Abstract

A finite element method based eigenvalue algorithm is developed for the analysis of band structures of two-dimensional non-diagonal anisotropic photonic crystals under the in-plane wave propagation. The characteristics of band structures for the square and triangular lattices consisting of anisotropic materials are examined in detail and the intrinsic effect of anisotropy on the construction of band structures is investigated. We discover some interesting relationships of band structures for certain directions of the wave vector in the first Brillouin zone and present a theoretical explanation for this phenomenon. The complete band structures can be conveniently constructed by means of this concept.

© 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. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984).
  8. I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993).
    [Crossref]
  9. 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]
  10. 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]
  11. 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]
  12. 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).
  13. 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]
  14. 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]
  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. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).
  17. 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]
  18. 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]

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 (2)

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]

1998 (1)

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]

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, 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).

Atkin, D. M.

Birks, T. A.

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. Express 12, 1397–1408 (2004).
[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]

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]

Gu, C.

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

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984).

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]

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]

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.

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).

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]

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. Express 12, 1397–1408 (2004).
[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).

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 (1)

Opt. Express (1)

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. (3)

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]

Other (4)

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984).

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

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

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Figures (9)

Fig. 1.
Fig. 1.

(a) Quadratic triangular element. (b) Curvilinear triangular element.

Fig. 2.
Fig. 2.

(a) A square unit cell and its corresponding PBCs. (b) A hexagonal unit cell and its corresponding PBCs.

Fig. 3.
Fig. 3.

Schematic definition of rotation angles for the LC molecule.

Fig. 4.
Fig. 4.

(a) The cross-section of a 2D PC with square lattice in the background of LCs. (b) The first BZ of (a).

Fig. 5.
Fig. 5.

Comparisons of band structures for the 2D PC with square lattice of (j>c = 30c between different sub-zones. (a) Γ-X-M and Γ-X′-M; (b) Γ-X-M and TΓX′-M′; (c) Γ-X-M and Γ-X′′-M′; (d) Γ-X′-M and Γ-X′-M′; (e) Γ-X′-M and Γ-X′′-M′; (f) Γ-X′-M′ and Γ-X′′-M′.

Fig. 6.
Fig. 6.

Explanations for the equivalence of band structures of the 2D PC with square lattice between certain directions of the wave vector in the first BZ.

Fig. 7.
Fig. 7.

The band structures of the TE mode for the 2D PC with square lattice of (a) φc = 0°, (b) φc = 30°, and (c) φc = 45°.

Fig. 8.
Fig. 8.

(a) The cross-section of a 2D PC with triangular lattice in the background of LCs. (b) The first BZ of (a).

Fig. 9.
Fig. 9.

The band structures of the TE mode for the 2D PC with triangular lattice of (a) φc = 0°, (b) φc = 30°, and (c) φc = 45°.

Equations (68)

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× 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 ] .
E y x E x y = j ω μ 0 μ zz H z
H z y = j ω ε 0 ( ε xx E x + ε xy E y )
H z x = j ω ε 0 ( ε yx E x + ε yy E y ) .
x ( ε yx ε yx ε xy ε xx ε yy H z y ) + x ( ε xx ε yx ε xy ε xx ε yy H z x )
y ( ε yy ε yy ε xx ε xy ε yx H z y ) y ( ε xy ε yy ε xx ε xy ε yx H z x ) = k 0 2 μ zz H z
E z y = j ω μ 0 ( μ xx H x + μ xy H y )
E z y = j ω μ 0 ( μ yx H x + μ yy H y )
H y x H x y = jωμε 0 ε zz E z .
x ( μ yx μ yx μ xy μ xx μ yy E z y ) + x ( μ xx μ yx μ xy μ xx μ yy E z x )
y ( μ yy μ yy μ xx μ xy μ yx E z y ) y ( μ xy μ yy μ xx μ xy μ yx E z x ) = k 0 2 ε zz E z .
x ( p yx p yx p xy p xx p yy φ z y ) + x ( p xx p yx p xy p xx p yy φ z x )
y ( p yy p yy p xx p xy p yx φ z y ) y ( p xy p yy p xx p xy p yx φ z x ) = k 0 2 q zz φ z
[ p ] = [ p xx p xy p xz p yx p yy p yz p zx p zy p zz ] = [ ε xx ε xy 0 ε yx ε yy 0 0 0 ε zz ]
[ q ] = [ q xx q xy q xz q yx q yy q yz q zx q zy q zz ] = [ μ xx μ xy 0 μ yx μ yy 0 0 0 μ zz ]
[ p ] = [ p xx p xy p xz p yx p yy p yz p zx p zy p zz ] = [ μ xx μ xy 0 μ yx μ yy 0 0 0 μ zz ]
[ q ] = [ q xx q xy q xz q yx q yy q yz q zx q zy q zz ] = [ ε xx ε xy 0 ε yx ε yy 0 0 0 ε zz ]
{ N } = [ N 1 N 2 N 3 N 4 N 5 N 6 ] = [ L 1 ( 2 L 1 1 ) L 2 ( 2 L 2 1 ) L 3 ( 2 L 3 1 ) 4 L 1 L 2 4 L 2 L 3 4 L 3 L 1 ]
L i = 1 2 Δ ( a i + b i x + c i y )
a i = x i + 1 y i + 2 x i + 2 y i + 1
b i = y i + l y i 1
c i = x i 1 x i + 1
Δ = 1 2 ( a 1 + a 2 + a 3 ) = 1 2 ( b i + 1 c i 1 b i 1 c i + 1 ) .
x = j = 1 6 N j x j
y = j = 1 6 N j y j
[ L 1 L 2 ] = [ J ] [ x y ]
[ J ] = [ x L 1 y L 1 x L 2 y L 2 ] .
∫∫ 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 ) dL 2 ] dL 1
[ K ] { φ z } k 0 2 [ M ] { φ z } = { ψ }
[ K ] = e ∫∫ ( p yx p yx p xy p xx p yy { N } x { N } T y p xx p yx p xy p xx p yy { N } x { N } T x
+ p yy p yy p xx p xy p yx { N } y { N } T y + p xy p yy p xx p xy p yx { N } y { N } T x ) dxdy
[ M ] = e ∫∫ ( q zz { N } { N } T ) dxdy
{ ψ } = e [ x ̂ ( p yx p yx p xy p xx p yy { N } φ z y p xx p yx p xy p xx p yy { N } φ z x )
+ y ̂ ( p yy p yy p xx p xy p yx { N } φ z y + p xy p yy p xx p xy p yx { N } φ z x ) n ̂ dl
φ z I = e j k x a φ z III
φ z x I = e j k x a φ z x III
φ z y I = e j k x a φ z y III
φ z II = e j k x a φ z IV
φ z y II = e j k x a φ z x IV
φ z y II = e j k x a φ z y IV
[ [ K ] 11 [ K ] 12 [ K ] 13 [ K ] 14 [ K ] 15 [ K ] 21 [ K ] 22 [ K ] 23 [ K ] 24 [ K ] 25 [ K ] 31 [ K ] 32 [ K ] 33 [ K ] 34 [ K ] 35 [ K ] 41 [ K ] 42 [ K ] 43 [ K ] 44 [ K ] 45 [ K ] 51 [ K ] 52 [ K ] 53 [ K ] 54 [ K ] 55 ] [ { φ Z } 0 { φ Z } I { φ Z } II { φ Z } III { φ Z } IV ]
K 0 2 [ [ M ] 11 [ M ] 12 [ M ] 13 [ M ] 14 [ M ] 15 [ M ] 21 [ M ] 22 [ M ] 23 [ M ] 24 [ M ] 25 [ M ] 31 [ M ] 32 [ M ] 33 [ M ] 34 [ M ] 35 [ M ] 41 [ M ] 42 [ M ] 43 [ M ] 44 [ M ] 45 [ M ] 51 [ M ] 52 [ M ] 53 [ M ] 54 [ M ] 55 ] [ { φ Z } 0 { φ Z } I { φ Z } II { φ Z } III { φ Z } IV ] = [ { 0 } { ψ } I { ψ } II { ψ } III { ψ } IV ]
{ ψ } I = e ( p yx p yx p xy p xx p yy { N } φ z y I p xx p yx p xy p xx p yy { N } φ z x I ) dl
{ ψ } II = e ( p yy p yy p xx p xy p yx { N } φ z y II + p xy p yy p xx p xy p yx { N } φ z x II ) dl
{ ψ } III = e ( p yx p yx p xy p xx p yy { N } φ z y III + p xx p yx p xy p xx p yy { N } φ z x III ) dl
{ ψ } IV = e ( p yy p yy p xx p xy p yx { N } φ z y IV p xy p yy p xx p xy p yx { N } φ z x IV ) dl .
[ [ K ] 11 [ K ] 14 + e j k x a [ K ] 12 [ K ] 41 + 1 e j k x a [ K ] 21 [ K ] 44 + e j k x a [ K ] 42 + 1 e j k x a ( [ K ] 24 + e j k x a [ K ] 22 ) [ K ] 51 + 1 e j k y a [ K ] 31 [ K ] 54 + e j k x a [ K ] 52 + 1 e j k y a ( [ K ] 34 + e j k x a [ K ] 32 )
[ K ] 15 + e j k y a [ K ] 13 [ K ] 45 + e j k y a [ K ] 43 + 1 e jk x a ( [ K ] 25 + e jk y a [ K ] 23 ) [ K ] 55 + e j k y a [ K ] 53 + 1 e j k y a ( [ K ] 35 + e j k y a [ K ] 33 ) ] [ { φ z } 0 { φ z } III { φ z } IV ]
k 0 2 [ [ M ] 11 [ M ] 14 + e j k x a [ M ] 12 [ M ] 41 + 1 e j k x a [ M ] 21 [ M ] 44 + e j k x a [ M ] 42 + 1 e j k x a ( [ M ] 24 + e j k x a [ M ] 22 ) [ M ] 51 + 1 e j k y a [ M ] 31 [ M ] 54 + e j k x a [ M ] 52 + 1 e j k y a ( [ M ] 34 + e j k x a [ M ] 32 )
[ M ] 15 + e j k y a [ M ] 13 [ M ] 45 + e j k y a [ M ] 43 + 1 e j k x a ( [ M ] 25 + e j k y a [ M ] 23 ) [ M ] 55 + e j k y a [ M ] 53 + 1 e j k y a ( [ M ] 35 + e j k y a [ M ] 33 ) ] [ { φ z } 0 { φ z } III { φ z } IV ] = [ { 0 } { 0 } { 0 } ] .
φ z I = e j ( k x 3 a 2 + k y a 2 ) φ z IV
φ z x I = e j ( k x 3 a 2 + k y a 2 ) φ z x IV
φ z y I = e j ( k x 3 a 2 + k y a 2 ) φ z y IV
φ z II = e j k y a φ z V
φ z x II = e j k y a φ z x V
φ z y II = e j k y a φ z y V
φ z III = e j ( k x 3 a 2 - k y a 2 ) φ z VI
φ z x III = e j ( k x 3 a 2 - k y a 2 ) φ z x VI
φ z y III = e j ( k x 3 a 2 - k y a 2 ) φ z y VI
ε xx = n o 2 + ( n e 2 n o 2 ) sin 2 θ c cos 2 φ c
ε xy = 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 2 θ c cos θ 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 2 θ c cos θ c sin φ c
ε zz = n o 2 + ( n e 2 n o 2 ) cos 2 θ c

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