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

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef]
  3. O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer-Verlag, 1984).
  4. J. Sokołwski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, 1992).
  5. B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids (Clarendon, 2001).
  6. M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).
  7. J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation (SIAM, 2003).
  8. S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
    [CrossRef]
  9. 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]
  10. D. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
    [CrossRef]
  11. 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]
  12. M. Richter, “Optimization of Photonic Band Structures,” http://digbib.ubka.uni-karlsruhe.de/volltexte/1000021317 .
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
    [CrossRef]
  19. 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).
  20. 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]
  21. 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]
  22. 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]
  23. 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]
  24. 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]

2012 (1)

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]

2011 (1)

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]

2010 (1)

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]

2009 (1)

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]

2006 (3)

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

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]

2005 (1)

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]

2003 (1)

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]

2002 (1)

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]

2000 (2)

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]

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]

1999 (2)

D. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[CrossRef]

1988 (1)

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]

1987 (2)

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]

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

Boffi, D.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Cheng, X. L.

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]

Christiansen, O.

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).

Conforti, M.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Cox, S. J.

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]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[CrossRef]

Dobson, D.

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]

D. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

Dobson, D. C.

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]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[CrossRef]

Gastaldi, L.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Gopalakrishnan, J.

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]

Haslinger, J.

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation (SIAM, 2003).

He, S. L.

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]

Huang, K.

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]

John, S.

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

Kao, C. Y.

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]

Kauf, P.

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]

Li, Y. P.

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]

Lie, J.

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]

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]

Lin, P.

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).

Liu, C. X.

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]

Lysaker, M.

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]

Mäkinen, R. A. E.

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation (SIAM, 2003).

Mohammadi, B.

B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids (Clarendon, 2001).

Osher, S.

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. 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]

Pasciak, J. E.

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]

Pironneau, O.

B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids (Clarendon, 2001).

O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer-Verlag, 1984).

Schmidt, K.

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]

Sethian, J. A.

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]

Shen, L. F.

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]

Shi, P.

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]

Sigmund, O.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

Skjaelaaen, I.

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).

Sokolwski, J.

J. Sokołwski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, 1992).

Tai, X. C.

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]

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).

Wu, Q. B.

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]

Xiao, S. S.

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]

Yablonovitch, E.

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]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Ye, Z.

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]

Zhang, Z. F.

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]

Zhu, S. F.

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]

Zolesio, J.-P.

J. Sokołwski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, 1992).

Appl. Phys. B (1)

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]

Comput. Methods Appl. Mech. Eng. (2)

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]

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]

IEEE Trans. Image Process (1)

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. Comput. Phys. (5)

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]

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]

Math. Comput. (1)

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]

Numer. Math. (1)

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Opt. Commun. (1)

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]

Phys. Rev. B (2)

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]

Phys. Rev. Lett. (2)

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]

SIAM J. Appl. Math. (1)

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[CrossRef]

Other (7)

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).

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).

M. Richter, “Optimization of Photonic Band Structures,” http://digbib.ubka.uni-karlsruhe.de/volltexte/1000021317 .

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Figures (11)

Fig. 1.
Fig. 1.

Brillouin zone, with labels for symmetry points.

Fig. 2.
Fig. 2.

The chosen elements for initial value searching.

Fig. 3.
Fig. 3.

Evolution of the ε(x)-distribution in the TM for the n=4 case: (a) describes the initial ε(x)-distribution, (b) describes the ε(x)-distribution after 30 iterations, (c) describes the ε(x)-distribution after 60 iterations, and (d) describes the optimized unit lattice.

Fig. 4.
Fig. 4.

Band gap between bands 4 and 5 for the optimal ε(x) in the TM.

Fig. 5.
Fig. 5.

Band gap versus the number of iterations.

Fig. 6.
Fig. 6.

3×3 array of the optimized unit lattice, for the TE, when m=4.

Fig. 7.
Fig. 7.

Band gap between bands 4 and 5 for the optimal ε(x) in the TE.

Fig. 8.
Fig. 8.

3×3 array of the optimized unit lattice, for the TE, when m=7.

Fig. 9.
Fig. 9.

Band gap between bands 7 and 8 for the optimal ε(x) in the TE.

Fig. 10.
Fig. 10.

3×3 array of the optimized unit lattice, complete band gaps when n=3 and m=2.

Fig. 11.
Fig. 11.

Band structure for the optimal ε(x) for complete band gaps when n=3 and m=2. The solid lines represent frequencies for the TM, while the dot lines are for the TE.

Tables (2)

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

1ε(x)·(E(x))=ω2c2E(x)
·(1ε(x)H(x))=ω2c2H(x).
[nE(x)]=[1ε(x)nH(x)]=0,
1ε(x)(+iα)·(+iα)Eα=ωTM2c2Eα
(+iα)·1ε(x)(+iα)Hα=ωTE2c2Hα
Eα(x+Ri)=Eα(x),
nEα(x+Ri)=nEα(x),
Hα(x+Ri)=Hα(x),
1ε(x+Ri)nHα(x+Ri)=1ε(x)nHα(x),
Ω(+iα)Eα·(+iα)F¯dσ=λTMΩε(x)EαF¯dσ,
Ω1ε(x)(+iα)Hα·(+iα)G¯dσ=λTEΩHαG¯dσ,
Ω(+iα)Eαh·(+iα)F¯dσ=λTMhΩεh(x)EαhF¯dσ,
Ω1εh(x)(+iα)Hαh·(+iα)G¯dσ=λTEhΩHαhG¯dσ,
Kαu=λM(Ξ)u,Kα(ϒ)u=λMu,
ΞΞ+δΞ,λλ+δλ,uu+δu.
Kα(u+δu)=(λ+δλ)M(Ξ+δΞ)(u+δu),
Kαδu=λM(Ξ)δu+λM(δΞ)u+δλM(Ξ)u.
δλ(M(Ξ)u,u)=λ(M(δΞ)u,u)
λTMhεT=λTMhT|Eαh|2dσ,
λTEhρT=T|(+iα)Hαh|2dσ
λTEhεT=1εT2T|(+iα)Hαh|2dσ.
supϕ(infαλTM(n+1)supαλTM(n)).
supϕ(infαλTE(m+1)supαλTE(m)).
supϕ(inf(infαλTM(n+1),infαλTE(m+1))sup(supαλTM(n),supαλTE(m))).
ϕ(x)={0xS,1xΩ\S,
ϕ¯=Pϕ={0ϕ<0.5,1ϕ0.5.
J|T=λTM(n+1)2T|Eα1h|2dσ+λTM(n)2T|Eα2h|2dσ.
J|T=12λTE(m+1)εT2T|(+iα1)Hα1h|2dσ+12λTE(m)εT2T|(+iα2)Hα2h|2dσ.

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