Abstract

Mie theory can be used to generate full-color simulations of atmospheric glories, but it offers no explanation for the formation of glories. Simulations using the Debye series indicate that glories are caused by rays that have suffered one internal reflection within spherical droplets of water. In 1947, van de Hulst suggested that backscattering (i.e., scattering angle θ = 180°) could be caused by surface waves, which would generate a toroidal wavefront due to spherical symmetry. Furthermore, he postulated that the glory is the interference pattern corresponding to this toroidal wavefront. Although van de Hulst’s explanation for the glory has been widely accepted, the author offers a slightly different explanation. Noting that surface waves shed radiation continuously around the droplet (not just at θ = 180°), scattering in a specific direction θ = 180° − δ can be considered as the vector sum of two surface waves: one deflecting the incident light by 180° + δ and the other by 180° + δ. The author suggests that the glory is the result of two-ray interference between these two surface waves. Simple calculations indicate that this model produces more accurate results than van de Hulst’s model.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. P. Laven, “Atmospheric glories: simulations and observations,” Appl. Opt. 44, 5667–5674 (2005).
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  5. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).
  6. 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]
  7. W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, Cambridge, UK, 2001).
  8. R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
    [CrossRef]
  9. H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16 (1947).
    [CrossRef]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition).
  11. T. S. Fahlen, H. C. Bryant, “Direct observation of surface waves on droplets,” J. Opt. Soc. Am. 56, 1635–1636 (1966).
    [CrossRef]
  12. H. C. Bryant, A. J. Cox, “Mie theory and the glory,” J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [CrossRef]
  13. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  14. H. Inada, “New calculation of surface wave contributions associated with Mie backscattering,” Appl. Opt. 12, 1516–1523 (1973).
    [CrossRef] [PubMed]
  15. H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
    [CrossRef]
  16. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  17. 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).
    [CrossRef]
  18. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [CrossRef]
  19. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  20. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, UK, 1992).
    [CrossRef]
  21. H. M. Nussenzveig, “Light tunneling in clouds,” Appl. Opt. 42, 1588–1593 (2003).
    [CrossRef] [PubMed]
  22. D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).
  23. R. Greenler, Rainbows, Halos and Glories (Cambridge University, Cambridge, UK, 1980).
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  25. H. M. Nussenzveig, “Does the glory have a simple explanation?,” Opt. Lett. 27, 1379–1381 (2002).
    [CrossRef]
  26. V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 1976). N.B. This reference may not be readily available, but the calculation method is summarized in Ref. 6.
  27. J. A. Lock, “Role of the tunneling ray in near-critical-angle scattering by a dielectric sphere,” J. Opt. Soc. Am. A 20, 499–507 (2003).
    [CrossRef]
  28. Ref. 10, Section 12.22.

2005 (1)

2003 (4)

2002 (1)

1998 (1)

1992 (1)

1979 (1)

1977 (1)

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

1974 (1)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

1973 (1)

1969 (3)

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

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).
[CrossRef]

1966 (2)

1947 (1)

1908 (2)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

Bohren, C. F.

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

Bryant, H. C.

Cox, A. J.

Dave, J. V.

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

Fahlen, T. S.

Gedzelman, S. D.

Grandy, W. T.

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, Cambridge, UK, 2001).

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge University, Cambridge, UK, 1980).

Hovenac, E. A.

Huffman, D. R.

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

Inada, H.

Jarmie, N.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

Khare, V.

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

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 1976). N.B. This reference may not be readily available, but the calculation method is summarized in Ref. 6.

Laven, P.

Lee, R. L.

Livingston, W.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

Lock, J. A.

Lynch, D. K.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Light tunneling in clouds,” Appl. Opt. 42, 1588–1593 (2003).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Does the glory have a simple explanation?,” Opt. Lett. 27, 1379–1381 (2002).
[CrossRef]

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

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, UK, 1992).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16 (1947).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition).

Ann. Phys. Leipzig (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (7)

J. Math. Phys. (2)

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).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Lett. (1)

Phys. Rev. Lett. (1)

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

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

Sci. Am. (1)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

Other (8)

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, Cambridge, UK, 2001).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition).

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

R. Greenler, Rainbows, Halos and Glories (Cambridge University, Cambridge, UK, 1980).

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

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, UK, 1992).
[CrossRef]

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 1976). N.B. This reference may not be readily available, but the calculation method is summarized in Ref. 6.

Ref. 10, Section 12.22.

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Figures (14)

Fig. 1
Fig. 1

Comparison of Mie-theory and Debye-series calculations for the scattering of sunlight by a spherical water drop with radius r = 10 μm (m// and m⊥ denote the Debye-series terms p = m for polarization-parallel and perpendicular to the scattering plane, respectively). The colored bars above the graph represent the brightness and color of the scattered light calculated using Mie theory, while the curves in the graph represent the saturated color of the scattered light.

Fig. 2
Fig. 2

Comparison of simulations of the glory caused by scattering of sunlight by water drops of radius r = 10 μm using Debye p = 2 term (left) and Mie theory (right) The width of this image corresponds to an angle of about ±5°.

Fig. 3
Fig. 3

Path geometry for p = 2 scattering by a sphere: the incident ray is deflected by angle θ = 180° + 2α − 4β. Note that b is the dimensionless impact parameter: b = 0 corresponds to a central ray (α = 0), while b = 1 and b = −1 correspond to edge rays with grazing incidence (α = 90°).

Fig. 4
Fig. 4

Graph of intensity versus scattering angle θ for geometric p = 2 rays with refractive index n = 1.333 (the values marked along the curves correspond to the impact parameter b).

Fig. 5
Fig. 5

van de Hulst’s surface wave associated with a p = 2 ray for a sphere with refractive index n = 1.333.

Fig. 6
Fig. 6

(a) Comparison of calculation methods for p = 1 scattering of light of wavelength λ = 650 nm from a sphere of radius r = 100 μm and refractive index n = 1.333. (b) As in (a) but with r = 10 μm.

Fig. 7
Fig. 7

Surface-wave paths (p = 1) resulting in scattering angle θ = 175°: the upper part of this diagram shows the “short” path, while the lower part shows the “long” path.

Fig. 8
Fig. 8

(a) As in Fig. 6(b), but showing the separate contributions from short-path and long-path surface waves (see Fig. 7). (b) As in (a), but showing the vector sum of the short-path and long-path surface wave contributions.

Fig. 9
Fig. 9

Comparison of Debye-series and surface-wave calculations for parallel polarization for p = 1 scattering (N.B., Amplitude from Khare’s formula multiplied by factor of 0.8).

Fig. 10
Fig. 10

Difference in phase between the Debye-series and the short-path surface-wave calculations for p = 1 scattering.

Fig. 11
Fig. 11

Comparison of Debye-series, surface-wave, and geometric-optics calculations for parallel polarization for p = 2 scattering (N.B., Amplitude from Khare’s formula multiplied by factor of 0.5).

Fig. 12
Fig. 12

Diagram showing p = 2 rays for that result in a scattering angle θ = 175° for a sphere with refractive index n = 1.333.

Fig. 13
Fig. 13

Comparison of Debye-series calculations with the vector sum of surface-wave and geometric-optics calculations for parallel polarization for p = 2 scattering (N.B., Amplitude from Khare’s formula multiplied by factor of 0.5 with a phase correction of +40°.)

Fig. 14
Fig. 14

(a) Comparison of Debye-series calculations for p = 1 scattering with calculations based on van de Hulst’s diffraction pattern. (b) Comparison of Debye-series calculations for p = 1 scattering with calculations based on two-ray interference between short- and long-path surface waves.

Equations (3)

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θ = ( p - 1 ) 180 ° + 2 α - 2 p β ,
θ = 180 ° + 2 α - 4 β .
I 1 = [ C 1 { J 1 ( u ) - J 2 ( u ) } + C 2 { J 1 ( u ) + J 2 ( u ) } ] 2 , I 2 = [ C 2 { J 1 ( u ) - J 2 ( u ) } + C 1 { J 1 ( u ) + J 2 ( u ) } ] 2 ,

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