Abstract

The spectral element method (SEM) is used to calculate band structures of two-dimensional photonic crystals (PCs) consisting of dispersive anisotropic materials. As in the conventional finite element method, for a dispersive PC, the resulting eigenvalue problem in the SEM is nonlinear and the eigenvalues are in general complex frequencies. We develop an efficient way of incorporating the dispersion in the system matrices. The band structures of a PC with a square lattice of dispersive cylindrical rods are first analyzed. The imaginary part of the complex frequency is the time-domain decay rate of the eigenmode, which is very useful for tracing a band from discrete numerical data. Modification of the band structure of TE mode by an external static magnetic field in the out-of-plane direction is explored for this square lattice. A plasmon resonance mode is found near the plasmon frequency when the magnetic field is nonzero. The band structure of a PC with a triangular lattice is also calculated with the SEM. Other types of lattices can also be treated readily by the SEM.

© 2009 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] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).
  4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  5. 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]
  6. T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
    [CrossRef]
  7. 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]
  8. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681-9691 (2007).
    [CrossRef] [PubMed]
  9. S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
    [CrossRef]
  10. R. L. Chern, “Surface plasmon modes for periodic lattices of plasmonic hole waveguides,” Phys. Rev. B 77, 045409 (2008).
    [CrossRef]
  11. P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
    [CrossRef]
  12. M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
    [CrossRef]
  13. G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).
  14. J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrödinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
    [CrossRef]
  15. J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
    [CrossRef]
  16. R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, 1973), Chaps. 3 and 4.
  17. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
    [CrossRef]
  18. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
    [CrossRef]
  19. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
    [CrossRef] [PubMed]
  20. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 316.
  21. J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
    [CrossRef]

2009

M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

2008

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

R. L. Chern, “Surface plasmon modes for periodic lattices of plasmonic hole waveguides,” Phys. Rev. B 77, 045409 (2008).
[CrossRef]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

2007

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681-9691 (2007).
[CrossRef] [PubMed]

2006

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

2005

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrödinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002

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]

2001

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]

1996

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

1987

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

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

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, 1973), Chaps. 3 and 4.

Chang, H. -C.

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Chern, R. L.

R. L. Chern, “Surface plasmon modes for periodic lattices of plasmonic hole waveguides,” Phys. Rev. B 77, 045409 (2008).
[CrossRef]

Chiang, P. -J.

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Chui, S. T.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

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]

Cohen, G. C.

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

Davanco, M.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Du, J.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

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]

Fan, S.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Giannakis, N.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

Green, A. A.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

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]

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]

Inglesfield, J. E.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

Istrate, E.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

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]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 316.

Joannopoulos, J. D.

J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

John, S.

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

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]

Lee, J. -H.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrödinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

Lehoucq, R. B.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

Li, Z.

M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Lin, Z.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Liu, Q. H.

M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrödinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

Liu, S.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Luo, M.

M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

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]

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]

Pitarke, J. M.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

Sakoda, K.

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, 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]

Sargent, E. H.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Shvets, G.

Sorensen, D. C.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

Urzhumov, Y.

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Wang, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Wu, R. X.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Xiao, T.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

Yablonovitch, E.

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

Yu, C. -P.

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Yu, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

IEEE Trans. Comput.-Aided Des.

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrödinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B

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]

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]

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

R. L. Chern, “Surface plasmon modes for periodic lattices of plasmonic hole waveguides,” Phys. Rev. B 77, 045409 (2008).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

Phys. Rev. E

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

M. Luo, Q. H. Liu, and Z. Li, “A spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Phys. Rev. Lett.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

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

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

SIAM J. Matrix Anal. Appl.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 316.

J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, 1973), Chaps. 3 and 4.

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

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Figures (9)

Fig. 1
Fig. 1

(a) Distribution of the SEM nodal points in the reference domain. (b) Meshing of the unit cell of a 2D square-lattice PC with a circular cylinder in the middle and the distribution of SEM nodal points. There are four nodal points lying along the thick curve.

Fig. 2
Fig. 2

(a) Band structure and (b) corresponding lifetime of the TM mode of a square lattice of dispersive circular cylinder rods with a radius of 0.3 a and a dispersive relative permittivity given by the Drude model in Eq. (17).

Fig. 3
Fig. 3

(a) Band structure and (b) corresponding lifetime of the TE mode of the same square lattice as in Fig. 2.

Fig. 4
Fig. 4

Relative errors of the band structure of the square-lattice PC in Fig. 1. Solid curve and line are the relative errors versus the order of SEM of the first band at point X for the (a) TM and (b) TE modes. The dashed curve is the relative error of one of the surface plasmon resonance TE modes of frequency ω a / 2 π c = 0.6807 0.0293 i .

Fig. 5
Fig. 5

Same plot as Fig. 3 but with a smaller region of frequency near surface plasmon frequency in (a). The curves in (a) and (b) with the same color belong to the same band. The two red (gray in print) and blue (black in print) dots belong to the same band.

Fig. 6
Fig. 6

Band structure of the TE mode of a square lattice of dispersive cylinder rods (a) without and (b) with an external static magnetic field along the out-of-plane direction. The two straight (red online) lines near the bottom of (b) indicate a small bandgap.

Fig. 7
Fig. 7

Field magnitude patterns of the 16 TE modes at point X. The frequency of each mode is marked at the top of each plot in the units of a ω / 2 π c .

Fig. 8
Fig. 8

SEM mesh of the unit cell of a 2D triangular-lattice PC with a circular cylinder in the middle and the distribution of nodal points.

Fig. 9
Fig. 9

Band structure of (a) TM and the (b) TE modes of a triangular lattice of dispersive cylinder rods with a radius of 0.3 a .

Equations (20)

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

× [ u 1 ( × φ z ̂ ) ] k 0 2 v φ z ̂ = 0 ,
φ ( x , y ) = U ( x , y ) e i k r ,
F ( φ ) = Γ d r { ( × φ z ̂ ) [ u 1 ( × φ z ̂ ) ] k 0 2 v φ φ } ,
ϕ j ( N ) ( ξ ) = 1 N ( N + 1 ) L N ( ξ j ) ( 1 ξ 2 ) L N ( ξ ) ( ξ ξ j ) .
U = j = 1 M u j Φ j ( N ) ( x , y ) ,
S U = k 0 2 M U ,
S = e = 1 N e S ( e ) ,     M = e = 1 N e M ( e ) ,
S j , k ( e ) = Γ e d r { [ ( i k ) × Φ j ( N ) z ̂ ] u 1 [ ( + i k ) × Φ k ( N ) z ̂ ] } ,
M j , k ( e ) = Γ e d r Φ j ( N ) v Φ k ( N ) ,
S ̃ U ̃ = k 0 2 U ̃ ,
U ̃ = M 1 / 2 U ,     S ̃ = M 1 / 2 S M 1 / 2 .
S = e = 1 N e n d S ( e ) + ( u 1 ) x x e = 1 N e d S ( e ) , d ( x x ) + ( u 1 ) x y e = 1 N e d S ( e ) , d ( x y ) + ( u 1 ) y x e = 1 N e d S ( e ) , d ( y x ) + ( u 1 ) y y e = 1 N e d S ( e ) , d ( y y ) = S n d + ( u 1 ) x x S x x d + ( u 1 ) x y S x y d + ( u 1 ) y x S y x d + ( u 1 ) y y S y y d ,
M = e = 1 N e n d M ( e ) + v e = 1 N e d M ( e ) , d = M n d + v M d ,
S j , k ( e ) , d ( m n ) = Γ e d r { [ ( i k ) × Φ j ( N ) z ̂ ] m [ ( + i k ) × Φ k ( N ) z ̂ ] n } ,
M j , k ( e ) , d = Γ e d r Φ j ( N ) Φ k ( N ) ,
[ S n d + ( u 1 ) x x S x x d + ( u 1 ) x y S x y d + ( u 1 ) y x S y x d + ( u 1 ) y y S y y d ] U = k 0 2 ( M n d + v M d ) U .
B ( ω ) = ω c k 0 closest ( ω ) ,
ε r ( ω ) = 1 ω P ω ( ω + i γ ) ,
λ = λ 0 | 1 + ε r ( ω ) ε r ( ω ) | ,
ε r B ( ω ) = [ ε d i ε f 0 i ε f ε d 0 0 0 ε r ] ,

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