Abstract

We have developed a plane-wave transfer-matrix method in combination with an interpolation method to calculate the whole diagram of photonic band structures in the first Brillouin zone by only looking at one single stacking direction of crystalline layers with fast numerical convergence. Using the interpolation method, one does not need to scan a considerable frequency range to accurately compute the exact value of eigenfrequency corresponding to a Bloch wave. The calculated photonic band structures by our proposed method are in good agreement with results obtained by means of the conventional plane-wave expansion method but with a better numerical convergency. The proposed approach can become an economical, flexible, and accurate numerical tool to understand and design different two-dimensional and three-dimensional photonic crystals.

© 2008 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, New Jersey, 1995).
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  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. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488-4492 (2000).
    [CrossRef]
  5. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143-149 (1997).
    [CrossRef]
  6. R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
    [CrossRef]
  7. M. Che and Z. Y. Li, “Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method,” Phys. Rev. B 77, 125138 (2008).
    [CrossRef]
  8. 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]
  9. 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]
  10. A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
    [CrossRef]
  11. H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
    [CrossRef]
  12. Y. C. Hsue and T. J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,” Phys. Rev. E 70, 016706 (2004).
    [CrossRef]
  13. Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
    [CrossRef]
  14. S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
    [CrossRef]
  15. J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209-229 (1994).
    [CrossRef]
  16. J. B. Pendry, “Calculating photonic band structure,” J. Phys. Condens. Matter 8, 1085-1108 (1996).
    [CrossRef]
  17. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
    [CrossRef]
  18. Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
    [CrossRef]
  19. Z. Y. Li and K. M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
    [CrossRef]
  20. C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
    [CrossRef]
  21. N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
    [CrossRef]
  22. S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E 73, 066609 (2006).
    [CrossRef]
  23. A. R. McGurn, “Green's-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059-7064 (1996).
    [CrossRef]
  24. A. I. Rahachou and I. V. Zozoulenko, “Light propagation in finite and infinite photonic crystals: The recursive Green's function technique,” Phys. Rev. B 72, 155117 (2005).
    [CrossRef]
  25. J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
    [CrossRef]
  26. D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
    [CrossRef]
  27. E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
    [CrossRef]
  28. E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
    [CrossRef]
  29. W. Axmann and P. Kuchment, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363-376 (1999).
    [CrossRef]
  30. D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
    [CrossRef]
  31. D. S. Gao and Z. P. Zhou, “Nonlinear equation method for band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 88, 163105 (2006).
    [CrossRef]
  32. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
    [CrossRef]
  33. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  34. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  35. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interface,” Rev. Mod. Phys. 78, 455-481 (2006).
    [CrossRef]
  36. 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]
  37. M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
    [CrossRef]

2008 (2)

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

M. Che and Z. Y. Li, “Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method,” Phys. Rev. B 77, 125138 (2008).
[CrossRef]

2007 (3)

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

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

M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
[CrossRef]

2006 (3)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interface,” Rev. Mod. Phys. 78, 455-481 (2006).
[CrossRef]

D. S. Gao and Z. P. Zhou, “Nonlinear equation method for band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 88, 163105 (2006).
[CrossRef]

S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E 73, 066609 (2006).
[CrossRef]

2005 (2)

Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
[CrossRef]

A. I. Rahachou and I. V. Zozoulenko, “Light propagation in finite and infinite photonic crystals: The recursive Green's function technique,” Phys. Rev. B 72, 155117 (2005).
[CrossRef]

2004 (1)

Y. C. Hsue and T. J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

2003 (4)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Z. Y. Li and K. M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

2002 (3)

E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
[CrossRef]

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

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

2000 (4)

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488-4492 (2000).
[CrossRef]

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (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 (1)

W. Axmann and P. Kuchment, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363-376 (1999).
[CrossRef]

1998 (1)

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 (2)

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

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

1996 (4)

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

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

A. R. McGurn, “Green's-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059-7064 (1996).
[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]

1995 (2)

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

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

1994 (1)

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

1992 (1)

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[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]

Abram, R. A.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Albert, J. P.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

Axmann, W.

W. Axmann and P. Kuchment, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363-376 (1999).
[CrossRef]

Bertho, D.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

Botten, L. C.

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

Brand, S.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Cassagne, D.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

Chan, C. T.

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

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

Chang, H. C.

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

Che, M.

M. Che and Z. Y. Li, “Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method,” Phys. Rev. B 77, 125138 (2008).
[CrossRef]

M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
[CrossRef]

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]

Cheng, B. Y.

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

Chiang, P. J.

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

Chutinan, A.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488-4492 (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]

Croucher, M. P.

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
[CrossRef]

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

Fan, S.

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]

Freeman, A. J.

Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
[CrossRef]

Gao, D. S.

D. S. Gao and Z. P. Zhou, “Nonlinear equation method for band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 88, 163105 (2006).
[CrossRef]

Gopalakrishnan, J.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Gu, B. Y.

Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
[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]

Hafner, C.

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

E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
[CrossRef]

Haus, J. W.

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Ho, K. M.

Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Z. Y. Li and K. M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

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

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

Hsue, Y. C.

Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
[CrossRef]

Y. C. Hsue and T. J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

Inguva, R.

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interface,” Rev. Mod. Phys. 78, 455-481 (2006).
[CrossRef]

Joannopoulos, J. D.

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]

J. D. Joannopoulos, R. D. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, New Jersey, 1995).

Jouanin, C.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

Kaliteevski, M. A.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Kuchment, P.

W. Axmann and P. Kuchment, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363-376 (1999).
[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]

Li, L.

Li, Z. Y.

M. Che and Z. Y. Li, “Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method,” Phys. Rev. B 77, 125138 (2008).
[CrossRef]

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
[CrossRef]

Z. Y. Li and K. M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[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, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, Z. F.

S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E 73, 066609 (2006).
[CrossRef]

Liu, R. J.

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
[CrossRef]

Liu, S. Y.

S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E 73, 066609 (2006).
[CrossRef]

McGurn, A. R.

A. R. McGurn, “Green's-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059-7064 (1996).
[CrossRef]

McPhedran, R. C.

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

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, New Jersey, 1995).

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]

Moreno, E.

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

E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
[CrossRef]

Moroz, A.

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

Nicorovici, N. A.

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

Noda, S.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488-4492 (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]

Pasciak, J. E.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Pendry, J. B.

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

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

Rahachou, A. I.

A. I. Rahachou and I. V. Zozoulenko, “Light propagation in finite and infinite photonic crystals: The recursive Green's function technique,” Phys. Rev. B 72, 155117 (2005).
[CrossRef]

Ruan, M.

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interface,” Rev. Mod. Phys. 78, 455-481 (2006).
[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]

Sözüer, H. S.

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[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]

Villeneuve, P. R.

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]

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]

Whittaker, D. M.

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

Winn, J.

J. D. Joannopoulos, R. D. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, New Jersey, 1995).

Yablonovitch, E.

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

Yang, T. J.

Y. C. Hsue and T. J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

Yu, C. P.

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

Yu, Q. L.

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

Zhang, D. Z.

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

Zhou, F.

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

Zhou, Z. P.

D. S. Gao and Z. P. Zhou, “Nonlinear equation method for band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 88, 163105 (2006).
[CrossRef]

Zozoulenko, I. V.

A. I. Rahachou and I. V. Zozoulenko, “Light propagation in finite and infinite photonic crystals: The recursive Green's function technique,” Phys. Rev. B 72, 155117 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

D. S. Gao and Z. P. Zhou, “Nonlinear equation method for band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 88, 163105 (2006).
[CrossRef]

J. Appl. Phys. (1)

R. J. Liu, M. Ruan, F. Zhou, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “Waveguide bend designs in three-dimensional woodpile photonic crystals,” J. Appl. Phys. 103, 034502 (2008).
[CrossRef]

J. Comput. Phys. (2)

W. Axmann and P. Kuchment, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363-376 (1999).
[CrossRef]

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

J. Mod. Opt. (1)

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. Condens. Matter (1)

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

Nature (1)

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

Phys. Rev. A (1)

M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystal,” Phys. Rev. A 76, 023809 (2007).
[CrossRef]

Phys. Rev. B (15)

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

A. R. McGurn, “Green's-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059-7064 (1996).
[CrossRef]

A. I. Rahachou and I. V. Zozoulenko, “Light propagation in finite and infinite photonic crystals: The recursive Green's function technique,” Phys. Rev. B 72, 155117 (2005).
[CrossRef]

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

M. Che and Z. Y. Li, “Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method,” Phys. Rev. B 77, 125138 (2008).
[CrossRef]

Y. C. Hsue, A. J. Freeman, and B. Y. Gu, “Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals,” Phys. Rev. B 72, 195118 (2005).
[CrossRef]

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Z. Y. Li and K. M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

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

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488-4492 (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]

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Phys. Rev. E (6)

Y. C. Hsue and T. J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

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

S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E 73, 066609 (2006).
[CrossRef]

E. Moreno, D. Erni, and C. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618 (2002).
[CrossRef]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

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

Phys. Rev. Lett. (3)

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]

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]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interface,” Rev. Mod. Phys. 78, 455-481 (2006).
[CrossRef]

Science (1)

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 (1)

J. D. Joannopoulos, R. D. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, New Jersey, 1995).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of 2D PC structure built in a square lattice of a dielectric rod embedded in the air background. (b) Schematic of 2D PC structure built in a triangle lattice of air holes etched into a high dielectric background. The 2D PC is periodic in the Y O Z plane and homogeneous along the x axis. The stacking direction of the cylinders is along the z axis.

Fig. 2
Fig. 2

(a) Plot of the real part of the determinant of the matrix P and (b) plot of the imaginary part of the determinant of the matrix P in the neighboring region of zero for a series of frequency points for the wave vector k y = 2 4 ( π a ) , k z = 2 4 ( π a ) in the first band along the Γ - X direction under the TM-mode excitation. The structure and the parameter of the 2D PC structure are the same as Fig. 1.

Fig. 3
Fig. 3

Whole diagram of photonic band structures for (a) 2D PCs made from a square lattice of dielectric cylinders in air under the TM-mode excitation and (b) triangle lattice of air holes etched into a high dielectric background under the TE-mode excitation. The open circles and the solid curves respond to our proposed theoretical approach based on the PWTMM in combination with the interpolation method and the CPWM calculation results, respectively.

Fig. 4
Fig. 4

Diagram of photonic band structures along some high-symmetry lines in the first BZ calculated by our proposed method and the 3D CPWM for a 3D FCC PC. The sphere has a radius of 0.2616a and a refractive index of 3.6. The open circles and the solid curves correspond to our proposed theoretical approach based on the PWTMM in combination with the interpolation method and the CPWM calculation results, respectively.

Fig. 5
Fig. 5

Diagram of photonic band structures along some high-symmetry lines in the first BZ calculated by our proposed method and the 3D CPWM for a 3D layer-by-layer PC. The crystal is of a rod-to-rod spacing a, a rod width and thickness of 0.25 a and 0.3125 a , and a refractive index of the rods n = 3.4 . The open circles and the solid curves respond to our proposed theoretical approach based on the PWTMM in combination with the interpolation method and the CPWM calculation results, respectively.

Equations (3)

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( Ω 1 + Ω 0 ) = S ( Ω 0 + Ω 1 ) = ( S 11 S 12 S 21 S 22 ) ( Ω 0 + Ω 1 ) ,
( S 11 0 S 21 I ) ( Ω 0 + Ω 0 ) = e i k a 3 ( I S 12 0 S 22 ) ( Ω 0 + Ω 0 ) ,
[ ( S 11 0 S 21 I ) e i k a 3 ( I S 12 0 S 22 ) ] ( Ω 0 + Ω 0 ) = P ( Ω 0 + Ω 0 ) = 0 .

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