Abstract

Glare points are the intensity maxima seen when a water drop illuminated by a wide beam is viewed from a certain direction and imaged. We show that good resolution in both the scattering angle and the glare point position can be achieved only if the size parameter x = 2πa/λ is ≫ 1 and that the positions of the glare points can be computed by a Fourier transform from the familiar Lorenz–Mie scattering function. Sample computations made with x = 10,000 and x = 20,000 are presented. Glare points corresponding to rays that have suffered as many as 15 internal reflections can be identified, in agreement with experimental findings.

© 1991 Optical Society of America

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References

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  1. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  2. J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” Appl. Opt. 28, 523–529 (1989).
    [CrossRef] [PubMed]
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957;H. C. van de Hulst, Light Scattering by Small ParticlesDover, New York, 1981).
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  7. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  8. J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237, 138–144 (July1977).
    [CrossRef]
  9. J. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242, 147–152 (June1980).
  10. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  11. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  12. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  13. 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]
  14. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [CrossRef]
  15. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  16. H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas, and Applications (Academic, New York, 1980).
  17. G. Gouesbet, B. Maheu, G. Gréhan, “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]
  18. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. Paris 19, 59–67 (1988).
    [CrossRef]
  19. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  20. H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
    [CrossRef]
  21. S. A. Schaub, D. R. Alexander, J. P. Barton, “Modeling of a coherent imaging system,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), pp. 239–250.

1991 (1)

1989 (1)

1988 (4)

1987 (1)

1984 (2)

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

1980 (1)

J. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242, 147–152 (June1980).

1979 (2)

1977 (1)

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237, 138–144 (July1977).
[CrossRef]

1976 (1)

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

1947 (1)

Alexander, D. R.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Modeling of a coherent imaging system,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), pp. 239–250.

Barton, J. P.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Modeling of a coherent imaging system,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), pp. 239–250.

Bohren, C. F.

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

Gouesbet, G.

Gréhan, G.

Huffman, D. R.

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

Kerker, M.

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

Khare, V.

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]

Lock, J. A.

Maheu, B.

Marston, P. L.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

Nussenzveig, H. M.

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, “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]

Nye, J. F.

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

Sassen, K.

Schaub, S. A.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Modeling of a coherent imaging system,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), pp. 239–250.

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

van de Hulst, H. C.

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
[CrossRef]

H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas, and Applications (Academic, New York, 1980).

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

Walker, J.

J. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242, 147–152 (June1980).

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237, 138–144 (July1977).
[CrossRef]

Walker, J. D.

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

Wang, R. T.

Woodruff, J. R.

Am. J. Phys. (1)

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

Appl. Opt. (4)

J. Opt. Paris (1)

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. Paris 19, 59–67 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Nature (London) (2)

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

Sci. Am. (2)

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237, 138–144 (July1977).
[CrossRef]

J. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242, 147–152 (June1980).

Other (6)

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

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

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

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, Multiple Light Scattering. Tables, Formulas, and Applications (Academic, New York, 1980).

S. A. Schaub, D. R. Alexander, J. P. Barton, “Modeling of a coherent imaging system,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), pp. 239–250.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the experiment and definition of the main parameters.

Fig. 2
Fig. 2

Symbols used in deriving the glare point equation.

Fig. 3
Fig. 3

Mie glare point images for a waterdrop at θ = 125°. The rays are identified by their order of p. Top: x = 10,000, b/r = 0.01; bottom: x = 20,000, b/r = 0.02.

Fig. 4
Fig. 4

Mie glare point images for a waterdrop over the full range of scattering angles. Polarization 1, m = 1.331, x = 10,000, b/r = 0.01. See the explanation in the text.

Fig. 5
Fig. 5

Part of the Mie glare point images for scattering angles of 30° ⩽ θ ⩽ 150° in the vicinity of both edges of the drop image (0.85 < |w| < 1.025). Polarization 1, m = 1.331, x = 20,000, b/r = 0.02. See the key in Fig. 6 and a further explanation in the text.

Fig. 6
Fig. 6

Map of the glare point positions read from Fig. 5: dots, clearly visible peaks; squares, clearly visible peaks outside the edge; crosses, suspected peaks. The curves correspond to ray optics with p − 1 internal reflections, and special points indicate E (edge) and R (rainbow). The fourth-order intersection points are encircled.

Equations (20)

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x = 2 π a / λ ,
w = α r / a ,
Δ θ = b / r .
r Δ α = a Δ w = r ( λ / 2 π ) / b ,
Δ θ Δ w = 1 / x .
1 f = 1 r + 1 s ,
cos τ = | w | , sin τ Δ τ = Δ w .
amplitude | amplitude | 2 TE mode is polarization 1 ( φ = 90 ° in Ref. 3 , Fig. 19 ) { E θ = H φ = 0 E φ = H θ i 1 ( θ ) , = S 1 ( θ ) exp ( ikr )
TM mode is polarization 2 ( φ = 0 ° in Ref. 3 , Fig. 19 ) { E θ = H φ = S 2 ( θ ) exp ( ikr ) i 2 ( θ ) , E φ = H θ = 0
O L = r + y 2 2 r , L Q = s + ( y y ) 2 2 s .
before passing the lens S 1 ( θ ) exp ( ikr ) exp ( i k y 2 / 2 r , after passing the lens S 1 ( θ ) exp ( ikr ) exp ( + i k y 2 / 2 s ,
A Q = exp [ i k ( r + s ) + i π / 4 ] ( λ s ) 1 / 2 × b b S 1 ( θ 0 y r ) exp ( i k y 2 2 s ) exp [ i k ( y y ) 2 2 s ] d y ,
θ = θ 0 y / r
y = s α = saw / r ,
A Q = e i σ r ( λ s ) 1 / 2 A 1 ( w ) ,
A 1 ( w ) = θ 0 b / r θ 0 + b / r S 1 ( θ ) exp [ ixw ( θ θ 0 ) ] d θ .
θ = 2 ( τ p τ ) = 2 π k + q θ ,
w = q cos τ .
w = + 0.666 , θ = 96.56 ° , p = 0 , 3 , 6 , 9 , , w = 0.666 , θ = 23.44 ° , p = 1 , 4 , 7 , 10 , , w = 0.666 , θ = 143.44 ° , p = 2 , 5 , 8 , 11 , ,
w = + 0.941 , θ = 39.51 ° , p = 0 , 4 , 8 , 12 , , w = 0.941 , θ = 50.49 ° , p = 1 , 5 , 9 , 13 , , w = 0.941 , θ = 140.49 ° , p = 2 , 6 , 10 , 14 , , w = 0.941 , θ = 129.51 ° , p = 3 , 7 , 11 , 15 , ,

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