## Defect solitons in parity-time symmetric superlattices |

Optics Express, Vol. 19, Issue 12, pp. 11457-11462 (2011)

http://dx.doi.org/10.1364/OE.19.011457

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### Abstract

We study defect solitons (DSs) in a parity-time (PT) symmetric superlattice with focusing Kerr nonlinearity. The properties of the DSs with a PT symmetrical potential are obviously different from those in a superlattice with a real refractive index. Unusual features stemming from PT symmetry can be found. Research results show that the solitons with a zero defect or a positive defect can exist and stably propagate in the semi-infinite gap, but they cannot exist in the first gap. For the case of a negative defect, the soliton can stably exist in both the semi-infinite gap and the first gap.

© 2011 OSA

## 1. Introduction

1. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. **80**, 5243 (1998). [CrossRef]

2. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. **100**, 030402 (2008). [CrossRef] [PubMed]

6. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. **6**, 192 (2010). [CrossRef]

2. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. **100**, 030402 (2008). [CrossRef] [PubMed]

7. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E **73**, 026609 (2006). [CrossRef]

13. 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]

14. A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. **30**(14), 1849–1851 (2005). [CrossRef] [PubMed]

15. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A **78**, 013847 (2008). [CrossRef]

16. A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. **93**, 171104 (2008). [CrossRef]

17. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. **35**, 2928–2930 (2010). [CrossRef] [PubMed]

18. H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express **19**(5), 4030 (2011). [CrossRef] [PubMed]

## 2. Theory

*μ*≤ −3.26 and the first, second gap is −2.72 ≤

*μ*≤ −1.60 and −0.17 ≤

*μ*≤ 1.85, respectively. The intensity distributions of PT superlattice with defect:

*ε*= 0,

*ε*= 0.5, and

*ε*= −0.5 are displayed in Fig. 1(b)–1(d), respectively.

*v,w*<< 1, and

***represents the complex conjugation. Substituting Eq. (5) into Eq. (1) and then linearing Eq. (1), we arrive at an eigenvalue equation

*Re*(

*δ*)[20

20. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. **277**(14), 6862–6876 (2008). [CrossRef]

*Re*(

*δ*) = 0, the slitons are linearly stable; otherwise, they are linearly unstable.

## 3. Numerical results and discussion

*ε*= −0.5 as a case of the negative defect in the PT superlattices and the corresponding power

*P*of defect solitons versus the propagation constant

*μ*is showed in Fig. 2(a). It can be found in this figure that defect solitons can exist in both the semi-infinite gap and the first gap. In the semi-infinite gap, the stable region of defect solitons is

*μ*≤ −4.03 and the unstable region is −4.03 <

*μ*≤ −3.26. In the first gap, the stable region of defect solitons is −2.72 ≤

*μ*≤ −1.75 and the unstable region is −1.75 <

*μ*≤ −1.61. Two stable examples:

*μ*= −6.0 (point A in Fig. 2(a)) and

*μ*= −2.0 (point B in Fig. 2(a)) have been used to certify soliton stability in the semi-infinite gap and the first gap. Their soliton profiles are showed in Fig. 2(c) and 2(d) and the corresponding soliton propagations are showed in Fig. 2(f) and 2(g), respectively. In the first gap, we select an unstable example:

*μ*= −1.74 (point C in Fig. 2(a)) and plot its soliton profile in Fig. 2(e). In Ref. [18

18. H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express **19**(5), 4030 (2011). [CrossRef] [PubMed]

*f*(

*x*) is even and the imaginary part is odd. The corresponding soliton propagation is showed in Fig. 2(h). Figure 2(b) shows the change of

*Re*(

*δ*) with propagation constant

*μ*. In the regions:

*μ*≤ −4.03 and −2.72 ≤

*μ*≤ −1.75, the slope of power curve is negative and the

*Re*(

*δ*) is equal to zero; while in the regions: −4.03

*< μ*≤ −3.26 and −1.75 <

*μ*≤ −1.61, the slope of power curve is positive and the

*Re*(

*δ*) is not equal to zero. From these results, we can conclude that the stability or instability of defect soliton in the both semi-infinite gap and the first gap is in accordance with the Vakhitov-Kolokolov (VK) criterion.

*P*versus propagation constant

*μ*for a positive defect (

*ε*= 0.5) in the PT superlattices is presented in Fig. 3(a). In this figure, we can see that the defect solitons can only exist in the semi-infinite gap. In these regions which we have considered, defect solitons can stably transmit. We choose two stable example:

*μ*= −6.5, −5.0 (points A and B in Fig. 3(a)) and show their soliton profiles in Fig. 3(b) and 3(c), respectively. Figure 3(d) and 3(e) present their corresponding soltion propagations. In the regions of the semi-infinite gap which we have considered, the negative slope of power curve and the

*Re*(

*δ*) = 0 indicate that the solitons stability is in accordance with the VK criterion.

*P*versus propagation constant

*μ*for the case of a zero defect (

*ε*= 0). This figure indicates that solitons only exist in the semi-infinite gap. In the regions of propagation constant: −4.07 ≤

*μ*≤ −3.50, and −3.36 ≤

*μ*, the solitons cannot stably propagate. One example:

*μ*= −3.3 (point B in Fig. 4(a)) has been introduced to certify the instability of soliton in these regions. Its solitons profile is showed in Fig. 4(d) and the corresponding soliton propagations is showed in Fig. 4(f). In the regions of propagation constant:

*μ*< −4.07 and −3.50 <

*μ*< −3.36, the solitons can stably transmit. The soliton profile of a stable example:

*μ*= −5.0 (point A in Fig. 4(a)) is shown in Fig. 4(c) and its corresponding propagation is showed in Fig. 4(e). To further verify the instability, we plot the change of the

*Re*(

*δ*) with propagation constant

*μ*in Fig. 4(b). For the negative slope of power curve (

*dP/dμ*< 0) and

*Re*(

*δ*) = 0 in the regions:

*μ*< −4.07 and the positive slope of power (

*dP/dμ*> 0) and

*Re*(

*δ*) > 0 in the regions: −3.36 ≤

*μ*, we can conclude that the solitons stability or instability in these regions is in accordance with the VK criterion.

*μ, ε*) of the PT solitons according to the relation between defect depth

*ε*and the propagation constant

*μ*. When increasing the defect depth of positive defect from

*ε*= 0.1 to 0.8 or negative defect from

*ε*= −0.1 to −0.7, the stable region of defect solitons in the semi-infinite gap will become narrow (see Fig. 5(a) and 5(b)). For the case of

*ε*= −0.8, the stable region will disappear. In the first gap, the stable region of defect soliton will firstly become large and then become narrow when increasing the defect depth from

*ε*= −0.1 to −0.6. The stable region is vanished for the case of

*ε*= −0.7 (see Fig. 5(c)). The stability of solitons propagation is mainly affected by light diffraction and self-focusing resulting from nonlinearity.Beacuse of the counteraction between light diffraction and nonlinearity, the soliton pulses can stably propagate. When diffraction cannot be suppressed by nonlinearity, soliton pulses will finally decay into linear diffractive waves. In the case of a negative defect, the defect site, which has lower light intensity, can leads to that the repusion from the negative defect will increase the light diffraction and then change the existence and stability of PT solitons. Thus, with the effect of a negative defect mode, the PT gap soliton cannot only exist, but also stably propagate in the first gap. In the case of a positive defect, the defect site, which has a higher intensity, is attractive to the light field. The attraction from the positive defect will decrease the light diffraction. As a result, the existence and stability of PT solitons will also be altered, justly like the case of the negative defect.

## 4. Conclusion

## Acknowledgments

## References and links

1. | C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. |

2. | Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. |

3. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A |

4. | O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. |

5. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. |

6. | C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. |

7. | J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E |

8. | Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. |

9. | X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express |

10. | X. Zhu, H. Wang, T. W. Wu, and L. X. Zheng, “Defect solitons in triangular optical lattices,” J. Opt. Soc. Am. B |

11. | 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 |

12. | Z. Lu and Z. Zhang, “Surface line defect solitons in square optical lattice,” Opt. Express |

13. | 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. |

14. | A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. |

15. | F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A |

16. | A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. |

17. | K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. |

18. | H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express |

19. | J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. |

20. | J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. 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: April 11, 2011

Revised Manuscript: May 7, 2011

Manuscript Accepted: May 7, 2011

Published: May 27, 2011

**Citation**

Zhien Lu and Zhi-Ming Zhang, "Defect solitons in parity-time symmetric superlattices," Opt. Express **19**, 11457-11462 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11457

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### References

- C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998). [CrossRef]
- Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef] [PubMed]
- K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010). [CrossRef]
- O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009). [CrossRef] [PubMed]
- K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef] [PubMed]
- C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010). [CrossRef]
- J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006). [CrossRef]
- Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007). [CrossRef] [PubMed]
- X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010). [CrossRef] [PubMed]
- X. Zhu, H. Wang, T. W. Wu, and L. X. Zheng, “Defect solitons in triangular optical lattices,” J. Opt. Soc. Am. B 28(3), 521 (2011). [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, 043824 (2009). [CrossRef]
- Z. Lu and Z. Zhang, “Surface line defect solitons in square optical lattice,” Opt. Express 19(3), 2410 (2011). [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]
- A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. 30(14), 1849–1851 (2005). [CrossRef] [PubMed]
- F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008). [CrossRef]
- A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008). [CrossRef]
- K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010). [CrossRef] [PubMed]
- H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express 19(5), 4030 (2011). [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. 277(14), 6862–6876 (2008). [CrossRef]

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