Abstract

We examine the behavior of the first-order rainbow for a coated sphere by using both ray theory and Aden–Kerker wave theory as the radius of the core a12 and the thickness of the coating δ are varied. As the ratio δ/a12 increases from 10−4 to 0.33, we find three classes of rainbow phenomena that cannot occur for a homogeneous-sphere rainbow. For δ/a12 ≲ 10−3, the rainbow intensity is an oscillatory function of the coating thickness, for δ/a12 ≈ 10−2, the first-order rainbow breaks into a pair of twin rainbows, and for δ/a12 ≈ 0.33, various rainbow-extinction transitions occur. Each of these effects is analyzed, and their physical interpretations are given. A Debye series decomposition of coated-sphere partial-wave scattering amplitudes is also performed and aids in the analysis.

© 1994 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 13.2
  2. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  3. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  4. V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
    [CrossRef]
  5. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  6. D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
    [CrossRef] [PubMed]
  7. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  8. B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).
  9. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  10. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A5, 2032–2044 (1988).
    [CrossRef]
  11. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  12. G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  13. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  14. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 5.1.
  15. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 8.1, App. B.
  16. Ref. 1, Secs. 12.34 and 13.24.
  17. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3.
  18. Ref. 17, Secs. 9.41 and 9.6.
  19. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11–160) and the paragraph following it.
  20. K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
    [CrossRef] [PubMed]
  21. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.
  22. R. L. Hightower, C. B. Richardson, “Resonant Mie scattering from a layered sphere,” Appl. Opt. 27, 4850–4855 (1988).
    [CrossRef] [PubMed]
  23. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Eqs. (6.40) and (6.47).
    [CrossRef]
  24. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
  25. Ref. 23, pp. 131-132, Fig. 3a.
  26. N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
    [CrossRef] [PubMed]
  27. P. L. Marston, “Colors observed when sunlight is scattered by bubble clouds in seawater,” Appl. Opt. 30, 3479–3484, 3549 (1991).
    [CrossRef] [PubMed]
  28. J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
    [CrossRef]
  29. D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970), claims that this interpretation of counterpropagating surface waves is incorrect. Numerical-Debye-series computations performed by us for p = 1 for a bubble in a homogeneous medium, however, bear out Nussenzveig's interpretation of the situation. See also Ref. 26, p. 1019.
    [CrossRef]
  30. J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
    [CrossRef]
  31. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 44–46.
  32. J. A. Lock, “Theory of the observation made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  33. H. C. van de Hulst, R. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  34. Ref. 17, Figs. 4.55 and 4.60.
  35. D. S. Falk, D. R. Brill, D. G. Stork, Seeing the Light (Harper & Row, Cambridge, 1986), Figs. 2.42–2.43.
  36. C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), Chap. 5.
  37. J. Burke, The Day the Universe Changed (Little, Brown, Boston, Mass., 1985), pp. 52–53.

1993 (2)

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

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

1992 (1)

1991 (4)

D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
[CrossRef] [PubMed]

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

P. L. Marston, “Colors observed when sunlight is scattered by bubble clouds in seawater,” Appl. Opt. 30, 3479–3484, 3549 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, R. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
[CrossRef] [PubMed]

1988 (4)

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

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A5, 2032–2044 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (1)

1980 (1)

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

1979 (1)

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

1977 (2)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1970 (1)

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970), claims that this interpretation of counterpropagating surface waves is incorrect. Numerical-Debye-series computations performed by us for p = 1 for a bubble in a homogeneous medium, however, bear out Nussenzveig's interpretation of the situation. See also Ref. 26, p. 1019.
[CrossRef]

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Eqs. (6.40) and (6.47).
[CrossRef]

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

1951 (1)

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

1937 (1)

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Aden, A. L.

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

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11–160) and the paragraph following it.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 8.1, App. B.

Boyer, C. B.

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), Chap. 5.

Bremmer, H.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Brill, D. R.

D. S. Falk, D. R. Brill, D. G. Stork, Seeing the Light (Harper & Row, Cambridge, 1986), Figs. 2.42–2.43.

Burke, J.

J. Burke, The Day the Universe Changed (Little, Brown, Boston, Mass., 1985), pp. 52–53.

Falk, D. S.

D. S. Falk, D. R. Brill, D. G. Stork, Seeing the Light (Harper & Row, Cambridge, 1986), Figs. 2.42–2.43.

Fiedler-Ferrari, N.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Fuller, K. A.

Gouesbet, G.

Grehan, G.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3.

Hightower, R. L.

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 8.1, App. B.

Jamison, J. M.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

Kerker, M.

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 5.1.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

Langley, D. S.

Lin, C.-Y.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

Lock, J. A.

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
[CrossRef]

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A5, 2032–2044 (1988).
[CrossRef]

J. A. Lock, “Theory of the observation made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

Ludwig, D.

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970), claims that this interpretation of counterpropagating surface waves is incorrect. Numerical-Debye-series computations performed by us for p = 1 for a bubble in a homogeneous medium, however, bear out Nussenzveig's interpretation of the situation. See also Ref. 26, p. 1019.
[CrossRef]

Maheu, B.

Marston, P. L.

P. L. Marston, “Colors observed when sunlight is scattered by bubble clouds in seawater,” Appl. Opt. 30, 3479–3484, 3549 (1991).
[CrossRef] [PubMed]

Morrell, M. J.

Nussenzveig, H. M.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

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

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Eqs. (6.40) and (6.47).
[CrossRef]

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

Richardson, C. B.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Stork, D. G.

D. S. Falk, D. R. Brill, D. G. Stork, Seeing the Light (Harper & Row, Cambridge, 1986), Figs. 2.42–2.43.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 44–46.

van de Hulst, H. C.

H. C. van de Hulst, R. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 13.2

van der Pol, B.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Walker, J. D.

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R.

Wiscombe, W. J.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Am. J. Phys. (2)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

Appl. Opt. (5)

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

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

J. Math. Phys. (3)

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

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Eqs. (6.40) and (6.47).
[CrossRef]

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970), claims that this interpretation of counterpropagating surface waves is incorrect. Numerical-Debye-series computations performed by us for p = 1 for a bubble in a homogeneous medium, however, bear out Nussenzveig's interpretation of the situation. See also Ref. 26, p. 1019.
[CrossRef]

J. Opt. Soc. Am. (2)

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A5, 2032–2044 (1988).
[CrossRef]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Philos. Mag. (1)

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Phys. Rev. A (1)

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Sci. Am. (2)

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

Other (15)

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 44–46.

Ref. 17, Figs. 4.55 and 4.60.

D. S. Falk, D. R. Brill, D. G. Stork, Seeing the Light (Harper & Row, Cambridge, 1986), Figs. 2.42–2.43.

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), Chap. 5.

J. Burke, The Day the Universe Changed (Little, Brown, Boston, Mass., 1985), pp. 52–53.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

V. Khare, H. M. Nussenzveig, “The theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1977), pp. 723–764.
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 13.2

Ref. 23, pp. 131-132, Fig. 3a.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 5.1.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 8.1, App. B.

Ref. 1, Secs. 12.34 and 13.24.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3.

Ref. 17, Secs. 9.41 and 9.6.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11–160) and the paragraph following it.

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Figures (13)

Fig. 1
Fig. 1

(a) Geometry of a coated sphere and the trajectory of a geometric light ray through it. (b) Two geometric rays α and β that dominate the first-order rainbow for a coated sphere. The flat interface Fresnel coefficients for the various transmissions and internal reflections are also indicated.

Fig. 2
Fig. 2

(a) When the core is large and the coating is thin, the core–coating–air interfaces are approximated by the thin-film geometry. (b) Comparison between Aden–Kerker wave theory (solid curve) and ray theory employing the α and the β rays (dashed curve) for the rainbow intensity as a function of coating thickness. The Aden–Kerker intensity was calculated for 2πδ/λ in increments of 0.1. The arrows denote the coating thicknesses for constructive and destructive interferences in ray theory as predicted by Eqs. (20) and (21), respectively. (c) Ray γ that makes three internal reflections within the coating. (d) Comparison between Aden–Kerker wave theory (solid curve) and ray theory when the α, β, γ, γ′, and γ″ rays (dashed curve) are used for the rainbow intensity as a function of coating thickness.

Fig. 3
Fig. 3

(a) Comparison between Aden–Kerker wave theory (data points) and ray theory (solid lines) for the shift in the angular positions of the twin first-order rainbows as a function of coating thickness for the α and the β rainbows of Fig. 1(b). (b) Scattered intensity for unpolarized incident light as a function of θ for x12 = 5000, (2πδ)/λ = 175, n1 = 1.333, n2 = 1.2, and n3 = 1.0, showing the twin first-order rainbows α and β of Fig. 1(b).

Fig. 4
Fig. 4

Ray trajectories with up to four interactions with the interfaces that contribute to geometric ray scattering by a coated sphere.

Fig. 5
Fig. 5

(a) Debye-series expansion of the core scattering amplitude Ql of Eq. (38). (b) Debye-series expansion of the coated-sphere partial-wave-scattering amplitudes al and bl of Eq. (37). The star symbol represents the infinite series of successive interactions of the partial wave with the coating–core interface beafore it heads back into the coating.

Fig. 6
Fig. 6

(a) Debye-series expansion of the partial-wave interior function Il of Eq. (43). (b) Debye-series expansion of the core partial-wave amplitudes cl and dl of Eq. (42). The filled square represents the infinite series of successive interactions of the partial wave with the coating–core interface before it finally ends up in the core.

Fig. 7
Fig. 7

(a) Debye-series expansion of the partial-wave coating incoming amplitudes Klin and Llin of Eq. (44). (b) Debye-series expansion of the partial-wave coating outgoing amplitudes Klout and Llout of Eq. (45).

Fig. 8
Fig. 8

Ray trajectories corresponding to the Debye-series term Tl32Rl212Rl232Rl212Tl23 for (a) a small core that permits rainbow formation, (b) a core that begins to obstruct the Descartes rainbow ray, causing a rainbow extinction transition, (c) a large core for which no rainbow can occur. The dashed lines in (a) indicate the partial-wave reflection coefficient Rl212 → 1.

Fig. 9
Fig. 9

Scattered intensity |S1(θ)|2 of Eq. (50) as a function of θ for x23 = 900, n1 = 1.333, n2 = 1.2, n3 = 1.0, and (a)x12 = 600 where rainbow formation occurs, (b) x12 = 700 near the rainbow extinction transition, (c) x12 = 750 where rainbow formation does not occur.

Fig. 10
Fig. 10

Ray trajectories that correspond to the Debye-series term Tl32Tl21Rl11Tl12Tl23 for (a) particlelike scattering for a large core, which permits rainbow formation, (b) the transition between particlelike and bubblelike scattering where rainbow extinction occurs, (c) bubblelike scattering for a small core where no rainbow can occur.

Fig. 11
Fig. 11

Scattering angle θ as a function of the angle of incidence θ3 for the ray trajectories of Fig. 10. The curves labeled a, b, and c correspond to the situations of Figs. 10(a), 10(b), and 10(c), respectively. Point R denotes the rainbow, and point S denotes total external reflection. The two curves labeled a correspond to two different coating thicknesses in the particlelike scattering regime.

Fig. 12
Fig. 12

Scattered intensity |S1(θ)|2 of Eq. (54) as a function of θ for x23 = 900, n1 = 1.33, n2 = 1.5, n3 = 1.0 and (a) x12 = 850 in the particle-like scattering regime, (b) x12 = 712.5 near the rainbow extinction transition, and (c) x12 = 630 in the bubblelike scattering regime.

Fig. 13
Fig. 13

Magnitude squared of the Fourier transform of the Aden–Kerker scattering amplitude S1(θ) of Eq. (6) as a function of spatial frequency for x12 = 10,000, (2πδ)/λ = 300, = 1.333, and n2 = 1.5. The scattering amplitude was computed over a 3° interval centered on θ = 137°. The center of the coated sphere corresponds to p = 0, and the outer edge corresponds to p = 174.53 deg−1. The β peaks are due to the β supernumerary rays of Fig. 1(b), and the α peak is due the α rays of Fig. 1(b).

Equations (54)

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a 23 = a 12 + δ .
x 12 = 2 π a 12 λ ,
x 23 = 2 π a 23 λ ,
I ( r , θ , ϕ ) = E 0 2 2 μ 0 c 1 k 2 r 2 [ | S 2 ( θ ) | 2 cos 2 ϕ + | S 1 ( θ ) | 2 sin 2 ϕ ] ,
k = 2 π λ
S 1 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l π l ( θ ) + b l τ l ( θ ) ] , S 2 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l τ l ( θ ) + b l π l ( θ ) ] .
π l ( θ ) = 1 sin θ P l 1 ( cos θ ) , τ l ( θ ) = d d θ P l 1 ( cos θ ) .
a l b l } = N l 23 N l 23 + i D l 23 ,
N l 23 = n 2 x 23 2 [ j l ( x 23 ) j l 1 ( n 2 x 23 ) n 2 j l 1 ( x 23 ) j l ( n 2 x 23 ) + ( n 2 2 1 ) l n 2 x 23 j l ( x 23 ) j l ( n 2 x 23 ) ]
+ A l n 2 x 23 2 [ j l ( x 23 ) n l 1 ( n 2 x 23 ) n 2 j l 1 ( x 23 ) n l ( n 2 x 23 ) + ( n 2 2 1 ) l n 2 x 23 j l ( x 23 ) n l ( n 2 x 23 ) ] ,
D l 23 = n 2 x 23 2 [ n l ( x 23 ) j l 1 ( n 2 x 23 ) n 2 n l 1 ( x 23 ) j l ( n 2 x 23 ) + ( n 2 2 1 ) l n 2 x 23 n l ( x 23 ) j l ( n 2 x 23 ) ] + A l n 2 x 23 2 [ n l ( x 23 ) n l 1 ( n 2 x 23 ) n 2 n l 1 ( x 23 ) × n l ( n 2 x 23 ) + ( n 2 2 1 ) l n 2 x 23 n l ( x 23 ) n l ( n 2 x 23 ) ] , A l = N l 12 D l 12 ,
N l 12 = n 1 n 2 x 12 2 [ n 2 j l ( n 2 x 12 ) j l 1 ( n 1 x 12 ) n 1 j l 1 ( n 2 x 12 ) × j l ( n 1 x 12 ) + ( n 1 2 n 2 2 ) l n 1 n 2 x 12 j l ( n 2 x 12 ) j l ( n 1 x 12 ) ] ,
D l 12 = n 1 n 2 x 12 2 [ n 2 n l ( n 2 x 12 ) j l 1 ( n 1 x 12 ) n 1 n l 1 ( n 2 x 12 ) × j l ( n 1 x 12 ) + ( n 1 2 n 2 2 ) l n 1 n 2 x 12 n l ( n 2 x 12 ) j l ( n 1 x 12 ) ]
N l 23 = n 2 x 23 2 [ n 2 j l ( x 23 ) j l 1 ( n 2 x 23 ) j l 1 ( x 23 ) j l ( n 2 x 23 ) ] + B l n 2 x 23 2 [ n 2 j l ( x 23 ) n l 1 ( n 2 x 23 ) j l 1 ( x 23 ) n l ( n 2 x 23 ) ] ,
D l 23 = n 2 x 23 2 [ n 2 n l ( x 23 ) j l 1 ( n 2 x 23 ) n l 1 ( x 23 ) j l ( n 2 x 23 ) ] + B l n 2 x 23 2 [ n 2 n l ( x 23 ) n l 1 ( n 2 x 23 ) n l 1 ( x 23 ) n l ( n 2 x 23 ) ] ,
B l = N l 12 D l 12 ,
N l 12 = n 1 n 2 x 12 2 [ n 1 j l ( n 2 x 12 ) j l 1 ( n 1 x 12 ) n 2 j l 1 ( n 2 x 12 ) j l ( n 1 x 12 ) ] ,
D l 12 = n 1 n 2 x 12 2 [ n 1 n l ( n 2 x 12 ) j l 1 ( n 1 x 12 ) n 2 n l 1 ( n 2 x 12 ) j l ( n 1 x 12 ) ]
cos θ 3 R = ( n 1 2 1 3 ) 1 / 2 .
4 π n 2 δ λ cos θ 2 R + ϕ 12 + ϕ 23 = 2 π N ,
4 π n 2 δ λ cos θ 2 R + ϕ 12 + ϕ 23 = 2 π ( N + ½ ) ,
n 3 sin θ 3 R = n 2 sin θ 2 R ,
ϕ 12 = { 0 if n 2 < n 1 π if n 2 > n 1 , ϕ 23 = { 0 if n 3 < n 2 π if n 3 > n 2 .
E α = T g 32 T g 21 R g 11 T g 12 T g 23 = 0.0941
E β = T g 32 T g 21 T g 12 R g 232 T g 21 T g 12 T g 23 = 0.3425
I max = 1 2 μ 0 c ( E α + E β ) 2 = 0.1906 ,
I min = 1 2 μ 0 c ( E α E β ) 2 = 0.0617 ,
θ 2 = θ 2 + arccos [ a 23 a 12 sin 2 θ 2 + cos θ 2 ( 1 a 23 2 a 12 2 sin 2 θ 2 ) 1 / 2 ] θ 2 + δ a 12 sin θ 3 ( n 2 2 sin 2 θ 3 ) 1 / 2 + O ( δ 2 a 12 2 ) .
= θ 2 θ 2 .
θ α = π + 2 θ 3 4 θ 2 + 2 ,
θ β = π + 2 θ 3 4 θ 2 + 4 .
θ α R ( δ a 12 ) = θ R ( 0 ) + 2 δ a 12 [ ( 4 n 1 2 3 n 2 2 + n 1 2 4 ) 1 / 2 ( 4 n 1 2 n 1 2 1 ) 1 / 2 ] + O ( δ 2 a 12 2 ) ,
θ β R ( δ a 12 ) = θ R ( 0 ) + 2 δ a 12 [ 2 ( 4 n 1 2 3 n 2 2 + n 1 2 4 ) 1 / 2 ( 4 n 1 2 n 1 2 1 ) 1 / 2 ] + O ( δ 2 a 12 2 ) ,
a l b l } = 1 2 ( 1 R l 22 T l 21 T l 12 1 R l 11 ) = 1 2 ( 1 R l 22 T l 21 T l 12 T l 21 R l 11 T l 12 ) .
E g = D + R g 33 + T g 32 ( Q g + 1 ) T g 23 1 R g 232 ( Q g + 1 ) ,
Q g = R g 212 + T g 21 T g 12 1 R g 11 ,
a l b l } = 1 2 ( 1 R l 33 T l 32 Q l T l 23 1 Q l R l 232 ) ,
Q l = R l 212 + T l 21 T l 12 1 R l 11 .
ψ scattered TE = l = 1 i l 2 l + 1 l ( l + 1 ) ( b l ) h l ( 1 ) ( k r ) P l 1 ( cos θ ) sin ϕ , ψ scattered TM = l = 1 i l 2 l + 1 l ( l + 1 ) ( a l ) h l ( 1 ) ( k r ) P l 1 ( cos θ ) cos ϕ .
ψ core TE = l = 1 i l 2 l + 1 l ( l + 1 ) ( n 1 d l ) j l ( n 1 k r ) P l 1 ( cos θ ) sin ϕ , ψ core TM = l = 1 i l 2 l + 1 l ( l + 1 ) ( n 1 c l ) j l ( n 1 k r ) P l 1 ( cos θ ) sin ϕ ,
ψ coating TE = l = 1 i l 2 l + 1 l ( l + 1 ) n 2 [ L l out h l ( 1 ) ( n 2 k r ) + L l in h l ( 2 ) ( n 2 k r ) ] × P l 1 ( cos θ ) sin ϕ , ψ coating TM = l = 1 i l 2 l + 1 l ( l + 1 ) n 2 [ K l out h l ( 1 ) ( n 2 k r ) + K l in h l ( 2 ) ( n 2 k r ) ] × P l 1 ( cos θ ) cos ϕ ,
c l d l } = n 3 n 1 T l 32 I l 1 Q l R l 232 ,
I l = T l 21 1 R l 11 ,
K l in L l in } = n 3 2 n 2 T l 32 1 Q l R l 232 ,
K l out L l out } = n 3 2 n 2 T l 32 Q l 1 Q l R l 232 .
ψ coating TE = l = 1 i l 2 l + 1 l ( l + 1 ) n 2 [ V l inc j l ( n 2 k r ) V l scatt h l ( 1 ) ( n 2 k r ) ] × P l 1 ( cos θ ) sin ϕ , ψ coating TM = l = 1 i l 2 l + 1 l ( l + 1 ) n 2 [ U l inc j l ( n 2 k r ) U l scatt h l ( 1 ) ( n 2 k r ) ] × P l 1 ( cos θ ) cos ϕ ,
U l inc V l inc } = n 3 n 2 T l 32 1 Q l R l 232 ,
U l scatt = ½ U l inc ( 1 Q l ) , V l scatt = ½ V l inc ( 1 Q l ) ,
c l = n 2 n 1 U l inc I l , d l = n 2 n 1 V l inc I l ,
a l b l } = T l 32 R l 212 R l 232 R l 212 T l 23 .
a 12 = a 23 sin θ 3 R n 2 .
a 12 = 1 n 1 a 23 .
a 12 a 23 > 1 n 1 ,
a l b l } = T l 32 T l 21 R l 11 T l 12 T l 23 ,

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