Abstract

For photonic crystals (PhCs) and related devices, it is useful to calculate the Dirichlet-to-Neumann (DtN) map of a unit cell, which maps the wave field to its normal derivative on the boundary. The DtN map can be used to avoid further calculations in the interiors of the unit cells and formulate mathematical problems on the cell boundaries. We develop a method to approximate the DtN map for two-dimensional PhCs involving anisotropic media, and we calculate band structures for PhCs involving liquid crystals. For band structures of triangular lattice PhCs, we also develop new eigenvalue problem formulations involving smaller matrices.

© 2009 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, 1995).
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    [CrossRef]
  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]
  4. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (1999).
    [CrossRef]
  5. S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
    [CrossRef]
  6. C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
    [CrossRef]
  7. Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
    [CrossRef]
  8. C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
    [CrossRef]
  9. M. J. Escuti, J. Qi, and G. P. Crawford, “Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals,” Opt. Lett. 28, 522-524 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  25. G. Alagappan, X. W. Sun, and P. Shum, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A 23, 2002-2013 (2006).
    [CrossRef]
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    [CrossRef]

2008 (3)

2007 (6)

2006 (3)

2004 (2)

S. F. Mingaleev, M. Schillinger, D. Hermann, and K. Busch, “Tunable photonic crystal circuits: concepts and designs based on single-pore infiltration,” Opt. Lett. 29, 2858-2860 (2004).
[CrossRef]

H. Takeda and K. Yoshino, “TE-TM mode coupling in two-dimensional photonic crystals composed of liquid-crystal rods,” Phys. Rev. E 70, 026601 (2004).
[CrossRef]

2003 (2)

M. J. Escuti, J. Qi, and G. P. Crawford, “Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals,” Opt. Lett. 28, 522-524 (2003).
[CrossRef] [PubMed]

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

2002 (1)

2001 (3)

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
[CrossRef]

2000 (1)

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

1999 (1)

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (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]

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]

Alagappan, G.

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

Busch, K.

S. F. Mingaleev, M. Schillinger, D. Hermann, and K. Busch, “Tunable photonic crystal circuits: concepts and designs based on single-pore infiltration,” Opt. Lett. 29, 2858-2860 (2004).
[CrossRef]

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

Chang, H. C.

Chen, M. M.

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]

Crawford, G. P.

Escuti, M. J.

Forchel, A.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[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]

Ha, Y. K.

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Hermann, D.

Hernández-Figueroa, H. E.

Hsu, S. M.

Hu, Z.

Huang, Y.

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

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

John, S.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

Johnson, S. G.

Kamp, M.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

Kee, C. S.

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Kim, J. E.

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Klopf, F.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

Leonard, S. W.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

Li, S.

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]

Lim, H.

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Lu, Y. Y.

Marrone, M.

Meade, R. D.

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

Mingaleev, S. F.

Mondia, J. P.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

Ozaki, M.

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
[CrossRef]

Park, H. Y.

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Qi, J.

Reithmaier, J. P.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

Rodriguez-Esquerre, V. F.

Schillinger, M.

Schuller, C.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

Shimoda, Y.

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
[CrossRef]

Shum, P.

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.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Takeda, H.

H. Takeda and K. Yoshino, “TE-TM mode coupling in two-dimensional photonic crystals composed of liquid-crystal rods,” Phys. Rev. E 70, 026601 (2004).
[CrossRef]

Toader, O.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

van Driel, H. M.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

Winn, J. N.

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

Wu, Y.

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]

Yoshino, K.

H. Takeda and K. Yoshino, “TE-TM mode coupling in two-dimensional photonic crystals composed of liquid-crystal rods,” Phys. Rev. E 70, 026601 (2004).
[CrossRef]

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
[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]

Yuan, J.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217-3222 (2006).
[CrossRef]

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]

Appl. Phys. Lett. (2)

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627-3629 (2001).
[CrossRef]

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

J. Comput. Phys. (1)

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

J. Lightwave Technol. (1)

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

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

Opt. Commun. (1)

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. B (3)

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

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389-R2392 (2000).
[CrossRef]

C. S. Kee, H. Lim, Y. K. Ha, J. E. Kim, and H. Y. Park, “Two-dimensional tunable metallic photonic crystals infiltrated with liquid crystals,” Phys. Rev. B 64, 085114 (2001).
[CrossRef]

Phys. Rev. E (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]

H. Takeda and K. Yoshino, “TE-TM mode coupling in two-dimensional photonic crystals composed of liquid-crystal rods,” Phys. Rev. E 70, 026601 (2004).
[CrossRef]

Phys. Rev. Lett. (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]

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

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

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Figures (4)

Fig. 1
Fig. 1

Square and hexagon unit cells for square and triangular lattices, respectively.

Fig. 2
Fig. 2

Left: irreducible Brillouin zone of square lattice. Right: irreducible Brillouin zone for a triangular lattice.

Fig. 3
Fig. 3

Band structure of a 2D PhC composed of liquid crystal cylinders on a square lattice for ϕ = π 6 .

Fig. 4
Fig. 4

Band structures of 2D photonic crystals involving a triangular lattice of circular liquid crystal cylinders in a silicon background. The angle between the optic axis of the liquid crystal and the x axis is ϕ = 0 (top), ϕ = π 6 (middle), and ϕ = π 4 (bottom).

Tables (2)

Tables Icon

Table 1 Computed Bloch Wave Vector Component for a Triangular Lattice of Liquid Crystal Cylinders in a Silicon Background with ω L ( 2 π c ) = 0.4 and ϕ = π 4

Tables Icon

Table 2 Normalized Frequency ω L ( 2 π c ) of the Second Band for ( α L , β L ) = ( 0 , 1.61856 ) and ϕ = π 4 Calculated Using a Plane-Wave Expansion Method

Equations (48)

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

ϵ = [ ϵ 11 ϵ 12 0 ϵ 21 ϵ 22 0 0 0 ϵ 3 ] ,
2 E z + k 0 2 ϵ 3 ( x ) E z = 0 ,
( ϵ * 1 ( x ) H z ) + k 0 2 H z = 0 ,
ϵ * = [ ϵ 22 ϵ 21 ϵ 12 ϵ 11 ] .
Λ H z = H z ν on Ω ,
H z ( x ) m = 1 K c m Φ m ( x ) ,
[ ϵ 11 ϵ 12 ϵ 21 ϵ 22 ] = Q [ ϵ 1 ϵ 2 ] Q T ,
Q = [ cos ϕ sin ϕ sin ϕ cos ϕ ] .
[ x y ] = [ r cos ( θ ϕ ) r sin ( θ ϕ ) ] = Q T [ x y ] .
( Q T ϵ * 1 Q H z ) + k 0 2 H z = 0 ,
Q T ϵ * 1 Q = [ ϵ 2 1 ϵ 1 1 ] .
x ( 1 ϵ 2 H z x ) + y ( 1 ϵ 1 H z y ) + k 0 2 H z = 0 .
2 H z x ̂ 2 + 2 H z y ̂ 2 + k 0 2 H z = 0 .
n ̂ 2 = ϵ 1 sin 2 θ + ϵ 2 cos 2 θ , tan θ ̂ = ϵ 1 ϵ 2 tan θ .
Φ m ( x ) = J m ( k 0 n ̂ r ) e i m θ ̂ , r < a ,
Φ m ( x ) = j = [ a j ( m ) J j ( k 0 n 0 r ) + b j ( m ) Y j ( k 0 n 0 r ) ] e i j θ , r > a ,
ν ϵ * 1 H z = cos θ ϵ 2 H z x + sin θ ϵ 1 H z y .
1 n 0 2 H z r r = a + = ( cos θ ϵ 2 H z x + sin θ ϵ 1 H z y ) r = a .
W m ( x ) = cos θ ϵ 2 Φ m x + sin θ ϵ 1 Φ m y = e i m θ ̂ [ k 0 n ̂ J m ( k 0 n ̂ r ) + i m ( ϵ 2 ϵ 1 ) sin ( 2 θ ) 2 r n ̂ 2 ϵ 1 ϵ 2 J m ( k 0 n ̂ r ) ] .
J m ( k 0 n ̂ a ) e i m θ ̂ = j = V ̂ j ( m ) e i j θ , W m r = a = j = W ̂ j ( m ) e i j θ ,
[ J j ( k 0 n 0 a ) Y j ( k 0 n 0 a ) J j ( k 0 n 0 a ) Y j ( k 0 n 0 a ) ] [ a j ( m ) b j ( m ) ] = [ V ̂ j ( m ) n 0 k 0 W ̂ j ( m ) ] .
H z ( x ) = Ψ ( x ) exp ( i α x + i β y ) ,
Λ [ u 0 v 0 u 1 v 1 ] = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] [ u 0 v 0 u 1 v 1 ] = [ y u 0 x v 0 y u 1 x v 1 ] ,
v 1 = ρ 1 v 0 , x v 1 = ρ 1 x v 0 ,
u 1 = ρ 2 u 0 , y u 1 = ρ 2 y u 0 ,
M [ u 0 u 1 ] = [ M 11 M 12 M 21 M 22 ] [ u 0 u 1 ] = [ y u 0 y u 1 ] ,
[ M 11 I M 21 0 ] [ u 0 y u 0 ] = ρ 2 [ M 12 0 M 22 I ] [ u 0 y u 0 ] ,
a 1 = ( 1 2 , 3 2 ) , a 2 = ( 1 , 0 ) , a 3 = ( 1 2 , 3 2 ) ,
Λ [ u 0 v 0 w 0 u 1 v 1 w 1 ] = [ ν u 0 ν v 0 ν w 0 ν u 1 ν v 1 ν w 1 ] ,
b 1 = ( 3 2 , 1 2 ) , b 2 = ( 0 , 1 ) , b 3 = ( 3 2 , 1 2 ) .
( α , β ) = γ j a j + δ j b j
u 1 = μ 1 u 0 , ν u 1 = μ 1 ν u 0 .
M [ v 0 w 0 v 1 w 1 ] = [ ν v 0 ν w 0 ν v 1 ν w 1 ] .
τ 1 v 0 = η 1 v 1 , τ 1 ν v 0 = η 1 ν v 1 ,
w 0 = τ 1 η 1 w 1 , ν w 0 = τ 1 η 1 ν w 1 ,
[ M 13 M 14 0 0 M 23 M 24 0 0 M 33 M 34 I 0 M 43 M 44 0 I ] [ v 1 w 1 ν v 1 ν w 1 ] = η 1 [ 1 τ 1 M 11 τ 1 M 12 1 τ 1 I 0 1 τ 1 M 21 τ 1 M 22 0 τ 1 I 1 τ 1 M 31 τ 1 M 32 0 0 1 τ 1 M 41 τ 1 M 42 0 0 ] [ v 1 w 1 ν v 1 ν w 1 ] ,
u 1 = τ 2 η 2 u 0 , v 1 = μ 2 v 0 , η 2 w 1 = τ 2 w 0 ,
ν u 1 = τ 2 η 2 ν u 0 , ν v 1 = μ 2 ν v 0 , η 2 ν w 1 = τ 2 ν w 0 ,
u 1 = τ 1 g , g = τ 1 u 0 , ν u 1 = τ 1 ν g , ν g = τ 1 ν u 0 .
A u = τ 1 B u ,
ϵ 11 = n o 2 sin 2 ϕ + n e 2 cos 2 ϕ ,
ϵ 12 = ϵ 21 = ( n e 2 n o 2 ) sin ϕ cos ϕ ,
ϵ 22 = n o 2 cos 2 ϕ + n e 2 sin 2 ϕ .
D 0 = μ 1 Λ 11 Λ 41 + μ 1 2 Λ 14 μ 1 Λ 44 ,
D j = D 0 1 ( Λ 4 j μ 1 Λ 1 j ) , j = 2 , 3 , 5 , 6 ,
M = [ Λ 22 Λ 23 Λ 25 Λ 26 Λ 32 Λ 33 Λ 35 Λ 36 Λ 52 Λ 53 Λ 55 Λ 56 Λ 62 Λ 63 Λ 65 Λ 56 ] + [ Λ 21 + μ 1 Λ 24 Λ 31 + μ 1 Λ 34 Λ 51 + μ 1 Λ 54 Λ 61 + μ 1 Λ 64 ] [ D 2 D 3 D 5 D 6 ] .
A = [ Λ 11 Λ 12 Λ 16 I 0 0 0 0 Λ 21 Λ 22 Λ 26 0 I 0 0 0 Λ 31 Λ 32 Λ 36 0 0 0 0 0 Λ 41 Λ 42 Λ 46 0 0 0 0 0 Λ 51 Λ 52 Λ 56 0 0 0 0 0 Λ 61 Λ 62 Λ 66 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I ] ,
B = [ 0 1 η 1 Λ 15 η 1 Λ 13 0 0 0 Λ 14 0 0 1 η 1 Λ 25 η 1 Λ 23 0 0 0 Λ 24 0 0 1 η 1 Λ 35 η 1 Λ 33 0 0 η 1 I Λ 34 0 0 1 η 1 Λ 45 η 1 Λ 43 0 0 0 Λ 44 I 0 1 η 1 Λ 55 η 1 Λ 53 0 1 η 1 I 0 Λ 54 0 0 1 η 1 Λ 65 η 1 Λ 63 0 0 0 Λ 64 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 ] .

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