Abstract

A simple and efficient method for computing bandgap structures of two-dimensional photonic crystals is presented. Using the Dirichlet-to-Neumann (DtN) map of the unit cell, the bandgaps are calculated as an eigenvalue problem for each given frequency, where the eigenvalue is related to the Bloch wave vector. A linear matrix eigenvalue problem is obtained even when the medium is dispersive. For photonic crystals composed of a square lattice of parallel cylinders, the DtN map is obtained by a cylindrical wave expansion. This leads to eigenvalue problems for relatively small matrices. Unlike other methods based on cylindrical wave expansions, sophisticated lattice sums techniques are not needed.

© 2006 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  3. K. M. Leung and Y. F. Liu, "Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media," Phys. Rev. Lett. 65, 2646-2649 (1990).
    [CrossRef] [PubMed]
  4. Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in periodic structures--Bloch wave solution of Maxwell's equations," Phys. Rev. Lett. 65, 2650-2653 (1990).
    [CrossRef] [PubMed]
  5. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  6. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
    [CrossRef]
  7. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  8. K. M. Leung and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Phys. Rev. B 48, 7767-7771 (1993).
    [CrossRef]
  9. N. A. Nicorovici and R. C. McPhedran, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135-1145 (1995).
    [CrossRef]
  10. K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
    [CrossRef]
  11. J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
    [CrossRef] [PubMed]
  12. J. B. Pendry, "Calculating photonic band structure," J. Phys.: Condens. Matter 8, 1085-1108 (1996).
    [CrossRef]
  13. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  14. H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
    [CrossRef]
  15. C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
    [CrossRef]
  16. S. Guo, F. Wu, S. Albin, and R. S. Rogowski, "Photonic band gap analysis using finite-difference frequency-domain method," Opt. Express 12, 1741-1746 (2004).
    [CrossRef] [PubMed]
  17. D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-379 (1999).
    [CrossRef]
  18. W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
    [CrossRef]
  19. C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
    [CrossRef]
  20. K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
    [CrossRef]
  21. T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
    [CrossRef]
  22. E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
    [CrossRef]
  23. M. Marrone, V. F. Rodriguez-Esquerre, and H. E. Hernández-Figueroa, "Novel numerical method for the analysis of 2D photonic crystals: the cell method," Opt. Express 10, 1299-1304 (2002).
    [PubMed]
  24. S. Jun, Y. S. Cho, and S. Im, "Moving least-square method for the band-structure calculation of 2D photonic crystals," Opt. Express 11, 541-551 (2003).
    [CrossRef] [PubMed]
  25. X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
    [CrossRef]
  26. A. Figotin and Y. A. Godin, "The computation of spectra of some 2D photonic crystals," J. Comput. Phys. 136, 585-598 (1997).
    [CrossRef]
  27. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
    [CrossRef]

2006 (1)

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

2004 (2)

C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

S. Guo, F. Wu, S. Albin, and R. S. Rogowski, "Photonic band gap analysis using finite-difference frequency-domain method," Opt. Express 12, 1741-1746 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

M. Marrone, V. F. Rodriguez-Esquerre, and H. E. Hernández-Figueroa, "Novel numerical method for the analysis of 2D photonic crystals: the cell method," Opt. Express 10, 1299-1304 (2002).
[PubMed]

2001 (4)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

1999 (2)

D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-379 (1999).
[CrossRef]

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

1998 (1)

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

1997 (1)

A. Figotin and Y. A. Godin, "The computation of spectra of some 2D photonic crystals," J. Comput. Phys. 136, 585-598 (1997).
[CrossRef]

1996 (2)

J. B. Pendry, "Calculating photonic band structure," J. Phys.: Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

1995 (2)

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

N. A. Nicorovici and R. C. McPhedran, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

1993 (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

K. M. Leung and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Phys. Rev. B 48, 7767-7771 (1993).
[CrossRef]

1992 (1)

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

1990 (3)

K. M. Leung and Y. F. Liu, "Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media," Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in periodic structures--Bloch wave solution of Maxwell's equations," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Albin, S.

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Amemiya, K.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Axmann, W.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Botten, L. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Chan, C. T.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Chang, H. C.

C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Checoury, X.

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

Cho, Y. S.

Chutinan, A.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Dobson, D. C.

D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-379 (1999).
[CrossRef]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Figotin, A.

A. Figotin and Y. A. Godin, "The computation of spectra of some 2D photonic crystals," J. Comput. Phys. 136, 585-598 (1997).
[CrossRef]

Godin, Y. A.

A. Figotin and Y. A. Godin, "The computation of spectra of some 2D photonic crystals," J. Comput. Phys. 136, 585-598 (1997).
[CrossRef]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Guo, S.

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Hernández-Figueroa, H. E.

Hirao, K.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Ho, K. M.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Im, S.

Ito, T.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Johnson, S. G.

Jun, S.

Kawai, N.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Kuchment, P.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Leung, K. M.

K. M. Leung and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Phys. Rev. B 48, 7767-7771 (1993).
[CrossRef]

K. M. Leung and Y. F. Liu, "Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media," Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Liu, Y. F.

K. M. Leung and Y. F. Liu, "Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media," Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Lourtioz, J. M.

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

MacKinnon, A.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Marrone, M.

Martijn de Sterke, C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

N. A. Nicorovici and R. C. McPhedran, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mitsuyu, T.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Moreno, E.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

N. A. Nicorovici and R. C. McPhedran, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

Noda, S.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Ohtaka, K.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Calculating photonic band structure," J. Phys.: Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Qiu, Y.

K. M. Leung and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Phys. Rev. B 48, 7767-7771 (1993).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Rodriguez-Esquerre, V. F.

Rogowski, R. S.

Sakoda, K.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy," Phys. Rev. B 64, 045116 (2001).
[CrossRef]

T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

Satpathy, S.

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in periodic structures--Bloch wave solution of Maxwell's equations," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guides, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Ueta, T.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, F.

Yang, H. Y. D.

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

Yu, C. P.

C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Yu, Q. L.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Zhang, Z.

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in periodic structures--Bloch wave solution of Maxwell's equations," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Computed band structure for example 1: E polarization.

Fig. 2
Fig. 2

Gap map of E polarization for example 1.

Fig. 3
Fig. 3

Computed band structure for example 2: E polarization.

Fig. 4
Fig. 4

Gap map of E polarization for example 2.

Fig. 5
Fig. 5

Computed band structure for example 2: H polarization.

Equations (38)

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2 u x 2 + 2 u y 2 + k 0 2 n 2 u = 0
x ( 1 n 2 u x ) + y ( 1 n 2 n y ) + k 0 2 u = 0
n ( x , y ) = n ( x + m 1 L , y + m 2 L )
u ( x , y ) = m = C m Φ m ( r , θ ) , Φ m ( r , θ ) = ϕ m ( r ) e i m θ ,
ϕ m ( r ) = { A m J m ( k 0 n 1 r ) , r < a B m J m ( k 0 n 2 r ) + Y m ( k 0 n 2 r ) , r > a } .
A m J m ( k 0 n 1 a ) B m J m ( k 0 n 2 a ) = Y m ( k 0 n 2 a ) ,
n 1 n 2 A m J m ( k 0 n 1 a ) B m J m ( k 0 n 2 a ) = Y m ( k 0 n 2 a )
A m J m ( k 0 n 1 a ) B m J m ( k 0 n 2 a ) = Y m ( k 0 n 2 a ) ,
n 2 n 1 A m J m ( k 0 n 1 a ) B m J m ( k 0 n 2 a ) = Y m ( k 0 n 2 a )
Λ [ u 0 v 0 u 1 v 1 ] = [ y u 0 x v 0 y u 1 x v 1 ] ,
u 0 = u ( x , 0 ) , u 1 = u ( x , L ) , v 0 = u ( 0 , y ) , v 1 = u ( L , y ) ,
y u 0 = y u y = 0 , y u 1 = y u y = L , x v 0 = x u x = 0 ,
x v 1 = x u x = L .
x j = y j = ( j 0.5 ) L N , j = 1 , 2 , , N ,
u ( x , y ) = m = 2 N 2 N 1 C m Φ m ( r , θ ) ,
Λ = Λ 2 Λ 1 1 .
u ( x , y ) = e i ( a x + β y ) Ψ ( x , y ) ,
u ( x , L ) = ρ β u ( x , 0 ) , u ( x , L ) y = ρ β u ( x , 0 ) y ,
u ( L , y ) = ρ α u ( 0 , y ) , u ( L , y ) x = ρ α u ( 0 , y ) x ,
[ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] [ u 0 v 0 u 1 v 1 ] = [ y u 0 x v 0 y u 1 x v 1 ] .
[ Λ 31 Λ 32 Λ 41 Λ 42 ] [ u 0 v 0 ] = [ M 11 M 12 M 13 M 14 ] [ u 0 v 0 ] ,
M 11 = ρ β 2 Λ 13 + ρ β ( Λ 11 Λ 33 ) ,
M 12 = ρ β ρ α Λ 14 ρ α Λ 34 + ρ β Λ 12 ,
M 21 = ρ β ρ α Λ 23 + ρ α Λ 21 ρ β Λ 43 ,
M 22 = ρ α 2 Λ 24 + ρ α ( Λ 22 Λ 44 ) .
( λ 2 A + λ B + C ) U = 0 ,
A = [ 0 0 0 Λ 24 ] , B = [ 0 Λ 34 Λ 14 Λ 21 Λ 23 Λ 44 Λ 22 ] ,
C = [ Λ 31 Λ 13 + Λ 33 Λ 11 Λ 32 Λ 12 Λ 41 + Λ 43 Λ 42 ] .
A = [ Λ 13 0 0 0 ] , B = [ Λ 33 Λ 11 Λ 14 Λ 12 Λ 43 + Λ 23 0 ] ,
C = [ Λ 31 Λ 32 Λ 34 Λ 41 + Λ 21 Λ 42 Λ 24 + Λ 22 Λ 44 ] .
A = [ Λ 13 Λ 14 Λ 23 Λ 24 ] , B = [ Λ 33 Λ 11 Λ 34 Λ 12 Λ 43 Λ 21 Λ 44 Λ 22 ] ,
C = [ Λ 31 Λ 32 Λ 41 Λ 42 ] .
λ [ A 0 0 I ] [ V U ] + [ B C I 0 ] [ V U ] = 0 ,
F ( ω ) = min { 1 λ , λ is an eigenvalue satisfying Eq. ( 17 ) and λ 1 } .
ϵ 1 ( ω ) = ϵ ( 1 ω p 2 ω 2 ) ,
ϵ = 1 , a = 0.472 L , ω p = 2 π c L ,
ϵ = 1 , a = 0.3 L , ω p = 2 π c L
ϵ 1 ( ω ) = ϵ [ 1 ω p 2 ( ω + i δ ) ( ω + i γ ) ] ,

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