## Defect solitons in kagome optical lattices |

Optics Express, Vol. 18, Issue 20, pp. 20786-20792 (2010)

http://dx.doi.org/10.1364/OE.18.020786

Acrobat PDF (1123 KB)

### Abstract

We report the existence and stability of solitons in kagome optical lattices with a defect in photorefractive crystal under focusing saturable nonlinearity. For different types of defects, solitons will exist in different gaps. For a positive defect, the solitons only exist in the semi-infinite gap and only stably exist in the low power region. For a negative defect, the solitons exist both in the semi-infinite gap and the first gap. With an increasing of the negative defect depth, the stable region in the semi-infinite will be narrowed, while solitons will be firstly unstable in the high power region of the first gap, and finally solitons will be not stable in the whole first 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. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. **27**(22), 2019–2021 (2002). [CrossRef]

6. T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**(10), 1293–1295 (2007). [CrossRef] [PubMed]

7. Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. **33**(24), 2949–2951 (2008). [CrossRef] [PubMed]

8. Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B **21**(5), 973–981 (2004). [CrossRef]

9. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. **92**(12), 123903 (2004). [CrossRef] [PubMed]

10. K. J. H. Law, H. Susanto, and P. G. Kevrekidis, “Solitons and vortices in honeycomb defocusing photonic lattices,” Phys. Rev. A **78**(3), 033802 (2008). [CrossRef]

11. K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, “Localized strctures in kagome lattices,” Phys. Rev. A **79**(5), 053818 (2009). [CrossRef]

12. J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A **77**(3), 033834 (2008). [CrossRef]

13. Z. Shi, J. Wang, Z. Chen, and J. Yang, “Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media,” Phys. Rev. A **78**(6), 063812 (2008). [CrossRef]

14. E. A. Ostrovskaya and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express **12**(1), 19–29 (2004). [CrossRef] [PubMed]

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

17. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express **18**(11), 10956–10961 (2010). [CrossRef] [PubMed]

18. 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]

11. K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, “Localized strctures in kagome lattices,” Phys. Rev. A **79**(5), 053818 (2009). [CrossRef]

## 2. The theoretical model

*Z*for the varying amplitude

*U*is described by the normalized 2D nonlinearity Schrödinger equation [1

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]

15. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment,” Stud. Appl. Math. **113**(4), 389–412 (2004). [CrossRef]

11. K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, “Localized strctures in kagome lattices,” Phys. Rev. A **79**(5), 053818 (2009). [CrossRef]

*I*is the intensity profile of kagome lattices with a defect that described by Here,

_{L}*p*= 3/2,

*k = 4π/d*,

*h*= (1 + 4

*p*/3) [11].

*Z*is the propagation distance, its real units is

*2k*,

_{1}D^{2}/π^{2}*D*is the lattice spacing, (

*x, y*) are the transverse distance, its real units is

*D/π*,

*k*,

_{1}= k_{0}n_{e}*k*is the wavenumber in vacuum,

_{0}*n*is the refractive index along the extraordinary axis,

_{e}*γ*is the electrooptic coefficient of the crystal, and

_{33}*E*is the applied DC field voltage, its real unit is

_{0}*π*[15

^{2}/(k_{0}^{2}n_{e}^{4}D^{2}γ_{33})15. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment,” Stud. Appl. Math. **113**(4), 389–412 (2004). [CrossRef]

*E*= 15 [19

_{0}19. J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A **76**(1), 013828 (2007). [CrossRef]

*d =*3π,

*V*= 0.375,

_{0}*D*= 20

*μm*,

*n*= 2.3,

_{e}*γ*= 280

_{33}*pm/V*, and λ

_{0}= 0.5

*μm*. Thus, one

*x*or

*y*unit corresponds to 6.4

*μm*, one

*z*unit corresponds 2.3

*mm*, one

*E*unit corresponds to 20

_{0}*V/mm*, and the uniform lattice peak intensity is 6.

*μ*≤6.33, and the first gap as 6.86≤

*μ*≤8.96. The bandgap diagram is shown in Fig. 1(a). Figure 1(b) is the intensity distribution of kagome lattices with a negative defect (ε = −0.5), while Fig. 1(c) is the intensity distribution of kagome lattices with a positive defect (ε = 0.5).

*μ*is the propagation constant and

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

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

## 3. Numerical results

*μ*. In the region of 3.24≤

*μ*≤6.25, where the power of solitons is moderate and the solitons are stable, Fig. 2(c) shows the profile (|u|) of soliton for

*μ*= 4.6 [point A in Fig. 2(a)]. Figures 2(d) and 2(e) show the profile (|u|) at z = 100 and z = 200, respectively. Obviously the soliton can stably transmit. In the region of 3.24≤

*μ*≤6.25, the slope of the power curve is negative, that is d

*p/dμ*<0. According to the VK criterion, solitons can stably exist. So the stability of gap solitons in the region is in accordance with the VK criterion. In the region of 6.26≤

*μ*≤6.32, where the power of solitons is low and the slope of power diagram changes, d

*p/dμ*>0. In the region, we find that the solitons are unstable. Figure 2(b) shows the Re(δ), Re(δ)>0 indicate the solitons are unstable. Figure 2(f) shows the profile (|u|) of gap soliton for

*μ*= 6.30 [point B in Fig. 2(a)]. Figures 2(g) and 2(h) show the profile (|u|) at z = 100 and z = 200, respectively. The soliton is not stable.

*μ*= 6.33 (very close to the band), soliton can stably propagate. Figure 3(d) shows the profile (|u|) of the soliton. Figures 3(e) and 3(f) show the profile (|u|) at z = 100 and z = 200, respectively.

*μ*<3.24, in the region, the power of solitons is high and exponentially grows with the decreasing of the propagation constant

*μ*. Solitons can’t stably propagate. This kind of instability is different from the VK instability caused by the slope of power curve is positive [21

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

*μ*= 2.5 [point C in Fig. 2(a)]. Figures 3(b) and 3(c) show the profile (|u|) at z = 100 and z = 200, respectively. The soliton changes the place along the propagation.

*μ*. In the semi-infinite gap, the stable region is 4.16≤

*μ*≤6.05, where the power of DSs is moderate. In the region,

*dp/dμ*<0, according to the VK criterion, DSs in the region are stable. Figure 4(c) shows the profile (|u|) of DS for

*μ*= 5.0 [point A in Fig. 4(a)]. Figures 4(d) and 4(e) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can stably propagate.

*μ*≤6.33, the power curve is not smooth. That means DSs in the region aren’t stable, Fig. 4(b) shows the Re(δ) in the region, where we can see Re(δ)>0 . Figure 4(f) shows the profile (|u|) of defect soliton for

*μ*= 6.16 [point B in Fig. 4(a)]. Figures 4(g) and 4(h) show the profile (|u|) at z = 100 and z = 200, respectively. DS can’t stably propagate.

*μ*<4.16, the power of DSs is high and exponentially grows with the decreasing of the propagation constant

*μ*, DSs cannot propagate stably. Figure 4(b) shows the Re(δ)>0 obviously in the region. Figure 5(a) shows the profile (|u|) of DS for

*μ*= 3.5 [point C in Fig. 4(a)]. Figures 5(b) and 5(c) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can’t stay at the same site along the propagation.

*μ*= 7.0 [point D in Fig. 4(a)]. Figures 5(e) and 5(f) show the profile (|u|) at z = 100 and z = 200, respectively. We can see the DS can stably transmit. Figure 5(g) shows the profile (|u|) of DS for

*μ*= 7.5 [point E in Fig. 4(a)]. Figures 5(h) and 5(i) show the profile (|u|) at z = 100 and z = 200, respectively. The DS is stable. Figure 4(a) show the power curve of DSs in the first gap, the power of DSs is gradually decreasing with the increasing of the propagation constant

*μ*, obviously

*dp/dμ*<0, so the stability of DSs in the first gap also is in accordance with the VK criterion.

*μ*≤6.01, but when ε = −0.7, the DSs in the whole semi-infinite gap are unstable. In the first gap, with an increasing of the negative defect depth, the DSs will not all stably exist, when ε = −0.8, DSs in the high power region are not stable, when ε = −1, DSs in the whole first gap are all unstable.

*μ*. For the positive defects, solitons can stably exist in the low power region, but cannot stably exist in the high power region, which is similar to the DSs in 2D square optical lattices [17

17. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express **18**(11), 10956–10961 (2010). [CrossRef] [PubMed]

*μ*>2.81, DSs are stable. When

*μ*>2.81, Re(δ) = 0, that also means solitons are stable. Figure 6(c) shows the profile (|u|) of DS for

*μ*= 3[point A in Fig. 6(a)]. Figures 6(d) and 6(e) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can stably propagate. In the region of

*μ*≤2.81, the power of DSs exponentially grows with the decreasing of the propagation constant

*μ*, DSs are not stable, Fig. 6(b) show the Re(δ) versus the propagation constant

*μ*. In the region of

*μ*≤2.81, Re(δ)>0, it means solitons can’t stably exist. Figure 6(f) shows the profile (|u|) of DS for

*μ*= 2.2 [point B in Fig. 6(a)]. Figures 6(g) and 6(h) show the profile (|u|) at z = 100 and z = 200, respectively. DS can’t maintain its original shape at z = 100 and z = 200.

*E*in this paper) can affect the properties of defect solitons in optical lattices. To compare the properties of defect solitons in kagome optical lattices with that in square optical lattices which has been reported recently [17

_{0}17. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express **18**(11), 10956–10961 (2010). [CrossRef] [PubMed]

*I*to 6 and

_{0}*E*to 15 in [17

_{0}**18**(11), 10956–10961 (2010). [CrossRef] [PubMed]

**18**(11), 10956–10961 (2010). [CrossRef] [PubMed]

*μ*≤5.53, the first gap: 5.76≤

*μ*≤9.21, and the second gap: 10.13≤

*μ*≤12.68. For positive defects, DSs in the two types of optical lattices all only exist in the semi-infinite gap, and are only stable in the low power region while not stable in the high power region. The stable region of DSs in kagome lattices is wider than that in square lattices. For negative defects, DSs in kagome lattices and square lattices can exist in the semi-infinite gap and the first gap at the same time. In the semi-infinite gap, DSs in the two types of lattices both stably exist in the moderate power region, the stable region of DSs in square lattices is wider than that in kagome lattices. In the first gap, with an increasing of the negative defect depth, DSs will not always be stable in the gap in the two kinds of lattices, with an increasing of the negative defect depth, for kagome lattices, at first DSs in the high power region will not be stable, and then DSs in the whole first gap are unstable, while for square lattices, the finding varies a little: Firstly, DSs in the low power region are unstable, and later DSs also are unstable in the whole first gap. In the first gap, the stable region of DSs in square lattices is wider than that in kagome lattices. For the uniform lattices, solitons in kagome lattices and square lattices both only exist in the semi-infinite gap. Solitons in kagome lattices and square lattices stably exist in the moderate power region, the stable region of solitons in square lattices is wider than that in kagome lattices. When propagation constant

*µ*is very close to the band, solitons in square lattices are not stable, which is a little different from kagome lattices.

*E*) and lattice peak intensity (the peak intensity of kagome lattices and the square lattices are the same, and

_{0}*E*in the two types of lattices are also the same), the similarities and differences between kagome lattices and square lattices will be different

_{0}## 4. Conclusions

## Acknowledgment

## 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. | 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. |

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. | N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. |

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

6. | T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. |

7. | Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. |

8. | Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B |

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

10. | K. J. H. Law, H. Susanto, and P. G. Kevrekidis, “Solitons and vortices in honeycomb defocusing photonic lattices,” Phys. Rev. A |

11. | K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, “Localized strctures in kagome lattices,” Phys. Rev. A |

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

13. | Z. Shi, J. Wang, Z. Chen, and J. Yang, “Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media,” Phys. Rev. A |

14. | E. A. Ostrovskaya and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express |

15. | J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment,” Stud. Appl. Math. |

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

17. | W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express |

18. | 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. |

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

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

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

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 11, 2010

Revised Manuscript: September 6, 2010

Manuscript Accepted: September 6, 2010

Published: September 15, 2010

**Citation**

Xing Zhu, Hong Wang, and Li-Xian Zheng, "Defect solitons in kagome optical lattices," Opt. Express **18**, 20786-20792 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20786

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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]
- 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]
- 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]
- N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]
- Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002). [CrossRef]
- T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. 32(10), 1293–1295 (2007). [CrossRef] [PubMed]
- Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. 33(24), 2949–2951 (2008). [CrossRef] [PubMed]
- Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B 21(5), 973–981 (2004). [CrossRef]
- D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). [CrossRef] [PubMed]
- K. J. H. Law, H. Susanto, and P. G. Kevrekidis, “Solitons and vortices in honeycomb defocusing photonic lattices,” Phys. Rev. A 78(3), 033802 (2008). [CrossRef]
- K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, “Localized strctures in kagome lattices,” Phys. Rev. A 79(5), 053818 (2009). [CrossRef]
- J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008). [CrossRef]
- Z. Shi, J. Wang, Z. Chen, and J. Yang, “Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media,” Phys. Rev. A 78(6), 063812 (2008). [CrossRef]
- E. A. Ostrovskaya and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express 12(1), 19–29 (2004). [CrossRef] [PubMed]
- J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment,” Stud. Appl. Math. 113(4), 389–412 (2004). [CrossRef]
- J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Wave in General Nonlinear Wave Equation,” Stud. Appl. Math. 118(2), 153–197 (2007). [CrossRef]
- W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express 18(11), 10956–10961 (2010). [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. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007). [CrossRef]
- J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227(14), 6862–6876 (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]

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