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. Express 12, 1397-1408 (2004).
    [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. Express 15, 5416-5430 (2007).
    [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 (2)

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]

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. Express 15, 5416-5430 (2007).
[CrossRef] [PubMed]

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]

2004 (1)

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 (3)

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)

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. 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]

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]

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]

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. Express (2)

Opt. Lett. (1)

Phys. Rev. B (4)

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]

I. H. H. Zabel and D. Stroud, "Photonic band structures of optically anisotropic periodic arrays," Phys. Rev. B 48, 5004-5012 (1993).
[CrossRef]

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]

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)

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]

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]

Other (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).

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

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

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

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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)

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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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