Abstract

The tertiary rainbow of acoustically levitated water drops was observed in the laboratory. Nontrivial caustics were observed for relatively small values of eccentricity. The angular locations of caustics were modeled with matrix methods of generalized ray tracing. Photographs of the scattering were in general agreement with models. Possible effects on the appearance of natural tertiary bow features are discussed.

© 1998 Optical Society of America

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References

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  1. C. Hartwell, “Description of a tertiary rainbow,” Am. J. Sci. 17, 2nd ser., 56–57 (1854).
  2. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  3. C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, New Jersey, 1987), pp. 247–249.
  4. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  5. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  6. C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991).
    [CrossRef] [PubMed]
  7. H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
    [CrossRef] [PubMed]
  8. G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
    [CrossRef] [PubMed]
  9. G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
    [CrossRef] [PubMed]
  10. P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander’s dark band,” Appl. Opt. 33, 4702–4713 (1994).
    [CrossRef] [PubMed]
  11. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  12. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  13. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 258–346.
    [CrossRef]
  14. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 52–148; see also P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1994), pp. 432–614.
  15. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London 438, 397–417 (1992).
    [CrossRef]
  16. A. W. Green, “An approximation for the shapes of large raindrops,” J. Acoust. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  17. W. P. Arnott, P. L. Marston, “Optical glory of freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
    [CrossRef]
  18. W. P. Arnott, P. L. Marston, “Unfolded optical glory of spheroids: backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429–3442 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 4.
  21. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 478–480.
  22. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  23. H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988); P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). See also Section 3.11 of Ref. 14.
  24. A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
    [CrossRef]
  25. S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032–3037 (1982).
    [CrossRef] [PubMed]
  26. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]

1994

1992

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London 438, 397–417 (1992).
[CrossRef]

1991

1988

1986

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

1985

1984

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

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

1982

1979

1976

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

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

1975

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

1972

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

1964

1854

C. Hartwell, “Description of a tertiary rainbow,” Am. J. Sci. 17, 2nd ser., 56–57 (1854).

Arnott, W. P.

Berry, M. V.

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 258–346.
[CrossRef]

Boyer, C. B.

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, New Jersey, 1987), pp. 247–249.

Dean, C. E.

Fraser, A. B.

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

Gedzelman, S. D.

Green, A. W.

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

Hartwell, C.

C. Hartwell, “Description of a tertiary rainbow,” Am. J. Sci. 17, 2nd ser., 56–57 (1854).

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 478–480.

Kaduchak, G.

Kneisly, J. A.

Marston, P. L.

G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
[CrossRef] [PubMed]

P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander’s dark band,” Appl. Opt. 33, 4702–4713 (1994).
[CrossRef] [PubMed]

C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991).
[CrossRef] [PubMed]

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
[CrossRef] [PubMed]

W. P. Arnott, P. L. Marston, “Unfolded optical glory of spheroids: backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429–3442 (1991).
[CrossRef] [PubMed]

W. P. Arnott, P. L. Marston, “Optical glory of freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
[CrossRef]

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

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

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 52–148; see also P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1994), pp. 432–614.

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London 438, 397–417 (1992).
[CrossRef]

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

Pedgley, D. E.

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 4.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 4.

Sassen, K.

Simpson, H. J.

G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
[CrossRef] [PubMed]

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
[CrossRef] [PubMed]

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988); P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). See also Section 3.11 of Ref. 14.

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]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 258–346.
[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]

Adv. Phys.

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Am. J. Phys.

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

Am. J. Sci.

C. Hartwell, “Description of a tertiary rainbow,” Am. J. Sci. 17, 2nd ser., 56–57 (1854).

Appl. Opt.

J. Acoust. Meteorol.

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

J. Atmos. Sci.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

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]

Opt. Lett.

Proc. R. Soc. London

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London 438, 397–417 (1992).
[CrossRef]

Weather

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Other

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, New Jersey, 1987), pp. 247–249.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 4.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 478–480.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 258–346.
[CrossRef]

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 52–148; see also P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1994), pp. 432–614.

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988); P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). See also Section 3.11 of Ref. 14.

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Figures (9)

Fig. 1
Fig. 1

Ray paths that create three reflections inside a water drop. For ellipsoidal drops, the plane of the figure corresponds to the equatorial plane of the drop. θ R , measured from the forward direction, is the largest scattering angle for parallel incident rays of this type. Two nearby rays entering on both sides of the Descartes ray emerge on the same side, and the Descartes ray lies on the caustic separating the two-ray and the zero-ray regions.

Fig. 2
Fig. 2

Geometry of an oblate ellipsoidal water drop illuminated by horizontal light rays. The drop’s equator is circular with diameter D, and its vertical height is H. Acoustically levitated drops and falling raindrops depart from this ideal shape, but have similar curvature near the equator. Light rays incident upon the surface at angle i are refracted at angle r in accordance with Snell’s law.

Fig. 3
Fig. 3

Angular locations of caustics that are due to vertical focusing of four-chord rays, according to Eqs. (6). Three categories of caustic are found for oblate drops of different aspect ratios. These caustics merge with the rainbow caustic for values of q, giving θ = θ R ≈ 42. The left and the right curves are double valued in some ranges of q, giving E 6 diffraction catastrophes near θ R at q ≈ 1.1 and 1.53. The middle curve indicates a hyperbolic umbilic catastrophe occurring at q ≈ 1.46. Two lip events occur for q near 1.51, as caustic cusps lie in the near-backward region.

Fig. 4
Fig. 4

Magnified view of the rainbow region for caustics governed by Eq. (6A). Two closely spaced cusp caustics are present for q in the 1.069–1.105 range. The E 6 diffraction catastrophe corresponds to the merging of these and the Descartes ray caustic at the upper limit of this range, giving strong scattering. Similar cusp caustics should occur for large natural raindrops.

Fig. 5
Fig. 5

Magnified view of the rainbow region for caustics governed by Eq. (6C). Two vertical-focusing caustics merge toward the Descartes ray caustic as q approaches 1.533, resulting in an E 6 diffraction catastrophe and bright far-zone scattering.

Fig. 6
Fig. 6

Top-view diagram of camera placement for recording drop profiles and tertiary rainbow scattering.

Fig. 7
Fig. 7

Tertiary rainbow patterns from a drop with different degrees of oblateness. The drop actually evaporated and became more spherical, but the sequence of photographs is reversed. (a) Airy pattern from a drop with 0.49-mm diameter and q ≈ 1.07; (b) cusp caustic pair entering the rainbow region as q increases slightly, in accordance with Fig. 4; (c) three caustics merging at q ≈ 1.1; (d), (e) vertical-focusing caustics beginning to separate from the scattering plane as q exceeds 1.105; (f), (g) vertical-focusing caustics vanish as q increases, and bright arcs become curved opposite those of the Airy pattern. The drop in (g) had an ∼0.72-mm diameter and q ≈ 1.29.

Fig. 8
Fig. 8

Scattering events in the tertiary bow region as drops become more highly oblate: (a) vanishing caustics graphed in Fig. 4; (b) the supernumerary arcs curved toward the dark side of the pattern; (c) the arcs begin to reverse their curvature again as q nears the condition for the hyperbolic umbilic diffraction catastrophe, seen in (d) for q ≈ 1.464; (e) the E 6 catastrophe pattern predicted at q ≈ 1.533.

Fig. 9
Fig. 9

Somewhat exaggerated appearance of features related to the tertiary bow when caustics that are due to vertical focusing by oblate drops are present, including bright cusps and anomalously curved arcs in the near-horizontal region. A dispersion of drop sizes and shapes tend to broaden and whiten these features, but enhanced brightness in the cusp region can still occur.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

sin   i = n   sin   r .
M t = 1 0   d 1 ,     M r = 1 - 2   cos   r R   0 1 , M i = 1 - γ R   0 1 - γ ,     M f = 1 γ R γ - 1   0 1 1 - γ
γ = cos   r - cos   i n = sin   i   cos   r - cos   i   sin   r sin   i = sin i - r sin   i .
M f M t M r M t N M i = M 11 M 21   M 12 M 22 ,
8 γ ρ 3 Q 2 - 4 ρ γ + ρ Q + 1 × 8 γ ρ 4 Q 2 - 4 ρ 2 2 γ + ρ Q + 3 ρ + γ = 0 ,  
q a q b q c q d = 1 2 ρ γ × 2 γ + ρ - 2 γ 2 - 2 γ ρ + ρ 2 1 / 2 1 / 2 γ + ρ + γ 2 + ρ 2 1 / 2 1 / 2 2 γ + ρ + 2 γ 2 - 2 γ ρ + ρ 2 1 / 2 1 / 2 γ + ρ - γ 2 + ρ 2 1 / 2 1 / 2 .
cos   i p = n 2 - 1 p 2 - 1 1 / 2 , cos   r p = p n cos   i p , γ p = p - 1 p cos   r p .

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