Abstract

Establishing a vector spherical harmonic expansion of the electromagnetic field propagating inside an arbitrary anisotropic medium, we extend Mie theory to the diffraction by an anisotropic sphere, with or without losses. The particular case of a uniaxial material leads to a simpler analysis. This work opens the way to the construction of a differential theory of diffraction by a three-dimensional object with arbitrary shape, filled by an arbitrary anisotropic material.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998), Chaps. 5 and 6.
    [CrossRef]
  2. Ref. , Chap. 7.
  3. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957, 1981).
  4. L. Lorenz, "Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge," Vidensk Selk. Skr. 6, 1-62 (1890).
  5. L. Lorenz, "Sur la lumière réfléchie et réfractée par une sphère transparente," Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, 1898).
  6. G. Mie, "Beiträge zur Optik Trüben Mefien speziell kolloidoaled Metallosungen," Ann. Phys. 25, 377-452 (1908).
    [CrossRef]
  7. J. Roth and M. J. Dignam, "Scattering and extinction cross sections for a spherical particle with an oriented molecular layer," J. Opt. Soc. Am. 63, 308-311 (1973).
    [CrossRef]
  8. A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
    [CrossRef]
  9. 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]
  10. Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
    [CrossRef]
  11. Ref. , Chap. 8.2.
  12. S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, "Scattering of a plane wave by a general anisotropic dielectric ellipsoid," J. Opt. Soc. Am. A 7, 991-997 (1990).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, 2002).
    [PubMed]
  14. 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]
  15. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).
  16. B. Stout, M. Nevière, and E. Popov, "Mie scattering by an anisotropic object. Part II. Arbitrary-shaped object: differential theory," J. Opt. Soc. Am. A 23, 1124-1134 (2006).
    [CrossRef]

2006 (1)

2005 (1)

2004 (1)

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

2002 (2)

A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, 2002).
[PubMed]

1998 (1)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998), Chaps. 5 and 6.
[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]

1973 (1)

1960 (1)

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

1908 (1)

G. Mie, "Beiträge zur Optik Trüben Mefien speziell kolloidoaled Metallosungen," Ann. Phys. 25, 377-452 (1908).
[CrossRef]

1898 (1)

L. Lorenz, "Sur la lumière réfléchie et réfractée par une sphère transparente," Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, 1898).

1890 (1)

L. Lorenz, "Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge," Vidensk Selk. Skr. 6, 1-62 (1890).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998), Chaps. 5 and 6.
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, 2002).
[PubMed]

Capsalis, C. N.

Dignam, M. J.

Edmonds, A. R.

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

Geng, Y. L.

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

Guan, B. R.

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998), Chaps. 5 and 6.
[CrossRef]

Kiselev, A. D.

A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
[CrossRef]

Li, L. W.

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

Lorenz, L.

L. Lorenz, "Sur la lumière réfléchie et réfractée par une sphère transparente," Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, 1898).

L. Lorenz, "Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge," Vidensk Selk. Skr. 6, 1-62 (1890).

Mie, G.

G. Mie, "Beiträge zur Optik Trüben Mefien speziell kolloidoaled Metallosungen," Ann. Phys. 25, 377-452 (1908).
[CrossRef]

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.

Reshetnyak, V. Yu.

A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
[CrossRef]

Roth, J.

Sluckin, T. J.

A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
[CrossRef]

Stout, B.

Uzunoglu, N. K.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957, 1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, 2002).
[PubMed]

Wu, Xin-Bao

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

Ann. Phys. (1)

G. Mie, "Beiträge zur Optik Trüben Mefien speziell kolloidoaled Metallosungen," Ann. Phys. 25, 377-452 (1908).
[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. (1)

J. Opt. Soc. Am. A (3)

Phys. Rev. E (2)

A. D. Kiselev, V. Yu. Reshetnyak, and T. J. Sluckin, "Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies," Phys. Rev. E 65, 056609-1-056609-16 (2002).
[CrossRef]

Y. L. Geng, Xin-Bao Wu, L. W. Li, and B. R. Guan, "Mie scattering by a uniaxial anisotropic sphere," Phys. Rev. E 70, 056609-1-056609-6 (2004).
[CrossRef]

Vidensk Selk. Skr. (1)

L. Lorenz, "Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge," Vidensk Selk. Skr. 6, 1-62 (1890).

Other (7)

L. Lorenz, "Sur la lumière réfléchie et réfractée par une sphère transparente," Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, 1898).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998), Chaps. 5 and 6.
[CrossRef]

Ref. , Chap. 7.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957, 1981).

Ref. , Chap. 8.2.

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

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, Cambridge, 2002).
[PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (2)

Tables Icon

Table 1 Discretization Corresponding to a Dipolar Representation ( n Max = 1 )

Tables Icon

Table 2 Discretization Corresponding to a Quadrupolar Representation for n Max = 2

Equations (175)

Equations on this page are rendered with MathJax. Learn more.

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 ] ,
ϵ ͌ = R ϵ ̿ R T
R = ( r ̂ x ̂ r ̂ y ̂ r ̂ z ̂ θ ̂ x ̂ θ ̂ y ̂ θ ̂ z ̂ φ ̂ x ̂ φ ̂ y ̂ φ ̂ z ̂ ) .
r ̂ = x ̂ cos φ sin θ + y ̂ sin φ sin θ + z ̂ cos θ ,
φ ̂ = x ̂ sin φ + y ̂ cos φ ,
θ ̂ = r ̂ × φ ̂ = x ̂ cos φ cos θ + y ̂ sin φ cos θ z ̂ sin θ ,
R = ( cos φ sin θ sin φ sin θ cos θ cos φ cos θ sin φ cos θ sin θ sin φ cos φ 0 ) .
curl curl E k 0 2 ϵ ͌ E = 0 ,
E ( k , r O M ) = A ( k ) exp ( i k r O M )
curl [ A exp ( i k r O M ) ] = i k × A exp ( i k r O M ) ,
k × ( k × A ) + k 0 2 ϵ ͌ A = 0 .
[ k 2 I ( k k ) k 0 2 ϵ ͌ ] A ( k ) = 0 ,
det [ k 2 I ( k k ) k 0 2 ϵ ͌ ] = 0 .
( k k ) = k 2 I 1 3
I 1 3 = ( 1 0 0 0 0 0 0 0 0 ) ,
det [ k 2 k 0 2 ( I I 1 3 ) ϵ ͌ ] = 0 ϵ r r ϵ r θ ϵ r φ ϵ θ r ϵ θ θ k ̑ 2 ϵ θ φ ϵ φ r ϵ φ θ ϵ φ φ k ̑ 2 = 0 ,
α k ̑ 4 β k ̑ 2 + γ = 0 ,
α = ϵ r r ,
β = ϵ r r ( ϵ θ θ + ϵ φ φ ) ϵ r θ ϵ θ r ϵ r φ ϵ φ r ,
γ = det ( ϵ ͌ ) .
k 1 k 0 = k ̑ 1 = ( k ̑ 2 ) = k ̑ 3 = k 3 k 0 ,
k 2 k 0 = k ̑ 2 = ( k ̑ 2 ) = k ̑ 4 = k 4 k 0 .
cos ( θ C ) = r ̂ C z ̂
cos ( φ C ) = r ̂ C x ̂ sin ( θ C )
ϵ r r A j , r + ϵ r θ A j , θ + ϵ r φ A j , φ = 0 , ϵ θ r A j , r + ( ϵ θ θ k ̑ j 2 ) A j , θ + ϵ θ φ A j , φ = 0 , ϵ φ r A j , r + ϵ φ θ A j , θ + ( ϵ φ φ k ̑ j 2 ) A j , φ = 0 .
A j , θ = ϵ r r ϵ r φ ϵ θ r ϵ θ φ ϵ r θ ϵ r φ ϵ θ θ k ̑ j 2 ϵ θ φ A j , r ,
A j , φ = ϵ r r ϵ r θ ϵ θ r k ̑ j 2 ϵ θ θ ϵ r θ ϵ r φ ϵ θ θ k ̑ j 2 ϵ θ φ A j , r .
A j = A ̃ j Γ j .
ϵ ̿ = ( ϵ x 0 0 0 ϵ x 0 0 0 ϵ z ) .
ϵ ͌ = ( ϵ x sin 2 θ + ϵ z cos 2 θ ( ϵ x ϵ z ) sin θ cos θ 0 ( ϵ x ϵ z ) sin θ cos θ ϵ z sin 2 θ + ϵ x cos 2 θ 0 0 0 ϵ x ) ,
ϵ r r A j , r + ϵ r θ A j , θ = 0 ,
ϵ θ r A j , r + ( ϵ θ θ k ̑ j 2 ) A j , θ = 0 ,
( ϵ φ φ k ̑ j 2 ) A j , φ = 0 .
ϵ φ φ = k ̑ 1 2 = 0 ,
A 1 , φ 0 .
A 1 = A ̃ 1 Γ 1 , with Γ 1 = φ ̂ and A ̃ 1 = A 1 , φ .
A j , φ A 2 , φ = 0 .
ϵ r r ϵ r θ ϵ θ r ϵ θ θ k ̑ 2 2 = 0 ,
1 k ̑ 2 2 ( θ ) = sin 2 θ ϵ z + cos 2 θ ϵ x .
A 2 , r = ϵ r θ ϵ r r A 2 , θ ( ϵ z ϵ x ) sin θ cos θ ϵ x sin 2 θ + ϵ z cos 2 θ A ̃ 2 A 2 = A ̃ 2 Γ 2 ,
with Γ 2 = ( ϵ z ϵ x ) sin θ cos θ ϵ x sin 2 θ + ϵ z cos 2 θ r ̂ + θ ̂ , A ̃ 2 = A 2 , θ .
A 1 = A ̃ 1 φ ̂ ,
A 2 = A ̃ 2 θ ̂ .
E ( r O M ) = j = 1 2 0 π sin θ d θ 0 2 π d φ A j ( θ , φ ) exp [ i k j ( θ , φ ) r ̂ . r O M ] ,
n θ = 1 , 2 , , 2 n Max + 1 ,
θ ν = π n θ 1 2 n Max .
θ ν = 0 , π 2 n Max , 2 π 2 n Max , , π
n φ = 1 , , n θ ,
φ ν = 2 π n φ n θ + 1
ν = n φ + n θ ( n θ 1 ) 2 .
ν = ( n Max + 1 ) ( n Max + 2 ) 2
n θ = Int ( 1 + 8 ν 7 2 ) ,
n φ = ν n θ ( n θ 1 ) 2 .
n φ = 1 , , 2 ( n Max + 1 ) n θ
φ ν = 2 π n φ 2 ( n Max + 1 ) n θ + 1 ,
ν = ( n Max + 1 ) 2 + n φ ( 2 n Max n θ + 3 ) ( 2 n Max n θ + 2 ) 2 ,
n θ > n Max + 1 , n φ 2 ( n Max + 1 ) n θ .
n θ = 2 ( n Max + 1 ) Int ( 1 + 8 ( n Max + 1 ) 2 8 ν + 1 2 ) ,
n φ = ν + ( 2 n Max n θ + 3 ) ( 2 n Max n θ + 2 ) 2 ( n Max + 1 ) 2 .
N ν = 2 N up + ( n Max + 1 ) = ( n Max + 1 ) 2 .
E ( r O M ) = j = 1 2 ν = 1 N ν A j , ν exp ( i k j , ν r ̂ ν r O M ) = j = 1 2 ν = 1 N ν A ̃ j , ν Γ j , ν exp ( i k j , ν r ̂ ν r O M ) ,
A ̃ j , ν = A ̃ j ( θ ν , φ ν ) sin θ ν ,
Γ j , ν = Γ j ( θ ν , φ ν ) ,
k j , ν = k j ( θ ν , φ ν ) ,
r ̂ ν = r ̂ ( θ ν , φ ν ) .
E ( r , θ , φ ) = n = 0 n Max 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 ) ] .
p = n ( n + 1 ) + m + 1 ,
p Max = ( n Max + 1 ) 2 ,
n ( p ) = Int p 1 ,
m ( p ) = p 1 n ( p ) [ n ( p ) + 1 ] ,
Γ j , ν exp ( i k j , ν r ̂ j , ν r O M )
= p = 1 p Max { a h , p , j , ν ( k j , ν r O M ) X p ( θ O M , φ O M ) + [ a p a e , p , j , ν j n ( k j , ν r O M ) k j , ν r O M + a o , p , j , ν j n ( k j , ν r O M ) ] Y p ( θ O M , φ O M ) + [ a e , p , j , ν ( k j , ν r O M j n ( k j , ν r O M ) ) k j , ν r O M + a p a o , p , j , ν j n ( k j , ν r O M ) k j , ν r O M ] Z p ( θ O M , φ O M ) } ,
a h , p , j , ν = 4 π i n X p * ( r ̂ ν ) Γ j , ν , a e , p , j , ν = 4 π i n 1 Z p * ( r ̂ ν ) Γ j , ν ,
a o , p , j , ν = 4 π i n 1 Y p * ( r ̂ ν ) Γ j , ν .
E = E ( i ) + E ( d ) ,
E ( i ) ( r O M ) = p = 1 p Max ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) { a h , p , j , ν j n ( k j , ν r O M ) X p ( θ O M , φ O M ) + [ a p a e , p , j , ν j n ( k j , ν r O M ) k j , ν r O M + a o , p , j , ν j n ( k j , ν r O M ) ] Y p ( θ O M , φ O M ) + [ a e , p , j , ν ( k j , ν r O M j n ( k j , ν r O M ) ) k j , ν r O M + a p a o , p , j , ν j n ( k j , ν r O M ) k j , ν r O M ] Z p ( θ O M , φ O M ) } ,
E ( d ) ( r O M ) = p = 1 p Max ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) { a h , p , j , ν h n + ( k j , ν r O M ) X p ( θ O M , φ O M ) + [ a p a e , p , j , ν h n + ( k j , ν r O M ) k j , ν r O M + a o , p , j , ν h n + ( k j , ν r O M ) ] Y p ( θ O M , φ O M ) + [ a e , p , j , ν ( k j , ν r O M h n + ( k j , ν r O M ) ) k j , ν r O M + a p a o , p , j , ν h n + ( k j , ν r O M ) k j , ν r O M ] Z p ( θ O M , φ O M ) } .
ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) a h , p , j , ν j n ( n out k 0 R ) + ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) a h , p , j , ν h n + ( n out k 0 R ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a h , p , j , ν j n ( k j , ν R ) , p = 2 , , p Max .
A h , p ( i ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) a h , p , j , ν ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) 4 π i n X p * ( r ̂ ν ) Γ j , ν ,
A e , p ( i ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) a e , p , j , ν ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( i ) 4 π i n 1 Z p * ( r ̂ ν ) Γ j , ν ,
B h , p = ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) a h , p , j , ν ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) 4 π i n X p * ( r ̂ ν ) Γ j , ν ,
B e , p = ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) a e , p , j , ν ν = 1 N ν 1 j = 1 2 A ̃ j , ν ( d ) 4 π i n 1 Z p * ( r ̂ ν ) Γ j , ν .
A h , p ( i ) j n ( n out k 0 R ) + B h , p h n + ( n out k 0 R ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a h , p , j , ν j n ( k j , ν R ) , p = 2 , p Max .
A e , p ( i ) [ n out k 0 R j n ( n out k 0 R ) ] n out k 0 R + B e , p [ n out k 0 R h n + ( n out k 0 R ) ] n out k 0 R = ν = 1 N ν 1 j = 1 2 A ̃ j , ν { a e , p , j , ν [ k j , ν R j n ( k j , ν R ) ] k j , ν R + a p a o , p , j , ν j n ( k j , ν R ) k j , ν R } , p = 2 , p Max .
A e , p ( i ) ψ n ( n out k 0 R ) n out k 0 R + B e , p ξ n ( n out k 0 R ) n out k 0 R = ν = 1 N ν 1 j = 1 2 A ̃ j , ν [ a e , p , j , ν ψ n ( k j , ν R ) k j , ν R + a p a o , p , j , ν j n ( k j , ν R ) k j , ν R ] ,
ψ n ( z ) = z j n ( z ) , ξ n ( z ) = z h n + ( z ) .
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 .
n out k 0 [ A e , p ( i ) j n ( n out k 0 R ) + B e , p h n + ( n out k 0 R ) ] = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a e , p , j , ν k j , ν j n ( k j , ν R ) .
A h , p ( i ) ψ n ( n out k 0 R ) + B h , p ξ n ( n out k 0 R ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a h , p , j , ν ψ n ( k j , ν R ) .
( Ψ 0 ) p q = δ p q ψ n ( n out k 0 R ) , ( Ψ 0 ) p q = δ p q ψ n ( n out k 0 R ) ,
( ξ 0 ) p q = δ p q ξ n ( n out k 0 R ) , ( ξ 0 ) p q = δ p q ξ n ( n out k 0 R ) ,
Ψ h , a = ( Ψ h , a 1 , Ψ h , a 2 ) , with ( Ψ h , a j ) p ν = a h , p , j , ν ψ n ( k j , ν R ) ,
Ψ h , a = ( Ψ h , a 1 , Ψ h , a 2 ) , with ( Ψ h , a j ) p ν = a h , p , j , ν ψ n ( k j , ν R ) ,
ξ h , a = ( ξ h , a 1 , ξ h , a 2 ) , with ( ξ h , a j ) p ν = a h , p , j , ν ξ n ( k j , ν R ) ,
ξ h , a = ( ξ h , a 1 , ξ h , a 2 ) , with ( ξ h , a j ) p ν = a h , p , j , ν ξ n ( k j , ν R ) ,
J o , a = ( J o , a 1 , J o , a 2 ) , with ( J o , a j ) p ν = a p a o , p , j , ν j n ( k j , ν R ) ,
( k out k in ) = ( ( k out k in , 1 ) 0 0 ( k out k in , 2 ) ) with ( k out k in , j ) τ ν = δ τ ν n out k 0 k j , ν .
Ψ 0 [ A h ( i ) ] + ξ 0 [ B h ] = Ψ h , a ( k out k in ) [ A ̃ ] ,
Ψ 0 [ A h ( i ) ] + ξ 0 [ B h ] = Ψ h , a [ A ̃ ] ,
Ψ 0 [ A e ( i ) ] + ξ 0 [ B e ] = ( Ψ e , a + J o , a ) ( k out k in ) [ A ̃ ] ,
Ψ 0 [ A e ( i ) ] + ξ 0 [ B e ] = Ψ e , a [ A ̃ ] ,
ψ n ( x ) ξ n ( x ) ψ n ( x ) ξ n ( x ) = i , n ,
i [ A h ( i ) ] = [ ξ 0 Ψ h , a ( k out k in ) ξ 0 Ψ h , a ] [ A ̃ ] .
i [ A e ( i ) ] = [ ξ 0 Ψ e , a ξ 0 ( Ψ e , a + J o , a ) ( k out k in ) ] [ A ̃ ] .
[ A ̃ ] = i U 1 ( [ A h ( i ) ] [ A e ( i ) ] ) ,
U = ( ξ 0 Ψ h , a ( k out k in ) ξ 0 Ψ h , a ξ 0 Ψ e , a ξ 0 ( Ψ e , a + J o , a ) ( k out k in ) ) .
i [ B h ] = [ Ψ 0 Ψ h , a Ψ 0 Ψ h , a ( k out k in ) ] [ A ̃ ] .
i [ B e ] = [ Ψ 0 ( Ψ e , a + J o , a ) ( k out k in ) Ψ 0 Ψ e , a ] [ A ̃ ] .
( [ B h ] [ B e ] ) = ( T h h T h e T e h T e e ) ( [ A h ( i ) ] [ A e ( i ) ] ) ,
T = ( T h h T h e T e h T e e ) = ( Ψ 0 Ψ h , a Ψ 0 Ψ h , a ( k out k in ) Ψ 0 ( Ψ e , a + J o , a ) ( k out k in ) Ψ 0 Ψ e , a ) ( ξ 0 Ψ h , a ( k out k in ) ξ 0 Ψ h , a ξ 0 Ψ e , a ξ 0 ( Ψ e , a + J o , a ) ( k out k in ) ) 1 .
A h , p ( 1 ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a h , p , j , ν ,
A e , p ( 1 ) = ν = 1 N ν 1 j = 1 2 A ̃ j , ν a e , p , j , ν .
( Ψ h , a [ A ̃ ] ) p = ψ n ( n 1 k 0 R ) A h , p ( 1 ) ,
( Ψ e , a [ A ̃ ] ) p = ψ n ( n 1 k 0 R ) A e , p ( 1 ) ,
i A h , p ( i ) = [ ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ] A h , p ( 1 ) ,
i A e , p ( i ) = [ ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ] A e , p ( 1 ) .
i B h , p = [ ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ] A h , p ( 1 ) ,
i B e , p = [ ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ] A e , p ( 1 ) .
( T h h ) p q = ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) δ p q ,
( T e e ) p q = ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 ψ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) ξ n ( n out k 0 R ) ψ n ( n 1 k 0 R ) n out n 1 δ p q .
T e h = T h e = 0 .
a h , p , 1 , ν = 4 π i n X p * ( r ̂ ν ) Γ 1 , ν = 4 π i n X p * ( θ ν , φ ν ) φ ̂ ν = 2 π α h , p , 1 ( θ ν ) exp [ i m ( p ) φ ν ] ,
a e , p , 1 , ν = 4 π i n 1 Z p * ( r ̂ ν ) Γ 1 , ν = 4 π i n 1 Z p * ( θ ν , φ ν ) . φ ̂ ν = 2 π α e , p , 1 ( θ ν ) exp [ i m ( p ) φ ν ] ,
a o , p , 1 , ν = 4 π i n 1 Y p * ( r ̂ ν ) Γ 1 , ν = 4 π i n 1 Y p * ( θ ν , φ ν ) φ ̂ ν = 0 ,
a h , p , 2 , ν = 4 π i n X p * ( r ̂ ν ) Γ 2 , ν = 4 π i n X p * ( θ n θ , φ ν ) . θ ̂ ν = 2 π α h , p , 2 ( θ ν ) exp [ i m ( p ) φ ν ] ,
a e , p , 2 , ν = 4 π i n 1 Z p * ( r ̂ ν ) Γ 2 , ν = 4 π i n 1 Z p * ( θ ν , φ ν ) θ ̂ ν = 2 π α e , p , 2 ( θ ν ) exp [ i m ( p ) φ ν ] ,
a o , p , 2 , ν = 4 π i n 1 Y p * ( r ̂ ν ) Γ 2 , ν = 4 π i n 1 Y p * ( θ ν , φ ν ) Γ 2 , ν , r = 2 π α o , p , 2 ( θ ν ) exp [ i m ( p ) φ ν ] .
α h , p , 1 ( θ ν ) = 2 i n a p d P ¯ n m ( cos θ ν ) d θ ν ,
α e , p , 1 ( θ ν ) = 2 i n m a p sin θ ν P ¯ n m ( cos θ ν ) ,
α h , p , 2 ( θ ν ) = 2 i n + 1 a p sin θ ν P ¯ n m ( cos θ ν ) ,
α e , p , 2 ( θ ν ) = 2 i n 1 a p d P ¯ n m ( cos θ ν ) d θ ν ,
α o , p , 2 ( θ ν ) = 2 i n 1 P ¯ n m ( cos θ ν ) ( ϵ z ϵ x ) sin θ ν cos θ ν ϵ x sin 2 θ ν + ϵ z cos 2 θ ν ,
A ̃ j , p , n θ = 1 2 π n φ = 1 n θ A ̃ j , ν ( n θ , n φ ) exp [ i m ( p ) φ n φ ] .
i A h , p ( i ) = n θ = 1 2 n Max j = 1 2 A ̃ j , p , n θ α h , p , j , n θ [ ξ n ( n out k 0 R ) ψ n ( k j , n θ R ) n out k 0 k j , n θ ξ n ( n out k 0 R ) ψ n ( k j , n θ R ) ] ,
i A e , p ( i ) = n θ = 1 2 n Max j = 1 2 A ̃ j , p , n θ { α e , p , j , n θ ξ n ( n out k 0 R ) ψ n ( k j , n θ R ) ξ n ( n out k 0 R ) [ α e , p , j , n θ ψ n ( k j , n θ R ) + a p α o , p , j , n θ j n ( k j , n θ R ) ] n out k 0 k j , n θ } .
exp ( i k r ) = q = 0 ( 2 q + 1 ) i q j q ( k r ) P q ( r ̂ k r ̂ ) ,
P q ( r ̂ k r ̂ ) = 4 π 2 q + 1 m = q q Y q m * ( r ̂ k ) Y q m ( r ̂ ) .
Y q m ( θ , φ ) = [ 2 m + 1 4 π ( q m ) ! ( q + m ) ! ] 1 2 P q m ( cos θ ) exp ( i m φ ) = P ¯ q m ( cos θ ) exp ( i m φ ) .
exp ( i k r ) = 4 π q = 0 m = q q i q j q ( k r ) Y q m ( r ̂ ) Y q m * ( r k ) .
χ ̂ 1 = 1 2 ( x ̂ + i y ̂ ) ,
χ ̂ 0 = z ̂ ,
χ ̂ 1 = 1 2 ( x ̂ i y ̂ ) .
μ = 1 1 χ ̂ μ * χ ̂ μ = I , χ ̂ μ * χ ̂ τ = δ μ τ .
Y n , q m = μ = 1 1 ( q , m μ ; 1 , μ n , m ) Y q , m μ χ ̂ μ ,
q = n 1 , n , n + 1 .
0 4 π d Ω Y n , q * m ( r ̂ ) Y n , q m ( r ̂ ) = δ n n δ m m δ q q ,
n , m , q Y n , q * m ( r ̂ ) Y n , q * m ( r ̂ ) = I δ Ω ( r ̂ , r ̂ ) .
I P q ( r ̂ k r ̂ ) = 4 π 2 q + 1 n , m Y n q * m ( r ̂ k ) Y n q m ( r ̂ ) ,
I exp ( i k r ) = 4 π n , m , q i q j q ( k r ) Y n q * m ( r ̂ k ) Y n q m ( r ̂ ) .
Γ exp ( i k r ) = I Γ exp ( i k r ) = 4 π n , m , q i q j q ( k r ) Y n q * m ( r ̂ k ) Y n q m ( r ̂ ) Γ .
X n m Y n , m ( m ) 1 i Y n , n m ,
Z n m Y n , m ( e ) ( n + 1 2 n + 1 ) 1 2 Y n , n 1 m + ( n 2 n + 1 ) 1 2 Y n , n + 1 m ,
Y n m Y n , m ( o ) ( n 2 n + 1 ) 1 2 Y n , n 1 m ( n + 1 2 n + 1 ) 1 2 Y n , n + 1 m ,
1 i Y n , n m = i X n m ,
Y n , n 1 m = ( n + 1 2 n + 1 ) 1 2 Z n m + ( n 2 n + 1 ) 1 2 Y n m ,
Y n , n + 1 m = ( n 2 n + 1 ) 1 2 Z n m ( n + 1 2 n + 1 ) 1 2 Y n m .
4 π n , m i n 1 j n 1 ( k r ) Y n , n 1 * m ( k ̂ ) Y n , n 1 m ( r ̂ ) = 4 π n , m i n 1 j n 1 ( k r ) { n + 1 2 n + 1 Z n m ( r ̂ ) Z n m * ( k ̂ ) + n 2 n + 1 Y ̂ n m ( r ̂ ) Y ̂ n m * ( k ̂ ) + n ( n + 1 ) 2 n + 1 [ Y ̂ n m ( r ̂ ) Z n m * ( k ̂ ) + Z n m ( r ̂ ) Y ̂ n m * ( k ̂ ) ] } ,
4 π n , m i n j n ( k r ) Y n , n * m ( k ̂ ) Y n , n m ( r ̂ ) = 4 π n , m i n j n ( k r ) X n m ( r ̂ ) X n m * ( k ̂ ) ,
4 π n , m i n + 1 j n + 1 ( k r ) Y n , n + 1 * m ( k ̂ ) Y n , n + 1 m ( r ̂ ) = 4 π n , m i n + 1 j n + 1 ( k r ) { n 2 n + 1 Z n m ( r ̂ ) Z n m * ( k ̂ ) + n + 1 2 n + 1 Y n m ( r ̂ ) Y n m * ( k ̂ ) n ( n + 1 ) 2 n + 1 [ Y n m ( r ̂ ) Z n m * ( k ̂ ) + Z n m ( r ̂ ) Y n m * ( k ̂ ) ] } .
I exp ( i k r ) = 4 π n , m i n j n ( k r ) X n m ( r ̂ ) X n m * ( k ̂ ) + 4 π n , m i n 1 2 n + 1 [ ( n + 1 ) j n 1 ( k r ) n j n + 1 ( k r ) ] Z n m ( r ̂ ) Z n m * ( k ̂ ) + 4 π n , m i n 1 2 n + 1 [ n j n 1 ( k r ) ( n + 1 ) j n + 1 ( k r ) ] Y n m ( r ̂ ) Y n m * ( k ̂ ) + 4 π n , m i n 1 2 n + 1 n ( n + 1 ) [ j n 1 ( k r ) + j n + 1 ( k r ) ] [ Y n m ( r ̂ ) Z n m * ( k ̂ ) + Z n m ( r ̂ ) Y n m * ( k ̂ ) ] .
j n 1 ( x ) + j n + 1 ( x ) = 2 n + 1 x j n ( x ) ,
n j n 1 ( x ) ( n + 1 ) j n + 1 ( x ) = ( 2 n + 1 ) j n ( x ) ,
( n + 1 ) j n 1 ( x ) n j n + 1 ( x ) = 2 n + 1 x [ x j n ( x ) ] ,
I exp ( i k r ) = 4 π n , m { i n j n ( k r ) X n m ( r ̂ ) X n m * ( k ̂ ) + i n 1 [ k r j n ( k r ) ] k r Z n m ( r ̂ ) Z n m * ( k ̂ ) + i n 1 j n ( k r ) Y n m ( r ̂ ) Y n m * ( k ̂ ) + i n 1 n ( n + 1 ) j n ( k r ) k r [ Y n m ( r ̂ ) Z n m * ( k ̂ ) + Z n m ( r ̂ ) Y n m * ( k ̂ ) ] } .
exp ( i k r ) Γ = n , m { a h , n m j n ( k r ) X n m ( r ̂ ) + [ a e , n m j n ( k r ) k r n ( n + 1 ) + a o , n m j n ( k r ) ] Y n m ( r ̂ ) + [ a e , n m [ k r j n ( k r ) ] k r + n ( n + 1 ) a o , n m j n ( k r ) k r ] Z n m ( r ̂ ) } ,
a h , n m = 4 π i n X n m * ( k ̂ ) Γ , a e , n m = 4 π i n 1 Z n m * ( k ̂ ) Γ ,
a o , n m = 4 π i n 1 Y n m * ( k ̂ ) Γ .
Y n m ( θ , φ ) = r ̂ P ¯ n m ( cos θ ) exp ( i m φ )
Z n m ( θ , φ ) = r n ( n + 1 ) grad [ P ¯ n m ( cos θ ) exp ( i m φ ) ] = exp ( i m φ ) n ( n + 1 ) [ φ ̂ i m sin θ + θ ̂ d d θ ] P ¯ n m ( cos θ )
X n m ( θ , φ ) = Z n m ( θ , φ ) × r ̂ = exp ( i m φ ) n ( n + 1 ) [ θ ̂ i m sin θ φ ̂ d d θ ] P ¯ n m ( cos θ ) .

Metrics