Abstract

Angular scattering measurements obtained with a polar nephelometer employing a linearly polarized laser source are used to examine the general scattering behavior and rainbow generation of pendant water drops, a type of near-spherical particle that has certain similarities to the shape of distorted raindrops. Comparison of the experimental data with theoretical predictions of spherical drop scattering reveals that in many respects the near-spherical particles behave like spheres when the measurements are performed in the horizontal scattering plane, the plane in which the drops display circular cross sections. Furthermore, the angular positions of the rainbow intensity maxima corresponding to the main rainbow peak and supernumerary bows are shown to be predicted accurately by the approximate Airy theory for both the primary and secondary rainbows. Pendant drops whose shapes are significantly elongated in the vertical direction are indicated to generate anomalously strong rainbows from three or more internal reflections. The implications of these findings to rainbow formation in the atmosphere are discussed.

© 1979 Optical Society of America

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  1. These dissimilar approaches to the treatment of the rainbow have also long been a source of conflict. The poets have been highly critical of attempts to quantify the rainbow in mathematical terms, while scientific observers have in turn drawn attention to inaccuracies in the artistic renderings of the rainbow. We would like to express our sentiment that the search for an understanding of the processes involved in rainbow formation and for the representation of their proper structure does not detract from the pleasure of observing a splendid rainbow. Au contraire!
  2. See H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977), for an interesting review of the history and significance of rainbow theories.
    [Crossref]
  3. G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).
  4. H. C. van de Hulst, Light Scattering From Small Particles (Wiley, New York, 1957).
  5. V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [Crossref]
  6. F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, edited by Weickmann Helmut (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.
  7. A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
    [Crossref]
  8. H. R. Pruppacher and K. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
    [Crossref]
  9. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).
  10. K. Sassen and K. N. Liou, “Scattering of polarized laser light by water droplet, mixed phase, and ice crystal clouds: I. Angular scattering patterns,” J. Atmos. Sci. 36, 838–851 (1979).
    [Crossref]
  11. K. Sassen, “Optical backscattering from near-spherical water, ice, and mixed phase drops,” Appl. Opt. 16, 1332–1341 (1977).
    [Crossref] [PubMed]
  12. F. Bashforth and J. C. Adams, An Attempt to Test the Theory of Capillary Action (Cambridge University, London, 1883).
  13. S. Fordham, “On the calculation of surface tension from measurements of pendant drops,” Proc. R. Soc. London Sect. A 194, 1–16 (1948).
    [Crossref]
  14. H. R. Pruppacher and R. L. Fitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
    [Crossref]
  15. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [Crossref]
  16. K. N. Liou and J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: A comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
    [Crossref]
  17. As described in Ref. 11, the backscattering efficiency of pendant drops from 178°–180° is typically somewhat enhanced in comparison with freely falling raindrops of the same diameter.
  18. W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929).
  19. Although van de Hulst (loc. cit.) states that values of h are constant for a given rainbow order, these values are a weak function of the incident light wavelength owing to the slight dependence of the refractive index of water on the visible wavelength (see Ref. 18).
  20. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

1979 (1)

K. Sassen and K. N. Liou, “Scattering of polarized laser light by water droplet, mixed phase, and ice crystal clouds: I. Angular scattering patterns,” J. Atmos. Sci. 36, 838–851 (1979).
[Crossref]

1977 (2)

K. Sassen, “Optical backscattering from near-spherical water, ice, and mixed phase drops,” Appl. Opt. 16, 1332–1341 (1977).
[Crossref] [PubMed]

See H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977), for an interesting review of the history and significance of rainbow theories.
[Crossref]

1975 (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[Crossref]

1974 (1)

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

1972 (1)

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[Crossref]

1971 (2)

K. N. Liou and J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: A comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[Crossref]

H. R. Pruppacher and R. L. Fitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

1970 (1)

H. R. Pruppacher and K. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[Crossref]

1948 (1)

S. Fordham, “On the calculation of surface tension from measurements of pendant drops,” Proc. R. Soc. London Sect. A 194, 1–16 (1948).
[Crossref]

1838 (1)

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Adams, J. C.

F. Bashforth and J. C. Adams, An Attempt to Test the Theory of Capillary Action (Cambridge University, London, 1883).

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Bashforth, F.

F. Bashforth and J. C. Adams, An Attempt to Test the Theory of Capillary Action (Cambridge University, London, 1883).

Beard, K.

H. R. Pruppacher and K. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[Crossref]

Fitter, R. L.

H. R. Pruppacher and R. L. Fitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

Fordham, S.

S. Fordham, “On the calculation of surface tension from measurements of pendant drops,” Proc. R. Soc. London Sect. A 194, 1–16 (1948).
[Crossref]

Fraser, A. B.

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[Crossref]

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[Crossref]

Hansen, J. E.

K. N. Liou and J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: A comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[Crossref]

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929).

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Liou, K. N.

K. Sassen and K. N. Liou, “Scattering of polarized laser light by water droplet, mixed phase, and ice crystal clouds: I. Angular scattering patterns,” J. Atmos. Sci. 36, 838–851 (1979).
[Crossref]

K. N. Liou and J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: A comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[Crossref]

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

Nussenzveig, H. M.

See H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977), for an interesting review of the history and significance of rainbow theories.
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Pruppacher, H. R.

H. R. Pruppacher and R. L. Fitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

H. R. Pruppacher and K. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[Crossref]

Sassen, K.

K. Sassen and K. N. Liou, “Scattering of polarized laser light by water droplet, mixed phase, and ice crystal clouds: I. Angular scattering patterns,” J. Atmos. Sci. 36, 838–851 (1979).
[Crossref]

K. Sassen, “Optical backscattering from near-spherical water, ice, and mixed phase drops,” Appl. Opt. 16, 1332–1341 (1977).
[Crossref] [PubMed]

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering From Small Particles (Wiley, New York, 1957).

Volz, F. E.

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, edited by Weickmann Helmut (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

Appl. Opt. (1)

J. Appl. Meteorol. (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[Crossref]

J. Atmos. Sci. (4)

K. N. Liou and J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: A comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[Crossref]

H. R. Pruppacher and R. L. Fitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[Crossref]

K. Sassen and K. N. Liou, “Scattering of polarized laser light by water droplet, mixed phase, and ice crystal clouds: I. Angular scattering patterns,” J. Atmos. Sci. 36, 838–851 (1979).
[Crossref]

Phys. Rev. Lett. (1)

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Proc. R. Soc. London Sect. A (1)

S. Fordham, “On the calculation of surface tension from measurements of pendant drops,” Proc. R. Soc. London Sect. A 194, 1–16 (1948).
[Crossref]

Q. J. R. Meteorol. Soc. (1)

H. R. Pruppacher and K. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[Crossref]

Sci. Am. (1)

See H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977), for an interesting review of the history and significance of rainbow theories.
[Crossref]

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Other (9)

H. C. van de Hulst, Light Scattering From Small Particles (Wiley, New York, 1957).

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, edited by Weickmann Helmut (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

These dissimilar approaches to the treatment of the rainbow have also long been a source of conflict. The poets have been highly critical of attempts to quantify the rainbow in mathematical terms, while scientific observers have in turn drawn attention to inaccuracies in the artistic renderings of the rainbow. We would like to express our sentiment that the search for an understanding of the processes involved in rainbow formation and for the representation of their proper structure does not detract from the pleasure of observing a splendid rainbow. Au contraire!

F. Bashforth and J. C. Adams, An Attempt to Test the Theory of Capillary Action (Cambridge University, London, 1883).

As described in Ref. 11, the backscattering efficiency of pendant drops from 178°–180° is typically somewhat enhanced in comparison with freely falling raindrops of the same diameter.

W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill, New York, 1929).

Although van de Hulst (loc. cit.) states that values of h are constant for a given rainbow order, these values are a weak function of the incident light wavelength owing to the slight dependence of the refractive index of water on the visible wavelength (see Ref. 18).

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

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Figures (4)

FIG. 1
FIG. 1

Comparison of the shapes of a freely falling raindrop (right) and a pendant drop suspended from the tip of a flat-ground needle (left), both for a ∼4-mm diameter drop. Despite their dissimilar appearances, both drops display circular cross sections through horizontal planes and a reduction in surface curvature along the vertical sides about the drop center.

FIG. 2
FIG. 2

Experimental angular scattering patterns (solid lines) of a 4.4-mm horizontal diameter pendant drop for vertically (left) and horizontally (right) polarized incident laser light. Dashed lines show predictions from geometrical optics theory for large water spheres. Numbers give linear depolarization ratios (in percent) at selected points.

FIG. 3
FIG. 3

Fine structure of the main and supernumerary rainbows generated by a 2.0-mm diameter pendant drop. Dashed lines give the locations of the angles of minimum or maximum deviation (θm) predicted from geometrical optics theory for the primary (n = 1) and secondary (n = 2) rainbows. The X symbols mark the relative intensity and angular position of the first ten maxima predicted by Airy’s theory (see Table III).

FIG. 4
FIG. 4

Fine structure of the primary (n = 1), secondary (n = 2), and sexary (n = 6) rainbows generated by a 2.2-mm diameter pendant drop. Angles of minimum or maximum deviation are shown as dashed lines, with X symbols giving Airy theory predictions of the first ten rainbow maxima. Arrows indicate the predicted angular positions of maxima i = 1–10 for the n = 6 rainbow.

Tables (3)

Tables Icon

TABLE I Angular positions in degrees of the primary (n = 1) to sexary (n = 6) rainbows for red light (λ = 0.6328 μm) derived from geometrical optics theory (θm, the angle of maximum or minimum deviation), and of the first maximum predicted from Airy’s theory [θ(Z1)] for a 1.0-mm radius drop (n = number of internal reflections)

Tables Icon

TABLE II Values for Zi and f2(Zi) derived from Airy’s theory for the first ten maxima (i) corresponding to the main (i = 1) and supernumerary bows, from Humphreys.18

Tables Icon

TABLE III Comparison of Airy theory predictions with measurements from Fig. 3 for the first ten maxima (i = 1–10) of the primary (n = 1) and secondary (n = 2) rainbows. Theoretical values for angular position ( θ i * ) and relative intensity [f2(Zi)*] of maxima calculated for a 1.0-mm radius drop with λ = 0.6328 μm

Equations (1)

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θ m θ i = ( Z i λ / 4 ) ( 4 h / 3 λ r 2 ) 1 / 3 ,