Abstract

To effectively investigate the fundamental characteristics of two-dimensional (2D) photonic crystals (PCs) with arbitrary 3D material anisotropy under the out-of-plane wave propagation, we establish a full-vectorial finite element method based eigenvalue algorithm to perform related analysis correctly. The band edge diagrams can be conveniently constructed from the band structures of varied propagation constants obtained from the algorithm, which is helpful for the analysis and design of photonic band gap (PBG) fibers. Several PCs are analyzed to demonstrate the correctness of this numerical model. Our analysis results for simple PCs are checked with others’ ones using different methods, including the transfer matrix method, the finite-difference frequency-domain (FDFD) method, and the plane-wave expansion method. And the validity of those for the most complex PC with arbitrary 3D anisotropy is supported by related liquid-crystal-filled PBG fiber mode analysis, which demonstrates the dependence of transmission properties on the PBGs, employing a full-vectorial finite element beam propagation method (FE-BPM).

© 2008 Optical Society of America

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2008 (1)

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[CrossRef]

2007 (3)

2006 (2)

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]

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

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)

2003 (1)

2001 (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 (1)

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 (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]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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]

1995 (1)

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

1994 (1)

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium," J. Modern Opt. 41, 275-284 (1994).
[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]

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.

Atkin, D. M.

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]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Bosc, D.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

Brilland, L.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Chang, H. C.

Chartier, T.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

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.

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]

Gong, Y.

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[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]

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.

Hu, J.

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[CrossRef]

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]

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]

Jouan, T.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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]

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.

Le Person, J.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

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]

Maradudin, A. A.

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium," J. Modern Opt. 41, 275-284 (1994).
[CrossRef]

McGurn, A. R.

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium," J. Modern Opt. 41, 275-284 (1994).
[CrossRef]

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]

Ren, G.

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[CrossRef]

Roberts, P. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Russell, P. St. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Saitoh, K.

Shepherd, T. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Shum, P.

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[CrossRef]

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]

Smektala, F.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

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.

Troles, J.

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

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]

Yablonovitch, E.

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]

Yu, C. P.

Yu, M. B.

Yu, X

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[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]

Electron. Lett. (1)

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

G. Ren, P. Shum, J. Hu, X Yu, and Y. Gong, "Study of polarization-dependent bandgap formation in liquid crystal filled photonic crystal fibers," IEEE Photonics Technol. Lett. 20, 602-604 (2008).
[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. (2)

J. Modern Opt. (1)

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium," J. Modern Opt. 41, 275-284 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

Mater. Res. Bull. (1)

J. Le Person, F. Smektala, T. Chartier, L. Brilland, T. Jouan, J. Troles, and D. Bosc, "Light guidance in new chalcogenide holey fibres from GeGaSbS glass," Mater. Res. Bull. 41, 1303-1309 (2006).
[CrossRef]

Opt. Express (4)

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. (3)

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]

Science (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Other (5)

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

C. P. Yu and H. C. Chang, "Applications of the finite difference frequency domain mode solution method to photonic crystal structures," in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, Ed. 351-400 (Marcel Dekker/CRC Press, Inc., Boca Raton, Florida, 2006).

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

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).

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Figures (11)

Fig. 1.
Fig. 1.

(a) The unit cell of a 2D triangle-arranged air-hole PC. (b) The first BZ of a 2D PC with triangular lattice.

Fig. 2.
Fig. 2.

The band structures of the 2D triangle-arranged air-hole PC with f=0.45 for (a) βa=8 and (b) βa=10.

Fig. 3.
Fig. 3.

The band edge diagrams of the 2D triangle-arranged air-hole PC with (a) f=0.45 and (b) f=0.7.

Fig. 4.
Fig. 4.

(a) The unit cell of a 2D PC with triangle-arranged LC-filled holes in the silica. (b) Schematic definition of rotation angles for the LC molecule.

Fig. 5.
Fig. 5.

(a) The band edge diagram and (b) the band gap map of the 2D PC with triangle-arranged LC-filled holes in the silica.

Fig. 6.
Fig. 6.

The band edge diagrams of the 2D PC with triangle-arranged LC-filled holes in the chalcogenide glass of (a) θc =0° and ϕc =30°, (b) θc =10° and ϕc =30°, (c) θc =20° and ϕc =30°, and (d) θc =90° and ϕc =30°.

Fig. 7.
Fig. 7.

(a) The cross-section of an LC-core PCF. (b) The schematic geometry of the core region for (a).

Fig. 8.
Fig. 8.

(a) The effective index and (b) the confinement loss for the LC-core PCF of a=2.26 µm with four rings of LC-filled holes.

Fig. 9.
Fig. 9.

(a) The effective index and (b) the confinement loss for the LC-core PCF of a=2.26 µm with six rings of LC-filled holes.

Fig. 10.
Fig. 10.

Ey field distribution of the y-polarized fundamental mode for the LC-core PCF of θc =0° and ϕc =30° with a=2.26µm and six rings of LC-filled holes. (a) λ=1.55µm. (b) λ=1.571µm.

Fig. 11.
Fig. 11.

(a) The effective index and (b) the confinement loss for the LC-core PCF of a=1.90 µm with six rings of LC-filled holes.

Equations (49)

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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 ] .
× ( [ 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 ) e j β z + z ̂ Φ z ( x , y ) e j β z
= 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 j β z ̂ × Φ t ) )
+ z ̂ × ( [ 0 0 p xz 0 0 p yz 0 0 0 ] ( j β t × Φ t ) + [ p xx p xy 0 p yx p yy 0 0 0 0 ] ( j β t × z ̂ Φ z β 2 z ̂ × Φ t ) )
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 j β z ̂ × Φ t ) )
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 .
Φ = Φ t e j β z + z ̂ Φ z e j β z = [ { U } T { ϕ t e } { V } T { ϕ t e } { N } T { ϕ z e } ] e j β z
[ K ] { ϕ } k 0 2 [ M ] { ϕ } = { ψ }
{ ϕ } = [ { ϕ t } { ϕ z } ] , { ψ } = [ { ψ t } { ψ 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
+ j β p zx { V } x { V } T j β p zy { V } x { U } T j β p zx { U } y { V } T + j β p zy { U } y { U } T
+ j β p yz { U } { V } T x j β p yz { U } { U } T y j β p xz { V } { V } T x + j β p xz { V } { U } T y
β 2 p xy { U } { V } T + β 2 p yy { U } { U } T + β 2 p xx { V } { V } T β 2 p xy { V } { U } T ] d x d y
[ K tz ] = e [ p zx { V } x { N } T y p zy { V } x { N } T x p zx { U } y { N } T y
+ p zy { U } y { N } T x + j β p yx { U } { N } T y j β p yy { U } { N } T x
j β p xx { V } { N } T y + j β p xy { V } { N } T x ] dx dy
[ K zt ] = e [ p xz { N } y { V } T x p xz { N } y { U } T y p yz { N } x { V } T x
+ p yz { N } x { U } T y + j β p xx { N } y { V } j β p xy { N } y { U }
j β p yx { N } x { V } + j β p yy { N } x { U } ] dx dy
[ K zz ] = e [ p xx { N } y { N } T y p xy { N } y { N } T x
p yx { N } x { N } T y + p yy { N } x { N } T x ] dx dy
[ M tt ] = e [ q xx { U } { U } T + q xy { U } { V } T + q yx { V } { U } T + q yy { V } { V } T ] dx dy
[ M tz ] = e [ q xz { U } { N } T + q yz { V } { N } T ] dx dy
[ M zt ] = e [ q zx { N } { U } T + q zy { N } { V } T ] dx dy
[ M zz ] = e [ q zz { N } { N } T ] dx dy
{ ψ t } = e [ x ̂ ( p zz { V } { V } T x { ϕ t } p zz { V } { U } T y { ϕ t } + p zx { V } { N } T y { ϕ z }
p zy { V } { N } T x { ϕ z } + j β p zx { V } { V } T { ϕ t } j β p zy { V } { U } T { ϕ t } )
+ y ̂ ( p zz { U } { V } T x { ϕ t } + p zz { U } { U } T y { ϕ t } p zx { U } { N } T y { ϕ z }
+ p zy { U } { N } T x { ϕ z } j β p zx { U } { V } T { ϕ t } + j β p zy { U } { U } T { ϕ t } ) ] · n ̂ dl
{ ψ z } = e [ x ̂ ( p yz { N } { V } T x { ϕ t } + p yz { N } { U } T y { ϕ t } p yx { N } { N } T y { ϕ z }
+ p yy { N } { N } T x { ϕ z } j β p yx { N } { V } T { ϕ t } + j β p yy { N } { U } T { ϕ t } )
+ y ̂ ( p xz { N } { V } T x { ϕ t } p xz { N } { U } T y { ϕ t } + p xx { N } { N } T y { ϕ z }
p xy { N } { N } T x { ϕ z } + j β p xx { N } { V } T { ϕ t } j β p xy { N } { U } T { ϕ t } ) ] · n ̂ dl
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 θ c cos ϕ c
ε xz = ε zx = ( n e 2 n o 2 ) sin 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 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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