Abstract

The surface integral equation method is introduced to characterize the scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped homogeneous dielectric particles. The incident Gaussian beam is represented by the Davis first-order approximation in Cartesian coordinates. For a numerical solution, the particles with arbitrary shapes are modeled by using surface triangular patches and the surface integral equations are discretized with the method of moments. The resulting matrix equations are solved by means of the parallel conjugate gradient method. The calculated results for a sphere and a spheroid are compared with those from the generalized Lorenz–Mie theory, and very good agreements are observed. We also present the numerical results for several selected irregular particles. These results can be used as a reference for other numerical methods to analyze the light scattering by irregular particles illuminated by Gaussian beam.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011

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 Applic. 25, 211–222 (2011).
[CrossRef]

2010

2009

P. Pawliuk and M. Yedlin, “Gaussian beam scattering from a dielectric cylinder, including the evanescent region,” J. Opt. Soc. Am. A 26, 2558–2566 (2009).
[CrossRef]

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

A. I. Mackenzie, S. M. Rao, and M. E. Baginski, “Electromagnetic scattering from arbitrarily shaped dielectric bodies using paired pulse vector basis functions and method of moments,” IEEE Trans. Antennas Propag. 57, 2076–2073 (2009).
[CrossRef]

2008

2007

2004

2003

2001

Y. P. Han and Z. S. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001).
[CrossRef]

Y. P. Han and Z. S. Wu, “The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,” IEEE Trans. Antennas Propag. 49, 615–620 (2001).
[CrossRef]

1999

1998

X. Q. Sheng, J. Ming, J. Jin, M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

1997

1995

1994

1990

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics 10, 407–421 (1990).
[CrossRef]

J. P. Chevaillier, J. Fabre, G. Gréhan, and G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

1989

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applic. 3, 1–15(1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and 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

J. P. Barton, D. R. Alexander, and 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]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

1986

G. Grehan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

K. Umashankar, A. Taflove, and S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

K. Umashankar, A. Taflove, and S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

1983

1982

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

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

1979

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

1978

1975

Alexander, D. R.

J. P. Barton, D. R. Alexander, and 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, and 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]

Baginski, M. E.

A. I. Mackenzie, S. M. Rao, and M. E. Baginski, “Electromagnetic scattering from arbitrarily shaped dielectric bodies using paired pulse vector basis functions and method of moments,” IEEE Trans. Antennas Propag. 57, 2076–2073 (2009).
[CrossRef]

Barton, J. P.

J. P. Barton, “Internal and near-surface electromagnetic fields for an infinite cylinder illuminated by an arbitrary focused beam,” J. Opt. Soc. Am. A 16, 160–166 (1999).
[CrossRef]

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
[CrossRef] [PubMed]

J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 8472–8473 (1995).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, and 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, and 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]

Cai, X.

Chevaillier, J. P.

Chew, W. C.

X. Q. Sheng, J. Ming, J. Jin, M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Corriveau, R.

Cui, Z. W.

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 Applic. 25, 211–222 (2011).
[CrossRef]

Dändliker, R.

Davis, L. W.

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

Doicu, A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Edmonds, A. R.

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

Engels, S. A.

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Ergül, Ö.

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

Fabre, J.

Glisson, A. W.

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

Gouesbet, G.

F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

L. Mees, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[CrossRef]

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. 28, 45–65 (1997).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

J. T. Hodges, G. Gréhan, G. Gouesbet, and C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
[CrossRef] [PubMed]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. 26, 225–239 (1995).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

J. P. Chevaillier, J. Fabre, G. Gréhan, and G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Grehan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

Grehan, G.

Gréhan, G.

F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

L. Mees, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

J. T. Hodges, G. Gréhan, G. Gouesbet, and C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, G. Gréhan, and G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

Gürel, L.

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

Han, Y. P.

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 Applic. 25, 211–222 (2011).
[CrossRef]

H. Y. Zhang and Y. P. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B 25, 131–135 (2008).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

Y. P. Han and Z. S. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001).
[CrossRef]

Y. P. Han and Z. S. Wu, “The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,” IEEE Trans. Antennas Propag. 49, 615–620 (2001).
[CrossRef]

Harrington, R. F.

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applic. 3, 1–15(1989).
[CrossRef]

R. F. Harrington, Field Computation by Moment Methods(Macmillan, 1968).

He, J. P.

Hodges, J. T.

Jin, J.

X. Q. Sheng, J. Ming, J. Jin, M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Karlsson, A.

Kim, J. S.

Kozaki, S.

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

Krattiger, B.

Lee, S. S.

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 Applic. 25, 211–222 (2011).
[CrossRef]

Lock, J. A.

Lu, C. C.

X. Q. Sheng, J. Ming, J. Jin, M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Mackenzie, A. I.

A. I. Mackenzie, S. M. Rao, and M. E. Baginski, “Electromagnetic scattering from arbitrarily shaped dielectric bodies using paired pulse vector basis functions and method of moments,” IEEE Trans. Antennas Propag. 57, 2076–2073 (2009).
[CrossRef]

Maheu, B.

Mees, L.

Ming, J.

X. Q. Sheng, J. Ming, J. Jin, M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Pawliuk, P.

Pogorzelski, R. J.

Presser, C.

Rao, S. M.

A. I. Mackenzie, S. M. Rao, and M. E. Baginski, “Electromagnetic scattering from arbitrarily shaped dielectric bodies using paired pulse vector basis functions and method of moments,” IEEE Trans. Antennas Propag. 57, 2076–2073 (2009).
[CrossRef]

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics 10, 407–421 (1990).
[CrossRef]

K. Umashankar, A. Taflove, and S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

K. Umashankar, A. Taflove, and S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

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

Ren, K. F.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and 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, and 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]

Sheng, X. Q.

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Figures (12)

Fig. 1
Fig. 1

Geometry of Cartesian coordinates of the particle, beam, and accessory.

Fig. 2
Fig. 2

Euler angles of rotation α, β, and γ for transforming the O x y z coordinate system into the O x y z coordinate system.

Fig. 3
Fig. 3

Geometry of an arbitrarily shaped homogeneous dielectric particle in the free-space medium.

Fig. 4
Fig. 4

Geometrical parameters associated with the nth edge.

Fig. 5
Fig. 5

Comparison of the DSCS for a spherical particle obtained from the SIE method and the GLMT: (a) E Plane, (b) H Plane.

Fig. 6
Fig. 6

Comparison of the DSCS for a spheroid obtained from the SIE method and the GLMT: (a) E Plane, (b) H Plane.

Fig. 7
Fig. 7

Illustration of an irregular particle containing two overlapping spheres: (a) Geometry of the particle, (b) 3D discretized model.

Fig. 8
Fig. 8

The differential scattering cross section for an irregular particle consists of two overlapping spheres: (a) E Plane, (b) H Plane.

Fig. 9
Fig. 9

Illustration of a prolate cylinder particle: (a) Geometry of the particle, (b) 3D discretized model.

Fig. 10
Fig. 10

The angular distributions of the DSCS for a prolate cylinder particle in different incident angles: (a) E Plane, (b) H Plane.

Fig. 11
Fig. 11

Illustration of a disk-like particle: (a) Cross section of the disk-like model, (b) 3D model.

Fig. 12
Fig. 12

The differential scattering cross section for a disk-like particle.

Equations (37)

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E x = E 0 Ψ 0 e i k 0 z , E y = 0 , E z = 2 x l Q E x
H x = 0 , H y = H 0 Ψ 0 e i k 0 z , H z = 2 y l Q H y ,
Ψ 0 = i Q exp [ i Q ( ξ 2 + η 2 ) ] ,
Q = 1 i 2 ζ
ξ = x ω 0 , η = y ω 0 , ζ = z l ,
[ x 0 y 0 z 0 ] = A 1 [ x 0 y 0 z 0 ] ,
A = [ cos α sin α 0 sin α cos α 0 0 0 1 ] [ cos β 0 sin β 0 1 0 sin β 0 cos β ] [ cos γ sin γ 0 sin γ cos γ 0 0 0 1 ] .
E x = E 0 Ψ 0 e i k 0 ( z z 0 ) , E y = 0 , E z = x x 0 l 2 Q E x
H x = 0 , H y = H 0 Ψ 0 e i k 0 ( z z 0 ) , H z = y y 0 l 2 Q H y ,
ξ = x x 0 ω 0 , η = y y 0 ω 0 , ζ = z z 0 l .
[ E x E y E z ] = A [ E x E y E z ] , [ H x H y H z ] = A [ H x H y H z ] ,
[ x y z ] = A 1 [ x y z ] .
E i = x ^ E x + y ^ E y + z ^ E z
H i = x ^ H x + y ^ H y + z ^ H z .
E 0 S = Z 0 L 0 ( J ) K 0 ( M )
H 0 S = K 0 ( J ) + 1 Z 0 L 0 ( M ) ,
L 0 ( X ) = i k 0 S [ X ( r ) + 1 k 0 2 ( · X ( r ) ) ] G 0 ( r , r ) d S
K 0 ( X ) = S X ( r ) × G 0 ( r , r ) d S ,
G 0 ( r , r ) = e i k 0 | r r | 4 π | r r | ,
E 1 S = Z 1 L 1 ( J ) K 1 ( M )
H 1 S = K 1 ( J ) + 1 Z 1 L 1 ( M ) ,
| Z 0 L 0 ( J ) + Z 1 L 1 ( J ) K 0 ( M ) K 1 ( M ) = E i | tan
| K 0 ( J ) + K 1 ( J ) + 1 Z 0 L 0 ( M ) + 1 Z 1 L 1 ( M ) = H i | tan ,
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 = n = 1 N J n f n
M = n = 1 N M n f n ,
[ Z J J Z J M Z M J Z M M ] { J M } = { b E b H } ,
Z J J , m n = S f m · [ Z 0 L 0 ( f n ) + Z 1 L 1 ( f n ) ] d S
Z J M , m n = S f m · [ K 0 ( f n ) + K 1 ( f n ) ] d S
Z M J , m n = S f m · [ K 0 ( f n ) + K 1 ( f n ) ] d S
Z M M , m n = S f m · [ 1 Z 0 L 0 ( f n ) + 1 Z 1 L 1 ( f n ) ] d S ,
b E , m = S f m · E i d S
b H , m = S f m · H i d S .
σ = lim r 4 π r 2 | E far sca | 2 | E 0 | 2 ,
E far sca ( r ) e i k 0 r 4 π r S [ i k 0 Z 0 r ^ × ( r ^ × J ( r ) ) + r ^ × M ( r ) ] e i k 0 r ^ · r d S .
r ( θ ) = { ( a b ) cos θ + b 2 ( a b ) 2 sin 2 θ , θ < θ 0 , b / sin θ , θ 0 θ < π θ 0 , ( a b ) cos θ + b 2 ( a b ) 2 sin 2 θ , π θ 0 < θ < π .
D ( x ) = [ 1 ( x / R 0 ) 2 ] 1 / 2 [ C 0 + C 2 ( x / R 0 ) 2 + C 2 ( x / R 0 ) 4 ] ,

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