Abstract

Although scattering of light by a coated sphere is much more complicated than scattering by a homogeneous sphere, each of the partial wave amplitudes for scattering of a plane wave by a coated sphere can be expanded in a Debye series. The Debye series can then be rearranged in terms of the various reflections that each partial wave undergoes inside the coated sphere. For a given number of internal reflections, it is found that many different Debye terms produce the same scattered intensity as a function of scattering angle. This is called path degeneracy. In addition, some of the ray trajectories are repeats of those occurring for a smaller number of internal reflections in the sense that they produce identical time delays as a function of scattering angle. These repeated paths, however, have a different intensity as a function of scattering angle than their predecessors. The degenerate paths and repeated paths considerably simplify the interpretation of scattering within the coated sphere, thus making it possible to catalog the contributions of the various paths.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
    [CrossRef]
  5. H. Hattori, H. Kakui, H. Kurniawan, and K. Kagawa, “Liquid refractometry by the rainbow method,” Appl. Opt. 37, 4123–4129 (1998).
    [CrossRef]
  6. C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
    [CrossRef]
  7. A. B. Pluchino, “Surface waves and the radiative properties of micron-sized particles,” Appl. Opt. 20, 2986–2992 (1981).
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    [CrossRef]
  10. J. A. Lock, “Interference enhancement of the internal fields at structural scattering resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
    [CrossRef]
  11. T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a coated sphere: investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33, 7789–7797 (1994).
    [CrossRef]
  12. T. M. Bambino and L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
    [CrossRef]
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  20. F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
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  21. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
    [CrossRef]
  22. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef]
  23. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef]
  24. L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef]
  25. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. III. Finely stratified sphere model,” J. Opt. Soc. Am. A 25, 2991–3000 (2008).
    [CrossRef]
  26. Y. Takano and K.-N. Liou, “Phase matrix for light scattering by concentrically stratified spheres: comparison of geometric optics and the “exact” theory,” Appl. Opt. 49, 3990–3996(2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. J. A. Lock and P. Laven, “Mie scattering in the time domain. Part 1. The role of surface waves,” J. Opt. Soc. Am. A 28, 1086–1095 (2011).
    [CrossRef]
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    [CrossRef]
  36. H. C. van de Hulst, “Solution of coefficients from boundary conditions” in Light Scattering by Small Particles (Dover, 1981), pp. 121–124.
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  38. J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594–5603 (2005).
    [CrossRef]
  39. F. Xu and J. A. Lock, “Debye series for light scattering by a coated nonspherical particle,” Phys. Rev. A 81 (063812), 1–16 (2010).
    [CrossRef]
  40. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  41. R. D. Mattuck, “The two-particle propagator and the particle-hole propagator,” in A Guide to Feynman Diagrams in the Many-Body Problem, 2nd. ed. (Dover, 1992), p 22.
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    [CrossRef]
  47. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980–2990 (2008).
    [CrossRef]
  48. R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260–1270 (2006).
    [CrossRef]
  49. R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt. 45, 6255–6262 (2006).
    [CrossRef]

2012 (1)

P. Laven and J. A. Lock, “Understanding scattering by a coated sphere. Part 2: Time domain analysis,” J. Opt. Soc. Am A 29, 1498–1507 (2012).
[CrossRef]

2011 (3)

2010 (2)

2008 (2)

2007 (1)

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

2006 (2)

2005 (1)

2003 (2)

2002 (1)

2001 (1)

1998 (1)

1997 (1)

1996 (1)

T. M. Bambino and L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

1995 (1)

1994 (4)

1993 (2)

T. Kaiser and G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
[CrossRef]

1990 (2)

1988 (2)

1986 (1)

1985 (1)

1984 (2)

1981 (2)

1976 (1)

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

1962 (1)

1951 (1)

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

1937 (1)

B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting earth, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

Ackerman, T. P.

Aden, A. L.

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

Adler, C. L.

Altenkirch, R. A.

Bambino, T. M.

Barber, P. W.

Barbosa, V. C.

Bhandari, R.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, “Coated sphere” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 483–489.

Breitschaft, A. M. S.

Bremmer, H.

B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting earth, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

Fuller, K. A.

Gouesbet, G.

Greenler, R.

R. Greenler, “The fata morgana” in Rainbows, Halos, and Glories (Cambridge University, 1980), pp. 165–169.

Gréhan, G.

Guimarães, L. G.

Guo, L. X.

Han, X.

Hattori, H.

Hickok, W.

Hightower, R. L.

Hill, S. C.

Hood, D. A.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, “Coated sphere” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 483–489.

Ishimaru, A.

Jamison, J. M.

Jiang, H.

Kagawa, K.

Kai, L.

Kaiser, T.

T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a coated sphere: investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33, 7789–7797 (1994).
[CrossRef]

T. Kaiser and G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

Kakui, H.

Kattawar, G. W.

Kerker, M.

Khaled, E. E. M.

Khare, V.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph. D. dissertation (University of Rochester, 1975).

Kratohvil, J. P.

Kuga, Y.

Kurniawan, H.

Lange, S.

Laven, P.

Li, R.

Lin, C.-Y.

Liou, K.-N.

Liu, L.

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Lock, J. A.

P. Laven and J. A. Lock, “Understanding scattering by a coated sphere. Part 2: Time domain analysis,” J. Opt. Soc. Am A 29, 1498–1507 (2012).
[CrossRef]

J. A. Lock and P. Laven, “Mie scattering in the time domain. Part II.The role of diffraction,” J. Opt. Soc. Am. A 28, 1096–1106 (2011).
[CrossRef]

J. A. Lock and P. Laven, “Mie scattering in the time domain. Part 1. The role of surface waves,” J. Opt. Soc. Am. A 28, 1086–1095 (2011).
[CrossRef]

F. Xu and J. A. Lock, “Debye series for light scattering by a coated nonspherical particle,” Phys. Rev. A 81 (063812), 1–16 (2010).
[CrossRef]

J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. III. Finely stratified sphere model,” J. Opt. Soc. Am. A 25, 2991–3000 (2008).
[CrossRef]

J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980–2990 (2008).
[CrossRef]

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594–5603 (2005).
[CrossRef]

C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
[CrossRef]

C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
[CrossRef]

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

J. A. Lock, “Interference enhancement of the internal fields at structural scattering resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
[CrossRef]

Mackowski, D. W.

Massoli, P.

Matijevic, E.

Mattuck, R. D.

R. D. Mattuck, “The two-particle propagator and the particle-hole propagator,” in A Guide to Feynman Diagrams in the Many-Body Problem, 2nd. ed. (Dover, 1992), p 22.

Menguc, M. P.

Nash, J. K.

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Onofri, F.

Pluchino, A. B.

Rafferty, I. P.

Ren, K. F.

Richardson, C. B.

Saunders, K. W.

Schweiger, G.

T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a coated sphere: investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33, 7789–7797 (1994).
[CrossRef]

T. Kaiser and G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

Shen, J.

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Smith, D. D.

Takano, Y.

Toon, O. B.

Tsang, L.

Twersky, V.

V. Twersky, “On propagation in random media of discrete scatterers,” in Proceedings of Symposia in Applied Mathematics, vol. XVI, Stochastic Processes in Mathematical Physics and Engineering, R. Bellman, ed. (Am. Math. Soc., 1964), pp. 84–116.

van de Hulst, H. C.

H. C. van de Hulst, “Amplitude functions” in Light Scattering by Small Particles (Dover, 1981), pp. 124–126.

H. C. van de Hulst, “The localization principle” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

H. C. van de Hulst, “Solution of coefficients from boundary conditions” in Light Scattering by Small Particles (Dover, 1981), pp. 121–124.

Van der Pol, B.

B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting earth, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

Wang, H.

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Wu, Z. S.

Xu, F.

F. Xu and J. A. Lock, “Debye series for light scattering by a coated nonspherical particle,” Phys. Rev. A 81 (063812), 1–16 (2010).
[CrossRef]

Xu, Y.

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Yu, B.

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Appl. Opt. (21)

G. W. Kattawar and D. A. Hood, “Electromagnetic scattering from a spherical polydispersion of coated spheres,” Appl. Opt. 15, 1996–1999 (1976).
[CrossRef]

A. B. Pluchino, “Surface waves and the radiative properties of micron-sized particles,” Appl. Opt. 20, 2986–2992 (1981).
[CrossRef]

O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
[CrossRef]

R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
[CrossRef]

R. L. Hightower and C. B. Richardson, “Resonant Mie scattering from a layered sphere,” Appl. Opt. 27, 4850–4855 (1988).
[CrossRef]

J. A. Lock, “Interference enhancement of the internal fields at structural scattering resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
[CrossRef]

L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef]

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Light scattering by a coated sphere illuminated with a Gaussian beam,” Appl. Opt. 33, 3308–3314 (1994).
[CrossRef]

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

T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a coated sphere: investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33, 7789–7797 (1994).
[CrossRef]

F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef]

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

H. Hattori, H. Kakui, H. Kurniawan, and K. Kagawa, “Liquid refractometry by the rainbow method,” Appl. Opt. 37, 4123–4129 (1998).
[CrossRef]

C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
[CrossRef]

C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
[CrossRef]

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594–5603 (2005).
[CrossRef]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260–1270 (2006).
[CrossRef]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt. 45, 6255–6262 (2006).
[CrossRef]

Y. Takano and K.-N. Liou, “Phase matrix for light scattering by concentrically stratified spheres: comparison of geometric optics and the “exact” theory,” Appl. Opt. 49, 3990–3996(2010).
[CrossRef]

P. Laven, “Time domain analysis of scattering by a water droplet,” Appl. Opt. 50, F29–F38 (2011).
[CrossRef]

China Particuol. (1)

L. Liu, H. Wang, B. Yu, Y. Xu, and J. Shen, “Improved algorithm of light scattering by a coated sphere,” China Particuol. 5, 230–236 (2007).
[CrossRef]

Comput. Phys. (1)

T. Kaiser and G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

J. Appl. Phys. (1)

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

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. Opt. Soc. Am A (1)

P. Laven and J. A. Lock, “Understanding scattering by a coated sphere. Part 2: Time domain analysis,” J. Opt. Soc. Am A 29, 1498–1507 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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

Opt. Lett. (2)

Philos. Mag. (1)

B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting earth, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

Phys. Rev. A (1)

F. Xu and J. A. Lock, “Debye series for light scattering by a coated nonspherical particle,” Phys. Rev. A 81 (063812), 1–16 (2010).
[CrossRef]

Phys. Rev. E (1)

T. M. Bambino and L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

Other (8)

C. F. Bohren and D. R. Huffman, “Coated sphere” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 483–489.

V. Twersky, “On propagation in random media of discrete scatterers,” in Proceedings of Symposia in Applied Mathematics, vol. XVI, Stochastic Processes in Mathematical Physics and Engineering, R. Bellman, ed. (Am. Math. Soc., 1964), pp. 84–116.

H. C. van de Hulst, “The localization principle” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

H. C. van de Hulst, “Solution of coefficients from boundary conditions” in Light Scattering by Small Particles (Dover, 1981), pp. 121–124.

H. C. van de Hulst, “Amplitude functions” in Light Scattering by Small Particles (Dover, 1981), pp. 124–126.

R. D. Mattuck, “The two-particle propagator and the particle-hole propagator,” in A Guide to Feynman Diagrams in the Many-Body Problem, 2nd. ed. (Dover, 1992), p 22.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph. D. dissertation (University of Rochester, 1975).

R. Greenler, “The fata morgana” in Rainbows, Halos, and Glories (Cambridge University, 1980), pp. 165–169.

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Figures (5)

Fig. 1.
Fig. 1.

Ray path A passes through the core, whereas ray B misses the core.

Fig. 2.
Fig. 2.

Ray paths for N3 internal reflections showing the (N, A, B) values, where A is the number of chords in the coating and B is the number of chords in the core.

Fig. 3.
Fig. 3.

Two degenerate (2, 4, 1) ray paths.

Fig. 4.
Fig. 4.

Ray path (1, 4, 2) is shown in green and the repeated path (3, 4, 2) is shown in blue.

Fig. 5.
Fig. 5.

Transverse-electric polarized scattering from polydisperse coated spheres with median value of xe=2πa23/λ=600, variance ve=1/9, a12/a23=0.8, m1=1.5 and m2=1.33 (as in Fig. 2 of [26]). Figure 5(a) compares the results of Aden–Kerker calculations (shown in red) with the sum of the 16 Debye terms for N3: the blue curve has been calculated using the degeneracy factors D listed in Table 1, while the green curve has been calculated assuming that D=1. The scattering contributions from individual terms of the Debye series terms in Fig. 5(b) take account of the fact that D=2 for the (2, 4, 1) and (2, 4, 3) terms, whereas the results in Fig. 5(c) incorrectly assume that D=1 for all of the terms.

Tables (1)

Tables Icon

Table 1. Debye Series Terms (N, A, B) for N7 Internal Reflections Showing the Number of Internal Reflections N232, N212 and N121, and the Degeneracy Factor Da

Equations (42)

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Nn12=αψn(m2ka12)ψn(m1ka12)βψn(m2ka12)ψn(m1ka12)
Dn12=αχn(m2ka12)ψn(m1ka12)βχn(m2ka12)ψn(m1ka12)
Nn23=γψn(m3ka23)ψn(m2ka23)δψn(m3ka23)ψn(m2ka23)
Dn23=γχn(m3ka23)ψn(m2ka23)δχn(m3ka23)ψn(m2ka23)
Pn23=γψn(m3ka23)χn(m2ka23)δψn(m3ka23)χn(m2ka23)
Qn23=γχn(m3ka23)χn(m2ka23)δχn(m3ka23)χn(m2ka23),
an,bn=(Dn12Nn23Nn12Pn23)/[(Dn12Nn23Nn12Pn23)+i(Dn12Dn23Nn12Qn23)].
I(r,θ)=[E02/(2μ0ck2r2)][|S1(θ)|2+|S2(θ)|2],
S1(θ)=n=1{(2n+1)/[n(n+1)]}[anπn(θ)+bnτn(θ)]
S2(θ)=n=1{(2n+1)/[n(n+1)]}[anτn(θ)+bnπn(θ)],
Pn12=αψn(m2ka12)χn(m1ka12)βψn(m2ka12)χn(m1ka12)
Qn12=αχn(m2ka12)χn(m1ka12)βχn(m2ka12)χn(m1ka12),
Rn323=(Nn23+Qn23+iDn23+iPn23)/(Nn23+Qn23+iDn23iPn23)
Rn232=(Nn23+Qn23iDn23iPn23)/(Nn23+Qn23+iDn23iPn23)
Tn32=2im3/(Nn23+Qn23+iDn23iPn23)
Tn23=2im2/(Nn23+Qn23+iDn23iPn23)
Rn212=(Nn12+Qn12+iDn12+iPn12)/(Nn12+Qn12+iDn12iPn12)
Rn121=(Nn12+Qn12iDn12iPn12)/(Nn12+Qn12+iDn12iPn12)
Tn21=2im2/(Nn12+Qn12+iDn12iPn12)
Tn12=2im1/(Nn12+Qn12+iDn12iPn12).
an,bn=[1Rn323Tn32WnTn23/(1WnRn232)]/2=[1Rn323q=1Tn23(WnRn232)q1WnTn23]/2,
Wn=Rn212+Tn21Tn12/(1Rn121)=Rn212+p=1Tn21(Rn121)p1Tn12.
N212+N121+N232=N.
(an,bn)N=0=(1Rn323Tn32Tn21Tn12Tn23)/2.
(an,bn)N=1=Tn32(Rn212+Tn21Rn121Tn12+Tn21Tn12Rn232Tn21Tn12)Tn23/2.
(an,bn)N=2=Tn32(Tn21Rn121Rn121Tn12+Rn212Rn232Tn21Tn12+Tn21Tn12Rn232Rn212+Tn21Tn12Rn232Tn21Rn121Tn12+Tn21Rn121Tn12Rn232Tn21Tn12+Tn21Tn12Rn232Tn21Tn12Rn232Tn21Tn12)Tn23/2.
m3sin(θ3i)=m2sin(θ2t)
m2sin(θ2i)=m1sin(θ1t),
a23sin(θ2t)=a12sin(θ2i).
N232=(A2)/2,
N121=(N+1A+B)/2,
N212=(N+1B)/2,
A=2N232+2,
B=N+12N212.
N232=Nj,N121=j,N212=0
N232=Nj,N121=0,N212=j
an,bn=(1Rn323N=0Tn32FnNTn23)/2,
FnN=N232=0NN212=0NmaxD(N,N232,N212)(Tn21Tn12)N232+1N212×(Rn232)N232(Rn212)N212(Rn121)N121+GnN,
GnN=(Rn232)(N1)/2(Rn212)(N+1)/2
D(N,N232,N212)=(N232+1)!(N2N212)!/[(N212)!(N232+1N212)!(NN232N212)!(N232N212)!].
an,bn=(1Rn212N=0Tn21FnNTn12)/2,
FnN=D(N)(Rn121)N

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