Abstract

We applied a multiple-multipole method to calculate the field enhancement of discrete metal nanosphere assemblies due to plasma resonance, thus performing the first full electromagnetic simulation of a variety of nanoparticle assemblies for efficient field focusing, including the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman. Our study captures electromagnetic resonance effects important for optimizing nanoparticle assemblies to achieve maximum electric field focusing. We predict optical frequency electric fields can be enhanced in gold nanoparticle assemblies in aqueous solution by the order of ~450, within a factor of 2 of that achievable in silver nanostructures. We find that both absorption and far-field scattering resonances of nanoparticle assemblies must be carefully interpreted when inferring near-field focusing properties.

© 2008 Optical Society of America

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References

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  1. L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).
  2. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006).
    [CrossRef]
  3. L. A. Sweatlock, S. A. Maier, and H. A. Atwater, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408 (2005).
    [CrossRef]
  4. K. Li. and M. I. Stockman, and D. J. Bergman, "Self-similar chain of metal nanospheres as an efficient nanolens," Phys. Rev. Lett. 91, 227402 (2003).
    [CrossRef] [PubMed]
  5. M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
    [CrossRef] [PubMed]
  6. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, "Controlling the surface enhanced Raman effect via the nanoshell geometry," Appl. Phys. Lett. 82, 257-259 (2003).
    [CrossRef]
  7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
    [CrossRef] [PubMed]
  8. F. Aldaye and H. F. Sleiman, "Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching," J. Am. Chem. Soc. 129, 4130-4131 (2007).
    [CrossRef] [PubMed]
  9. Y. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995).
    [CrossRef] [PubMed]
  10. D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London Ser. A 433, 599-614 (1991).
    [CrossRef]
  11. J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I - Multipole Expansion and Ray-Optical Solutions," IEEE Tran.Antennas Propag. AP-19, 378-390 (1971).
    [CrossRef]
  12. Y. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
    [CrossRef]
  13. F. J. Garcia de Abajo, "Multiple scattering of radiation in clusters of dielectrics," Phys. Rev. B 60, 6086-6102 (1999).
    [CrossRef]
  14. G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, "Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems," Mat. Sci. Eng. C 27, 1347-1350 (2007).
    [CrossRef]
  15. H. Xu, "Calculation of the near field of aggregates of arbitrary spheres," J. Opt. Soc. Am. A 21, 804-809 (2004).
    [CrossRef]
  16. R.-L. Chern, X.-X. Liu and C.-C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, 016609 (2007).
    [CrossRef]
  17. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches," Nanotechnology 17, 1437-1445 (2006).
    [CrossRef]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  20. G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th. ed., (Academic, Orlando, 2005).
  21. D. W. Mackowski, "Calculation of total cross sections of multiple-sphere clusters," J. Opt. Soc. Am. A 11, 2851-2861 (1994).
    [CrossRef]
  22. A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).
  23. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  24. L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802 (2007).
    [CrossRef] [PubMed]
  25. F. Wang and Y. Ron Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806 (2006).
    [CrossRef] [PubMed]
  26. I. H. El-Sayed, X. Huang and M. A. El-Sayed, "Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles," Cancer Lett. 239, 129-135 (2006).
    [CrossRef]
  27. E. Prodan, C. Radloff, N. J. Halas and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419 - 422 (2003).
    [CrossRef] [PubMed]
  28. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
    [CrossRef] [PubMed]
  29. B. J. Messinger, K. U. von Raben, R. K. Chang and P. W. Barber, "Local fields at the surface of noble-metal microspheres," Phys. Rev. B 24, 649-657 (1981).
    [CrossRef]

2007

F. Aldaye and H. F. Sleiman, "Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching," J. Am. Chem. Soc. 129, 4130-4131 (2007).
[CrossRef] [PubMed]

G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, "Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems," Mat. Sci. Eng. C 27, 1347-1350 (2007).
[CrossRef]

R.-L. Chern, X.-X. Liu and C.-C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, 016609 (2007).
[CrossRef]

L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

2006

F. Wang and Y. Ron Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

I. H. El-Sayed, X. Huang and M. A. El-Sayed, "Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles," Cancer Lett. 239, 129-135 (2006).
[CrossRef]

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches," Nanotechnology 17, 1437-1445 (2006).
[CrossRef]

2005

B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
[CrossRef] [PubMed]

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408 (2005).
[CrossRef]

2004

M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

H. Xu, "Calculation of the near field of aggregates of arbitrary spheres," J. Opt. Soc. Am. A 21, 804-809 (2004).
[CrossRef]

2003

J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, "Controlling the surface enhanced Raman effect via the nanoshell geometry," Appl. Phys. Lett. 82, 257-259 (2003).
[CrossRef]

K. Li. and M. I. Stockman, and D. J. Bergman, "Self-similar chain of metal nanospheres as an efficient nanolens," Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

E. Prodan, C. Radloff, N. J. Halas and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419 - 422 (2003).
[CrossRef] [PubMed]

2002

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

1999

F. J. Garcia de Abajo, "Multiple scattering of radiation in clusters of dielectrics," Phys. Rev. B 60, 6086-6102 (1999).
[CrossRef]

1996

Y. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
[CrossRef]

1995

1994

1991

D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London Ser. A 433, 599-614 (1991).
[CrossRef]

1981

B. J. Messinger, K. U. von Raben, R. K. Chang and P. W. Barber, "Local fields at the surface of noble-metal microspheres," Phys. Rev. B 24, 649-657 (1981).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1971

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I - Multipole Expansion and Ray-Optical Solutions," IEEE Tran.Antennas Propag. AP-19, 378-390 (1971).
[CrossRef]

Antennas Propag.

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I - Multipole Expansion and Ray-Optical Solutions," IEEE Tran.Antennas Propag. AP-19, 378-390 (1971).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, "Controlling the surface enhanced Raman effect via the nanoshell geometry," Appl. Phys. Lett. 82, 257-259 (2003).
[CrossRef]

Cancer Lett.

I. H. El-Sayed, X. Huang and M. A. El-Sayed, "Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles," Cancer Lett. 239, 129-135 (2006).
[CrossRef]

J. Am. Chem. Soc.

F. Aldaye and H. F. Sleiman, "Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching," J. Am. Chem. Soc. 129, 4130-4131 (2007).
[CrossRef] [PubMed]

J. Comput. Phys.

Y. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Mat. Sci. Eng. C

G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, "Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems," Mat. Sci. Eng. C 27, 1347-1350 (2007).
[CrossRef]

Nano Lett.

B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
[CrossRef] [PubMed]

Nanotechnology

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches," Nanotechnology 17, 1437-1445 (2006).
[CrossRef]

Phys. Rev. B

F. J. Garcia de Abajo, "Multiple scattering of radiation in clusters of dielectrics," Phys. Rev. B 60, 6086-6102 (1999).
[CrossRef]

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408 (2005).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

B. J. Messinger, K. U. von Raben, R. K. Chang and P. W. Barber, "Local fields at the surface of noble-metal microspheres," Phys. Rev. B 24, 649-657 (1981).
[CrossRef]

Phys. Rev. E

R.-L. Chern, X.-X. Liu and C.-C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, 016609 (2007).
[CrossRef]

Phys. Rev. Lett.

L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

F. Wang and Y. Ron Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

K. Li. and M. I. Stockman, and D. J. Bergman, "Self-similar chain of metal nanospheres as an efficient nanolens," Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London Ser. A 433, 599-614 (1991).
[CrossRef]

Science

E. Prodan, C. Radloff, N. J. Halas and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419 - 422 (2003).
[CrossRef] [PubMed]

Other

A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th. ed., (Academic, Orlando, 2005).

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Figures (26)

Fig. 1.
Fig. 1.

Depiction of the general problem. Each sphere is characterized by a radius aj and refractive index mj (left). The rotation scheme of a bisphere jl to the overall coordinate system (right).

Fig. 2.
Fig. 2.

An example showing the convergence of near-field enhancement at the focal point for varying maximum degree N of spherical harmonics included in the simulation (Eqs. (4), (5) specifically). The structure employed is a self-similar series of 12 nanospheres with a geometric growth factor κ=0.831 (see section 3.3 for definition). Both the cases of gold and silver particles are illustrated.

Fig. 3.
Fig. 3.

Linear chain geometries of identical gold or silver nanospheres.

Fig. 4.
Fig. 4.

Enhancement of near-field at focal point among linear chains with varying number of gold nanospheres.

Fig. 5.
Fig. 5.

Enhancement of near-field at focal point among linear chains with varying number of silver nanospheres.

Fig. 6.
Fig. 6.

A 4-sphere with varying geometric progression growth factors κ=1, 0.5, 0.3, 0.2, 0.143 and 0.111.

Fig. 7.
Fig. 7.

A 6-sphere with varying geometric progression growth factors κ=1, 0.768, 0.558, 0.456, and 0.393.

Fig. 8.
Fig. 8.

Enhancement of near-field at focal point among various geometric progressions of 4 gold nanospheres as the geometric parameter κ is varied.

Fig. 9.
Fig. 9.

Enhancement of near-field at focal point among various geometric progressions of 4 silver nanospheres. as the geometric parameter κ is varied.

Fig. 10.
Fig. 10.

Enhancement of nearfield at focal point among various geometric progressions of 6 gold nanospheres as the geometric growth parameter κ is varied.

Fig. 11.
Fig. 11.

Enhancement of nearfield at focal point among various geometric progressions of 6 silver nanospheres as the geometric growth parameter κ is varied.

Fig. 12.
Fig. 12.

Geometric progression of nanospheres in self-similar structures whose total lengths are fixed to that of a 16-sphere linear chain (167.5nm).

Fig. 13.
Fig. 13.

Focal point electric fieldenhancement for gold self-similar structures equal in length to a 16-sphere linear chain (167.5nm).

Fig. 14.
Fig. 14.

Focal point electric fieldenhancement for silver selfsimilar structures equal in length to a 16-sphere linear chain (167.5nm).

Fig. 15.
Fig. 15.

Focal point electric fieldenhancement for gold self-similar structures equal in length to a 20-sphere linear chain (209.5nm).

Fig. 16.
Fig. 16.

Focal point electric fieldenhancement for silver selfsimilar structures equal in length to a 20-sphere linear chain (209.5nm).

Fig. 17.
Fig. 17.

The electric field enhancement in the xz-plane for self-similar gold assemblies of length 167.5nm.

Fig. 18.
Fig. 18.

The electric field enhancement in the xy-plane for self-similar gold assemblies of length 167.5nm.

Fig. 19.
Fig. 19.

The time-average Poynting vector magnitude in the xz-plane for self-similar gold assemblies of length 167.5nm. The reference incident power is situated at -2.58=log (1/η).

Fig. 20.
Fig. 20.

Electric field enhancement in the xz-plane for the silver Ns =8-self-similar structure at three different wavelengths. a) 542nm, b) 474nm c) 440nm.

Fig. 21.
Fig. 21.

Absorption efficiency among various geometric progressions of gold nanospheres whose lengths are fixed to that of a 16-sphere linear chain.

Fig. 22.
Fig. 22.

Absorption efficiency among various geometric progressions of silver nanospheres whose lengths are fixed to that of a 16-sphere linear chain.

Fig. 23.
Fig. 23.

Scattering efficiency among various geometric progressions of gold nanospheres whose lengths are fixed to that of a 16-sphere linear chain.

Fig. 24.
Fig. 24.

Scattering efficiency among various geometric progressions of silver nanospheres whose lengths are fixed to that of a 16-sphere linear chain.

Fig. 29.
Fig. 29.

Discrete approximation of a) cone, b) pyramid and c) triangular truss.

Fig. 30.
Fig. 30.

Enhancement of nearfield at focal point among the cone, pyramid and truss geometries.

Tables (1)

Tables Icon

Table 1: Summary of resonance peaks for self-similar structures of length 167.5nm taken from data plotted in Figs. 13,14,21,22. Near-field distributions are plotted at near-field enhancement peaks.

Equations (36)

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P S ( v S ) E focal 4 E 0 4
E sca j = n = 1 m = n n i E mn [ a mn j N mn ( 3 ) ( j ) + b mn j M mn ( 3 ) ( j ) ]
E inc j = n = 1 m = n n i E mn [ p mn j N mn ( 1 ) ( j ) + q mn j M mn ( 1 ) ( j ) ]
E int j = n = 1 m = n n i E mn [ d mn j N mn ( 1 ) ( j ) + c mn j M mn ( 1 ) ( j ) ]
H sca j = 1 η n = 1 m = n n E mn [ b mn j N mn ( 3 ) ( j ) + a mn j M mn ( 3 ) ( j ) ]
H inc j = 1 η n = 1 m = n n E mn [ q mn j N mn ( 1 ) ( j ) + p mn j M mn ( 1 ) ( j ) ]
H int j = 1 η j n = 1 m = n n E mn [ c mn j N mn ( 1 ) ( j ) + d mn j M mn ( 1 ) ( j ) ]
E sca = i N s E sca j .
p 1 n 0 = q 1 n 0 = exp ( i Z j ) 2 , p 1 n 0 = q 1 n 0 = exp ( i Z j ) 2 n ( n + 1 )
a mn j = α n j ( p mn j j = 1 j l N s v = 1 μ = v v E μ v E m n [ a μ v l A mn μ v ( R jl ) + b μ v l B mn μ v ( R jl ) ] )
b mn j = β n j ( q mn j j = 1 j l N s v = 1 μ = v v E μ v E m n [ a μ v l B mn μ v ( R jl ) + b μ v l A mn μ v ( R jl ) ] )
a j + i = 1 i j N s T ji a i = p j or ( 1 + T ) a = p
α n j = m j ψ n ( x j ) ψ n ( m j x j ) ψ n ( x j ) ψ n ( m j x j ) m j ξ n ( x j ) ψ n ( m j x j ) ξ n ( x j ) ψ n ( m j x j )
β n j = ψ n ( x j ) ψ n ( m j x j ) m j ψ n ( x j ) ψ n ( m j x j ) ξ n ( x j ) ψ n ( m j x j ) m j ξ n ( x j ) ψ n ( m j x j )
M mn ( 3 ) ( j ) = v = 1 μ = v v [ A μ v mn ( R jl ) M μ v ( 1 ) ( l ) + B μ v mn ( R jl ) N μ v ( 1 ) ( l ) ]
N mn ( 3 ) ( j ) = v = 1 μ = v v [ B μ v mn ( R jl ) M μ v ( 1 ) ( l ) + A μ v mn ( R jl ) N μ v ( 1 ) ( l ) ] .
A μ v mn ( R jl ) ( R μ m ( n ) ) 1 A μ v mn ( 0 , 0 , R jl ) R μ m ( n )
B μ v mn ( R jl ) ( R μ m ( n ) ) 1 B μ v mn ( 0 , 0 , R jl ) R μ m ( n )
R μ m ( n ) ( α , β , γ ) = E mn E μ n F μ n F mn D μ m ( n ) ( α , β , γ )
E mn = E 0 i n ( 2 n + 1 ) ( n m ) ! ( n + m ) !
F mn = ( 1 ) m ( 2 n + 1 ) ( n m ) ! ( 4 π ( n + m ) ! )
D μ m ( n ) ( α , β , γ ) = e ik γ d μ m ( n ) ( β ) e im α
d μ m ( n ) ( β ) = [ ( n + μ ) ! ( n μ ) ! ( n + m ) ! ( n m ) ! ] 1 2 ( cos β 2 ) μ + m ( sin β 2 ) μ m P n μ ( μ m , μ + m ) ( cos β )
P n ( α , β ) ( x ) = 2 n v = 0 n ( n + α v ) ( n + β n v ) ( x 1 ) n v ( x + 1 ) v .
A mn mv = ( 1 ) m + n 2 n + 1 2 n ( n + 1 ) p = n v n + v ( 1 ) q [ n ( n + 1 ) + v ( v + 1 ) p ( p + 1 ) ] a ( m , n , m , v , p ) h p ( 1 ) ( R jl )
B mn mv = ( 1 ) m + n 2 n + 1 2 n ( n + 1 ) 2 im R jl p = n v n + v ( 1 ) q a ( m , n , m , v , p ) h p ( 1 ) ( R jl )
q = ( n + v p ) 2
α p 3 a p 4 ( α p 2 + α p 1 4 m 2 ) a p 2 + α p a p = 0
α p = [ ( n + v + 1 ) 2 p 2 ] [ p 2 ( n v ) 2 ] 4 p 2 1
a n + v = ( 2 n 1 ) ! ! ( 2 v 1 ) ! ! ( 2 n + 2 v 1 ) ! ! ( n + v ) ! ( n m ) ! ( v + m ) !
a n + v 2 = ( 2 n + 2 v 3 ) ( 2 n 1 ) ( 2 v 1 ) ( n + v ) [ n v m 2 ( 2 n + 2 v 1 ) ] a n + v
Q sca = 4 π G k 2 n = 1 m = n n n ( n + 1 ) ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ( a mn T 2 + b mn T 2 )
Q ext = 4 π G k 2 n = 1 m = n n n ( n + 1 ) ( 2 n + 1 ) ( n m ) ! ( n + m ) ! Re { p mn 0 * a mn T + q mn 0 * b mn T }
Q abs = Q ext Q sca
a mn T = j = 1 N s v = 1 μ = v v [ A mn μ v ( R jl ) a μ v j + B mn μ v ( R jl ) b μ v j ]
b mn T = j = 1 N s v = 1 μ = v v [ A mn μ v ( R jl ) b μ v j + B mn μ v ( R jl ) a μ v j ] .

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