Abstract

We propose a holographic design of five-beam symmetric umbrella configuration, where there are a central beam and four ambient beams symmetrically scattered around the central one with the same apex angle, for fabrication of three-dimensional photonic crystals with tetragonal or cubic symmetries, and systematically analyzed the band gap properties of resultant photonic crystals when the apex angle is continuously increased. Our calculations reveal that large complete photonic band gaps exist in a wide range of apex angle for a relatively low refractive index contrast. Specifically, the face-centered cubic structure with a relative band gap of 25.1% for ε = 11.9 can be obtained with this recording geometry conveniently where all the beams are incident from the same half-space. These results will provide us with more understanding of this important recording geometry and give guidelines to its use in experiments.

© 2006 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. M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
    [CrossRef]
  4. Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
    [CrossRef]
  5. H. B. Sun, S. M. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786-788 (1999).
    [CrossRef]
  6. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
    [CrossRef]
  7. G. Wang, C. Tan, Y. Yi, and H. Shan, "Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals with a single continuous wavelength laser beam," J. Mod. Opt. 50, 2155-2161 (2003).
  8. S. Shoji, H. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003).
    [CrossRef]
  9. C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
    [CrossRef]
  10. D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003).
    [CrossRef]
  11. C. K. Ullal, M. Maldovan, M. Wohlgemuth, C. A. White, S. Yang, and E. L. Thomas, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
    [CrossRef]
  12. L. Z. Cai, X. L. Yang, and Y. R. Wang, "All fourteen Bravais lattices can be formed by interference of four noncoplanar beams," Opt. Lett. 27, 900-902 (2002).
    [CrossRef]
  13. X. L. Yang, L. Z. Cai, Y. R. Wang, and Q. Liu, "Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices," Opt. Lett. 28, 453-455 (2003).
    [CrossRef] [PubMed]
  14. X. L. Yang, L. Z. Cai, and Q. Liu, "Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams," Opt. Express. 11, 1050-1055 (2003).
    [CrossRef] [PubMed]
  15. I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
    [CrossRef]
  16. Yu. V. Miklyaev, D. C. Meisel, A. Blanco, and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
    [CrossRef]
  17. D. C. Meisel, M. Wegener, and K. Busch, "Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: symmetries and complete photonic band gaps," Phys. Rev. B 70, 165104 (2004).
    [CrossRef]
  18. M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002).
    [CrossRef]
  19. O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92, 043905 (2004).
    [CrossRef] [PubMed]
  20. T. Y. M. Chan, O. Toader, and S. John, "Photonic band gap templating using optical interference lithography," Phys. Rev. E. 71, 046605 (2005).
    [CrossRef]
  21. Y. Lin, P. R. Herman, and K. Darmawikarta, "Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals," Appl. Phys. Lett. 86, 071117 (2005).
    [CrossRef]
  22. W. Y. Tam, "Woodpile and diamond structures by optical interference holography," (2006), http://arxiv.org/ftp/physics/papers/0607/0607092.
  23. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  24. 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]
  25. B. Gralak and M. de Dood, "Theoretical study of photonic band gaps in woodpile crystals," Phys. Rev. E 67, 066601 (2003).
    [CrossRef]

2005 (2)

T. Y. M. Chan, O. Toader, and S. John, "Photonic band gap templating using optical interference lithography," Phys. Rev. E. 71, 046605 (2005).
[CrossRef]

Y. Lin, P. R. Herman, and K. Darmawikarta, "Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals," Appl. Phys. Lett. 86, 071117 (2005).
[CrossRef]

2004 (4)

O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef] [PubMed]

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
[CrossRef]

D. C. Meisel, M. Wegener, and K. Busch, "Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: symmetries and complete photonic band gaps," Phys. Rev. B 70, 165104 (2004).
[CrossRef]

2003 (9)

G. Wang, C. Tan, Y. Yi, and H. Shan, "Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals with a single continuous wavelength laser beam," J. Mod. Opt. 50, 2155-2161 (2003).

S. Shoji, H. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003).
[CrossRef]

X. L. Yang, L. Z. Cai, Y. R. Wang, and Q. Liu, "Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices," Opt. Lett. 28, 453-455 (2003).
[CrossRef] [PubMed]

X. L. Yang, L. Z. Cai, and Q. Liu, "Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams," Opt. Express. 11, 1050-1055 (2003).
[CrossRef] [PubMed]

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
[CrossRef]

Yu. V. Miklyaev, D. C. Meisel, A. Blanco, and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003).
[CrossRef]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, C. A. White, S. Yang, and E. L. Thomas, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

B. Gralak and M. de Dood, "Theoretical study of photonic band gaps in woodpile crystals," Phys. Rev. E 67, 066601 (2003).
[CrossRef]

2002 (2)

L. Z. Cai, X. L. Yang, and Y. R. Wang, "All fourteen Bravais lattices can be formed by interference of four noncoplanar beams," Opt. Lett. 27, 900-902 (2002).
[CrossRef]

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002).
[CrossRef]

2001 (1)

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[CrossRef]

2000 (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

1999 (1)

H. B. Sun, S. M. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786-788 (1999).
[CrossRef]

1998 (1)

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

1990 (1)

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]

1987 (2)

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]

Blanco, A.

Yu. V. Miklyaev, D. C. Meisel, A. Blanco, and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Bo, X. Z.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[CrossRef]

Busch, K.

D. C. Meisel, M. Wegener, and K. Busch, "Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: symmetries and complete photonic band gaps," Phys. Rev. B 70, 165104 (2004).
[CrossRef]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Cai, L. Z.

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

Carter, W. C.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002).
[CrossRef]

Chan, C. T.

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]

Chan, T. Y. M.

T. Y. M. Chan, O. Toader, and S. John, "Photonic band gap templating using optical interference lithography," Phys. Rev. E. 71, 046605 (2005).
[CrossRef]

O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef] [PubMed]

Chen, G.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
[CrossRef]

Crespi, V. H.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
[CrossRef]

Darmawikarta, K.

Y. Lin, P. R. Herman, and K. Darmawikarta, "Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals," Appl. Phys. Lett. 86, 071117 (2005).
[CrossRef]

de Dood, M.

B. Gralak and M. de Dood, "Theoretical study of photonic band gaps in woodpile crystals," Phys. Rev. E 67, 066601 (2003).
[CrossRef]

Denning, R. G.

D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

Divliansky, I.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
[CrossRef]

Gralak, B.

B. Gralak and M. de Dood, "Theoretical study of photonic band gaps in woodpile crystals," Phys. Rev. E 67, 066601 (2003).
[CrossRef]

Han, Y.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
[CrossRef]

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

Herman, P. R.

Y. Lin, P. R. Herman, and K. Darmawikarta, "Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals," Appl. Phys. Lett. 86, 071117 (2005).
[CrossRef]

Ho, K. 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]

Holliday, K. S.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
[CrossRef]

Ippen, E. P.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

Joannopoulos, J. D.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

John, S.

T. Y. M. Chan, O. Toader, and S. John, "Photonic band gap templating using optical interference lithography," Phys. Rev. E. 71, 046605 (2005).
[CrossRef]

O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef] [PubMed]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

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

Johnson, S. G.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

Kawata, S.

S. Shoji, H. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003).
[CrossRef]

Lidorikis, E.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

Lin, Y.

Y. Lin, P. R. Herman, and K. Darmawikarta, "Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals," Appl. Phys. Lett. 86, 071117 (2005).
[CrossRef]

Liu, Q.

X. L. Yang, L. Z. Cai, and Q. Liu, "Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams," Opt. Express. 11, 1050-1055 (2003).
[CrossRef] [PubMed]

X. L. Yang, L. Z. Cai, Y. R. Wang, and Q. Liu, "Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices," Opt. Lett. 28, 453-455 (2003).
[CrossRef] [PubMed]

Maldovan, M.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
[CrossRef]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, C. A. White, S. Yang, and E. L. Thomas, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002).
[CrossRef]

Matsuo, S. M.

H. B. Sun, S. M. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786-788 (1999).
[CrossRef]

Mayer, T. S.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, "Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography," Appl. Phys. Lett. 82, 1667-1669 (2003).
[CrossRef]

Meisel, D. C.

D. C. Meisel, M. Wegener, and K. Busch, "Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: symmetries and complete photonic band gaps," Phys. Rev. B 70, 165104 (2004).
[CrossRef]

Yu. V. Miklyaev, D. C. Meisel, A. Blanco, and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Miklyaev, Yu. V.

Yu. V. Miklyaev, D. C. Meisel, A. Blanco, and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Misawa, H.

H. B. Sun, S. M. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786-788 (1999).
[CrossRef]

Norris, D. J.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[CrossRef]

Qi, M.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

Rakich, P. T.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[CrossRef]

Shan, H.

G. Wang, C. Tan, Y. Yi, and H. Shan, "Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals with a single continuous wavelength laser beam," J. Mod. Opt. 50, 2155-2161 (2003).

Sharp, D. N.

D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

Shoji, S.

S. Shoji, H. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003).
[CrossRef]

Smith, H. I.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature (London) 429, 538-542 (2004).
[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]

Sturm, J. C.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[CrossRef]

Sun, H.

S. Shoji, H. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003).
[CrossRef]

Sun, H. B.

H. B. Sun, S. M. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786-788 (1999).
[CrossRef]

Tan, C.

G. Wang, C. Tan, Y. Yi, and H. Shan, "Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals with a single continuous wavelength laser beam," J. Mod. Opt. 50, 2155-2161 (2003).

Thomas, E. L.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, "Photonic crystals through holographic lithography: Simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434-5436 (2004).
[CrossRef]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, C. A. White, S. Yang, and E. L. Thomas, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002).
[CrossRef]

Toader, O.

T. Y. M. Chan, O. Toader, and S. John, "Photonic band gap templating using optical interference lithography," Phys. Rev. E. 71, 046605 (2005).
[CrossRef]

O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef] [PubMed]

Turberfield, A. J.

D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature (London) 404, 53-56 (2000).
[CrossRef]

Ullal, C. K.

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Supplementary Material (1)

» Media 1: GIF (155 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Symmetric umbrella recording geometry and the coordinate system used for calculations.

Fig. 2.
Fig. 2.

Evolution of irreducible Brillouin zone (Movie, 155K). The irreducible Brillouin zones of tetragonal structures for θ = 80° (a), θ = 90° (b), and θ = 100° (c).

Fig. 3.
Fig. 3.

(a). The real fcc structure formed by five-beam symmetric umbrella configuration when θ = 70.53° and I t = 1.39 in Eq. (4), the corresponding filling ration is 21.7%; (b) the primitive cell of the fcc structure shown in (a); (c) the rhombohedral structure with fcc symmetry fabricated by four-beam symmetric umbrella configuration when the apex angle is 38.94°; and (d) the primitive cell of the structure shown in (c).

Fig. 4.
Fig. 4.

Relative band gap of optimized structures as a function of apex angle for 50°< θ < 115° when ε = 11.9. The solid symbols are the date for fcc and bcc structures when θ = 70.53° and θ = 90°, respectively. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained.

Fig. 5.
Fig. 5.

Photonic band structure for the fcc structure with θ = 70.53°. The position of the high symmetry points together with the irreducible Brillouin zone are shown in the inset.

Fig. 6.
Fig. 6.

Variation of relative band gaps with different filling ratios for different refractive index contrasts for the fcc symmetric structure with the apex angle θ = 70.53°.

Equations (12)

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K 1 = ( 2 π λ ) ( sin θ , 0 , cos θ ) , K 2 = ( 2 π λ ) ( 0 , sin θ , cos θ ) ,
K 3 = ( 2 π λ ) ( sin θ , 0 , cos θ ) , K 4 = ( 2 π λ ) ( 0 , sin θ , cos θ ) ,
K c = ( 2 π λ ) ( 0 , 0 , 1 ) .
e 1 = e 3 = ( 0 , 1 , 0 ) , e 2 = e 4 = ( 1 , 0 , 0 ) , e c = 1 2 ( 1 , i , 0 ) .
I ( r ) = 2 E C 2 + 4 E A 2 + 2 E C E A { sin [ 2 π λ ( x sin θ + ( 1 cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 cos θ ) z ) ]
+ sin [ 2 π λ ( x sin θ + ( 1 cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 cos θ ) z ) ] } .
Δ I ( r ) = sin [ 2 π λ ( x sin θ + ( 1 cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 cos θ ) z ) ]
+ sin [ 2 π λ ( x sin θ + ( 1 cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 cos θ ) z ) ] .

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