Abstract

In this paper, an introduction to electromagnetic scattering is presented. We introduce the basic concepts needed to face a scattering problem, including the scattering, absorption, and extinction cross sections. We define the vector harmonics and we present some of their properties. Finally, we tackle the two canonical problems of the scattering by an infinitely long circular cylinder, and by a sphere, showing that the introduction of the vector wave function makes the imposition and solution of the boundary conditions particularly simple.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  12. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).
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    [Crossref]
  15. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
    [Crossref]
  16. F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
    [Crossref]
  17. F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015).
    [Crossref]
  18. M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
    [Crossref]
  19. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
    [Crossref]
  20. F. Frezza and F. Mangini, “Electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a multilayered sphere,” J. Electromagn. Waves Appl. 30, 492–504 (2016).
    [Crossref]
  21. F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
    [Crossref]
  22. C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
    [Crossref]
  23. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
  24. S.-C. Lee and A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
    [Crossref]
  25. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
    [Crossref]
  26. F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
    [Crossref]
  27. S. N. Samaddar, “Scattering of plane waves from an infinitely long cylinder of anisotropic materials at oblique incidence with an application to an electronic scanning antenna,” Appl. Sci. Res. B 10, 385–411 (1962).
  28. X. B. Wu and W. Ren, “Wave-function solution of plane-wave scattering by an anisotropic circular cylinder,” Microwave Opt. Technol. Lett. 8, 39–42 (1995).
    [Crossref]
  29. Y.-L. Geng, X.-B. Wu, L.-W. Li, and B.-R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
    [Crossref]
  30. S.-C. Mao and Z.-S. Wu, “Scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions and Fourier series,” J. Opt. Soc. Am. A 25, 2925–2931 (2008).
    [Crossref]
  31. F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
    [Crossref]
  32. T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
    [Crossref]
  33. W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1934).
    [Crossref]
  34. X.-S. Zhou, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).
  35. G. Han, Y. Han, and H. Zhang, “Relations between cylindrical and spherical vector wavefunctions,” J. Opt. A 10, 015006 (2008).
    [Crossref]
  36. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  37. G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
    [Crossref]
  38. A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  39. F. Frezza, A Primer on Electromagnetic Fields (Springer, 2015).
  40. E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (Scitech, 2004).
  41. A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).
  42. F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. A 33, 947–953 (2016).
    [Crossref]
  43. T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [Crossref]
  44. E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed. (Cambridge University, 1990).
  45. T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
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    [Crossref]
  47. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).

2017 (1)

2016 (3)

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. A 33, 947–953 (2016).
[Crossref]

F. Frezza and F. Mangini, “Electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a multilayered sphere,” J. Electromagn. Waves Appl. 30, 492–504 (2016).
[Crossref]

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

2015 (2)

C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015).
[Crossref]

2014 (2)

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
[Crossref]

2013 (3)

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

2012 (1)

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

2011 (1)

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

2010 (1)

2008 (2)

2004 (1)

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

1998 (2)

1996 (1)

1995 (1)

X. B. Wu and W. Ren, “Wave-function solution of plane-wave scattering by an anisotropic circular cylinder,” Microwave Opt. Technol. Lett. 8, 39–42 (1995).
[Crossref]

1993 (1)

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

1988 (1)

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[Crossref]

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1962 (1)

S. N. Samaddar, “Scattering of plane waves from an infinitely long cylinder of anisotropic materials at oblique incidence with an application to an electronic scanning antenna,” Appl. Sci. Res. B 10, 385–411 (1962).

1934 (1)

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1934).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

1881 (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73), 81–101 (1881).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Bai, L.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1940).

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, 1969).

Cincotti, G.

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Ding, K. H.

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[Crossref]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Fiaz, M. A.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

Frezza, F.

F. Frezza and F. Mangini, “Electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a multilayered sphere,” J. Electromagn. Waves Appl. 30, 492–504 (2016).
[Crossref]

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. A 33, 947–953 (2016).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015).
[Crossref]

C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
[Crossref]

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[Crossref]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[Crossref]

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

F. Frezza, A Primer on Electromagnetic Fields (Springer, 2015).

Furnò, F.

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Geng, Y.-L.

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

Gori, F.

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[Crossref]

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Grzesik, A.

Guan, B.-R.

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

Han, G.

G. Han, Y. Han, and H. Zhang, “Relations between cylindrical and spherical vector wavefunctions,” J. Opt. A 10, 015006 (2008).
[Crossref]

Han, Y.

G. Han, Y. Han, and H. Zhang, “Relations between cylindrical and spherical vector wavefunctions,” J. Opt. A 10, 015006 (2008).
[Crossref]

Hansen, W. W.

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1934).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1940).

Kerker, M.

M. Kerker, The Scattering of Light (Academic, 1969).

Knott, E. F.

E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (Scitech, 2004).

Kong, J. A.

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).

Lee, S.-C.

Li, H.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Li, L.-W.

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

Li, Z.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Lord Rayleigh,

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73), 81–101 (1881).
[Crossref]

Lorenz, L.

L. Lorenz, Videnskabernes Selskab Skrifter (1890), Vol. 6, p. 142. [Reprinted in L. Lorenz, Oeuvres Scientifiques, Librairie Lehmann, Copenhagen, Vol. 1, p. 405, 1896 (Reprinted by Johnson, New York, 1964).]

Mangini, F.

Mao, S.-C.

Mie, G.

G. Mie, “Beiträge zur Optik trüber medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Osipov, A. V.

A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).

Pajewski, L.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[Crossref]

Ponti, C.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[Crossref]

Qu, T.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Ren, W.

X. B. Wu and W. Ren, “Wave-function solution of plane-wave scattering by an anisotropic circular cylinder,” Microwave Opt. Technol. Lett. 8, 39–42 (1995).
[Crossref]

Rother, T.

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).

Samaddar, S. N.

S. N. Samaddar, “Scattering of plane waves from an infinitely long cylinder of anisotropic materials at oblique incidence with an application to an electronic scanning antenna,” Appl. Sci. Res. B 10, 385–411 (1962).

Santarsiero, M.

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[Crossref]

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Santini, C.

C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
[Crossref]

Schettini, G.

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[Crossref]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[Crossref]

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, 1969).

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E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (Scitech, 2004).

Shang, Q.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Sihvola, A.

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tedeschi, N.

F. Mangini and N. Tedeschi, “Scattering of an electromagnetic plane wave by a sphere embedded in a cylinder,” J. Opt. Soc. Am. A 34, 760–769 (2017).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015).
[Crossref]

C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
[Crossref]

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

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J. J. Thomson, Recent Researches in Electricity and Magnetism (Oxford University, 1893).

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A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).

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L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).

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E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (Scitech, 2004).

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J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, 1969).

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J. G. Van Bladel, Electromagnetic Fields, 2nd ed. (Wiley, 2007).

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T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[Crossref]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Wu, X. B.

X. B. Wu and W. Ren, “Wave-function solution of plane-wave scattering by an anisotropic circular cylinder,” Microwave Opt. Technol. Lett. 8, 39–42 (1995).
[Crossref]

Wu, X.-B.

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

Wu, Z.

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Wu, Z.-S.

Zhang, H.

G. Han, Y. Han, and H. Zhang, “Relations between cylindrical and spherical vector wavefunctions,” J. Opt. A 10, 015006 (2008).
[Crossref]

Zhou, X.-S.

X.-S. Zhou, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Appl. Sci. Res. B (1)

S. N. Samaddar, “Scattering of plane waves from an infinitely long cylinder of anisotropic materials at oblique incidence with an application to an electronic scanning antenna,” Appl. Sci. Res. B 10, 385–411 (1962).

IEEE Geosci. Remote Sens. Lett. (1)

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical-wave approach,” IEEE Geosci. Remote Sens. Lett. 10, 179–183 (2013).
[Crossref]

IEEE Trans. Antennas Propag. (2)

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering of a circular cylinder buried under a rough surface: the cylindrical wave approach,” IEEE Trans. Antennas Propag. 60, 2834–2842 (2012).
[Crossref]

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[Crossref]

J. Appl. Geophys. (1)

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium,” J. Appl. Geophys. 97, 55–59 (2013).
[Crossref]

J. Electromagn. Waves Appl. (2)

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014).
[Crossref]

F. Frezza and F. Mangini, “Electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a multilayered sphere,” J. Electromagn. Waves Appl. 30, 492–504 (2016).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

J. Opt. A (1)

G. Han, Y. Han, and H. Zhang, “Relations between cylindrical and spherical vector wavefunctions,” J. Opt. A 10, 015006 (2008).
[Crossref]

J. Opt. Soc. Am. A (9)

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. A 33, 947–953 (2016).
[Crossref]

F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014).
[Crossref]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
[Crossref]

S.-C. Lee and A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[Crossref]

S.-C. Mao and Z.-S. Wu, “Scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions and Fourier series,” J. Opt. Soc. Am. A 25, 2925–2931 (2008).
[Crossref]

F. Frezza, F. Mangini, L. Pajewski, G. Schettini, and N. Tedeschi, “Spectral domain method for the electromagnetic scattering by a buried sphere,” J. Opt. Soc. Am. A 30, 783–790 (2013).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015).
[Crossref]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[Crossref]

F. Mangini and N. Tedeschi, “Scattering of an electromagnetic plane wave by a sphere embedded in a cylinder,” J. Opt. Soc. Am. A 34, 760–769 (2017).
[Crossref]

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

T. Qu, Z. Wu, Q. Shang, Z. Li, L. Bai, and H. Li, “Scattering of an anisotropic sphere by an arbitrarily incident Hermite-Gaussian beam,” J. Quant. Spectrosc. Radiat. Transfer 170, 117–130 (2016).
[Crossref]

Microwave Opt. Technol. Lett. (1)

X. B. Wu and W. Ren, “Wave-function solution of plane-wave scattering by an anisotropic circular cylinder,” Microwave Opt. Technol. Lett. 8, 39–42 (1995).
[Crossref]

Opt. Commun. (4)

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

C. Santini, F. Frezza, and N. Tedeschi, “Plane-wave expansion of elliptic cylindrical functions,” Opt. Commun. 349, 185–192 (2015).
[Crossref]

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[Crossref]

G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Philos. Mag. (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73), 81–101 (1881).
[Crossref]

Phys. Rev. (1)

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1934).
[Crossref]

Phys. Rev. E (1)

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

Other (19)

X.-S. Zhou, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).

F. Frezza, A Primer on Electromagnetic Fields (Springer, 2015).

E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (Scitech, 2004).

A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).

L. Lorenz, Videnskabernes Selskab Skrifter (1890), Vol. 6, p. 142. [Reprinted in L. Lorenz, Oeuvres Scientifiques, Librairie Lehmann, Copenhagen, Vol. 1, p. 405, 1896 (Reprinted by Johnson, New York, 1964).]

J. J. Thomson, Recent Researches in Electricity and Magnetism (Oxford University, 1893).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, 1969).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

J. G. Van Bladel, Electromagnetic Fields, 2nd ed. (Wiley, 2007).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1940).

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

M. Kerker, The Scattering of Light (Academic, 1969).

E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed. (Cambridge University, 1990).

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1944).

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Figures (6)

Fig. 1.
Fig. 1. Representation of an obliquely incident plane wave on a dielectric infinite circular cylinder.
Fig. 2.
Fig. 2. Scattering cross section of a PEC cylinder with radius 0.25 m, when the plane wave at normal incidence is linearly TM-polarized in the range of frequencies from 0.1 MHz to 8.0 GHz. The cross section is computed (solid line) implementing on MATLAB the formula (54) and (dashed line) simulating the scattering problem on a software based on the finite-element method.
Fig. 3.
Fig. 3. Scattering cross section of a PEC cylinder with radius 0.25 m, when the plane wave at normal incidence is linearly TE-polarized in the range of frequencies from 0.1 MHz to 8.0 GHz. The cross section is computed (solid line) implementing on MATLAB the formula (54) and (dashed line) simulating the scattering problem on a software based on the finite-element method.
Fig. 4.
Fig. 4. Representation of a plane wave incident on a sphere.
Fig. 5.
Fig. 5. Scattering cross section of a PEC sphere with radius 0.25 m for an incident plane wave in the range of frequencies from 0.1 MHz to 10.0 GHz. The cross section is computed (solid line) implementing on MATLAB the formula (96) and (dashed line) simulating the scattering problem on a software based on the finite-element method. As we can see, the asymptotic limit of the graph is 2, meaning that in the large particle limit, twice as much energy is removed as expected based on the geometric cross section. This is contrary to intuition and is referred to as the extinction paradox.
Fig. 6.
Fig. 6. Scattering cross section of a dielectric sphere with radius 0.25 m and relative permittivity ϵ2=4 for an incident plane wave in the range of frequencies from 0.1 MHz to 10.0 GHz.

Equations (114)

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

Ei=E0e(ik·riωt),
Hi=H0e(ik·riωt),
S=12Re[E×H*].
S=12Re[Ei×Hi*]+12Re[Es×Hs*]+12Re[Ei×Hs*+Es×Hi*]=Si+Ss+Se,
Wa=Ssn^·SdS,
Wa=Sn^·SdS=WiWs+We,
Wi=Sn^·SidS,Ws=Sn^·SsdS,We=Sn^·SedS.
We=Wa+Ws.
σs=Ws|Si|σa=Wa|Si|σe=We|Si|.
σe=σa+σs.
Qs=σsGQa=σaGQe=σeG,
×E+Bt=0,
×H+Dt=J,
2Cμϵ2Ct2μσCt=0,
2C+k2C=0,
2ψ+k2ψ=0.
L=ψ;M=×(a^ψ);N=1k×M,
2M+k2M=×[a^(2ψ+k2ψ)].
M=1k×N;·L=2ψ=k2ψ.
A=1iωn=0+(anMn+bnNn+cnLn).
E=n=0+(anMn+bnNn),
H=kiωμn=0+(anNn+bnMn).
ψ(r)=eik·r,
L=iψk;M=iψk×a^;N=1kψ(k×a0)×k.
ψ(r)=++g(kx,ky)eik·rdkxdky=g(α,β)eik·rdαdβ,
L=ig(α,β)k(α,β)eik·rdβdα,M=ig(α,β)k(α,β)×a^eik·rdβdα,N=1kg(α,β)[k(α,β)×a^]×k(α,β)eik·rdβdα.
1ρr(rψρ)+1ρ22ψϕ2+(k2kz2)ψ=0.
ψm(ρ,ϕ,z)=AeimϕZm(kρρ)eikzz,
Lm(r)=lm(kρρ)eimϕeikzz,
Mm(r)=mm(kρρ)eimϕeikzz,
Nm(r)=nm(kρρ)eimϕeikzz,
lm(kρρ)=Zm(kρρ)ρρ^imρZm(kρρ)ϕ^+ikzZm(kρρ)z^,
mm(kρρ)=imZm(kρρ)ρρ^kρZm(kρρ)ρϕ^,
nm(kρρ)=ikzkρkZm(kρρ)ρρ^mkzkZm(kρρ)ρϕ^+kρ2kZm(kρρ)z^.
lm(kρρ)=lρ(kρρ)ρ^+lϕ(kρρ)ϕ^+lz(kρρ)z^,
mm(kρρ)=mρ(kρρ)ρ^+mϕ(kρρ)ϕ^,
nm(kρρ)=nρ(kρρ)ρ^+nϕ(kρρ)ϕ^+nz(kρρ)z^.
Ei(r)=[Eviv^(ϑi,ϕi)+Ehih^(ϑi,ϕi)]eik1·r,
Ei(r)=m=+[amM(1)(r)+bmN(1)(r)],
am=Ehik1ρim+1eimϕibm=Evik1ρimeimϕi,
k1z=k1cosϑik1ρ=k1sinϑi.
Esc(r)=m=+[cmMm(3)(r)+dmNm(3)(r)],
Ecy(r)=m=+[emMm(1)(r)+fmNm(1)(r)].
(Ei+EscEcy)×ρ^=0for  ρ=a,
[×(Ei+EscEcy)]×ρ^=0for  ρ=a.
mm(kρρ)×ρ^=mϕm(kρρ)z^,
nm(kρρ)×ρ^=nϕm(kρρ)z^nzm(kρρ)ϕ^,
[×mm(kρρ)]×ρ^=knϕm(kρρ)z^knzm(kρρ)ϕ^,
[×nm(kρρ)]×ρ^=kmϕm(kρρ)z^.
{cmmϕm(3)(k1ρa)+dmnϕm(3)(k1ρa)emmϕm(1)(k2ρa)fmnϕm(1)(k2ρa)=ammϕm(1)(k1ρa)bmnϕm(1)(k1ρa)cmk1nϕm(3)(k1ρa)dmk1mϕm(3)(k1ρa)emk2nϕm(1)(k2ρa)+fmk2mϕm(1)(k2ρa)=amk1nϕm(1)(k1ρa)+bmk1mϕm(1)(k1ρa)dmnzm(3)(k1ρa)fmnzm(1)(k2ρa)=bmnzm(1)(k1ρa)cmk1mzm(3)(k1ρa)emk2mzm(1)(k2ρa)=amk1mzm(1)(k1ρa).
(Ei+Esc)×ρ^=0for  ρ=a.
{cm=ammϕm(1)(k1ρa)mϕm(3)(k1ρa)bm[nϕm(1)(k1ρa)mϕm(3)(k1ρa)nzm(1)(k1ρa)nϕm(3)(k1ρa)nzm(3)(k1ρa)mϕm(3)(k1ρa)]dm=bmnzm(1)(k1ρa)nzm(3)(k1ρa).
{cm=amMϕm(1)(k1ρa)Mϕm(3)(k1ρa)dm=bmNzm(1)(k1ρa)Nzm(3)(k1ρa).
{cm=amJ˙m(k1ρa)H˙m(1)(k1ρa)dm=bmJm(k1ρa)Hm(1)(k1ρa).
Cs=4πk1m=1+(|cmam|2+|dmbm|2).
Ce=4πk1m=1+Re[|cmam|2+|dmbm|2].
1r2r(r2ψr)+1r2sinϑϑ(sinϑψϑ)+1r2sin2ϑ2ψϕ2+k2ψ=0.
ψ(r)=Azn(kr)Pnm(cosϑ)eimϕ,
Mmn(r)=×[rr^ψmn(r)],
Mmn(r)=imsinϑzn(kr)Pnm(cosϑ)eimϕϑ^zn(kr)Pnm(cosϑ)ϑeimϕϕ^,
Nmn(r)=zn(kr)krn(n+1)Pnm(cosϑ)eimϕr^+1kr[rzn(kr)]rPnm(cosϑ)ϑeimϕϑ^+1kr[rzn(kr)]rimsinϑPnm(cosϑ)eimϕϕ^.
πmn(ϑ)=mPnm(cosϑ)sinϑ,
τmn(ϑ)=dPnm(cosϑ)dϑ.
Mmn=zn(kr)[iπmn(cosϑ)ϑ^τmn(cosϑ)ϕ^]eimϕ,
Nmn={n(n+1)zn(kr)rPnm(cosϑ)r^+1kr[rzn(kr)]r[τmn(cosϑ)ϑ^+iπmn(cosϑ)ϕ^]}eimϕ.
mmn(ϑ,ϕ)=eimϕ[iπmn(cosϑ)ϑ^τmn(cosϑ)ϕ^],
nmn(ϑ,ϕ)=eimϕ[τmn(cosϑ)ϑ^+iπmn(cosϑ)ϕ^],
pmn(ϑ,ϕ)=eimϕn(n+1)Pnm(cosϑ)r^.
Mmn(r,ϑ,ϕ)=zn(r)mmn(ϑ,ϕ),
Nmn(r,ϑ,ϕ)=zn(r)rpmn(ϑ,ϕ)+1r[rzn(r)]rnmn(ϑ,ϕ).
Ei(r)=epoleiki·r=(Eϑiϑ^i+Eϕiϕ^i)eiki·r,
ki=k1k^i=k1(sinϑicosϕix^+sinϑisinϕiy^+cosϑiz^),
ϕ^i=z^×k^i|z^×k^i|=sinϕix^+cosϕiy^,
ϑ^i=ϕ^×k^i=cosϑicosϕix^+cosϑisinϕiy^sinϑiz^.
Ei(r)=n=1+m=nn[amnMmn(1)(r)+bmnNmn(1)(r)],
amn=(1)min2n+1n(n+1)(nm)!(n+m)!epol·mmn*(ϑi,ϕi),
bmn=(1)min12n+1n(n+1)(nm)!(n+m)!epol·nmn*(ϑi,ϕi).
Es(r)=n=1+m=nn[cmnMmn(3)(r)+dmnNmn(3)(r)].
(Ei+Es)×r^=0for  r=a.
Ei(r)=n=1+m=nn[amnmmn(ϑ,ϕ)jn(k1r)+bmnnmn(ϑ,ϕ)j˙n(k1r)+bmnpmn(ϑ,ϕ)jn(k1r)k1r],
Es(r)=n=1+m=nn[cmnmmn(ϑ,ϕ)hn(1)(k1r)+dmnnmn(ϑ,ϕ)h˙n(1)(k1r)+dmnpmn(ϑ,ϕ)hn(1)(k1r)k1r],
z˙n(kr)=1kr[xzn(kx)]x|x=r
mmn(ϑ,ϕ)×r^=nmn(ϑ,ϕ),
nmn(ϑ,ϕ)×r^=mmn(ϑ,ϕ),
pmn(ϑ,ϕ)×r^=0.
n=1+m=nn{mmn(ϑ,ϕ)[bmnj˙n(k1a)+dmnh˙n(1)(k1a)]nmn(ϑ,ϕ)[amnjn(k1a)+cmnhn(1)(k1a)]}=0.
{amnjn(k1a)+cmnhn(1)(k1a)=0bmnj˙n(k1a)+dmnh˙n(1)(k1a)=0,
{cmn=amnjn(k1a)hn(1)(k1a)dmn=bmnj˙n(k1a)h˙n(1)(k1a).
Ep(r)=n=1+m=nn[emnMmn(1)(r)+fmnNmn(1)(r)].
(Ei+EsEp)×r^=0for  r=a,
[×(Ei+EsEp)]×r^=0for  r=a.
n=1+m=nn{mmn(ϑ,ϕ)[bmnj˙n(k1a)+dmnh˙n(1)(k1a)fmnj˙n(k2a)]nmn(ϑ,ϕ)[amnjn(k1a)+cmnhn(1)(k1a)emnjn(k2a)]}=0,
n=1+m=nn{mmn(ϑ,ϕ)[k1amnj˙n(k1a)+k1cmnh˙n(1)(k1a)k2emnj˙n(k2a)]nmn(ϑ,ϕ)[k1bmnjn(k1a)+k1dmnhn(1)(k1a)k2fmnjn(k2a)]}=0.
{amnjn(k1a)+cmnhn(1)(k1a)emnjn(k2a)=0bmnj˙n(k1a)+dmnh˙n(1)(k1a)fmnj˙n(k2a)=0k1amnj˙n(k1a)+k1cmnh˙n(1)(k1a)k2emnj˙n(k2a)=0k1bmnjn(k1a)+k1dmnhn(1)(k1a)k2fmnjn(k2a)=0.
cmn=amnj˙n(k1a)jn(k2a)χjn(k1a)j˙n(k2a)h˙n(1)(k1a)jn(k2a)χhn(1)(k1a)j˙n(k2a),
dmn=bmnjn(k1a)j˙n(k2a)χj˙n(k1a)jn(k2a)hn(1)(k1a)j˙n(k2a)χh˙n(1)(k1a)jn(k2a),
Cs=2πk12n=1+(2n+1)(|c1na1n|2+|d1nb1n|2).
Ce=2πk12n=1+(2n+1)Re[|c1na1n|2+|d1nb1n|2].
Jn(z)Yn(z)Jn(z)Yn(z)=2πz.
02π0πmmn·nmn*sinϑdϑdϕ=0,
02π0πpmn·mmn*sinϑdϑdϕ=0,
02π0πpmn·nmn*sinϑdϑdϕ=0,
02π0πmmn·mmn*sinϑdϑdϕ=4πn(n+1)2n+1(n+m)!(nm)!δmmδnn,
02π0πnmn·nmn*sinϑdϑdϕ=4πn(n+1)2n+1(n+m)!(nm)!δmmδnn,
02π0πpmn·pmn*sinϑdϑdϕ=4π[n(n+1)]22n+1(n+m)!(nm)!δmmδnn.
limϑ0πmn(cosϑ)=limϑ0mPnm(cosϑ)sinϑ.
Pnm(cosϑ)=(1)msinmϑdmPn(cosϑ)d(cosϑ)m.
limϑ0dmPn(cosϑ)d(cosϑ)m=limx1dmPn(x)dxm=1·3··(2m1)limx1Cnm(m12)(x)=(n+m)!(nm)!,
limϑ0τmn(cosϑ)=limϑ0dPnm(cosϑ)dϑ.
limϑ0τmn(cosϑ)=limϑ0sinϑdPn(cosϑ)d(cosϑ)=0,
limϑ0τmn(cosϑ)=limϑ0[cosϑdPn(cosϑ)d(cosϑ)sin2ϑd2Pn(cosϑ)d(cosϑ)2]=n(n+1).
limϑ0τmn(cosϑ)=(1)mlimϑ0[msinm1ϑcosϑdmPn(cosϑ)d(cosϑ)msinm+1ϑdm+1Pn(cosϑ)d(cosϑ)m+1]=0.
limϑ0πmn(cosϑ)=limϑ0τmn(cosϑ)=n(n+1)δ1m.
jn(z)y˙n(z)j˙n(z)yn(z)=1z2.

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