapone, W. Chinaglia, P. Di Trapani, and A. Berzanskis, “Pixellike parametric generator based on controlled spatial-soliton formation,” Opt. Lett. **25**, 326 (2000). [CrossRef]

10. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. **27**, 2019 (2002). [CrossRef]

11. J. Petter, J. Schröder, D. Träger, and C. Denz, “Optical control of arrays of photorefractive screening solitons,” Opt. Lett. **28**, 438 (2003). [CrossRef] [PubMed]

*in-phase*spatial solitons created by the amplitude modulation, every pixel of the lattice induces a waveguide which can be manipulated by an external steering beam [11

11. J. Petter, J. Schröder, D. Träger, and C. Denz, “Optical control of arrays of photorefractive screening solitons,” Opt. Lett. **28**, 438 (2003). [CrossRef] [PubMed]

12. M. Petrović, D. Träger, A. Strinić, M. Belić, J. Schröder, and C. Denz, “Solitonic lattices in photorefractive crystals,” Phys. Rev. E **68**, 055601R (2003). [CrossRef]

*out-of-phase*spatial solitons were demonstrated to be robust in the isotropic saturable model [13

13. Ya. V. Kartashov, V. A. Visloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E **68**, 015603 (2003). [CrossRef]

*π*-phase jumps between the neighboring sites.

14. A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. **30**, 869 (2005). [CrossRef] [PubMed]

15. 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]

14. A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. **30**, 869 (2005). [CrossRef] [PubMed]

*a square geometry*, using

*an anisotropic nonlocal model*of photorefractive nonlinearity, in both self-focusing and self-defocusing nonlinear media. We then expand our analysis and introduce novel symmetry-types of two-dimensional nonlinear modes with edge-type phase dislocations and three-fold symmetry,

*triangular lattices*. Similar to the two distinct orientations for square lattices, we distinguish the triangular patterns with one of the three dislocation lines oriented parallel or perpendicular to the crystal axis. Again the differences appear in the distribution of the refractive index, and they became more pronounced for larger intensities of the lattice wave, i.e., in the regime of higher saturation. We demonstrate experimentally how this orientational dependence changes the guiding properties of the photonic lattice.

*c*-axis of the crystal. In Section 5 we describe theoretically and experimentally the formation and nonlinear propagation of lattices with nontrivial vortex-like phase patterns.

## 2. Phase-modulated lattices: theoretical background

*stationary periodic nonlinear waves*, and they include well studied

*cnoidal waves*, described by the

*cn*and

*dn*Jacobi elliptic functions as the stationary solutions of the generalized nonlinear Schrödinger (NLS) equation [1],

*I*≡ |

*E*|

^{2}is the light intensity. Similar nonlinear waves appear as periodic solutions of different nonlinear models, including quadratic, Kerr-type saturable, and photorefractive anisotropic nonlocal models.

24. 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 (1997). [CrossRef]

^{2}

*r*

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

*r*

_{eff}and externally applied bias electrostatic field ℰ. The dimensionless parameter σ = ± 1 indicates the polarity of the applied voltage that changes the character of the photorefractive screening nonlinearity: self-focusing, for σ = +1, or self-defocusing otherwise. The electrostatic potential φ of the optically-induced space-charge field pattern satisfies the equation:

_{⊥}for the two-dimensional gradient operator such that

*I*is measured in units of the background (dark) illumination intensity, required for the formation of spatial solitons in this medium. The physical variables

*x̃*,

*ỹ*, and

*z̃*correspond to their dimensionless counterparts as (

*x̃,ỹ*) =

*x*

_{0}(

*x,y*) and

*z̃*= 2κ

*z*, here

*x*

_{0}is the transverse scale factor and κ = 2

*π*

*n*

_{0}/λ is the carrier wave vector with the linear refractive index

*n*

_{0}. Stationary solutions to the system (1)-(3) are sought in the standard form,

*E*(

*x,y,z*) =

*U*(

*x,y*) exp(

*ikz*), where the field envelope

*U*satisfies the equation

14. A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. **30**, 869 (2005). [CrossRef] [PubMed]

*U*(

*X,Y*) =

*U*(

*X*+ 2

*π,Y*+ 2

*π*), and solve Eqs. (3), (4) using the relaxation technique [24

24. 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 (1997). [CrossRef]

*U*

_{lin}(

*X, Y*), depending on its spatial orientation:

*a square lattice*parallel to the

*c*-axis with

*X*=

*x*and

*Y*=

*y*, and

*a diamond lattice*oriented diagonally, in the latter case

*X*= (

*x*+

*y*)/√2 and

*Y*= (

*x*-

*y*)√2. Fig. 1 and Fig. 2 show the field and refractive index for the three values of the lattice intensity, corresponding to the low, moderate, and high saturation regimes for both families of periodic modes, respectively. In a general case Γ ≠ 1, the existence region of these two families occupy a band

*k*∈ [-2,Γ - 2] with the amplitude

*A*(

*k*) and power density

*P*(

*k*) vanishing in the linear limit

*k*→ -2, see Fig. 1(d) and Fig. 2(d). Here the power density is defined as the power of a unit cell,

*P*= 4

*U*

^{2}

*dXdY*. The main difference between the solutions with different orientations comes from the structure of the refractive index, as is clearly seen comparing Fig. 1(b) and Fig. 2(b). In the regime of high saturation of nonlinearity the regions with the effective focusing lenses are well separated for the diamond lattice [see Fig. 2(a-d)], and fuse effectively to vertical lines for the square lattice [see Fig. 1(a-d)]; this happens due to larger nonlocality in the limit of strong nonlinearity saturation. In Fig. 1(d) and Fig. 2(d), we plot the maximum and minimum values (extrema) of the refractive index, Extr(∂φ/∂

*x*).

*sn*-type one-dimensional cnoidal waves. The corresponding families of such periodic solutions [Fig. 1(c) and Fig. 2(c)] are compared to the four-fold symmetry lattices in a self-focusing medium. We notice that, while the field distribution is almost identical in both cases, the refractive index is opposite: the positive maxima in the self-focusing case (focusing lenses) is inverted for σ = - 1, and it represents defocusing lenses (induced potential maxima). Parameters of two families for σ = ±1 in Fig. 1(d) and Fig. 2(d) are almost symmetric with respect to the linear solution at

*k*= -2, however, these two families correspond to different existence domains.

## 3. Experimental approach

**30**, 869 (2005). [CrossRef] [PubMed]

*π*. The output of the modulator is then imaged by a high numerical aperture telescope onto the front face of our 20 mm long Strontium Barium Niobate (SBN) photorefractive crystal. The polarization of the beam is orientated parallel to the crystalline

*c*-axis, thus the beam will experience a strong photorefractive nonlinearity (the electro-optic coefficient

*r*

_{33}≈ 200 pm/V). The crystal is biased by an externally applied electric field and uniformly illuminated with a white-light source to control the dark irradiance. Either the front face or the back face of the crystal can be imaged with a lens onto the CCD-camera. The periodic pattern generated by the modulator is filtered in its Fourier plane by an iris diaphragm. The input then represents a non-diffracting wave, and it experiences robust linear propagation inside the crystal (at zero bias voltage). When bias voltage is applied onto the crystal, the output intensity pattern changes and nonlinear index change is induced in the crystal. In order to probe this refractive index change and establish correspondence to the numerical simulations of the refractive index, the modulator can be switched off so that a broad plane wave illuminates the crystal. Due to the slow response of the photorefractive nonlinearity, we can quickly monitor the output of the plane wave without modifying the induced refractive index change. The plane wave is guided by the periodic array of waveguides induced in the medium and the output intensity pattern of the guided beam qualitatively maps the induced refractive index modulation.

25. G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of nonlinear photonic lattices,” Phys. Rev. Lett. **94**, 163902 (2005) [CrossRef] [PubMed]

26. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. **92**, 093901 (2004). [CrossRef] [PubMed]

*higher-order*bands of the transmission spectrum. Thus, the four side beams, visible in Fig. 4(e), indicate the central points of the

*second band from the extended Brillouin zone*.

*k*= -1.5. In contrast, the diamond pattern shows well-pronounced 2D array of waveguiding channels, see Fig. 4(d,bottom) and Fig. 2(b), and its Fourier image contains all four side beams.

## 4. Triangular lattices

*U*(

*x,y*), then the only type of the phase modulations allowed is the edge-type

*π*phase jumps that produce the lines of zero intensity in the lattice. The simplest pattern of this kind with the three-fold symmetry is a triangular lattice. The corresponding linear mode can be constructed by interference of six plane waves, and it can be presented in the following form:

*X*-axis, so that two orientations of this pattern with respect to the crystal

*c*-axis in Eq. (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 (2003). [CrossRef] [PubMed]

*X, Y*) = (

*x,y*) for the parallel orientation, and, e.g., (

*X, Y*) = (

*y,x*), for the perpendicular orientation. For the linear wave, the value of

*k*is given by

*U*

_{3}/

*U*

_{3}= -16/3 ≈= - 5.3. The periodicity of the refractive index pattern is defined by the intensity distribution, not by the field, and in our notation this distance between the closest neighboring peaks of the intensity is

*π*/2.

*k*> -5.(3). We summarize our numerical and experimental data in Fig. 5, where we compare numerical results with experimental images, for both parallel and perpendicular orientations, and for low and high saturations. Experimentally, for switching from low to high saturation, the total power of the lattice-governing beam as well as the intensity of the background illumination are changed. In the nonlinear output, the lattice period is about 48 μm.

24. 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 (1997). [CrossRef]

*reduced symmetry*with respect to the lattice. Indeed, instead of six side-beams forming the lattice wave, there is only four side-beams in the guided wave meaning that the induced potential is unable to trap two side beams (top and bottom), and the guided-mode symmetry is reduced.

*c*-axis, and the guided wave has an essentially two-dimensional modulation. Even in the high-saturation regime when the refractive index profile resembles vertical stripes [cf. the square pattern in Fig. 1], the guided wave still has a visible modulation in the vertical direction. This is in contrast to the square pattern shown in Fig. 4 where the guided wave modulation also has the reduced symmetry.

## 5. Vortex lattices

21. A. Desyatnikov, D. Neshev, and Yu. Kivshar, “Vortex solitons, soliton clusters, and vortex lattices,” Ukr. J. Phys. Optics **6**, 71 (2005). [CrossRef]

*vortex lattices*. Theoretically, these nonlinear waves bifurcate from the corresponding linear modes, that can be easily found from the linear paraxial equation 1 without nonlinear term, i.e.

*n*(

*I*) = 0. A linear superposition of two azimuthal modes sinφ and cosφ with azimuthal coordinate φ is known to give rise to a vortex beam with a phase dislocation, ~

*x*+

*iy*=

*r*exp(

*i*φ). Similarly, superimposing two linear modes

*U*

_{lin}from Eq. (5), we obtain a vortex lattice of a square geometry,

18. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. **198**, 21 (2001). [CrossRef]

7. D. Neshev, Yu. S. Kivshar, H. Martin, and Z. G. Chen, “Soliton stripes in two-dimensional nonlinear photonic lattices,” Opt. Lett. **29**, 486 (2004). [CrossRef] [PubMed]

*π*). The number of such steps depends on the symmetry of the lattice [see Fig. 6 and Fig. 7], so that each lattice site has a fairly well defined phase, as shown by different colors. Indeed, the phase steps for the square vortex lattice are equal 2

*π*/4 =

*π*/2, thus there are four colors in the phase profiles of Fig. 6, dark blue (0), light blue (

*π*/2), yellow (

*π*), and red (3

*π*/2). In contrast, the phase step for the hexagonal lattice is 2

*π*/3 (each vortex is “built” with three lattice sites), so there is almost no yellow. This phase behavior resembles the structure of soliton clusters [27

27. A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. **88**, 053901 (2002). [CrossRef] [PubMed]

28. A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. **95**, 203904 (2005). [CrossRef] [PubMed]

29. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Yu. 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]

30. 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 attices,” Phys. Rev. Lett. **92**, 123904 (2004). [CrossRef] [PubMed]

21. A. Desyatnikov, D. Neshev, and Yu. Kivshar, “Vortex solitons, soliton clusters, and vortex lattices,” Ukr. J. Phys. Optics **6**, 71 (2005). [CrossRef]

## 6. Conclusions

*c*-axis of the crystal. We have demonstrated that the highly anisotropic periodic refractive index induced by the lattices differs significantly from its isotropic counterpart, and it depends strongly on the lattice orientation. We have expanded our theoretical approach to the periodic lattices with nested arrays of vortex-type phase dislocations, or vortex lattices, and have generated experimentally vortex lattices with two different symmetries. The square vortex lattice acquires a diamond-like intensity pattern, while for the six-fold symmetry we have demonstrated that a hexagonal intensity pattern can be generated with the vortices arranged in a honeycomb lattice.

## Acknowledgments

## References and links

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

2. | S. F. Mingaleev, Yu. S. Kivshar, and R. Sammut, in: |

3. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photore-fractive 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, Yu. S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

6. | A. S. Desyatnikov, E. A. Ostrovskaya, Yu. S. Kivshar, and C. Denz, “Composite band-gap solitons in nonlinear optically induced lattices,” Phys. Rev. Lett. |

7. | D. Neshev, Yu. S. Kivshar, H. Martin, and Z. G. Chen, “Soliton stripes in two-dimensional nonlinear photonic lattices,” Opt. Lett. |

8. | 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. |

9. | S. Minardi, S. Sapone, W. Chinaglia, P. Di Trapani, and A. Berzanskis, “Pixellike parametric generator based on controlled spatial-soliton formation,” Opt. Lett. |

10. | Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. |

11. | J. Petter, J. Schröder, D. Träger, and C. Denz, “Optical control of arrays of photorefractive screening solitons,” Opt. Lett. |

12. | M. Petrović, D. Träger, A. Strinić, M. Belić, J. Schröder, and C. Denz, “Solitonic lattices in photorefractive crystals,” Phys. Rev. E |

13. | Ya. V. Kartashov, V. A. Visloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E |

14. | A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in anisotropic nonlocal self-focusing media,” Opt. Lett. |

15. | 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. |

16. | A. Sukhorukov, “Soliton dynamics in deformable nonlinear lattices,” arXiv:nlin.PS/0507050 (2005). |

17. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in |

18. | J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. |

19. | D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. |

20. | A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B |

21. | A. Desyatnikov, D. Neshev, and Yu. Kivshar, “Vortex solitons, soliton clusters, and vortex lattices,” Ukr. J. Phys. Optics |

22. | D. Jović, D. Arsenović, A. Strinić, M. Belić, and M. Petrović, “Counterpropagat-ing optical vortices in photorefractive crystals,” Opt. Express |

23. |
See, e.g.,
N. R. Cooper, E. H. Rezayi, and S. H. Simon, “Vortex lattices in rotating atomic Bose gases with dipolar interactions,” Phys. Rev. Lett. |

24. | 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 |

25. | G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of nonlinear photonic lattices,” Phys. Rev. Lett. |

26. | A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. |

27. | A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. |

28. | A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. |

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

30. | 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 attices,” Phys. Rev. Lett. |

**OCIS Codes**

(090.7330) Holography : Volume gratings

(190.5330) Nonlinear optics : Photorefractive optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 3, 2006

Revised Manuscript: March 12, 2006

Manuscript Accepted: March 23, 2006

Published: April 3, 2006

**Citation**

Anton S. Desyatnikov, Nina Sagemerten, Robert Fischer, Bernd Terhalle, Denis Träger, Dragomir N. Neshev, Alexander Dreischuh, Cornelia Denz, Wieslaw Krolikowski, and Yuri S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express **14**, 2851-2863 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2851

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### References

- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003), 540 pp., and references therein.
- S. F. Mingaleev, Yu. S. Kivshar, and R. Sammut, in: Soliton-driven Photonics, A. D. Boardman and A. P. Sukhorukov, Eds. (Kluwer, Dordrecht, Netherlands, 2001), pp. 487-504. [CrossRef]
- 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, 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]
- D. Neshev, E. Ostrovskaya, Yu. S. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003). [CrossRef] [PubMed]
- A. S. Desyatnikov, E. A. Ostrovskaya, Yu. S. Kivshar, and C. Denz, "Composite band-gap solitons in nonlinear optically induced lattices, " Phys. Rev. Lett. 91, 153902 (2003). [CrossRef] [PubMed]
- D. Neshev, Yu. S. Kivshar, H. Martin, and Z. G. Chen, "Soliton stripes in two-dimensional nonlinear photonic lattices," Opt. Lett. 29, 486 (2004). [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]
- S. Minardi, S. Sapone, W. Chinaglia, P. Di Trapani, and A. Berzanskis, "Pixellike parametric generator based on controlled spatial-soliton formation," Opt. Lett. 25, 326 (2000). [CrossRef]
- Z. Chen, and K. McCarthy, "Spatial soliton pixels from partially incoherent light," Opt. Lett. 27, 2019 (2002). [CrossRef]
- J. Petter, J. Schröder, D. Träger, and C. Denz, "Optical control of arrays of photorefractive screening solitons," Opt. Lett. 28, 438 (2003). [CrossRef] [PubMed]
- M. Petrović, D. Träger, A. Strinić, M. Belić, J. Schröder, and C. Denz, "Solitonic lattices in photorefractive crystals," Phys. Rev. E 68, 055601R (2003). [CrossRef]
- Ya. V. Kartashov, V. A. Visloukh, and L. Torner, "Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media," Phys. Rev. E 68, 015603 (2003). [CrossRef]
- A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30, 869 (2005). [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]
- A. Sukhorukov, "Soliton dynamics in deformable nonlinear lattices," arXiv:nlin.PS/0507050 (2005).
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