Abstract

The differential theory of diffraction by an arbitrary-shaped body made of arbitrary anisotropic material is developed. The electromagnetic field is expanded on the basis of vector spherical harmonics, and the Maxwell equations in spherical coordinates are reduced to a first-order differential set. When discontinuities of permittivity exist, we apply the fast numerical factorization to find the link between the electric field vector and the vector of electric induction, developed in a truncated basis. The diffraction problem is reduced to a boundary-value problem by using a shooting method combined with the S-matrix propagation algorithm, formulated for the field components instead of the amplitudes.

© 2006 Optical Society of America

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References

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  1. B. Stout, M. Nevière, and E. Popov, "Light diffraction by a three-dimensional object: differential theory," J. Opt. Soc. Am. A 22, 2385-2404 (2005).
    [CrossRef]
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    [CrossRef]
  5. J. C. Monzon, "Three-dimensional field expansion in the most general rotationally symmetric anisotropic material: application to the scattering by a sphere," IEEE Trans. Antennas Propag. 37, 728-735 (1989).
    [CrossRef]
  6. W. Ren and X. B. Wu, "Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder," J. Phys. D 28, 1031-1039 (1995).
    [CrossRef]
  7. A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
    [CrossRef]
  8. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).
  9. B. Stout, M. Nevière, and E. Popov, "Mie scattering by an anisotropic object. Part I. Homogeneous sphere," J. Opt. Soc. Am. A 23, 1111-1123 (2006).
    [CrossRef]
  10. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  11. E. Popov, M. Nevière, and N. Bonod, "Factorization of products of discontinuous functions applied to Fourier-Bessel basis," J. Opt. Soc. Am. A 21, 46-51 (2004).
    [CrossRef]
  12. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).
  13. Y. L. Xu, "Fast evaluation of the Gaunt coefficients," Math. Comput. 65, 1601-1612 (1996).
    [CrossRef]

2006 (1)

2005 (1)

2004 (2)

E. Popov, M. Nevière, and N. Bonod, "Factorization of products of discontinuous functions applied to Fourier-Bessel basis," J. Opt. Soc. Am. A 21, 46-51 (2004).
[CrossRef]

B. T. Draine, "Interstellar dust," in Origin and Evolution of the Elements, A.McWilliams and M.Rauch, eds. (Cambridge U. Press, 2004), p. 230.

2003 (1)

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

2002 (1)

A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
[CrossRef]

1996 (2)

1995 (1)

W. Ren and X. B. Wu, "Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder," J. Phys. D 28, 1031-1039 (1995).
[CrossRef]

1990 (1)

1989 (1)

J. C. Monzon, "Three-dimensional field expansion in the most general rotationally symmetric anisotropic material: application to the scattering by a sphere," IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

1983 (1)

M. Gottlied, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Defection, Optical Engineering Series (Marcel Dekker, New York, 1983).

1960 (1)

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

Bonod, N.

Capsalis, C. N.

Draine, B. T.

B. T. Draine, "Interstellar dust," in Origin and Evolution of the Elements, A.McWilliams and M.Rauch, eds. (Cambridge U. Press, 2004), p. 230.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

Gottlied, M.

M. Gottlied, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Defection, Optical Engineering Series (Marcel Dekker, New York, 1983).

Ireland, C. L. M.

M. Gottlied, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Defection, Optical Engineering Series (Marcel Dekker, New York, 1983).

Kiselev, A. D.

A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
[CrossRef]

Ley, J. M.

M. Gottlied, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Defection, Optical Engineering Series (Marcel Dekker, New York, 1983).

Li, L.

Monzon, J. C.

J. C. Monzon, "Three-dimensional field expansion in the most general rotationally symmetric anisotropic material: application to the scattering by a sphere," IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

Nevière, M.

Papadakis, S. N.

Popov, E.

Ren, W.

W. Ren and X. B. Wu, "Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder," J. Phys. D 28, 1031-1039 (1995).
[CrossRef]

Reshetnyaba, V. Yu.

A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
[CrossRef]

Sluckain, T. J.

A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
[CrossRef]

Stout, B.

Uzunoglu, N. K.

Wu, X. B.

W. Ren and X. B. Wu, "Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder," J. Phys. D 28, 1031-1039 (1995).
[CrossRef]

Xu, Y. L.

Y. L. Xu, "Fast evaluation of the Gaunt coefficients," Math. Comput. 65, 1601-1612 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. C. Monzon, "Three-dimensional field expansion in the most general rotationally symmetric anisotropic material: application to the scattering by a sphere," IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Phys. D (1)

W. Ren and X. B. Wu, "Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder," J. Phys. D 28, 1031-1039 (1995).
[CrossRef]

Math. Comput. (1)

Y. L. Xu, "Fast evaluation of the Gaunt coefficients," Math. Comput. 65, 1601-1612 (1996).
[CrossRef]

Phys. Rev. E (1)

A. D. Kiselev, V. Yu. Reshetnyaba, and T. J. Sluckain, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609 (2002).
[CrossRef]

Other (4)

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

B. T. Draine, "Interstellar dust," in Origin and Evolution of the Elements, A.McWilliams and M.Rauch, eds. (Cambridge U. Press, 2004), p. 230.

M. Gottlied, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Defection, Optical Engineering Series (Marcel Dekker, New York, 1983).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

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Figures (3)

Fig. 1
Fig. 1

Depiction of the diffracting object and notation.

Fig. 2
Fig. 2

Finite-length cylinder and notation.

Fig. 3
Fig. 3

Anisotropic brick and notation.

Equations (124)

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f ( r , θ , φ ) = 0 ,
r = g ( θ , φ ) , θ [ 0 , π ] .
ϵ ̿ = [ ϵ x x ϵ x y ϵ x z ϵ y x ϵ y y ϵ y z ϵ z x ϵ z y ϵ z z ] ,
R = [ i ̂ x ̂ i ̂ y ̂ i ̂ z ̂ j ̂ x ̂ j ̂ y ̂ j ̂ z ̂ l ̂ x ̂ l ̂ y ̂ l ̂ z ̂ ] .
ϵ ͌ i j ( r , θ , φ ) = n = 0 m = n n ϵ i j , n m ( r ) Y n m ( θ , φ ) ,
ϵ i j , n m ( r ) = 0 2 π d φ 0 π ϵ ͌ i j ( r , θ , φ ) Y n m * ( θ , φ ) sin θ d θ 0 4 π ϵ ͌ i j ( r , θ , φ ) Y n m * ( θ , φ ) d Ω .
E ( r , θ , φ ) = n = 0 m = n n [ E Y n m ( r ) Y n m ( θ , φ ) + E X n m ( r ) X n m ( θ , φ ) + E Z n m ( r ) Z n m ( θ , φ ) ] .
n = Int p 1 ,
m = p 1 n ( n + 1 ) .
W p ( η ) = { Y p , η = 1 X p , η = 2 Z p , η = 3 } .
E ( r , θ , φ ) = p = 1 p Max η = 1 3 E η p ( r ) W p ( η ) ( θ , φ ) .
ϵ ͌ i j ( r , θ , φ ) = p = 1 p Max ϵ i j , p ( r ) Y p ( θ , φ ) .
a p E X , p r = i ω μ 0 H Y , p ,
a p E Y , p r E Z , p r d E Z , p d r = i ω μ 0 H X , p ,
E X , p r + d E X , p d r = i ω μ 0 H Z , p ,
a p H X , p r = i ω D Y , p ,
a p H Y , p r H Z , p r d H Z , p d r = i ω D X , p ,
H X , p r + d H X , p d r = i ω D Z , p ,
( [ D Y ] [ D X ] [ D Z ] ) = ϵ 0 Q ϵ ( [ E Y ] [ E X ] [ E Z ] ) ,
[ E X ] = ( E X , 2 E X , p Max ) .
d d r [ F ] = M ( r ) [ F ] ,
[ F ] = ( [ E X ] [ E Z ] [ H ̃ X ] [ H ̃ Z ] )
M 11 = I r , M 12 = M 13 = 0 , M 14 = i μ k 0 I ,
M 21 = a r Q ϵ , Y Y 1 Q ϵ , Y X , M 22 = I r a r Q ϵ , Y Y 1 Q ϵ , Y Z ,
M 23 = i ( a Q ϵ , Y Y 1 a k 0 r 2 μ k 0 I ) , M 24 = 0 ,
M 31 = i k 0 ( Q ϵ , Z Y Q ϵ , Y Y 1 Q ϵ , Y X Q ϵ , Z X ) ,
M 32 = i k 0 ( Q ϵ , Z Y Q ϵ , Y Y 1 Q ϵ , Y Z Q ϵ , Z Z ) ,
M 33 = 1 r ( Q ϵ , Z Y Q ϵ , Y Y 1 a I ) , M 34 = 0 ,
M 41 = i ( k 0 Q ϵ , X X k 0 Q ϵ , X Y Q ϵ , Y Y 1 Q ϵ , Y X a 2 μ k 0 r 2 ) ,
M 42 = i k 0 ( Q ϵ , X Z Q ϵ , X Y Q ϵ , Y Y 1 Q ϵ , Y Z ) ,
M 43 = Q ϵ , X Y Q ϵ , Y Y 1 a r , M 44 = I r .
N ̂ ( θ , φ ) = grad f grad f f = 0 .
N ̂ ( r , θ , φ ) N ̂ ( θ , φ ) , r [ R 1 , R 2 ] .
T ̂ 1 = N ̂ × φ ̂ N ̂ × φ ̂ N θ N r 2 + N θ 2 r ̂ N r N r 2 + N θ 2 θ ̂ ,
T ̂ 2 = T ̂ 1 × N ̂ N r N θ N r 2 + N θ 2 r ̂ N θ N φ N r 2 + N θ 2 θ ̂ + 1 N φ 2 N r 2 + N θ 2 φ ̂ ,
T ̂ 1 = r ̂ ,
T ̂ 2 = T ̂ 1 × N ̂ θ ̂ .
F ϵ ( F ϵ , 1 F ϵ , 2 F ϵ , 3 ) = ( E T 1 1 ϵ 0 D N E T 2 ) = def ( E T ̂ 1 1 ϵ 0 D N ̂ E T ̂ 2 )
F ϵ = A ϵ E ,
1 ϵ 0 D N = 1 ϵ 0 D N ̂ = N ̂ ( ϵ ͌ E ) = N r ( ϵ r r E r + ϵ r θ E θ + ϵ r φ E φ ) + N θ ( ϵ θ r E r + ϵ θ θ E θ + ϵ θ φ E φ ) + N φ ( ϵ φ r E r + ϵ φ θ E θ + ϵ φ φ E φ ) .
A ϵ = [ T 1 , r T 1 , θ 0 ( N r ϵ r r + N θ ϵ θ r + N φ ϵ φ r ) ( N r ϵ r θ + N θ ϵ θ θ + N φ ϵ φ θ ) ( N r ϵ r φ + N θ ϵ θ φ + N φ ϵ φ φ ) T 2 , r T 2 , θ T 2 , φ ] ,
A ϵ = [ T 1 , r T 1 , θ 0 ( N ̂ ϵ ͌ ) r ( N ̂ ϵ ͌ ) θ ( N ̂ ϵ ͌ ) φ T 2 , r T 2 , θ T 2 , φ ] .
C = def A ϵ 1 = 1 ξ 0 [ [ ( N ̂ ϵ ͌ ) × T ̂ 2 ] r N r [ T ̂ 1 × ( N ̂ ϵ ͌ ) ] r [ ( N ̂ ϵ ͌ ) × T ̂ 2 ] θ N θ [ T ̂ 1 × ( N ̂ ϵ ͌ ) ] θ [ ( N ̂ ϵ ͌ ) × T ̂ 2 ] φ N φ [ T ̂ 1 × ( N ̂ ϵ ͌ ) ] φ ] ,
C = ( T 1 , r ξ 1 ξ 0 N r 1 ξ 0 N r T 2 , r ξ 2 ξ 0 N r T 1 , θ ξ 1 ξ 0 N θ 1 ξ 0 N θ T 2 , θ ξ 2 ξ 0 N θ T 1 , φ ξ 1 ξ 0 N φ 1 ξ 0 N φ T 2 , φ ξ 2 ξ 0 N φ ) = ( C 1 , C 2 , C 3 ) ,
E = C F ϵ ,
D = ϵ 0 ϵ ͌ E = ϵ 0 ϵ ͌ C F ϵ .
p , η D η , p ( r ) W p ( η ) ( θ , φ ) = ϵ 0 ϵ ͌ ( r , θ , φ ) C ( r , θ , φ ) p ( F ϵ , 1 , p ( r ) F ϵ , 2 , p ( r ) F ϵ , 3 , p ( r ) ) Y p ( θ , φ ) ,
( ϵ ͌ C ) i J = j = r , θ , φ ϵ ͌ i j C j J .
p , η D η , p W p ( η ) = ϵ 0 J , p ( ϵ ͌ C J ) F ϵ , J , p Y p .
0 4 π d Ω W p ( η ) ( θ , φ ) W q * ( τ ) ( θ , φ ) = δ p q δ η τ .
D τ , q ( r ) = ϵ 0 J , p W q ( τ ) ( ϵ ͌ C J ) Y p F ϵ , J , p ( r ) = ϵ 0 J , p { ϵ ͌ C } τ , J , q p F ϵ , J , p ( r ) ,
{ ϵ ͌ C } τ J , q p = W q ( τ ) ( ϵ ̃ ̃ C J ) Y p = 0 4 π d Ω W q * ( τ ) ( θ , φ ) ( ϵ ͌ C J ) Y p ( θ , φ ) ,
{ ϵ ͌ C } Y J , p q = Y p ( ϵ ͌ C ) r J Y q ,
{ ϵ ͌ C } X J , p q = i = θ , φ X p , i ( ϵ ͌ C ) i J Y q ,
{ ϵ ͌ C } Z J , p q = i = θ , φ Z p , i ( ϵ ͌ C ) i J Y q .
E τ , q ( r ) = J , p C τ J , q p F ϵ , J , p ( r ) ,
C τ J , q p = W q ( τ ) ( C J ) Y p = 0 4 π d Ω W q * ( τ ) ( θ , φ ) ( C J ) Y p ( θ , φ )
[ F ϵ ] = { C } 1 [ E ] .
[ D ] = ϵ 0 { ϵ ͌ C } { C } 1 [ E ] .
Q ϵ = { ϵ ͌ C } { C } 1 .
{ C } = ( { T ̂ 1 } , { 1 ϵ } { N ̂ } , { T ̂ 2 } )
{ C } 1 = ( { T ̂ 1 } T { 1 ϵ } 1 { N ̂ } T { T ̂ 2 } T ) .
{ ϵ ͌ C } = ( { ϵ } { T ̂ 1 } , { N ̂ } , { ϵ } { T ̂ 2 } )
D = ϵ 0 ϵ ͌ E η , p D η p ( r ) W p ( η ) ( θ , φ ) = ϵ 0 ϵ ͌ ( r , θ , φ ) τ , q E τ q ( r ) W q ( τ ) ( θ , φ ) .
Q ϵ = { ϵ ͌ } ,
{ ϵ ͌ } τ η , p q ( r ) = W p ( τ ) ( ϵ ͌ ) W q ( η ) = 0 4 π d Ω W p * ( τ ) ( θ , φ ) ϵ ͌ ( r , θ , φ ) W q ( η ) ( θ , φ ) .
W p * ( τ ) ϵ ͌ W q ( η ) = i , j W p , i * ( τ ) ϵ ͌ i j W q , j ( η ) ,
W n m ( η ) ( θ , φ ) j ̂ = ν = 1 + 1 b n m , μ ν , j ( η ) Y n + ν , m + μ ( θ , φ ) .
a ¯ ( { ν , μ } , { ν , μ } , { n , m } ) = 0 2 π d φ 0 π sin θ d θ Y ν , μ ( θ , φ ) Y ν , μ ( θ , φ ) Y n m ( θ , φ ) ,
[ F ( R j ) ] = Ψ aniso ( R 1 ) [ A ̃ ]
[ A ̃ ] = ( [ A ̃ 1 ] [ A ̃ 2 ] ) = ( A ̃ 1 , ν A ̃ 2 , ν ) ,
Ψ aniso = ( Ψ 1 , aniso Ψ 2 , aniso ) ,
Ψ j , aniso ( r ) = ( [ 1 k j , ν r a h , p , j , ν ψ n ( k j , ν r ) ] [ 1 k j , ν r [ a e , p , j , ν ψ n ( k j , ν r ) + a p a o , p , j , ν j n ( k j , ν r ) ] ] [ 1 i k 0 r a e , p , j , ν ψ n ( k j , ν r ) ] [ 1 i k 0 r a h , p , j , ν ψ n ( k j , ν r ) ] ) .
ψ n ( z ) = z j n ( z ) , ξ n ( z ) = z h n + ( z ) .
T ( R 2 , R 1 ) = [ F integ ] ,
[ F ( R 2 ) ] = T ( R 2 , R 1 ) [ A ̃ ] .
E X , p ( r = R 2 ) = [ A h , p ( i ) j n ( n out k 0 R 2 ) + B h , p h n + ( n out k 0 R 2 ) ] ,
E Z , p ( r = R 2 ) = 1 n out k 0 R 2 [ A e , p ( i ) ψ n ( n out k 0 R 2 ) + B e , p ξ n ( n out k 0 R 2 ) ] ,
H ̃ X , p ( r = R 2 ) = 1 i k 0 R 2 [ A e , p ( i ) ψ n ( n out k 0 R 2 ) + B e , p ξ n ( n out k 0 R 2 ) ] ,
H ̃ Z , p ( r = R 2 ) = 1 i k 0 R 2 [ A h , p ( i ) ψ n ( n out k 0 R 2 ) + B h , p ξ n ( n out k 0 R 2 ) ] ,
[ F ( R 2 ) ] = Ψ iso ( n out k 0 R 2 ) ( A e , p ( i ) ψ n ( n out k 0 R 2 ) n out k 0 R 2 A h , p ( i ) ψ n ( n out k 0 R 2 ) n out k 0 R 2 B e , p ξ n ( n out k 0 R 2 ) n out k 0 R 2 B h , p ξ n ( n out k 0 R 2 ) n out k 0 R 2 ) ,
Ψ iso = ( 0 I 0 I I 0 I 0 i n out p ( n out k 0 R 2 ) 0 i n out q ( n out k 0 R 2 ) 0 0 i n out p ( n out k 0 R 2 ) 0 i n out q ( n out k 0 R 2 ) )
p p , q ( z ) = δ p , q ψ n ( z ) ψ n ( z ) , q p , q ( z ) = δ p , q ξ n ( z ) ξ n ( z ) .
ϵ ͌ = ( ϵ x 0 0 0 ϵ x 0 0 0 ϵ z ) .
N ̂ = ρ ̂ , T ̂ 1 = z ̂ , T ̂ 2 = φ ̂ ,
C = ( 0 1 ϵ x 0 0 0 1 1 0 0 ) , ( ϵ ͌ C ) = ( 0 1 0 0 0 ϵ x ϵ z 0 0 ) .
N ̂ = z ̂ , T ̂ 1 = ρ ̂ , T ̂ 2 = φ ̂ ,
C = ( 1 0 0 0 0 1 0 1 ϵ z 0 ) , ( ϵ ͌ C ) = ( ϵ x 0 0 0 0 ϵ x 0 1 0 ) .
( C ) i j ( θ ) = n = 0 ( C ) i j , n Y n 0 ( θ , φ ) n = 0 ( C ) i j , n P ¯ n 0 ( cos θ ) ,
( ϵ ͌ C ) i j ( θ ) = n = 0 ( ϵ ͌ C ) i j , n Y n 0 ( θ , φ ) n = 0 ( ϵ ͌ C ) i j , n P ¯ n 0 ( cos θ ) .
P ¯ n 0 = 2 n + 1 4 π P n 0 ,
θ 1 θ 2 P n 0 ( cos θ ) sin θ d θ = 1 n [ cos θ P n 0 ( cos θ ) P n + 1 0 ( cos θ ) ] θ 1 θ 2 .
R = ( χ ̂ 1 ρ ̂ χ ̂ 1 φ ̂ χ ̂ 1 z ̂ χ ̂ 0 ρ ̂ χ ̂ 0 φ ̂ χ ̂ 0 z ̂ χ ̂ + 1 ρ ̂ χ ̂ + 1 φ ̂ χ ̂ + 1 z ̂ )
= ( 1 2 exp ( i φ ) i 2 exp ( i φ ) 0 0 0 1 1 2 exp ( i φ ) i 2 exp ( i φ ) 0 ) .
W n m * ( τ ) ( θ , φ ) ( C J ) Y n m ( θ , φ ) = μ , ν = 1 + 1 j = ρ , φ , z n = 0 b n m , μ ν , j * ( τ ) Y n + ν , m + μ * ( C ) j J , n Y n 0 Y n m .
T ̂ 1 = z ̂ , N ̂ = x ̂ , T ̂ 2 = y ̂ ,
ξ 0 = ϵ x x , ξ 1 = ϵ x z , ξ 2 = ϵ x y .
C ( C 1 , C 2 , C 3 ) = { ( z ̂ ϵ x z ϵ x x x ̂ , 1 ϵ x x x ̂ , y ̂ ϵ x y ϵ x x x ̂ ) , inside the parallelepiped ( z ̂ , 1 ϵ out x ̂ , y ̂ ) , outside the parallelepiped } .
R = ( χ ̂ 1 x ̂ χ ̂ 1 y ̂ χ ̂ 1 z ̂ χ ̂ 0 x ̂ χ ̂ 0 y ̂ χ ̂ 0 z ̂ χ ̂ + 1 x ̂ χ ̂ + 1 y ̂ χ ̂ + 1 z ̂ ) = ( 1 2 i 2 0 0 0 1 1 2 i 2 0 ) .
W n m * ( τ ) ( C J ) Y n m = μ , ν = 1 + 1 j = x y , z n = 0 m = n n b n m , μ ν , j * ( τ ) Y n + ν , m + μ * ( ϵ ͌ C ) j J , n m Y n m Y n m .
χ ̂ 1 = 1 2 ( x ̂ i y ̂ ) ,
χ ̂ 0 = z ̂ ,
χ ̂ + 1 = 1 2 ( x ̂ + i y ̂ ) .
Y n m χ ̂ μ = ( n 2 n + 1 Y n , n 1 m n + 1 2 n + 1 Y n , n + 1 m ) χ μ = n 2 n + 1 μ = 1 1 ( n 1 , m μ ; 1 , μ n , m ) Y n 1 , m μ χ ̂ μ χ ̂ μ n + 1 2 n + 1 μ = 1 1 ( n + 1 , m μ ; 1 , μ n , m ) Y n + 1 , m μ χ ̂ μ χ ̂ μ ,
X n m χ ̂ μ = 1 i Y n , n m χ ̂ μ = i μ = 1 1 ( n , m μ ; 1 , μ n , m ) Y n , m μ χ ̂ μ χ ̂ μ .
Z n m χ ̂ μ = ( n + 1 2 n + 1 Y n , n 1 m + n 2 n + 1 Y n , n + 1 m ) χ ̂ μ = n + 1 2 n + 1 μ = 1 1 ( n 1 , m μ ; 1 , μ n , m ) Y n 1 , m μ χ ̂ μ χ ̂ μ + n 2 n + 1 μ = 1 1 ( n + 1 , m μ ; 1 , μ n , m ) Y n + 1 , m μ χ ̂ μ χ ̂ μ .
χ ̂ μ χ ̂ ν * = δ μ ν ,
χ ̂ 1 χ ̂ 1 = 0 , χ ̂ 1 χ ̂ 0 = 0 ,
χ ̂ 1 χ ̂ + 1 = 1 ,
χ ̂ 0 χ ̂ 0 = 1 , χ ̂ 1 χ ̂ 0 = 0 ,
χ ̂ + 1 χ ̂ + 1 = 0 .
Y n m χ ̂ 1 = n 2 n + 1 ( n 1 , m 1 ; 1 , 1 n , m ) Y n 1 , m 1 + n + 1 2 n + 1 ( n + 1 , m 1 ; 1 , 1 n , m ) Y n + 1 , m 1 ,
Y n m χ ̂ 0 = n 2 n + 1 ( n 1 , m ; 1 , 0 n , m ) Y n 1 , m n + 1 2 n + 1 ( n + 1 , m ; 1 , 0 n , m ) Y n + 1 , m ,
Y n m χ ̂ 1 = n 2 n + 1 ( n 1 , m + 1 ; 1 , 1 n , m ) Y n 1 , m + 1 + n + 1 2 n + 1 ( n + 1 , m + 1 ; 1 , 1 n , m ) Y n + 1 , m + 1 .
X n m χ ̂ 1 = i ( n , m 1 ; 1 , 1 n , m ) Y n , m 1 ,
X n m χ ̂ 0 = i ( n , m ; 1 , 0 n , m ) Y n , m ,
X n m χ ̂ 1 = i ( n , m + 1 ; 1 , 1 n , m ) Y n , m + 1 .
Z n m χ ̂ 1 = n + 1 2 n + 1 ( n 1 , m 1 ; 1 , 1 n , m ) Y n 1 , m 1 n 2 n + 1 ( n + 1 , m 1 ; 1 , 1 n , m ) Y n + 1 , m 1 ,
Z n m χ ̂ 0 = n + 1 2 n + 1 ( n 1 , m ; 1 , 0 n , m ) Y n 1 , m + n 2 n + 1 ( n + 1 , m ; 1 , 0 n , m ) Y n + 1 , m ,
Z n m χ ̂ 1 = n + 1 2 n + 1 ( n 1 , m + 1 ; 1 , 1 n , m ) Y n 1 , m + 1 n 2 n + 1 ( n + 1 , m + 1 ; 1 , 1 n , m ) Y n + 1 , m + 1 .
W n m ( η ) χ ̂ μ = ν = 1 1 b n m , μ , ν , χ ( η ) Y n + ν , m + μ .
W n m ( η ) j ̂ μ = ν = 1 1 b n m , μ ν , j ( η ) Y n + ν , m + μ .
W n m ( η ) j ̂ μ j ̂ μ W n m ( η ) = μ R μ μ χ ̂ μ W n m ( η ) .
b n m , μ , ν , j ( η ) = μ R μ μ b n m , μ , ν , χ ( η ) .

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