## Maximizing band gaps in two-dimensional photonic crystals in square lattices |

JOSA A, Vol. 30, Issue 11, pp. 2314-2319 (2013)

http://dx.doi.org/10.1364/JOSAA.30.002314

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### Abstract

This paper is devoted to a numerical algorithm for the maximization of band gaps in two-dimensional photonic crystals in square lattices. We first apply the finite element method to solve the eigenvalue problem, then use the piecewise constant level set (PCLS) method to maximize the band gaps. The PCLS method is very powerful for representing and modeling regions of different structures. Extremely large gaps are realized with gallium arsenide material, for transverse magnetic field (TM), transverse electric field (TE), and for complete band gaps. When the mean gap frequency is below 1, the biggest gap is about 0.2922 for the TE.

© 2013 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Materials

**History**

Original Manuscript: April 17, 2013

Revised Manuscript: September 17, 2013

Manuscript Accepted: September 18, 2013

Published: October 22, 2013

**Citation**

Xiao-liang Cheng and Jing Yang, "Maximizing band gaps in two-dimensional photonic crystals in square lattices," J. Opt. Soc. Am. A **30**, 2314-2319 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-11-2314

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### References

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef]
- O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer-Verlag, 1984).
- J. Sokołwski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, 1992).
- B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids (Clarendon, 2001).
- M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).
- J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation (SIAM, 2003).
- S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999). [CrossRef]
- S. J. Cox and D. C. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comput. Phys. 158, 214–224 (2000). [CrossRef]
- D. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999). [CrossRef]
- D. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668–679 (2000). [CrossRef]
- M. Richter, “Optimization of Photonic Band Structures,” http://digbib.ubka.uni-karlsruhe.de/volltexte/1000021317 .
- C. Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B 81, 235–244 (2005). [CrossRef]
- S. F. Zhu, C. X. Liu, and Q. B. Wu, “Binary level set methods for topology and shape optimization of a two-density inhomogeneous drum,” Comput. Methods Appl. Mech. Eng. 199, 2970–2986 (2010). [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. B 68, 035109 (2003). [CrossRef]
- L. F. Shen, S. L. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66, 165315 (2002). [CrossRef]
- P. Shi, K. Huang, and Y. P. Li, “Photonic crystal with complex unit cell for large complete band gap,” Opt. Commun. 285, 3128–3132 (2012). [CrossRef]
- D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006). [CrossRef]
- X. C. Tai, O. Christiansen, P. Lin, and I. Skjaelaaen, “A remark on the mbo scheme and some piecewise constant level set methods,” Tech. Rep., UCLA, Applied Mathematics, (2005).
- J. Lie, M. Lysaker, and X. C. Tai, “A binary level set model and some applications to Mumford-Shah image segmentation,” IEEE Trans. Image Process 15, 1171–1181 (2006). [CrossRef]
- J. Lie, M. Lysaker, and X. C. Tai, “A variant of the level set method and applications to image segmentation,” Math. Comput. 75, 1155–1174 (2006). [CrossRef]
- S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988). [CrossRef]
- Z. F. Zhang and X. L. Cheng, “A boundary piecewise constant level set method for boundary control of eigenvalue optimization problems,” J. Comput. Phys. 230, 458–473 (2011). [CrossRef]
- K. Schmidt and P. Kauf, “Computation of the band structure of two-dimensional photonic crystals with hp finite elements,” Comput. Methods Appl. Mech. Eng. 198, 1249–1259 (2009). [CrossRef]

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