## The optimal structure of two dimensional photonic crystals with the large absolute band gap |

Optics Express, Vol. 19, Issue 20, pp. 19346-19353 (2011)

http://dx.doi.org/10.1364/OE.19.019346

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### Abstract

This paper reports a new designed square lattice GaAs structure of two-dimensional photonic crystals with absolute band gap approach to 0.1623 (2*πc/a*), where *a* is the period of the square lattice. The optimal structure is obtained by combining the Geometry Projection Method and Finite Element Method. Both gradient information and symmetric control points are introduced to reduce the calculation cost. For benefit to the fabrication in reality, the structure is simplified by the combination of triangle and rectangular geometry. Through parameter optimization, the absolute band gap of the new structure is improved to 0.1735 (*2πc/a*), which is much larger than those reported before. The new PC structure is convenient and stab for fabrication, and may be found applications in the future optical devices.

© 2011 OSA

## 1. Introduction

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**(23), 2486–2489 (1987). [CrossRef] [PubMed]

*2πc/a*) and 0.11(

*2πc/a*) respectively [2

2. L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B **66**(16), 165315 (2002). [CrossRef]

*2πc/a*) can be get by modifying the radius and positions of the dielectric rods [3

3. H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun. **282**(14), 3012–3017 (2009). [CrossRef]

*2πc/a*) by reducing the symmetry of the PC structure [4

4. S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt. **55**(18), 2971–2980 (2008). [CrossRef]

*2πc/a*) by using a novel optimization algorithms [5

5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B **68**(3), 035109 (2003). [CrossRef]

6. M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B **17**(6), 1027–1030 (2000). [CrossRef]

8. F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express **16**(16), 12278–12289 (2008). [CrossRef] [PubMed]

5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B **68**(3), 035109 (2003). [CrossRef]

9. O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. **100**(15), 153904 (2008). [CrossRef] [PubMed]

10. H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **83**(4), 046703 (2011). [CrossRef] [PubMed]

*ε*= 1) and dielectric material (

_{air}*ε*= 11.4 corresponding to GaAs). For introducing symmetric control points to improve the calculation efficiency, we assume that the unit cell of the 2D PC is symmetric about

_{r}*x*= 0,

*y*= 0 and

*x = y*(as shown in Fig. 1 ). One-eighth unit cell is selected to study in this work, even though the relaxation of symmetry may increase gap size [13

13. S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett. **86**(6), 061111 (2005). [CrossRef]

## 2. Method

14. O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim. **16**(1), 68–75 (1998). [CrossRef]

15. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. **32**(1), 77–79 (2007). [CrossRef] [PubMed]

*πc/a*) as an example (as shown in Fig. 2 ), and one-eighth of the structure is constructed as the initial distribution by GPM. The GPM is easy to be understood by following procedures [11

11. J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng. **60**(14), 2289–2312 (2004). [CrossRef]

*x*and

*x*are the coordinates of the refined mesh of the design domain (100 × 100 grids are considered in our study) and the mesh of the control points respectively. And the details of solving the coefficients

_{p}*c*,

*c*and

_{0}*λ*are given in [16

_{p}16. G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph. **21**(4), 855–873 (2002). [CrossRef]

*ξ*is used to control the intermediate dielectric at the interface between different dielectric material regions, and we can set available

*ξ*to make the intermediate dielectric vanish (

*ξ*= 100 in this paper).

*x*is the coordinate of the intersection curve, and

_{0}*d*(

*x*) is minimum distance from any point to the nearest intersection curve. So we can construct the initial distribution of the dielectric material by giving the special heights to the control points, and adjusting those heights will change the distribution.

_{φ=wn+1−wn}denotes the absolute band gap (n = 5 for TE wave and n = 9 for TM wave in current work). The sensitivity distribution is shown in Fig. 4 , the positive value means that increasing the dielectric material distribution at that region is energetic to improve the band gap, and the negative region show the contrary. So we only design the control points which located at the positive domain for increasing the band gap. This treatment will reduce the variables from 210 (mentioned in step 1) to about 100. The calculation precision and efficiency still can meet demand, even though the variables are very less.

17. H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett. **20**(22), 1830–1832 (2008). [CrossRef]

18. T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B **18**(3), 1109–1115 (2009). [CrossRef]

19. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett. **84**(12), 2022–2024 (2004). [CrossRef]

## 3. Application

*πc/a*) at a midfrequency of 0.8690 (2

*πc/a*).The absolute band gap of the new structure (as shown in Fig. 6 ) is about 4.6 times larger than that for the initial structure. The method demonstrates a great capacity in improving an existing absolute band gap. The 1681 plane waves are employed in the plane wave expansion method to verify the optimization result, and it shows that the relative error is about 10

^{−4}.

9. O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. **100**(15), 153904 (2008). [CrossRef] [PubMed]

20. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B **61**(20), 13458–13464 (2000). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

2. | L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B |

3. | H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun. |

4. | S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt. |

5. | L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B |

6. | M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B |

7. | W. L. Liu and T. J. Yang, “Engineering the band-gap of a two-dimensional photonic crystal with slender dielectric veins,” Phys. Lett. A |

8. | F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express |

9. | O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. |

10. | H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

11. | J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng. |

12. | W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. |

13. | S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett. |

14. | O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim. |

15. | W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. |

16. | G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph. |

17. | H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett. |

18. | T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B |

19. | J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett. |

20. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 15, 2011

Manuscript Accepted: September 1, 2011

Published: September 20, 2011

**Citation**

Donglin Wang, Zhongyuan Yu, Yumin Liu, Pengfei Lu, Lihong Han, Hao Feng, Xiaotao Guo, and Han Ye, "The optimal structure of two dimensional photonic crystals with the large absolute band gap," Opt. Express **19**, 19346-19353 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19346

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### References

- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B66(16), 165315 (2002). [CrossRef]
- H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun.282(14), 3012–3017 (2009). [CrossRef]
- S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt.55(18), 2971–2980 (2008). [CrossRef]
- L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68(3), 035109 (2003). [CrossRef]
- M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B17(6), 1027–1030 (2000). [CrossRef]
- W. L. Liu and T. J. Yang, “Engineering the band-gap of a two-dimensional photonic crystal with slender dielectric veins,” Phys. Lett. A369(5-6), 518–523 (2007). [CrossRef]
- F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express16(16), 12278–12289 (2008). [CrossRef] [PubMed]
- O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett.100(15), 153904 (2008). [CrossRef] [PubMed]
- H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(4), 046703 (2011). [CrossRef] [PubMed]
- J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng.60(14), 2289–2312 (2004). [CrossRef]
- W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(3), 033102 (2008). [CrossRef]
- S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett.86(6), 061111 (2005). [CrossRef]
- O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim.16(1), 68–75 (1998). [CrossRef]
- W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett.32(1), 77–79 (2007). [CrossRef] [PubMed]
- G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph.21(4), 855–873 (2002). [CrossRef]
- H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett.20(22), 1830–1832 (2008). [CrossRef]
- T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B18(3), 1109–1115 (2009). [CrossRef]
- J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12), 2022–2024 (2004). [CrossRef]
- E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B61(20), 13458–13464 (2000). [CrossRef]

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