Abstract

Effective permittivities for the two-dimensional Finite-Difference Time-Domain (FDTD) method are derived using a contour path approach that accounts for the boundary conditions of the electromagnetic field at dielectric interfaces. A phenomenological formula for the effective permittivities is also proposed as an effective and simpler alternative to the previous result. Our schemes are validated using Mie theory for the scattering of a dielectric cylinder and they are compared to the usual staircase and the widely used volume-average approximations. Significant improvements in terms of accuracy and error fluctuations are demonstrated, especially in the calculation of resonances.

© 2005 Optical Society of America

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References

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  1. Proceedings of the Fifth International Symposium on Photonic and Electromagnetic Crystal Structures (PECSV) (Kyoto, Japan, March 7-11, 2004); H. Benisty, S. Kawakami, D.J. Norris, and C.M. Soukoulis, eds, Phot. Nanostructures Fund. Appl. 2, 57-159 (2004); C. Jagadish, D.G. Deppe, S. Noda, T.F. Krauss, and O.J. Painter, eds, IEEE J. Sel. Top. Area Commun. 23, 1305-1423 (2005).
    [CrossRef]
  2. Special issue on nanostructured optical meta-materials: beyond photonic band gap effects, N. Zheludev, and V. Shalaev, eds., J. Opt. A: Pure and Applied Optics, 7, S1-S254 (2005).
    [CrossRef]
  3. Proceedings of the EOS Topical Meeting on Advanced Optical Imaging Techniques, (London, UK, June 29 - July 1, 2005).
  4. M.V.K. Chari, and S.J. Salon, Numerical methods in electromagnetism (Academic Press, San Diego, CA, 2000)
  5. K.S. Yee, "Numerical Solution of Initial Boundary Value Problems involving Maxwell's Equations in Isotropic Media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  6. A. Taflove, and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).
  7. K.K. Mei, A. Cangellaris, and D.J. Angelakos, "Conformal Time Domain Finite-Difference Method," Radio Sci. 19, 1145-1147 (1984).
    [CrossRef]
  8. R. Holland, "Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983).
    [CrossRef]
  9. M. Fusco, "FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 38, 76-89 (1990).
    [CrossRef]
  10. V. Shankar, A. Mohammadian, andW.F. Hall, "A Time-Domain Finite-Volume Treatment for the Maxwell Equations," Electromagnetics 10, 127-145 (1990).
    [CrossRef]
  11. N.K. Madsen, and R.W. Ziolkowski, "A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations," Electromagnetics 10, 147-161 (1990).
    [CrossRef]
  12. 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, 741-476 (1992).
    [CrossRef]
  13. T.G. Jurgens, A. Taflove, K. Umashankar, and T.G. Moore, "Finite-Difference Time-Domain Modeling of Curved Surfaces," IEEE Trans. Antennas Propag. 40, 357-365 (1992).
    [CrossRef]
  14. T.G. Jurgens, and A. Taflove, "Three-Dimensional Contour FDTD Modeling of Scattering from Single and Multiple Bodies," IEEE Trans. Antennas Propag. 41, 1703-1708 (1993).
    [CrossRef]
  15. C.J. Railton, I.J. Craddock, and J.B. Schneider, "Improved locally distorted CPFDTD algorithm with provable stability," Electron. Lett. 31, 1585-1586 (1995).
    [CrossRef]
  16. Y. Hao, and C.J. Railton, "Analyzing Electromagnetic Structures with Curved Boundaries on Cartesian FDTD Meshes," IEEE Trans. Microwave Theory Tech. 46, 82-88 (1998).
    [CrossRef]
  17. T.I. Kosmanis, and T.D. Tsiboukis, "A Systematic and Topologically Stable Conformal Finite-Difference Time- Domain Algorithm for Modeling Curved Dielectric Interfaces in Three Dimensions," IEEE Trans. Microwave Theory Tech. 51, 839-847 (2003).
    [CrossRef]
  18. 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]
  19. S.S. Zivanovic, K.S. Yee, and K.K. Mei, "A Subgridding Method for the Time-Domain Finite-Difference Method to Solve Maxwell's Equations," IEEE Trans. Microwave Theory Tech. 39, 471-479 (1991).
    [CrossRef]
  20. J.G. Maloney, and G.S. Smith, "The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time- Domain (FDTD) Method," IEEE Trans. Antennas Propag. 40, 323-330 (1992).
    [CrossRef]
  21. 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]
  22. 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]
  23. K.-P. Hwang, and A.C. Cangellaris, "Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces," IEEE Microwave Wireless Comp. Lett. 11, 158-160 (2001).
    [CrossRef]
  24. S. Dey, and R. Mittra, "A Conformal Finite-Difference Time-Domain Technique for Modeling Cylindrical Dielectric Resonators," IEEE Trans. Microwave Theory Tech. 47, 1737-1739 (1999).
    [CrossRef]
  25. W. Yu, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microw. Opt. Technol. Lett. 24, 151-155 (2000).
    [CrossRef]
  26. P. Yang, G.W. Kattawar, K.-N. Liou, and J.Q. Lu, "Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles," Appl. Opt. 43, 4611-4624 (2004).
    [CrossRef] [PubMed]
  27. 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]
  28. W. Yu, and R. Mittra, "A Conformal Finite Difference Time Domain Technique for Modeling Curved Dielectric Surfaces," IEEE Microwave Wireless Comp. Lett. 11, 25-27 (2001).
    [CrossRef]
  29. W. Sun, and Q. Fu "Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices," Appl. Opt. 39, 5569 (2000).
    [CrossRef]
  30. J.-Y. Lee, and N.-H. Myung, "Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces," Microw. Opt. Technol. Lett. 23, 245-249 (1999).
    [CrossRef]
  31. J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, "A 3-D Tensor FDTD-Formulation for Treatment of Slopes Interfaces in Electrically Inhomogeneous Media," IEEE Trans. Antennas Propag. 51, 1760- 1770 (2003).
    [CrossRef]
  32. 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 Propag. 49, 749-756 (2001).
    [CrossRef]
  33. A. Ditkowski, K. Dridi, and J.S. Hesthaven, "Convergent Cartesian Grid Methods for Maxwell's Equations in Complex Geometries," J. Comp. Phys. 170, 39-80 (2001).
    [CrossRef]
  34. M. Fujii, D. Lukashevich, I. Sakagami, and P. Russer, "Convergence of FDTD andWavelet-Collocation Modeling of Curved Dielectric Interface with the Effective Dielectric Constant Technique," IEEE Microwave Wireless Comp. Lett. 13, 469-471 (2003).
    [CrossRef]
  35. T. Xiao, and Q.H. Liu, "A Staggered Upwind Embedded Boundary (SUEB) Method to Eliminate the FDTD Staircasing Error," IEEE Trans. Antennas Propag. 52, 730-740 (2004).
    [CrossRef]
  36. C.F. Bohren, and D.R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).
  37. A. Bossavit, "Generalized finite differences in computational electromagnetics," Progress in Electromagnetic Research, PIER 32, 45-64 (2001).
    [CrossRef]
  38. K.L. Shlager, J.B. Schneider, "Comparison of the Dispersion Properties of Several Low-Dispersion Finite- Difference Time-Domain Algorithms," IEEE Trans. Antennas Propag. 51, 642-652 (2003).
    [CrossRef]
  39. J.A. Roden, and S.D. Gedney, "Convolutional PML (CPML): An efficient FDTD implementation of the CFSPML for arbitrary media," Microw. Opt. Technol. Lett. 27, 334-339 (2000).
    [CrossRef]
  40. A. Kirchner, K. Busch, and C.M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
    [CrossRef]

Appl. Opt. (3)

Electromagnetics (2)

V. Shankar, A. Mohammadian, andW.F. Hall, "A Time-Domain Finite-Volume Treatment for the Maxwell Equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

N.K. Madsen, and R.W. Ziolkowski, "A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations," Electromagnetics 10, 147-161 (1990).
[CrossRef]

Electron. Lett. (1)

C.J. Railton, I.J. Craddock, and J.B. Schneider, "Improved locally distorted CPFDTD algorithm with provable stability," Electron. Lett. 31, 1585-1586 (1995).
[CrossRef]

IEEE J. Sel. Top. Area Commun. (1)

Proceedings of the Fifth International Symposium on Photonic and Electromagnetic Crystal Structures (PECSV) (Kyoto, Japan, March 7-11, 2004); H. Benisty, S. Kawakami, D.J. Norris, and C.M. Soukoulis, eds, Phot. Nanostructures Fund. Appl. 2, 57-159 (2004); C. Jagadish, D.G. Deppe, S. Noda, T.F. Krauss, and O.J. Painter, eds, IEEE J. Sel. Top. Area Commun. 23, 1305-1423 (2005).
[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 Microwave Wireless Comp. Lett. (3)

K.-P. Hwang, and A.C. Cangellaris, "Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces," IEEE Microwave Wireless Comp. Lett. 11, 158-160 (2001).
[CrossRef]

M. Fujii, D. Lukashevich, I. Sakagami, and P. Russer, "Convergence of FDTD andWavelet-Collocation Modeling of Curved Dielectric Interface with the Effective Dielectric Constant Technique," IEEE Microwave Wireless Comp. Lett. 13, 469-471 (2003).
[CrossRef]

W. Yu, and R. Mittra, "A Conformal Finite Difference Time Domain Technique for Modeling Curved Dielectric Surfaces," IEEE Microwave Wireless Comp. Lett. 11, 25-27 (2001).
[CrossRef]

IEEE Trans. Antennas Propag. (9)

T. Xiao, and Q.H. Liu, "A Staggered Upwind Embedded Boundary (SUEB) Method to Eliminate the FDTD Staircasing Error," IEEE Trans. Antennas Propag. 52, 730-740 (2004).
[CrossRef]

M. Fusco, "FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 38, 76-89 (1990).
[CrossRef]

K.L. Shlager, J.B. Schneider, "Comparison of the Dispersion Properties of Several Low-Dispersion Finite- Difference Time-Domain Algorithms," IEEE Trans. Antennas Propag. 51, 642-652 (2003).
[CrossRef]

J.G. Maloney, and G.S. Smith, "The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time- Domain (FDTD) Method," IEEE Trans. Antennas Propag. 40, 323-330 (1992).
[CrossRef]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, "A 3-D Tensor FDTD-Formulation for Treatment of Slopes Interfaces in Electrically Inhomogeneous Media," IEEE Trans. Antennas Propag. 51, 1760- 1770 (2003).
[CrossRef]

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 Propag. 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 Propag. AP-14, 302-307 (1966).

T.G. Jurgens, A. Taflove, K. Umashankar, and T.G. Moore, "Finite-Difference Time-Domain Modeling of Curved Surfaces," IEEE Trans. Antennas Propag. 40, 357-365 (1992).
[CrossRef]

T.G. Jurgens, and A. Taflove, "Three-Dimensional Contour FDTD Modeling of Scattering from Single and Multiple Bodies," IEEE Trans. Antennas Propag. 41, 1703-1708 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (7)

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, 741-476 (1992).
[CrossRef]

Y. Hao, and C.J. Railton, "Analyzing Electromagnetic Structures with Curved Boundaries on Cartesian FDTD Meshes," IEEE Trans. Microwave Theory Tech. 46, 82-88 (1998).
[CrossRef]

T.I. Kosmanis, and T.D. Tsiboukis, "A Systematic and Topologically Stable Conformal Finite-Difference Time- Domain Algorithm for Modeling Curved Dielectric Interfaces in Three Dimensions," IEEE Trans. Microwave Theory Tech. 51, 839-847 (2003).
[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]

S.S. Zivanovic, K.S. Yee, and K.K. Mei, "A Subgridding Method for the Time-Domain Finite-Difference Method to Solve Maxwell's Equations," IEEE Trans. Microwave Theory Tech. 39, 471-479 (1991).
[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]

S. Dey, and R. Mittra, "A Conformal Finite-Difference Time-Domain Technique for Modeling Cylindrical Dielectric Resonators," IEEE Trans. Microwave Theory Tech. 47, 1737-1739 (1999).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

R. Holland, "Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983).
[CrossRef]

J. Comp. Phys. (1)

A. Ditkowski, K. Dridi, and J.S. Hesthaven, "Convergent Cartesian Grid Methods for Maxwell's Equations in Complex Geometries," J. Comp. Phys. 170, 39-80 (2001).
[CrossRef]

J. Opt. A: Pure and Applied Optics (1)

Special issue on nanostructured optical meta-materials: beyond photonic band gap effects, N. Zheludev, and V. Shalaev, eds., J. Opt. A: Pure and Applied Optics, 7, S1-S254 (2005).
[CrossRef]

Microw. Opt. Technol. Lett. (3)

W. Yu, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microw. Opt. Technol. Lett. 24, 151-155 (2000).
[CrossRef]

J.-Y. Lee, and N.-H. Myung, "Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces," Microw. Opt. Technol. Lett. 23, 245-249 (1999).
[CrossRef]

J.A. Roden, and S.D. Gedney, "Convolutional PML (CPML): An efficient FDTD implementation of the CFSPML for arbitrary media," Microw. Opt. Technol. Lett. 27, 334-339 (2000).
[CrossRef]

Phys. Rev. B (1)

A. Kirchner, K. Busch, and C.M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Progress in Electromagnetic Research (1)

A. Bossavit, "Generalized finite differences in computational electromagnetics," Progress in Electromagnetic Research, PIER 32, 45-64 (2001).
[CrossRef]

Radio Sci. (1)

K.K. Mei, A. Cangellaris, and D.J. Angelakos, "Conformal Time Domain Finite-Difference Method," Radio Sci. 19, 1145-1147 (1984).
[CrossRef]

Other (4)

A. Taflove, and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).

Proceedings of the EOS Topical Meeting on Advanced Optical Imaging Techniques, (London, UK, June 29 - July 1, 2005).

M.V.K. Chari, and S.J. Salon, Numerical methods in electromagnetism (Academic Press, San Diego, CA, 2000)

C.F. Bohren, and D.R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

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Figures (8)

Fig. 1.
Fig. 1.

(a) FDTD mesh showing the staircasing effect for a curved interface (blue line); (b) location of the field components for H-modes and integration lines for Ampére law (blue segment) and Faraday law (red segment) for Ex | i,j-1/2. ∆x and ∆y are the cell dimensions (mesh’s pitch); (i, j) refers to the position of Hz , while Ex and Ey are at (i, j - 1/2) and (i -1/2, j), respectively. Notice that the cells associated to the different field components partially overlap.

Fig. 2.
Fig. 2.

Partially filled cells: (a) interface parallel to the field component, (b) interface orthogonal to the field component, (c) inclusion without crossing the integration lines, (d) inclusion crossing both integration lines, (e) inclusion crossing only the integration line of Ampére law, (f) inclusion crossing only the integration line of Faraday law. d,f represent the line filling factors, 1 and 2 mean media with ε1 and ε2 respectively.

Fig. 3.
Fig. 3.

Partially filled cells with tilted interfaces: (a) tilted interface crossing only the integration line of Ampère law, (b) tilted interface crossing only the integration line of Faraday law, (c) tilted interface crossing both integration lines, (d) curved interface crossing only the integration line of Ampère law, (e) curved interface crossing only the integration line of Faraday law, (f) curved interface crossing both integration lines. n,m represent unit vectors normal to the interface, d, f represent the line filling factors, 1 and 2 mean media with ε 1 and ε 2, respectively.

Fig. 4.
Fig. 4.

Layout of the FDTD calculation: CPML layers (gray), cylindrical scatterer (orange), the H-polarized incident plane wave is excited using the total field / scattered field method, the integration line is for the calculation of the total SCS.

Fig. 5.
Fig. 5.

Accuracy on the total SCS: (a) total SCS for Nλ = 25, (b) relative error on the total SCS for Nλ = 25, (c) average relative error on the total SCS. Nλ is the number of divisions for the shortest wavelength in the cylinder; i. e. Nλ = (400nm/∆)/√ε. The phase-velocity error in (c) is computed for Nλ . Parameters: ε = 3, r = 400nm and ∆λ = 1nm.

Fig. 6.
Fig. 6.

Accuracy on the total SCS and the resonant wavelengths: (a) total SCS for Nλ =25, (b) average relative error on the total SCS, (c) relative error on the wavelength of the resonance λ 1 = 675.8nm, (d) relative error on the wavelength of the resonance λ 2 = 5323nm. Nλ is the number of divisions for λ 2 in the cylinder; i. e. Nλ = (λ 2/∆)/√ε. The phase-velocity error in (b)-(d) is computed for Nλ . Parameters: ε = 12, r = 150nm and ∆λ = 0.25nm.

Fig. 7.
Fig. 7.

Accuracy on resonant wavelengths: (a) resonant peak for Nλ = 11 and (b) Nλ = 19, (c) relative error on the wavelength of the resonance λo = 679 4nm. Nλ is the number of divisions for λo in the cylinder; i. e. Nλ = (λo /∆)/√ε. The phase-velocity error in (c) is computed for Nλ . Parameters: ε = 20, r = 120nm and ∆ λ = 0.2nm.

Fig. 8.
Fig. 8.

Color maps of the EPs: (a) staircase, (b) V-EP, (c) VP-EP, (d) CP-EP.

Equations (14)

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t D · n d s = H · d l , t B · n d s = E · d l .
t ( j 1 ) Δ y j Δ y D x i , y d y = H z i , j H z i , j 1 , t ( i 1 ) Δ x i Δ x D y | x , j d x = H z | i 1 , j H z | i , j ,
t ( i 1 / 2 ) Δ x ( i + 1 / 2 ) Δ x ( j 1 / 2 ) Δ y ( j + 1 / 2 ) Δ y H z x , y d x d y = ( i 1 / 2 ) Δ x ( i + 1 / 2 ) Δ x ( E x x , j 1 / 2 E x x , j + 1 / 2 ) d x +
( j 1 / 2 ) Δ y ( j + 1 / 2 ) Δ y
( E y i + 1 / 2 , y E y i 1 / 2 , y ) d y ,
( j 1 ) Δ y j Δ y D x i , y d y = ( j 1 ) Δ y ( j 1 ) Δ y + d D x i , y d y + ( j 1 ) Δ y + d j Δ y D x i , y d = [ d ε 2 + ( Δ y d ) ε 1 ] E x i , j 1 / 2 ,
( i 1 / 2 ) Δ x ( i + 1 / 2 ) Δ x ( E x x , j 1 / 2 E x x , j + 1 / 2 ) d x = ( i 1 / 2 ) Δ x ( i 1 / 2 ) Δ x + f ( E x x , j 1 / 2 E x x , j + 1 / 2 ) d x +
( i 1 / 2 ) Δ x + f ( i + 1 / 2 ) Δ x ε 2 ε 1
( E x x , j 1 / 2 E x x , j + 1 / 2 ) d x
= [ f + ε 2 ε 1 ( Δ x f ) ] ( E x i , j 1 / 2 E x i , j + 1 / 2 ) ,
ε E x i , j 1 / 2 = ε E x i , j 1 / 2 f / Δ x + ( 1 f / Δ x ) ε 2 / ε 1 = ε ε ε 2 E x i , j 1 / 2 .
( j 1 ) Δ y j Δ y D x i , y d y = { d [ ε 1 n 2 + ε 2 ( 1 n 2 ) ] + ( Δ y d ) ε 1 } E x i , j 1 / 2 .
( i 1 / 2 ) Δ x ( i + 1 / 2 ) Δ x ( E x x , j 1 / 2 E x x , j + 1 / 2 ) d x = { ( Δ x f ) + [ ε 1 ε 2 n 2 + ( 1 n 2 ) ] f } ( E x i , j 1 / 2 ) .
ε eff = ε ( 1 n 2 ) + ε n 2 ,

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