Abstract

We examine the reflection symmetries of the electromagnetic wave inside of a uniform spherical particle and identify the consequences of the symmetries for the Stokes parameters describing the polarization state of the far-field scattered wave. The connection between the two waves is described from a microphysical perspective that illustrates the wavelet-superposition origin of the scattered wave. In contrast to more conventional approaches, this microphysical perspective yields new insight into the physical character of the scattering of a plane wave by a sphere. The results of simulations are presented, which graphically demonstrate the relation between the symmetries present in the internal wave and the polarization state of the scattered wave.

© 2008 Optical Society of America

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  1. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.
  2. J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
    [CrossRef]
  3. C. Li, G. W. Kattawar, and P. W. Zhai, "Electric and mangnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave," Opt. Express 13, 4554-4559 (2005).
    [CrossRef] [PubMed]
  4. N. Velesco, T. Kaiser, and G. Schweiger, "Computation of the internal field of a large spherical particle by use of the geometrical-optics approximation," Appl. Opt. 36, 8724-8728 (1997).
    [CrossRef]
  5. L. Kai and A. D'Alessio, "Internal-field characteristics of spherical particles," Part. Part. Syst. Charact. 12, 237-241 (1995).
    [CrossRef]
  6. E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Internal electric energy in a spherical particle illuminated with a plane wave or off-axis Gaussian beam," Appl. Opt. 33, 524-532 (1994).
    [CrossRef] [PubMed]
  7. G. Chen, D. Q. Chowdhury, and R. K. Chang, "Laser-induced radiation leakage from microdroplets," J. Opt. Soc. Am. B 10, 620-632 (1993).
    [CrossRef]
  8. H. M. Lai, P. T. Leung, K. L. Poon, and K. Young, "Characterization of the internal energy density in Mie scattering," J. Opt. Soc. Am. A 8, 1553-1558 (1991).
    [CrossRef]
  9. C. C. Dobson and J. W. Lewis, "Survey of the Mie problem source function," J. Opt. Soc. Am. A 6, 463-466 (1989).
    [CrossRef]
  10. P. Chýlek, J. D. Pendleton, and R. G. Pinnick, "Internal and near-surface scattered fields of a spherical particle at resonant conditions," Appl. Opt. 24, 3940-3942 (1985).
    [CrossRef] [PubMed]
  11. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  12. P. W. Dusel, M. Kerker, and D. D. Cooke, "Distribution of absorption centers within irradiated spheres," J. Opt. Soc. Am. 69, 55-59 (1979).
    [CrossRef]
  13. C. E. Baum and N. H. Kritikos, Electromagnetic Symmetry (Taylor & Francis, 1995).
  14. S. H. Yueh, R. Kwok, and S. V. Nghiem, "Polarimetric scattering and emission properties of targets with reflection symmetry," Radio Sci. 29, 1409-1420 (1994).
    [CrossRef]
  15. C. H. Hu, G. W. Kattawar, M. E. Parkin, and P. Herb, "Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a nonspherical dielectric scatterer," Appl. Opt. 26, 4159-4173 (1987).
    [CrossRef] [PubMed]
  16. K. F. Ren, G. Gréhan, and G. Gouesbet, "Symmetry relations in generalized Lorentz-Mie theory," J. Opt. Soc. Am. A 11, 1812-1817 (1994).
    [CrossRef]
  17. F. M. Schultz, K. Stamnes, and J. J. Stamnes, "Point-group symmetries in electromagnetic scattering," J. Opt. Soc. Am. A 16, 853-865 (1999).
    [CrossRef]
  18. J. W. Hovenier, "Symmetry relations for forward and backward scattering by randomly oriented particles," J. Quant. Spectrosc. Radiat. Transf. 60, 483-492 (1998).
    [CrossRef]
  19. S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
    [CrossRef]
  20. J. W. Hovenier, "Principles of symmetry for polarization studies of planets," Astron. Astrophys. 7, 86-90 (1970).
  21. J. W. Hovenier, "Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-489 (1969).
    [CrossRef]
  22. P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic symmetry," Phys. Rev. D 3, 825-839 (1971).
    [CrossRef]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  24. M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transf. 100, 268-276 (2006).
    [CrossRef]
  25. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  26. B. Chu, Laser Light Scattering (Academic, 1991).
  27. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).
  28. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  29. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  30. J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London, Ser. A 389, 279-290 (1983).
    [CrossRef]
  31. J. F. Nye, "Line singularities in wave fields," Philos. Trans. R. Soc. London, Ser. A 355, 2065-2069 (1997).
    [CrossRef]

2007

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

2006

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transf. 100, 268-276 (2006).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

2005

2002

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.

1999

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

F. M. Schultz, K. Stamnes, and J. J. Stamnes, "Point-group symmetries in electromagnetic scattering," J. Opt. Soc. Am. A 16, 853-865 (1999).
[CrossRef]

1998

J. W. Hovenier, "Symmetry relations for forward and backward scattering by randomly oriented particles," J. Quant. Spectrosc. Radiat. Transf. 60, 483-492 (1998).
[CrossRef]

1997

1995

C. E. Baum and N. H. Kritikos, Electromagnetic Symmetry (Taylor & Francis, 1995).

L. Kai and A. D'Alessio, "Internal-field characteristics of spherical particles," Part. Part. Syst. Charact. 12, 237-241 (1995).
[CrossRef]

1994

1993

1992

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

1991

1990

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

1989

1987

1985

1983

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

J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London, Ser. A 389, 279-290 (1983).
[CrossRef]

1981

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

1979

1971

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic symmetry," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

1970

J. W. Hovenier, "Principles of symmetry for polarization studies of planets," Astron. Astrophys. 7, 86-90 (1970).

1969

J. W. Hovenier, "Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-489 (1969).
[CrossRef]

Barber, P. W.

Baum, C. E.

C. E. Baum and N. H. Kritikos, Electromagnetic Symmetry (Taylor & Francis, 1995).

Bohren, C. F.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Chang, R. K.

Chen, G.

Chowdhury, D. Q.

Chu, B.

B. Chu, Laser Light Scattering (Academic, 1991).

Chýlek, P.

Cooke, D. D.

D'Alessio, A.

L. Kai and A. D'Alessio, "Internal-field characteristics of spherical particles," Part. Part. Syst. Charact. 12, 237-241 (1995).
[CrossRef]

Dobson, C. C.

Dusel, P. W.

Gouesbet, G.

Gréhan, G.

Herb, P.

Hill, S. C.

Hovenier, J. W.

J. W. Hovenier, "Symmetry relations for forward and backward scattering by randomly oriented particles," J. Quant. Spectrosc. Radiat. Transf. 60, 483-492 (1998).
[CrossRef]

J. W. Hovenier, "Principles of symmetry for polarization studies of planets," Astron. Astrophys. 7, 86-90 (1970).

J. W. Hovenier, "Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-489 (1969).
[CrossRef]

Hu, C. H.

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, Classical Electrodynamics (Wiley, 1999).

Kai, L.

L. Kai and A. D'Alessio, "Internal-field characteristics of spherical particles," Part. Part. Syst. Charact. 12, 237-241 (1995).
[CrossRef]

Kaiser, T.

Kattawar, G. W.

Kerker, M.

Khaled, E. E. M.

Kritikos, N. H.

C. E. Baum and N. H. Kritikos, Electromagnetic Symmetry (Taylor & Francis, 1995).

Kwok, R.

S. H. Yueh, R. Kwok, and S. V. Nghiem, "Polarimetric scattering and emission properties of targets with reflection symmetry," Radio Sci. 29, 1409-1420 (1994).
[CrossRef]

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.

Lai, H. M.

Leung, P. T.

Lewis, J. W.

Li, C.

Li, F. K.

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transf. 100, 268-276 (2006).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.

Muinonen, K.

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

Nghiem, S. V.

S. H. Yueh, R. Kwok, and S. V. Nghiem, "Polarimetric scattering and emission properties of targets with reflection symmetry," Radio Sci. 29, 1409-1420 (1994).
[CrossRef]

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

Nye, J. F.

J. F. Nye, "Line singularities in wave fields," Philos. Trans. R. Soc. London, Ser. A 355, 2065-2069 (1997).
[CrossRef]

J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London, Ser. A 389, 279-290 (1983).
[CrossRef]

Parkin, M. E.

Pendleton, J. D.

Pinnick, R. G.

Poon, K. L.

Ren, K. F.

Schultz, F. M.

Schweiger, G.

Stamnes, J. J.

Stamnes, K.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.

Tyynelä, J.

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Velesco, N.

Videen, G.

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

Waterman, P. C.

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic symmetry," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Young, K.

Yueh, S. H.

S. H. Yueh, R. Kwok, and S. V. Nghiem, "Polarimetric scattering and emission properties of targets with reflection symmetry," Radio Sci. 29, 1409-1420 (1994).
[CrossRef]

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

Zhai, P. W.

Zubko, E.

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

J. W. Hovenier, "Principles of symmetry for polarization studies of planets," Astron. Astrophys. 7, 86-90 (1970).

J. Atmos. Sci.

J. W. Hovenier, "Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles," J. Atmos. Sci. 26, 488-489 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transf.

J. Tyynelä, E. Zubko, G. Videen, and K. Muinonen, "Inter-relating angular scattering characteristics to internal electric fields for wavelength-sized spherical particles," J. Quant. Spectrosc. Radiat. Transf. 106, 520-534 (2007).
[CrossRef]

J. W. Hovenier, "Symmetry relations for forward and backward scattering by randomly oriented particles," J. Quant. Spectrosc. Radiat. Transf. 60, 483-492 (1998).
[CrossRef]

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transf. 100, 268-276 (2006).
[CrossRef]

Opt. Express

Part. Part. Syst. Charact.

L. Kai and A. D'Alessio, "Internal-field characteristics of spherical particles," Part. Part. Syst. Charact. 12, 237-241 (1995).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

J. F. Nye, "Line singularities in wave fields," Philos. Trans. R. Soc. London, Ser. A 355, 2065-2069 (1997).
[CrossRef]

Phys. Rev. D

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic symmetry," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Proc. R. Soc. London, Ser. A

J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London, Ser. A 389, 279-290 (1983).
[CrossRef]

Radio Sci.

S. V. Nghiem, S. H. Yueh, R. Kwok, and F. K. Li, "Symmetry properties in polarimetric remote sensing," Radio Sci. 27, 693-711 (1992).
[CrossRef]

S. H. Yueh, R. Kwok, and S. V. Nghiem, "Polarimetric scattering and emission properties of targets with reflection symmetry," Radio Sci. 29, 1409-1420 (1994).
[CrossRef]

Other

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), freely available in the pdf format at http://www.giss.nasa.gov/~crmim/books.html.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

C. E. Baum and N. H. Kritikos, Electromagnetic Symmetry (Taylor & Francis, 1995).

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

B. Chu, Laser Light Scattering (Academic, 1991).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

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Figures (12)

Fig. 1
Fig. 1

Scattering arrangement. Shown at the tip of r is the scattered electric field E sca and the polarization ellipse discussed in Section 4.

Fig. 2
Fig. 2

Sketch showing the Π x , Π y , and Π z planes and C h and C v contours. The planes pass through the origin of the coordinate system and the center of the sphere.

Fig. 3
Fig. 3

Electric field inside and surrounding a sphere with k R = 12 and m = 1.33 + 0 i for points in the Π x plane, recall Fig. 2. The sphere’s surface is outlined. The normalized field magnitude is shown by the color shades in log scale. Contour lines are also shown to give relief to the plot when not viewed in color. No field vectors are shown since the field is normal to this plane, see Eqs. (15, 16).

Fig. 4
Fig. 4

Same as Fig. 3 except for the Π y plane.

Fig. 5
Fig. 5

Same as Figs. 3, 4 except for the Π z plane.

Fig. 6
Fig. 6

Sketch of the hemispheres used to derive Eqs. (27, 32). The sphere is separated to delineate the hemispheres. The internal locations r j and r k are mirror points about the Π x plane (top sketch) and the Π y plane (bottom sketch).

Fig. 7
Fig. 7

Sketch of the internal points and quadrants of the sphere used to derive Eqs. (33, 34, 35, 36). The incident wave travels through the page toward the reader, and the sketch shows the Π z slice through the sphere depicted in Fig. 2. The locations r j ( II ) , r j ( III ) , r j ( IV ) and r k ( II ) , r k ( III ) , r k ( IV ) are generated from the locations r j ( I ) and r k ( I ) in V I through successive reflections about the Π x and Π y planes.

Fig. 8
Fig. 8

The polarization state of the scattered wave for a sphere with k R = 4 and m = 1.25 + 0 i . The plot shows the vibration ellipses at various points on S l , recall Fig. 1. The bold red ellipses correspond to left-handed rotation of the field, whereas thin blue ellipses correspond to right-handed rotation. The ellipticity is shown in gray shades on S l . Darker (lighter) shades indicate more elliptical (linear) polarization.

Fig. 9
Fig. 9

Intensity of the scattered wave on S l . The intensity is normalized to the forward direction ( θ = 0 ) , and gray shades are assigned in log scale as indicated. The sphere is the same as in Fig. 8.

Fig. 10
Fig. 10

Ellipticity of the scattered wave on S l . The gray scale is the same as in Fig. 8, and the sphere is the same as in Figs. 8, 9. The dashed lines show the intersection of the Π x and Π y planes with S l . These lines also indicate the location of four of the seven L -lines, that separate the the angular regions of opposing rotation in Fig. 11. One of the four points of circular polarization, or C -points, is indicated by the gray (red online) dot in the upper left of the plot. See the text for a discussion of L -lines and C -points.

Fig. 11
Fig. 11

Rotation of the scattered wave on S l . Black indicates left-handed rotation and white indicates right-handed rotation. The sphere is the same as in Figs. 8, 9, 10. Notice the correlation of the angular structure with that of Fig. 10.

Fig. 12
Fig. 12

This diagram illustrates the wavelet-component cancellation that is responsible for the linear polarization state of the scattered wave on the C h contour. A similar picture applies for the C v contour. Internal mirror points r j and r k about the Π x plane are shown along with an example of the internal electric field vectors E int ( r j ) and E int ( r k ) that satisfy the symmetry conditions of Eqs. (12, 13, 14, 15, 16). Each of these internal wavelets contribute E j and E k to the scattered field, where the cancellation of their θ-components is evident.

Equations (52)

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E inc ( r ) = E o inc exp ( i k r r ̂ n ̂ inc ) x ̂ ,
B inc ( r ) = k ω n ̂ inc × E inc ( r ) ,
E int ( r ) = n = 1 E n [ c n M n ( r ) i d n N n ( r ) ] , r V int ,
B int ( r ) = 1 ω × E int ( r ) , r V int .
R n ( r ) = E n [ c n M n ( r ) i d n N n ( r ) ] r ̂ ,
Θ n ( r ) = E n [ c n M n ( r ) i d n N n ( r ) ] θ ̂ ,
Φ n ( r ) = E n [ c n M n ( r ) i d n N n ( r ) ] ϕ ̂ ,
X n ( r ) = R n ( r ) sin θ cos ϕ + Θ n ( r ) cos θ cos ϕ Φ n ( r ) sin ϕ ,
Y n ( r ) = R n ( r ) sin θ sin ϕ + Θ n ( r ) cos θ sin ϕ + Φ n ( r ) cos ϕ ,
Z n ( r ) = R n ( r ) cos θ Θ n ( r ) sin θ .
E int ( r ) = n = 1 [ X n ( r ) x ̂ + Y n ( r ) y ̂ + Z n ( r ) z ̂ ] .
X n ( x , y , z ) = X n ( x , y , z ) = X n ( x , y , z ) ,
Y n ( x , y , z ) = Y n ( x , y , z ) = Y n ( x , y , z ) ,
Z n ( x , y , z ) = Z n ( x , y , z ) = Z n ( x , y , z ) .
Y n ( r ) = 0 , for x = 0 or y = 0 ,
Z n ( r ) = 0 , for x = 0 .
E sca ( r ) = exp ( i k r ) r k 2 4 π ( m 2 1 ) ( I r ̂ r ̂ ) V int E int ( r ) exp ( i k r r ̂ ) d r ,
B sca ( r ) = k ω r ̂ × E sca ( r ) .
E sca ( r ) = exp ( i k r ) r k 2 4 π ( m 2 1 ) lim Δ V 0 i ( I r ̂ r ̂ ) E int ( r i ) exp ( i k r i r ̂ ) Δ V ,
z i θ ( r ̂ ) = c o { [ E x int ( r i ) cos ϕ + E y int ( r i ) sin ϕ ] cos θ E z int ( r i ) sin θ } exp ( i k r i r ̂ ) ,
z i ϕ ( r ̂ ) = c o [ E y int ( r i ) cos ϕ E x int ( r i ) sin ϕ ] exp ( i k r i r ̂ ) ,
E sca ( r ) = exp ( i k r ) r i [ z i θ ( r ̂ ) θ ̂ + z i ϕ ( r ̂ ) ϕ ̂ ] ,
z i θ ( r ̂ ) = c o [ E z int ( r i ) sin θ E y int ( r i ) cos θ ] exp ( i k r i r ̂ ) ,
z i ϕ ( r ̂ ) = c o E x int ( r i ) exp ( i k r i r ̂ ) ,
i z i θ ( r ̂ ) = j x z j θ ( r ̂ ) + k x < z k θ ( r ̂ ) ,
z j θ ( r ̂ ) = z k θ ( r ̂ ) .
E sca ( R l r ̂ ) = c o exp ( i k R l ) R l i E x int ( r i ) exp ( i k r i r ̂ ) ϕ ̂ ,
r C h ,
z i θ ( r ̂ ) = c o [ E z int ( r i ) sin θ E x int ( r i ) cos θ ] exp ( i k r i r ̂ ) ,
z i ϕ ( r ̂ ) = ± c o E y int ( r i ) exp ( i k r i r ̂ ) ,
i z i ϕ ( r ̂ ) = j y z j ϕ ( r ̂ ) + k y < z k ϕ ( r ̂ ) ,
z j ϕ ( r ̂ ) = z k ϕ ( r ̂ ) ,
E sca ( R l r ̂ ) = c o exp ( i k R l ) R l i [ E z int ( r i ) sin θ E x int ( r i ) cos θ ] exp ( i k r i r ̂ ) θ ̂ , r C v ,
I ( r ̂ ) = 1 2 ϵ o μ o j , k ( I ) α , β { z j θ ( α ) ( r ̂ ) [ z k θ ( β ) ( r ̂ ) ] * + z j ϕ ( α ) ( r ̂ ) [ z k ϕ ( β ) ( r ̂ ) ] * } ,
Q ( r ̂ ) = 1 2 ϵ o μ o j , k ( I ) α , β { z j θ ( α ) ( r ̂ ) [ z k θ ( β ) ( r ̂ ) ] * z j ϕ ( α ) ( r ̂ ) [ z k ϕ ( β ) ( r ̂ ) ] * } ,
U ( r ̂ ) = 1 2 ϵ o μ o j , k ( I ) α , β { z j θ ( α ) ( r ̂ ) [ z k ϕ ( β ) ( r ̂ ) ] * + z j ϕ ( α ) ( r ̂ ) [ z k θ ( β ) ( r ̂ ) ] * } ,
V ( r ̂ ) = i 2 ϵ o μ o j , k ( I ) α , β { z j ϕ ( α ) ( r ̂ ) [ z k θ ( β ) ( r ̂ ) ] * z j θ ( α ) ( r ̂ ) [ z k ϕ ( β ) ( r ̂ ) ] * } ,
I ( θ , ϕ ) = I ( θ , π ϕ ) = I ( θ , 2 π ϕ ) ,
Q ( θ , ϕ ) = Q ( θ , π ϕ ) = Q ( θ , 2 π ϕ ) ,
U ( θ , ϕ ) = U ( θ , π ϕ ) = U ( θ , 2 π ϕ ) ,
V ( θ , ϕ ) = V ( θ , π ϕ ) = V ( θ , 2 π ϕ ) .
U ( θ , 0 ) = U ( θ , π 2 ) = U ( θ , π ) = U ( θ , 3 π 2 ) = 0 ,
V ( θ , 0 ) = V ( θ , π 2 ) = V ( θ , π ) = V ( θ , 3 π 2 ) = 0 .
V ( r ̂ ) = 0 , for E int ( r ) = E x int x ̂ ,
z j θ ( I ) ( r ̂ ) = c o [ E x ( r j ) cos θ cos ϕ + E y ( r j ) cos θ sin ϕ E z ( r j ) sin θ ] × exp [ i k ( r j x sin θ cos ϕ + r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j θ ( II ) ( r ̂ ) = c o [ E x ( r j ) cos θ cos ϕ E y ( r j ) cos θ sin ϕ E z ( r j ) sin θ ] × exp [ i k ( r j x sin θ cos ϕ r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j θ ( III ) ( r ̂ ) = c o [ E x ( r j ) cos θ cos ϕ + E y ( r j ) cos θ sin ϕ + E z ( r j ) sin θ ] × exp [ i k ( r j x sin θ cos ϕ r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j θ ( IV ) ( r ̂ ) = c o [ E x ( r j ) cos θ cos ϕ E y ( r j ) cos θ sin ϕ + E z ( r j ) sin θ ] × exp [ i k ( r j x sin θ cos ϕ + r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j ϕ ( I ) ( r ̂ ) = c o [ E y ( r j ) cos ϕ E x ( r j ) sin ϕ ] exp [ i k ( r j x sin θ cos ϕ + r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j ϕ ( II ) ( r ̂ ) = c o [ E y ( r j ) cos ϕ E x ( r j ) sin ϕ ] × exp [ i k ( r j x sin θ cos ϕ r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j ϕ ( III ) ( r ̂ ) = c o [ E y ( r j ) cos ϕ E x ( r j ) sin ϕ ] × exp [ i k ( r j x sin θ cos ϕ r j y sin θ sin ϕ + r j z cos θ ) ] ,
z j ϕ ( IV ) ( r ̂ ) = c o [ E y ( r j ) cos ϕ E x ( r j ) sin ϕ ] × exp [ i k ( r j x sin θ cos ϕ + r j y sin θ sin ϕ + r j z cos θ ) ] ,

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