Abstract

A photonic crystal waveguide network can form in a three-dimensional layer-by-layer photonic crystal along three orthogonal directions. We investigate guided-mode band structures for different waveguide configurations by numerical calculations that combine a plane-wave expansion method with a supercell technique. The in-plane waveguide network composed of waveguides along the (100) and (010) directions of the crystal is located in a single layer of the photonic crystal. This structural design is easy to achieve experimentally. One can create these in-plane waveguides either by removing one single rod or by breaking a segment in each of an array of parallel rods. One produces the off-plane waveguide by removing some segments of rods along the (001) stacking direction of the crystal. Single-mode operation of a waveguide can be achieved by appropriate adjustment of geometrical parameters such as the location and the size of the waveguide. Intrinsic lattice symmetries in these photonic crystal waveguides have been fully employed to reduce the complexity of the numerical solution of guided-mode band structures significantly.

© 2003 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. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
    [CrossRef]
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef] [PubMed]
  4. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
    [CrossRef] [PubMed]
  5. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
    [CrossRef]
  6. M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
    [CrossRef]
  7. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
    [CrossRef] [PubMed]
  8. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
    [CrossRef] [PubMed]
  9. D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
    [CrossRef]
  10. T. F. Krauss, “Photonic crystals for integrated optics,” in Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 89–98 (2001).
    [CrossRef]
  11. Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
    [CrossRef]
  12. A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
    [CrossRef]
  13. M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
    [CrossRef]
  14. M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
    [CrossRef]
  15. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
    [CrossRef]
  16. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zurbrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
    [CrossRef]
  17. J. G. Fleming and S. Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 μm,” Opt. Lett. 24, 49–51 (1999).
    [CrossRef]
  18. 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]
  19. 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]
  20. Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
    [CrossRef]
  21. Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
    [CrossRef]
  22. A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. 75, 3739–3741 (1999).
    [CrossRef]
  23. A. Chutinan and S. Noda, “Design for waveguides in three-dimensional photonic crystals,” Jpn. J. Appl. Phys. 39, 2353–2356 (2000).
    [CrossRef]

2002 (2)

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

2001 (3)

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

T. F. Krauss, “Photonic crystals for integrated optics,” in Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 89–98 (2001).
[CrossRef]

2000 (8)

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef] [PubMed]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[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]

Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Design for waveguides in three-dimensional photonic crystals,” Jpn. J. Appl. Phys. 39, 2353–2356 (2000).
[CrossRef]

1999 (2)

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

J. G. Fleming and S. Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 μm,” Opt. Lett. 24, 49–51 (1999).
[CrossRef]

1998 (3)

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

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
[CrossRef]

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

1994 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[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 (1)

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

Agio, M.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

Allenman, A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Asakawa, K.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Bayindir, M.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
[CrossRef]

Biswas, R.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

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

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Bouadma, N.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

Bur, J.

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

Carlsson, N.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[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]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Chow, E.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[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]

A. Chutinan and S. Noda, “Design for waveguides in three-dimensional photonic crystals,” Jpn. J. Appl. Phys. 39, 2353–2356 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Fleming, J. G.

J. G. Fleming and S. Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 μm,” Opt. Lett. 24, 49–51 (1999).
[CrossRef]

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

Gu, B. Y.

Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
[CrossRef]

Haus, H.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[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. Zurbrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).
[CrossRef]

Hietala, V.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Ho, K. M.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

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

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[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]

Ikeda, N.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Imada, M.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef] [PubMed]

Inoue, K.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Joannopoulos, J. D.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Johnson, S. G.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Kafesaki, M.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

Kawai, N.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Kosaka, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[CrossRef]

Krauss, T. F.

T. F. Krauss, “Photonic crystals for integrated optics,” in Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 89–98 (2001).
[CrossRef]

Kuchinsky, S. A.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Kurtz, S. R.

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

Le Gouezigou, L.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

Li, Z. Y.

Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
[CrossRef]

Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
[CrossRef]

Lin, S. Y.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

J. G. Fleming and S. Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 μm,” Opt. Lett. 24, 49–51 (1999).
[CrossRef]

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

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Lonar, M.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Mikhailov, M. D.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Nedeljkovi, D.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Noda, S.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[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]

A. Chutinan and S. Noda, “Design for waveguides in three-dimensional photonic crystals,” Jpn. J. Appl. Phys. 39, 2353–2356 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

Ozbay, E.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
[CrossRef]

Pearsall, T. P.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Scherer, A.

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

Sigalas, M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Sigalas, M. M.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

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

Soukoulis, C. M.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[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]

Sugimoto, Y.

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Talneau, A.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

Temelkuran, B.

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
[CrossRef]

Tokushima, M.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[CrossRef]

Tomita, A.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[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]

Vawter, G. A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Villineneve, P. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Wang, J.

Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
[CrossRef]

Wendt, J. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Yablonovitch, E.

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

Yamada, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[CrossRef]

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]

Zhang, Z. Q.

Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
[CrossRef]

Zubrzycki, W.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Zurbrzycki, W.

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

AIP Conf. Proc. (2)

D. Nedeljkovi, T. P. Pearsall, S. A. Kuchinsky, M. D. Mikhailov, M. Lonar, and A. Scherer, “Planar photonic crystal,” in, Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 107–114 (2001).
[CrossRef]

T. F. Krauss, “Photonic crystals for integrated optics,” in Nanoscale Linear and Nonlinear Optics: International School on Quantum Electronics M. Bertolitti, C. M. Bowden, and C. Sibilia, eds., AIP Conf. Proc. 560, 89–98 (2001).
[CrossRef]

Appl. Phys. Lett. (4)

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[CrossRef]

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm,” Appl. Phys. Lett. 80, 547–549 (2002).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77, 3902–3904 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

J. Appl. Phys. (1)

Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. 91, 922–929 (2002).
[CrossRef]

Jpn. J. Appl. Phys. (1)

A. Chutinan and S. Noda, “Design for waveguides in three-dimensional photonic crystals,” Jpn. J. Appl. Phys. 39, 2353–2356 (2000).
[CrossRef]

Nature (4)

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef] [PubMed]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villineneve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Haus, and A. Allenman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (4)

Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58, 3721–3729 (1998).
[CrossRef]

Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
[CrossRef]

M. Bayindir, E. Ozbay, B. Temelkuran, M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Guiding, bending, and splitting of electromagnetic waves in highly confined photonic crystal waveguides,” Phys. Rev. B 63, 081107 (2001).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

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

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[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]

Science (2)

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[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]

Solid State Commun. (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Top-view schematic configuration of (a) a layer-by-layer 3D photonic crystal made from rectangular dielectric rods in an air background and (b) an X waveguide created by removal of one single rod along the (100) direction of the photonic crystal.

Fig. 2
Fig. 2

Photonic band structure of guided modes in the X waveguide shown in Fig. 1(b). The shaded background pattern represents the body’s photonic bands projected onto the waveguide direction. The waveguide is formed in a photonic crystal made from silicon rods with a filling fraction f=0.25 and an aspect ratio c0/a=1.25, where c0 and a are lattice spacings along the (001) and (100) directions, respectively.

Fig. 3
Fig. 3

Schematic configuration of Y waveguides created in a layer-by-layer photonic crystal by removal of one wx-wide segment of the x-axis rods along the (010) direction. (a) Type I Y waveguide with its central axis located in a mirror-reflection symmetric plane with respect to the x axis. (b) Type II Y waveguide with its central axis shifted by a/4 relative to that of a type I Y waveguide and located in a central-inversion symmetric position.

Fig. 4
Fig. 4

Photonic band structure of guided modes in (a) the type I Y waveguide shown in Fig. 3(a) for wx=0.7a and (b) the type II Y waveguide shown in Fig. 3(b) for wx=0.4a. The photonic crystal has parameters of f=0.25 and c0/a=1.25.

Fig. 5
Fig. 5

Schematic configuration of Z waveguides created in a layer-by-layer photonic crystal by removal of some segments of rods consecutively every four layers along the (001) direction. (a) Type I Z waveguide with its central axis located at the intersection of two mirror-reflection symmetric planes with respect to the x and y axes. (b) Type II Z waveguide with its central axis shifted by a/4 relative to that of a type I Z waveguide and located in a position with both central-inversion symmetry and mirror-reflection symmetry with respect to the y axis.

Fig. 6
Fig. 6

Photonic band structure of guided modes in the type I Z waveguide shown in Fig. 5(a) for (a) wx=0.7a, wy=0 and (b) wx=1.5a, wy=0. The photonic crystal has f=0.25 and c0/a=1.25.

Fig. 7
Fig. 7

Photonic band structure of guided modes in the type I Z waveguide shown in Fig. 5(a) for wx=wy=0.7a.

Fig. 8
Fig. 8

Photonic band structure of guided modes in the type II Z waveguide shown in Fig. 5(b) for (a) wx=0.4a and (b) wx=1.5a. The photonic crystal has f=0.25 and c0/a=1.25.

Equations (32)

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

×1(r) ×H(r)=ω2c2H(r),
H(r)=K(h1Ke^1K+h2Ke^2K)exp(iKr),
(r)=G Gexp(iGr).
HKKλλh1Kh2K=ω2c2h1Kh2K,
HKKλλ=e^2Ke^2K-e^2Ke^1K-e^1Ke^2Ke^1Ke^1KKK|K||K|
H001102H01110000H0022002H01212H01222H01110H1111+H1211H1112-H12120000H1121+H1221H1122-H12220002H012100H1111-H1211H1112+H121202H012200H1121-H1221H1122+H1222 g1K0g2K0g1K1g2K1g1K2g2K2=ω2c2g1K0g2K0g1K1g2K1g1K2g2K2.
hλK0hλK1hλK2=SgλK0gλK1gλK2=I00S2 gλK0gλK1gλK2,
S2=121010010110-100-101,
g1K0=h1K0,g2K0=h2K0,
g1K1=(h1K1+h1K2)/2,
g2K1=(h2K1-h2K2)/2,
g1K2=(h1K1-h1K2)/2,
g2K2=(h2K1+h2K2)/2.
H00λλH01λλH02λλH10λλH11λλH12λλH20λλH21λλH22λλ hλK0hλK1hλK2=ω2c2hλK0hλK1hλK2.
H00112H011102H0111H1111+H1211H1112-H12120H1121+H1221H1122-H1222 g1K0g1K1g2K1
=ω2c2g1K0g1K1g2K1,
H00222H01212H01222H0121H1111-H1211H1112+H12122H0122H1121-H1221H1122+H1222 g2K0g1K2g2K2
=ω2c2g2K0g2K2g1K2,
H00112H011102H0111H1111+H1211H1112-H12120H1112-H1212H1122-H1222 g1K0g1K1g2K1
=ω2c2g1K0g1K1g2K1,
H00222H01212H01222H0121H1111-H1211H1112+H12122H0122H1112+H1212H1122+H1222 g2K0g1K2g2K2
=ω2c2g2K0g1K2g2K2.
h1K0=g1K0,h2K0=0,
h1K1=g1K1/2,h2K1=g2K1/2,
h1K2=g1K1/2,h2K2=-g2K1/2
h1K0=0,h2K0=g2K0,
h1K1=g1K2/2,h2K1=g2K2/2,
h1K2=-g1K2/2,h2K2=g2K2/2
H(r)=K0(h1K0e^1K0+h2K0e^2K0)exp(iK0r)+K1(h1K1e^1K1+h2K1e^2K1)exp(iK1r)+K2(h1K2e^1K2+hK2e^2k2)exp(iK2r),
=K0g1K0e^1K0exp(iK0r)+K112 {g1K1[e^1K1exp(iK1r)+e^1K2exp(iK2r)]+g2K1[e^2K1exp(iK1r)-e^2K2exp(iK2r)]}.
Hx(r)=K1sin(K1xx)Vx(y, z, K1),
Hy(z)(r)=K0Uy(z)(y, z, K0)+K1cos(K1xx)Vy(z)(y, z, K1),

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