Abstract

The Fourier modal method for crossed gratings with square symmetry is reformulated by use of a group-theoretic approach that we developed recently. In the new formulation, a crossed-grating problem is decomposed into six symmetrical basis problems whose field distributions are the symmetry modes of the grating. Then the symmetrical basis problems are solved with symmetry simplifications, whose solutions are superposed to get the solution of the original problem. Theoretical and numerical results show that when the grating is at some Littrow mountings, the computation efficiency can be improved effectively: The memory occupation is reduced by 34 and the computation time is reduced by a factor from 25.6 to 64 in different incident cases. Numerical examples are given to show the effectiveness of the new formulation.

© 2006 Optical Society of America

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References

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  1. B. Bai and L. Li, 'Reduction of computation time for crossed-grating problems: a group-theoretic approach,' J. Opt. Soc. Am. A 21, 1886-1894 (2004).
    [CrossRef]
  2. Ph. Lalanne and D. Lemercier-Lalanne, 'On the effective medium theory of subwavelength periodic structures,' J. Mod. Opt. 43, 2063-2085 (1996).
    [CrossRef]
  3. Ph. Lalanne, 'Improved formulation of the coupled-wave method for two-dimensional gratings,' J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  4. C. Zhou and L. Li, 'Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,' J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
    [CrossRef]
  5. Z. Y. Li and K. M. Ho, 'Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,' Phys. Rev. B 68, 245117 (2003).
    [CrossRef]
  6. R. Bräuer and O. Bryngdahl, 'Electromagnetic diffraction analysis of two-dimensional gratings,' Opt. Commun. 100, 1-5 (1993).
    [CrossRef]
  7. E. Noponen and J. Turunen, 'Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,' J. Opt. Soc. Am. A 11, 2494-2502 (1994).
    [CrossRef]
  8. L. Li, 'New formulation of the Fourier modal method for crossed surface-relief gratings,' J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  9. C. Hammond, The Basics of Crystallography and Diffraction (Oxford U. Press, 2001).
  10. B. Bai and L. Li, 'Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case,' J. Opt. Soc. Am. A 22, 654-661 (2005).
    [CrossRef]
  11. B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,' J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
    [CrossRef]
  12. B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,' J. Mod. Opt. 52, 1619-1634 (2005).
    [CrossRef]
  13. J.F.Cornwell, ed., Group Theory in Physics: An Introduction (Academic, 1997), App. C, pp. 299-318.
  14. L. Li, 'Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,' J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  15. L. Li, 'Note on the S-matrix propagation algorithm,' J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]

2005 (3)

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,' J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,' J. Mod. Opt. 52, 1619-1634 (2005).
[CrossRef]

B. Bai and L. Li, 'Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case,' J. Opt. Soc. Am. A 22, 654-661 (2005).
[CrossRef]

2004 (2)

B. Bai and L. Li, 'Reduction of computation time for crossed-grating problems: a group-theoretic approach,' J. Opt. Soc. Am. A 21, 1886-1894 (2004).
[CrossRef]

C. Zhou and L. Li, 'Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,' J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

2003 (2)

Z. Y. Li and K. M. Ho, 'Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,' Phys. Rev. B 68, 245117 (2003).
[CrossRef]

L. Li, 'Note on the S-matrix propagation algorithm,' J. Opt. Soc. Am. A 20, 655-660 (2003).
[CrossRef]

1997 (2)

1996 (2)

L. Li, 'Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,' J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

Ph. Lalanne and D. Lemercier-Lalanne, 'On the effective medium theory of subwavelength periodic structures,' J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

1994 (1)

1993 (1)

R. Bräuer and O. Bryngdahl, 'Electromagnetic diffraction analysis of two-dimensional gratings,' Opt. Commun. 100, 1-5 (1993).
[CrossRef]

Bai, B.

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,' J. Mod. Opt. 52, 1619-1634 (2005).
[CrossRef]

B. Bai and L. Li, 'Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case,' J. Opt. Soc. Am. A 22, 654-661 (2005).
[CrossRef]

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,' J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

B. Bai and L. Li, 'Reduction of computation time for crossed-grating problems: a group-theoretic approach,' J. Opt. Soc. Am. A 21, 1886-1894 (2004).
[CrossRef]

Bräuer, R.

R. Bräuer and O. Bryngdahl, 'Electromagnetic diffraction analysis of two-dimensional gratings,' Opt. Commun. 100, 1-5 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer and O. Bryngdahl, 'Electromagnetic diffraction analysis of two-dimensional gratings,' Opt. Commun. 100, 1-5 (1993).
[CrossRef]

Hammond, C.

C. Hammond, The Basics of Crystallography and Diffraction (Oxford U. Press, 2001).

Ho, K. M.

Z. Y. Li and K. M. Ho, 'Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,' Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, 'Improved formulation of the coupled-wave method for two-dimensional gratings,' J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

Ph. Lalanne and D. Lemercier-Lalanne, 'On the effective medium theory of subwavelength periodic structures,' J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

Lemercier-Lalanne, D.

Ph. Lalanne and D. Lemercier-Lalanne, 'On the effective medium theory of subwavelength periodic structures,' J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

Li, L.

Li, Z. Y.

Z. Y. Li and K. M. Ho, 'Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,' Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Noponen, E.

Turunen, J.

Zhou, C.

C. Zhou and L. Li, 'Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,' J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

J. Mod. Opt. (2)

Ph. Lalanne and D. Lemercier-Lalanne, 'On the effective medium theory of subwavelength periodic structures,' J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,' J. Mod. Opt. 52, 1619-1634 (2005).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

B. Bai and L. Li, 'Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,' J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

C. Zhou and L. Li, 'Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,' J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

R. Bräuer and O. Bryngdahl, 'Electromagnetic diffraction analysis of two-dimensional gratings,' Opt. Commun. 100, 1-5 (1993).
[CrossRef]

Phys. Rev. B (1)

Z. Y. Li and K. M. Ho, 'Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,' Phys. Rev. B 68, 245117 (2003).
[CrossRef]

Other (2)

C. Hammond, The Basics of Crystallography and Diffraction (Oxford U. Press, 2001).

J.F.Cornwell, ed., Group Theory in Physics: An Introduction (Academic, 1997), App. C, pp. 299-318.

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Figures (6)

Fig. 1
Fig. 1

Representative plane pattern of plane group p 4 m m . The periods of the pattern in its two principal periodic directions are both d and the width of each white square is w. Lines O x , O y , O a , and O b are the mirror lines for four reflection operations.

Fig. 2
Fig. 2

Crossed grating with square symmetry illuminated by a linearly polarized plane wave. A rectangular Cartesian coordinate system is set up so that its x and y axes are along the two periodic directions, and the z axis is parallel to the normal of the grating plane.

Fig. 3
Fig. 3

Schematic illustration of the eight symmetry modes of a crossed grating with square symmetry. In each mode, the symmetrical distribution of the electric vectors projected onto the O x y plane is shown by arrows. The solid (open) arrows indicate the vectors with the same z components, which have the opposite sign to those indicated by the open (solid) arrows.

Fig. 4
Fig. 4

Truncated reciprocal lattice with square symmetry. All the points represent Fourier terms retained for the incident case 1 (the origin of the k space is at point O), and those within the double-dashed–dotted boundary are for the incident case 2 (the origin of the k space is at point O ). Two sets of symmetrical points are marked by dashed circles [for case 1] and solid circles [for case 2]. The points within other close boundaries represent the Fourier terms retained in numerical computation for four cases: dashed for E ̃ x [ 3 ] , solid for E ̃ y [ 3 ] , dashed–dotted for E ̃ x [ 8 ] , and dotted for E ̃ y [ 8 ] .

Fig. 5
Fig. 5

Convergence of the ( 1 , 3 ) th , ( 1 , 0 ) th , and ( 4 , + 1 ) th reflected orders of the metallic crossed grating with square symmetry, which were computed by both NA and OA.

Fig. 6
Fig. 6

Comparison of the total computation time costs of NA and OA.

Tables (2)

Tables Icon

Table 1 Character Table of Point Group C 4 ν a

Tables Icon

Table 2 Reduction of Computation Time for Different Incident Cases

Equations (76)

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T ( 3 ) ( g j ) = ± 1 ,
T ( 5 ) ( e ) = [ 1 0 0 1 ] , T ( 5 ) ( c 2 ) = [ 1 0 0 1 ] ,
T ( 5 ) ( c 4 1 ) = [ 0 1 1 0 ] , T ( 5 ) ( c 4 3 ) = [ 0 1 1 0 ] ,
T ( 5 ) ( σ x ) = [ 1 0 0 1 ] , T ( 5 ) ( σ y ) = [ 1 0 0 1 ] ,
T ( 5 ) ( σ a ) = [ 0 1 1 0 ] , T ( 5 ) ( σ b ) = [ 0 1 1 0 ] .
a [ 1 ] = 2 4 ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) ,
a [ 2 ] = 2 4 ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) ,
a [ 3 ] = 2 4 ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) ,
a [ 4 ] = 2 4 ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) ,
a [ 5 ] = 1 2 ( 1 , 1 , 0 , 0 , 1 , 1 , 0 , 0 ) ,
a [ 6 ] = 1 2 ( 0 , 0 , 1 , 1 , 0 , 0 , 1 , 1 ) ,
a [ 7 ] = 1 2 ( 0 , 0 , 1 , 1 , 0 , 0 , 1 , 1 ) ,
a [ 8 ] = 1 2 ( 1 , 1 , 0 , 0 , 1 , 1 , 0 , 0 ) ,
E 0 ( r ) = 2 4 a j = 1 4 E [ j ] ( r ) + 1 2 a [ E [ 5 ] ( r ) + E [ 8 ] ( r ) ] .
T = T ( 1 ) T ( 2 ) T ( 3 ) T ( 4 ) T ( 5 ) T ( 5 ) ,
M ( e ) = I , M ( c 2 ) = 1 1 1 ,
M ( c 4 1 ) = [ 0 1 1 0 ] 1 , M ( c 4 3 ) = [ 0 1 1 0 ] 1 ,
M ( σ x ) = 1 1 1 , M ( σ y ) = 1 1 1 ,
M ( σ a ) = [ 0 1 1 0 ] 1 , M ( σ b ) = [ 0 1 1 0 ] 1 .
M ( g n ) E [ i ] [ M ( g n ) 1 r ] = j = 1 8 T j i ( g n ) E [ j ] ( r ) ( i , n = 1 , , 8 ) .
E x [ 3 ] ( x , y ) = E x [ 3 ] ( x , y ) = E y [ 3 ] ( y , x ) = E y [ 3 ] ( y , x ) = E x [ 3 ] ( x , y ) = E x [ 3 ] ( x , y ) = E y [ 3 ] ( y , x ) = E y [ 3 ] ( y , x ) ,
E y [ 3 ] ( x , y ) = E y [ 3 ] ( x , y ) = E x [ 3 ] ( y , x ) = E x [ 3 ] ( y , x ) = E y [ 3 ] ( x , y ) = E y [ 3 ] ( x , y ) = E x [ 3 ] ( y , x ) = E x [ 3 ] ( y , x ) ;
E x [ 8 ] ( x , y ) = E x [ 8 ] ( x , y ) = E x [ 8 ] ( x , y ) = E x [ 8 ] ( x , y ) ,
E y [ 8 ] ( x , y ) = E y [ 8 ] ( x , y ) = E y [ 8 ] ( x , y ) = E y [ 8 ] ( x , y ) ,
E x [ 7 ] ( x , y ) = E y [ 8 ] ( y , x ) , E y [ 7 ] ( x , y ) = E x [ 8 ] ( y , x ) ,
E ρ [ 3 ] ( r ) = m , n { E ρ m n [ 3 , 1 ] ( z ) exp [ i ( α m x + β n y ) ] + E ρ ( m ) ( n ) [ 3 , 2 ] ( z ) exp [ i ( α m x β n y ) ] + E ρ ( n ) m [ 3 , 3 ] ( z ) exp [ i ( β n x + α m y ) ] + E ρ n ( m ) [ 3 , 4 ] ( z ) exp [ i ( β n x α m y ) ] + E ρ m ( n ) [ 3 , 5 ] ( z ) exp [ i ( α m x β n y ) ] + E ρ ( m ) n [ 3 , 6 ] ( z ) exp [ i ( α m x + β n y ) ] + E ρ n m [ 3 , 7 ] ( z ) exp [ i ( β n x + α m y ) ] + E ρ ( n ) ( m ) [ 3 , 8 ] ( z ) exp [ i ( β n x α m y ) ] } ,
α m = α 2 s m = β s t + m = β s t m ,
β n = β 2 t n = α s t n = α s + t + n .
α m = α 2 s m 1 = β s t + m = β s t m 1 ,
β n = β 2 t n 1 = α s t n 1 = α s + t + n .
E ρ [ 3 ] ( r ) = n = L 2 L 2 + m = L 1 L 1 + E ρ m n [ 3 ] ( z ) exp [ i ( α m x + β n y ) ] ,
E ρ m n [ 3 ] E ρ m n [ 3 , 1 ] + E ρ ( 2 s + m ) ( 2 t + n ) [ 3 , 2 ] + E ρ ( s + t + m ) ( s + t + n ) [ 3 , 3 ] + E ρ ( s t + m ) ( s + t + n ) [ 3 , 4 ] + E ρ m ( 2 t + n ) [ 3 , 5 ] + E ρ ( 2 s + m ) n [ 3 , 6 ] + E ρ ( s t + m ) ( s + t + n ) [ 3 , 7 ] + E ρ ( s + t + m ) ( s + t + n ) [ 3 , 8 ]
E x m n [ 3 ] = E x ( 2 s m ) ( 2 t n ) [ 3 ] = E y ( s + t + n ) ( s t m ) [ 3 ] = E y ( s t n ) ( s t + m ) [ 3 ] = E x m ( 2 t n ) [ 3 ] = E x ( 2 s m ) n [ 3 ] = E y ( s + t + n ) ( s t + m ) [ 3 ] = E y ( s t n ) ( s t m ) [ 3 ] ,
E y m n [ 3 ] = E y ( 2 s m ) ( 2 t n ) [ 3 ] = E x ( s + t + n ) ( s t m ) [ 3 ] = E x ( s t n ) ( s t + m ) [ 3 ] = E y m ( 2 t n ) [ 3 ] = E y ( 2 s m ) n [ 3 ] = E x ( s + t + n ) ( s t + m ) [ 3 ] = E x ( s t n ) ( s t m ) [ 3 ] ,
E ρ [ 3 ] ( r ) = exp ( i γ 00 ( + 1 ) z ) { I ρ [ 3 , 1 ] exp [ i ( α 0 x + β 0 y ) ] + I ρ [ 3 , 2 ] exp [ i ( α 2 s x + β 2 t y ) ] + I ρ [ 3 , 3 ] exp [ i ( α s t x + β s t y ) ] + I ρ [ 3 , 4 ] exp [ i ( α s + t x + β s t y ) ] + I ρ [ 3 , 5 ] exp [ i ( α 0 x + β 2 t y ) ] + I ρ [ 3 , 6 ] exp [ i ( α 2 s x + β 0 y ) ] + I ρ [ 3 , 7 ] exp [ i ( α s + t x + β s t y ) ] + I ρ [ 3 , 8 ] exp [ i ( α s t x + β s t y ) ] } + n = L 2 L 2 + m = L 1 L 1 + R ρ m n [ 3 ] exp [ i ( α m x + β n y + γ m n ( + 1 ) z ) ]
( ρ = x , y ; z > h ) ,
E ρ [ 3 ] ( r ) = n = L 2 L 2 + n = L 1 L 1 + T ρ m n [ 3 ] exp [ i ( α m x + β n y γ m n ( 1 ) z ) ]
( ρ = x , y ; z < 0 ) ,
I x [ 3 , 1 ] = I x [ 3 , 2 ] = I y [ 3 , 3 ] = I y [ 3 , 4 ] = I x [ 3 , 5 ] = I x [ 3 , 6 ] = I y [ 3 , 7 ] = I y [ 3 , 8 ] = 2 4 ( cos ψ cos θ cos φ + sin ψ sin φ ) ,
I y [ 3 , 1 ] = I y [ 3 , 2 ] = I x [ 3 , 3 ] = I x [ 3 , 4 ] = I y [ 3 , 5 ] = I y [ 3 , 6 ] = I x [ 3 , 7 ] = I x [ 3 , 8 ] = 2 4 ( cos ψ cos θ sin φ sin ψ cos φ ) .
( F G μ k 0 2 γ 2 ) [ E ̃ x j [ 3 ] E ̃ y j [ 3 ] ] = 0 ,
F = [ U ̃ 11 ( i , j ) μ k 0 2 δ i j + U ̃ 12 ( i , j ) μ k 0 2 δ i j + U ̃ 21 ( i , j ) U ̃ 22 ( i , j ) ] ,
G = [ α i β i δ i j + V ̃ 11 ( i , j ) α i 2 δ i j + V ̃ 12 ( i , j ) β i 2 δ i j + V ̃ 21 ( i , j ) α i β i δ i j + V ̃ 22 ( i , j ) ] ,
E ρ [ 8 ] ( r ) = n = L 2 L 2 + m = L 1 L 1 + [ E ρ m n [ 8 , 1 ] ( z ) + E ρ ( 2 s + m ) ( 2 t + n ) [ 8 , 2 ] ( z ) + E ρ m ( 2 t + n ) [ 8 , 5 ] ( z ) + E ρ ( 2 s + m ) n [ 8 , 6 ] ( z ) ] exp [ i ( α m x + β n y ) ] n = L 2 L 2 + m = L 1 L 1 + E ρ m n [ 8 ] ( z ) exp [ i ( α m x + β n y ) ] .
E x m n [ 8 ] = E x ( 2 s m ) ( 2 t n ) [ 8 ] = E x m ( 2 t n ) [ 8 ] = E x ( 2 s m ) n [ 8 ] ,
E y m n [ 8 ] = E y ( 2 s m ) ( 2 t n ) [ 8 ] = E y m ( 2 t n ) [ 8 ] = E y ( 2 s m ) n [ 8 ] ,
E ρ [ 8 ] ( r ) = exp ( i γ 00 ( + 1 ) z ) { I ρ [ 8 , 1 ] exp [ i ( α 0 x + β 0 y ) ] + I ρ [ 8 , 1 ] exp [ i ( α 2 s x + β 2 t y ) ] ± I ρ [ 8 , 1 ] exp [ i ( α 0 x + β 2 t y ) ] ± I ρ [ 8 , 1 ] exp [ i ( α 2 s x + β 0 y ) ] } + m = L 1 L 1 + n = L 2 L 2 + R ρ m n [ 8 ] exp [ i ( α m x + β n y + γ m n ( + 1 ) x 3 ) ]
( ρ x , y ; z > h ) ,
E ρ [ 8 ] ( r ) = n = L 2 L 2 + m = L 1 L 1 + T ρ m n [ 8 ] exp [ i ( α m x + β n y γ m n ( 1 ) z ) ]
( ρ = x , y ; z < 0 ) ,
I x [ 8 , 1 ] = 1 2 ( cos ψ cos θ cos φ + sin ψ sin φ ) ,
I y [ 8 , 1 ] = 1 2 ( cos ψ cos θ sin φ sin ψ cos φ ) ;
U ̃ p 1 ( i , j ) = U p 1 ( i , j ) U p 1 ( i , j 1 ) U p 2 ( i , j 2 ) + U p 2 ( i , j 3 ) ,
V ̃ 12 ( i , j ) = V 12 ( i , j ) V 12 ( i , j 1 ) ,
V ̃ 22 ( i , j ) = V 21 ( i , j 2 ) V 21 ( i , j 3 ) ;
U ̃ p 1 ( i , j ) = U p 1 ( i , j ) U p 1 ( i , j 1 ) + U p 1 ( i , j 2 ) U p 1 ( i , j 3 ) + U p 2 ( i , j ) U p 2 ( i , j 1 ) U p 2 ( i , j 2 ) + U p 2 ( i , j 3 ) ,
V ̃ 12 ( i , j ) = α i β i δ i j + V 12 ( i , j ) V 12 ( i , j 1 ) + V 12 ( i , j 2 ) V 12 ( i , j 3 ) ,
V ̃ 22 ( i , j ) = V 21 ( i , j ) + V 21 ( i , j 1 ) + V 21 ( i , j 2 ) V 21 ( i , j 3 ) ;
U ̃ p 1 ( i , j ) = U p 1 ( i , j ) U p 1 ( i , j 1 ) + U p 1 ( i , j 4 ) U p 1 ( i , j 5 ) U p 2 ( i , j 2 ) + U p 2 ( i , j 3 ) + U p 2 ( i , j 6 ) U p 2 ( i , j 7 ) ,
U ̃ p 2 ( i , j ) = U p 2 ( i , j ) + U p 1 ( i , j 2 ) U p 1 ( i , j 3 ) + U p 1 ( i , j 6 ) U p 1 ( i , j 7 ) U p 2 ( i , j 1 ) U p 2 ( i , j 4 ) + U p 2 ( i , j 5 ) ,
V ̃ 11 ( i , j ) = V 12 ( i , j 2 ) + V 12 ( i , j 3 ) V 12 ( i , j 6 ) + V 12 ( i , j 7 ) ,
V ̃ 12 ( i , j ) = V 12 ( i , j ) V 12 ( i , j 1 ) + V 12 ( i , j 4 ) V 12 ( i , j 5 ) ,
V ̃ 21 ( i , j ) = V 21 ( i , j ) V 21 ( i , j 1 ) V 21 ( i , j 4 ) + V 21 ( i , j 5 ) ,
V ̃ 22 ( i , j ) = V 21 ( i , j 2 ) V 21 ( i , j 3 ) V 21 ( i , j 6 ) + V 21 ( i , j 7 ) .
U 11 ( i , j ) = α i ϵ i , j 1 β j ,
U 12 ( i , j ) = α i ϵ i , j 1 α j ,
U 21 ( i , j ) = β i ϵ i , j 1 β j ,
U 22 ( i , j ) = β i ϵ i , j 1 α j ,
V 12 ( i , j ) = μ k 0 2 ϵ i , j ,
V 21 ( i , j ) = μ k 0 2 ϵ i , j ;
W ̃ p q ( i , j ) = W p q ( i , j ) ;
W ̃ p q ( i , j ) = W p q ( i , j ) + W p q ( i , j 1 ) ;
U ̃ p 1 ( i , j ) = U p 1 ( i , j ) + U p 1 ( i , j 1 ) U p 1 ( i , j 4 ) U p 1 ( i , j 5 ) ,
U ̃ p 2 ( i , j ) = U p 2 ( i , j ) + U p 2 ( i , j 1 ) + U p 2 ( i , j 4 ) + U p 2 ( i , j 5 ) ,
V ̃ 12 ( i , j ) = V 12 ( i , j ) + V 12 ( i , j 1 ) V 12 ( i , j 4 ) V 12 ( i , j 5 ) ,
V ̃ 21 ( i , j ) = V 21 ( i , j ) + V 21 ( i , j 1 ) + V 21 ( i , j 4 ) + V 21 ( i , j 5 ) .

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