Abstract

Accuracy degradation at a dielectric interface in simulations using the finite-difference time-domain method can be prevented by assigning suitable effective permittivities at the nodes in the vicinity of the interface. The effective permittivities with exact second-order accuracy at the interface inclined to the Yee-lattice axis are analytically derived for what we believe to be the first time. We discuss two interfaces with different inclined angles between their normal and the Yee-lattice axis in the case of two-dimensional TE polarization. The tangent of the angle is 1 for one interface and 1/2 for the other. With the derived effective permittivities, reflection and transmission at the interface are simulated with second-order accuracy with respect to cell size. The accuracy is demonstrated by numerical examples.

© 2010 Optical Society of America

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References

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  1. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
    [CrossRef]
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  3. T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
    [CrossRef]
  4. P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
    [CrossRef]
  5. I. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations,” IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990).
    [CrossRef]
  6. K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
    [CrossRef]
  7. M. Celuch-Marcysiak and W. K. Gwarek, “Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits,” in Proceedings of the 24th European Microwave Conference (Wiley, 1994), pp. 1530-1535.
    [CrossRef]
  8. N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
    [CrossRef]
  9. P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727-3737 (2000).
    [CrossRef]
  10. M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
    [CrossRef]
  11. A. Mohammadi, H. Nadgaran, and M. Agio, “Contour-path effective permittivities for the two-dimensional finite-difference time-domain method,” Opt. Express 13, 10367-10381(2005).
    [CrossRef] [PubMed]
  12. T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
    [CrossRef]
  13. T. Hirono and Y. Yoshikuni, “Accurate modeling of dielectric interfaces by the effective permittivities for the fourth-order symplectic finite-difference time-domain method,” Appl. Opt. 46, 1514-1524 (2007).
    [CrossRef] [PubMed]
  14. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974), pp. 7-17.
  15. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972-2974 (2006).
    [CrossRef] [PubMed]
  16. G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal, anisotropic dielectrics,” J. Comput. Phys. 226, 1085-1101 (2007).
    [CrossRef]

2007 (2)

2006 (1)

2005 (1)

2003 (1)

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

2001 (1)

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
[CrossRef]

2000 (2)

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727-3737 (2000).
[CrossRef]

1997 (1)

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

1992 (2)

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
[CrossRef]

1990 (1)

I. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations,” IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

Agio, M.

Bermel, P.

Burr, G. W.

Cary, J. R.

G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal, anisotropic dielectrics,” J. Comput. Phys. 226, 1085-1101 (2007).
[CrossRef]

Celuch-Marcysiak, M.

M. Celuch-Marcysiak and W. K. Gwarek, “Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits,” in Proceedings of the 24th European Microwave Conference (Wiley, 1994), pp. 1530-1535.
[CrossRef]

Ditkowski, A.

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
[CrossRef]

Dridi, K. H.

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
[CrossRef]

Farjadpour, A.

Fujii, M.

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

Gao, B.-C.

Gwarek, W. K.

M. Celuch-Marcysiak and W. K. Gwarek, “Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits,” in Proceedings of the 24th European Microwave Conference (Wiley, 1994), pp. 1530-1535.
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Harms, P. H.

P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
[CrossRef]

Hesthaven, J. S.

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
[CrossRef]

Hirono, T.

T. Hirono and Y. Yoshikuni, “Accurate modeling of dielectric interfaces by the effective permittivities for the fourth-order symplectic finite-difference time-domain method,” Appl. Opt. 46, 1514-1524 (2007).
[CrossRef] [PubMed]

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

Hoefer, W. J. R.

I. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations,” IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990).
[CrossRef]

Houshmand, B.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Ibanescu, M.

Itoh, T.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Jurgens, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

Kaneda, N.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Kim, I. S.

I. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations,” IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990).
[CrossRef]

Lee, J. F.

P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
[CrossRef]

Liou, K. N.

Lui, W. W.

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

Lukashevich, D.

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974), pp. 7-17.

Mishchenko, M. I.

Mittra, R.

P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
[CrossRef]

Mohammadi, A.

Moore, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

Nadgaran, H.

Rodriguez, A.

Roundy, D.

Russar, P.

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

Sakagami, I.

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

Seki, S.

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

Shibata, Y.

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

Taflove, A.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Umashankar, K.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

Werner, G. R.

G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal, anisotropic dielectrics,” J. Comput. Phys. 226, 1085-1101 (2007).
[CrossRef]

Yang, P.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

Yoshikuni, Y.

T. Hirono and Y. Yoshikuni, “Accurate modeling of dielectric interfaces by the effective permittivities for the fourth-order symplectic finite-difference time-domain method,” Appl. Opt. 46, 1514-1524 (2007).
[CrossRef] [PubMed]

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

Appl. Opt. (2)

IEEE Microw. Wireless Compon. Lett. (1)

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russar, “Convergence of FDTD and wavelet-collocation modeling of curved dielectric interface with the effective dielectric constant technique,” IEEE Microw. Wireless Compon. Lett. 13, 469-471 (2003).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359-361(2000).
[CrossRef]

IEEE Trans. Antennas Propagat. (3)

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propagat. 49, 749-756 (2001).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357-366 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

P. H. Harms, J. F. Lee, and R. Mittra, “A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities,” IEEE Trans. Microwave Theory Tech. 40, 741-746 (1992).
[CrossRef]

I. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations,” IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990).
[CrossRef]

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

J. Comput. Phys. (1)

G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal, anisotropic dielectrics,” J. Comput. Phys. 226, 1085-1101 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (3)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974), pp. 7-17.

M. Celuch-Marcysiak and W. K. Gwarek, “Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits,” in Proceedings of the 24th European Microwave Conference (Wiley, 1994), pp. 1530-1535.
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the analyzed y z plane for the effective permittivities used for the dielectric interface orthogonal to the ( 2 1 / 2 , 2 1 / 2 ) direction.

Fig. 2
Fig. 2

Schematic drawing of the analyzed y z plane for the effective permittivities used for the dielectric interface orthogonal to the ( ( 4 / 5 ) 1 / 2 , 5 1 / 2 ) direction.

Fig. 3
Fig. 3

Phase velocity errors in the simulations of wave propagation in the TM 2 and TM 5 modes of a 2D step-profile waveguide. The effective permittivities ε i 1 ( 1 1 ) and ε i 2 ( 1 1 ) were assigned to the nodes in the vicinity of the core–cladding interfaces. The offset ratio of one core–cladding interface is 0. The offset ratio at the other interface is described in the legend.

Fig. 4
Fig. 4

Phase velocity errors in the simulations of wave propagation in the TM 2 and TM 5 modes of a 2D step-profile waveguide. The effective permittivities ε i 1 ( 1 2 ) , ε i 2 ( 1 2 ) , and ε i 3 ( 1 2 ) were assigned to the nodes in the vicinity of the core–cladding interfaces. The offset ratio of one core–cladding interface is 0. The offset ratio at the other interface is described in the legend.

Fig. 5
Fig. 5

Phase velocity errors in the simulations of wave propagation in the TM 1 mode of a 2D step-profile waveguide. The method by Kaneda et al. [8], the VP-EP scheme [11], and the present method were applied to the nodes in the vicinity of the core–cladding interfaces. The offset ratio at the interfaces is 0.4.

Fig. 6
Fig. 6

Simulated reflectivity at a waveguide facet as a function of the number of cells per wavelength in the core region. Here, d is the offset ratio at the facet.

Fig. 7
Fig. 7

Difference in the simulated reflectivity at a waveguide facet Δ r , 2 / 3 as a function of the number of cells per wavelength in the core region.

Equations (155)

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H x n ( j , k ) - H x n 1 ( j , k ) Δ t = - E z n - 1 / 2 ( j + 1 / 2 , k ) - E z n - 1 / 2 ( j - 1 / 2 , k ) Δ + E y n - 1 / 2 ( j , k + 1 / 2 ) - E y n - 1 / 2 ( j , k - 1 / 2 ) Δ ,
ε j , k + 1 / 2 E y n + 1 / 2 ( j , k + 1 / 2 ) - E y n - 1 / 2 ( j , k + 1 / 2 ) Δ t = H x n ( j , k + 1 ) - H x n ( j , k ) Δ ,
ε j + 1 / 2 , k E z n + 1 / 2 ( j + 1 / 2 , k ) - E z n - 1 / 2 ( j + 1 / 2 , k ) Δ t = - H x n ( j + 1 , k ) - H x n ( j , k ) Δ ,
H x n ( j , k ) = exp ( i n Δ t ω ) H x s ( j Δ , k Δ ) ,
E y n + 1 / 2 ( j , k + 1 / 2 ) = exp ( i ( n + 1 / 2 ) Δ t ω ) E y s ( j Δ , ( k + 1 / 2 ) Δ ) ,
E z n + 1 / 2 ( j + 1 / 2 , k ) = exp ( i ( n + 1 / 2 ) Δ t ω ) E z s ( ( j + 1 / 2 ) Δ , k Δ ) .
ε j + 1 / 2 , k = ε m ( 1 1 ) ( m = j + k + 1 / 2 ) ,
ε j , k + 1 / 2 = ε m ( 1 1 ) ( m = j + k + 1 / 2 ) .
H x s ( j Δ , k Δ ) = exp ( - i 2 k t k - j 2 Δ ) H x s ( j + k 2 Δ , j + k 2 Δ ) ,
E y s ( j Δ , ( k + 1 2 ) Δ ) = exp ( - i 2 k t ( k - j 2 + 1 4 ) Δ ) E y s ( ( j + k 2 + 1 4 ) Δ , ( j + k 2 + 1 4 ) Δ ) ,
E z s ( ( j + 1 2 ) Δ , k Δ ) = exp ( - i 2 k t ( k - j 2 - 1 4 ) Δ ) E z s ( ( j + k 2 + 1 4 ) Δ , ( j + k 2 + 1 4 ) Δ ) .
H x s ( j + k 2 Δ , j + k 2 Δ ) = H x ( 1 1 ) ( j + k ) ,
E y s ( ( j + k 2 + 1 4 ) Δ , ( j + k 2 + 1 4 ) Δ ) = E y ( 1 1 ) ( j + k + 1 2 ) ,
E z s ( ( j + k 2 + 1 4 ) Δ , ( j + k 2 + 1 4 ) Δ ) = E z ( 1 1 ) ( j + k + 1 2 ) .
H x ( 1 1 ) ( m + 1 ) = i W ε m + 1 / 2 ( 1 1 ) Δ 2 cos ( k t Δ 2 ) [ ( i W Δ + 2 i W ε m + 1 / 2 ( 1 1 ) Δ ) H x ( 1 1 ) ( m ) + exp ( i k t Δ 2 2 ) E y ( 1 1 ) ( m - 1 / 2 ) - exp ( - i k t Δ 2 2 ) E z ( 1 1 ) ( m - 1 / 2 ) ] ,
E y ( 1 1 ) ( m + 1 / 2 ) = 1 2 cos ( k t Δ 2 ) { i [ exp ( - i k t Δ 2 2 ) ( W Δ - 2 W ε m + 1 / 2 ( 1 1 ) Δ ) + 2 cos ( k t Δ 2 ) exp ( i k t Δ 2 2 ) W ε m + 1 / 2 ( 1 1 ) Δ ] H x ( 1 1 ) ( m ) + E y ( 1 1 ) ( m - 1 / 2 ) - exp ( - i k t Δ 2 ) E z ( 1 1 ) ( m - 1 / 2 ) } ,
E z ( 1 1 ) ( m + 1 / 2 ) = - 1 2 cos ( k t Δ 2 ) { i [ exp ( i k t Δ 2 2 ) ( W Δ - 2 W ε m + 1 / 2 ( 1 1 ) Δ ) + 2 cos ( k t Δ 2 ) exp ( - i k t Δ 2 2 ) W ε m + 1 / 2 ( 1 1 ) Δ ] H x ( 1 1 ) ( m ) + exp ( i k t Δ 2 ) E y ( 1 1 ) ( m - 1 / 2 ) - E z ( 1 1 ) ( m - 1 / 2 ) } ,
W = sin ( ω Δ t 2 ) / ( Δ t 2 ) .
[ H x ( 1 1 ) ( m + 1 ) E y ( 1 1 ) ( m + 1 / 2 ) E z ( 1 1 ) ( m + 1 / 2 ) ] = G ( 1 1 ) ( ε m + 1 / 2 ( 1 1 ) ) [ H x ( 1 1 ) ( m ) E y ( 1 1 ) ( m - 1 / 2 ) E z ( 1 1 ) ( m - 1 / 2 ) ] .
s = k t / k 0 .
s ε sin θ ,
y + z = d Δ .
ε m ( 1 1 ) = ε 1 ( m < - 1 ) ,
ε - 1 / 2 ( 1 1 ) = ε i 1 ( 1 1 ) ,
ε 1 / 2 ( 1 1 ) = ε i 2 ( 1 1 ) ,
ε m ( 1 1 ) = ε 2 ( m > 1 ) .
k t k 0 ε 1 sin θ i .
F 1 , f ( 1 1 ) ( m ) = [ H x ( 1 1 ) ( m ) E y ( 1 1 ) ( m 1 / 2 ) E z ( 1 1 ) ( m 1 / 2 ) ] ( 1 i ε 1 cos θ i k 0 Δ m 2 ) [ 1 ( cos θ i + sin θ i ) ( 1 + i ε 1 cos θ i k 0 Δ / 2 3 / 2 ) / 2 ε 1 ( cos θ i sin θ i ) ( 1 + i ε 1 cos θ i k 0 Δ / 2 3 / 2 ) / 2 ε 1 ] .
F 1 , b ( 1 1 ) ( m ) = [ H x ( 1 1 ) ( m ) E y ( 1 1 ) ( m - 1 / 2 ) E z ( 1 1 ) ( m - 1 / 2 ) ] ( 1 + i ε 1 cos θ i k 0 Δ m 2 ) [ 1 ( cos θ i - sin θ i ) ( 1 - i ε 1 cos θ i k 0 Δ / 2 3 / 2 ) / 2 ε 1 - ( cos θ i + sin θ i ) ( 1 - i ε 1 cos θ i k 0 Δ / 2 3 / 2 ) / 2 ε 1 ] .
F 2 , f ( 1 1 ) ( m ) = [ H x ( 1 1 ) ( m ) E y ( 1 1 ) ( m - 1 / 2 ) E z ( 1 1 ) ( m - 1 / 2 ) ] ( 1 - i ε 2 cos θ t k 0 Δ m 2 ) [ 1 - ( cos θ t + sin θ t ) ( 1 + i ε 2 cos θ t k 0 Δ / 2 3 / 2 ) / 2 ε 2 ( cos θ t - sin θ t ) ( 1 + i ε 2 cos θ t k 0 Δ / 2 3 / 2 ) / 2 ε 2 ] ,
sin θ t ε 1 ε 2 sin θ i .
R exact = cos θ i ε 1 - cos θ t ε 2 cos θ i ε 1 + cos θ t ε 2 ,
T exact = 2 cos θ i ε 1 cos θ i ε 1 + cos θ t ε 2 .
F 1 ( 1 1 ) ( m ) = F 1 , f ( 1 1 ) ( m - d ) + R exact F 1 , b ( 1 1 ) ( m - d ) .
T exact F 2 , f ( 1 1 ) ( 2 - d ) G ( 1 1 ) ( ε 2 ) G ( 1 1 ) ( ε i 2 ( 1 1 ) ) G ( 1 1 ) ( ε i 1 ( 1 1 ) ) F 1 ( 1 1 ) ( - 1 ) .
ε i 1 ( 1 1 ) + ε i 2 ( 1 1 ) = ( 1 + d ) ε 1 + ( 1 - d ) ε 2 ,
1 ε i 1 ( 1 1 ) + 1 ε i 2 ( 1 1 ) = ( 1 + d ) 1 ε 1 + ( 1 - d ) 1 ε 2 .
ε i 1 ( 1 1 ) = ε am - ( ε 2 - ε 1 ) ( 1 - d 2 ) ε am ε hm 2 ε 1 ε 2 ,
ε i 2 ( 1 1 ) = ε am + ( ε 2 - ε 1 ) ( 1 - d 2 ) ε am ε hm 2 ε 1 ε 2 ,
ε am = ( 1 + d ) ε 1 + ( 1 - d ) ε 2 2 ,
1 ε hm = 1 + d ε 1 + 1 - d ε 2 2 .
ε j + 1 / 2 , k = ε m ( 1 2 ) ( m = 2 ( j + 1 / 2 ) + k ) ,
ε j , k + 1 / 2 = ε m ( 1 2 ) ( m = 2 j + ( k + 1 / 2 ) ) .
H x s ( j Δ , k Δ ) = exp ( - i 5 k t - j + 2 k 5 Δ ) H x s ( 4 j + 2 k 5 Δ , 2 j + k 5 Δ ) ,
E y s ( j Δ , ( k + 1 2 ) Δ ) = exp ( - i 5 k t - j + 2 k + 1 5 Δ ) E y s ( 4 j + 2 k + 1 5 Δ , 2 j + k + 1 / 2 5 Δ ) ,
E z s ( ( j + 1 2 ) Δ , k Δ ) = exp ( - i 5 k t - j + 2 k - 1 / 2 5 Δ ) E z s ( 4 j + 2 k + 2 5 Δ , 2 j + k + 1 5 Δ ) .
H x s ( 4 j + 2 k 5 Δ , 2 j + k 5 Δ ) = H x ( 1 2 ) ( 2 j + k ) ,
E y s ( 4 j + 2 k + 1 5 Δ , 2 j + k + 1 / 2 5 Δ ) = E y ( 1 2 ) ( 2 j + k + 1 2 ) ,
E z s ( 4 j + 2 k + 2 5 Δ , 2 j + k + 1 5 Δ ) = E z ( 1 2 ) ( 2 j + k + 1 ) .
H x ( 1 2 ) ( m + 1 ) = exp ( - 3 i k t Δ 5 ) H x ( 1 2 ) ( m ) - i W ε m ( 1 2 ) Δ exp ( - i k t Δ 2 5 ) E z ( 1 2 ) ( m ) - i W ε m - 1 / 2 ( 1 2 ) Δ exp ( - 2 i k t Δ 5 ) E y ( 1 2 ) ( m - 1 / 2 ) ,
E z ( 1 2 ) ( m + 1 ) = - i exp ( - i k t Δ 2 5 ) ( W Δ + exp ( - 5 i k t Δ ) - 1 W ε m + 1 / 2 ( 1 2 ) Δ ) H x ( 1 2 ) ( m ) - ε m ( 1 2 ) ε m + 1 / 2 ( 1 2 ) exp ( - 3 i k t Δ 5 ) E z ( 1 2 ) ( m ) - ( exp ( i k t Δ 2 5 ) + ε m - 1 / 2 ( 1 2 ) ε m + 1 / 2 ( 1 2 ) exp ( - 9 i k t Δ 2 5 ) ) E y ( 1 2 ) ( m - 1 / 2 ) + exp ( - i k t Δ 5 ) E z ( 1 2 ) ( m - 1 ) ,
E y ( 1 2 ) ( m + 1 / 2 ) = exp ( - 4 i k t Δ 5 ) - exp ( i k t Δ 5 ) i W ε m + 1 / 2 ( 1 2 ) Δ H x ( 1 2 ) ( m ) - ε m ( 1 2 ) ε m + 1 / 2 ( 1 2 ) exp ( - 3 i k t Δ 2 5 ) E z ( 1 2 ) ( m ) - ε m - 1 / 2 ( 1 2 ) ε m + 1 / 2 ( 1 2 ) exp ( - 3 i k t Δ 5 ) E y ( 1 2 ) ( m - 1 / 2 ) .
[ H x ( 1 2 ) ( m + 1 ) E z ( 1 2 ) ( m + 1 ) E y ( 1 2 ) ( m + 1 2 ) E z ( 1 2 ) ( m ) ] = G ( 1 2 ) ( ε m - 1 / 2 ( 1 2 ) , ε m ( 1 2 ) , ε m + 1 / 2 ( 1 2 ) ) [ H x ( 1 2 ) ( m ) E z ( 1 2 ) ( m ) E y ( 1 2 ) ( m - 1 2 ) E z ( 1 2 ) ( m - 1 ) ] .
s ε sin θ ,
2 y + z = ( 1 2 + d ) Δ .
ε m ( 1 2 ) = ε 1 ( m - 1 2 ) ,
ε 0 ( 1 2 ) = ε i 1 ( 1 2 ) ,
ε 1 / 2 ( 1 2 ) = ε i 2 ( 1 2 ) ,
ε 1 ( 1 2 ) = ε i 3 ( 1 2 ) ,
ε m ( 1 2 ) = ε 2 ( m 3 2 ) .
k t k 0 ε 1 sin θ i .
F 1 , f ( 1 2 ) ( m ) = [ H x ( 1 2 ) ( m ) E z ( 1 2 ) ( m ) E y ( 1 2 ) ( m - 1 / 2 ) E z ( 1 2 ) ( m - 1 ) ] ( 1 i ε 1 cos θ i k 0 Δ m 5 ) [ 1 2 cos θ i sin θ i 5 ε 1 2 sin θ i + cos θ i 5 ε 1 ( 1 + i ε 1 cos θ i k 0 Δ 2 5 ) 2 cos θ i - sin θ i 5 ε 1 ( 1 + i ε 1 cos θ i k 0 Δ 5 ) ] .
F 1 , b ( 1 2 ) ( m ) = [ H x ( 1 2 ) ( m ) E z ( 1 2 ) ( m ) E y ( 1 2 ) ( m - 1 / 2 ) E z ( 1 2 ) ( m - 1 ) ] ( 1 + i ε 1 cos θ i k 0 Δ m 5 ) [ 1 - 2 cos θ i + sin θ i 5 ε 1 - 2 sin θ i + cos θ i 5 ε 1 ( 1 - i ε 1 cos θ i k 0 Δ 2 5 ) - 2 cos θ i - sin θ i 5 ε 1 ( 1 - i ε 1 cos θ i k 0 Δ 5 ) ] .
F 1 , p p ( 1 2 ) ( m ) = [ H x ( 1 2 ) ( m ) E z ( 1 2 ) ( m ) E y ( 1 2 ) ( m - 1 / 2 ) E z ( 1 2 ) ( m - 1 ) ] ( 3 + 5 2 ) m ( 1 i ε 1 sin θ i k 0 Δ m 5 ) [ i ε 1 k 0 Δ 5 1 1 + 5 2 + i ( 5 5 ) ε 1 sin θ i k 0 Δ 20 3 5 2 i ( 5 + 3 5 ) ε 1 sin θ i k 0 Δ 10 ] .
F 1 , p n ( 1 2 ) ( m ) = [ H x ( 1 2 ) ( m ) E z ( 1 2 ) ( m ) E y ( 1 2 ) ( m - 1 / 2 ) E z ( 1 2 ) ( m - 1 ) ] ( 3 + 5 2 ) m ( 1 i ε 1 sin θ i k 0 Δ m 5 ) [ i ε 1 k 0 Δ 5 1 1 5 2 i ( 5 + 5 ) ε 1 sin θ i k 0 Δ 20 3 + 5 2 i ( 5 + 3 5 ) ε 1 sin θ i k 0 Δ 10 ] .
sin θ t ε 1 ε 2 sin θ i .
F 2 ( 1 2 ) ( m ) = a 2 , f F 2 , f ( 1 2 ) ( m - ( 1 2 + d ) ) + a 2 , b F 2 , b ( 1 2 ) ( m - ( 1 2 + d ) ) + a 2 , p p F 2 , p p ( 1 2 ) ( m - ( 1 2 + d ) ) + a 2 , p n F 2 , p n ( 1 2 ) ( m - ( 1 2 + d ) ) ,
Q ( m ) = P ( m ) - 1 ,
P ( m ) = [ F 2 , f ( 1 2 ) ( m 1 ) F 2 , b ( 1 2 ) ( m 1 ) F 2 , p p ( 1 2 ) ( m 1 ) F 2 , p n ( 1 2 ) ( m 1 ) ] ,
m 1 = m - ( 1 2 + d ) ,
[ a 2 , f a 2 , b a 2 , p p a 2 , p n ] = Q ( m ) F 2 ( 1 2 ) ( m ) .
F 1 , i ( 1 2 ) ( m ) = F 1 , f ( 1 2 ) ( m - ( 1 2 + d ) ) + R F 1 , b ( 1 2 ) ( m - ( 1 2 + d ) ) + a 1 , p p F 1 , p p ( 1 2 ) ( m - ( 1 2 + d ) ) ,
F 2 , t ( 1 2 ) ( m ) = T F 2 , f ( 1 2 ) ( m - ( 1 2 + d ) ) + a 2 , p n F 2 , p n ( 1 2 ) ( m - ( 1 2 + d ) ) ,
F 2 , t ( 1 2 ) ( 2 ) = L F 1 , i ( 1 2 ) ( 0 ) ,
L = G ( 1 2 ) ( ε i 2 ( 1 2 ) , ε i 3 ( 1 2 ) , ε 2 ) G ( 1 2 ) ( ε 1 , ε i 1 ( 1 2 ) , ε i 2 ( 1 2 ) ) .
[ Q ( 2 ) L F 1 , i ( 1 2 ) ( 0 ) ] 2 = 0 ,
[ Q ( 2 ) L F 1 , i ( 1 2 ) ( 0 ) ] 3 = 0.
T = [ Q ( 2 ) F 2 , t ( 1 2 ) ( 2 ) ] 1 = [ Q ( 2 ) L F 1 , i ( 1 2 ) ( 0 ) ] 1 ( 1 + i ε 2 cos θ t k 0 Δ ( 3 2 - d ) 5 ) e 1 L F 1 , i ( 1 2 ) ( 0 ) .
R = R exact + R 1 Δ + O ( Δ 2 ) ,
T = T exact + T 1 Δ + O ( Δ 2 ) .
( 5 - 3 5 ) ( ε 1 - ε 2 ) d 4 { ( 1 + 5 ) ( ε i 1 ( 1 2 ) ε i 2 ( 1 2 ) + ε i 2 ( 1 2 ) ε i 3 ( 1 2 ) + ε i 3 ( 1 2 ) ε i 1 ( 1 2 ) ) + 2 [ ε 1 ( ε i 2 ( 1 2 ) + ε i 3 ( 1 2 ) ) + ε 2 ( ε i 1 ( 1 2 ) + ε i 2 ( 1 2 ) ) ] + ( - 1 + 5 ) ε 1 ε 2 } + { - ε 2 [ ( 15 + 3 5 ) ε 1 - 2 5 ε 2 ] ε i 1 ( 1 2 ) + 2 5 ( ε 1 + ε 2 ) 2 ε i 2 ( 1 2 ) + ε 1 [ 2 5 ε 1 - ( 15 + 3 5 ) ε 2 ] ε i 3 ( 1 2 ) } / ( 5 + 5 ) + ( ε 1 + ε 2 ) ( ε i 1 ( 1 2 ) ε i 2 ( 1 2 ) + ε i 2 ( 1 2 ) ε i 3 ( 1 2 ) + ε i 3 ( 1 2 ) ε i 1 ( 1 2 ) ) - 6 5 ε 1 ε 2 ( ε 1 + ε 2 ) 5 + 5 = 0 ,
100 ε i 1 ( 1 2 ) ε i 2 ( 1 2 ) ε i 3 ( 1 2 ) + 2 [ ( - 10 + 3 5 ) ε 1 + ( - 35 + 8 5 ) ε 2 ] ε i 1 ( 1 2 ) ε i 2 ( 1 2 ) + 2 [ ( - 35 + 8 5 ) ε 1 + ( - 10 + 3 5 ) ε 2 ] ε i 2 ( 1 2 ) ε i 3 ( 1 2 ) + 2 ( - 10 + 3 5 ) ( ε 1 + ε 2 ) ε i 3 ( 1 2 ) ε i 1 ( 1 2 ) + ( - 5 + 13 5 ) 41 { ε 2 [ ( 4 + 35 5 ) ε 1 - 41 ε 2 ] ε i 1 ( 1 2 ) + [ - 41 ( ε 1 2 + ε 2 2 ) + ( - 107 + 17 5 ) ε 1 ε 2 ] ε i 2 ( 1 2 ) + ε 1 [ - 41 ε 1 + ( 4 + 35 5 ) ε 2 ] ε i 3 ( 1 2 ) } + ( - 25 + 19 5 ) ( ε 1 + ε 2 ) ε 1 ε 2 = 0 ,
2 ε i 1 ( 1 2 ) ε i 2 ( 1 2 ) + 2 ε i 2 ( 1 2 ) ε i 3 ( 1 2 ) - 8 ε i 3 ( 1 2 ) ε i 1 ( 1 2 ) + ( 4 - 2 5 ) ( ε 2 ε i 1 ( 1 2 ) + ε 1 ε i 3 ( 1 2 ) ) + ( - 1 + 5 ) ( ε 1 + ε 2 ) ε i 2 ( 1 2 ) + 2 ( - 1 + 5 ) ε 1 ε 2 = 0.
ε i 1 ( 1 2 ) = ( - 1 + 5 ) ε 1 [ - 20 d 2 ( ε 1 - ε 2 ) - 10 d ( ε 1 - ε 2 ) + ( 11 + 3 5 ) ε 1 + ( 19 + 7 5 ) ε 2 ] 2 [ 20 d 2 ( ε 1 - ε 2 ) - 10 5 d ( ε 1 - ε 2 ) + ( - 1 + 3 5 ) ε 1 + ( 11 + 7 5 ) ε 2 ] ,
ε i 2 ( 1 2 ) = - 5 ( 5 + 5 ) ε 1 ε 2 [ 4 d ( ε 1 - ε 2 ) + ( 1 + 5 ) ( ε 1 + ε 2 ) ] 2 [ 20 d 2 ( ε 1 - ε 2 ) 2 + 10 5 d ( ε 1 2 - ε 2 2 ) - ( 11 + 7 5 ) ( ε 1 2 + ε 2 2 ) - ( 28 + 16 5 ) ε 1 ε 2 ] ,
ε i 3 ( 1 2 ) = ( - 1 + 5 ) ε 2 [ 20 d 2 ( ε 1 - ε 2 ) - 10 d ( ε 1 - ε 2 ) + ( 19 + 7 5 ) ε 1 + ( 11 + 3 5 ) ε 2 ] 2 [ 20 d 2 ( - ε 1 + ε 2 ) - 10 5 d ( ε 1 - ε 2 ) + ( 11 + 7 5 ) ε 1 + ( - 1 + 3 5 ) ε 2 ] .
λ wg = 2 π β ,
Δ r , 2 / 3 ( Δ ) = | r ( Δ ) r ( 2 Δ 3 ) | .
Δ r , 2 / 3 ( Δ ) = O ( Δ 2 ) .
n t ( 1 1 ) = [ 1 / 2 1 / 2 ] .
H x s ( r + c t n t ( 1 1 ) ) = exp ( i k t c t ) H x s ( r ) ,
E y s ( r + c t n t ( 1 1 ) ) = exp ( i k t c t ) E y s ( r ) ,
E z s ( r + c t n t ( 1 1 ) ) = exp ( i k t c t ) E z s ( r ) ,
n n ( 1 1 ) = [ 1 / 2 1 / 2 ] .
[ y z ] = c n n n ( 1 1 ) + c t n t ( 1 1 ) ,
c n = ( y + z ) / 2 ,
c t = ( y + z ) / 2 .
H x s ( y , z ) = H x s ( c n n n ( 1 1 ) + c t n t ( 1 1 ) ) = exp ( i k t c t ) H x s ( c n n n ( 1 1 ) ) = exp ( i k t ( y + z ) / 2 ) H x s ( ( y + z ) / 2 , ( y + z ) / 2 ) ,
E y s ( y , z ) = E y s ( c n n n ( 1 1 ) + c t n t ( 1 1 ) ) = exp ( i k t c t ) E y s ( c n n n ( 1 1 ) ) = exp ( i k t ( y + z ) / 2 ) E y s ( ( y + z ) / 2 , ( y + z ) / 2 ) ,
E z s ( y , z ) = E z s ( c n n n ( 1 1 ) + c t n t ( 1 1 ) ) = exp ( i k t c t ) E z s ( c n n n ( 1 1 ) ) = exp ( i k t ( y + z ) / 2 ) E z s ( ( y + z ) / 2 , ( y + z ) / 2 ) .
G ( 1 1 ) ( ε m + 1 / 2 ( 1 1 ) ) = [ g 1 , 1 ( 1 1 ) g 1 , 2 ( 1 1 ) g 1 , 3 ( 1 1 ) g 2 , 1 ( 1 1 ) g 2 , 2 ( 1 1 ) g 2 , 3 ( 1 1 ) g 3 , 1 ( 1 1 ) g 3 , 2 ( 1 1 ) g 3 , 3 ( 1 1 ) ] ,
g 1 , 1 ( 1 1 ) 1 ,
g 1 , 2 ( 1 1 ) i 2 ε m + 1 / 2 ( 1 1 ) k 0 Δ ,
g 1 , 3 ( 1 1 ) i 2 ε m + 1 / 2 ( 1 1 ) k 0 Δ ,
g 2 , 1 ( 1 1 ) s 2 ε m + 1 / 2 ( 1 1 ) + i k 0 Δ 2 ( 1 s 2 2 ε m + 1 / 2 ( 1 1 ) ) ,
g 2 , 2 ( 1 1 ) 1 2 ,
g 2 , 3 ( 1 1 ) 1 2 ( 1 i s k 0 Δ 2 ) ,
g 3 , 1 ( 1 1 ) s 2 ε m + 1 / 2 ( 1 1 ) + i k 0 Δ 2 ( s 2 2 ε m + 1 / 2 ( 1 1 ) 1 ) ,
g 3 , 2 ( 1 1 ) 1 2 ( 1 + i s k 0 Δ 2 ) ,
g 3 , 3 ( 1 1 ) 1 2 .
G ( 1 2 ) ( ε m 1 / 2 ( 1 2 ) , ε m ( 1 2 ) , ε m + 1 / 2 ( 1 2 ) ) = [ g 1 , 1 ( 1 2 ) g 1 , 2 ( 1 2 ) g 1 , 3 ( 1 2 ) g 1 , 4 ( 1 2 ) g 2 , 1 ( 1 2 ) g 2 , 2 ( 1 2 ) g 2 , 3 ( 1 2 ) g 2 , 4 ( 1 2 ) g 3 , 1 ( 1 2 ) g 3 , 2 ( 1 2 ) g 3 , 3 ( 1 2 ) g 3 , 4 ( 1 2 ) g 4 , 1 ( 1 2 ) g 4 , 2 ( 1 2 ) g 4 , 3 ( 1 2 ) g 4 , 4 ( 1 2 ) ] ,
g 1 , 1 ( 1 2 ) 1 i 3 s k 0 Δ 5 ,
g 1 , 2 ( 1 2 ) i ε m ( 1 2 ) k 0 Δ ,
g 1 , 3 ( 1 2 ) i ε m 1 / 2 ( 1 2 ) k 0 Δ ,
g 1 , 4 ( 1 2 ) = 0 ,
g 2 , 1 ( 1 2 ) 5 s ε m + 1 / 2 ( 1 2 ) + i k 0 Δ ( 1 + 3 s 2 ε m + 1 / 2 ( 1 2 ) ) ,
g 2 , 2 ( 1 2 ) ε m ( 1 2 ) ε m + 1 / 2 ( 1 2 ) ( 1 i 3 s k 0 Δ 5 ) ,
g 2 , 3 ( 1 2 ) 1 ε m 1 / 2 ( 1 2 ) ε m + 1 / 2 ( 1 2 ) + i s k 0 Δ 2 5 ( 1 + 9 ε m 1 / 2 ( 1 2 ) ε m + 1 / 2 ( 1 2 ) ) ,
g 2 , 4 ( 1 2 ) 1 i s k 0 Δ 5 ,
g 3 , 1 ( 1 2 ) 5 s ε m + 1 / 2 ( 1 2 ) + i 3 s 2 k 0 Δ 2 ε m + 1 / 2 ( 1 2 ) ,
g 3 , 2 ( 1 2 ) ε m ( 1 2 ) ε m + 1 / 2 ( 1 2 ) ( 1 i 3 s k 0 Δ 2 5 ) ,
g 3 , 3 ( 1 2 ) ε m 1 / 2 ( 1 2 ) ε m + 1 / 2 ( 1 2 ) ( 1 i 3 s k 0 Δ 5 ) ,
g 3 , 4 ( 1 2 ) = 0 ,
g 4 , 1 ( 1 2 ) = 0 ,
g 4 , 2 ( 1 2 ) = 1 ,
g 4 , 3 ( 1 2 ) = 0 ,
g 4 , 4 ( 1 2 ) = 0.
Q ( m ) [ ( 1 + i ε 2 cos θ t k 0 Δ [ m ( 1 2 + d ) ] 5 ) e 1 ( 1 i ε 2 cos θ t k 0 Δ [ m ( 1 2 + d ) ] 5 ) e 2 ( 3 + 5 2 ) m + 1 2 + d ( 1 + i ε 2 sin θ t k 0 Δ [ m ( 1 2 + d ) ] 5 ) e 3 ( 3 + 5 2 ) m + 1 2 + d ( 1 + i ε 2 sin θ t k 0 Δ [ m ( 1 2 + d ) ] 5 ) e 4 ] ,
e 1 = [ e 1 , 1 e 1 , 2 e 1 , 3 e 1 , 4 ] ,
e 2 = [ e 2 , 1 e 2 , 2 e 2 , 3 e 2 , 4 ] ,
e 3 = [ e 3 , 1 e 3 , 2 e 3 , 3 e 3 , 4 ] ,
e 4 = [ e 4 , 1 e 4 , 2 e 4 , 3 e 4 , 4 ] ,
e 1 , 1 = 1 2 i ε 2 ( cos θ t + 2 sin θ t ) k 0 Δ 4 5 ,
e 1 , 2 = ε 2 2 5 cos θ t [ 1 i ε 2 2 5 ( 2 cos θ t sin θ t ) k 0 Δ ] ,
e 1 , 3 = ε 2 2 5 ( 1 cos θ t + i 2 ε 2 5 k 0 Δ ) ,
e 1 , 4 = ε 2 2 5 cos θ t ( 1 i ε 2 2 5 sin θ t k 0 Δ ) ,
e 2 , 1 = 1 2 + i ε 2 ( cos θ t 2 sin θ t ) k 0 Δ 4 5 ,
e 2 , 2 = ε 2 2 5 cos θ t [ 1 + i ε 2 2 5 ( 2 cos θ t + sin θ t ) k 0 Δ ] ,
e 2 , 3 = ε 2 2 5 ( 1 cos θ t i 2 5 ε 2 k 0 Δ ) ,
e 2 , 4 = ε 2 2 5 cos θ t ( 1 i ε 2 2 5 sin θ t k 0 Δ ) ,
e 3 , 1 = ( 5 + 5 ) sin θ t 10 ε 2 + i [ 3 5 + ( 5 + 4 5 ) cos ( 2 θ t ) ] k 0 Δ 50 ,
e 3 , 2 = 3 + 5 10 i ( 2 + 5 ) ε 2 sin θ t k 0 Δ 25 ,
e 3 , 3 = 1 5 + 1 5 i ( 8 + 5 ) ε 2 sin θ t k 0 Δ 25 ,
e 3 , 4 = 1 5 + i ( 1 + 5 ) ε 2 sin θ t k 0 Δ 50 ,
e 4 , 1 = ( 5 + 5 ) sin θ t 10 ε 2 + i [ 3 5 + ( 5 4 5 ) cos ( 2 θ t ) ] k 0 Δ 50 ,
e 4 , 2 = 3 5 10 i ( 2 + 5 ) ε 2 sin θ t k 0 Δ 25 ,
e 4 , 3 = 1 5 1 5 i ( 8 + 5 ) ε 2 sin θ t k 0 Δ 25 ,
e 4 , 4 = 1 5 + i ( 1 + 5 ) ε 2 sin θ t k 0 Δ 50 .
( u 1 , 1 u 1 , 2 u 2 , 1 u 2 , 2 ) [ R a 1 , p p ] = [ u 1 , 0 u 2 , 0 ] ,
u 1 , 1 e 2 L F 1 , b ( 1 2 ) ( ( 1 / 2 + d ) ) ,
u 1 , 2 e 2 L F 1 , p p ( 1 2 ) ( ( 1 / 2 + d ) ) ,
u 2 , 1 e 3 L F 1 , b ( 1 2 ) ( ( 1 / 2 + d ) ) ,
u 2 , 2 e 3 L F 1 , p p ( 1 2 ) ( ( 1 / 2 + d ) ) ,
u 1 , 0 e 2 L F 1 , f ( 1 2 ) ( ( 1 / 2 + d ) ) ,
u 2 , 0 e 3 L F 1 , f ( 1 2 ) ( ( 1 / 2 + d ) ) .

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