Abstract

In this paper, some generalizations of electromagnetic scattering problems by elementary shapes are presented. In particular, the aim of the paper is to provide solutions to the scattering problem by multiple objects with simple shapes, either in concentric configuration or arbitrarily distributed in the space. The vector harmonics, representing the fields, and their properties are applied in order to solve five different problems: the electromagnetic scattering by an infinitely long circular stratified cylinder, by a multilayered sphere, by an ensemble of parallel cylinders, by an ensemble of multi-spheres, and ultimately by a sphere embedded in a circular cylinder. Numerical results in particularly important configurations are shown.

© 2020 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Frezza, F. Mangini, and N. Tedeschi, “Introduction to electromagnetic scattering: tutorial,” J. Opt. Soc. Am. A 35, 163–173 (2018).
    [Crossref]
  2. Z. Xue-Song, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).
  3. J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968).
    [Crossref]
  4. O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
    [Crossref]
  5. S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
    [Crossref]
  6. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [Crossref]
  7. T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [Crossref]
  8. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).
  9. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  10. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
  11. A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).
  12. S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.
  13. S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
    [Crossref]
  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1940).
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  16. J. G. Van Bladel, Electromagnetic Fields, 2nd ed. (Wiley, 2007).
  17. T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
  18. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).
  19. 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]
  20. K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994).
    [Crossref]
  21. H. He, N. Zeng, W. Li, T. Yun, R. Liao, Y. He, and H. Ma, “Two-dimensional backscattering Mueller matrix of sphere-cylinder scattering medium,” Opt. Lett. 35, 2323–2325 (2010).
    [Crossref]
  22. 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]
  23. F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
    [Crossref]
  24. H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaf, and J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
    [Crossref]
  25. E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.
  26. H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express 18, 15104–15112 (2010).
    [Crossref]
  27. F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (2015).
    [Crossref]
  28. V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
    [Crossref]
  29. Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
    [Crossref]
  30. R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
    [Crossref]
  31. F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .
  32. B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
    [Crossref]
  33. A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
    [Crossref]
  34. V. I. Ovod, “Modeling of multiple scattering from an ensemble of spheres in a laser beam,” Part. Syst. Charact. 16, 106–112 (1999).
    [Crossref]
  35. Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
    [Crossref]
  36. H. Zhong, A. Redo-Sanchez, and X. C. Zhang, “Identification and classification of chemicals using terahertz reflective spectroscopic focal-plane imaging system,” Opt. Express 14, 9130–9141 (2006).
    [Crossref]
  37. S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
    [Crossref]
  38. H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941).
    [Crossref]
  39. A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  40. 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]
  41. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover1972).
  42. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  43. 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]
  44. 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]
  45. 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]
  46. F. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (2016).
    [Crossref]
  47. G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1–7 (1995).
    [Crossref]
  48. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).
  49. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2000).
  50. D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018).
    [Crossref]
  51. G. Mie, “Beiträge zur Optik trüber medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [Crossref]
  52. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).
    [Crossref]
  53. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).

2020 (1)

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

2019 (1)

F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
[Crossref]

2018 (2)

F. Frezza, F. Mangini, and N. Tedeschi, “Introduction to electromagnetic scattering: tutorial,” J. Opt. Soc. Am. A 35, 163–173 (2018).
[Crossref]

D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018).
[Crossref]

2017 (2)

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[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]

2016 (2)

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. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (2016).
[Crossref]

2015 (3)

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]

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (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]

2011 (1)

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

2010 (2)

2006 (2)

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

H. Zhong, A. Redo-Sanchez, and X. C. Zhang, “Identification and classification of chemicals using terahertz reflective spectroscopic focal-plane imaging system,” Opt. Express 14, 9130–9141 (2006).
[Crossref]

2005 (1)

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

2003 (1)

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

1999 (1)

V. I. Ovod, “Modeling of multiple scattering from an ensemble of spheres in a laser beam,” Part. Syst. Charact. 16, 106–112 (1999).
[Crossref]

1998 (1)

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

1995 (2)

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

G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1–7 (1995).
[Crossref]

1994 (1)

K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994).
[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]

1990 (2)

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[Crossref]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaf, and J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[Crossref]

1984 (1)

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[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]

1968 (1)

J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968).
[Crossref]

1962 (1)

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

1941 (1)

H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941).
[Crossref]

1908 (1)

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

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover1972).

Ao, C. O.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).

Balanis, C. A.

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

Batool, S.

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

Benedetti, A.

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

Bisceglia, B.

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

Bohren, C. F.

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

Borghese, F.

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

Born, M.

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

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968).
[Crossref]

Choi, W.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

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]

Clifford, S.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

Cole, B.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Cruzan, O. R.

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

De Flaviis, F.

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

De Leo, R.

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

de Mul, F. F. M.

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaf, and J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[Crossref]

E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.

Denti, P.

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[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]

Di Gregorio, P. P.

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

Ding, K. H.

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

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).

Dinia, L.

F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
[Crossref]

Dinwiddie, C.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

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).

Farrell, D.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

Fen, M. Q.

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

Ferrara, V.

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

Figueiras, E.

E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.

Frezza, F.

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Introduction to electromagnetic scattering: tutorial,” J. Opt. Soc. Am. A 35, 163–173 (2018).
[Crossref]

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[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]

F. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (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]

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (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]

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]

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

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]

Gori, 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]

Graaf, R.

Greve, J.

Grimm, R. E.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

He, H.

He, Y.

Heggy, E.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

Hermsen, R. G. A. M.

Huffman, D. R.

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

Humeau, A.

E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.

Jentink, H. W.

Jeong, S.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Jofre, L.

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

Joo, J. H.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Kang, S.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Kemp, M.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Kim, Y. J.

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

Ko, H.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Kong, J. A.

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

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2000).

Kramers, H. A.

H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941).
[Crossref]

La Russa, M. F.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[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).

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).

Lee, J.-S.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Lee, S.-C.

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[Crossref]

Li, W.

Liao, R.

Lim, Y.-S.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Lo, T.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Lo, Y. T.

J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968).
[Crossref]

Ma, H.

Macchia, A.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

Mangini, F.

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
[Crossref]

F. Frezza, F. Mangini, and N. Tedeschi, “Introduction to electromagnetic scattering: tutorial,” J. Opt. Soc. Am. A 35, 163–173 (2018).
[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]

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[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]

F. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (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]

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (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]

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

McGinnis, R.

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[Crossref]

Meriakri, V.

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

Mie, G.

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

Mishchenko, M. I.

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

Muzi, M.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (2015).
[Crossref]

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

Osipov, A. V.

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

Ovod, V. I.

V. I. Ovod, “Modeling of multiple scattering from an ensemble of spheres in a laser beam,” Part. Syst. Charact. 16, 106–112 (1999).
[Crossref]

Pajewski, L.

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

Park, Q.-H.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Pastore, A. P.

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

Polatin, P. F.

K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994).
[Crossref]

Redo-Sanchez, A.

Requicha Ferreira, L. F.

E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.

Ricca, M.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

Rivaroli, L.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

Rother, T.

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

Ruffolo, S. A.

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

Saija, R.

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

Santarsiero, M.

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]

Sarabandi, K.

K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994).
[Crossref]

Schettini, 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]

Shen, Y. C.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Sihvola, A.

D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018).
[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. 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]

Simeoni, P.

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

Sindoni, O. I.

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover1972).

Stoja, E.

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (2015).
[Crossref]

Stratton, J. A.

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

Taday, P.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Tedeschi, N.

Toscano, G.

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

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).

Tretyakov, S. A.

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

Tribe, W.

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Troiani, F.

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

Tsang, L.

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

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2000).

Tzarouchis, D.

D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018).
[Crossref]

Van Bladel, J. G.

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

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).

Videen, G.

G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1–7 (1995).
[Crossref]

von Gratowski, S.

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

Wannier, G. H.

H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941).
[Crossref]

Wolf, E.

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

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

Wriedt, T.

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).

Xu, Y. L.

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

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

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

Xue-Song, Z.

Z. Xue-Song, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).

Yang, T. D.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Yun, T.

Zeng, N.

Zhang, X. C.

Zhong, H.

Aerosol Sci. Technol. (1)

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

Ann. Phys. (1)

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

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[Crossref]

Appl. Sci. (1)

D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018).
[Crossref]

IEEE Trans. Antennas Propag. (2)

K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994).
[Crossref]

Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003).
[Crossref]

J. Appl. Phys. (1)

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[Crossref]

J. Biol. Phys. (1)

F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (2015).
[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. Geophys. Res. (1)

R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006).
[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. Microwave Power Electromagn. Energy (1)

B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011).
[Crossref]

J. Opt. Soc. Am. A (4)

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

S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020).
[Crossref]

J. Telecommun. Inf. Technol. (1)

F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019).
[Crossref]

Math. Mech. Complex Syst. (1)

F. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (2016).
[Crossref]

Nat. Photonics (1)

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015).
[Crossref]

Opt. Commun. (3)

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

G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1–7 (1995).
[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]

Opt. Express (2)

Opt. Lett. (1)

Part. Syst. Charact. (1)

V. I. Ovod, “Modeling of multiple scattering from an ensemble of spheres in a laser beam,” Part. Syst. Charact. 16, 106–112 (1999).
[Crossref]

Phys. Rev. (1)

H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941).
[Crossref]

Proc. IEEE (1)

J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968).
[Crossref]

Prog. Org. Coat. (1)

A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017).
[Crossref]

Quart. Appl. Math. (1)

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

Other (19)

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).

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

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

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

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

S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.

Z. Xue-Song, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).

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

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

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

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

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

E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.

F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf .

V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4.
[Crossref]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover1972).

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

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1. Geometry of the scattering problem by a multilayer cylinder.
Fig. 2.
Fig. 2. (a) Scattered field by a two-layered cylinder at the point $(2{a_1},0,0)$. In particular, solid line represents the two-layer case, and the dashed line represents the single-layer case. (b) Scattered field by a two-layer cylinder at the point $(2{a_1},0,0)$ for several ${\varepsilon _1}$ values from two to six.
Fig. 3.
Fig. 3. Field map of the scattered electric field $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ in the same scenario as Fig. 2, at a wavelength of 600 nm.
Fig. 4.
Fig. 4. Geometry of the scattering problem by a multilayer sphere.
Fig. 5.
Fig. 5. (a) Scattered field by a two-layer sphere at the point $(2{a_1},0,0)$. In particular, the solid line represents the two-layer case, and the dashed line represents the single-layer case. (b) Scattered field by a two-layer sphere at the same point for several ${\varepsilon _1}$ values from two to six.
Fig. 6.
Fig. 6. Field map of the scattered electric field $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ in the same scenario as Fig. 5, at a wavelength of 600 nm.
Fig. 7.
Fig. 7. Geometry of the scattering problem by several parallel cylinders arbitrarily posed in free space.
Fig. 8.
Fig. 8. (a) Profile of the scattered electric field by (b) eight equidistant parallel cylinders with the centers on a circumference with radius 200 nm, computed on the point $(2a,0,0)$ in the visible frequency range for five different dielectric constant values (${\varepsilon _1} = [2,3,4,5,6]$).
Fig. 9.
Fig. 9. Field map of the scattered electric field $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ in the same scenario as Fig. 8, at a wavelength of 600 nm.
Fig. 10.
Fig. 10. Statement of the scattering problem by multiple spheres in free space.
Fig. 11.
Fig. 11. (a) Profile of the scattered electric field by (b) eight equidistant spheres with their centers on a circumference with radius 200 nm, computed on the point $(2a,0,0)$ in the visible frequency range for five different dielectric constant values (${\varepsilon _1} = [2,3,4,5,6]$).
Fig. 12.
Fig. 12. Field map of the scattered electric field $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ in the same scenario as Fig. 11, at a wavelength of 600 nm.
Fig. 13.
Fig. 13. Geometry of the scattering problem by a sphere embedded inside a circular cylinder.
Fig. 14.
Fig. 14. Magnitude of the scattered electric field as a function of frequency in the visible range, computed at the point $(2{a_c},0,0)$ for different values of the cylinder permittivity ${\varepsilon _c} = [2,3,4,5,6]$.
Fig. 15.
Fig. 15. Field map of the scattered electric field $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ in the same scenario as Fig. 14, at a wavelength of 600 nm.

Equations (141)

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

E i ( k i r ) = m = + [ a m M m ( 1 ) ( k i r ) + b m N m ( 1 ) ( k i r ) ] ,
a m = E h i k i ρ i m + 1 e i m φ i b m = E v i k i ρ i m e i m φ i ,
k i z = k i cos ϑ i k i ρ = k i sin ϑ i ,
M m ( r ) = m m ( k i ρ ρ ) e i m φ e i k i z z ,
N m ( r ) = n m ( k i ρ ρ ) e i m φ e i k i z z ,
m m ( k i ρ ρ ) = i m Z m ( k i ρ ρ ) ρ ρ k i ρ Z m ( k i ρ ρ ) ρ φ ,
n m ( k i ρ ρ ) = i k i z k i ρ k Z m ( k i ρ ρ ) ρ ρ m k i z k i Z m ( k i ρ ρ ) ρ φ + k i ρ 2 k i Z m ( k i ρ ρ ) z .
m m ( k i ρ ρ ) = m ρ ( k i ρ ρ ) ρ + m φ ( k i ρ ρ ) φ ,
n m ( k i ρ ρ ) = n ρ ( k i ρ ρ ) ρ + n φ ( k i ρ ρ ) φ + n z ( k i ρ ρ ) z .
E s ( k i r ) = m = + [ e m M m ( 3 ) ( k i r ) + f m N m ( 3 ) ( k i r ) ] .
E j ( k j r ) = m = + [ r m j M m ( 1 ) ( k j r ) + s m j N m ( 1 ) ( k j r ) ] + m = + [ u m j M m ( 2 ) ( k j r ) + v m j N m ( 2 ) ( k j r ) ] ,
E N ( k N r ) = m = + [ r m N M m ( 1 ) ( k N r ) + s m N N m ( 1 ) ( k N r ) ] .
( E i j + E s j E i j + 1 E s j + 1 ) × ρ i = 0 for ρ = a j ,
( H i j + H s j H i j + 1 H s j + 1 ) × ρ i = 0 for ρ = a j .
( E i j + E s j E i j + 1 E s j + 1 ) × ρ i = 0 for ρ = a j ,
[ × ( E i j + E s j E i j + 1 E s j + 1 ) ] ρ i = 0 for ρ = a j .
m m ( k ρ ρ ) × ρ = m φ m ( k ρ ρ ) z ,
n m ( k ρ ρ ) × ρ = n φ m ( k ρ ρ ) z n z m ( k ρ ρ ) φ ,
[ × m m ( k ρ ρ ) ] × ρ = k n φ m ( k ρ ρ ) z k n z m ( k ρ ρ ) φ ,
[ × n m ( k ρ ρ ) ] × ρ = k m φ m ( k ρ ρ ) z .
{ s m j + 1 n φ m ( 1 ) ( k j + 1 ρ j ) + v m j + 1 n φ m ( 2 ) ( k j + 1 ρ j ) + r m j + 1 m φ m ( 1 ) ( k j + 1 ρ j ) + u m j + 1 m φ m ( 2 ) ( k j + 1 ρ j ) = s m j n φ m ( 1 ) ( k j ρ j ) + v m j n φ m ( 2 ) ( k j ρ j ) + r m j m φ m ( 1 ) ( k j ρ j ) + u m j m φ m ( 2 ) ( k j ρ j ) k j + 1 [ s m j + 1 m φ m ( 1 ) ( k j + 1 ρ j ) v m j + 1 m φ m ( 2 ) ( k j + 1 ρ j ) + r m j + 1 n φ m ( 1 ) ( k j + 1 ρ j ) + u m j + 1 n φ m ( 2 ) ( k j + 1 ρ j ) ] = k j [ s m j m φ m ( 1 ) ( k j ρ j ) v m j m φ m ( 2 ) ( k j ρ j ) + r m j n φ m ( 1 ) ( k j ρ j ) + u m j n φ m ( 2 ) ( k j ρ j ) ] s m j + 1 n z m ( 1 ) ( k j + 1 ρ j ) + v m j + 1 n z m ( 2 ) ( k j + 1 ρ j ) = s m j n z m ( 1 ) ( k j ρ j ) + v m j n z m ( 2 ) ( k j ρ j ) k j + 1 [ r m j + 1 n z m ( 1 ) ( k j + 1 ρ j ) + u m j + 1 n z m ( 2 ) ( k j + 1 ρ j ) ] = k j [ r m j n z m ( 1 ) ( k j ρ j ) + u m j n z m ( 2 ) ( k j ρ j ) ] .
[ n z m ( 1 ) ( k j + 1 ρ j ) n z m ( 2 ) ( k j + 1 ρ j ) 0 0 n φ m ( 1 ) ( k j + 1 ρ j ) n φ m ( 2 ) ( k j + 1 ρ j ) m φ m ( 1 ) ( k j + 1 ρ j ) m φ m ( 2 ) ( k j + 1 ρ j ) ζ j m φ m ( 1 ) ( k j + 1 ρ j ) ζ j m φ m ( 2 ) ( k j + 1 ρ j ) ζ j n φ m ( 1 ) ( k j + 1 ρ j ) ζ j n φ m ( 2 ) ( k j + 1 ρ j ) 0 0 ζ j n z m ( 1 ) ( k j + 1 ρ j ) ζ j n z m ( 2 ) ( k j + 1 ρ j ) ] ( s m j + 1 v m j + 1 r m j + 1 u m j + 1 ) = [ n z m ( 1 ) ( k j ρ j ) n z m ( 2 ) ( k j ρ j ) 0 0 n φ m ( 1 ) ( k j ρ j ) n φ m ( 2 ) ( k j ρ j ) m φ m ( 1 ) ( k j ρ j ) m φ m ( 2 ) ( k j ρ j ) m φ m ( 1 ) ( k j ρ j ) m φ m ( 2 ) ( k j ρ j ) n φ m ( 1 ) ( k j ρ j ) n φ m ( 2 ) ( k j ρ j ) 0 0 n z m ( 1 ) ( k j ρ j ) n z m ( 2 ) ( k j ρ j ) ] ( s m j v m j r m j u m j ) ,
( s m j + 1 v m j + 1 r m j + 1 u m j + 1 ) = [ A 1 B ] ( s m j v m j r m j u m j ) = [ M ] ( s m j v m j r m j u m j ) .
( s m N 0 r m N 0 ) = i = N 1 1 [ M i ] ( e m a m f m b m ) ,
{ M 11 e m + M 12 a m + M 13 f m + M 14 b m = s m n N M 21 e m + M 22 a m + M 23 f m + M 24 b m = 0 M 31 e m + M 32 a m + M 33 f m + M 34 b m = r m n N M 41 e m + M 42 a m + M 43 f m + M 44 b m = 0.
E i ( r ) = ( E i H ϑ 0 + E i E φ 0 ) e i k i r ,
k i = k i ( sin ϑ i cos φ i x i + sin ϑ i sin φ i y i + cos ϑ i z i ) ,
ϑ 0 = cos ϑ i cos φ i x i + cos ϑ i sin φ i y i sin ϑ i z i ,
φ 0 = sin φ i x i + cos φ i y i ,
E i ( r ) = ( E i H ϑ 0 + E i E φ 0 i ) e i k i ( sin ϑ i x i + cos ϑ i z i ) = n = 1 m = n n [ a m n M m n ( 1 ) ( k i r ) + b m n N m n ( 1 ) ( k i r ) ] ,
M m n ( 1 ) ( k i r ) = j n ( k i r ) m m n ( ϑ , φ ) ,
N m n ( 1 ) ( k i r ) = j n ( k i r ) k i r p m n ( ϑ , φ ) + 1 k i r d [ r j n ( k i r ) ] d r n m n ( ϑ , φ ) ,
m m n ( ϑ , φ ) = e i m φ [ i π n m ( cos ϑ ) ϑ i τ n m ( cos ϑ ) φ i ] ,
n m n ( ϑ , φ ) = e i m φ [ τ n m ( cos ϑ ) ϑ i + i π n m ( cos ϑ ) φ i ] ,
p m n ( ϑ , φ ) = e i m φ n ( n + 1 ) P n m ( cos ϑ ) r i ,
π n m ( cos ϑ ) = m sin ϑ P n m ( cos ϑ ) ,
τ n m ( cos ϑ ) = d P n m ( cos ϑ ) d ϑ .
a m n = i n 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! ( E i H ϑ 0 + E i E φ 0 ) m m n ( ϑ i , φ i ) ,
b m n = i n 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! ( E i H ϑ 0 + E i E φ 0 ) n m n ( ϑ i , φ i ) ,
m m n ( ϑ i , φ i ) = e i m φ i [ i m sin ϑ i P n m ( cos ϑ i ) ϑ i dP n m ( cos ϑ i ) d ϑ φ i ] ,
n m n ( ϑ i , φ i ) = e i m φ i [ dP n m ( cos ϑ i ) d ϑ ϑ i + i m sin ϑ i P n m ( cos ϑ i ) φ i ] .
E s ( k i r ) = n = 1 + m = n n [ e m n M m n ( 3 ) ( k i r ) + f m n N m n ( 3 ) ( k i r ) ] ,
E j ( k j r ) = E i j ( k j r ) + E s j ( k j r ) = n = 1 + m = n n [ r m n j M m n ( 1 ) ( k j r ) + s m n j N m n ( 1 ) ( k j r ) ] + n = 1 + m = n n [ u m n j M m n ( 2 ) ( k j r ) + v m n j N m n ( 2 ) ( k j r ) ] ,
E N ( k N r ) = n = 1 + m = n n [ r m n N M m n ( 1 ) ( k N r ) + s m n N N m n ( 1 ) ( k N r ) ] .
( E i j + E s j E i j + 1 E s j + 1 ) × r i = 0 for r = a j ,
[ × ( E i j + E s j E i j + 1 E s j + 1 ) ] × r i = 0 for r = a j .
E i ( r ) = n = 1 + m = n n { a m n j m m n ( ϑ , φ ) j n ( k j r ) + b m n j [ n m n ( ϑ , φ ) j n ( k j r ) + p m n ( ϑ , φ ) j n ( k j r ) k j r ] } ,
E i ( r ) = n = 1 + m = n n [ a m n j M m n ( 1 ) ( k j r ) + b m n j N m n ( 1 ) ( k j r ) ] ,
m m n ( ϑ , φ ) × r i = n m n ( ϑ , φ ) ,
n m n ( ϑ , φ ) × r i = m m n ( ϑ , φ ) ,
× M m n ( k r , ϑ , φ ) = k N m n ( k r , ϑ , φ ) = k m m n ( ϑ , φ ) z n ( k r ) ,
× N m n ( k r , ϑ , φ ) = k M m n ( k r , ϑ , φ ) = k n m n ( ϑ , φ ) z n ( k r ) .
{ r m n j j n ( k j r j ) + u m n j y n ( k j r j ) = r m n j + 1 j n ( k j + 1 r j ) + u m n j + 1 y n ( k j + 1 r j ) k j [ r m n j j n ( k j r j ) + u m n j y n ( k j r j ) ] = k j + 1 [ r m n j + 1 j n ( k j + 1 r j ) + u m n j + 1 y n ( k j + 1 r j ) ] s m n j j n ( k j r j ) + v m n j y n ( k j r j ) = s m n j + 1 j n ( k j + 1 r j ) + v m n j + 1 y n ( k j + 1 r j ) k j [ s m n j j n ( k j r j ) + v m n j y n ( k j r j ) ] = k j + 1 [ s m n j + 1 j n ( k j + 1 r j ) + v m n j + 1 y n ( k j + 1 r j ) ] .
( r m n j + 1 u m n j + 1 ) = 1 A j + 1 [ y n ( k j + 1 r j ) y n ( k j + 1 r j ) ζ j j n ( k j + 1 r j ) j n ( k j + 1 r j ) ζ j ] × [ j n ( k j r j ) y n ( k j r j ) j n ( k j r j ) y n ( k j r j ) ] ( r m n j u m n j ) ,
( s m n j + 1 v m n j + 1 ) = 1 A j + 1 [ y n ( k j + 1 r j ) y n ( k j + 1 r j ) ζ j j n ( k j + 1 r j ) j n ( k j + 1 r j ) ζ j ] × [ j n ( k j r j ) y n ( k j r j ) j n ( k j r j ) y n ( k j r j ) ] ( s m n j v m n j ) ,
A j + 1 = j n ( k j + 1 r j ) y n ( k j + 1 r j ) y n ( k j + 1 r j ) j n ( k j + 1 r j ) ,
z n ( z ) = z n 1 ( z ) n + 1 z f n ( z ) ,
z n ( z ) = z n + 1 ( z ) n z f n ( z ) .
A j + 1 = 1 ( k j + 1 r j ) 2 .
( r m n j + 1 u m n j + 1 ) = [ M j ] ( r m n j u m n j ) ,
( s m n j + 1 v m n j + 1 ) = [ N j ] ( s m n j v m n j ) ,
[ M j ] = 1 A j + 1 [ j n ( k j r j ) y n ( k j + 1 r j ) j n ( k j r j ) y n ( k j + 1 r j ) ζ j y n ( k j r j ) y n ( k j + 1 r j ) y n ( k j r j ) y n ( k j + 1 r j ) ζ j j n ( k j r j ) j n ( k j + 1 r j ) + j n ( k j r j ) j n ( k j + 1 r j ) ζ j y n ( k j r j ) j n ( k j + 1 r j ) + y n ( k j r j ) j n ( k j + 1 r j ) ζ j ] ,
[ N j ] = 1 A j + 1 [ j n ( k j r j ) y n ( k j + 1 r j ) + j n ( k j r j ) y n ( k j + 1 r j ) ζ j y n ( k j r j ) y n ( k j + 1 r j ) + y n ( k j r j ) y n ( k j + 1 r j ) ζ j j n ( k j r j ) j n ( k j + 1 r j ) j n ( k j r j ) j n ( k j + 1 r j ) ζ j y n ( k j r j ) j n ( k j + 1 r j ) y n ( k j r j ) j n ( k j + 1 r j ) ζ j ] ,
( r m n N 0 ) = i = N 1 1 [ M i ] ( a m n 1 e m n 1 ) ,
( s m n N 0 ) = i = N 1 1 [ N i ] ( b m n 1 f m n 1 ) .
{ M 11 a m n + M 12 e m n = r m n N M 21 a m n + M 22 e m n = 0 ,
{ N 11 b m n + N 12 f m n = s m n N N 21 b m n + N 22 f m n = 0 ,
{ e m n = a m n M 21 M 22 f m n = b m n N 21 N 22 ,
{ r m n N = a m n det [ M ] M 22 s m n N = b m n det [ N ] N 22 ,
det [ M j ] = A j ζ j A j + 1 ,
det [ N j ] = ζ j A j A j + 1 .
E ex q = E i + p = 1 p q L E s p .
E i ( k i ρ q ) = [ E v 0 v + E h 0 h ] e i k i ρ q e i k i ( ρ ρ q ) = m = + [ a ~ m M m ( 1 ) ( k i , ρ ρ q ) + b ~ m N m ( 1 ) ( k i , ρ ρ q ) ] ,
a ~ m = a m e i k i ρ q ,
b ~ m = b m e i k i ρ q
E ex q ( k i ρ q ) = m = + [ w m q M m ( 1 ) ( k i , ρ ρ q ) + v m q N m ( 1 ) ( k i , ρ ρ q ) ] ,
E s p ( k i ρ p ) = m = + [ T m M w m p M m ( 3 ) ( k i , ρ ρ p ) + T m N v m p N m ( 3 ) ( k i , ρ ρ p ) ] ,
T m M = J m ( k i ρ a ) H m ( 1 ) ( k i ρ a ) ,
T m N = J m ( k i ρ a ) H m ( 1 ) ( k i ρ a ) .
M m ( 3 ) ( k , ρ ρ p ) = m = + A m m M m ( 1 ) ( k , ρ ρ q ) ,
N m ( 3 ) ( k , ρ ρ p ) = m = + A m m N m ( 1 ) ( k , ρ ρ q ) ,
M m ( 1 ) ( k , ρ ρ p ) = m = + B m m M m ( 1 ) ( k , ρ ρ q ) ,
N m ( 1 ) ( k , ρ ρ p ) = m = + B m m N m ( 1 ) ( k , ρ ρ q ) ,
A m m = H m m ( 1 ) ( k | ρ p ρ q | ) e i ( m m ) φ p q ,
B m m = J m m ( 1 ) ( k | ρ p ρ q | ) e i ( m m ) φ p q .
w m q = a ~ m + m = + p = 1 p q A m m T m M w m p ,
v m q = b ~ m + m = + p = 1 p q A m m T m N v m p .
E s q = m = + [ e m q M m ( 3 ) ( k i , ρ ρ q ) + f m q N m ( 3 ) ( k i , ρ ρ q ) ] ,
e m q = T m M w m q ,
f m q = T m N v m q .
E s = q = 1 L E s q .
E ex q = E i + p = 1 p q L E s p .
E i ( k i r ) = n = 1 + m = n + n [ a m n M m n ( 1 ) ( k i r ) + b m n N m n ( 1 ) ( k I r ) ] ,
a m n = i n 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! ( E i H ϑ 0 + E i E φ 0 ) m m n ( ϑ i , φ i ) ,
b m n = i n 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! ( E i H ϑ 0 + E i E φ 0 ) n m n ( ϑ i , φ i ) .
E ex q ( k I r q ) = n = 1 + m = n + n [ w m n q M m n ( 1 ) ( k i , r r q ) + v m n q N m n ( 1 ) ( k I , r r q ) ] ,
E s p ( k i r p ) = n = 1 + m = n + n [ T m M w m p M m n ( 3 ) ( k i , r r p ) + T m N v m p N m n ( 3 ) ( k i , r r p ) ] ,
T m n M = a m n j ˙ n ( k i a ) j n ( k 1 a ) χ j n ( k i a ) j ˙ n ( k 1 a ) h ˙ n ( 1 ) ( k i a ) j n ( k 1 a ) χ h n ( 1 ) ( k i a ) j ˙ n ( k 1 a ) ,
T m n N = b m n j n ( k i a ) j ˙ n ( k 1 a ) χ j ˙ n ( k i a ) j n ( k 1 a ) h n ( 1 ) ( k i a ) j ˙ n ( k 1 a ) χ h ˙ n ( 1 ) ( k i a ) j n ( k 1 a ) ,
M m n ( 1 ) ( k r ) = n = 1 + m = n + n [ A μ ν m n 11 ( k r i ) M μ ν ( 1 ) ( k r ) + B μ ν m n 11 ( k r i ) N μ ν ( 1 ) ( k r ) ] N m n ( 1 ) ( k r ) = n = 1 + m = n + n [ A μ ν m n 11 ( k r i ) N μ ν ( 1 ) ( k r ) + B μ ν m n 11 ( k r i ) M μ ν ( 1 ) ( k r ) ] for r i > r ,
M m n ( 3 ) ( k r ) = n = 1 + m = n + n [ A μ ν m n 31 ( k r i ) M μ ν ( 1 ) ( k r ) + B μ ν m n 31 ( k r i ) N μ ν ( 1 ) ( k r ) ] N m n ( 3 ) ( k r ) = n = 1 + m = n + n [ A μ ν m n 31 ( k r i ) N μ ν ( 1 ) ( k r ) + B μ ν m n 31 ( k r i ) M μ ν ( 1 ) ( k r ) ] for r i < r .
m n w m n q M m n ( 1 ) ( k i , r r q ) + v m n q N m n ( 1 ) ( k i , r r q ) = μ ν m n a μ ν [ A μ ν m n 11 ( k i , r r q ) M μ ν ( 1 ) ( k i , r r q ) + B μ ν m n 11 ( k i , r r q ) N μ ν ( 1 ) ( k i , r r q ) ] + b μ ν [ B μ ν m n 11 ( k i , r r q ) M μ ν ( 1 ) ( k i , r r q ) + A μ ν m n 11 ( k i , r r q ) N μ ν ( 1 ) ( k i , r r q ) ] + p q μ ν m n w m n p T ν M p [ A μ ν m n 31 ( k i , r r q ) M μ ν ( 1 ) ( k i , r r q ) + B μ ν m n 31 ( k i , r r q ) N μ ν ( 1 ) ( k i , r r q ) ] + v m n p T ν M p [ B μ ν m n 31 ( k i , r r q ) M μ ν ( 1 ) ( k i , r r q ) + A μ ν m n 31 ( k i , r r q ) N μ ν ( 1 ) ( k i , r r q ) ] ,
w μ ν q = μ ν a ~ μ ν + μ ν p = 1 p q [ w μ ν p T ν M q A μ ν m n ( 31 ) + v μ ν p T ν N q B μ ν m n ( 31 ) ] ,
v μ ν q = μ ν b ~ μ ν + μ ν p = 1 p q [ w μ ν p T ν M q B μ ν m n ( 31 ) + v μ ν p T ν N q A μ ν m n ( 31 ) ] ,
a ~ μ ν = a μ ν A ν μ m n ( 11 ) + b μ ν B ν μ m n ( 11 ) , b ~ μ ν = a μ ν B ν μ m n ( 11 ) + b μ ν A ν μ m n ( 11 ) .
E s q ( k r ) = j = 1 L n = 1 + m = n + n [ e m n j M m n ( 3 ) ( k , r r j ) + f m n j N m n ( 3 ) ( k , r r j ) ] ,
( a m n j b m n j ) = [ T M 0 0 T N ] ( w m n j v m n j ) .
ρ = cos φ x + sin φ y ,
φ = sin φ x + cos φ y ,
r = sin ϑ cos φ x + sin ϑ sin φ y + cos ϑ z ,
ϑ = cos ϑ cos φ x + cos ϑ sin φ y + sin ϑ z .
k i = k i k i = k i ( sin ϑ cos φ x + sin ϑ sin φ y + cos ϑ z ) ,
E i ( r ) = m = + [ c m M m ( 1 ) ( r ) + d m N m ( 3 ) ( k i ρ , k i z , r ) ] .
E sc ( r ) = m = + [ c m M m ( 3 ) ( k e , r ) + d m N m ( 3 ) ( k e , r ) ] ,
E ic ( r ) = m = + [ e m M m ( 1 ) ( k c , r ) + f m N m ( 1 ) ( k c , r ) ] .
E ss ( r ) = n = 1 + m = n n [ g m n M m n ( 3 ) ( k s f r ) + l m n N m n ( 3 ) ( k s f r ) ] ,
E is ( r ) = n = 1 + m = n n [ p m n M m n ( 1 ) ( k s f r ) + q m n N m n ( 1 ) ( k s f r ) ] .
ρ i × ( E i + E sc E c y ) = 0 per ρ = a c ,
ρ i × [ × ( E i + E sc μ e E ic + E ss μ c ) ] = 0 per ρ = a c ,
r i × ( E ic + E ss E is ) = 0 per r = a s ,
r i × [ × ( E ic + E ss μ c E is μ s ) ] = 0 per r = a s .
M m n ( p ) ( r ) = i m n 1 2 k C [ τ m n ( cos α ) M m ( p ) ( r ) + π m n ( cos α ) N m ( p ) ] d α ,
N m n ( p ) ( r ) = i m n 1 2 k C [ π m n ( cos α ) M m ( p ) ( r ) + τ m n ( cos α ) N m ( p ) ] d α ,
M m ( p ) ( r ) = n = m + γ m n [ τ m n ( cos ϑ i ) M m ( p ) ( r ) + i π m n ( cos ϑ i ) N m ( p ) ] ,
N m ( p ) ( r ) = n = m + γ m n [ i π m n ( cos ϑ i ) M m ( p ) ( r ) + τ m n ( cos ϑ i ) N m ( p ) ] ,
γ m n = k i n m + 1 ( 2 n + 1 ) ( n m ) ! n ( n + 1 ) ( n + m ) ! sin ϑ i .
E ic ( r ) = m = 1 + n = m + γ m n { e m [ τ m n ( cos ϑ i ) M m ( 1 ) ( r ) + i π m n ( cos ϑ i ) N m ( 1 ) ] + f m [ i π m n ( cos ϑ i ) M m ( 1 ) ( r ) + τ m n ( cos ϑ i ) N m ( 1 ) ] } ,
E ss ( r ) = n = 1 + m = n n i m n 1 2 k g m n C [ τ m n ( cos α ) M m ( 3 ) ( r ) + π m n ( cos α ) N m ( 3 ) ] d α + l m n C [ π m n ( cos α ) M m ( 3 ) ( r ) + τ m n ( cos α ) N m ( 3 ) ] ,
b m n k e ρ 2 k e J m ( k e ρ a c ) d m k e ρ 2 k e H m ( 1 ) ( k e ρ a c ) + f m k c ρ 2 k c J m ( k c ρ a c ) + n = 1 + i m n 1 k c ρ 2 2 k c 2 H m ( 1 ) ( k c ρ a c ) [ e m Θ m n ( ϑ ic ) + f m n Φ m n ( ϑ ic ) ] = 0 ,
a m J m ( k e ρ a c ) b m m k z k e a c J m ( k e ρ a c ) c m H m ( 1 ) ( k e ρ a c ) + d m m k z k e a c H m ( 1 ) ( k e ρ a c ) + e m J m ( k c ρ a c ) + f m m k z k c a c J m ( k c ρ a c ) + n = 1 + i m n 1 2 k c { H m ( 1 ) ( k c ρ a c ) [ e m Ψ m n ( ϑ ic ) + i f m n Θ m n ( ϑ ic ) ] + m k z k c a c H m ( 1 ) ( k c ρ a c ) [ e m Θ m n ( ϑ ic ) + i f m n Φ m n ( ϑ ic ) ] = 0 ,
a m n k e ρ 2 k e J m ( k e ρ a c ) c m k e ρ 2 k e H m ( 1 ) ( k e ρ a c ) + ζ 12 e m k c ρ 2 k c J m ( k c ρ a c ) + ζ 12 n = 1 + i m n 1 k c ρ 2 2 k c 2 H m ( 1 ) ( k c ρ a c ) [ e m Ψ m n ( ϑ ic ) + i f m n Θ m n ( ϑ ic ) ] = 0 ,
b m J m ( k e ρ a c ) a m m k z k e a c J m ( k e ρ a c ) d m H m ( 1 ) ( k e ρ a c ) + c m m k z k e a c H m ( 1 ) ( k e ρ a c ) + ζ 12 f m J m ( k c ρ a c ) + ζ 12 e m m k z k c a c J m ( k c ρ a c ) + ζ 12 n = 1 + i m n 1 2 k c { H m ( 1 ) ( k c ρ a c ) [ e m Θ m n ( ϑ ic ) + f m n Φ m n ( ϑ ic ) ] + m k z k c a c H m ( 1 ) ( k c ρ a c ) [ e m Ψ m n ( ϑ ic ) + i f m n Θ m n ( ϑ ic ) ] = 0 ,
Θ m n ( ϑ ic ) = π m n ( cos ϑ ic ) τ m n ( cos ϑ ic ) ( G m n + i L m n ) ,
Θ m n ( ϑ ic ) = π m n ( cos ϑ ic ) τ m n ( cos ϑ ic ) ( G m n i L m n ) ,
Φ m n ( ϑ ic ) = i G m n π m n 2 ( cos ϑ ic ) + L m n τ m n 2 ( cos ϑ ic ) ,
Ψ m n ( ϑ ic ) = G m n τ m n 2 ( cos ϑ ic ) + i L m n π m n 2 ( cos ϑ ic ) ,
G m n = γ m n ζ c s j n ( k s a s ) j n ( k c a s ) j n ( k s a s ) j n ( k c a s ) j n ( k s a s ) h n ( 1 ) ( k c a s ) ζ c s j n ( k s a s ) h n ( 1 ) ( k c a s ) ,
L m n = γ m n ζ c s j n ( k s a s ) j n ( k c a s ) j n ( k s a s ) j n ( k c a s ) j n ( k s a s ) h n ( 1 ) ( k c a s ) ζ c s j n ( k s a s ) h n ( 1 ) ( k c a s ) ,
P m n = γ m n j n ( k s a s ) ) h n ( 1 ) ( k c a s ) j n ( k s a s ) ) h n ( 1 ) ( k c a s ) j n ( k s a s ) h n ( 1 ) ( k c a s ) ζ c s j n ( k s a s ) h n ( 1 ) ( k c a s ) ,
Q m n = γ m n j n ( k s a s ) h n ( 1 ) ( k c a s ) j n ( k s a s ) h n ( 1 ) ( k c a s ) j n ( k s a s ) h n ( 1 ) ( k c a s ) ζ c s j n ( k s a s ) h n ( 1 ) ( k c a s ) .
E sc ( r ) = n = 1 + m = n + n [ c m n M m ( 3 ) ( k i ρ , k i z , r ) + d m n N m ( 3 ) ( k i ρ , k i z , r ) ] .