Abstract

The scattering of an electromagnetic wave by a set of dielectric and metallic spheres is a well-known physical problem. We show a mathematical simplification of the multiple-scattering theory. In this paper, we will establish the multiple-scattering equation in two different ways. Through the study of the equation form, we can choose the simplest spherical wave expansion for calculations. Then, we propose concise expressions of the Mie scattering coefficients and translation coefficients for both polarizations. With these simplified expressions, large spheres are studied without loss of accuracy. Far-field expressions, cross-sections, and the scattering matrix are also simplified. Thus, we obtain formulas that can be easily understood from a physical point of view.

© 2012 Optical Society of America

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References

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  1. L. Rayleigh, “On the scattering of light by small particles,” Philos. Mag. 41, 447–454 (1871).
  2. L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Œuvre Scientifiques de Lorenz, H. Valentiner ed. (Lehman et Stage, 1898), pp. 405–529.
  3. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  4. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
  5. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  6. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. A 433, 599–614 (1991).
    [CrossRef]
  7. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef]
  8. L. Oyhenart and V. Vignéras, “Study of finite periodic structures using the generalized Mie theory,” Eur. Phys. J. Appl. Phys. 39, 95–100 (2007).
    [CrossRef]
  9. L. Oyhenart and V. Vignéras, “Overview of computational methods for photonic crystals,” in Photonic Crystals, A. Massaro, ed. (Intech, 2012), pp. 267–290.
  10. N. Eaton, “Comet dust—applications of Mie scattering,” Vistas Astron. 27, 111–129 (1984).
    [CrossRef]
  11. G. E. Thomas and K. Samnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).
  12. M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15, 2822–2836 (2007).
    [CrossRef]
  13. J. A. Stratton, Théorie de l’électromagnétisme (Dunod, 1961).
  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  15. A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  16. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  17. M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
    [CrossRef]
  18. M. I. Mishchenko, “Light scattering by random oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  19. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  20. X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
    [CrossRef]
  21. J. D. Jackson, Electrodynamique Classique (Dunod, 2001).
  22. J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13, 392–400 (1947).
    [CrossRef]
  23. W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
    [CrossRef]
  24. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).
  25. A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
    [CrossRef]
  26. L. Tsang and J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
    [CrossRef]
  27. Y. L. Xu and Bo A. S. Gustafson, “Experimental and theoretical results of light scattering by aggregate of spheres,” Appl. Opt. 36, 8026–8030 (1997).
    [CrossRef]
  28. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Phys. 36, 9496–9508 (1997).
  29. Y. L. Xu and Bo A. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
    [CrossRef]

2007 (2)

L. Oyhenart and V. Vignéras, “Study of finite periodic structures using the generalized Mie theory,” Eur. Phys. J. Appl. Phys. 39, 95–100 (2007).
[CrossRef]

M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15, 2822–2836 (2007).
[CrossRef]

2001 (1)

Y. L. Xu and Bo A. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

1998 (1)

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

1997 (2)

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

Y. L. Xu and Bo A. S. Gustafson, “Experimental and theoretical results of light scattering by aggregate of spheres,” Appl. Opt. 36, 8026–8030 (1997).
[CrossRef]

1995 (2)

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

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

1993 (1)

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

1991 (2)

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

M. I. Mishchenko, “Light scattering by random oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

1984 (1)

N. Eaton, “Comet dust—applications of Mie scattering,” Vistas Astron. 27, 111–129 (1984).
[CrossRef]

1982 (1)

L. Tsang and J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

1975 (1)

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

1954 (1)

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

1947 (1)

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13, 392–400 (1947).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

1871 (1)

L. Rayleigh, “On the scattering of light by small particles,” Philos. Mag. 41, 447–454 (1871).

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).

Asano,

Bohren, C. F.

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

Cairns, B.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Defos du Rau, M.

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).

Eaton, N.

N. Eaton, “Comet dust—applications of Mie scattering,” Vistas Astron. 27, 111–129 (1984).
[CrossRef]

Gustafson, Bo A. S.

Y. L. Xu and Bo A. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y. L. Xu and Bo A. S. Gustafson, “Experimental and theoretical results of light scattering by aggregate of spheres,” Appl. Opt. 36, 8026–8030 (1997).
[CrossRef]

Harmon, B. N.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Electrodynamique Classique (Dunod, 2001).

Kohn, W.

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Kong, J. A.

L. Tsang and J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).

Korringa, J.

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13, 392–400 (1947).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Liu, L.

Lorenz, L. V.

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Œuvre Scientifiques de Lorenz, H. Valentiner ed. (Lehman et Stage, 1898), pp. 405–529.

Mackowski, D. W.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

Moroz, A.

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

Oyhenart, L.

L. Oyhenart and V. Vignéras, “Study of finite periodic structures using the generalized Mie theory,” Eur. Phys. J. Appl. Phys. 39, 95–100 (2007).
[CrossRef]

L. Oyhenart and V. Vignéras, “Overview of computational methods for photonic crystals,” in Photonic Crystals, A. Massaro, ed. (Intech, 2012), pp. 267–290.

Parneix, J. P.

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

Pessan, F.

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the scattering of light by small particles,” Philos. Mag. 41, 447–454 (1871).

Rostoker, N.

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Ruffie, G.

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

Samnes, K.

G. E. Thomas and K. Samnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).

Stein, A.

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Stratton, J. A.

J. A. Stratton, Théorie de l’électromagnétisme (Dunod, 1961).

Thomas, G. E.

G. E. Thomas and K. Samnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Tsang, L.

L. Tsang and J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).

Videen, G.

Vignéras, V.

L. Oyhenart and V. Vignéras, “Study of finite periodic structures using the generalized Mie theory,” Eur. Phys. J. Appl. Phys. 39, 95–100 (2007).
[CrossRef]

L. Oyhenart and V. Vignéras, “Overview of computational methods for photonic crystals,” in Photonic Crystals, A. Massaro, ed. (Intech, 2012), pp. 267–290.

Vigneras-Lefebvre, V.

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wang, X.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Xu, Y. L.

Y. L. Xu and Bo A. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

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

Y. L. Xu and Bo A. S. Gustafson, “Experimental and theoretical results of light scattering by aggregate of spheres,” Appl. Opt. 36, 8026–8030 (1997).
[CrossRef]

Xu, Y.-L.

Yamamoto, G.

Yu, Q.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Zhang, X.-G.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

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

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Eur. Phys. J. Appl. Phys. (2)

M. Defos du Rau, F. Pessan, G. Ruffie, V. Vigneras-Lefebvre, and J. P. Parneix, “Scattering and coupling effects of electromagnetic waves in 3D networks of spheres,” Eur. Phys. J. Appl. Phys. 1, 45–52 (1998).
[CrossRef]

L. Oyhenart and V. Vignéras, “Study of finite periodic structures using the generalized Mie theory,” Eur. Phys. J. Appl. Phys. 39, 95–100 (2007).
[CrossRef]

J. Appl. Phys. (1)

L. Tsang and J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

Y. L. Xu and Bo A. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Opt. Express (1)

Philos. Mag. (1)

L. Rayleigh, “On the scattering of light by small particles,” Philos. Mag. 41, 447–454 (1871).

Phys. Rev. (1)

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Phys. Rev. B (2)

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Physica (1)

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13, 392–400 (1947).
[CrossRef]

Proc. R. Soc. A (1)

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

Q. Appl. Math. (2)

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Vistas Astron. (1)

N. Eaton, “Comet dust—applications of Mie scattering,” Vistas Astron. 27, 111–129 (1984).
[CrossRef]

Other (8)

G. E. Thomas and K. Samnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

J. D. Jackson, Electrodynamique Classique (Dunod, 2001).

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Œuvre Scientifiques de Lorenz, H. Valentiner ed. (Lehman et Stage, 1898), pp. 405–529.

L. Oyhenart and V. Vignéras, “Overview of computational methods for photonic crystals,” in Photonic Crystals, A. Massaro, ed. (Intech, 2012), pp. 267–290.

J. A. Stratton, Théorie de l’électromagnétisme (Dunod, 1961).

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

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves (Wiley, 2001).

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Figures (7)

Fig. 1.
Fig. 1.

Diagram of the interactions associated with the multiple-scattering equation.

Fig. 2.
Fig. 2.

Comparison of shifted electrical fields. The red arrows are the origin of the coordinate system. Plot (a) is the scattered field by a sphere of permittivity 16. Plot (b) is the shifted field of the previous field with the correct Cruzan formula. Plot (c) is the shifted field with the sign error in the Cruzan formula.

Fig. 3.
Fig. 3.

Definition of the scattering plane.

Fig. 4.
Fig. 4.

Comparison of scattering intensities between GMM and MSCATT. The GMM curves are under the MSCATT curves. The plots (a) and (b) are, respectively, the scattering intensities i1 and i2. The target is a 5×5 square sphere array. The size parameter is x=k0r=5.03, and the complex refractive index of spheres is n=1.615+i0.008. Neighboring spheres are in contact.

Fig. 5.
Fig. 5.

Comparison of scattering intensities between experiment values, MoM and MSCATT methods. The MoM curves are under the MSCATT curves. The plots (a) and (b) are respectively the scattering intensities i1 and i2. The target is the same as Fig. 4.

Fig. 6.
Fig. 6.

Comparison of scattering intensities between MoM and MSCATT methods. The target is the 5×5 square sphere array with a size parameter x=10.48. The complex refractive index of sphere is n=1.615+i0.008. The plots (a) and (b) are, respectively, the scattering intensities i1 and i2.

Fig. 7.
Fig. 7.

Comparison of scattering intensity i1 between MoM and MSCATT methods for two targets. The first target [plot (a)] is a linear chain of three touching identical spheres with a size parameter x=36.47 and a complex refractive index of sphere n=1.615+i0.008. The second target [plot (b)] is a 5×5 square sphere array with a size parameter x=0.79, a radius of spheres r=0.25a (a is the lattice constant), and a complex refractive index of spheres n=4.

Tables (1)

Tables Icon

Table 1. Comparison of Absolute Value of the Largest Coefficient Eσ,l,ms in the Expansion of the Field Scattered by a Sphere (Frequency/c = 1, Refraction Index = 2, Angle of Incidence θ=30°, φ=30°, TM-Polarization)

Equations (68)

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

ψmnoe(r,θ,φ)=cosmφsinmφPnm(cosθ)Zn(kr),M=r×ψ,N=1k×M.
ψmn(r,θ,φ)=eimφPnm(cosθ)Zn(kr),M=r×ψ.N=1k×M.
Mmn=[imsinθPnm(cosθ)e^θddθPnm(cosθ)e^ϕ]Zn(kr)eimφ,Nmn=n(n+1)Pnm(cosθ)Zn(kr)kreimφe^r+1kr[ddθPnm(cosθ)e^θ+imsinθPnm(cosθ)e^ϕ]d[rZn(kr)]dreimφ.
Ylm(r^)=(1)m2l+14π(lm)!(l+m)!Plm(cosθ)eimφ.
L^=1i(r×),Xlm(r^)=4πl(l+1)L^Ylm(r^).
|Xlm(r^)=Xlm(r^),Xlm(r^)|=Xlm*(r^).
|Xlm(r^)=X1lm(r^)e^θ+X1lm(r^)e^φ.
|ZM=1,l,m(r)=zl(kr)|Xlm(r^),|ZE=1,l,m(r)=ik0×|ZM,l,m(r),zl1(kr)=iljl(kr),zl3(kr)=ilhl(kr).
×[×E(r)]k02E(r)=k02[εr(r)1]E(r).
×[×Γ0(r,r)]k02Γ0(r,r)=δ(rr)I
Γ0(r,r)=(I+1k02)eik0|rr||rr|.
E(r)=Γ0(r,r)k02[εr(r)1]E(r)d3r.
E(r)=E0(r)+d3rΓ0(r,r)k02[εr(r)1]E(r).
Γ0(r,r)=|ik04πσ,l,m|Zσ,l,m(1)j(r)Zσ,l,m(3)j(r)|r>rik04πσ,l,m|Zσ,l,m(3)j(r)Zσ,l,m(1)j(r)|r<r,E0(r)=σ,l,mEσ,l,m0,j|Zσ,l,m(1)j(r),E(r)=σ,l,mEσ,l,mj[|Zσ,l,m(1)j(r)+Sjσ,l|Zσ,l,m(3)j(r)]r<r.
Eσ,l,m0,i=j,σ,l,m[δijδσσδllδmmσ,l,mGσ,l,mσ,l,mij·Sσ,l,mj]Eσ,l,mj.
E=E0+G·S·E.
E=(IG.S)1·E0.
E1=E0+G·S·E0.
En+1=E0+G·S·En.
Es,n+1=S·En+1=S·(E0+G·Es,n).
Es,1=S·E0,Es,n+1=S·(E0+G·Es,n).
E0TM=n=1m=nnin+12n+1n(n+1)(nm)!(n+m)!eimφ0[ddθPnm(cosθ0)Mmn(1)+msinθPnm(cosθ0)Nmn(1)].
E0TM=σ=1,1l=1m=llXσ,l,m(k^0)|Zσ,l,m(1).
E0TM=σ,l,mEσ,l,m0,TM|Zσ,l,m(1)withEσ,l,m0,TM=σXσ,l,m*(k^0).
Z0H0TM=σ,l,mHσ,l,m0,TM|Zσ,l,m(1)withHσ,l,m0,TM=Xσ,l,m*(k^0).
{E0TE=Z0H0TMZ0H0TE=E0TM.
E0j=eik0·(rjr0)E0.
(E0+EsEa)×e^r=0,(H0+HsHa)×e^r=0.
{Es=σ,l,mEσ,l,ms|Zσ,l,m(3)withEσ,l,ms=SσlEσ,l,m0Z0Hs=σ,l,mHσ,l,ms|Zσ,l,m(3)withHσ,l,ms=SσlHσ,l,m0,{Ea=σ,l,mAσlEσ,l,m0|Zσ,l,m(1)ZaHa=σ,l,mAσlHσ,l,m0|Zσ,l,m(1).
Es=S·E0.
|Zσ,l,m(J)(r)=σ,l,mGσ,l,mσ,l,m(J)(r0)|Zσ,l,m(1)(r)r<r0,|Zσ,l,m(J)(r)=σ,l,mGσ,l,mσ,l,m(1)(r0)|Zσ,l,m(J)(r)r<r0.
G(J)σ,l,mσ,l,m(r0)=(1)mβllp=|ll||l+l|ασσ·Cm,ml,l,p·zp(J)(k0r0)·Ypmm(r^0),ασσ=[l(l+1)+l(l+1)p(p+1)]C0,0l,l,p/2p+1,ασσ=iσ[(p2(ll)2)((l+l+1)2p2)]1/2C0,0l,l,p1/2p1,βll=π2l+1l(l+1)2l+1l(l+1).
Wa=A12Re(Ea×Ha*)·dA,
Wa=Wa+=A+S0+Ssca+Sext·dA.
S0=12Re(E0×H0*),Ssca=12Re(Es×Hs*),Sext=12Re(E0×Hs*+Es×H0*).
Csca=ASsca·dAI0,Cext=ASext·dAI0withI0=12k0ωμ=12Z0.
Csca=ARe(Es×Z0Hs*)·dA.
Csca=σ,l,mσ,l,mRe(Eσ,l,ms·Hσ,l,ms*Zσ,l,m(3)|×|Zσ,l,m(3)).
Csca=4πk02σ,l,mσEσ,l,ms·Hσ,l,ms*.
Csca=4πk02σ,l,mEσ,l,ms·Eσ,l,ms*.
Cext=ARe(E0×Z0Hs*+Es×Z0H0*)·dA.
Cext=σ,l,mRe(Eσ,l,m0·Hσ,l,ms*Zσ,l,m(3)|×|Zσ,l,m(1)+Eσ,l,ms·Hσ,l,m0*Zσ,l,m(1)|×|Zσ,l,m(3)).
Cext=4πk02σ,l,mRe(Eσ,l,m0·Eσ,l,ms*).
Cextall=j=1nbspheik0·(rjr0)Cextj.
|Z1,l,m(r)=zl(ρ)|Xlm(r^),|Z1,l,m(r)=4πl(l+1)zl(ρ)ρYlm(r^)eri(ρzl(ρ))ρer×|Xlm(r^).
zl(3)(ρ)=il·hl(1)(ρ)eiρiρ,(ρzl(3)(ρ))eiρ.
|Z1,l,m(3)(r)eiρiρ|Xlm(r^),|Z1,l,m(3)(r)eiρiρer×|Xlm(r^).
Es(r)=eik0rik0rl,mE1,l,ms|Xl,m(r^)+E1,l,mser×|Xl,m(r^).
[EsEs]=eik0(rz)ik0r[S2S3S4S1][E0E0].
E0TM=E0e0+E0e0=eik0z[cos(φφ0)e0+sin(φφ0)e0].
[EsTMEsTM]=eik0rik0rei(φφ0)[S2S4].
EsTM=eik0rik0rσ,l,m|σEσ,l,msXσ,l,meθEσ,l,msXσ,l,meφ.
{S2=ei(φφ0)σ,l,mσEσ,l,msXσ,l,m(r^)S4=ei(φφ0)σ,l,mEσ,l,msXσ,l,m(r^).
Sxall=j=1nbspheik0·(rjr0)Sxj.
Es=eik0rik0rE0VwithV=ei(φφ0)(S2eθS4eφ).
Csca=1k02Ω|V|2·dΩ.
V=σ,l,mEσ,l,ms(σXσ,l,m(r^)eθXσ,l,m(r^)eφ).
V=l,mE1,l,ms|Xl,m(r^)+E1,l,mser×|Xl,m(r^).
Csca=1k02l,m|E1,l,ms|2Xl,m(r^)|Xl,m(r^)+|E1,l,ms|2er×Xl,m(r^)|er×Xl,m(r^).
Cext=4πk02Re(p·V(r^=k^0)).
Cext=|4πk02σ,l,mσRe(Eσ,l,msXσ,l,m(k^0))TM-polarization4πk02σ,l,mRe(Eσ,l,msXσ,l,m(k^0))TE-polarization.
RCS=limr4πr2|Es|2=4πk02(i1+i2).
ψ1|ψ2=ψ1*(x,y,z)·ψ2(x,y,z)dxdydz,ψ1|A|ψ2=ψ1*(x,y,z)·A·ψ2(x,y,z)dxdydz.
Xlm|Xlm=4πXlm*(r^)·Xlm(r^)dΩ=δllδmm4π,WithdΩ=dAr2=sin(θ)dθdφ,er×Xlm|er×Xlm=δllδmm4π,Xlm|er×Xlm=0,Xlm|er=0,Xlm|×|er=4π(Xlm*(r^)×er)·erdΩ=0,Xlm|×|er×Xlm=4πXlm*(r^)·Xlm(r^)dΩ=δllδmm4π.
Zσlm|×|Zσlm=A(Zσlm*(r)×Zσlm(r))·dA=0,withdA=dAer=r2sin(θ)dθdφer.Ifρ=k0r,Z1lm(J)|×|Z1lm(K)=iρ*zl*(J)(ρ)(ρ·zl(K)(ρ))k02δllδmm4π,Z1lm(J)|×|Z1lm(K)=iρzl(K)(ρ)(ρ·zl(J)(ρ))*k02δllδmm4π.
hl(ρ)=jl(ρ)+iyl(ρ),ψl(ρ)=ρ·jl(ρ),χl(ρ)=ρ·yl(ρ),ξl(ρ)=ρ·hl(ρ)=ψl(ρ)+iχl(ρ).
ψl·χlχl·ψl=1.
Re(iξl*ξl)=Re(iξlξl*)=1,Re(i(ψlξl*+ξlψl*))=Re(i(ψlξl*+ξlψl*))=1,Re(i(ψlξl*ξl*ψl))=Re(i(ψlξl*ψl*ξl))=1.

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