Abstract

This study develops an efficient mode solver based on pseudospectral eigenvalue algorithm to analyze liquid crystal waveguides with full 3 × 3 anisotropic permittivity tensors. Present formulation yields a cubic eigenvalue matrix equation with an eigenvalue of the propagation constant, and they are solved using an iterative approach following the transformation of the matrix equation to a standard linear eigenvalue equation. The proposed scheme significantly reduces the memory storage and computational time by using only transverse magnetic field components. Although the proposed scheme requires an iterative procedure, the convergent eigenvalues are achieved after performing only four iterations. Therefore, for this scheme, computational efforts remain greatly lower than those for other reported schemes that used at least three field components. For solving the modes of nematic liquid crystal waveguides, the numerical results obtained by the proposed scheme are in good agreement with those calculated by using the finite-element and the finite-difference frequency-domain schemes, thus verifying the applicability of the proposed approach. Furthermore, the mode patterns of liquid crystal waveguides under arbitrary molecular orientations are also characterized in detail.

© 2011 OSA

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  6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009).
    [CrossRef]
  7. P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895–10909 (2009).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
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    [CrossRef]
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    [CrossRef]
  26. J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
    [CrossRef]
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2010 (3)

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010).
[CrossRef] [PubMed]

2009 (7)

M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009).
[CrossRef] [PubMed]

C. C. Huang, “Improved pseudospectral mode solver by prolate spheroidal wave functions for optical waveguides with step-index,” J. Lightwave Technol. 27(5), 597–605 (2009).
[CrossRef]

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895–10909 (2009).
[CrossRef] [PubMed]

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895–10909 (2009).
[CrossRef] [PubMed]

J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009).
[CrossRef]

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
[CrossRef]

B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
[CrossRef]

2008 (3)

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008).
[CrossRef]

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express 16(22), 17915–17934 (2008).
[CrossRef] [PubMed]

2007 (2)

S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express 15(24), 15797–15811 (2007).
[CrossRef] [PubMed]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026703 (2007).
[CrossRef] [PubMed]

2006 (1)

A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006).
[CrossRef]

2005 (2)

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005).
[CrossRef]

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

2002 (1)

2001 (1)

2000 (2)

E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000).
[CrossRef]

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[CrossRef]

1999 (1)

G. Tartarini and H. Renner, “Efficient finite-element analysis of tilted open anisotropic optical channel waveguides,”, ” IEEE Microw. Guid. Wave Lett. 9(10), 389–391 (1999).
[CrossRef]

1996 (1)

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[CrossRef]

1994 (1)

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, 199–205 (1994).
[CrossRef]

1993 (1)

T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. 14(3), 594–605 (1993).
[CrossRef]

1986 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986).
[CrossRef]

1982 (1)

M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982).
[CrossRef]

1973 (1)

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending function interpolation over arbitrary curved element domains,” Numer. Math. 21(2), 109–129 (1973).
[CrossRef]

Abo el Maaty, M. I.

Al-Begain, K.

Ando, T.

Asquini, R.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006).
[CrossRef]

Beccherelli, R.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
[CrossRef]

A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006).
[CrossRef]

Beeckman, J.

Bellini, B.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
[CrossRef]

Cambournac, C.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005).
[CrossRef]

Chang, H.-

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Chang, H. C.

Chen, J. C.

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, 199–205 (1994).
[CrossRef]

Chen, M. Y.

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 Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026703 (2007).
[CrossRef] [PubMed]

Chiang, P.-J.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

D’Alessandro, A.

A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006).
[CrossRef]

Dálessandro, A.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

De Cort, W.

Donisi, B. D.

A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006).
[CrossRef]

Donisi, D.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

Edahiro, T.

M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982).
[CrossRef]

Elston, S. J.

Fernandez, F. A.

Fernandez, F. A. Í.

Gilardi, G.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

Gordon, W. J.

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending function interpolation over arbitrary curved element domains,” Numer. Math. 21(2), 109–129 (1973).
[CrossRef]

Haelterman, M.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005).
[CrossRef]

Hall, C. A.

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending function interpolation over arbitrary curved element domains,” Numer. Math. 21(2), 109–129 (1973).
[CrossRef]

Hameed, M. F. O.

Hayata, K.

M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986).
[CrossRef]

Hsu, S. M.

Huang, C. C.

C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010).
[CrossRef] [PubMed]

C. C. Huang, “Improved pseudospectral mode solver by prolate spheroidal wave functions for optical waveguides with step-index,” J. Lightwave Technol. 27(5), 597–605 (2009).
[CrossRef]

C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express 16(22), 17915–17934 (2008).
[CrossRef] [PubMed]

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

Hutsebaut, X.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005).
[CrossRef]

James, R.

Jüngling, S.

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, 199–205 (1994).
[CrossRef]

Kawachi, M.

M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982).
[CrossRef]

Koshiba, M.

K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(3), 405–413 (2001).
[CrossRef]

M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986).
[CrossRef]

Kriezis, E. E.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008).
[CrossRef]

E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000).
[CrossRef]

Lehoucq, R. B.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[CrossRef]

Nakano, H.

Nakayama, H.

Nasr, A. M.

Neyts, K.

Numata, S.

Obayya, S. S. A.

Renner, H.

G. Tartarini and H. Renner, “Efficient finite-element analysis of tilted open anisotropic optical channel waveguides,”, ” IEEE Microw. Guid. Wave Lett. 9(10), 389–391 (1999).
[CrossRef]

Saitoh, K.

Selleri, S.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[CrossRef]

Shibata, N.

M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982).
[CrossRef]

Sorensen, D. C.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[CrossRef]

Sun, X. H.

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986).
[CrossRef]

Tang, T.

T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. 14(3), 594–605 (1993).
[CrossRef]

Tartarini, G.

G. Tartarini and H. Renner, “Efficient finite-element analysis of tilted open anisotropic optical channel waveguides,”, ” IEEE Microw. Guid. Wave Lett. 9(10), 389–391 (1999).
[CrossRef]

Teng, C.-H.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Trotta, M.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010).
[CrossRef]

Vanbrabant, P. J. M.

Vincetti, L.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[CrossRef]

Wu, C.-L.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Xiao, J. B.

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

Yamauchi, J.

Yang, C.-S.

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Figures (11)

Fig. 1
Fig. 1

The mesh division of an arbitrary interior subdomain r

Fig. 2
Fig. 2

(a) Cross-sectional view of the square LC waveguide with the permittivity tensor [ε] in the core region and refractive index ns of the surrounding substrate. (b) Schematic diagram of the orientation n ^ of the LC director. (c) Division of the computational domain for the LC waveguide.

Fig. 3
Fig. 3

Convergent effective index versus iteration time for different terms of the basis functions n.

Fig. 4
Fig. 4

Mode profiles of (a) |Hx | and (b) |Hy | of the fundamental mode of the square nematic LC waveguide.

Fig. 5
Fig. 5

(a) Cross-sectional view of the nematic LC-core channel waveguide with the core permittivity tensor [ε], substrate refractive index ng , and air refractive index na . (b) Schematic diagram of the twist angle φc and the tilt angle θc of the LC director n ^ .

Fig. 6
Fig. 6

Calculated effective indices versus different values of φc for tilt angles (a) θc = 30° and (b) θc = 60°.

Fig. 7
Fig. 7

|Hx | and |Hy | mode patterns of mode 1 for the tilt angle θc = 30°

Fig. 8
Fig. 8

|Hx | and |Hy | mode patterns of mode 1 for the tilt angle θc = 60°.

Fig. 9
Fig. 9

|Hx | and |Hy | mode patterns of the higher-order modes for the tilt angle θc = 30°.

Fig. 10
Fig. 10

|Hx | and |Hy | mode patterns of the higher-order modes for the tilt angle θc = 60°.

Fig. 11
Fig. 11

Calculated effective indices versus different values of θc for twisted angles (a) φc = 30° and (b) φc = 60°.

Equations (48)

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× ( [ ε ] 1 × H ) ω 2 μ 0 H = 0 ,
[ ε ] = ε 0 [ ε r ] = ε 0 [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] ,
[ P x x ( β ) P x y ( β ) P y x ( β ) P y y ( β ) ] [ H x H y ] = β 2 [ η y y η y x η x y η x x ] [ H x H y ] ,
P x x ( β ) H x = η y y 2 H x x 2 + η z z 2 H x y 2 η y x 2 H x y x + k 0 2 H x + j β ( η y z + η z y ) H x y ...                   + j β ( η z x y η z y x ) 2 H x y x ,
P x y ( β ) H y = ( η y y η z z ) 2 H y x y η y x 2 H y y 2 j β ( η y z H y x + η z x H y y ) ...                    + j β ( η z x y η z y x ) 2 H y y 2 ,
P y x ( β ) H x = ( η x x η z z ) 2 H x y x η x y 2 H x x 2 j β ( η x z H x y + η z y H x x ) ...                    + j β ( η z y x η z x y ) 2 H x x 2 ,
P y y ( β ) H y = η z z 2 H y x 2 + η x x 2 H y y 2 + η x y 2 H y x y + k 0 2 H y + j β ( η x z + η z x ) H y x ...                    + j β ( η z y x η z x y ) 2 H y x y ,
η x x = ( ε y y ε z z ε y z ε z y ) / Δ ,
η x y = ( ε x z ε z y ε x y ε z z ) / Δ ,
η x z = ( ε x y ε y z ε x z ε y y ) / Δ ,
η y x = ( ε y z ε z x ε y x ε z z ) / Δ ,
η y y = ( ε x x ε z z ε x z ε z x ) / Δ ,
η y z = ( ε x z ε y x ε x x ε y z ) / Δ ,
η z x = ( ε y x ε z y ε y y ε z x ) / Δ ,
η z y = ( ε x y ε z x ε x x ε z y ) / Δ ,
η z z = ( ε x x ε y y ε x y ε y x ) / Δ ,
Δ = ε x x ε y y ε z z + ε x y ε y z ε z x + ε x z ε y x ε z y ε x z ε y y ε z x ε x y ε y x ε z z ε x x ε y z ε z y .
H z = 1 j β ( H x x + H y y ) ,
E z = η z x [ ( 1 β ) y ( H x x + H y y ) + β H y ] + η z y [ ( 1 β ) x ( H x x + H y y ) β H x ] + j η z z ( H x y H y x ) ,
H x r ( x , y ) = p = 0 n x q = 0 n y θ p r ( x ) ψ q r ( y ) H x , p q r ,
H y r ( x , y ) = p = 0 n x q = 0 n y θ p r ( x ) ψ q r ( y ) H y , p q r ,
[ P x x r ( β ) P x y r ( β ) P y x r ( β ) P y y r ( β ) ] [ H x r H y r ] = β 2 [ η y y r η y x r η x y r η x x r ] [ H x r H y r ]
P u v r ( β ) = [ ( D ˜ 00 n x 1 , 1 ) u v r ( D ˜ 01 n x 1 , 1 ) u v r ( D ˜ 0 n y n x 1 , 1 ) u v r ( D ˜ 00 n x 1 , 2 ) u v r ( D ˜ 01 n x 1 , 2 ) u v r ( D ˜ 0 n y n x 1 , 2 ) u v r ( D ˜ 00 n x 1 , n y 1 ) u v r ( D ˜ 01 n x 1 , 2 ) u v r ( D ˜ 0 n y n x 1 , 2 ) u v r ] , u v = x x , x y , y x ,   and   y y ,
( D ˜ 0 s n x 1 , t ) u v r = [ ( D 0 s 1 , t ) u v r ( D 1 s 1 , t ) u v r ( D n x s 1 , t ) u v r ( D 0 s 2 , t ) u v r ( D 1 s 2 , t ) u v r ( D n x s 2 , 1 ) u v r ( D 0 s n x 1 , t ) u v r ( D 1 s n x 1 , t ) u v r     ( D n x s n x 1 , t ) u v r ] , s = 0 , 1 , ... , n y , t = 0 , 1 , ... , n x ,
( D p q i , j ) x x r = η y y r θ p r ( 2 ) ( x i r ) ψ q r ( y j r ) + η z z r θ p r ( x i r ) ψ q r ( 2 ) ( y j r ) η y x r θ p r ( 1 ) ( x i r ) ψ q r ( 1 ) ( y j r ) + k 0 2 θ p r ( x i r ) ψ q r ( y j r ) ...                + j β ( η y z r + η z y r ) θ p r ( x i r ) ψ q r ( 1 ) ( y j r ) + j β ( η z x r θ p r ( 1 ) ( x i r ) ψ q r ( 2 ) ( y j r ) η z y r θ p r ( 2 ) ( x i r ) ψ q r ( 1 ) ( y j r ) ) ,
( D p q i , j ) x y r = ( η y y r η z z r ) θ p r ( 1 ) ( x i r ) ψ q r ( 1 ) ( y j r ) η y x r θ p r ( x i r ) ψ q r ( 2 ) ( y j r ) j β ( η y z r θ p r ( 1 ) ( x i r ) ψ q r ( y j r ) ...                + η z x r θ p r ( x i r ) ψ q r ( 1 ) ( y j r ) ) + j β ( η z x r θ p r ( x i r ) ψ q r ( 3 ) ( y j r ) η z y r θ p r ( 1 ) ( x i r ) ψ q r ( 2 ) ( y j r ) ) ,
( D p q i , j ) y x r = ( η x x r η z z r ) θ p r ( 1 ) ( x i r ) ψ q r ( 1 ) ( y j r ) η x y r θ p r ( 2 ) ( x i r ) ψ q r ( y j r ) j β ( η x z r θ p r ( x i r ) ψ q r ( 1 ) ( y j r ) ...                + η z y r θ p r ( 1 ) ( x i r ) ψ q r ( y j r ) ) + j β ( η z y r θ p r ( 3 ) ( x i r ) ψ q r ( y j r ) η z x r θ p r ( 2 ) ( x i r ) ψ q r ( 1 ) ( y j r ) ) ,
( D p q i , j ) y y r = η z z r θ p r ( 2 ) ( x i r ) ψ q r ( y j r ) + η x x r θ p r ( x i r ) ψ q r ( 2 ) ( y j r ) + η x y r θ p r ( 1 ) ( x i r ) ψ q r ( 1 ) ( y j r ) + k 0 2 θ p r ( x i r ) ψ q r ( y j r ) ...                 + j β ( η x z r + η z x r ) θ p r ( 1 ) ( x i r ) ψ q r ( y j r ) + j β ( η z y r θ p r ( 2 ) ( x i r ) ψ q r ( 1 ) ( y j r ) η z x r θ p r ( 1 ) ( x i r ) ψ q r ( 2 ) ( y j r ) ) ,
[ Q 1 ( β ) 0 0 0 0 Q 2 ( β ) 0 0 0 0 0 0 0 0 Q m ( β ) ] [ H 1 H 2 H m ] = β 2 [ η 1 0 0 0 0 η 2 0 0 0 0 0 0 0 0 η m ] [ H 1 H 2 H m ]
Q r ( β ) = [ P x x r ( β ) P x y r ( β ) P y x r ( β ) P y y r ( β ) ] , Η r = [ H x r H y r ] ,     η r = [ η y y r η y x r η x y r η x x r ] , ( r = 1 , 2 , 3... m ) .
[ A x + A y + ] [ H x + H y + ] = [ A x A y ] [ H x H y ] ,
A k ± = [ A k ± ( ψ 0 ( y 0 ± ) ) A k ± ( ψ 1 ( y 0 ± ) ) A k ± ( ψ n y ( y 0 ± ) ) A k ± ( ψ 0 ( y 1 ± ) ) A k ± ( ψ 1 ( y 1 ± ) ) A k ± ( ψ n y ( y 1 ± ) ) A k ± ( ψ 0 ( y n y ± ) ) A k ± ( ψ 1 ( y n y ± ) ) A k ± ( ψ n y ( y n y ± ) ) ] , k = x , y ,
A x + ( ψ i ( y j + ) ) = [ θ 0 ( 1 ) ( x 0 + ) ψ i ( y j + ) θ 1 ( 1 ) ( x 0 + ) ψ i ( y j + ) θ n x ( 1 ) ( x 0 + ) ψ i ( y j + ) ] ,
A y + ( ψ i ( y j + ) ) = [ θ 0 ( x 0 + ) ψ i ( 1 ) ( y j + ) θ 1 ( x 0 + ) ψ i ( 1 ) ( y j + ) θ n x ( x 0 + ) ψ i ( 1 ) ( y j + ) ] ,
A x ( ψ i ( y j ) ) = [ θ 0 ( 1 ) ( x n x ) ψ i ( y j ) θ 1 ( 1 ) ( x n x ) ψ i ( y j ) θ n x ( 1 ) ( x n x ) ψ i ( y j ) ] ,
A y ( ψ i ( y j ) ) = [ θ 0 ( x n x ) ψ i ( 1 ) ( y j ) θ 1 ( x n x ) ψ i ( 1 ) ( y j ) θ n x ( x n x ) ψ i ( 1 ) ( y j ) ] ,
[ B x + B y + ] [ H x + H y + ] = [ B x B y ] [ H x H y ] ,
B k ± = [ B k ± ( ψ 0 ( y 0 ± ) ) B k ± ( ψ 1 ( y 0 ± ) )     B k ± ( ψ n y ( y 0 ± ) )   B k ± ( ψ n y ( y 0 ± ) )   B k ± ( ψ 1 ( y 1 ± ) )     B k ± ( ψ n y ( y 1 ± ) ) B k ± ( ψ 0 ( y n y ± ) ) B k ± ( ψ 1 ( y n y ± ) )     B k ± ( ψ n y ( y n y ± ) ) ] , k = x , y ,
B x + ( ψ i ( y j + ) ) = [ η z x ( 1 β ) θ 0 ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) + η z y ( 1 β ) θ 0 ( 2 ) ( x 0 + ) ψ i ( y j + ) β η z y δ i j + j η z z θ 0 ( x 0 + ) ψ i ( 1 ) ( y j + ) η z x ( 1 β ) θ 1 ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) + η z y ( 1 β ) θ 1 ( 2 ) ( x 0 + ) ψ i ( y j + ) β η z y δ i j + j η z z θ 1 ( x 0 + ) ψ i ( 1 ) ( y j + ) η z x ( 1 β ) θ n x ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) + η z y ( 1 β ) θ n x ( 2 ) ( x 0 + ) ψ i ( y j + ) β η z y δ i j + j η z z θ n x ( x 0 + ) ψ i ( 1 ) ( y j + ) ] T ,
B y + ( ψ i ( y j + ) ) = [ η z x ( 1 β ) θ 0 ( x 0 + ) ψ i ( 2 ) ( y j + ) + β η z x δ i j + η z y ( 1 β ) θ 0 ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) j η z z θ 0 ( 1 ) ( x 0 + ) ψ i ( y j + ) η z x ( 1 β ) θ 1 ( x 0 + ) ψ i ( 2 ) ( y j + ) + β η z x δ i j + η z y ( 1 β ) θ 1 ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) j η z z θ 1 ( 1 ) ( x 0 + ) ψ i ( y j + ) η z x ( 1 β ) θ n x ( x 0 + ) ψ i ( 2 ) ( y j + ) + β η z x δ i j + η z y ( 1 β ) θ n x ( 1 ) ( x 0 + ) ψ i ( 1 ) ( y j + ) j η z z θ n x ( 1 ) ( x 0 + ) ψ i ( y j + ) ] T ,
B x ( ψ i ( y j ) ) = [ η z x ( 1 β ) θ 0 ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) + η z y ( 1 β ) θ 0 ( 2 ) ( x n x ) ψ i ( y j ) β η z y δ i j + j η z z θ 0 ( x n x ) ψ i ( 1 ) ( y j ) η z x ( 1 β ) θ 1 ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) + η z y ( 1 β ) θ 1 ( 2 ) ( x n x ) ψ i ( y j ) β η z y δ i j + j η z z θ 1 ( x n x ) ψ i ( 1 ) ( y j ) η z x ( 1 β ) θ n x ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) + η z y ( 1 β ) θ n x ( 2 ) ( x n x ) ψ i ( y j ) β η z y δ i j + j η z z θ n x ( x n x ) ψ i ( 1 ) ( y j ) ] T ,
B y ( ψ i ( y j ) ) = [ η z x ( 1 β ) θ 0 ( x n x ) ψ i ( 2 ) ( y j ) + β η z x δ i j + η z y ( 1 β ) θ 0 ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) j η z z θ 0 ( 1 ) ( x n x ) ψ i ( y j ) η z x ( 1 β ) θ 1 ( x n x ) ψ i ( 2 ) ( y j ) + β η z x δ i j + η z y ( 1 β ) θ 1 ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) j η z z θ 1 ( 1 ) ( x n x ) ψ i ( y j ) η z x ( 1 β ) θ n x ( x n x ) ψ i ( 2 ) ( y j ) + β η z x δ i j + η z y ( 1 β ) θ n x ( 1 ) ( x n x ) ψ i ( 1 ) ( y j ) j η z z θ n x ( 1 ) ( x n x ) ψ i ( y j ) ] T ,
θ p ( x ) = ( 1 ) p + 1 ( 1 x 2 ) T v ' ( x ) c p n 2 ( x x p ) ,
c p = { 2 ,   if  p =0,   N 1,  if 1 p N -1
θ p ( α x ) = x L v ( α x ) α ( x x p ) [ x L v ' ( α x ) ] | x = x p e α ( x x p ) / 2 ,
α = max 0 i n { x i } M
[ ε ] = ε 0 [ n o 2 0 0 0 n o 2 + Δ ε sin 2 θ c Δ ε sin θ c cos θ c 0 Δ ε sin θ c cos θ c n o 2 + Δ ε cos 2 θ c ]
[ ε ] = ε 0 [ n o 2 + Δ ε sin 2 θ c cos 2 φ c Δ ε sin 2 θ c sin φ c cos φ c Δ ε sin θ c cos θ c cos φ c Δ ε sin 2 θ c sin φ c cos φ c n o 2 + Δ ε sin 2 θ c sin 2 φ c Δ ε sin θ c cos θ c sin φ c Δ ε sin θ c cos θ c cos φ c Δ ε sin θ c cos θ c sin φ c n o 2 + Δ ε cos 2 θ c ] ,

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