## Counterpropagating surface solitons in two-dimensional photorefractive lattices

Optics Express, Vol. 17, Issue 24, pp. 21515-21521 (2009)

http://dx.doi.org/10.1364/OE.17.021515

Acrobat PDF (212 KB)

### Abstract

We study interaction of counterpropagating beams in truncated two-dimensional photonic lattices induced optically in photorefractive crystals, and demonstrate the existence of counterpropagating surface solitons localized in the lattice corners and at the edges. We display intriguing dynamical properties of such composite optical beams and reveal that the lattice surface provides a strong stabilization effect on the beam propagation. We also observe dynamical instabilities for stronger coupling and longer propagation distances in the form of beam splitting. No such instabilities exist in the single beam surface propagation.

© 2009 Optical Society of America

## 1. Introduction

1. S. Suntsov*et al*., “Observation of one- and two-dimensional discrete surface spatial solitons,” J. Nonlinear Opt. Phys. Mater. **16**, 401 (
2007). [CrossRef]

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

3. 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]

4. 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]

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E **71**, 016601 (
2005). [CrossRef]

6. Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. **48**, 1541 (
1982). [CrossRef]

7. M. Haelterman, A. P. Sheppard, and A. W. Snyder, “Bimodal counterpropagating spatial solitary waves,” Opt. Commun. **103**, 145 (
1993). [CrossRef]

8. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. **89**, 133901 (
2002). [CrossRef] [PubMed]

9. M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E **68**, 025601 (
2003). [CrossRef]

10. See, e.g., a recent review: M.S. Petrović, M.R. Belić, C. Denz, and Yu.S. Kivshar, “Counterpropagating optical beams and solitons,” to appear in Laser & Photonics Reviews ( 2010). http://www.arXiv.org, Preprint [physics.optics] arXiv:0910.4700

11. E. Smirnov, M. Stepić, C. E. Rüter, V. Shandarov, and D. Kip, “Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array,” Opt. Lett. **32**, 512 (
2007). [CrossRef] [PubMed]

12. S. Koke, D. Träger, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, “Stabilization of counterpropagating solitons by photonic lattices,” Opt. Express **15**, 6279 (
2007). [CrossRef] [PubMed]

*three different regimes*: stable propagation of vector solitons at low power, instability for intermediate powers, with a transverse shift of the solitons, and an irregular dynamical behavior of the two beams at high input powers. Nevertheless, both theoretical and experimental results suggest that spatiotemporal soliton instabilities are suppressed with the increasing strength of the optical lattice [11

11. E. Smirnov, M. Stepić, C. E. Rüter, V. Shandarov, and D. Kip, “Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array,” Opt. Lett. **32**, 512 (
2007). [CrossRef] [PubMed]

12. S. Koke, D. Träger, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, “Stabilization of counterpropagating solitons by photonic lattices,” Opt. Express **15**, 6279 (
2007). [CrossRef] [PubMed]

*counterpropagating surface solitons*, localized in the lattice corners or at its edges (see Fig. 1). We also study extensively the dynamical properties of such composite solitons and demonstrate that the lattice surface produces a strong stabilizing effect on such vectorial solitons.

## 2. Model and basic equations

3. 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]

9. M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E **68**, 025601 (
2003). [CrossRef]

13. K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E **68**, 066611 (
2003). [CrossRef]

*F*and

*B*are the forward and backward propagating beam envelopes, Δ is the transverse Laplacian, Γ is the dimensionless coupling constant, and

*E*is the homogenous part of SCF. The relaxation time of the crystal

*τ*also depends on the total intensity,

*τ*=

*τ*

_{0}(1+

*I*)

^{-1}. The total intensity

*I*=|

*F*|

^{2}+|

*B*|

^{2}is measured in units of the background intensity. A scaling

*x*/

*x*

_{0}→

*x*,

*y*/

*x*

_{0}→

*y*,

*z*/

*L*→

_{D}*z*, is utilized for the dimensionless equations, where

*x*

_{0}is the typical FWHM beam waist and

*L*is the diffraction length. We assume that mutually incoherent CP components interact through the intensity-dependent saturable SCF.

_{D}*I*, optically induced in the crystal

_{g}*I*=

_{g}*I*

_{0}

*sin*

^{2}[

*π*(

*x*+

*y*)/(

*d*√2)]

*sin*

^{2}[

*π*(

*x*-

*y*)/(

*d*√2)], where

*d*is the lattice spacing. The propagation equations are solved numerically, concurrently with the temporal equation, in the manner described in Ref. [14

14. M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, 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 (
2004). [CrossRef] [PubMed]

3. 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]

*L*, and the lattice and beam intensities. However, we confine our attention here to the cases with fixed lattice and input beam intensities. We choose the lattice intensity comparable but stronger than the beam intensities, in accordance with the experiment [3

**98**, 123903 (
2007). [CrossRef] [PubMed]

**98**, 123903 (
2007). [CrossRef] [PubMed]

## 3. Spatially localized surface states

*corner*states. Two mutually incoherent Gaussian beams of the same intensity are launched head-on from the opposite faces of a photorefractive crystal, in which an optically induced 2D photonic lattice is established [see Fig. 1(a)]. The beams are launched in the center of the corner unit cell. Some characteristic outcomes of the surface modes are presented in Fig. 2, after steady state is reached. The upper row in the figure displays intensity distributions of the forward beam at the exit face of the crystal. Insets depict the same situation in the inverse space; added squaresmark the first Brillouin zone (BZ) of the full lattice. It is seen that with the increasing coupling constant, the beams focus into well defined CP solitons, close to the corner lattice site. As they focus in the direct space (the bottom row), they spread in the inverse space, spilling over the first BZ (insets). For larger Γ, the influence of the neighboring sites on the beam distribution is lost.

9. M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E **68**, 025601 (
2003). [CrossRef]

13. K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E **68**, 066611 (
2003). [CrossRef]

14. M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, 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 (
2004). [CrossRef] [PubMed]

*splitup*transition. Here it appears in the localized modes in the corner of photonic lattice, and represents the simplest form of the dynamical beam instability. Such instabilities cannot occur in the single beam surface states.

*L*and Γ. The input intensities are kept fixed, as in Fig. 2. Three regions are visible in the diagram. For small coupling constants there exists a narrow region where no conversion to trapped surface modes is observed. We term these states the diffraction modes. The beams spread upon propagation, and overlap with the neighboring lattice sites (

*cf*. Fig. 5). This regime is similar to the single beam experimental results at low bias field [3

**98**, 123903 (
2007). [CrossRef] [PubMed]

*L*the region of stable corner solitons is observed. For still higher values of the parameters, the region of unstable modes is reached.

**98**, 123903 (
2007). [CrossRef] [PubMed]

*L*, we observe the development of instabilities, in the form of one or two subsequent splitup transitions. After the splitup transitions, the intertwined Gaussian-like beams are again strongly pinned to the lattice site. Characteristic cases from Fig. 4 are also presented as movies in the lower part of Fig. 5.

## 4. Soliton instabilities

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. Suntsov |

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

3. | 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. |

4. | 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. |

5. | A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E |

6. | Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. |

7. | M. Haelterman, A. P. Sheppard, and A. W. Snyder, “Bimodal counterpropagating spatial solitary waves,” Opt. Commun. |

8. | O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. |

9. | M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E |

10. | See, e.g., a recent review: M.S. Petrović, M.R. Belić, C. Denz, and Yu.S. Kivshar, “Counterpropagating optical beams and solitons,” to appear in Laser & Photonics Reviews ( 2010). http://www.arXiv.org, Preprint [physics.optics] arXiv:0910.4700 |

11. | E. Smirnov, M. Stepić, C. E. Rüter, V. Shandarov, and D. Kip, “Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array,” Opt. Lett. |

12. | S. Koke, D. Träger, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, “Stabilization of counterpropagating solitons by photonic lattices,” Opt. Express |

13. | K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E |

14. | M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express |

**OCIS Codes**

(190.5330) Nonlinear optics : Photorefractive optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 8, 2009

Revised Manuscript: October 25, 2009

Manuscript Accepted: November 4, 2009

Published: November 10, 2009

**Citation**

Dragana Jovic, Yuri S. Kivshar, Raka Jovanovic, and Milivoj Belic, "Counterpropagating surface solitons in two-dimensional photonic lattices," Opt. Express **17**, 21515-21521 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21515

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### References

- S. Suntsov et al., "Observation of one- and two-dimensional discrete surface spatial solitons," J. Nonlinear Opt. Phys. Mater. 16, 401 (2007). [CrossRef]
- Yu. S. Kivshar, "Nonlinear Tamm states and surface effects in periodic photonic structures," Laser Phys. Lett. 5, 703 (2008). [CrossRef]
- 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. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007). [CrossRef]
- A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005). [CrossRef]
- Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982). [CrossRef]
- M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993). [CrossRef]
- O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002). [CrossRef] [PubMed]
- M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003). [CrossRef]
- See, e.g., a recent review: M. S. Petrovic, M.R. Belic, C. Denz, and Yu. S. Kivshar, "Counterpropagating optical beams and solitons," to appear in Laser & Photonics Reviews (2010). http://www.arXiv.org, Preprint [physics.optics] arXiv:0910.4700
- E. Smirnov,M. Stepic, C. E. Ruter, V. Shandarov, and D. Kip, "Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array," Opt. Lett. 32, 512 (2007). [CrossRef] [PubMed]
- S. Koke, D. Trager, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express 15, 6279 (2007). [CrossRef] [PubMed]
- K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003). [CrossRef]
- M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, 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 (2004). [CrossRef] [PubMed]

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