Abstract

In this paper, we introduce an efficient numerical method based on surface integral equations to characterize the scattering of an arbitrarily incident Gaussian beam by arbitrarily shaped particles with multiple internal inclusions. The incident Gaussian beam is described by the Davis–Barton fifth-order approximation in combination with rotation Euler angles. For numerical purposes, the surfaces of the host particle and the inclusions are modeled using small triangular patches and the established surface integral equations are discretized with the method of moments. The resultant matrix equation is solved by using a parallel implementation of conjugate gradient method on distributed-memory architectures. Some numerical results are included to illustrate the validity and capability of the developed method. These results are also expected to provide useful insights into the scattering of Gaussian beam by composite particles.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  32. Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011).
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    [CrossRef]
  36. Z. W. Cui, Y. P. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A 28(11), 2200–2208 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2011

H. Y. Zhang and T. Q. Liao, “Scattering of Gaussian beam by a spherical particle with a spheroidal inclusion,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1486–1491 (2011).
[CrossRef]

B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011).
[CrossRef]

Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011).
[CrossRef]

J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A 28(1), 24–39 (2011).
[CrossRef] [PubMed]

J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011).
[CrossRef]

Z. W. Cui, Y. P. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A 28(11), 2200–2208 (2011).
[CrossRef] [PubMed]

Z. W. Cui, Y. P. Han, and H. Y. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B 28(11), 2625–2632 (2011).
[CrossRef]

2010

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18(15), 15876–15886 (2010).
[CrossRef] [PubMed]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010).
[CrossRef]

2009

M. Mikrenska and P. Koulev, “Simulation of light scattering by large particles with randomly distributed spherical or cubic inclusions,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1411–1417 (2009).
[CrossRef]

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009).
[CrossRef]

B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009).
[CrossRef]

D. K. Wu and Y. P. Zhou, “Forward scattering light of droplets containing different size inclusions,” Appl. Opt. 48(15), 2957–2965 (2009).
[CrossRef] [PubMed]

2008

2007

2006

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transf. 100(1–3), 237–249 (2006).
[CrossRef]

2005

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

T. Weigel, J. Schulte, and G. Schweiger, “Inelastic scattering on particles with inclusions,” J. Opt. Soc. Am. A 22(6), 1048–1052 (2005).
[CrossRef] [PubMed]

2002

2001

2000

G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47(5), 821–837 (2000).

1996

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99(1), 94–112 (1996).
[CrossRef]

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101(D18), 23311–23316 (1996).
[CrossRef]

1995

1994

1993

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).
[CrossRef]

1992

1991

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991).
[CrossRef]

1989

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989).
[CrossRef]

1985

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).
[CrossRef]

1982

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989).
[CrossRef]

Barber, P. W.

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989).
[CrossRef]

Borghese, F.

Cairns, B.

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101(D18), 23311–23316 (1996).
[CrossRef]

Cha, C. C.

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991).
[CrossRef]

Chrissoulidis, D. P.

Chylek, P.

Chýlek, P.

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99(1), 94–112 (1996).
[CrossRef]

Cravey, R. L.

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991).
[CrossRef]

Cui, Z. W.

Davies, M.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[CrossRef]

Denti, P.

Dunavant, D. A.

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).
[CrossRef]

García-Tuñón, I.

Glisson, A. W.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Gouesbet, G.

J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011).
[CrossRef]

J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A 28(1), 24–39 (2011).
[CrossRef] [PubMed]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47(5), 821–837 (2000).

Graglia, R. D.

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).
[CrossRef]

Gréhan, G.

Hai-Hua, W.

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009).
[CrossRef]

Han, G. X.

Han, X. E.

B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009).
[CrossRef]

Han, Y. P.

J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A 28(1), 24–39 (2011).
[CrossRef] [PubMed]

Z. W. Cui, Y. P. Han, and H. Y. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B 28(11), 2625–2632 (2011).
[CrossRef]

Z. W. Cui, Y. P. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A 28(11), 2200–2208 (2011).
[CrossRef] [PubMed]

J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011).
[CrossRef]

Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010).
[CrossRef]

G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25(12), 2064–2072 (2008).
[CrossRef]

Hill, S. C.

Ioannidou, M. P.

Jin, S.

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009).
[CrossRef]

Khaled, E. E.

Koulev, P.

M. Mikrenska and P. Koulev, “Simulation of light scattering by large particles with randomly distributed spherical or cubic inclusions,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1411–1417 (2009).
[CrossRef]

Landesa, L.

Li, M. L.

Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011).
[CrossRef]

Liao, T. Q.

H. Y. Zhang and T. Q. Liao, “Scattering of Gaussian beam by a spherical particle with a spheroidal inclusion,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1486–1491 (2011).
[CrossRef]

Liu, C. H.

B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011).
[CrossRef]

Liu, J. Y.

Macke, A.

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101(D18), 23311–23316 (1996).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transf. 100(1–3), 237–249 (2006).
[CrossRef]

Mikrenska, M.

M. Mikrenska and P. Koulev, “Simulation of light scattering by large particles with randomly distributed spherical or cubic inclusions,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1411–1417 (2009).
[CrossRef]

Mishchenko, M. I.

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101(D18), 23311–23316 (1996).
[CrossRef]

Moneda, A. P.

Ngo, D.

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99(1), 94–112 (1996).
[CrossRef]

G. Videen, D. Ngo, P. Chylek, and R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12(5), 922–928 (1995).
[CrossRef]

Obelleiro, F.

Pinnick, R. G.

Prabhu, D. R.

Rao, S. M.

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991).
[CrossRef]

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Ren, K. F.

B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009).
[CrossRef]

Rivero, J.

Saengkaew, S.

Saija, R.

Sarvas, J.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

Schulte, J.

Schweiger, G.

Sindoni, O. I.

Skaropoulos, N. C.

Taboada, J. M.

Taskinen, M.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

Videen, G.

Wang, J. J.

Wan-Qiang, L.

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009).
[CrossRef]

Weigel, T.

Wilkes, D.

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991).
[CrossRef]

Wilton, D. R.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Wu, D. K.

Xian-Ming, S.

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009).
[CrossRef]

Xu, Q.

Yan, B.

B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011).
[CrossRef]

B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009).
[CrossRef]

Ylä-Oijala, P.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

Zhang, H. Y.

Z. W. Cui, Y. P. Han, and H. Y. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B 28(11), 2625–2632 (2011).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry of Cartesian coordinates of the beam and particle.

Fig. 2
Fig. 2

Configuration of an arbitrarily shaped particle with multiple internal inclusions.

Fig. 3
Fig. 3

Geometrical parameters associated with the nth edge.

Fig. 4
Fig. 4

Comparison of the DSCSs for a spherical particle with a concentric spherical inclusion obtained from the present method and that from the GLMT.

Fig. 5
Fig. 5

Geometry of a spheroidal particle with two spherical inclusions.

Fig. 6
Fig. 6

DSCSs for a spheroidal particle with two spherical inclusions illuminated by a plane wave and a Gaussian beam: (a) E-plane, (b) H-plane.

Fig. 7
Fig. 7

Geometry of a spherical particle with four inclusions of different shape.

Fig. 8
Fig. 8

DSCSs for a spherical particle containing four different inclusions with the location of the beam waist center as a varied parameter: (a) E-plane, (b) H-plane.

Fig. 9
Fig. 9

Illustration of a cubic particle containing 27 randomly distributed spherical inclusions.

Fig. 10
Fig. 10

Angular distributions of the DSCS for a cubic particle containing 27 randomly distributed spherical inclusions in the E-plane with Euler angles α=γ= 0 o and β as a parameter.

Fig. 11
Fig. 11

Illustration of a hexagonal prism with a fractal aggregate of spherical inclusions: (a) host hexagonal prism, (b) internal fractal aggregate.

Fig. 12
Fig. 12

Differential scattering cross sections for a hexagonal prism with a fractal aggregate of spherical inclusions.

Equations (36)

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[ x x 0 y y 0 z z 0 ]=A[ u v w ]
A=[ cosα sinα 0 sinα cosα 0 0 0 1 ][ cosβ 0 sinβ 0 1 0 sinβ 0 cosβ ][ cosγ sinγ 0 sinγ cosγ 0 0 0 1 ]
[ E x E y E z ]=A[ E u E v E w ], [ H x H y H z ]=A[ H u H v H w ]
E u = E 0 { 1+ s 2 ( ρ 2 Q 2 +i ρ 4 Q 3 2 Q 2 ξ 2 ) + s 4 [ 2 ρ 4 Q 4 i3 ρ 6 Q 5 0.5 ρ 8 Q 6 +( 8 ρ 2 Q 4 i2 ρ 4 Q 5 ) ξ 2 ] } ψ 0 e ikw
E v = E 0 { s 2 ( 2 Q 2 ξη )+ s 4 [ ( 8 ρ 2 Q 4 i2 ρ 4 Q 5 )ξη ] } ψ 0 e ikw
E w = E 0 { s( 2Qξ ) + s 3 [ ( 6 ρ 2 Q 3 i2 ρ 4 Q 4 )ξ ] + s 5 [ ( 20 ρ 4 Q 5 +i10 ρ 6 Q 6 + ρ 8 Q 7 )ξ ] ψ 0 e ikw
H u = H 0 { s 2 ( 2 Q 2 ξη )+ s 4 [ ( 8 ρ 2 Q 4 i2 ρ 4 Q 5 )ξη ] } ψ 0 e ikw
H v = H 0 { 1+ s 2 ( ρ 2 Q 2 +i ρ 4 Q 3 2 Q 2 η 2 ) + s 4 [ 2 ρ 4 Q 4 i3 ρ 6 Q 5 0.5 ρ 8 Q 6 +( 8 ρ 2 Q 4 i2 ρ 4 Q 5 ) η 2 ] } ψ 0 e ikw
H w = H 0 { s( 2Qη ) + s 3 [ ( 6 ρ 2 Q 3 i2 ρ 4 Q 4 )η ] + s 5 [ ( 20 ρ 4 Q 5 +i10 ρ 6 Q 6 + ρ 8 Q 7 )η ] ψ 0 e ikw
s= 1 k ω 0
Q= 1 i+2ζ
ρ= ξ 2 + η 2
ψ 0 =iQexp( i ρ 2 Q )
ξ= u ω 0 ,η= v ω 0 ,ζ= w k ω 0 2
E 0 sca = Z 0 L 0 S p ( J p ) K 0 S p ( M p )
H 0 sca = K 0 S p ( J p )+ 1 Z 0 L 0 S p ( M p )
L 0 S p (X)=i k 0 S p [ X( r' )+ 1 k 0 2 ( 'X( r' ) ) ] G 0 ( r,r' )dS'
K 0 S p (X)= S p X( r' )× G 0 ( r,r' )dS'
G 0 ( r,r' )= e -i k 0 | r-r' | 4π| r-r' |
E p sca =[ Z p L p S p ( J p ) K p S p ( M p ) ]+ i=1 m [ Z p L p S i ( J i ) K p S i ( M i ) ]
H p sca =[ K p S p ( J p )+ 1 Z p L p S p ( M p ) ]+ i=1 m [ K p S i ( J i )+ 1 Z p L p S i ( M i ) ]
E i sca = Z i L i S i ( J i ) K i S i ( M i )
H i sca = K i S i ( J i )+ 1 Z i L i S i ( M i )
( E 0 sca + E inc ) | tan( S p ) = E p sca | tan( S p )
( H 0 sca + H inc ) | tan( S p ) = H p sca | tan( S p )
E p sca | tan( S i ) = E i sca | tan( S i )
H p sca | tan( S i ) = H i sca | tan( S i )
f n (r)={ l n 2 A n + ρ n + ,r in T n + l n 2 A n ρ n ,r in T n 0,otherwise
J p = n=1 N p J p,n f n , M p = n=1 N p M p,n f n
J i = n=1 N i J i,n f n , M i = n=1 N i M i,n f n
[ Z J p J p Z J p M p Z J p J 1 Z J p J m Z J p M 1 Z J p M m Z M p J p Z M p M p Z M p J 1 Z M p J m Z M p M 1 Z M p M m Z J 1 J p Z J 1 M p Z J 1 J 1 Z J 1 J m Z J 1 M 1 Z J 1 M m Z J m J p Z J m M p Z J m J 1 Z J m J m Z J m M 1 Z J m M m Z M 1 J p Z M 1 M p Z M 1 J 1 Z M 1 J m Z M 1 M 1 Z M 1 M m Z M m J p Z M m M p Z M m J 1 Z M m J m Z M m M 1 Z M m M m ]{ J p M p J 1 J m M 1 M m }={ b E b H 0 0 0 0 }
{ J t }={ J 1 J m },{ M t }={ M 1 M m }
[ Z J p J p Z J p M p Z J p J t Z J p M t Z M p J p Z M p M p Z M p J t Z M p M t Z J t J p Z J t M p Z J t J t Z J t M t Z M t J p Z M t M p Z M t J t Z M t M t ]{ J p M p J t M t }={ b E b H 0 0 }
E far sca ( r )=i k 0 e i k 0 r 4πr S p [ Z 0 ( θ ^ θ ^ + ϕ ^ ϕ ^ ) J p ( r' )( θ ^ ϕ ^ ϕ ^ θ ^ ) M p ( r' ) ] e i k 0 k ^ r' dS'
H far sca ( r )=i k 0 e i k 0 r 4πr S p [ ( ϕ ^ θ ^ θ ^ ϕ ^ ) J p ( r' )+ 1 Z 0 ( θ ^ θ ^ + ϕ ^ ϕ ^ ) M p ( r' ) ] e i k 0 k ^ r' dS'
σ= lim r 4π r 2 | E far sca | 2 | E 0 | 2

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