Abstract

Based on the scattering theory of a chiral sphere, rainbow phenomenon of a chiral sphere is numerically analyzed in this paper. For chiral spheres illuminated by a linearly polarized wave, there are three first-order rainbows, with whose rainbow angles varying with the chirality parameter. The spectrum of each rainbow structure is presented and the ripple frequencies are found associated with the size and refractive indices of the chiral sphere. Only two rainbow structures remain when the chiral sphere is illuminated by a circularly polarized plane wave. Finally, the rainbows of chiral spheres with slight chirality parameters are found appearing alternately in E-plane and H-plane with the variation of the chirality.

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  1. V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt.33(21), 4677–4690 (1994).
    [CrossRef] [PubMed]
  6. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.36(21), 5188–5198 (1997).
    [CrossRef] [PubMed]
  7. G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt.33(21), 4691–4696 (1994).
    [CrossRef] [PubMed]
  8. J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt.35(13), 2259–2266 (1996).
    [CrossRef] [PubMed]
  9. J. P. A. J. van Beeck, Rainbow Phenomena: Development of a Laser-Based, Non-Intrusive Technique for Measuring Droplet Size, Temperature and Velocity (Technische Universiteit Eindhoven, 1997).
  10. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a Liquid Jet by Rainbow Sizing,” Appl. Opt.37(36), 8498–8503 (1998).
    [CrossRef] [PubMed]
  11. J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  13. J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
    [CrossRef]
  24. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt.51(27), 6661–6668 (2012).
    [CrossRef] [PubMed]
  25. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express21(7), 8677–8688 (2013).
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  26. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
    [CrossRef]
  27. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991).
    [CrossRef]
  28. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
    [CrossRef]
  29. Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
    [CrossRef]
  30. X. e. Han, “Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet,” thesis (Rouen University, 2000).

2013 (1)

2012 (1)

2011 (1)

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
[CrossRef]

2009 (1)

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
[CrossRef]

2004 (1)

2001 (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001).
[CrossRef]

1998 (1)

1997 (2)

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

1996 (2)

J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt.35(13), 2259–2266 (1996).
[CrossRef] [PubMed]

L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A5(4), 417–443 (1996).
[CrossRef]

1995 (1)

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

1994 (2)

1993 (1)

1992 (1)

1991 (2)

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt.30(1), 106–117 (1991).
[CrossRef] [PubMed]

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991).
[CrossRef]

1988 (2)

1986 (1)

1979 (1)

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).

1974 (1)

F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Bai, L.

Bassiri, S.

Bohren, F.

F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
[CrossRef]

Corbin, F.

Engheta, N.

Geng, Y. L.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
[CrossRef]

Gong, L.

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

Han, X.

Han, Y.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
[CrossRef]

Jaggard, D.

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).

Jaggard, D. L.

Jamison, J. M.

John, L.

L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A5(4), 417–443 (1996).
[CrossRef]

Kaduchak, G.

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Lakhtakia, A.

Le-Wei, L.

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

Li, Z.-J.

Lin, C. Y.

Lock, J. A.

Marston, P. L.

Mickelson, A.

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).

Mook-Seng, L.

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

Pang-Shyan, K.

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

Papas, C.

S. Bassiri, C. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A5(9), 1450–1459 (1988).
[CrossRef]

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).

Qiu, C. W.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
[CrossRef]

Qu, T.

Ren, K. F.

Riethmuller, M. L.

Saengkaew, S.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
[CrossRef]

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

Shang, Q.-C.

Silverman, M.

Simpson, H. J.

Sun, X.

Tat-Soon, Y.

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

van Beeck, J. P.

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt.35(13), 2259–2266 (1996).
[CrossRef] [PubMed]

van de Hulst, H. C.

Varadan, V. K.

Varadan, V. V.

Vetrano, M. R.

Wang, J.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
[CrossRef]

Wang, R. T.

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991).
[CrossRef]

Wu, Z.

Wu, Z. S.

Wu, Z.-S.

Yuan, N.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
[CrossRef]

Zimmer, L.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001).
[CrossRef]

Appl. Opt. (8)

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt.30(1), 106–117 (1991).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt.33(21), 4677–4690 (1994).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt.36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt.33(21), 4691–4696 (1994).
[CrossRef] [PubMed]

J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt.35(13), 2259–2266 (1996).
[CrossRef] [PubMed]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a Liquid Jet by Rainbow Sizing,” Appl. Opt.37(36), 8498–8503 (1998).
[CrossRef] [PubMed]

M. R. Vetrano, J. P. van Beeck, and M. L. Riethmuller, “Global Rainbow Thermometry: Improvements in the Data Inversion Algorithm and Validation Technique in Liquid-Liquid Suspension,” Appl. Opt.43(18), 3600–3607 (2004).
[CrossRef] [PubMed]

Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt.51(27), 6661–6668 (2012).
[CrossRef] [PubMed]

Appl. Phys., A Mater. Sci. Process. (1)

D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci. Process.18, 211–216 (1979).

Chem. Phys. Lett. (1)

F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
[CrossRef]

Exp. Fluids (1)

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids51(1), 149–159 (2011).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions in radially multilayered chiral media,” IEEE Trans. Antenn. Propag.43(3), 232–238 (1995).
[CrossRef]

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag.57(2), 572–576 (2009).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Particle & Particle Systems Characterization (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization18(4), 196–204 (2001).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

Pure and Applied Optics: Journal of the European Optical Society Part A (1)

L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical Society Part A5(4), 417–443 (1996).
[CrossRef]

Radio Sci. (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991).
[CrossRef]

Other (6)

X. e. Han, “Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet,” thesis (Rouen University, 2000).

L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer, 1989), Vol. 335.

V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

J. P. A. J. van Beeck, Rainbow Phenomena: Development of a Laser-Based, Non-Intrusive Technique for Measuring Droplet Size, Temperature and Velocity (Technische Universiteit Eindhoven, 1997).

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Figures (6)

Fig. 1
Fig. 1

Rainbow phenomenon of a chiral sphere. (a) κ = 0.10; (b) κ = 0.01.

Fig. 2
Fig. 2

Effects of chirality on (a) Peak angle; (b) Peak intensity.

Fig. 3
Fig. 3

Rainbow structures and spectrums.

Fig. 4
Fig. 4

Rainbow structures for circularly polarized wave incidences. (a) κ = 0.05; (b) κ = −0.05.

Fig. 5
Fig. 5

Rainbows for sphere with slight chirality. (a) κ = 5 × 10−5; (b) κ = −1.5 × 10−4.

Fig. 6
Fig. 6

Intensity at peak angle versus chirality.

Equations (27)

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

E ip = E 0 n=1 m=n n [ a mn ip M mn (1) (r,k)+ b mn ip N mn (1) (r,k) ] ,
H ip = k E 0 iωμ n=1 m=n n [ a mn ip N mn (1) (r,k)+ b mn ip M mn (1) (r,k) ] ,
E s = E 0 n=1 m=n n [ A mn s M mn (3) (r,k)+ B mn s N mn (3) (r,k) ] ,
H s = k E 0 iωμ n=1 m=n n [ A mn s N mn (3) (r,k)+ B mn s M mn (3) (r,k) ] ,
E int = n=1 m=n n [ A mn M mn (1) (r, k 1 )+ A mn N mn (1) (r, k 1 ) + B mn M mn (1) (r, k 2 ) B mn N mn (1) (r, k 2 ) ],
H int =i ε r ε 0 μ r μ 0 n=1 m=n n [ A mn N mn (1) (r, k 1 )+ A mn M mn (1) (r, k 1 )+ B mn N mn (1) (r, k 2 ) B mn M mn (1) (r, k 2 ) ] ,
A mn s = A n sa a mn ip + A n sb b mn ip , B mn s = B n sa a mn ip + B n sb b mn ip ,
A n sa = ψ n ( x 0 ) ξ n ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 0 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + D n (1) ( x 2 ) η r D n (1) ( x 0 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) η r D n (3) ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + η r D n (3) ( x 0 ) D n (1) ( x 2 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) ,
A n sb = ψ n ( x 0 ) ξ n ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 0 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) η r D n (1) ( x 2 ) D n (1) ( x 0 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) η r D n (3) ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + η r D n (3) ( x 0 ) D n (1) ( x 2 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) ,
B n sa = A n sb ,
B n sb = ψ n ( x 0 ) ξ n ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 0 ) D n (1) ( x 1 ) η r D n (3) ( x 0 ) + η r D n (1) ( x 2 ) D n (1) ( x 0 ) D n (1) ( x 2 ) η r D n (3) ( x 0 ) D n (3) ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 1 ) η r D n (3) ( x 0 ) + D n (3) ( x 0 ) η r D n (1) ( x 2 ) D n (1) ( x 2 ) η r D n (3) ( x 0 ) .
E θ s = E 0 exp(ikr) kr n=1 m=n n (i) n [ A mn s m π mn + B mn s τ mn ] e imϕ ,
E ϕ s =i E 0 exp(ikr) kr n=1 m=n n (i) n [ A mn s τ mn + B mn s m π mn ] e imϕ ,
π mn = P n m (cosθ) sinθ , τ mn = d P n m (cosθ) dθ .
a mn ix = i n+1 2n+1 2n(n+1) δ m,1 + i n+1 2n+1 2 δ m,1 , b mn ix = i n+1 2n+1 2n(n+1) δ m,1 i n+1 2n+1 2 δ m,1 ,
E θ s =i E 0 exp(ikr) kr n=1 2n+1 n(n+1) [ cosϕ( A n sa π n + B n sb τ n )+isinϕ( A n sb π n + B n sa τ n ) ] ,
E ϕ s = E 0 exp(ikr) kr n=1 2n+1 n(n+1) [ cosϕ( B n sa π n + A n sb τ n )+isinϕ( B n sb π n + A n sa τ n ) ] ,
π n = π 1n = P n 1 (cosθ) sinθ , τ n = τ 1n = d P n 1 (cosθ) dθ .
× n=1 m=n n [ a mn ix M mn (1) (r,k)+ b mn ix N mn (1) (r,k) ] =×( e ikz x ^ ).
a mn iR = a mn ix +i a mn iy = a mn ix + b mn ix , b mn iR = b mn ix +i b mn iy = b mn ix + a mn ix .
a mn iR = b mn iR = i n+1 2n+1 n(n+1) δ m,1
E θ s = E 0 exp(ikr) kr n=1 [ i e iϕ 2n+1 n(n+1) ( A n sa π n + A n sb π n + B n sa τ n + B n sb τ n ) ] ,
E ϕ s = E 0 exp(ikr) kr n=1 [ e iϕ 2n+1 n(n+1) ( A n sa τ n + A n sb τ n + B n sa π n + B n sb π n ) ] .
a mn iL = b mn iL = i n+1 (2n+1) δ m,1 .
E θ s = E 0 exp(ikr) kr n=1 [ i 2n+1 n(n+1) e iϕ ( A n sa π n A n sb π n B n sa τ n + B n sb τ n ) ] ,
E ϕ s = E 0 exp(ikr) kr n=1 [ 2n+1 n(n+1) e iϕ ( A n sa τ n A n sb τ n B n sa π n + B n sb π n ) ] .
I s = lim r k 2 r 2 ( | E θ s | 2 + | E ϕ s | 2 )/ | E 0 | 2 .

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