Abstract

The goal of this work is to analyze three-dimensional dispersive metallic photonic crystals (PCs) and to find a structure that can provide a bandgap and a high cutoff frequency. The determination of the band structure of a PC with dispersive materials is an expensive nonlinear eigenvalue problem; in this work we propose a rational-polynomial method to convert such a nonlinear eigenvalue problem into a linear eigenvalue problem. The spectral element method is extended to rapidly calculate the band structure of three-dimensional PCs consisting of realistic dispersive materials modeled by Drude and Drude–Lorentz models. Exponential convergence is observed in the numerical experiments. Numerical results show that, at the low frequency limit, metallic materials are similar to a perfect electric conductor, where the simulation results tend to be the same as perfect electric conductor PCs. Band structures of the scaffold structure and semi-woodpile structure metallic PCs are investigated. It is found that band structures of semi-woodpile PCs have a very high cutoff frequency as well as a bandgap between the lowest two bands and the higher bands.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  6. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
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  7. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  8. M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
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    [CrossRef]
  14. G. C. Cohen, Higher-order Numerical Methods for Transient Wave Equations (Springer, 2001).
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    [CrossRef]
  16. 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]
  17. J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
    [CrossRef]
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  19. M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
    [CrossRef]
  20. M. Luo and Q. H. Liu, “A spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
    [CrossRef]
  21. M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
  23. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
    [CrossRef]
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    [CrossRef]
  27. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
    [CrossRef] [PubMed]
  28. B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
    [CrossRef]

2009 (4)

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

M. Luo and Q. H. Liu, “A spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

P. S. Light, F. Couny, Y. Y. Wang, N. V. Wheeler, P. J. Roberts, and F. Benabid, “Double photonic bandgap hollow-core photonic crystal fiber,” Opt. Express 17, 16238–16243 (2009).
[CrossRef] [PubMed]

2007 (2)

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

2006 (3)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

A. Tip, “Some mathematical properties of Maxwell’s equations for macroscopic dielectrics,” J. Math. Phys. 47, 012902 (2006).
[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]

2005 (2)

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

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

2004 (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

2003 (2)

H. van der Lem, A. Tip, and A. Moroz, “Band structure of absorptive two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 1334–1341 (2003).
[CrossRef]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

2002 (2)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

2000 (2)

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

1999 (2)

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

1998 (2)

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

1996 (2)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

1984 (1)

A. T. Patera, “A spectral element method for fluid dynamics: Laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

Benabid, F.

Biswas, R.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Chutinan, A.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

Cohen, G. C.

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

Combes, J. -M.

J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Waves in Periodic and Random Media, Vol. 339 of Contemporary Mathematics, P.Kuchment, ed. (American Mathematical Society, 2003), pp. 1–13.
[CrossRef]

Couny, F.

Crouch, D. D.

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

de Dood, M.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Djurisic, A. B.

Efros, A. L.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Elazar, J. M.

El-Kady, I.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

Enoch, S.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Fan, S.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Fleming, J. G.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Golubev, V. G.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Gralak, B.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Waves in Periodic and Random Media, Vol. 339 of Contemporary Mathematics, P.Kuchment, ed. (American Mathematical Society, 2003), pp. 1–13.
[CrossRef]

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Ho, K. M.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Kamaev, V.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Kurdyukov, D. A.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lee, J. -H.

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[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 Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848–1858 (2005).
[CrossRef]

Li, C. Y.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Li, Z.

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

Light, P. S.

Lin, S. Y.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Liu, Q. H.

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “A spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

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

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[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 Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848–1858 (2005).
[CrossRef]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Luo, M.

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

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “A spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

Majewski, M. L.

Maystre, D.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Moroz, A.

H. van der Lem, A. Tip, and A. Moroz, “Band structure of absorptive two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 1334–1341 (2003).
[CrossRef]

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999).
[CrossRef]

Noda, S.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Patera, A. T.

A. T. Patera, “A spectral element method for fluid dynamics: Laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

Pokrovsky, A. L.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Rakic, A. D.

Roberts, P. J.

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Tayeb, G.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Tip, A.

A. Tip, “Some mathematical properties of Maxwell’s equations for macroscopic dielectrics,” J. Math. Phys. 47, 012902 (2006).
[CrossRef]

H. van der Lem, A. Tip, and A. Moroz, “Band structure of absorptive two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 1334–1341 (2003).
[CrossRef]

J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Waves in Periodic and Random Media, Vol. 339 of Contemporary Mathematics, P.Kuchment, ed. (American Mathematical Society, 2003), pp. 1–13.
[CrossRef]

Tomoda, K.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

van der Lem, H.

Vardeny, Z. V.

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Wang, Y. Y.

Wheeler, N. V.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

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.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Yamamoto, N.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Comput.-Aided Des. (1)

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

IEEE Trans. Microwave Theory Tech. (2)

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, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

J. Chem. Phys. (2)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold [J. Chem. Phys. 125, 164705 (2006)],” J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

J. Comput. Phys. (1)

A. T. Patera, “A spectral element method for fluid dynamics: Laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

J. Math. Phys. (1)

A. Tip, “Some mathematical properties of Maxwell’s equations for macroscopic dielectrics,” J. Math. Phys. 47, 012902 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nature (2)

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (4)

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

A. L. Pokrovsky, V. Kamaev, C. Y. Li, Z. V. Vardeny, A. L. Efros, D. A. Kurdyukov, and V. G. Golubev, “Theoretical and experimental studies of metal-infiltrated opals,” Phys. Rev. B 71, 165114 (2005).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, and D. D. Crouch, “Waveguides in three-dimensional metallic photonic band-gap materials,” Phys. Rev. B 60, 4426–4429 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Phys. Rev. E (3)

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

M. Luo and Q. H. Liu, “A spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999).
[CrossRef]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar,“Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000).
[CrossRef] [PubMed]

Other (2)

J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Waves in Periodic and Random Media, Vol. 339 of Contemporary Mathematics, P.Kuchment, ed. (American Mathematical Society, 2003), pp. 1–13.
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

The PC structure of 2 × 2 × 2   unit cells under investigation. (a) Scaffold structure with a circular cylinder in the middle. (b) Semi-woodpile structure.

Fig. 2
Fig. 2

Band structure of the scaffold structure PC in Fig. 1a with the Drude model dispersive materials with (a) k P a / 2 π = 1 , (b) k P a / 2 π = 2 , (c) k P a / 2 π = 3 , and (d) PEC, where a is the length of a unit cell size. The x-axis is the Bloch wave vector that scales through the high symmetry points in the first Brillouin zone denoted by Γ, X, M, R, T, and Z, where the normalized wave number vectors are given by k = ( 0 , 0 , 0 ) , (1/2,0,0), (1/2,1/2,0), (1/2,1/2,1/2), (0,1/2,1/2), and (0,0,1/2), respectively.

Fig. 3
Fig. 3

Band structure of the scaffold structure PC in Fig. 1a with gold as the material. The sizes of the unit cells are (a) a = 400   nm and (b) a = 800   nm .

Fig. 4
Fig. 4

Band structure of the semi-woodpile structure PC in Fig. 1b with the dispersive materials being Drude model material with (a) k P a / 2 π = 1 , (b) k P a / 2 π = 2 , (c) k P a / 2 π = 3 , and (d) PEC, where a is the length of a unit cell size.

Fig. 5
Fig. 5

Band structure of the semi-woodpile structure PC in Fig. 1b with gold as the material. The sizes of the unit cells are (a) a = 400   nm and (b) a = 800   nm .

Fig. 6
Fig. 6

Relative error of the SEM result at the R point of the band structure of (a) PEC semi-woodpile PC in Fig. 4d and (b) gold semi-woodpile PC with lattice size a = 800   nm in Fig. 5b, versus the order of the SEM.

Equations (27)

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× [ μ r 1 ( × E ) ] k 0 2 ε r E = 0 ,
× [ ε r 1 ( × H ) ] k 0 2 μ r H = 0 ,
E ( r + R ) = E ( r ) e j k R ,
H ( r + R ) = H ( r ) e j k R ,
R = l a 1 + m a 2 + n a 3 .
ε r ( k 0 ) = 1 k P 2 k 0 2 i Γ P k 0 ,
ε r ( k 0 ) = ε inf k P 2 k 0 2 i Γ P k 0 + j = 1 , 2 A j 2 [ 1 i k j k 0 Γ j i + 1 + i k j + k 0 + Γ j i ] ,
E = j = 1 N E j Φ j ,
S E = k 0 2 M E ,
S = e = 1 N e S ( e ) ,     M = e = 1 N e M ( e ) ,
S j , k ( e ) = Ω e d r ( × Φ j ) μ r 1 ( × Φ k ) ,
M j , k ( e ) = Ω e d r Φ j ε r Φ k ,
M = M n + ε r d ( k 0 ) M d ,
M n = e n = 1 N e n M ( e n ) ,     M d = e d = 1 N e d M ( e d ) / ε r d ( k 0 ) ,
S E = k 0 2 M n E + k 0 2 ε r d ( k 0 ) M d E .
ε r ( k 0 ) = 1 k P 2 k 0 2 .
( S + k P 2 M d ) E = k 0 2 ( M n + M d ) E .
[ i Γ P S k 0 ( k P 2 M d + S ) i Γ P k 0 2 ( M n + M d ) + k 0 3 ( M n + M d ) ] E = 0.
[ 0 I 0 0 0 I i Γ P S ( k P 2 M d + S ) i Γ P ( M n + M d ) ] [ E F G ] = k 0 [ I 0 0 0 I 0 0 0 M n + M d ] [ E F G ] ,
[ 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I i T U i V W i X Y i Z ] [ E F G H J K L ] = k 0 [ I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 M n ] [ E F G H J K L ] ,
T = Γ P χ 1 χ 2 S ,
U = ( χ 1 χ 2 + 2 Γ P ψ ) S + χ 1 χ 2 k P 2 M d ,
V = ( Γ P κ + 2 ψ ) S Γ P χ 1 χ 2 M n 2 ( A 2 Γ P χ 1 φ 2 + A 1 Γ P χ 2 φ 1 + ψ k P 2 ) M d ,
W = ( 2 Γ P Γ L + κ ) S ( χ 1 χ 2 + 2 Γ P ψ ) M n [ 2 A 1 ( χ 2 φ 1 + Γ P ϑ 2 ) + 2 A 2 ( χ 1 φ 2 + Γ P ϑ 1 ) + κ k P 2 ] M d ,
X = ( Γ P + 2 Γ L ) S + ( Γ P κ + 2 ψ ) M n + 2 { A 1 [ ϑ 2 + Γ P ( φ 1 + 2 Γ 2 ) ] + A 2 [ ϑ 1 + Γ P ( φ 2 + 2 Γ 1 ) ] + Γ L k P 2 } M d ,
Y = S + ( κ + 2 Γ P Γ L ) M n + [ 2 A 1 ( Γ P + 2 Γ 2 + φ 1 ) + 2 A 2 ( Γ P + 2 Γ 1 + φ 2 ) + k P 2 ] M d ,
Z = ( Γ P + 2 Γ L ) M n 2 ( A 1 + A 2 ) M d .

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