Abstract

Irregularities in the perimeter of a water droplet adhering to a vertical pane of glass cause perturbations in the curvature of the droplet surface. When laser light passes through such a droplet, the perturbations produce a far field refraction caustic, which is a section of the caustic known as the parabolic umbilic in the catastrophe theory classification. As the water evaporates and the droplet surface curvature changes, the section of the parabolic umbilic caustic on the viewing screen also changes. We determine the evolution of curvature of the droplet surface by observing the evolution of the far field caustic and the locations on the droplet responsible for the various features of the caustic.

© 1990 Optical Society of America

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References

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  1. M. Herzberger, “Light Distribution in an Optical Image,” J. Opt. Soc. Am. 37, 485–493 (1947).
    [CrossRef] [PubMed]
  2. M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), p. 156.
  3. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 159.
  4. J. B. Keller, H. B. Keller, “Determination of Reflected and Transmitted Fields by Geometrical Optics,” J. Opt. Soc. Am. 40, 48–52 (1950).
    [CrossRef]
  5. D. G. Burkhard, D. L. Shealy, “Flux Density for Ray Propagation in Geometrical Optics,” J. Opt. Soc. Am. 63, 299–304 (1973).
    [CrossRef]
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    [CrossRef]
  7. D. G. Burkhard, D. L. Shealy, “Simplified Formula for the Illuminance in an Optical System,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef] [PubMed]
  8. M. V. Berry, “Waves and Thom’s Theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  9. M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
    [CrossRef]
  10. J. F. Nye, “Optical Caustics in the Near Field from Liquid Drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
    [CrossRef]
  11. J. F. Nye, “Optical Caustics from Liquid Drops Under Gravity: Observations of the Parabolic and Symbolic Umbilics,” Philos. Trans. R. Soc. London Ser. A 292, 4–44 (1979).
  12. P. L. Marston, E. H. Trinh, “Hyperbolic Umbilic Diffraction Catastrophe and Rainbow Scattering from Spheroidal Drops,” Nature London 312, 529–531 (1984).
    [CrossRef]
  13. J. F. Nye, “Rainbow Scattering from Spheroidal Drops—an Explanation of the Hyperbolic Umbilic Foci,” Nature London 312, 531–532 (1984).
    [CrossRef]
  14. J. F. Nye, “The Catastrophe Optics of Liquid Drop Lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
    [CrossRef]
  15. M. V. Berry, “Disruption of Images: the Caustic Touching Theorem,” J. Opt. Soc. Am. A 4, 561–569 (1987).
    [CrossRef]
  16. G. Dangelmayr, F. J. Wright, “On the Validity of the Paraxial Eikonal in Catastrophe Optics,” J. Phys. A Gen. Phys. 17, 99–108 (1984).
    [CrossRef]
  17. E. Bochove, “Geometrical Optics Field of General Optical Systems,” J. Opt. Soc. Am. 69, 891–897 (1979).
    [CrossRef]
  18. P. S. Theocaris, J. G. Michopoulos, “Generalization of the Theory of Far-Field Caustics by the Catastrophe Theory,” Appl. Opt. 21, 1080–1091 (1982).
    [CrossRef] [PubMed]
  19. J. D. Walker, “Caustics: Mathematical Curves Generated by Light Shined Through Rippled Plastic,” Sci. Am. 249(3), 190–201 (1983).
    [CrossRef]
  20. J. A. Lock, J. H. Andrews, “Caustics Associated with Cubic Phase Functions,” J. Opt. Soc. Am., submitted.
  21. J. A. Lock, J. D. Walker, “In Search of the Transverse Swallowtail Caustic,” J. Opt. Soc. Am., submitted.
  22. R. Thom, Structural Stability and Morphogenesis (Benjamin, Reading, MA, 1975), pp. 81–90 and 189–191.
  23. J. D. Walker, “A Drop of Water Becomes a Gateway into the World of Catastrophe Optics,” Sci. Am. 261(3), 176–179 (1989).
    [CrossRef]
  24. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), pp. 167–169.
  25. M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
    [CrossRef]

1989 (1)

J. D. Walker, “A Drop of Water Becomes a Gateway into the World of Catastrophe Optics,” Sci. Am. 261(3), 176–179 (1989).
[CrossRef]

1987 (1)

1986 (1)

J. F. Nye, “The Catastrophe Optics of Liquid Drop Lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

1984 (3)

G. Dangelmayr, F. J. Wright, “On the Validity of the Paraxial Eikonal in Catastrophe Optics,” J. Phys. A Gen. Phys. 17, 99–108 (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]

J. F. Nye, “Rainbow Scattering from Spheroidal Drops—an Explanation of the Hyperbolic Umbilic Foci,” Nature London 312, 531–532 (1984).
[CrossRef]

1983 (1)

J. D. Walker, “Caustics: Mathematical Curves Generated by Light Shined Through Rippled Plastic,” Sci. Am. 249(3), 190–201 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

1979 (3)

J. F. Nye, “Optical Caustics from Liquid Drops Under Gravity: Observations of the Parabolic and Symbolic Umbilics,” Philos. Trans. R. Soc. London Ser. A 292, 4–44 (1979).

M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
[CrossRef]

E. Bochove, “Geometrical Optics Field of General Optical Systems,” J. Opt. Soc. Am. 69, 891–897 (1979).
[CrossRef]

1978 (1)

J. F. Nye, “Optical Caustics in the Near Field from Liquid Drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

1976 (2)

1973 (1)

1950 (1)

1947 (1)

Andrews, J. H.

J. A. Lock, J. H. Andrews, “Caustics Associated with Cubic Phase Functions,” J. Opt. Soc. Am., submitted.

Berry, M. V.

M. V. Berry, “Disruption of Images: the Caustic Touching Theorem,” J. Opt. Soc. Am. A 4, 561–569 (1987).
[CrossRef]

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
[CrossRef]

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

Bochove, E.

Burkhard, D. G.

Dangelmayr, G.

G. Dangelmayr, F. J. Wright, “On the Validity of the Paraxial Eikonal in Catastrophe Optics,” J. Phys. A Gen. Phys. 17, 99–108 (1984).
[CrossRef]

Fronczak, R. C.

Herzberger, M.

M. Herzberger, “Light Distribution in an Optical Image,” J. Opt. Soc. Am. 37, 485–493 (1947).
[CrossRef] [PubMed]

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), p. 156.

Keller, H. B.

Keller, J. B.

Lock, J. A.

J. A. Lock, J. D. Walker, “In Search of the Transverse Swallowtail Caustic,” J. Opt. Soc. Am., submitted.

J. A. Lock, J. H. Andrews, “Caustics Associated with Cubic Phase Functions,” J. Opt. Soc. Am., submitted.

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]

Michopoulos, J. G.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), pp. 167–169.

Nye, J. F.

J. F. Nye, “The Catastrophe Optics of Liquid Drop Lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

J. F. Nye, “Rainbow Scattering from Spheroidal Drops—an Explanation of the Hyperbolic Umbilic Foci,” Nature London 312, 531–532 (1984).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
[CrossRef]

J. F. Nye, “Optical Caustics from Liquid Drops Under Gravity: Observations of the Parabolic and Symbolic Umbilics,” Philos. Trans. R. Soc. London Ser. A 292, 4–44 (1979).

J. F. Nye, “Optical Caustics in the Near Field from Liquid Drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

Shealy, D. L.

Stavroudis, O. N.

O. N. Stavroudis, R. C. Fronczak, “Caustic Surfaces and the Structure of the Geometrical Image,” J. Opt. Soc. Am. 66, 795–800 (1976).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 159.

Theocaris, P. S.

Thom, R.

R. Thom, Structural Stability and Morphogenesis (Benjamin, Reading, MA, 1975), pp. 81–90 and 189–191.

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,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

Walker, J. D.

J. D. Walker, “A Drop of Water Becomes a Gateway into the World of Catastrophe Optics,” Sci. Am. 261(3), 176–179 (1989).
[CrossRef]

J. D. Walker, “Caustics: Mathematical Curves Generated by Light Shined Through Rippled Plastic,” Sci. Am. 249(3), 190–201 (1983).
[CrossRef]

J. A. Lock, J. D. Walker, “In Search of the Transverse Swallowtail Caustic,” J. Opt. Soc. Am., submitted.

Wright, F. J.

G. Dangelmayr, F. J. Wright, “On the Validity of the Paraxial Eikonal in Catastrophe Optics,” J. Phys. A Gen. Phys. 17, 99–108 (1984).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
[CrossRef]

Adv. Phys. (1)

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

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

J. Phys. A Gen. Phys. (1)

G. Dangelmayr, F. J. Wright, “On the Validity of the Paraxial Eikonal in Catastrophe Optics,” J. Phys. A Gen. Phys. 17, 99–108 (1984).
[CrossRef]

Nature London (2)

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]

Philos. Trans. R. Soc. London (1)

M. V. Berry, J. F. Nye, F. J. Wright, “The Elliptic Umbilic Diffraction Catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

J. F. Nye, “Optical Caustics from Liquid Drops Under Gravity: Observations of the Parabolic and Symbolic Umbilics,” Philos. Trans. R. Soc. London Ser. A 292, 4–44 (1979).

Proc. R. Soc. London Ser. A (2)

J. F. Nye, “Optical Caustics in the Near Field from Liquid Drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

J. F. Nye, “The Catastrophe Optics of Liquid Drop Lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

Sci. Am. (2)

J. D. Walker, “A Drop of Water Becomes a Gateway into the World of Catastrophe Optics,” Sci. Am. 261(3), 176–179 (1989).
[CrossRef]

J. D. Walker, “Caustics: Mathematical Curves Generated by Light Shined Through Rippled Plastic,” Sci. Am. 249(3), 190–201 (1983).
[CrossRef]

Other (6)

J. A. Lock, J. H. Andrews, “Caustics Associated with Cubic Phase Functions,” J. Opt. Soc. Am., submitted.

J. A. Lock, J. D. Walker, “In Search of the Transverse Swallowtail Caustic,” J. Opt. Soc. Am., submitted.

R. Thom, Structural Stability and Morphogenesis (Benjamin, Reading, MA, 1975), pp. 81–90 and 189–191.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), pp. 167–169.

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), p. 156.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 159.

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Figures (8)

Fig. 1
Fig. 1

Evolution of the caustic produced by an evaporating water droplet. Photographs (a)–(f) represent successively later times. After photograph (f), the evolution of the caustic reversed, finally returning to a form resembling photograph (a). (The photographs were taken with different droplets, with slightly different magnifications.)

Fig. 2
Fig. 2

Location on the viewing screen of the initial cusp point (IC), the bottom of the fold caustic (F), and the diffraction star/elliptic umbilic (DS/C) as a function of time. The points labeled af represent the stages of the evolution of the caustic shown in Figs. 1(a)–(f). Shortly after the start of the reversal of the caustic, the cusp inside the fold temporarily disappeared and is thus not shown in the figure. Its return as a dim, featureless patch of light is indicated by the dashed segment. The time at which the dim patch became a recognizable diffraction star corresponds to the end of the dashed segment.

Fig. 3
Fig. 3

Approximate location of the light rays passing through the water droplet which produce the initial cusp point (triangle), the bottom of the fold (circle), and the diffraction star/elliptic umbilic (star) of the caustic in Fig. 1. The droplet cross sections labeled (a)–(f) correspond to the stages of the evolution of the caustic shown in Figs. 1(a)–(f).

Fig. 4
Fig. 4

(a) Finger perimeter irregularity and (b) an indentation perimeter irregularity both shown greatly exaggerated. The double lines R represent surface ridges and the dashed lines r represent surface ravines. The surface perturbation y p 4 / 4 + η x p 2 y p with < 0 and η > 0 is shown in (c) as a function of xp and yp. The double lines M are relative maxima and the dashed lines m are relative minima.

Fig. 5
Fig. 5

Path of the first half of the evolution in the γ-δ plane of the caustic produced by the evaporating water droplet. Points af correspond to the stages of the evolution of the caustic shown in Figs. 1(a)–(f).

Fig. 6
Fig. 6

The H(ϕ) = 0 curves in the xp-yp plane on the droplet surface for the parabolic umbilic phase function of Eq. (5). The graphs labeled (a)–(f) correspond to the stages of the evolution of the caustic shown in Figs. 1(a)–(f). The points labeled 1–3 are the solutions of Eq. (6) for x = 0.

Fig. 7
Fig. 7

Sections through the parabolic umbilic caustic in α-β space. The graphs labeled (a)–(f) are identical to the observed caustic in Figs. 1(a)–(f). The points labeled 1–3 are the images of the analogous points in Fig. 6.

Fig. 8
Fig. 8

Path of the second half of the evolution in the γ-δ plane of the caustic produced by the evaporating water droplet. The dashed portion of the path can only be inferred since it corresponds to the time interval during which the cusp caustic temporarily disappeared as indicated in Fig. 2.

Equations (21)

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ϕ ( x , y , x o , y 0 , z 0 ) = 2 π λ [ ( n - 1 ) t ( x , y ) + x 2 2 z 0 + y 2 2 z 0 - x x 0 z 0 - y y 0 z 0 ]
E ( x 0 , y 0 , z 0 ) = dxdy exp [ i ϕ ( x , y , x 0 , y 0 , z 0 ) ] .
H ( ϕ ) ( 2 ϕ x 2 ) ( 2 ϕ y 2 ) - ( 2 ϕ x y ) 2 = 0.
ϕ x = 0 ,             ϕ y = 0.
δ ( x , y ) = 4 y 4 + η x 2 y + γ x 2 + δ y 2 - α x - β y .
( η y + γ ) ( 3 y 2 + 2 δ ) = 2 η 2 x 2 .
α = 2 x ( η y + γ ) ,
β = η x 2 + y ( y 2 + 2 δ ) ,
x = x p ,             y = y p + K .
t c ( x , y ) = t 0 + ( r c 2 - x 2 - y 2 ) 1 / 2 - r c ,
f c = r c n - 1 .
t p ( x p , y p ) = 4 ( n - 1 ) y p 4 + η ( n - 1 ) x p 2 y p - x p 2 2 r x - y p 2 2 r y ,
α = x 0 z 0 ,
β = y 0 z 0 + K f c - K z 0 ,
γ = 1 2 [ 1 z 0 - 1 f c - ( n - 1 ) r x ] ,
δ = 1 2 [ 1 z 0 - 1 f c - ( n - 1 ) r y ] .
y 0 ( β - K f c ) z 0 .
y 0 β z 0 .
y 0 - 4 3 δ ( 2 δ 2 ) 1 / 2 z 0 ,
y 0 - γ η ( - γ 2 η 2 + 2 δ ) z 0 .
γ = ( 2 η 2 δ 3 ) 1 / 2

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