Abstract

A full-vectorial finite element method based eigenvalue algorithm is developed to analyze the band structures of two-dimensional (2D) photonic crystals (PCs) with arbitray 3D anisotropy for in-planewave propagations, in which the simple transverse-electric (TE) or transverse-magnetic (TM) modes may not be clearly defined. By taking all the field components into consideration simultaneously without decoupling of the wave modes in 2D PCs into TE and TM modes, a full-vectorial matrix eigenvalue equation, with the square of the wavenumber as the eigenvalue, is derived. We examine the convergence behaviors of this algorithm and analyze 2D PCs with arbitrary anisotropy using this algorithm to demonstrate its correctness and usefulness by explaining the numerical results theoretically.

© 2007 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).
  4. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
    [Crossref] [PubMed]
  5. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
    [Crossref]
  6. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [Crossref]
  7. M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
    [Crossref]
  8. L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).
  9. C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express12, 1397–1408 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-7-1397
    [Crossref] [PubMed]
  10. 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]
  11. I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
    [Crossref]
  12. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
    [Crossref]
  13. C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B 72, 045133 (2005).
    [Crossref]
  14. S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416
    [Crossref] [PubMed]
  15. G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A 23, 2002–2013 (2006).
    [Crossref]
  16. G. E. Antilla and N. G. Alexopoulos, “Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach,” J. Opt. Soc. Am. A 11, 1445–1457 (1994).
    [Crossref]
  17. L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics 19, 225–239 (1999).
    [Crossref]
  18. D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
    [Crossref]
  19. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).
  20. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
    [Crossref]
  21. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
    [Crossref]
  22. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

2007 (1)

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]

2006 (1)

2005 (1)

C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B 72, 045133 (2005).
[Crossref]

2000 (2)

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[Crossref]

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[Crossref]

1999 (4)

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics 19, 225–239 (1999).
[Crossref]

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

1998 (2)

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

1996 (2)

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

1994 (1)

1993 (1)

I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
[Crossref]

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]

Alagappan, G.

Alexopoulos, N. G.

L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics 19, 225–239 (1999).
[Crossref]

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

G. E. Antilla and N. G. Alexopoulos, “Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach,” J. Opt. Soc. Am. A 11, 1445–1457 (1994).
[Crossref]

Antilla, G. E.

Atkin, D. M.

Birks, T. A.

Broas, R. F. J.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

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]

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express12, 1397–1408 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-7-1397
[Crossref] [PubMed]

S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416
[Crossref] [PubMed]

Chen, L. W.

C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B 72, 045133 (2005).
[Crossref]

Chen, M. M.

S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416
[Crossref] [PubMed]

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]

den Engelsen, D.

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Gu, B. Y.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

He, S.

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[Crossref]

Hsu, S. M.

S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416
[Crossref] [PubMed]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).

John, S.

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

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Knight, J. C.

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Koshiba, M.

Li, Z. Y.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Liu, C. Y.

C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B 72, 045133 (2005).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).

Qiu, M.

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[Crossref]

Russell, P. St. J.

Shum, P.

Sievenpiper, D.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

Stroud, D.

I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
[Crossref]

Sun, X. W.

Tsuji, Y.

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).

Yablonovitch, E.

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

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

Yang, G. Z.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

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]

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express12, 1397–1408 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-7-1397
[Crossref] [PubMed]

Yu, M. B.

Zabel, I. H. H.

I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
[Crossref]

Zhang, L.

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics 19, 225–239 (1999).
[Crossref]

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

Electromagnetics (1)

L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics 19, 225–239 (1999).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[Crossref]

in 1999 IEEE MTT-S Dig. (1)

L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. 4, 1703–1706 (1999).

J. Appl. Phys. (1)

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[Crossref]

J. Lightwave Technol. (1)

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

Opt. Lett. (1)

Phys. Rev. B (4)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
[Crossref]

C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B 72, 045133 (2005).
[Crossref]

Phys. Rev. E (1)

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. (4)

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]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Other (5)

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416
[Crossref] [PubMed]

J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express12, 1397–1408 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-7-1397
[Crossref] [PubMed]

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Figures (10)

Fig. 1.
Fig. 1.

The cross-section of a 2D PC and the corresponding unit cell for (a) the square lattice and (b) the triangular lattice.

Fig. 2.
Fig. 2.

Curvilinear hybrid edge/nodal element.

Fig. 3.
Fig. 3.

The first BZ of a 2D PC with (a) square lattice and (b) triangular lattice.

Fig. 4.
Fig. 4.

The band structures for the 2D isotropic PC formed by (a) square-arranged alumina rods with ε=8.9 and r=0.2a in the air and (b) triangle-arranged dielectric cylinders with ε=11.4 and r=0.2a in the air.

Fig. 5.
Fig. 5.

The convergence behaviors of the full-vectorial algorithm for the 2D isotropic PC with square lattice compared with the scalar algorithm as k is fixed at the X point. (a) TE first band; (b) TE third band; (c) TM first band; (d) TM third band.

Fig. 6.
Fig. 6.

The convergence behaviors of the full-vectorial algorithm for the 2D isotropic PC with triangular lattice compared with the scalar algorithm as k is fixed at the M point. (a) TE first band; (b) TE third band; (c) TM first band; (d) TM third band.

Fig. 7.
Fig. 7.

(a) Cross-section of the 2D anisotropic PC with triangular lattice. (b) Schematic definition of rotation angles for the LC molecule.

Fig. 8.
Fig. 8.

The band structures of the 2D anisotropic PC with triangular lattice of (a) θ c =0° and ϕ c =30°, (b) θ c =30° and ϕ c =30°, (c) θ c =45° and ϕ c =30°, (d) θ c =60° and ϕ c =30°, and (e) θ c =90° and ϕ c =30°.

Fig. 9.
Fig. 9.

The normalized frequency range of the absolute band gap for the 2D anisotropic PC with triangular lattice versus ϕ c for five different θ c ’s from 0° to 90°.

Fig. 10.
Fig. 10.

The variation of the absolute band gap limits for the 2D anisotropic PC with triangular lattice versus ϕ c for five different θ c ’s from 0° to 90°.

Tables (1)

Tables Icon

Table 1. Vector- and scalar-based shape functions

Equations (58)

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

× E = j ω μ 0 [ μ r ] H
× H = j ω ε 0 [ ε r ] E
[ μ r ] = [ μ xx μ xy μ xz μ yx μ yy μ yz μ zx μ zy μ zz ]
[ ε r ] = [ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ] .
× ( [ p ] × Φ ) k 0 2 [ q ] Φ = 0
[ p ] = [ p xx p xy p xz p yx p yy p yz p zx p zy p zz ] = [ μ xx μ xy μ xz μ yx μ yy μ yz μ zx μ zy μ zz ] 1
[ q ] = [ q xx q xy q xz q yx q yy q yz q zx q zy q zz ] = [ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ]
[ p ] = [ p xx p xy p xz p yx p yy p yz p zx p zy p zz ] = [ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ] 1
[ q ] = [ q xx q xy q xz q yx q yy q yz q zx q zy q zz ] = [ μ xx μ xy μ xz μ yx μ yy μ yz μ zx μ zy μ zz ]
Φ ( x , y , z ) = Φ t ( x , y ) + z ̂ Φ z ( x , y )
= t + z
t × ( [ 0 0 0 0 0 0 0 0 p zz ] t × Φ t + [ 0 0 0 0 0 0 p zx p zy 0 ] t × z ̂ Φ z )
k 0 2 ( [ q xx q xy 0 q yx q yy 0 0 0 0 ] Φ t + [ 0 0 q xz 0 0 q yz 0 0 0 ] Φ z ) = 0
t × ( [ 0 0 p xz 0 0 p yz 0 0 0 ] t × Φ t + [ p xx p xy 0 p yx p yy 0 0 0 0 ] t × z ̂ Φ z )
k 0 2 ( [ 0 0 0 0 0 0 q zx q zy 0 ] Φ t + [ 0 0 0 0 0 0 0 0 q zz ] Φ z ) = 0 .
[ J ] = [ x L 1 y L 1 x L 2 y L 2 ]
[ L 1 L 2 ] = [ J ] [ x y ]
e f ( x , y ) dxdy = 0 1 [ 0 1 L 1 f ( L 1 , L 2 , L 3 ) J ( L 1 , L 2 , L 3 ) d L 2 ] d L 1 .
Φ = Φ t + z ̂ Φ z = [ { U } T { ϕ t e } { V } T { ϕ t e } { N } T { ϕ z e } ]
Φ I = e j k x a Φ III
Φ x I = e j k x a Φ x III
Φ y I = e j k x a Φ y III
Φ II = e j k y a Φ IV
Φ x II = e j k y a Φ x IV
Φ y II = e j k y a Φ y IV
Φ I = e j ( k x 3 a 2 + k y a 2 ) Φ IV
Φ x I = e j ( k x 3 a 2 + k y a 2 ) Φ x IV
Φ y I = e j ( k x 3 a 2 + k y a 2 ) Φ y IV
Φ II = e j k y a Φ V
Φ x II = e j k y a Φ x V
Φ y II = e j k y a Φ y V
Φ III = e j ( k x 3 a 2 k y a 2 ) Φ VI
Φ x III = e j ( k x 3 a 2 k y a 2 ) Φ x VI
Φ y III = e j ( k x 3 a 2 k y a 2 ) Φ y VI
[ K ] { ϕ } k 0 2 [ M ] { ϕ } = { 0 }
{ ϕ } = [ { ϕ t } { ϕ z } ]
[ K ] = [ [ K tt ] [ K tz ] [ K zt ] [ K zz ] ]
[ M ] = [ [ M tt ] [ M tz ] [ M zt ] [ M zz ] ]
[ K tt ] = e [ p zz { V } x { V } T x p zz { V } x { U } T y
p zz { U } y { V } T x + p zz { U } y { U } T y ] dxdy
[ K tz ] = e [ p zx { V } x { N } T x p zy { V } x { N } T y
p zx { U } y { N } T x + p zy { U } y { N } T y ] dxdy
[ K zt ] = e [ p xz { N } x { V } T x p xz { N } x { U } T y
p yz { N } y { V } T x + p yz { N } y { U } T y ] dxdy
[ K zz ] = e [ p xx { N } x { N } T x p xy { N } x { N } T y
p yx { N } y { N } T x + p yy { N } y { N } T y ] dxdy
[ M tt ] = e [ q xx { U } { U } T + q xy { U } { V } T
+ q yx { V } { U } T + q yy { V } { V } T ] dxdy
[ M tz ] = e [ q xz { U } { N } T + q yz { V } { N } T ] dxdy
[ M zt ] = e [ q zx { N } { U } T + q zy { N } { V } T ] dxdy
[ M zz ] = e [ q zz { N } { N } T ] dxdy .
ε xx = n o 2 + ( n e 2 n o 2 ) sin 2 θ c cos 2 ϕ c
ε xx = ε yx = ( n e 2 n o 2 ) sin 2 θ c sin ϕ c cos ϕ c
ε xz = ε zx = ( n e 2 n o 2 ) sin θ c cos ϕ c cos ϕ c
ε yy = n o 2 + ( n e 2 n o 2 ) sin 2 θ c sin 2 ϕ c
ε yz = ε zy = ( n e 2 n o 2 ) sin θ c cos θ c sin ϕ c
ε zz = n o 2 + ( n e 2 n o 2 ) cos 2 θ c
ω a 2 π c normalized = ω a 2 π c ω a 2 π c max + ω a 2 π c min 2

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