Abstract

Simulation of light propagation within nematic liquid-crystal (LC) devices is considered, of which the director is aligned normal to the z axis. A three-dimensional full-vector finite-difference beam propagation method for an anisotropic medium is presented and an alternating direction implicit scheme is adopted. Simulations of light propagation in a bulk polarization converter, a waveguide with a LC covering layer, and an integrated polarization splitter and optical switch are presented. Comparison with an existing simulation method is carried out for beam behavior within the bulk polarization converter. The effect of strong surface anchoring of a LC cell on the beam behaviors within the integrated switch is also demonstrated.

© 2006 Optical Society of America

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  1. N. A. Riza and S. Yuan, "Reconfigurable wavelength add-drop filtering based on a Banyan network topology and ferroelectric liquid crystal fiber-optic switches," J. Lightwave Technol. 17, 1575-1584 (1999).
    [CrossRef]
  2. K. Wu, J. Liu, and Y. Chen, "Optical attenuator using polarization modulation and a feedback controller," U.S. patent 5,963,291 (5 October 1999).
  3. K. Hirabayashi and C. Amano, "Liquid-crystal polarization controller arrays on planar waveguide circuits," IEEE Photon. Technol. Lett. 14, 504-506 (2002).
    [CrossRef]
  4. Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
    [CrossRef]
  5. W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
    [CrossRef]
  6. K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).
  7. D. B. Walker, E. N. Glytsis, and T. K. Gaylord, "Ferroelectric liquid-crystal waveguide modulation based on a switchable uniaxial-uniaxial interface" Appl. Opt. 35, 3016-3030 (1996).
    [CrossRef] [PubMed]
  8. C. Y. Liu and L. W. Chen, "Tunable photonic-crystal waveguide Mach-Zehnder interferometer achieved by nematic liquid-crystal phase modulation," Opt. Express 12, 2616-2624 (2004).
    [CrossRef] [PubMed]
  9. A. Fratalocchi, R. Asquini, and G. Assanto, "Integrated electro-optic switch in liquid crystals," Opt. Express 13, 32-37 (2005).
    [CrossRef] [PubMed]
  10. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley Interscience, 1999).
  11. D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation," J. Opt. Soc. Am. 62, 505-510 (1972).
    [CrossRef]
  12. B. Witzigmann, P. Regli, and W. Fichtner, "Rigorous electromagnetic simulation of liquid crystals," J. Opt. Soc. Am. A 15, 753-757 (1998).
    [CrossRef]
  13. E. E. Kriezis and S. J. Elston, "Finite-difference time domain method for light wave propagation within liquid crystal devices," Opt. Commun. 165, 99-105 (1999).
    [CrossRef]
  14. E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2D finite-difference time-domain method," Opt. Commun. 177, 69-77 (2000).
    [CrossRef]
  15. E. E. Kriezis and S. J. Elston, "Wide-angle beam propagation method for liquid-crystal device calculations," Appl. Opt. 39, 5707-5714 (2000).
    [CrossRef]
  16. C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
    [CrossRef]
  17. Y. L. Hsueh, M. C. Yang, and H. C. Chang, "Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 17, 2389-2397 (1999).
    [CrossRef]
  18. G. R. Hadley, "Transparent boundary condition for beam propagation," IEEE J. Quantum Electron. 28, 363-370 (1992).
    [CrossRef]
  19. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
    [CrossRef]
  20. S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
    [CrossRef]
  21. Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
    [CrossRef]

2005 (1)

2004 (2)

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

C. Y. Liu and L. W. Chen, "Tunable photonic-crystal waveguide Mach-Zehnder interferometer achieved by nematic liquid-crystal phase modulation," Opt. Express 12, 2616-2624 (2004).
[CrossRef] [PubMed]

2002 (1)

K. Hirabayashi and C. Amano, "Liquid-crystal polarization controller arrays on planar waveguide circuits," IEEE Photon. Technol. Lett. 14, 504-506 (2002).
[CrossRef]

2001 (1)

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

2000 (2)

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2D finite-difference time-domain method," Opt. Commun. 177, 69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Wide-angle beam propagation method for liquid-crystal device calculations," Appl. Opt. 39, 5707-5714 (2000).
[CrossRef]

1999 (3)

1998 (1)

1996 (3)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).

D. B. Walker, E. N. Glytsis, and T. K. Gaylord, "Ferroelectric liquid-crystal waveguide modulation based on a switchable uniaxial-uniaxial interface" Appl. Opt. 35, 3016-3030 (1996).
[CrossRef] [PubMed]

1995 (1)

W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
[CrossRef]

1994 (2)

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

1992 (1)

G. R. Hadley, "Transparent boundary condition for beam propagation," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

1972 (1)

D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation," J. Opt. Soc. Am. 62, 505-510 (1972).
[CrossRef]

Amano, C.

K. Hirabayashi and C. Amano, "Liquid-crystal polarization controller arrays on planar waveguide circuits," IEEE Photon. Technol. Lett. 14, 504-506 (2002).
[CrossRef]

Asquini, R.

Assanto, G.

Berreman, D. W.

D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation," J. Opt. Soc. Am. 62, 505-510 (1972).
[CrossRef]

Chang, H. C.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Chen, J. C.

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

Chen, L. W.

Chen, Y.

K. Wu, J. Liu, and Y. Chen, "Optical attenuator using polarization modulation and a feedback controller," U.S. patent 5,963,291 (5 October 1999).

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Chuang, W. C.

K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).

Dovgalets, S.

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

Elston, S. J.

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2D finite-difference time-domain method," Opt. Commun. 177, 69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Wide-angle beam propagation method for liquid-crystal device calculations," Appl. Opt. 39, 5707-5714 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Finite-difference time domain method for light wave propagation within liquid crystal devices," Opt. Commun. 165, 99-105 (1999).
[CrossRef]

Farrell, G.

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

Fichtner, W.

Fratalocchi, A.

Gaylord, T. K.

Glytsis, E. N.

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley Interscience, 1999).

Hadley, G. R.

G. R. Hadley, "Transparent boundary condition for beam propagation," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

He, S.

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

Hirabayashi, K.

K. Hirabayashi and C. Amano, "Liquid-crystal polarization controller arrays on planar waveguide circuits," IEEE Photon. Technol. Lett. 14, 504-506 (2002).
[CrossRef]

Hsueh, Y. L.

Huang, N.

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Jungling, S.

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

Kriezis, E. E.

E. E. Kriezis and S. J. Elston, "Wide-angle beam propagation method for liquid-crystal device calculations," Appl. Opt. 39, 5707-5714 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2D finite-difference time-domain method," Opt. Commun. 177, 69-77 (2000).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Finite-difference time domain method for light wave propagation within liquid crystal devices," Opt. Commun. 165, 99-105 (1999).
[CrossRef]

Lee, W. Y.

K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).

W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
[CrossRef]

Lin, J. S.

W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
[CrossRef]

Lin, K. C.

K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).

Liu, C. Y.

Liu, J.

K. Wu, J. Liu, and Y. Chen, "Optical attenuator using polarization modulation and a feedback controller," U.S. patent 5,963,291 (5 October 1999).

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

Panarin, Y.

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

Regli, P.

Riza, N. A.

Semenova, Y.

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

Walker, D. B.

Wang, Q.

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

Wang, S. Y.

W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
[CrossRef]

Witzigmann, B.

Wu, K.

K. Wu, J. Liu, and Y. Chen, "Optical attenuator using polarization modulation and a feedback controller," U.S. patent 5,963,291 (5 October 1999).

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Yang, M. C.

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley Interscience, 1999).

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

Yu, F.

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

Yuan, S.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

G. R. Hadley, "Transparent boundary condition for beam propagation," IEEE J. Quantum Electron. 28, 363-370 (1992).
[CrossRef]

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer (PML) for the beam propagation method," IEEE Photon. Technol. Lett. 8, 649-651 (1996).
[CrossRef]

K. Hirabayashi and C. Amano, "Liquid-crystal polarization controller arrays on planar waveguide circuits," IEEE Photon. Technol. Lett. 14, 504-506 (2002).
[CrossRef]

J. Lightwave Technol. (5)

N. A. Riza and S. Yuan, "Reconfigurable wavelength add-drop filtering based on a Banyan network topology and ferroelectric liquid crystal fiber-optic switches," J. Lightwave Technol. 17, 1575-1584 (1999).
[CrossRef]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Y. L. Hsueh, M. C. Yang, and H. C. Chang, "Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 17, 2389-2397 (1999).
[CrossRef]

W. Y. Lee, J. S. Lin, and S. Y. Wang, "A novel vertical Δκ directional coupler switch using liquid crystals," J. Lightwave Technol. 13, 49-54 (1995).
[CrossRef]

K. C. Lin, W. C. Chuang, and W. Y. Lee, "Proposal and analysis of an ultrashort directional coupler polarization splitter with an NLC coupling layer," J. Lightwave Technol. 14, 2517-2553 (1996).

J. Opt. Soc. Am. (1)

D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation," J. Opt. Soc. Am. 62, 505-510 (1972).
[CrossRef]

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

Mol. Cryst. Liq. Cryst. (1)

Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, "Liquid crystal based optical switches," Mol. Cryst. Liq. Cryst. 413, 2521-2534 (2004).
[CrossRef]

Opt. Commun. (2)

E. E. Kriezis and S. J. Elston, "Finite-difference time domain method for light wave propagation within liquid crystal devices," Opt. Commun. 165, 99-105 (1999).
[CrossRef]

E. E. Kriezis and S. J. Elston, "Light wave propagation in liquid crystal displays by the 2D finite-difference time-domain method," Opt. Commun. 177, 69-77 (2000).
[CrossRef]

Opt. Eng. (1)

Q. Wang, S. He, F. Yu, and N. Huang, "Iterative finite-difference method for calculating the distribution of a liquid-crystal director," Opt. Eng. 40, 2552-2557 (2001).
[CrossRef]

Opt. Express (2)

Other (2)

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley Interscience, 1999).

K. Wu, J. Liu, and Y. Chen, "Optical attenuator using polarization modulation and a feedback controller," U.S. patent 5,963,291 (5 October 1999).

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Figures (5)

Fig. 1
Fig. 1

Normalized power of the polarization component versus the propagation distance within a twisted nematic LC cell.

Fig. 2
Fig. 2

Full-vector eigenmodes of a waveguide with a parallel-aligned LC covering layer: (a) E x of the TE mode, (b) E y of the TE mode, (c) E x of the TM mode, (d) E y of the TM mode.

Fig. 3
Fig. 3

Schematic configuration of an integrated polarization splitter and optical switch.

Fig. 4
Fig. 4

Light propagation within the directional coupler for three cases: (a) TE mode and (b) TM mode for case one; (c) TE mode and (d) TM mode for case two; (e) TE mode and (f) TM mode for case three.

Fig. 5
Fig. 5

Normalized power in waveguide B versus propagation distance for the three cases.

Equations (25)

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

× × E k 2 ϵ ̂ E = 0 ,
ϵ ̂ = [ ϵ x x ϵ x y 0 ϵ y x ϵ y y 0 0 0 ϵ z z ] = [ n o 2 + ( n e 2 n o 2 ) cos 2 φ ( n e 2 n o 2 ) cos φ sin φ 0 ( n e 2 n o 2 ) cos φ sin φ n o 2 + ( n e 2 n o 2 ) sin 2 φ 0 0 0 n o 2 ] ,
2 E x y 2 + 2 E x z 2 + k 2 [ ϵ x x E x + ϵ x y E y ] = 2 E y x y x [ 1 ϵ z z x [ ϵ x x E x + ϵ x y E y ] ] x [ 1 ϵ z z y [ ϵ y x E x + ϵ y y E y ] ] ,
2 E y x 2 + 2 E y z 2 + k 2 [ ϵ y x E x + ϵ y y E y ] = 2 E x x y y [ 1 ϵ z z x [ ϵ x x E x + ϵ x y E y ] ] y [ 1 ϵ z z y [ ϵ y x E x + ϵ y y E y ] ] .
z ( [ E ̂ x E ̂ y ] ) = [ P x x P x y P y x P y y ] [ E ̂ x E ̂ y ] ,
P x x E ̂ x = j 2 k n 0 { 2 E ̂ x y 2 + x [ 1 ϵ z z x ( ϵ x x E ̂ x ) ] + x [ 1 ϵ z z y ( ϵ y x E ̂ x ) ] + k 2 [ ( ϵ x x n 0 2 ) E ̂ x ] } ,
P x y E ̂ y = j 2 k n 0 { k 2 [ ϵ x y E ̂ y ] + x [ E ̂ y y + 1 ϵ z z y ( ϵ y y E ̂ y ) + 1 ϵ z z x ( ϵ x y E ̂ y ) ] } ,
P y x E ̂ x = j 2 k n 0 { k 2 [ ϵ y x E ̂ x ] + y [ E ̂ x x + 1 ϵ z z x ( ϵ x x E ̂ x ) + 1 ϵ z z y ( ϵ y x E ̂ x ) ] } ,
P y y E ̂ y = j 2 k n 0 { 2 E ̂ y x 2 + y [ 1 ϵ z z y ( ϵ y y E ̂ y ) ] + y [ 1 ϵ z z x ( ϵ x y E ̂ y ) ] + k 2 [ ( ϵ y y n 0 2 ) E ̂ y ] } .
z ( [ E ̂ x E ̂ y ] ) = [ P x x x + P x x y + C x y P x y P y x P y y x + P y y y + C y x ] [ E ̂ x E ̂ y ] ,
P x x x E ̂ x = x [ 1 ϵ z z x ( ϵ x x E ̂ x ) ] + k 2 2 [ ( ϵ x x n 0 2 ) E ̂ x ] ,
P x x y E ̂ x = 2 E ̂ x y 2 + k 2 2 [ ( ϵ x x n 0 2 ) E ̂ x ] ,
C x y E ̂ x = x [ 1 ϵ z z y ( ϵ y x E ̂ x ) ] .
P y y x E ̂ y = 2 E ̂ y x 2 + k 2 2 [ ( ϵ y y n 0 2 ) E ̂ y ] ,
P y y y E ̂ y = y [ 1 ϵ z z y ( ϵ y y E ̂ t ) ] + k 2 2 [ ( ϵ y y n 0 2 ) E ̂ y ] ,
C y x E ̂ y = y [ 1 ϵ z z x ( ϵ x y E ̂ y ) ] .
1 Δ z { [ E ̂ x E ̂ y ] n + 1 [ E ̂ x E ̂ y ] n } = 1 2 { [ P x x x P x y 0 P y y x ] + [ P x x y 0 P y x P y y y ] + [ C x y 0 0 C y x ] } { [ E ̂ x E ̂ y ] n + 1 [ E ̂ x E ̂ y ] n } .
1 Δ z { [ E ̂ x E ̂ y ] n + 1 [ E ̂ x E ̂ y ] n } = 1 2 { [ P x x x P x y 0 P y y x ] + [ P x x y 0 P y x P y y y ] } { [ E ̂ x E ̂ y ] n + 1 + [ E ̂ x E ̂ y ] n } + [ C x y 0 0 C y x ] [ E ̂ x E ̂ y ] n .
[ E ̂ x E ̂ y ] n + 1 = ( 1 + Δ z 2 [ P x x y 0 P y x P y y y ] ) ( 1 + Δ z 2 [ P x x x P x y 0 P y y x ] ) ( 1 Δ z 2 [ P x x y 0 P y x P y y y ] ) ( 1 + Δ z 2 [ P x x x P x y 0 P y y x ] ) [ E ̂ x E ̂ y ] n ¯ ,
[ E ̂ x E ̂ y ] n ¯ = ( 1 + Δ z [ C x y 0 0 C y x ] ) [ E ̂ x E ̂ y ] n .
E ̂ y n + ( 1 2 ) Δ z 2 P y y x E ̂ y n + ( 1 2 ) = E ̂ y n ¯ + Δ z 2 P y y y E ̂ y n ¯ + Δ z 2 P y x E ̂ x n ¯ ,
E ̂ x n + 1 2 Δ z 2 P x x x E x n + ( 1 2 ) = E ̂ x n ¯ + Δ z 2 P x x y E ̂ y n ¯ + Δ z 2 P x y E ̂ y n + ( 1 2 ) ,
E ̂ x n + 1 Δ z 2 P x x y E ̂ y n + 1 = E ̂ x n + ( 1 2 ) + Δ z 2 P x x x E x n + ( 1 2 ) + Δ z 2 P x y E ̂ y n + ( 1 2 ) ,
E ̂ y n + 1 Δ z 2 P y y y E ̂ y n + 1 = E ̂ y n + ( 1 2 ) + Δ z 2 P y y x E ̂ y n + ( 1 2 ) + Δ z 2 P y x E ̂ x n + 1 .
x [ 1 ϵ z z y ( ϵ y x E x ) ] , y [ 1 ϵ z z x ( ϵ x y E y ) ] , x [ 1 ϵ z z x ( ϵ x y E y ) ] , y [ 1 ϵ z z y ( ϵ y x E x ) ]

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