Abstract

The theory of optical forces on spherical scatterers is here generalized to arbitrary incident fields. The interaction between spherical harmonics of different order, and the degree and azimuthal parity, is studied in detail. The resulting force from all the contributing components is presented in analytical form. A further generalization of this formulation to nonspherical scatterers is also discussed.

© 2012 Optical Society of America

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  1. J. C. Maxwell, Treatise on Electricity and Magnetism (Oxford, 1881).
  2. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  3. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics, 1st English ed. (Pergamon, 1960).
  4. R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
    [CrossRef]
  5. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles. attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20, 1201–1209(2003).
    [CrossRef]
  6. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. A 362, 2889–2890 (2004).
    [CrossRef]
  7. V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
    [CrossRef]
  8. S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
    [CrossRef]
  9. J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
    [CrossRef]
  10. G. Mie, “Beitrage zur optik Triiber medien speziel kolloidaler Metallösungen,” Ann. Phys. 330, 377 (1908).
    [CrossRef]
  11. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
    [CrossRef]
  12. C.F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  13. M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011).
    [CrossRef]
  14. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
    [CrossRef]
  15. J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
    [CrossRef]
  16. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
    [CrossRef]
  17. G. Arfken, Mathematical Methods for Physicists, 3rd ed.(Academic Press, 1985).
  18. J. A. Stratton, Electromagnetic Theory (Wiley-IEEE, 2007).
  19. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
    [CrossRef]
  20. D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
    [CrossRef]

2011 (2)

2010 (2)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
[CrossRef]

2009 (1)

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

2008 (1)

D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
[CrossRef]

2007 (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

2006 (1)

V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

2005 (2)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[CrossRef]

2004 (1)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. A 362, 2889–2890 (2004).
[CrossRef]

2003 (1)

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur optik Triiber medien speziel kolloidaler Metallösungen,” Ann. Phys. 330, 377 (1908).
[CrossRef]

Afanas'ev, A. A.

D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
[CrossRef]

Albaladejo, S.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed.(Academic Press, 1985).

Arias-Gonzalez, J. R.

Bohren, C.F.

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

Chan, C. T.

J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Chantada, L.

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. A 362, 2889–2890 (2004).
[CrossRef]

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

Gaida, L. S.

D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
[CrossRef]

Gómez-Medina, R.

Guzatov, D. V.

D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
[CrossRef]

Heckenberg, N. R

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics, 1st English ed. (Pergamon, 1960).

Laroche, M.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics, 1st English ed. (Pergamon, 1960).

Lin, Z.

J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[CrossRef]

Lin, Z. F.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Marques, M. I.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, Treatise on Electricity and Magnetism (Oxford, 1881).

Mie, G.

G. Mie, “Beitrage zur optik Triiber medien speziel kolloidaler Metallösungen,” Ann. Phys. 330, 377 (1908).
[CrossRef]

Ng, J.

J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Nieminen, T. A.

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

Nieto-Vesperinas, M.

Pfeifer, R. N. C.

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

Rahmani, A.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. A 362, 2889–2890 (2004).
[CrossRef]

Ratner, M.

V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

Rubinsztein-Dunlop, H.

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

Saenz, J. J.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Sáenz, J. J.

Shen, J.

J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
[CrossRef]

Sheng, P.

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (Wiley-IEEE, 2007).

Wong, V.

V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

Ann. Phys. (2)

G. Mie, “Beitrage zur optik Triiber medien speziel kolloidaler Metallösungen,” Ann. Phys. 330, 377 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nat. Photon. (1)

J. Shen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phil. Trans. R. Soc. A (1)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. A 362, 2889–2890 (2004).
[CrossRef]

Phys. Rev. B (2)

V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Quantum Electron. (1)

D. V. Guzatov, L. S. Gaida, and A. A. Afanas'ev, “Theoretical study of the light pressure force acting on a spherical dielectric particle of an arbitrary size in the interference field of two plane monochromatic electromagnetic waves,” Quantum Electron. 38, 1155 (2008).
[CrossRef]

Rev. Mod. Phys. (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[CrossRef]

Other (6)

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

J. C. Maxwell, Treatise on Electricity and Magnetism (Oxford, 1881).

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics, 1st English ed. (Pergamon, 1960).

G. Arfken, Mathematical Methods for Physicists, 3rd ed.(Academic Press, 1985).

J. A. Stratton, Electromagnetic Theory (Wiley-IEEE, 2007).

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Equations (99)

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× × E k 2 E = 0.
ψ e m n = cos ( m φ ) P n m ( cos θ ) z n ( k r ) , ψ o m n = sin ( m φ ) P n m ( cos θ ) z n ( k r ) .
M e m n = × ( r ^ r ψ e m n ) , M o m n = × ( r ^ r ψ o m n ) , N e m n = × M e m n k , N o m n = × M o m n k ,
if z n ( k r ) = j n ( k r ) M e m n , M o m n , N e m n , N o m n , if z n ( k r ) = h n ( 1 , 2 ) ( k r ) M e m n ( 1 , 2 ) , M o m n ( 1 , 2 ) , N e m n ( 1 , 2 ) , N o m n ( 1 , 2 ) .
E i = n = 1 m = 0 n ( a e n m N e n m + a o n m N o n m + b e n m M e n m + b o n m M o n m ) .
H i = ε b i η 0 n = 1 m = 0 n ( a e n m M e n m + a o n m M o n m + b e n m N e n m + b o n m N o n m ) ,
a ( e / o ) n m = 0 2 π 0 π E i · N ( e / o ) n m sin ( θ ) d θ d φ 0 2 π 0 π | N ( e / o ) n m | 2 sin ( θ ) d θ d φ ,
b ( e / o ) n m = 0 2 π 0 π E i · M ( e / o ) n m sin ( θ ) d θ d φ 0 2 π 0 π | M ( e / o ) n m | 2 sin ( θ ) d θ d φ .
E c = n = 1 m = 0 n ( a e n m c N e n m + a o n m c N o n m + b e n m c M e n m + b o n m c M o n m ) ,
H c = ε c i η 0 n = 1 m = 0 n ( a e n m c M e n m + a o n m c M o n m + b e n m c N e n m + b o n m c N o n m ) ,
E s = n = 1 m = 0 n ( a e n m s N e n m ( 1 ) + a o n m s N o n m ( 1 ) + b e n m s M e n m ( 1 ) + b o n m s M o n m ( 1 ) ) ,
H s = ε b i η 0 n = 1 m = 0 n ( a e n m s M e n m ( 1 ) + a o n m s M o n m ( 1 ) + b e n m s N e n m ( 1 ) + b o n m s N o n m ( 1 ) ) .
{ a ( o / e ) n m s = ε b χ c j n ( χ b ) j n ( χ c ) j n ( χ c ) [ ( ε c ε b ) j n ( χ b ) + ε c χ b j n ( χ b ) ] ε c χ b j n ( χ c ) h n ( 1 ) ( χ b ) + h n ( 1 ) ( χ b ) [ ( ε c ε b ) j n ( χ c ) ε b χ c j n ( χ c ) ] a ( o / e ) n m , b ( o / e ) n m s = ε c j n ( χ b ) j n ( χ c ) ε b j n ( χ c ) j n ( χ b ) ε c h n ( 1 ) ( χ b ) j n ( χ c ) ε b j n ( χ c ) h n ( 1 ) ( χ b ) b ( o / e ) n m , χ b = k 0 ε b r 0 , χ c = k 0 ε c r 0 ,
F = 1 2 Re S T ¯ · n ^ d S .
T ¯ = ε 0 ε b EE * + μ 0 HH * 1 2 ( ε 0 ε b E · E * + μ 0 H · H * ) I ¯ ,
T ¯ = { ε 0 ε b E i E i * + μ 0 H i H i * 1 2 I ¯ [ ε 0 ε b E i · E i * + μ 0 H i · H i * ] } + { ε 0 ε b E s E s * + μ 0 H s H s * 1 2 I ¯ [ ε 0 ε b E s · E s * + μ 0 H s · H s * ] } + { ε 0 ε b E i E s * + μ 0 H i H s * 1 2 I ¯ [ ε 0 ε b E i · E s * + μ 0 H i · H s * ] } + { ε 0 ε b E s E i * + μ 0 H s H i * 1 2 I ¯ [ ε 0 ε b E s · E i * + μ 0 H s · H i * ] } .
F = F s s + F i s + F s i ,
F s s = 1 2 Re S [ ( ε 0 ε b E s E s * + μ 0 H s H s * ) · r ^ 1 2 ( ε 0 ε b E s · E s * + μ 0 H s · H s * ) r ^ ] d S ,
F i s = 1 2 Re S [ ( ε 0 ε b E i E s * + μ 0 H i H s * ) · r ^ 1 2 ( ε 0 ε b E i · E s * + μ 0 H i · H s * ) r ^ ] d S ,
F s i = 1 2 Re S [ ( ε 0 ε b E s E i * + μ 0 H s H i * ) · r ^ 1 2 ( ε 0 ε b E s · E i * + μ 0 H s · H i * ) r ^ ] d S .
F s s = 1 4 lim r S [ ( ε 0 ε b | E s | 2 + μ 0 | H s | 2 ) r ^ ] d S , F i s + F s i = lim r Re S [ 1 2 ( ε 0 ε b E i · E s * + μ 0 H i · H s * ) r ^ ] d S .
F = ε b c 1 2 Re lim r S [ ( E s + E i ) × ( H s * + H i * ) E i × H i * ] d S = ε b c 1 2 Re [ lim r S ( S S i ) d S ] ,
F s s = 1 4 Re n 1 = 1 m 1 = 0 n 1 n 2 = 1 m 2 = 0 n 2 ε 0 ε b × [ ( a e n 1 m 1 s a e n 2 m 2 s * + b e n 1 m 1 s b e n 2 m 2 s * ) [ 2 S N e n 1 m 1 ( 1 ) N e n 2 m 2 ( 1 ) * · r ^ d S S ( N e n 1 m 1 ( 1 ) · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( M e n 1 m 1 ( 1 ) · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a o n 1 m 1 s a o n 2 m 2 s * + b o n 1 m 1 s b o n 2 m 2 s * ) [ 2 S N o n 1 m 1 ( 1 ) N o n 2 m 2 ( 1 ) * · r ^ d S S ( N o n 1 m 1 ( 1 ) · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( M o n 1 m 1 ( 1 ) · M o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b e n 1 m 1 s a o n 2 m 2 s * + a e n 1 m 1 s b o n 2 m 2 s * ) [ 2 S M e n 1 m 1 ( 1 ) N o n 2 m 2 ( 1 ) * · r ^ d S S ( M e n 1 m 1 ( 1 ) · N o n 2 m 2 ( 1 ) * ) r ^ d S - S ( N e n 1 m 1 ( 1 ) · M o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b o n 1 m 1 s a e n 2 m 2 s * + a o n 1 m 1 s b e n 2 m 2 s * ) [ 2 S M o n 1 m 1 ( 1 ) N e n 2 m 2 ( 1 ) * · r ^ d S S ( M o n 1 m 1 ( 1 ) · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( N o n 1 m 1 ( 1 ) · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a e n 1 m 1 s a o n 2 m 2 s * + b e n 1 m 1 s b o n 2 m 2 s * ) [ 2 S N e n 1 m 1 ( 1 ) N o n 2 m 2 ( 1 ) * · r ^ d S S ( N e n 1 m 1 ( 1 ) · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( M e n 1 m 1 ( 1 ) · M o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a o n 1 m 1 s a e n 2 m 2 s * + b o n 1 m 1 s b e n 2 m 2 s * ) [ 2 S N o n 1 m 1 ( 1 ) N e n 2 m 2 ( 1 ) * · r ^ d S S ( N o n 1 m 1 ( 1 ) · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( M o n 1 m 1 ( 1 ) · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b e n 1 m 1 s a e n 2 m 2 s * + a e n 1 m 1 s b e n 2 m 2 s * ) [ 2 S M e n 1 m 1 ( 1 ) N e n 2 m 2 ( 1 ) * · r ^ d S S ( M e n 1 m 1 ( 1 ) · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( N e n 1 m 1 ( 1 ) · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b o n 1 m 1 s a o n 2 m 2 s * + a o n 1 m 1 s b o n 2 m 2 s * ) [ 2 S M o n 1 m 1 ( 1 ) N o n 2 m 2 ( 1 ) * · r ^ d S S ( M o n 1 m 1 ( 1 ) · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( N o n 1 m 1 ( 1 ) · M o n 2 m 2 ( 1 ) * ) r ^ d S ] ] .
F x = F · x ^ = ( F s s + F i s + F s i ) · x ^ , F y = F · y ^ = ( F s s + F i s + F s i ) · y ^ , F z = F · z ^ = ( F s s + F i s + F s i ) · z ^ ,
F s s · x ^ = π ε 0 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 + δ m , 1 ) Im ( a e n m s a e ( n + 1 ) ( m 1 ) s * + b e n m s b e ( n + 1 ) ( m - 1 ) s * ) + ( 1 δ m , 1 ) Im ( a o n m s a o ( n + 1 ) ( m - 1 ) s * + b o n m s b o ( n + 1 ) ( m - 1 ) s * ) ] ( n + m + 2 ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( a e n m s a e ( n + 1 ) ( m + 1 ) s * + b e n m s b e ( n + 1 ) ( m + 1 ) s * ) + ( 1 δ m , 0 ) Im ( a o n m s a o ( n + 1 ) ( m + 1 ) s * + b o n m s b o ( n + 1 ) ( m + 1 ) s * ) ] + ( n + m + 1 ) ! n m 1 ) ! 1 ( 2 n + 1 ) ( 1 + δ m , 0 ) Im ( b e n m s a o n ( m + 1 ) s * + a e n m s b o n ( m + 1 ) s * ) + ( n + m ) ! ( n m ) ! 1 ( 2 n + 1 ) ( 1 δ m , 0 ) ( 1 δ m , 1 ) Im ( b e n m s * a o n ( m 1 ) s + a e n m s * b o n ( m 1 ) s ) } ,
( F i s + F s i ) · x ^ = π ε 0 2 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 + δ m , 1 ) Im ( a e n m a e ( n + 1 ) ( m - 1 ) s * + a e n m s a e ( n + 1 ) ( m - 1 ) * + b e n m b e ( n + 1 ) ( m - 1 ) s * + b e n m s b e ( n + 1 ) ( m - 1 ) * ) + ( 1 + δ m , 1 ) Im ( a e n m a e ( n + 1 ) ( m - 1 ) s * + a e n m s a e ( n + 1 ) ( m - 1 ) * + b e n m b e ( n + 1 ) ( m - 1 ) s * + b e n m s b e ( n + 1 ) ( m - 1 ) * ) ] ( n + m + 2 ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( a e n m a e ( n + 1 ) ( m + 1 ) s * + a e n m s a e ( n + 1 ) ( m + 1 ) * + b e n m b e ( n + 1 ) ( m + 1 ) s * + b e n m s b e ( n + 1 ) ( m + 1 ) * ) + ( 1 δ m , 0 ) Im ( a o n m a o ( n + 1 ) ( m + 1 ) s * + a o n m s a o ( n + 1 ) ( m + 1 ) * + b o n m b o ( n + 1 ) ( m + 1 ) s * + b o n m s b o ( n + 1 ) ( m + 1 ) * ) ] + ( n + m + 1 ) ! ( n m 1 ) ! 1 ( 2 n + 1 ) ( 1 + δ m , 0 ) Im ( b e n m a o n ( m + 1 ) s * + b e n m s a o n ( m + 1 ) * + a e n m b o n ( m + 1 ) s * + a e n m s b o n ( m + 1 ) * ) + ( n + m ) ! ( n - m ) ! 1 ( 2 n + 1 ) + ( 1 δ m , 0 ) ( 1 δ m , 1 ) Im ( b e n m s * a o n ( m 1 ) + b e n m * a o n ( m 1 ) s + a e n m s * b o n ( m 1 ) + a e n m * b o n ( m 1 ) s ) } ,
F s s · y ^ = π ε 0 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 δ m , 1 ) Im ( a e n m s a o ( n + 1 ) ( m 1 ) s * + b e n m s b o ( n + 1 ) ( m 1 ) s * ) ] + ( n + m + 2 ) ! ( n - m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 + δ m , 0 ) Im ( a e n m s a o ( n + 1 ) ( m + 1 ) s * + b e n m s b o ( n + 1 ) ( m + 1 ) s * ) + ( n + m + 1 ) ! ( n m 1 ) ! 1 ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( b e n m s a e n ( m + 1 ) s * + a e n m s b e n ( m + 1 ) s * ) + ( 1 δ m , 0 ) Im ( b o n m s a o n ( m + 1 ) s * + a o n m s b o n ( m + 1 ) s * ) ] + ( n + m ) ! ( n m ) ! 1 ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 + δ m , 1 ) Im ( b e n m s * a e n ( m 1 ) s + a e n m s * b e n ( m 1 ) s ) + ( 1 δ m , 1 ) Im ( b o n m s * a o n ( m 1 ) s + a o n m s * b o n ( m 1 ) s ) ] } ,
( F i s + F s i ) · y ^ = π ε 0 2 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 δ m , 1 ) Im ( a e n m s a o ( n + 1 ) ( m - 1 ) * + b e n m s b o ( n + 1 ) ( m 1 ) * + a e n m a o ( n + 1 ) ( m 1 ) s * + b e n m b o ( n + 1 ) ( m 1 ) s * ) ] + ( n + m + 2 ) ! ( n m ) ! n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) ( 1 + δ m , 0 ) Im ( a e n m s a o ( n + 1 ) ( m + 1 ) * + b e n m s b o ( n + 1 ) ( m + 1 ) * + a e n m a o ( n + 1 ) ( m + 1 ) s * + b e n m b o ( n + 1 ) ( m + 1 ) s * ) + ( n + m + 1 ) ! ( n - m - 1 ) ! 1 ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( b e n m a e n ( m + 1 ) s * + a e n m b e n ( m + 1 ) s * + b e n m s a e n ( m + 1 ) * + a e n m s b e n ( m + 1 ) * ) + ( 1 δ m , 0 ) Im ( b o n m a o n ( m + 1 ) s * + a o n m b o n ( m + 1 ) s * + b o n m s a o n ( m + 1 ) * + a o n m s b o n ( m + 1 ) * ) ] + ( n + m ) ! ( n m ) ! 1 ( 2 n + 1 ) ( 1 δ m , 0 ) [ ( 1 + δ m , 1 ) Im ( b e n m s * a e n ( m 1 ) + a e n m s * b e n ( m 1 ) + b e n m * a e n ( m 1 ) s + a e n m * b e n ( m 1 ) s ) + ( 1 δ m , 1 ) Im ( b o n m s * a o n ( m 1 ) + a o n m s * b o n ( m 1 ) + b o n m * a o n ( m 1 ) s + a o n m * b o n ( m 1 ) s ) ] } ,
F s s · z ^ = π ε 0 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! 2 n ( n + 2 ) ( n + m + 1 ) ( 2 n + 3 ) ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( a e n m s a e ( n + 1 ) m s * + b e n m s b e ( n + 1 ) m s * ) + ( 1 δ m , 0 ) Im ( a o n m s a o ( n + 1 ) m s * + b o n m s b o ( n + 1 ) m s * ) ] + ( n + m ) ! ( n m ) ! 2 m ( 2 n + 1 ) Im ( b e n m s a o n m s * + a e n m s b o n m s * ) } ,
( F i s + F s i ) · z ^ = π ε 0 2 k 0 2 n = 1 m = 0 n { ( n + m ) ! ( n m ) ! 2 n ( n + 2 ) ( n + m + 1 ) ( 2 n + 3 ) ( 2 n + 1 ) [ ( 1 + δ m , 0 ) Im ( a e n m a e ( n + 1 ) m s * + a e n m s a e ( n + 1 ) m * + b e n m b e ( n + 1 ) m s * + b e n m s b e ( n + 1 ) m * ) + ( 1 δ m , 0 ) Im ( a o n m a o ( n + 1 ) m s * + a o n m s a o ( n + 1 ) m * + b o n m b o ( n + 1 ) m s * + b o n m s b o ( n + 1 ) m * ) ] + ( n + m ) ! ( n m ) ! 2 m ( 2 n + 1 ) Im ( b e n m a o n m s * + b e n m s a o n m * + a e n m b o n m * + a e n m s b o n m s * ) } .
δ a b = { 0 , a b , 1 , a = b .
p = α E E i | r = 0 .
E s = 1 4 π ε 0 { k 0 2 ( r ^ × p ) × r ^ e i k 0 r r + [ 3 r ^ ( r ^ · p ) p ] ( 1 r 3 i k 0 r 2 ) e i k 0 r } , H s = c k 0 2 4 π ( r ^ × p ) e i k 0 r r ( 1 1 i k 0 r ) .
E s = a e 10 s N e 10 ( 1 ) + a o 11 s N o 11 ( 1 ) + a e 11 s N e 11 ( 1 ) , H s = 1 i η 0 [ a e 10 s M e 10 ( 1 ) + a o 11 s M o 11 ( 1 ) + a e 11 s M e 11 ( 1 ) ] .
p = α E E i | r = 0 = i 4 π ε 0 k 0 3 [ x ^ a e 11 s + y ^ a o 11 s z ^ a e 10 s ] .
E i | r = 0 = 2 3 ( x ^ a e 11 + y ^ a o 11 z ^ a e 10 ) .
{ a e 11 s = i k 0 3 α E 6 π ε 0 a e 11 , a o 11 s = i k 0 3 α E 6 π ε 0 a o 11 , a e 10 s = i k 0 3 α E 6 π ε 0 a e 10 .
F x = π k 0 2 [ 6 5 Im ( a e 10 s a e 21 * ) + 2 3 Im ( b e 10 a o 11 s * + a e 10 s b o 11 * ) + 2 5 Im ( a e 11 s a e 20 * ) 12 5 Im ( a e 11 s a e 22 * + a o 11 s a o 22 * ) ] ,
F y = π k 0 2 [ 6 5 Im ( a e 10 s a o 21 * ) + 12 5 Im ( a e 11 s a o 22 * ) + 4 3 Im ( b e 10 a e 11 s * + a e 10 s b e 11 * ) ] ,
F z = π k 0 2 [ 4 5 Im ( a e 10 s a e 20 * ) + 2 3 Im ( b e 11 a o 11 s * + a e 11 s b o 11 * ) + 6 5 Im ( a e 11 s a e 21 * + a o 11 s a o 21 * ) ] .
F x = k 0 Re [ α E ( a e 10 a e 21 * 5 + b e 10 * a o 11 a e 10 b o 11 * 9 a e 11 a e 20 * 15 + 2 a e 11 a e 22 * + 2 a o 11 a o 22 * 5 ) ] ,
F y = k 0 Re [ α E ( a e 10 a o 21 * 5 + 2 a e 11 a o 22 * 5 + 2 a e 10 b e 11 * + 2 b e 10 * a e 11 9 ) ] ,
F z = k 0 Re [ α E ( 2 a e 10 a e 20 * 15 b e 11 * a o 11 + a e 11 b o 11 * 9 a e 11 a e 21 * + a o 11 a o 21 * 5 ) ] .
F = 1 4 α r | E i | 2 | r = 0 + k 0 α i ε 0 [ Re ( E i × H i * ) 2 c + ε 0 4 k 0 i × ( E i × E i * ) ] r = 0 .
a ( e / o ) n m s = 0 2 π 0 π E S · N ( e / o ) n m ( 2 ) sin ( θ ) d θ d φ 0 2 π 0 π N ( e / o ) n m ( 1 ) · N ( e / o ) n m ( 2 ) sin ( θ ) d θ d φ ,
b ( e / o ) n m s = 0 2 π 0 π E S · M ( e / o ) n m ( 2 ) sin ( θ ) d θ d φ 0 2 π 0 π M ( e / o ) n m ( 1 ) · M ( e / o ) n m ( 2 ) sin ( θ ) d θ d φ .
F s i = 1 4 Re n 1 = 1 m 1 = 0 n 1 n 2 = 1 m 2 = 0 n 2 ε 0 × [ ( a e n 1 m 1 s a e n 2 m 2 * + b e n 1 m 1 s b e n 2 m 2 * ) [ 2 S N e n 1 m 1 ( 1 ) N e n 2 m 2 * · r ^ d S S ( N e n 1 m 1 ( 1 ) · N e n 2 m 2 * ) r ^ d S S ( M e n 1 m 1 ( 1 ) · M e n 2 m 2 * ) r ^ d S ] + ( a o n 1 m 1 s a o n 2 m 2 * + b o n 1 m 1 s b o n 2 m 2 * ) [ 2 S N o n 1 m 1 ( 1 ) N o n 2 m 2 * · r ^ d S S ( N o n 1 m 1 ( 1 ) · N o n 2 m 2 * ) r ^ d S S ( M o n 1 m 1 ( 1 ) · M o n 2 m 2 * ) r ^ d S ] + ( a e n 1 m 1 s a o n 2 m 2 * + b e n 1 m 1 s b o n 2 m 2 * ) [ 2 S N e n 1 m 1 ( 1 ) N o n 2 m 2 * · r ^ d S S ( N e n 1 m 1 ( 1 ) · N o n 2 m 2 * ) r ^ d S S ( M e n 1 m 1 ( 1 ) · M o n 2 m 2 * ) r ^ d S ] + ( a o n 1 m 1 s a e n 2 m 2 * + b o n 1 m 1 s b e n 2 m 2 * ) [ 2 S N o n 1 m 1 ( 1 ) N e n 2 m 2 * · r ^ d S S ( N o n 1 m 1 ( 1 ) · N e n 2 m 2 * ) r ^ d S S ( M o n 1 m 1 ( 1 ) · M e n 2 m 2 * ) r ^ d S ] + ( b e n 1 m 1 s a e n 2 m 2 * + a e n 1 m 1 s b e n 2 m 2 * ) [ 2 S M e n 1 m 1 ( 1 ) N e n 2 m 2 * · r ^ d S S ( M e n 1 m 1 ( 1 ) · N e n 2 m 2 * ) r ^ d S S ( N e n 1 m 1 ( 1 ) · M e n 2 m 2 * ) r ^ d S ] + ( b e n 1 m 1 s a o n 2 m 2 * + a e n 1 m 1 s b o n 2 m 2 * ) [ 2 S M e n 1 m 1 ( 1 ) N o n 2 m 2 * · r ^ d S S ( M e n 1 m 1 ( 1 ) · N o n 2 m 2 * ) r ^ d S ] + ( b o n 1 m 1 s a e n 2 m 2 * + a o n 1 m 1 s b e n 2 m 2 * ) [ 2 S M o n 1 m 1 ( 1 ) N e n 2 m 2 * · r ^ d S S ( M o n 1 m 1 ( 1 ) · N e n 2 m 2 * ) r ^ d S ] + ( b o n 1 m 1 s a o n 2 m 2 * + a o n 1 m 1 s b o n 2 m 2 * ) [ 2 S M o n 1 m 1 ( 1 ) N o n 2 m 2 * · r ^ d S S ( M o n 1 m 1 ( 1 ) · N o n 2 m 2 * ) r ^ d S S ( N o n 1 m 1 ( 1 ) · M o n 2 m 2 * ) r ^ d S ] - ( a o n 1 m 1 s b e n 2 m 2 * + b o n 1 m 1 s a e n 2 m 2 * ) S ( N o n 1 m 1 ( 1 ) · M e n 2 m 2 * ) r ^ d S ( a e n 1 m 1 s b o n 2 m 2 * + b e n 1 m 1 s a o n 2 m 2 * ) S ( N e n 1 m 1 ( 1 ) · M o n 2 m 2 * ) r ^ d S ] ,
F i s = 1 4 Re n 1 = 1 m 1 = 0 n 1 n 2 = 1 m 2 = 0 n 2 ε 0 × [ ( a e n 1 m 1 a e n 2 m 2 s * + b e n 1 m 1 b e n 2 m 2 s * ) [ 2 S N e n 1 m 1 N e n 2 m 2 ( 1 ) * · r ^ d S S ( N e n 1 m 1 · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( M e n 1 m 1 · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a o n 1 m 1 a o n 2 m 2 s * + b o n 1 m 1 b o n 2 m 2 s * ) [ 2 S N o n 1 m 1 N o n 2 m 2 ( 1 ) * · r ^ d S S ( N o n 1 m 1 · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( M o n 1 m 1 · M o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a e n 1 m 1 a o n 2 m 2 s * + b e n 1 m 1 b o n 2 m 2 s * ) [ 2 S N e n 1 m 1 N o n 2 m 2 ( 1 ) * · r ^ d S S ( N e n 1 m 1 · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( M e n 1 m 1 · M o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( a o n 1 m 1 a e n 2 m 2 s * + b o n 1 m 1 b e n 2 m 2 s * ) [ 2 S N o n 1 m 1 N e n 2 m 2 ( 1 ) * · r ^ d S S ( N o n 1 m 1 · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( M o n 1 m 1 · M e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b e n 1 m 1 a e n 2 m 2 s * + a e n 1 m 1 b e n 2 m 2 s * ) [ 2 S M e n 1 m 1 N e n 2 m 2 ( 1 ) * · r ^ d S S ( M e n 1 m 1 · N e n 2 m 2 ( 1 ) * ) r ^ d S S ( N e n 1 m 1 · M e n 2 m 2 ( 1 ) * r ^ d S ] + ( b e n 1 m 1 a o n 2 m 2 s * + a e n 1 m 1 b o n 2 m 2 s * ) [ 2 S M e n 1 m 1 N o n 2 m 2 ( 1 ) * · r ^ d S S ( M e n 1 m 1 · N o n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b o n 1 m 1 a e n 2 m 2 s * + a o n 1 m 1 b e n 2 m 2 s * ) [ 2 S M o n 1 m 1 N e n 2 m 2 ( 1 ) * · r ^ d S S ( M o n 1 m 1 · N e n 2 m 2 ( 1 ) * ) r ^ d S ] + ( b o n 1 m 1 a o n 2 m 2 s * + a o n 1 m 1 b o n 2 m 2 s * ) [ 2 S M o n 1 m 1 N o n 2 m 2 ( 1 ) * · r ^ d S S ( M o n 1 m 1 · N o n 2 m 2 ( 1 ) * ) r ^ d S S ( N o n 1 m 1 · M o n 2 m 2 ( 1 ) * ) r ^ d S ] - ( a o n 1 m 1 b e n 2 m 2 s * + b o n 1 m 1 a e n 2 m 2 s * ) S ( N o n 1 m 1 · M e n 2 m 2 ( 1 ) * ) r ^ d S ( a e n 1 m 1 b o n 2 m 2 s * + b e n 1 m 1 a o n 2 m 2 s * ) S ( N e n 1 m 1 · M o n 2 m 2 ( 1 ) * ) r ^ d S ] .
S ( N e m 1 n 1 · N e m 2 n 2 * ) ( r ^ · x ^ ) d S = ( 1 + δ m 1 0 + δ 0 m 2 ) [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 - m 2 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 m 2 ) ! ] × S ( N e 0 n 1 · N e 0 n 2 * ) ( r ^ · z ^ ) d S ,
S ( N e m 1 n 1 · N e m 2 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( N e m 1 n 1 · N e m 2 n 2 * ) ( r ^ · z ^ ) d S = π δ m 1 m 2 k 0 2 z n 1 ( k 0 r ) z n 2 * ( k 0 r ) ( δ m 1 m 2 + δ m 1 0 δ 0 m 2 ) × [ 2 n 1 ( 1 + n 1 ) 2 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 1 ) ( 2 n 1 + 3 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 n 1 2 ( n 1 2 1 ) ( n 1 - m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 - 1 ) ( n 1 + m 1 ) ! ( n 1 - m 1 ) ! δ ( n 1 1 ) n 2 ] + π δ m 1 m 2 k 0 2 d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] d d ( k 0 r ) [ k 0 r z n 2 * ( k 0 r ) ] ( δ m 1 m 2 + δ m 1 0 δ 0 m 2 ) × [ 2 n 1 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] .
S ( N o m 1 n 1 · N o m 2 n 2 * ) ( r ^ · x ^ ) d S = ( 1 δ m 1 0 δ 0 m 2 ) [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 m 2 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 - m 2 ) ! ] × S ( N e 0 n 1 · N e 0 n 2 * ) ( r ^ · z ^ ) d S ,
S ( N o m 1 n 1 · N o m 2 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( N o m 1 n 1 · N o m 2 n 2 * ) ( r ^ · z ^ ) d S = π δ m 1 m 2 k 0 2 z n 1 ( k 0 r ) z n 2 * ( k 0 r ) × ( δ m 1 m 2 δ m 1 0 δ 0 m 2 ) × [ 2 n 1 ( 1 + n 1 ) 2 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 1 ) ( 2 n 1 + 3 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 n 1 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] + π δ m 1 m 2 k 0 2 d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] d d ( k 0 r ) [ k 0 r z n 2 * ( k 0 r ) ] × ( δ m 1 m 2 δ m 1 0 δ 0 m 2 ) × [ 2 n 1 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 ( n 1 2 - 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] .
S ( N e m 1 n 1 · N o m 2 n 2 * ) ( r ^ · x ^ ) d S = 0 ,
S ( N e m 1 n 1 · N o m 2 n 2 * ) ( r ^ · y ^ ) d S = ( 1 + δ m 1 0 δ 0 m 2 ) [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 m 2 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 - m 2 ) ! ] × S ( N e 0 n 1 · N e 0 n 2 * ) ( r ^ · z ^ ) d S ,
S ( N e m 1 n 1 · N o m 2 n 2 * ) ( r ^ · z ^ ) d S = 0.
S ( N e m 1 n 1 · M e m 2 n 2 * ) ( r ^ · x ^ ) d S = 0 ,
S ( N e m 1 n 1 · M e m 2 n 2 * ) ( r ^ · y ^ ) d S = π r k 0 z n 1 * ( k 0 r ) 1 k d d r [ k 0 r z n 2 ( k 0 r ) ] × δ n 1 n 2 [ ( δ m 1 ( m 2 + 1 ) + δ m 1 1 δ m 2 0 ) 1 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! + ( δ m 2 ( m 1 + 1 ) + δ m 2 1 δ m 1 0 ) 1 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ] ,
S ( N e m 1 n 1 · M e m 2 n 2 * ) ( r ^ · z ^ ) d S = 0.
S ( N o m 1 n 1 · M o m 2 n 2 * ) ( r ^ · x ^ ) d S = 0 ,
S ( N o m 1 n 1 · M o m 2 n 2 * ) ( r ^ · y ^ ) d S = π r k 0 z n 1 * ( k 0 r ) 1 k d d r [ k 0 r z n 2 ( k 0 r ) ] δ n 1 n 2 [ ( δ m 1 ( m 2 + 1 ) δ m 1 1 δ m 2 0 ) 1 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 - m 1 ) ! ( δ m 2 ( m 1 + 1 ) δ m 2 1 δ m 1 0 ) 1 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ] ,
S ( N o m 1 n 1 · M o m 2 n 2 * ) ( r ^ · z ^ ) d S = 0.
S ( M e m 1 n 1 · M e m 1 n 2 * ) ( r ^ · x ^ ) d S = S ( M e 0 n 1 · M e 0 n 2 * ) ( r ^ · z ^ ) d S × ( 1 + δ m 1 0 + δ 0 m 2 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 1 m 1 1 ) ( n 1 m 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 - m 1 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( M e m 1 n 1 · M e m 1 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( M e m 1 n 1 · M e m 1 n 2 * ) ( r ^ · z ^ ) d S = π δ m 1 m 2 z n 1 ( k 0 r ) z n 2 * ( k 0 r ) r 2 [ 2 n 1 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] .
S ( M o m 1 n 1 · M o m 1 n 2 * ) ( r ^ · x ^ ) d S = S ( M e 0 n 1 · M e 0 n 2 * ) ( r ^ · z ^ ) d S × ( 1 δ m 1 0 δ 0 m 2 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 1 m 1 1 ) ( n 1 m 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( M o m 1 n 1 · M o m 1 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( M o m 1 n 1 · M o m 1 n 2 * ) ( r ^ · z ^ ) d S = [ 2 n 1 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] .
S ( M e m 1 n 1 · M o m 1 n 2 * ) ( r ^ · x ^ ) d S = 0 ,
S ( M e m 1 n 1 · M o m 1 n 2 * ) ( r ^ · y ^ ) d S = S ( M e 0 n 1 · M e 0 n 2 * ) ( r ^ · z ^ ) d S × ( 1 + δ m 1 0 δ 0 m 2 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 1 m 1 1 ) ( n 1 m 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( M e m 1 n 1 · M o m 1 n 2 * ) ( r ^ · z ^ ) d S = 0.
S ( N e m 1 n 1 · M o m 2 n 2 * ) ( r ^ · x ^ ) d S = π r k 0 d d ( k 0 r ) [ k 0 r z n 2 * ( k 0 r ) ] z n 1 ( k 0 r ) δ n 1 n 2 ( 1 δ 0 m 2 ) × [ 2 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ m 1 ( m 2 + 1 ) + 2 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ( δ m 1 ( m 2 1 ) + δ m 1 0 δ m 2 1 ) ] ,
S ( N e m 1 n 1 · M o m 2 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( N e m 1 n 1 · M o m 2 n 2 * ) ( r ^ · z ^ ) d S = π δ m 1 m 2 1 k 0 r d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] z n 1 * ( k 0 r ) r 2 m 1 2 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ n 1 n 2 .
S ( N o m 1 n 1 · M e m 2 n 2 * ) ( r ^ · x ^ ) d S = π r k 0 d d ( k 0 r ) [ k 0 r z n 2 * ( k 0 r ) ] z n 1 ( k 0 r ) δ n 1 n 2 ( 1 δ 0 m 1 ) × [ 2 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! ( δ m 1 ( m 2 + 1 ) + δ m 1 1 δ m 2 0 ) + 2 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! δ m 2 ( m 1 + 1 ) ] ,
S ( N o m 1 n 1 · M e m 2 n 2 * ) ( r ^ · y ^ ) d S = 0 ,
S ( N o m 1 n 1 · M e m 2 n 2 * ) ( r ^ · z ^ ) d S = π δ m 1 m 2 1 k 0 r d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] z n 1 * ( k 0 r ) r 2 m 1 2 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ n 1 n 2 .
S ( N e m 1 n 1 N e m 2 n 2 * · r ^ ) · x ^ d S = S ( N e 0 n 1 N e 0 n 2 * · r ^ ) · z ^ d S ( 1 + δ m 1 0 + δ 0 m 2 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 1 m 1 ) ( n 1 m 1 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( N e m 1 n 1 N e m 2 n 2 * · r ^ ) · y ^ d S = 0 ,
S ( N e m 1 n 1 N e m 2 n 2 * · r ^ ) · z ^ d S = z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 2 π δ m 1 m 2 × [ 2 n 1 ( n 1 + 1 ) 2 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 n 1 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] z n 2 * ( k 0 r ) k 0 2 π δ m 1 m 2 × { [ 2 n 1 ( n 1 + 1 ) ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 - m 1 ) ! δ ( n 1 + 1 ) n 2 2 n 1 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] } .
S ( N o m 1 n 1 N o m 2 n 2 * · r ^ ) · x ^ d S = S ( N e 0 n 1 N e 0 n 2 * · r ^ ) · z ^ d S × ( 1 δ m 1 0 δ 0 m 2 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 - m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 1 m 1 ) ( n 1 m 1 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( N o m 1 n 1 N o m 2 n 2 * · r ^ ) · y ^ d S = 0 ,
S ( N o m 1 n 1 N o m 2 n 2 * · r ^ ) · z ^ d S = z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 2 π δ m 1 m 2 × [ 2 n 1 ( n 1 + 1 ) 2 ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 + 2 n 1 2 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] d d ( k 0 r ) [ k 0 r z n 1 ( k 0 r ) ] z n 2 * ( k 0 r ) k 0 2 π δ m 1 m 2 × { [ 2 n 1 ( n 1 + 1 ) ( n 1 + 2 ) ( n 1 + m 1 + 1 ) ( 2 n 1 + 3 ) ( 2 n 1 + 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 + 1 ) n 2 2 n 1 ( n 1 2 1 ) ( n 1 m 1 ) ( 2 n 1 + 1 ) ( 2 n 1 1 ) ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ ( n 1 1 ) n 2 ] } .
S ( M o m 1 n 1 N e m 2 n 2 * · r ^ ) · x ^ d S = ( 1 δ m 1 0 ) r z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 π × [ 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! ( δ m 1 ( m 2 + 1 ) + δ m 1 1 δ 0 m 2 ) + 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! δ m 2 ( m 1 + 1 ) ] ,
S ( M o m 1 n 1 N e m 2 n 2 * · r ^ ) · y ^ d S = 0 ,
S ( M o m 1 n 1 N e m 2 n 2 * · r ^ ) · z ^ d S = r z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 m 1 π δ m 1 m 2 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ n 1 n 2 .
S ( M e m 1 n 1 N o m 2 n 2 * · r ^ ) · x ^ d S = ( 1 δ m 2 0 ) r z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 π × [ 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ m 1 ( m 2 + 1 ) + 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ( δ m 2 ( m 1 + 1 ) + δ m 2 1 δ 0 m 1 ) ] ,
S ( M e m 1 n 1 N o m 2 n 2 * · r ^ ) · y ^ d S = 0 ,
S ( M e m 1 n 1 N o m 2 n 2 * · r ^ ) · z ^ d S = r z n 1 ( k 0 r ) z n 2 * ( k 0 r ) k 0 m 1 π δ m 1 m 2 2 n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! δ n 1 n 2 .
S ( N e m 1 n 1 N o m 2 n 2 * · r ^ ) · x ^ d S = 0 ,
S ( N e m 1 n 1 N o m 2 n 2 * · r ^ ) · y ^ d S = S ( N e 0 n 1 N e 0 n 2 * · r ^ ) · z ^ d S × ( 1 + δ m 1 0 δ m 2 0 ) × [ δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 1 ( n 1 m 1 ) ! + δ n 1 ( n 2 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 1 ( n 2 m 2 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 + 1 ) ( n 1 + m 1 ) ! 4 n 2 ( n 1 m 1 ) ! δ n 2 ( n 1 + 1 ) δ m 1 ( m 2 1 ) ( n 2 + m 2 ) ! 4 n 2 ( n 2 m 2 ) ! ] ,
S ( N e m 1 n 1 N o m 2 n 2 * · r ^ ) · z ^ d S = 0.
S ( M e m 1 n 1 N e m 2 n 2 * · r ^ ) · x ^ d S = 0 ,
S ( M e m 1 n 1 N e m 2 n 2 * · r ^ ) · y ^ d S = δ n 1 n 2 π r k 0 z n 1 ( k 0 r ) z n 2 * ( k 0 r ) × [ n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! ( δ m 1 ( m 2 + 1 ) + δ m 1 1 δ m 2 0 ) + n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ( δ m 2 ( m 1 + 1 ) + δ m 2 1 δ m 1 0 ) ] ,
S ( M e m 1 n 1 N e m 2 n 2 * · r ^ ) · z ^ d S = 0.
S ( M o m 1 n 1 N o m 2 n 2 * · r ^ ) · x ^ d S = 0 ,
S ( M o m 1 n 1 N o m 2 n 2 * · r ^ ) · y ^ d S = δ n 1 n 2 π r k 0 z n 1 ( k 0 r ) z n 2 * ( k 0 r ) × [ n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 1 ) ! ( n 1 m 1 ) ! ( δ m 1 ( m 2 + 1 ) δ m 1 1 δ m 2 0 ) + n 1 ( n 1 + 1 ) 2 n 1 + 1 ( n 1 + m 2 ) ! ( n 1 m 2 ) ! ( δ m 2 ( m 1 + 1 ) δ m 2 1 δ m 1 0 ) ] ,
S ( M o m 1 n 1 N o m 2 n 2 * · r ^ ) · z ^ d S = 0.

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