## Vortex solitons at the boundaries of photonic lattices |

Optics Express, Vol. 19, Issue 27, pp. 26232-26238 (2011)

http://dx.doi.org/10.1364/OE.19.026232

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### Abstract

Using numerical analysis we demonstrate the existence of vortex solitons at the edge and in the corners of two-dimensional triangular photonic lattice. We develop a concise picture of their behavior in both single-propagating and counterpropagating beam geometries. In the single-beam geometry, we observe stable surface vortex solitons for long propagation distances only in the form of discrete six-lobe solutions at the edge of the photonic lattice. Other observed solutions, in the form of ring vortex and discrete solitons with two or three lobes, oscillate during propagation in a way indicating the exchange of power between neighboring lobes. For higher beam powers we observe dynamical instabilities of surface vortex solitons and study orbital angular momentum transfer of such vortex states. In the two-beam counterpropagating geometry, all kinds of vortex solutions are stable for propagation distances of the order of typical experimental crystal lengths.

© 2011 OSA

## 1. Introduction

1. Yu. S. Kivshar, “Nonlinear Tamm states and surface effects in periodic photonic structures,” Laser Phys. Lett. **5**(10), 703–713 (2008). [CrossRef]

2. S. Suntsov, K. G. Makris, G. A. Siviloglou, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, H. Yang, G. Salamo, M. Volatier, V. Aimez, R. Ars, M. Sorel, Y. Min, W. Sohler, X. Wang, A. Bezryadina, and Z. Chen“Observation of one- and two-dimensional discrete surface spatial solitons,” J. Nonlinear Opt. Phys. Mater. **16**(4), 401–426 (2007). [CrossRef]

3. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache“Discrete surface solitons,” Opt. Lett. **30**, 2466–2468 (2005). [CrossRef] [PubMed]

5. Y. V. Kartashov, V. A. Vysloukh, and L. Torner“Soliton emission in amplifying lattice surfaces,” Opt. Lett. **32**, 2061–2063 (2007). [CrossRef] [PubMed]

6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. **96**, 063901 (2006). [CrossRef] [PubMed]

7. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. **97**, 083901 (2006). [CrossRef] [PubMed]

8. D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. **32**, 3173–3175 (2007). [CrossRef] [PubMed]

9. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman“Observation of two-dimensional surface solitons,” Phys. Rev. Lett. **98**, 123903 (2007). [CrossRef] [PubMed]

10. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. **98**, 173903 (2007). [CrossRef]

11. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner“Observation of two-dimensional lattice interface solitons,” Opt. Lett. **33**, 663–665 (2008). [CrossRef] [PubMed]

12. M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner“Two-dimensional solitons at interfaces between binary superlattices and homogeneous lattices,” Phys. Rev. A **80**, 063832 (2009). [CrossRef]

13. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner“Surface vortex solitons,” Opt. Express **14**, 4049–4057 (2006). [CrossRef] [PubMed]

14. D. Song, C. Lou, K. J. H. Law, L. Tang, Z. Ye, P. G. Kevrekidis, J. Xu, and Z. Chen“Self-trapping of optical vortices at the surface of an induced semi-infinite photonic lattice,” Opt. Express **18**, 5873–5878 (2010). [CrossRef] [PubMed]

*linear*in their nature and require widely different media at the interface, e.g. a metal and a dielectric.

15. M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express **12**, 708–716 (2004). [CrossRef]

16. M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar“Counterpropagating optical beams and solitons,” Laser. Photon. Rev. **5**, 214–233 (2011). [CrossRef]

14. D. Song, C. Lou, K. J. H. Law, L. Tang, Z. Ye, P. G. Kevrekidis, J. Xu, and Z. Chen“Self-trapping of optical vortices at the surface of an induced semi-infinite photonic lattice,” Opt. Express **18**, 5873–5878 (2010). [CrossRef] [PubMed]

17. Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Yu. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B **22**, 1395–1405 (2005). [CrossRef]

18. M. S. Petrović, D. M. Jović, Milivoj R. Belić, and S. Prvanović, “Angular momentum transfer in optically induced photonic lattices,” Phys. Rev. A **76**, 023820 (2007). [CrossRef]

## 2. Model and basic equations

15. M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express **12**, 708–716 (2004). [CrossRef]

16. M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar“Counterpropagating optical beams and solitons,” Laser. Photon. Rev. **5**, 214–233 (2011). [CrossRef]

*z*is the propagation distance,

*F*and

*B*are the envelopes of the counterpropagating beams, Δ is the transverse Laplacian, Γ is the dimensionless beam coupling constant, and

*I*= |

*F*|

^{2}+|

*B*|

^{2}is the total laser light intensity measured in units of the background intensity

*I*. Typically, in photorefractive media laser beams interact incoherently, through the change in the index of refraction, caused by the light intensity. Here,

_{d}*I*is the intensity distribution of the optically induced truncated photonic lattice. In this paper, we concentrate on the triangular lattice.

_{g}16. M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar“Counterpropagating optical beams and solitons,” Laser. Photon. Rev. **5**, 214–233 (2011). [CrossRef]

19. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. **47**, 291–391 (2005). [CrossRef]

20. M. S. Petrović“Counterpropagating mutually incoherent vortex-induced rotating structures in optical photonic lattices,” Opt. Express **14**, 9415–9420 (2006). [CrossRef]

21. D. M. Jović, S. Prvanović, R. D. Jovanović, and M. S. Petrović“Gaussian-induced rotation in periodic photonic lattices,” Opt. Lett. **32**, 1857–1859 (2007). [CrossRef]

*F*=

*u*(

*x*,

*y*)

*cosθe*,

^{iμz}*B*=

*u*(

*x*,

*y*)

*sinθe*

^{−iμz}, where

*μ*is the propagation constant, and

*θ*is an arbitrary projection angle, which controls the relative size of beam components. For the simple CP geometry, we take

*θ*=

*π*/4; for the single-beam geometry, we take

*θ*= 0. After the substitution of the presumed solitonic solution in Eqs. (1) and (2), they transform into one, degenerate equation:

*x*,

*y*) plane. In solving this problem, i.e., in finding the solitonic solutions with the corresponding propagation constants, we utilize the modified Petviashvili’s iteration method [22, 23

23. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen“Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory an experiment,” Studies Appl. Math. **113**, 389–412 (2004). [CrossRef]

15. M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express **12**, 708–716 (2004). [CrossRef]

## 3. Vortex surface states

*μ*, but the existence domain of corner solitons is a bit broader than that of the edge solitons. The corresponding power diagrams are presented in Fig. 1(c), with the characteristic outcomes shown as insets. The beam power for vortex solitons is given by the formula

*can*exist for narrow input beam vortices. However, such surface states with symmetric uniform ring profiles exist only at the edge of the lattice and for lower values of the propagation constant. Asymmetric nonuniform surface vortex states are commonly observed in both corner and edge geometries. Both kinds of narrow ring vortex surface solitons are unstable during propagation in the single-beam geometry, similar to the experimental results [14

14. D. Song, C. Lou, K. J. H. Law, L. Tang, Z. Ye, P. G. Kevrekidis, J. Xu, and Z. Chen“Self-trapping of optical vortices at the surface of an induced semi-infinite photonic lattice,” Opt. Express **18**, 5873–5878 (2010). [CrossRef] [PubMed]

## 4. Soliton instabilities

17. Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Yu. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B **22**, 1395–1405 (2005). [CrossRef]

18. M. S. Petrović, D. M. Jović, Milivoj R. Belić, and S. Prvanović, “Angular momentum transfer in optically induced photonic lattices,” Phys. Rev. A **76**, 023820 (2007). [CrossRef]

*z*component of the orbital AM is adopted,

## 5. Conclusion

## Acknowledgments

## References and links

1. | Yu. S. Kivshar, “Nonlinear Tamm states and surface effects in periodic photonic structures,” Laser Phys. Lett. |

2. | S. Suntsov, K. G. Makris, G. A. Siviloglou, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, H. Yang, G. Salamo, M. Volatier, V. Aimez, R. Ars, M. Sorel, Y. Min, W. Sohler, X. Wang, A. Bezryadina, and Z. Chen“Observation of one- and two-dimensional discrete surface spatial solitons,” J. Nonlinear Opt. Phys. Mater. |

3. | K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache“Discrete surface solitons,” Opt. Lett. |

4. | K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. |

5. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner“Soliton emission in amplifying lattice surfaces,” Opt. Lett. |

6. | S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. |

7. | C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. |

8. | D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. |

9. | X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman“Observation of two-dimensional surface solitons,” Phys. Rev. Lett. |

10. | A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. |

11. | A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner“Observation of two-dimensional lattice interface solitons,” Opt. Lett. |

12. | M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner“Two-dimensional solitons at interfaces between binary superlattices and homogeneous lattices,” Phys. Rev. A |

13. | Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner“Surface vortex solitons,” Opt. Express |

14. | D. Song, C. Lou, K. J. H. Law, L. Tang, Z. Ye, P. G. Kevrekidis, J. Xu, and Z. Chen“Self-trapping of optical vortices at the surface of an induced semi-infinite photonic lattice,” Opt. Express |

15. | M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express |

16. | M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar“Counterpropagating optical beams and solitons,” Laser. Photon. Rev. |

17. | Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Yu. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B |

18. | M. S. Petrović, D. M. Jović, Milivoj R. Belić, and S. Prvanović, “Angular momentum transfer in optically induced photonic lattices,” Phys. Rev. A |

19. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. |

20. | M. S. Petrović“Counterpropagating mutually incoherent vortex-induced rotating structures in optical photonic lattices,” Opt. Express |

21. | D. M. Jović, S. Prvanović, R. D. Jovanović, and M. S. Petrović“Gaussian-induced rotation in periodic photonic lattices,” Opt. Lett. |

22. | V. I. Petviashvili“Equation of an extraordinary solution,” Plasma Phys. |

23. | J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen“Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory an experiment,” Studies Appl. Math. |

**OCIS Codes**

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(190.5330) Nonlinear optics : Photorefractive optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 2, 2011

Revised Manuscript: October 25, 2011

Manuscript Accepted: November 20, 2011

Published: December 8, 2011

**Citation**

Dragana Jović, Cornelia Denz, and Milivoj Belić, "Vortex solitons at the boundaries of photonic lattices," Opt. Express **19**, 26232-26238 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26232

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### References

- Yu. S. Kivshar, “Nonlinear Tamm states and surface effects in periodic photonic structures,” Laser Phys. Lett.5(10), 703–713 (2008). [CrossRef]
- S. Suntsov, K. G. Makris, G. A. Siviloglou, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, H. Yang, G. Salamo, M. Volatier, V. Aimez, R. Ars, M. Sorel, Y. Min, W. Sohler, X. Wang, A. Bezryadina, and Z. Chen“Observation of one- and two-dimensional discrete surface spatial solitons,” J. Nonlinear Opt. Phys. Mater.16(4), 401–426 (2007). [CrossRef]
- K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache“Discrete surface solitons,” Opt. Lett.30, 2466–2468 (2005). [CrossRef] [PubMed]
- K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett.31, 3125–3127 (2006). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner“Soliton emission in amplifying lattice surfaces,” Opt. Lett.32, 2061–2063 (2007). [CrossRef] [PubMed]
- S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett.96, 063901 (2006). [CrossRef] [PubMed]
- C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett.97, 083901 (2006). [CrossRef] [PubMed]
- D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett.32, 3173–3175 (2007). [CrossRef] [PubMed]
- X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman“Observation of two-dimensional surface solitons,” Phys. Rev. Lett.98, 123903 (2007). [CrossRef] [PubMed]
- A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett.98, 173903 (2007). [CrossRef]
- A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner“Observation of two-dimensional lattice interface solitons,” Opt. Lett.33, 663–665 (2008). [CrossRef] [PubMed]
- M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner“Two-dimensional solitons at interfaces between binary superlattices and homogeneous lattices,” Phys. Rev. A80, 063832 (2009). [CrossRef]
- Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner“Surface vortex solitons,” Opt. Express14, 4049–4057 (2006). [CrossRef] [PubMed]
- D. Song, C. Lou, K. J. H. Law, L. Tang, Z. Ye, P. G. Kevrekidis, J. Xu, and Z. Chen“Self-trapping of optical vortices at the surface of an induced semi-infinite photonic lattice,” Opt. Express18, 5873–5878 (2010). [CrossRef] [PubMed]
- M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express12, 708–716 (2004). [CrossRef]
- M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar“Counterpropagating optical beams and solitons,” Laser. Photon. Rev.5, 214–233 (2011). [CrossRef]
- Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Yu. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B22, 1395–1405 (2005). [CrossRef]
- M. S. Petrović, D. M. Jović, Milivoj R. Belić, and S. Prvanović, “Angular momentum transfer in optically induced photonic lattices,” Phys. Rev. A76, 023820 (2007). [CrossRef]
- A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt.47, 291–391 (2005). [CrossRef]
- M. S. Petrović“Counterpropagating mutually incoherent vortex-induced rotating structures in optical photonic lattices,” Opt. Express14, 9415–9420 (2006). [CrossRef]
- D. M. Jović, S. Prvanović, R. D. Jovanović, and M. S. Petrović“Gaussian-induced rotation in periodic photonic lattices,” Opt. Lett.32, 1857–1859 (2007). [CrossRef]
- V. I. Petviashvili“Equation of an extraordinary solution,” Plasma Phys.2, 469 (1976).
- J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen“Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory an experiment,” Studies Appl. Math.113, 389–412 (2004). [CrossRef]

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