Abstract

Based on the generalized multiparticle Mie theory and the Fourier transformation approach, electromagnetic (EM) scattering of two interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes is investigated. By introducing the Fourier transformation, the EM fields in the uniaxial anisotropic spheres are expanded in terms of the spherical vector wave functions. The interactive scattering coefficients and the expansion coefficients of the internal fields are derived through the continuous boundary conditions on which the interaction of the bispheres is considered. Some selected calculations on the effects of the size parameter, the uniaxial anisotropic absorbing dielectric, and the sphere separation distance are described. The backward radar cross section of two uniaxial anisotropic spheres with a complex permittivity tensor changing with the sphere separation distance is numerically studied. The authors are hopeful that the work in this paper will help provide an effective calibration for further research on the scattering characteristic of an aggregate of anisotropic spheres or other shaped anisotropic particles.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (1)

2009 (1)

2007 (1)

C. W. Qiu, L. W. Li, and T. S. Yeo, “Scattering by rotationally symmetric anisotropic spheres: potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

2006 (3)

2004 (1)

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

2003 (2)

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Y. L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105(2003).
[CrossRef]

1999 (1)

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

1997 (2)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112(1997).
[CrossRef]

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

1996 (3)

D. W. Mackowski, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2277 (1996).
[CrossRef]

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

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antenn. Propag. 41, 994–995 (1993).
[CrossRef]

1991 (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond., Ser. A 433, 599–614 (1991).
[CrossRef]

1990 (1)

1989 (2)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

1988 (2)

1984 (1)

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

1979 (1)

1971 (2)

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

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

Blum, J.

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

Bohren, C. F.

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

Borghese, F.

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Brunding, J. H.

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

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

Capsalis, C. N.

Chen, H. T.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” in IEE Proceedings H: Microwaves, Antennas and Propagation (IEEE, 1992), pp. 314–318.
[CrossRef]

Denti, P.

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Dou, L. X.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microw. Opt. Technol. Lett. 48, 2083–2090(2006).
[CrossRef]

Fuller, K. A.

Geng, Y. L.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Giovane, F.

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

Graglia, R. D.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Gustafson, B. Å. S.

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112(1997).
[CrossRef]

Hudson, S.

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antenn. Propag. 41, 994–995 (1993).
[CrossRef]

Huffman, D. R.

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

Kattawar, G. W.

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

Li, L. W.

C. W. Qiu, L. W. Li, and T. S. Yeo, “Scattering by rotationally symmetric anisotropic spheres: potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Li, Z. J.

Lo, Y. T.

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

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

Lubber, R. J.

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923 (1996).
[CrossRef]

Mackowski, D. W.

Nevière, M.

Papadakis, S. N.

Peng, Y.

Popov, E.

Qiu, C. W.

C. W. Qiu, L. W. Li, and T. S. Yeo, “Scattering by rotationally symmetric anisotropic spheres: potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

Saija, R.

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112(1997).
[CrossRef]

Schneider, J.

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antenn. Propag. 41, 994–995 (1993).
[CrossRef]

Schuster, J.

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923 (1996).
[CrossRef]

Sebak, A. R.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microw. Opt. Technol. Lett. 48, 2083–2090(2006).
[CrossRef]

Sindoni, O. I.

Stout, B.

Tehranian, S.

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

Toscano, G.

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[CrossRef] [PubMed]

Uslenghi, P. L. E.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Uzunoglu, N. K.

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

Wong, K. L.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” in IEE Proceedings H: Microwaves, Antennas and Propagation (IEEE, 1992), pp. 314–318.
[CrossRef]

Wu, X. B.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Wu, Z. S.

Xu, Y. L.

Y. L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105(2003).
[CrossRef]

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

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

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef] [PubMed]

Yeo, T. S.

C. W. Qiu, L. W. Li, and T. S. Yeo, “Scattering by rotationally symmetric anisotropic spheres: potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

Yuan, Q. K.

Zich, R. S.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Aerosol Sci. Tech. (1)

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tech. 3, 227–235 (1984).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antenn. Propag. (1)

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antenn. Propag. 41, 994–995 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

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

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

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. A (8)

Microw. Opt. Technol. Lett. (1)

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microw. Opt. Technol. Lett. 48, 2083–2090(2006).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (4)

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Y. L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365(1999).
[CrossRef]

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112(1997).
[CrossRef]

C. W. Qiu, L. W. Li, and T. S. Yeo, “Scattering by rotationally symmetric anisotropic spheres: potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

Proc. IEEE (1)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Proc. R. Soc. Lond., Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond., Ser. A 433, 599–614 (1991).
[CrossRef]

Radio Sci. (2)

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923 (1996).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Other (2)

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” in IEE Proceedings H: Microwaves, Antennas and Propagation (IEEE, 1992), pp. 314–318.
[CrossRef]

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

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Figures (9)

Fig. 1
Fig. 1

Geometry for light scattering of a plane wave by uniaxial anisotropic bispheres.

Fig. 2
Fig. 2

Distribution of the normalized RCS compared with that obtained by CST Microwave Studio.

Fig. 3
Fig. 3

Normalized RCS values versus the scattering angle for two close-packed uniaxial anisotropic spheres along the z axis with different size parameters.

Fig. 4
Fig. 4

Normalized RCS values versus the scattering angle for two close-packed anisotropic spheres along the x axis with different size parameters.

Fig. 5
Fig. 5

Effects of complex uniaxial electric anisotropy on RCS for uniaxial anisotropic bispheres along the z axis.

Fig. 6
Fig. 6

Effects of complex uniaxial electric anisotropy on RCS for uniaxial anisotropic bispheres along the x axis.

Fig. 7
Fig. 7

Normalized RCS values versus the scattering angle for TiO 2 bispheres along the z axis with different sphere separation distances.

Fig. 8
Fig. 8

Backward RCS values versus the sphere separation distance for isotropic bispheres.

Fig. 9
Fig. 9

Backward RCS values versus the sphere separation distance for uniaxial anisotropic bispheres with complex permit tivity tensor.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

E i = E 0 e i k 0 r e ^ x ,
E i = n = 1 m = n n E m n [ a m n i M m n ( 1 ) ( r , k 0 ) + b m n i N m n ( 1 ) ( r , k 0 ) ] , H i = k 0 i w μ 0 n = 1 m = n n E m n [ a m n i N m n ( 1 ) ( r , k 0 ) + b m n i M m n ( 1 ) ( r , k 0 ) ] ,
E m n = E 0 i n [ ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! ] 1 / 2 .
a m n i = i 2 n + 1 2 ( δ m , 1 + δ m , 1 ) , b m n i = i 2 n + 1 2 ( δ m , 1 δ m , 1 ) ,
a j m n i = e i k 0 r j a m n i , b j m n i = e i k 0 r j b m n i ,
E j s = n = 1 m = n n E m n [ a j m n s M m n ( 3 ) ( r j , k 0 ) + b j m n s N m n ( 3 ) ( r j , k 0 ) ] , H j s = k 0 i ω μ 0 n = 1 m = n n E m n [ a j m n s N m n ( 3 ) ( r j , k 0 ) + b j m n s M m n ( 3 ) ( r j , k 0 ) ] .
ε ¯ j = ε 0 [ ε j t 0 0 0 ε j t 0 0 0 ε j z ] , μ ¯ j = μ 0 [ μ j t 0 0 0 μ j t 0 0 0 μ j z ] .
× ( μ ¯ 1 · × E ) ω 2 ε ¯ · E = 0.
E j I ( r j ) = q = 1 2 n = 1 m = n n n = 1 2 π G j m n q 0 π [ A j m n q e M m n ( 1 ) ( r j , k j q ) + B j m n q e N m n ( 1 ) ( r j , k j q ) + C j m n q e L m n ( 1 ) ( r j , k j q ) ] p n m ( cos θ k j ) k j q 2 sin θ k j d θ k j , H j I ( r j ) = q = 1 2 n = 1 m = n n n = 1 2 π G j m n q 0 π [ A j m n q h M m n ( 1 ) ( r j , k j q ) + B j m n q h N m n ( 1 ) ( r j , k j q ) + C j m n q h L m n ( 1 ) ( r j , k j q ) ] p n m ( cos θ k j ) k j q 2 sin θ k j d θ k j .
E j I | t = E j i t | t + E j s | t , H j I | t = H j i t | t + H j s | t ( r j = a j ) ,
E j i t = E j i + E l , j s , H j i t = H j i + H l , j s ,
E j i t = n = 1 m = n n E m n [ f j m n i t M m n ( 1 ) ( r j , k 0 ) + g j m n i t N m n ( 1 ) ( r j , k 0 ) ] , H j i t = k 0 i ω μ 0 n = 1 m = n n E m n [ f j m n i t N m n ( 1 ) ( r j , k 0 ) + g j m n i t M m n ( 1 ) ( r j , k 0 ) ] .
f j m n i t = a j m n i + ( l j ) L v = 1 μ = v v [ a l μ v s A m n μ υ ( l , j ) + b l μ v s B m n μ υ ( l , j ) ] ( l j ) , g j m n i t = b j m n i + ( l j ) L v = 1 μ = v v [ a l μ v s B m n μ υ ( l , j ) + b l μ v s A m n μ υ ( l , j ) ] ( l j ) ,
f j m n i t j n ( k 0 r j ) + a j m n s h n ( 1 ) ( k 0 r j ) = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π A j m n q e j n ( k j q r j ) p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k ( r j = a j ) ,
g j m n i t 1 k 0 r j d ( r j j n ( k 0 r j ) ) d r j + b j m n s 1 k 0 r j d ( r j h n ( 1 ) ( k 0 r j ) ) d r j = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π [ B j m n q e 1 k j q r j d ( r j j n ( k j q r j ) ) d r j + C j m n q e j n ( k j q r j ) r j ] · p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k ( r j = a j ) ,
k 0 i ω μ 0 f j m n i t 1 k 0 r j d ( r j j n ( k 0 r j ) ) d r j + k 0 i ω μ 0 a j m n s 1 k 0 r j d ( r j h n ( 1 ) ( k 0 r j ) ) d r j = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π [ B j m n q h 1 k j q r j d ( r j j n ( k j q r j ) ) d r j + C j m n q h j n ( k j q r j ) r j ] · p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k ( r j = a j ) ,
k 0 i ω μ 0 g j m n i t j n ( k 0 r j ) + k 0 i ω μ 0 b j m n s h n ( 1 ) ( k 0 r j ) = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π A j m n q h j n ( k j q r j ) × p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k ( r j = a j ) .
f j m n i t 1 k 0 r j [ j n ( k 0 r j ) d ( r j h n ( 1 ) ( k 0 r j ) ) d r j h n ( 1 ) ( k 0 r j ) d ( r j j n ( k 0 r j ) ) d r j ] = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π V j m n q p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k ,
g j m n i t 1 k 0 r j [ j n ( k 0 r j ) d ( r h n ( 1 ) ( k 0 r j ) ) d r j h n ( 1 ) ( k 0 r j ) d ( r j j n ( k 0 r j ) ) d r j ] = 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π W j m n q P n ' m ( cos θ j k ) k j q 2 sin θ j k d θ j k ,
V j m n q = { A j m n q e 1 k 0 r j d ( r j h n ( 1 ) ( k 0 r j ) ) d r j j n ( k j q r j ) i ω μ 0 k 0 [ B j m n q h 1 k j q r j d ( r j j n ( k j q r j ) ) d r j + C j m n q h j n ( k j q r j ) r j ] h n ( 1 ) ( k 0 r j ) } ,
W j m n q = { i ω μ 0 k 0 A j m n q h 1 k 0 r j d ( r j h n ( 1 ) ( k 0 r j ) ) d r j [ B j m n q e 1 k j q r j d ( r j j n ( k q r j ) ) d r j + C j m n q e j n ( k j q r j ) r j ] h n ( 1 ) ( k 0 r j ) } .
a j m n s = 1 h n ( 1 ) ( k 0 a j ) [ 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π A j m n q e j n ( k j q r j ) p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k f j m n i t j n ( k 0 a j ) ] ,
b j m n s = 1 h n ( 1 ) ( k 0 a j ) i ω μ 0 k 0 [ 1 E m n q = 1 2 n = 1 2 π G j m n q 0 π A j m n q h j n ( k q r j ) p n m ( cos θ j k ) k j q 2 sin θ j k d θ j k g j m n i t j n ( k 0 a j ) ] .
E s t = E 1 s + E 2 s , H s t = H 1 s + H 2 s .
E s t = n = 1 m = n n E m n [ a m n s t M m n ( 3 ) ( r 0 , k 0 ) + b m n s t N m n ( 3 ) ( r 0 , k 0 ) ] , H s t = k 0 i ω μ 0 n = 1 m = n n E m n [ a m n s t N m n ( 3 ) ( r 0 , k 0 ) + b m n s t M m n ( 3 ) ( r 0 , k 0 ) ] ,
a m n s t = j = 1 2 a j m n s exp ( i k 0 Δ j ) , b m n s t = j = 1 2 b j m n s exp ( i k 0 Δ j ) ,
σ = lim r ( 4 π r 2 | E s t | 2 / | E i | 2 ) ,

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