Abstract

Through an application of our previously derived single spherical particle–arbitrary beam interaction theory, an iterative procedure has been developed for the determination of the electromagnetic field for a beam incident on two adjacent spherical particles. The two particles can differ in size and composition and can have any positioning relative to each other and relative to the focal point and propagation direction of the incident beam. Example calculations of internal and near-field normalized source function (∼|E|2) distributions are presented. Also presented are calculations demonstrating the effect of the relative positioning of the second adjacent particle on far-field scattering patterns.

© 1991 Optical Society of America

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References

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  1. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [CrossRef]
  2. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
    [CrossRef]
  3. G. W. Kattawar, C. E. Dean, “Electromagnetic scattering from two dielectric spheres: comparison between theory and experiment,” Opt. Lett. 8, 48–50 (1983).
    [CrossRef] [PubMed]
  4. K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
    [CrossRef] [PubMed]
  5. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [CrossRef] [PubMed]
  6. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef] [PubMed]
  7. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  8. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  9. J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  10. D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.
  11. D. R. Alexander, J. G. Armstrong, “Explosive vaporization of aerosol drops under irradiation by a CO2 laser beam,” Appl. Opt. 26, 533–538 (1987).
    [CrossRef] [PubMed]
  12. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  13. N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
    [CrossRef]
  14. W-C. Tsai, R. J. Pogorzelski, “Eigenfunction solution of the scattering of beam radiation by spherical objects,” J. Opt. Soc. Am. 65, 1457–1463 (1975).
    [CrossRef]
  15. C. Yeh, S. Colak, P. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
    [CrossRef] [PubMed]
  16. G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]

1989 (3)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

1988 (3)

1987 (1)

1986 (2)

1983 (1)

1982 (1)

1975 (1)

1971 (2)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

D. R. Alexander, J. G. Armstrong, “Explosive vaporization of aerosol drops under irradiation by a CO2 laser beam,” Appl. Opt. 26, 533–538 (1987).
[CrossRef] [PubMed]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

Armstrong, J. G.

Barber, P.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

Colak, S.

Dean, C. E.

Emanuel, M. A.

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

Fuller, K. A.

Gouesbet, G.

Grehan, G.

Kattawar, G. W.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Maheu, B.

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Nakanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Pogorzelski, R. J.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

Tanaka, T.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Tsai, W-C.

Wang, R. T.

Yamasaki, T.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Yeh, C.

Zhang, J.

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (3)

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II—Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

J. Appl. Phys. (4)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (3)

Other (1)

D. R. Alexander, J. P. Barton, S. A. Schaub, M. A. Emanuel, J. Zhang, “Experimental and theoretical analysis of the interaction of laser radiation with fluid cylinders and spheres,” in Proceedings, of the 1987 U.S. Army Scientific Conference on Obscuration and Aerosol Research, Aberdeen Proving Grounds, Md., CRDEC-SP-88031 (1987), pp. 251–272.

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Figures (15)

Fig. 1
Fig. 1

Geometrical arrangement for the two spherical particle– arbitrary beam interaction theory.

Fig. 2
Fig. 2

Normalized source function distribution in the xz plane for a linearly polarized (electric-field polarization perpendicular to the xz plane) Gaussian beam propagating in the θbd = 0° direction incident on two equal size, same composition spheres (the x, z coordinates are normalized relative to the radius of particle 1): α1 =α2 = 13.337, n ¯ 1 = n ¯ 2 = 1.179 + 0.072 i, z 12 = 3.556, w 0 = 2.222, x 0 = y 0 = 0.0, z 0 = 1.778, θbd = 0°, ϕbd = 90°, ∊ext = 1.0, xref = yref = 0.0, zref = 1.778 [10.6-μm wavelength (CO2 laser), 100m waist diameter beam incident on two 45-μm diameter water droplets separated along the propagation axis by 35 μm surface to surface].

Fig. 3
Fig. 3

Pulsed laser (Δt = 10 ns) image of the explosive fragmentation of the shadow side water droplet due to the “focusing” of incident CO2 laser radiation by the illuminated side water droplet. The CO2 laser beam is incident from left to right. The droplets are 45 μm in diameter and are falling from top to bottom through the focal point of the continuous CO2 laser beam.

Fig. 4
Fig. 4

Normalized source function distribution in the xz plane for a linearly polarized (electric-field polarization perpendicular to the xz plane) Gaussian beam propagating in the θbd = 30° direction incident on two unequal size, dissimilar composition spheres (the x, z coordinates are normalized relative to the radius of particle 1): α1 = 14.82, α2 =11.86, n ¯ 1 = 1.395 + 0.0163 i, n ¯ 2 = 1.179 + 0.072 i, z 12 = 2.60, w 0 = 0.80, x 0 = 1.0, y 0 = 0.0, z 0 = 0.0, θbd = 30°, ϕbd = 90°, ∊ext = 1.0, xref = yref = 0.0, zref = 1.40 [10.6-μmwavelength (CO2 laser), 40-μm waist diameter beam incident on a 50-μm diam methanol droplet and a 40-μm diameter water droplet separated by 20 μm surface to surface].

Fig. 5
Fig. 5

Number of iterations required for a three-digit convergence for nonresonant (α = 30.286) and resonant (α = 30.4770) spheres as a function of surface-to-surface separation distance:, n ¯ 1 = n ¯ 2 = 1.334 + 1.2 × 10 9 i, z 12 = 2.0 + d sep, w 0 = 0.4007, x 0 = y 0 = 0.0, z 0 = 1 . 165, θbd = 90°, ϕbd = 0°, ∊ext = 1.0.

Fig. 6
Fig. 6

Normalized source function distribution in the xz plane for a focused beam incident on two adjacent resonant particles: α1 = α2 = 30.4770, n ¯ 1 = n ¯ 2 = 1.334 + 1.2 × 10 9 i, w 0 = 0.4007, x 0 = y 0 = 0.0, z 0 = 1.165, θbd = 90°, ϕbd = 0°, ∊ext = 1.0, z 12 = 2.2, x ref = y ref = 0.0, z ref = 1.10 [0.5145-μm wavelength (argon-ion laser), 2-μm waist diameter beam incident on two 4.9912-μm diameter water droplets (thirty-fifth mode electric wave resonance)].

Fig. 7
Fig. 7

Normalized source function distribution in the xz plane for a focused beam incident on two adjacent nonresonant particles: α1 = α2 = 30.286, n ¯ 1 = n ¯ 2 = 1.334 + 1.2 × 10 9 i, w 0 = 0.4007, x 0 = y 0 = 0.0, z 0 = 1.165, θbd = 90°, ϕbd = 0°, ∊ext = 1.0, z 12 = 2.2, x ref = y ref = 0.0, z ref = 1.10 [0.5145-μm wavelength (argon-ion laser), 2m waist diameter beam incident on two 4.96-μm diameter water droplets (nonresonance)].

Fig. 8
Fig. 8

Geometrical arrangement for far-field scattering calculations. Particle 1 is fixed in position at the focal point of the beam, while the position of particle 2 is varied. The electric-field polarization of the incident beam is in the xb-axis direction. θff is the far-field-scattering angle relative to the beam propagation axis. θbd2 is the orientation angle of particle 2 relative to the beam propagation axis. α1 = α2 = 11.0, n ¯ = 1.334 + 1.2 × 10 9 i, w 0 = 2.22, x 0 = y 0 = z 0 = 0.0, ∊ext = 1.0 [0.5145-μm wavelength (argon-ion laser), 4-μm waist diam beam incident on two adjacent 1.8-μm diam water droplets].

Fig. 9
Fig. 9

Normalized scattering intensity as a function of the far-field-scattering angle for a single particle ( z 12 ): a.) xb-zb plane, beam incidence; b.) xb-zb plane, plane wave incidence; c.) yb-zb plane, beam incidence; d.) yb-zb plane, plane wave incidence.

Fig. 10
Fig. 10

Normalized scattering intensity in the xbzb plane as a function of the far-field-scattering angle. Particle 2 is in the xbzb plane and positioned one diameter surface to surface from particle 1 ( z 12 = 4.0 ): a.) θbd2 = 0°, b.) θbd2 = 30°, c.) θbd2 = 60°, d:) θbd2 = 90°, e.) θbd2 = 120°, f.) θbd2 = 150°, g.) θbd2 = 180°.

Fig. 11
Fig. 11

Normalized scattering intensity in the ybzb plane as a function of the far-field-scattering angle. Particle 2 is in the xbzb plane and positioned one diameter surface to surface from particle 1 z 12 = 4.0: a.) θbd2 = 0°, b.) θbd2 = 30°, c.) θbd2 = 60°, d.) θbd2 = 90°, e.) θbd2 = 120°, f.) θbd2 = 150°, g.) θbd2 = 180°.

Fig. 12
Fig. 12

Normalized scattering intensity in the xbzb plane as a function of the far-field-scattering angle. Particle 2 is positioned perpendicular to the beam propagation axis (θbd2 = 90°): a.) z 12 = 4.0, b.) z 12 = 6.0, c.) z 12 = 12.0, d.) z 12 = 18.0, e.) z 12 .

Fig. 13
Fig. 13

Normalized scattering intensity in the ybzb plane as a function of the far-field-scattering angle. Particle 2 is positioned perpendicular to the beam propagation axis (θbd2 = 90°): a.) z 12 = 4.0, b.) z 12 = 6.0, c.) z 12 = 12.0, d.) z 12 = 18.0, e.) z 12 .

Fig. 14
Fig. 14

Normalized scattering intensity in the xbzb plane as a function of the far-field-scattering angle. Particle 2 is positioned on the beam propagation axis (θbd2 = 0°): a.) z 12 = 4.0, b.) z 12 = 10.0, c.) z 12 = 20.0, d.) z 12 = 50.0, e.) z 12 = 100.0, f.) z 12 = .

Fig. 15
Fig. 15

Normalized scattering intensity in the ybzb plane as a function of the far-field-scattering angle. Particle 2 is positioned on the beam propagation axis (θbd2 = 0°): a.) z 12 = 4.0, b.) z 12 = 10.0, c.) z 12 = 20.0, d.) z 12 = 50.0, e.) z 12 = 100.0, f.) z 12 .

Equations (36)

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l = 1 m = l l ,
a l m = ψ l ( n ¯ α ) ψ l ( α ) n ¯ ψ l ( n ¯ α ) ψ l ( α ) n ¯ ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) A l m ,
b l m = n ¯ ψ l ( n ¯ α ) ψ l ( α ) ψ l ( n ¯ α ) ψ l ( α ) ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) n ¯ ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) B l m ,
c l m = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) n ¯ 2 ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) n ¯ ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) A l m ,
d l m = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) n ¯ ψ l ( n ¯ α ) ξ l ( 1 ) ( α ) B l m ,
A l m = 1 l ( l + 1 ) ψ l ( α ) 0 2 π 0 π sin θ E r ( i ) ( a , θ , ϕ ) Y l m * ( θ , ϕ ) d θ d ϕ ,
B l m = 1 l ( l + 1 ) ψ l ( α ) 0 2 π 0 π sin θ H r ( i ) ( a , θ , ϕ ) Y l m * ( θ , ϕ ) d θ d ϕ .
E ext ( r , θ , ϕ ) = E ( b ) ( r b , θ b , ϕ b ) + E ( 1 s ) ( r 1 , θ 1 , ϕ 1 ) + E ( 2 s ) ( r 2 , θ 2 , ϕ 2 ) ,
H ext ( r , θ , ϕ ) = H ( b ) ( r b , θ b , ϕ b ) + H ( 1 s ) ( r 1 , θ 1 , ϕ 1 ) + H ( 2 s ) ( r 2 , θ 2 , ϕ 2 ) ,
S r = lim r r 2 c 8 π Re ( E × H ) r c 8 π | E 0 | 2 ,
A l m ( 1 b ) = 1 l ( l + 1 ) ψ l ( α 1 ) 0 2 π 0 π E r 1 ( b ) ( a 1 , θ 1 , ϕ 1 ) × Y l m * ( θ 1 , ϕ 1 ) sin θ 1 d θ 1 d ϕ 1 ,
B l m ( 1 b ) = 1 l ( l + 1 ) ψ l ( α 1 ) 0 2 π 0 π H r 1 ( b ) ( a 1 , θ 1 , ϕ 1 ) × Y l m * ( θ 1 , ϕ 1 ) sin θ 1 d θ 1 d ϕ 1 ,
A l m ( 2 b ) = 1 l ( l + 1 ) ψ l ( α 2 ) 0 2 π 0 π E r 2 ( b ) ( a 2 , θ 2 , ϕ 2 ) × Y l m * ( θ 2 , ϕ 2 ) sin θ 2 d θ 2 d ϕ 2 ,
B l m ( 2 b ) = 1 l ( l + 1 ) ψ l ( α 2 ) 0 2 π 0 π H r 2 ( b ) ( a 2 , θ 2 , ϕ 2 ) × Y l m * ( θ 2 , ϕ 2 ) sin θ 2 d θ 2 d ϕ 2 .
a l m 1 = ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) A l m ( 1 b ) ,
b l m 1 = n ¯ 1 ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) B l m ( 1 b ) .
A l m ( 12 ) = 2 π l ( l + 1 ) ψ l ( α 2 ) 0 π E r ( 12 ) ( m , θ 2 ) P l m ( θ 2 ) sin θ 2 d θ 2 ,
B l m ( 12 ) = 2 π l ( l + 1 ) ψ l ( α 2 ) 0 π H r ( 12 ) ( m , θ 2 ) P l m ( θ 2 ) sin θ 2 d θ 2 ,
E r ( 12 ) ( m , θ 2 ) = l = 1 { 1 r 1 2 l ( l + 1 ) a l m 1 ξ l ( 1 ) ( α 1 , r 1 ) P l m ( θ 1 ) cos ( θ 2 θ 1 ) + α 1 r 1 [ a l m 1 ξ l ( 1 ) ( α 1 , r 1 ) d P l m ( θ 1 ) d θ 1 m b l m 1 ξ l ( 1 ) ( α 1 , r 1 ) P l m ( θ 1 ) sin θ 1 ] sin ( θ 2 θ 1 ) } ,
H r ( 12 ) ( m , θ 2 ) = l = 1 { 1 r 1 2 l ( l + 1 ) b l m 1 ξ l ( 1 ) ( α 1 , r 1 ) P l m ( θ 1 ) cos ( θ 2 θ 1 ) + α 1 r 1 [ b l m 1 ξ l ( 1 ) ( α 1 , r 1 ) d P l m ( θ 1 ) d θ 1 + m a l m 1 ξ l ( 1 ) ( α 1 , r 1 ) P l m ( θ 1 ) sin θ 1 ] sin ( θ 2 θ 1 ) } ,
θ 1 = tan 1 ( α 2 / α 1 sin θ 2 α 2 / α 1 cos θ 2 + z 12 ) ,
r 1 = [ ( z 12 + α 2 / α 1 cos θ 2 ) 2 + ( α 2 / α 1 sin θ 2 ) 2 ] 1 / 2 .
a l m 2 = ψ l ( n ¯ 2 α 2 ) ψ l ( α 2 ) n ¯ 2 ψ l ( n ¯ 2 α 2 ) ψ l ( α 2 ) n ¯ 2 ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) [ A l m ( 12 ) + A l m ( 2 b ) ] ,
b l m 2 = n ¯ 2 ψ l ( n ¯ 2 α 2 ) ψ l ( α 2 ) ψ l ( n ¯ 2 α 2 ) ψ l ( α 2 ) ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) n ¯ 2 ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) [ B l m ( 12 ) + B l m ( 2 b ) ] .
A l m ( 21 ) = 2 π l ( l + 1 ) ψ l ( α 1 ) 0 π E r ( 21 ) ( m , θ 1 ) P l m ( θ 1 ) sin θ 1 d θ 1 ,
B l m ( 21 ) = 2 π l ( l + 1 ) ψ l ( α 1 ) 0 π H r ( 21 ) ( m , θ 1 ) P l m ( θ 1 ) sin θ 1 d θ 1 ,
E r ( 21 ) ( m , θ 1 ) = l = 1 { 1 r 2 2 l ( l + 1 ) a l m 2 ξ l ( 1 ) ( α 2 , r 2 ) P l m ( θ 2 ) cos ( θ 2 θ 1 ) + α 2 r 2 [ a l m 2 ξ l ( 1 ) ( α 2 , r 2 ) d P l m ( θ 2 ) d θ 2 m b l m 2 ξ l ( 1 ) ( α 2 , r 2 ) P l m ( θ 2 ) sin θ 2 ] sin ( θ 1 θ 2 ) } ,
H r ( 21 ) ( m , θ 1 ) = l = 1 { 1 r 2 2 l ( l + 1 ) b l m 2 ξ l ( 1 ) ( α 2 , r 2 ) P l m ( θ 2 ) cos ( θ 1 θ 2 ) + α 2 r 2 [ b l m 2 ξ l ( 1 ) ( α 2 , r 2 ) d P l m ( θ 2 ) d θ 2 + m a l m 2 ξ l ( 1 ) ( α 2 , r 2 ) P l m ( θ 2 ) sin θ 2 ] sin ( θ 1 θ 2 ) } ,
θ 2 = π tan 1 ( sin θ 1 z 12 cos θ 1 ) ,
r 2 = α 1 / α 2 [ ( z 12 cos θ 1 ) 2 + ( sin θ 1 ) 2 ] 1 / 2 .
a l m 1 = ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ψ 1 ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) [ A l m ( 21 ) + A l m ( 1 b ) ] ,
b l m 1 = n ¯ 1 ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) ψ l ( n ¯ 1 α 1 ) ψ l ( α 1 ) ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) [ B l m ( 21 ) + B l m ( 1 b ) ] .
c l m 1 = ξ l ( 1 ) ( α 1 ) ψ l ( α 1 ) ξ l ( 1 ) ( α 1 ) ψ l ( α 1 ) n ¯ 1 2 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) [ A l m ( 21 ) + A l m ( 1 b ) ] ,
d l m 1 = ξ l ( 1 ) ( α 1 ) ψ l ( α 1 ) ξ l ( 1 ) ( α 1 ) ψ l ( α 1 ) ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) n ¯ 1 ψ l ( n ¯ 1 α 1 ) ξ l ( 1 ) ( α 1 ) [ B l m ( 21 ) + B l m ( 1 b ) ] ,
c l m 2 = ξ l ( 1 ) ( α 2 ) ψ l ( α 2 ) ξ l ( 1 ) ( α 2 ) ψ l ( α 2 ) n ¯ 2 2 ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) n ¯ 2 ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) [ A l m ( 12 ) + A l m ( 2 b ) ] ,
d l m 2 = ξ l ( 1 ) ( α 2 ) ψ l ( α 2 ) ξ l ( 1 ) ( α 2 ) ψ l ( α 2 ) ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) n ¯ 2 ψ l ( n ¯ 2 α 2 ) ξ l ( 1 ) ( α 2 ) [ B l m ( 12 ) + B l m ( 2 b ) ] ,

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