## Defect solitons in two-dimensional optical lattices

Optics Express, Vol. 18, Issue 11, pp. 10956-10962 (2010)

http://dx.doi.org/10.1364/OE.18.010956

Acrobat PDF (1008 KB)

### Abstract

We report on the existence and stability of solitons in a defect embedded in a square optical lattice based on a photorefractive crystal with focusing saturable nonlinearity. These solitons exist in different bandgaps due to the change of defect intensity. For a positive defect, the solitons only exist in the semi-infinite gap and can be stable in the low power region but not the high power region. For a negative defect, the solitons can exist not only in the semi-infinite gap, but also in the first gap. With increasing the defect depth, these solitons are stable within a moderate power region in the first gap while unstable in the entire semi-infinite gap.

© 2010 OSA

## 1. Introduction

1. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

5. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(2), 026609 (2006). [CrossRef] [PubMed]

6. Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. **18**(14), 1147–1149 (1993). [CrossRef] [PubMed]

8. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. **96**(7), 073901 (2006). [CrossRef] [PubMed]

5. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(2), 026609 (2006). [CrossRef] [PubMed]

9. W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B **24**(10), 2584–2588 (2007). [CrossRef]

10. 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**(2), 023902 (2003). [CrossRef] [PubMed]

13. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**(21), 2094–2096 (2003). [CrossRef] [PubMed]

1. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

14. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. **98**(10), 103901 (2007). [CrossRef] [PubMed]

16. 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**(12), 123903 (2007). [CrossRef] [PubMed]

17. J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A **76**(1), 013828 (2007). [CrossRef]

18. I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. **96**(22), 223903 (2006). [CrossRef] [PubMed]

19. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. **34**(6), 797–799 (2009). [CrossRef] [PubMed]

## 2. The model

1. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

10. 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**(2), 023902 (2003). [CrossRef] [PubMed]

*U*is the slowly varying amplitude of the probe beam and

*I*is the transverse intensity distribution of the lattice beam with a defect as described byHere

_{L}*I*is the lattice peak intensity,

_{0}*z*is the propagation distance (in units of

*n*is the unperturbed refractive index along the extraordinary axis,

_{e}*D*is the lattice spacing,

*x*and

*y*are the transverse distances (in units of

*E*is the applied dc field voltage [in units of

_{0}*ε*is the modulation parameter for the defect intensity. For a positive defect

*ε*>0, the lattice intensity

*I*at the defect site is higher than that in the surrounding regions. For a negative defect

_{L}*ε*<0, the lattice intensity

*I*at the defect site is lower than that in the surrounding regions. The intensity distributions of optical lattices with a negative defect (

_{L}*ε*= −0.5) and a positive defect (

*ε*= 0.5) are displayed in Figs. 1(b) and 1(c), respectively. In this paper, we choose the physical parameters as

*x*or

*y*unit corresponds to 6.4

*μm*, one

*z*unit corresponds to 2.3

*mm*, and one

*E*unit corresponds to 20

_{0}*V/mm*. We take

*I*= 3 and

_{0}*E*= 6, which are typical experimental conditions [11

_{0}11. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. **27**(22), 2019–2021 (2002). [CrossRef]

*u*(

*x*,

*y*) possesses the same periodicity as the lattices,

*μ*is a real propagation constant,

*k*and

_{x}*k*are wave numbers in the first Brillouin zone. We calculate this equation by the plane wave expansion method to obtain the bandgap diagram as shown in Fig. 1(a). Using above given parameters, we can obtain the boundaries of the allowed bands and provide the range of the semi-infinite gap and the first gap as

_{y}*u*(

*x*,

*y*) is a real-valued function and satisfies the nonlinear equationThe soliton solutions

*u*(

*x*,

*y*) can be calculated by the modified squared-operator iteration method [20

20. J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. **118**(2), 153–197 (2007). [CrossRef]

21. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. **227**(14), 6862–6876 (2008). [CrossRef]

## 3. Numerical results

*z*direction by 200 units distance, where

*σ*is equal to 10% of the input soliton amplitude).

*ε*= −0.5 as a typical case for the negative defect lattice. The DSs exist not only in the semi-infinite gap, but also in the first gap. Figure 2(a) shows the power of DSs versus propagation constant

*μ*. In the semi-infinite gap, the DSs can stably exist in the range of

*μ*= 2.8 [point A in Fig. 2(a)]. The DSs are trapped at the defect site and maintain their profiles at z = 100 [Fig. 2(d)] and z = 200 [Fig. 2(e)]. The VK criterion tells us that the solitons are stable if dp/dμ<0 and unstable if dp/dμ>0 [5

5. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(2), 026609 (2006). [CrossRef] [PubMed]

22. J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. **228**(18), 7007–7024 (2009). [CrossRef]

*μ*= 3.4 [point B in Fig. 2(a)]. Figures 2(g) and 2(h) show the profiles of DS at z = 100 and z = 200, respectively, where the DS cannot maintain its profile upon propagation. When

**73**(2), 026609 (2006). [CrossRef] [PubMed]

**73**(2), 026609 (2006). [CrossRef] [PubMed]

*μ*= 2.4 [point C in Fig. 2(a)] as shown in Fig. 3(a) . The DS cannot be trapped at the defect site at z = 100 [Fig. 3(b)] and z = 200 [Fig. 3(c)]. To further verify solitons’ instability, we numerically calculate Eq. (5) to obtain the perturbation growth rate Re(δ) as shown in Fig. 2(b). In the first gap the DSs can stably exist. The power of DSs is gradually decreasing with propagation constant, where dp/dμ<0 as shown in Fig. 2(a), which implies that the DSs are stable according to VK criterion. Taking

*μ*= 4.5 [point D in Fig. 2(a)], Figs. 3(d), 3(e), and 3(f) show the profiles of DS at z = 0, z = 100, and z = 200, respectively. When

*μ*= 4.7 [point E in Fig. 2(a)], we also give the profiles of DS at z = 0, z = 100, and z = 200 as shown in Figs. 3(g), 3(h), and 3(i), respectively. In the first gap these solitons can maintain their shapes with most of their energies concentrated on the defect site along propagation.

*ε*= −1 as an example. Figure 4(a) shows the power of the solitons versus propagation constant. The DSs can stably exist only in

*μ*= 5.0. Figures 4(d) and 4(e) are the profiles of DS at z = 100 and z = 200, respectively. In the first gap, near the first band (

*μ*= 4.68 [Fig. 4(f)] and its profiles at z = 100 [Fig. 4(g)] and z = 200 [Fig. 4(h)], respectively. The DSs finally break up. In the lower power region we find that the power of DSs isn’t monotonically decreasing with the propagation constant. When

*μ*= 5.08, Figs. 5(a) , 5(b), and 5(c) show the profiles of DS at z = 0, z = 100, and z = 200, respectively. We can see that the DS completely breaks at z = 200. In the semi-infinite gap, although we can obtain the soliton solutions, they are unstable. As an example, we also give the profile of DS for

*μ*= 3.0 [Fig. 5(d)] and its profiles at z = 100 [Fig. 5(e)] and z = 200 [Fig. 5(f)], respectively. The DSs cannot be trapped at defect site along propagation.

*ε*= 0.5 as a typical case for the positive defect lattice. Figure 6(a) shows the power of DSs versus propagation constant

*μ*. The DSs only exist in the semi-infinite gap. Moreover, the DSs are stable in the low power region but unstable in the high power region [Fig. 6(a)], which is similar to that of defect solitons in 1D optical lattices [5

**73**(2), 026609 (2006). [CrossRef] [PubMed]

*μ*= 3.0 [point B in Fig. 6(a)] and

*μ*= 1.4 [point A in Fig. 6(a)] as shown in Figs. 6(c) and (f), respectively. For

*μ*= 3.0, the DS can keep its profiles at z = 100 [Fig. 6(d)] and z = 200 [Fig. 6(e)], but for

*μ*= 1.4 the DS cannot keep its original shape at z = 100 [Fig. 6(g)] and z = 200 [Fig. 6(h)]. The perturbation growth rate Re(δ) is not zero that means the DSs are unstable in the case

23. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A **80**(4), 043824 (2009). [CrossRef]

## 4. Summary

## Acknowledgments

## References and links

1. | 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 |

2. | M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

3. | W. H. Chen, Y. J. He, and H. Z. Wang, “Defect superlattice solitons,” Opt. Express |

4. | J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A |

5. | J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

6. | Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. |

7. | C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. |

8. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. |

9. | W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B |

10. | 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. |

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

12. | Y. V. Kartashov, V. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express |

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

14. | O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. |

15. | L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. |

16. | 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. |

17. | J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A |

18. | I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. |

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

20. | J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. |

21. | J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. |

22. | J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. |

23. | Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 26, 2010

Revised Manuscript: April 30, 2010

Manuscript Accepted: April 30, 2010

Published: May 10, 2010

**Citation**

W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, "Defect solitons in two-dimensional optical lattices," Opt. Express **18**, 10956-10962 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-10956

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### References

- 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(6928), 147–150 (2003). [CrossRef] [PubMed]
- M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006). [CrossRef] [PubMed]
- W. H. Chen, Y. J. He, and H. Z. Wang, “Defect superlattice solitons,” Opt. Express 15(22), 14498–14503 (2007). [CrossRef] [PubMed]
- J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008). [CrossRef]
- J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026609 (2006). [CrossRef] [PubMed]
- Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. 18(14), 1147–1149 (1993). [CrossRef] [PubMed]
- C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96(7), 073901 (2006). [CrossRef] [PubMed]
- W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B 24(10), 2584–2588 (2007). [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(2), 023902 (2003). [CrossRef] [PubMed]
- Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002). [CrossRef]
- Y. V. Kartashov, V. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12(13), 2831–2837 (2004). [CrossRef] [PubMed]
- J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). [CrossRef] [PubMed]
- O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007). [CrossRef] [PubMed]
- L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (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(12), 123903 (2007). [CrossRef] [PubMed]
- J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007). [CrossRef]
- I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006). [CrossRef] [PubMed]
- A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009). [CrossRef] [PubMed]
- J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007). [CrossRef]
- J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227(14), 6862–6876 (2008). [CrossRef]
- J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228(18), 7007–7024 (2009). [CrossRef]
- Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009). [CrossRef]

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