Abstract

An exact analytical solution is obtained for the scattering of electromagnetic waves from a plane wave with arbitrary directions of propagation and polarization by an aggregate of interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes. The expansion coefficients of a plane wave with arbitrary directions of propagation and polarization, for both TM and TE modes, are derived in terms of spherical vector wave functions. The effects of the incident angle α and the polarization angle β on the radar cross sections (RCSs) of several types of collective uniaxial anisotropic spheres are numerically analyzed in detail. The characteristics of the forward and backward RCSs in relation to the incident wavelength are also numerically studied. Selected results on the forward and backward RCSs of several types of square arrays of SiO2 spheres illuminated by a plane wave with different incident angles are described. The accuracy of the expansion coefficients of the incident fields is verified by comparing them with the results obtained from references when the plane wave is degenerated to a z-propagating and x- or y-polarized plane wave. The validity of the theory is also confirmed by comparing the numerical results with those provided by a CST simulation.

© 2012 Optical Society of America

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References

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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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2011 (1)

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 (2)

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

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. homogeneous sphere,” J. Opt. Soc. Am. A 23, 1111–1123 (2006).
[CrossRef]

2004 (1)

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70056609 (2004).
[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 (1)

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

1996 (3)

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923–929 (1996).
[CrossRef]

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

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]

1995 (1)

1994 (1)

1993 (1)

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

1992 (1)

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc. H 139, 314–318 (1992).
[CrossRef]

1991 (1)

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

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 (2)

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part I: general theory,” IEEE Trans. Antennas Propag. 32, 867–869 (1984).
[CrossRef]

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

1980 (1)

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. Tecnol. 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 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]

Chen, H. T.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc. H 139, 314–318 (1992).
[CrossRef]

Denti, P.

F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Tecnol. 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,” Microwave 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 70056609 (2004).
[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]

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part I: general theory,” IEEE Trans. Antennas Propag. 32, 867–869 (1984).
[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. Antennas 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, H. Y.

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 70056609 (2004).
[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, 923–929 (1996).
[CrossRef]

Mackowski, D. W.

Nevière, M.

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. Tecnol. 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. Antennas 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, 923–929 (1996).
[CrossRef]

Sebak, A. R.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave 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. Tecnol. 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]

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part I: general theory,” IEEE Trans. Antennas Propag. 32, 867–869 (1984).
[CrossRef]

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]

Wiscombe, W. J.

Wong, K. L.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc. H 139, 314–318 (1992).
[CrossRef]

Wu, X. B.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70056609 (2004).
[CrossRef]

Wu, Z. S.

Xu, Y.

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

Xu, Y. L.

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,” 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. Tecnol. (1)

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

Appl. Opt. (3)

IEE Proc. H (1)

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc. H 139, 314–318 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

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

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part I: general theory,” IEEE Trans. Antennas Propag. 32, 867–869 (1984).
[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]

J. Comput. Phys. (1)

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 (6)

Microwave Opt. Technol. Lett. (1)

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

Opt. Lett. (2)

Phys. Rev. E (4)

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 70056609 (2004).
[CrossRef]

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, 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]

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. A (1)

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

Radio Sci. (1)

J. Schuster and R. J. Lubber, “Finite difference time domain analysis of arbitrarily biased magnetized ferrites,” Radio Sci. 31, 923–929 (1996).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

An aggregate of uniaxial anisotropic spheres illuminated by a plane wave.

Fig. 2
Fig. 2

Normalized RCS values versus scattering angles for different values of β. (a)  α = 0 ° , (b)  α = 30 ° .

Fig. 3
Fig. 3

Normalized RCS values versus scattering angles for different values of α. (a) E-plane, (b) H-plane.

Fig. 4
Fig. 4

Normalized RCS values versus scattering angles for a 4 × 4 square array of 16 SiO 2 spheres. (a) E-plane, (b) H-plane.

Fig. 5
Fig. 5

Normalized values of (a) forward and (b) backward RCSs versus incident wavelength λ for a 4 × 4 square array of 16 SiO 2 spheres.

Fig. 6
Fig. 6

Normalized values of (a) forward and (b) backward RCSs versus incident wavelength λ for a 4 × 4 square array of 16 SiO 2 spheres for different observing angles.

Fig. 7
Fig. 7

Normalized values of backward RCS versus incident wavelength λ for four types of aggregates of SiO 2 spheres: (a)  4 × 4 square array, (b)  8 × 8 square array, (c)  2 × 4 × 4 cuboid array, (d)  4 × 4 × 4 cube-shaped array.

Equations (44)

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

k 0 = k ( e ^ x sin α cos β + e ^ y sin α sin β + e ^ z cos α ) ,
E i = E 0 ( e ^ x cos β cos α + e ^ y sin β cos α e ^ z sin α ) e i k 0 r .
E i = E 0 ( e ^ x sin β e ^ y cos β ) e i k 0 r ,
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 ω 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 .
e ^ x e i k 0 · r = n = 1 m = n n { a m n x M m n ( 1 ) + b m n x N m n ( 1 ) } , e ^ y e i k 0 · r = n = 1 m = n n { a m n y M m n ( 1 ) + b m n y N m n ( 1 ) } , e ^ z e i k 0 · r = n = 1 m = n n { a m n z M m n ( 1 ) + b m n z N m n ( 1 ) } .
E i = E 0 n = 1 m = n n [ ( a m n x cos β cos α + a m n y sin β cos α a m n z sin α ) M m n ( 1 ) + ( b m n x cos β cos α + b m n y sin β cos α b m n z sin α ) N m n ( 1 ) ]
E i = E 0 m n [ ( a m n x sin β + a m n y cos β ) M m n ( 1 ) + ( b m n x sin β + b m n y cos β ) N m n ( 1 ) ]
a m n i = E 0 ( a m n x cos β cos α + a m n y sin β cos α a m n z sin α ) / E m n , b m n i = E 0 ( b m n x cos β cos α + b m n y sin β cos α b m n z sin α ) / E m n
a m n i = E 0 ( a m n x sin β + a m n y cos β ) / E m n , b m n i = E 0 ( b m n x sin β + b m n y cos β ) / E m n
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 ] .
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 + ( l j ) L E l , j s , H j i t = H j i + ( l j ) L 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 ) ,
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 = ( j = 1 ) L E j s , H s t = ( j = 1 ) L H j 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 L a j m n s exp ( i k 0 Δ j ) , b m n s t = j = 1 L 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 ) ,
σ λ 2 = 1 π { | n = 1 m = n n E m n ( i ) n e i m ϕ [ m a m n s t p n m ( cos θ ) sin θ b m n s t d p n m ( cos θ ) d θ ] | 2 + | n = 1 m = n n E m n ( i ) n + 1 e i m ϕ [ a m n s t d p n m ( cos θ ) d θ m b m n s t p n m ( cos θ ) sin θ ] | 2 } .
a m n c = e ^ c e i k 0 · r , M m n ( 1 ) / M m n ( 1 ) , M m n ( 1 ) , b m n c = e ^ c e i k 0 · r , N m n ( 1 ) / N m n ( 1 ) , N m n ( 1 ) ,
a m n x ( α , β ) ( m 0 ) = i n + 1 2 n + 1 2 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ ( n + m ) ( n m + 1 ) P n m 1 ( cos α ) e i ( m 1 ) β + P n m + 1 ( cos α ) e i ( m + 1 ) β ] ,
a m n y ( α , β ) ( m 0 ) = i n 2 n + 1 2 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ ( n + m ) ( n m + 1 ) P n m 1 ( cos α ) e i ( m 1 ) β P n m + 1 ( cos α ) e i ( m + 1 ) β ] ,
a m n z ( α , β ) ( m 0 ) = i n + 1 m ( 2 n + 1 ) n ( n + 1 ) ( n m ) ! ( n + m ) ! P n m ( cos α ) e i m β ,
b m n x ( α , β ) ( m 0 ) = i n + 1 ( 2 n + 1 ) 2 n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) ( n m ) ! ( n + m ) ! { ( 2 n + 3 ) ( n + 1 ) [ ( n + m ) ( n + m 1 ) P n 1 m 1 ( cos α ) e i ( m 1 ) β P n 1 m + 1 ( cos α ) e i ( m + 1 ) β ] ( 2 n 1 ) n [ P n + 1 m + 1 ( cos α ) e i ( m + 1 ) β + ( n m + 2 ) ( n m + 1 ) P n + 1 m 1 ( cos α ) e i ( m 1 ) β ] } ,
b m n y ( α , β ) ( m 0 ) = i n ( 2 n + 1 ) 2 n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) ( n m ) ! ( n + m ) ! { ( 2 n + 3 ) ( n + 1 ) [ ( n + m ) ( n + m 1 ) P n 1 m 1 ( cos α ) e i ( m 1 ) β + P n 1 m + 1 ( cos α ) e i ( m + 1 ) β ] + ( 2 n 1 ) n [ P n + 1 m + 1 ( cos α ) e i ( m + 1 ) β + ( n m + 2 ) ( n m + 1 ) P n + 1 m 1 ( cos α ) e i ( m 1 ) β ] } ,
b m n z ( α , β ) ( m 0 ) = i n + 1 ( 2 n + 1 ) n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) ( n m ) ! ( n + m ) ! * [ n ( n m + 1 ) ( 2 n 1 ) P n + 1 m ( cos α ) ( n + 1 ) ( n + m ) ( 2 n + 3 ) P n 1 m ( cos α ) ] e i m β ,
a m n x ( α , β ) ( m 0 ) = ( 1 ) m + 1 i n + 1 2 n + 1 2 n ( n + 1 ) [ ( n + m ) ( n m + 1 ) p n m 1 ( cos α ) e i ( m 1 ) β + p n m + 1 ( cos α ) e i ( m + 1 ) β ] ,
a m n y ( α , β ) ( m 0 ) = ( 1 ) m i n 2 n + 1 2 n ( n + 1 ) [ ( n + m ) ( n m + 1 ) P n m 1 ( cos α ) e i ( m 1 ) β P n m + 1 ( cos α ) e i ( m + 1 ) β ] ,
a m n z ( α , β ) ( m 0 ) = ( ) m + 1 e 2 i m β i n + 1 m ( 2 n + 1 ) n ( n + 1 ) P n m ( cos α ) e i m β ,
b m n x ( α , β ) ( m 0 ) = i n + 1 ( 2 n + 1 ) ( 1 ) m 2 n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) { ( 2 n + 3 ) ( n + 1 ) [ ( n + m ) ( n + m 1 ) P n 1 m 1 ( cos α ) e i ( m 1 ) β P n 1 m + 1 ( cos α ) e i ( m + 1 ) β ( 2 n 1 ) n [ P n + 1 m + 1 ( cos α ) e i ( m + 1 ) β + ( n m + 2 ) ( n m + 1 ) P n + 1 m 1 ( cos α ) e i ( m 1 ) β ] } ,
b m n y ( α , β ) ( m 0 ) = ( 1 ) m + 1 i n ( 2 n + 1 ) 2 n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) { ( 2 n + 3 ) ( n + 1 ) [ ( n + m ) ( n + m 1 ) P n 1 m 1 ( cos α ) e i ( m 1 ) β + P n 1 m + 1 ( cos α ) e i ( m + 1 ) β + ( 2 n 1 ) n [ P n + 1 m + 1 ( cos α ) e i ( m + 1 ) β + ( n m + 2 ) ( n m + 1 ) P n + 1 m 1 ( cos α ) e i ( m 1 ) β ] } ,
b m n z ( α , β ) ( m 0 ) = ( 1 ) m e 2 i m β i n + 1 ( 2 n + 1 ) n ( n + 1 ) ( 4 n 2 + 4 n + 3 ) [ n ( n m + 1 ) ( 2 n 1 ) P n + 1 m ( cos α ) ( n + 1 ) ( n + m ) ( 2 n + 3 ) P n 1 m ( cos α ) ] e i m β ,
P n m ( 1 ) = 0 ; m 0 ; and P n ( 1 ) = 1 .
a m n x ( α , β ) ( m = 1 ) = i n + 1 2 n + 1 2 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ ( n + m ) ( n m + 1 ) P n m 1 ( 1 ) + P n m + 1 ( 1 ) ] = i n + 1 2 n + 1 2 n ( n + 1 ) ,
a m n x ( α , β ) ( m = 1 ) = i n + 1 2 n + 1 2 ,
b m n x ( α , β ) ( m = 1 ) = i n + 1 ( 2 n + 1 ) 2 n ( n + 1 ) , b m n x ( α , β ) ( m = 1 ) = i n + 1 ( 2 n + 1 ) 2 .
a m n i = E 0 a m n x E m n = { i n + 1 2 n + 1 2 n ( n + 1 ) m = 1 i n + 1 2 n + 1 2 m = 1 , b m n i = E 0 b m n x E m n = { i n + 1 2 n + 1 2 n ( n + 1 ) m = 1 i n + 1 2 n + 1 2 m = 1 .
a m n i = { i n 2 n + 1 2 n ( n + 1 ) m = 1 i n 2 n + 1 2 m = 1 , b m n i = { i n 2 n + 1 2 n ( n + 1 ) m = 1 i n 2 n + 1 2 m = 1 .

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