Abstract

Scattering of electromagnetic radiation by an infinite array of perfectly conducting cylinders is studied by a combination of a finite-element method and a boundary-solution procedure. The cylinders, which have an arbitrary cross section, are embedded periodically in an inhomogeneous and lossy dielectric. The obliquely incident radiation is of linear polarization with either of its fields parallel to the axis of the cylinders. The method of solution is simple in nature and involves no iterative calculations. Numerical results are obtained for gratings with circular elements. The convergence and the accuracy of the solution are tested by the conservation-of-power criterion and by comparing the numerical results with those of the exact solution and other published data. Further numerical examples show some of the transmission characteristics of the circular gratings embedded in an inhomogeneous or lossy dielectric.

© 1988 Optical Society of America

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References

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  1. M. G. Hutley, Diffraction Gratings (Academic, London, 1982).
  2. K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
    [CrossRef]
  3. P. S. Henry, J. T. Ruscio, “A low loss diffraction grating frequency multiplexer,” IEEE Trans. Microwave Theor. Tech. MTT-26, 428–433 (1978).
    [CrossRef]
  4. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II.
  5. B. S. Baldwin, A. E. Heins, “On the diffraction of a plane wave by an infinite plane grating,” Math. Scand. 2, 103–118 (1954).
  6. Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).
  7. V. Twersky, “On scattering of waves by an infinite grating of circular cylinders,” IRE Trans. Antennas Propag. AP-10, 737–765(1962).
    [CrossRef]
  8. R. Petit, M. Cadilhac, “Form of the electromagnetic field in the groove region of a perfectly conducting echelette grating,” J. Opt. Soc. Am. 73, 963–965 (1983).
    [CrossRef]
  9. M. G. Moharam, T. K. Gaylord, “Rigorous coupled wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  10. H. A. Kalhor, A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,”J. Opt. Soc. Am. 61, 43–48 (1971).
    [CrossRef]
  11. H. A. Kalhor, M. K. Moaveni, “Analysis of diffraction gratings by finite-difference coupling technique,” J. Opt. Soc. Am. 63, 1584–1588 (1973).
    [CrossRef]
  12. M. K. Moaveni, “Analysis of transmission gratings by the method of finite elements,” Proc. Inst. Electr. Eng. 126, 35–40 (1979).
    [CrossRef]
  13. C. H. Tsao, R. Mittra, “A spectral-iteration approach for analyzing scattering from frequency selective surfaces,”IEEE Trans. Antennas Propag. AP-30, 303–308 (1982).
    [CrossRef]
  14. K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
    [CrossRef]
  15. P. Silvester, M. S. Hsieh, “Finite-element solution of two dimensional exterior field problems,” Proc. Inst. Electr. Eng. 118, 1743–1747 (1971).
    [CrossRef]
  16. B. H. McDonald, A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theor. Tech. MTT-20, 841–847 (1972).
    [CrossRef]
  17. O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
    [CrossRef]
  18. J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  19. P. M. Morse, H. Feshbach, The Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  20. W. Kinsner, E. Delia Torro, “An iterative approach to the finite-element method in field problems,” IEEE Trans. Microwave Theor. Tech. MTT-22, 221–228 (1974).
    [CrossRef]
  21. R. C. Johnson, H. Jasik, Antenna Engineering Handbook (McGraw-Hill, New York, 1984), Chap. 46, p. 3.
  22. W. Wasylkiwskyj, “On the transmission coefficient of an infinite grating of parallel perfectly conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-19, 704–708 (1971).
    [CrossRef]
  23. H. A. Kalhor, A. Armand, “Scattering of waves by gratings of conducting cylinders,” Proc. Inst. Electr. Eng. 122, 245–48 (1975).
    [CrossRef]

1987 (1)

K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
[CrossRef]

1986 (1)

1983 (1)

1982 (1)

C. H. Tsao, R. Mittra, “A spectral-iteration approach for analyzing scattering from frequency selective surfaces,”IEEE Trans. Antennas Propag. AP-30, 303–308 (1982).
[CrossRef]

1981 (1)

1979 (1)

M. K. Moaveni, “Analysis of transmission gratings by the method of finite elements,” Proc. Inst. Electr. Eng. 126, 35–40 (1979).
[CrossRef]

1978 (2)

K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
[CrossRef]

P. S. Henry, J. T. Ruscio, “A low loss diffraction grating frequency multiplexer,” IEEE Trans. Microwave Theor. Tech. MTT-26, 428–433 (1978).
[CrossRef]

1977 (1)

O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
[CrossRef]

1975 (1)

H. A. Kalhor, A. Armand, “Scattering of waves by gratings of conducting cylinders,” Proc. Inst. Electr. Eng. 122, 245–48 (1975).
[CrossRef]

1974 (1)

W. Kinsner, E. Delia Torro, “An iterative approach to the finite-element method in field problems,” IEEE Trans. Microwave Theor. Tech. MTT-22, 221–228 (1974).
[CrossRef]

1973 (1)

1972 (1)

B. H. McDonald, A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theor. Tech. MTT-20, 841–847 (1972).
[CrossRef]

1971 (3)

W. Wasylkiwskyj, “On the transmission coefficient of an infinite grating of parallel perfectly conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-19, 704–708 (1971).
[CrossRef]

H. A. Kalhor, A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,”J. Opt. Soc. Am. 61, 43–48 (1971).
[CrossRef]

P. Silvester, M. S. Hsieh, “Finite-element solution of two dimensional exterior field problems,” Proc. Inst. Electr. Eng. 118, 1743–1747 (1971).
[CrossRef]

1962 (2)

Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).

V. Twersky, “On scattering of waves by an infinite grating of circular cylinders,” IRE Trans. Antennas Propag. AP-10, 737–765(1962).
[CrossRef]

1954 (1)

B. S. Baldwin, A. E. Heins, “On the diffraction of a plane wave by an infinite plane grating,” Math. Scand. 2, 103–118 (1954).

Agronovitch, Z. S.

Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).

Armand, A.

H. A. Kalhor, A. Armand, “Scattering of waves by gratings of conducting cylinders,” Proc. Inst. Electr. Eng. 122, 245–48 (1975).
[CrossRef]

Baldwin, B. S.

B. S. Baldwin, A. E. Heins, “On the diffraction of a plane wave by an infinite plane grating,” Math. Scand. 2, 103–118 (1954).

Bettress, P.

O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
[CrossRef]

Cadilhac, M.

Delia Torro, E.

W. Kinsner, E. Delia Torro, “An iterative approach to the finite-element method in field problems,” IEEE Trans. Microwave Theor. Tech. MTT-22, 221–228 (1974).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, The Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gaylord, T. K.

Heins, A. E.

B. S. Baldwin, A. E. Heins, “On the diffraction of a plane wave by an infinite plane grating,” Math. Scand. 2, 103–118 (1954).

Henry, P. S.

P. S. Henry, J. T. Ruscio, “A low loss diffraction grating frequency multiplexer,” IEEE Trans. Microwave Theor. Tech. MTT-26, 428–433 (1978).
[CrossRef]

Hsieh, M. S.

P. Silvester, M. S. Hsieh, “Finite-element solution of two dimensional exterior field problems,” Proc. Inst. Electr. Eng. 118, 1743–1747 (1971).
[CrossRef]

Hugonin, J. P.

Hutley, M. G.

M. G. Hutley, Diffraction Gratings (Academic, London, 1982).

Jasik, H.

R. C. Johnson, H. Jasik, Antenna Engineering Handbook (McGraw-Hill, New York, 1984), Chap. 46, p. 3.

Johnson, R. C.

R. C. Johnson, H. Jasik, Antenna Engineering Handbook (McGraw-Hill, New York, 1984), Chap. 46, p. 3.

Kalhor, H. A.

Kelley, K. W.

O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
[CrossRef]

Kinsner, W.

W. Kinsner, E. Delia Torro, “An iterative approach to the finite-element method in field problems,” IEEE Trans. Microwave Theor. Tech. MTT-22, 221–228 (1974).
[CrossRef]

Knop, K.

K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
[CrossRef]

Marchenks, V. S.

Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).

Matsunaga, T.

K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
[CrossRef]

McDonald, B. H.

B. H. McDonald, A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theor. Tech. MTT-20, 841–847 (1972).
[CrossRef]

Mittra, R.

C. H. Tsao, R. Mittra, “A spectral-iteration approach for analyzing scattering from frequency selective surfaces,”IEEE Trans. Antennas Propag. AP-30, 303–308 (1982).
[CrossRef]

Moaveni, M. K.

M. K. Moaveni, “Analysis of transmission gratings by the method of finite elements,” Proc. Inst. Electr. Eng. 126, 35–40 (1979).
[CrossRef]

H. A. Kalhor, M. K. Moaveni, “Analysis of diffraction gratings by finite-difference coupling technique,” J. Opt. Soc. Am. 63, 1584–1588 (1973).
[CrossRef]

Moharam, M. G.

Morse, P. M.

P. M. Morse, H. Feshbach, The Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Neureuther, A. R.

Noda, T.

K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
[CrossRef]

Petit, R.

Rayleigh, Lord

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II.

Ruscio, J. T.

P. S. Henry, J. T. Ruscio, “A low loss diffraction grating frequency multiplexer,” IEEE Trans. Microwave Theor. Tech. MTT-26, 428–433 (1978).
[CrossRef]

Shestopalov, V. P.

Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).

Silvester, P.

P. Silvester, M. S. Hsieh, “Finite-element solution of two dimensional exterior field problems,” Proc. Inst. Electr. Eng. 118, 1743–1747 (1971).
[CrossRef]

Tsao, C. H.

C. H. Tsao, R. Mittra, “A spectral-iteration approach for analyzing scattering from frequency selective surfaces,”IEEE Trans. Antennas Propag. AP-30, 303–308 (1982).
[CrossRef]

Twersky, V.

V. Twersky, “On scattering of waves by an infinite grating of circular cylinders,” IRE Trans. Antennas Propag. AP-10, 737–765(1962).
[CrossRef]

Uchida, K.

K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
[CrossRef]

Wasylkiwskyj, W.

W. Wasylkiwskyj, “On the transmission coefficient of an infinite grating of parallel perfectly conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-19, 704–708 (1971).
[CrossRef]

Wexler, A.

B. H. McDonald, A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theor. Tech. MTT-20, 841–847 (1972).
[CrossRef]

Zienkiewicz, O. C.

O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

C. H. Tsao, R. Mittra, “A spectral-iteration approach for analyzing scattering from frequency selective surfaces,”IEEE Trans. Antennas Propag. AP-30, 303–308 (1982).
[CrossRef]

K. Uchida, T. Noda, T. Matsunaga, “Spectral domain analysis of electromagnetic wave scattering by an infinite plane metallic grating,” IEEE Trans. Antennas Propag. AP-35, 46–52 (1987).
[CrossRef]

W. Wasylkiwskyj, “On the transmission coefficient of an infinite grating of parallel perfectly conducting circular cylinders,” IEEE Trans. Antennas Propag. AP-19, 704–708 (1971).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (3)

W. Kinsner, E. Delia Torro, “An iterative approach to the finite-element method in field problems,” IEEE Trans. Microwave Theor. Tech. MTT-22, 221–228 (1974).
[CrossRef]

B. H. McDonald, A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theor. Tech. MTT-20, 841–847 (1972).
[CrossRef]

P. S. Henry, J. T. Ruscio, “A low loss diffraction grating frequency multiplexer,” IEEE Trans. Microwave Theor. Tech. MTT-26, 428–433 (1978).
[CrossRef]

Int. J. Numer. Methods Eng. (1)

O. C. Zienkiewicz, K. W. Kelley, P. Bettress, “The coupling of finite-element method and boundary solution procedures,” Int. J. Numer. Methods Eng. 11, 355–375 (1977).
[CrossRef]

IRE Trans. Antennas Propag. (1)

V. Twersky, “On scattering of waves by an infinite grating of circular cylinders,” IRE Trans. Antennas Propag. AP-10, 737–765(1962).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Math. Scand. (1)

B. S. Baldwin, A. E. Heins, “On the diffraction of a plane wave by an infinite plane grating,” Math. Scand. 2, 103–118 (1954).

Opt. Commun. (1)

K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
[CrossRef]

Proc. Inst. Electr. Eng. (3)

M. K. Moaveni, “Analysis of transmission gratings by the method of finite elements,” Proc. Inst. Electr. Eng. 126, 35–40 (1979).
[CrossRef]

P. Silvester, M. S. Hsieh, “Finite-element solution of two dimensional exterior field problems,” Proc. Inst. Electr. Eng. 118, 1743–1747 (1971).
[CrossRef]

H. A. Kalhor, A. Armand, “Scattering of waves by gratings of conducting cylinders,” Proc. Inst. Electr. Eng. 122, 245–48 (1975).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

Z. S. Agronovitch, V. S. Marchenks, V. P. Shestopalov, “The diffraction of electromagnetic waves from plane metallic lattices,” Sov. Phys. Tech. Phys. 7, 277–286 (1962).

Other (4)

M. G. Hutley, Diffraction Gratings (Academic, London, 1982).

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II.

R. C. Johnson, H. Jasik, Antenna Engineering Handbook (McGraw-Hill, New York, 1984), Chap. 46, p. 3.

P. M. Morse, H. Feshbach, The Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the transmission grating of an arbitrary cross section having a period a. A single cell is divided into three regions.

Fig. 2
Fig. 2

Detail of a single cell, showing the path of integration around region R2.

Fig. 3
Fig. 3

Discretization of the gap region and its boundary into triangular and linear elements.

Fig. 4
Fig. 4

Transmission coefficients of three circular gratings with elemental diameters 0.2λ, 0.4λ, and 0.6λ plotted against the relative spacing a/λ for perpendicular polarization at normal incidence. (a) a/λ ≤ 1; (b) a/λ > 1.

Fig. 5
Fig. 5

Transmission coefficients of three circular gratings with elemental diameters 0.2λ, 0.4λ, and 0.6λ plotted against the relative spacing a/λ for parallel polarization at normal incidence. (a) a/λ ≤ 1; (b) a/λ > 1.

Fig. 6
Fig. 6

Transmitted power in the zero-order mode plotted against the relative dielectric constant r at a/λ = 1.0, b/λ = 0.2, and δ = 0.0 at normal incidence and perpendicular polarization.

Fig. 7
Fig. 7

Transmitted power in the zero-order mode plotted against the relative spacing a/λ. The dielectric is polystyrene (r = 2.53 and δ = 0.0004) at normal incidence and b/λ = 0.2.

Fig. 8
Fig. 8

Transmitted power in the zero-order mode plotted against the relative spacing a/λ. The dielectric is inhomogeneous, with r = exp(−8.047x) and δ = 0.0, at normal incidence and b/λ = 0.2.

Fig. 9
Fig. 9

Normalized transmitted power in orders 0 and −1 plotted against the angle of incidence for perpendicular polarization. The dielectric is Bakelite (r = 4.75 and δ = 0.013), a/λ = 1.0, and b/λ = 0.2.

Tables (3)

Tables Icon

Table 1 Accuracy Test by Comparison with the Exact Solutiona

Tables Icon

Table 2 Convergence Test by Comparison with the Exact Solutiona

Tables Icon

Table 3 Convergence Test by the Conservation-of-Power Criteriona

Equations (52)

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2 ϕ ( x , y ) + k 0 2 ϕ ( x , y ) = 0 ,
2 E ( x , y ) + k 2 E ( x , y ) = ( E · c c )
2 H ( x , y ) + k 2 H ( x , y ) = j ω ( c × E )
ϕ = 0
ϕ n ̂ = 0
ϕ i = exp ( j β 0 x + j γ 0 y ) â z ,
ϕ r = n = A n exp ( j β n x j γ n y ) ,
ϕ t = n = B n exp [ j β n x + j γ n ( y + b ) ] ,
β n = β 0 + 2 π n a ,
γ n = ( k 0 2 β n 2 ) 1 / 2 .
χ = 1 2 R 2 ( | ϕ s | 2 k 2 ϕ s 2 ) dR 2
δ χ = R 2 ( 2 ϕ s + k 2 ϕ s ) δ ϕ s dR 2 + c δ ϕ s ϕ s n ̂ d l ,
δ χ c δ ϕ s ϕ s n ̂ d l = 0 .
δ χ + c 1 ϕ s y δ ϕ s d x + c 2 ϕ s y δ ϕ s d x = 0 ,
ϕ s | y = 0 = ϕ i | y = 0 + ϕ r | y = 0 = exp ( j β 0 x ) + n = A n exp ( j β n x ) ,
ϕ s y | y = 0 = j γ 0 exp ( j β 0 x ) j n = γ n A n exp ( j β n x ) .
ϕ s | y = b = ϕ t | y = b = n = B n exp ( j β n x ) ,
ϕ s y | y = b = j n = γ n B n exp ( j β n x ) .
δ ϕ s = m = δ A m exp ( j β m x ) on c 1 ,
δ ϕ s = m = δ B m exp ( j β m x ) on c 2 ,
δ A m = 1 a c 1 δ ϕ s exp ( j β m x ) d x ,
δ B m = 1 a c 2 δ ϕ s exp ( j β m x ) d x .
δ x + n = γ 0 ( β 0 + β n ) δ A n { exp [ j ( β 0 + β n ) a ] 1 } n = m = γ 0 ( β n + β m ) { exp [ j ( β n + β m ) a ] 1 } × ( A n δ A m B n δ B m ) = 0 .
N x = 3 2 a b N y .
| T | 2 = | B 0 | 2 = 1 1 1 + [ 2 a λ ln ( a π b ) ] 2
| T | 2 = | B 0 | 2 = 1 ( π 2 b 2 2 λ a ) 2 1 + ( π 2 b 2 2 λ a ) 2
r = exp ( 8.047 x ) .
ϕ r | y = 0 = n = A n exp ( j β n x ) ;
A n = 1 a 0 a ϕ r | y = 0 exp ( j β n x ) d x = 1 a 0 a [ ϕ s | y = 0 ϕ i | y = 0 ] exp ( j β n x ) d x = 1 a 0 a exp [ j ( β n β 0 ) x ] d x + 1 a 0 a ϕ s | y = 0 exp ( j β n x ) d x .
ϕ s = [ N l , N k ] [ ϕ l ϕ k ] ,
N l = x k x h , N k = x x l h ,
[ A ] = [ EA ] + [ E ] [ Φ 1 ] ,
E A n = { 1 a { exp [ j ( β 0 β n ) a ] 1 β 0 β n } , n 0 1 , n = 0 .
E n l = 1 a h [ g n l f n ( l 1 ) ] ,
g n l = exp ( j β n x l + 1 ) β n 2 + exp ( j β n x l ) ( j h β n + 1 β n 2 )
f n l = exp ( j β n x l + 1 ) ( j h β n + 1 β n 2 ) exp ( j β n x l ) β n 2 .
[ B ] = [ E ] [ Φ 2 ] ,
L 1 i = F 1 i + n = C in A in ,
F 1 i = m = γ 0 β 0 + β m { exp [ j ( β n + β 0 ) a ] 1 } E mi , C in = m = γ n β n + β m { exp [ j ( β n + β m ) a ] 1 } E mi ,
[ L 1 ] = [ F 1 ] + [ C ] [ A ] .
[ L 1 ] = [ F 1 ] + [ F 2 ] + [ D 1 ] [ Φ 1 ] ,
[ L 2 ] = [ D 2 ] [ Φ 2 ] ,
C in = m = γ m β n + β m { exp [ j ( β n + β m ) a ] 1 } E mi ,
ϕ s e = [ N i , N l , N k ] [ ϕ i ϕ l ϕ k ] ,
N i = 1 2 Δ ( a i + b i x + c i y ) , a i = x l y k x k y l , b i = y l y k , c i = x k x l .
χ e ϕ i = 1 ( 2 Δ ) 2 Δ { b i [ B ] t [ Φ e ] + c i [ C ] t [ Φ e ] k 2 [ N ] t [ Φ e ] N i } d x d y ,
χ e ϕ = [ χ e / ϕ i χ e / ϕ l χ e / ϕ k ] = [ S ] e [ Φ e ] ,
S r s e = { b r b s + c r c s 4 Δ k 2 Δ 6 if r = s b r b s + c r c s 4 Δ k 2 Δ 12 if r s .
χ ϕ i = all elements χ e ϕ i = j = 1 g S i j ϕ i .
[ χ ϕ ] R 2 = [ S ] [ Φ ] R 2 ,
[ S T ] [ Φ ] = [ F ] ,
S T i j = S i j D 1 i j D 2 i j , F i = F 1 i + F 2 i .

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