Abstract

Critical-angle scattering of laser light from bubbles was photographed, and angular structures in the far-zone intensity are compared with models. The radii of the freely rising bubbles were in the 26.4–814-μm range. Deviation from sphericity for the rising bubbles is considered. Diffraction effects make simple geometric optics inadequate near the critical angle. Coarse-structure intensity oscillations show agreement with a physical-optics approximation and with Mie theory. Measurements agree with derived expressions for the angular spacings of fine structure and contrast modulations. Uses of these structures in different bubble-sizing methods are discussed. The locations of scattered-wave virtual sources are modeled.

© 1984 Optical Society of America

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References

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  1. P. L. Marston, J. Opt. Soc. Am. 69, 1205 (1979); J. Opt. Soc. Am. 70, 353 (E) (1980).
    [CrossRef]
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    [CrossRef]
  3. C. Pulfrich, Ann. Phys. Chem. Leipzig 33, 209 (1888).
  4. G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
    [CrossRef]
  5. P. L. Marston, D. L. Kingsbury, J. Opt. Soc. Am. 71, 192 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
    [CrossRef]
  6. D. L. Kingsbury, P. L. Marston, J. Opt. Soc. Am. 71, 358 (1981).
    [CrossRef]
  7. P. L. Marston, D. S. Langley, D. L. Kingsbury, Appl. Sci. Res. 38, 373 (1982).
    [CrossRef]
  8. G. E. Davis, J. Opt. Soc. Am. 45, 572 (1955).
    [CrossRef]
  9. J.-C. Ravey, P. Mazeron, J. Opt. Paris 13, 273 (1982).
    [CrossRef]
  10. B. D. Johnson, R. C. Cooke, J. Geophys. Res. 84, 3761 (1979); F. Avellan, F. Resch, Int. J. Multiphase Flow 9, 649 (1983).
    [CrossRef]
  11. W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
    [CrossRef]
  12. D. L. Kingsbury, P. L. Marston, Appl. Opt. 20, 2348 (1981).
    [CrossRef] [PubMed]
  13. P. L. Marston, D. L. Kingsbury, J. Acoust. Soc. Am. 70, 1488 (1981).
    [CrossRef]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 499.
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  18. W. L. Haberman, R. K. Morton, Trans. Am. Soc. Civ. Eng. 121, 227 (1956).
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    [CrossRef]
  20. G. C. Farnell, in The Theory of the Photographic Process, C. E. K. Mees, T. H. James, Eds. (Macmillan, New York, 1966), Chap. 6, p. 72.
  21. Reciprocity failure may alter the H-D curve shape at different incident light intensities. Tests indicate this effect was negligible under the conditions of this experiment.
  22. S. Hartland, R. Hartley, Axisymmetric Fluid–Liquid Interfaces (Elsevier, Amsterdam, 1976).
  23. W. Siemes, Chem. Ing. Tech. 26, 614 (1954).
    [CrossRef]
  24. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), Chap. 7.
  25. D. S. Langley, P. L. Marston, Phys. Rev. Lett. 47, 913 (1981).
    [CrossRef]
  26. S. D. H. Andreasson, S. E. Gustafsson, N. O. Halling, J. Opt. Soc. Am. 61, 595 (1971).
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  27. W. J. Wiscombe, Appl. Opt. 19, 1505 (1980).
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  28. P. L. Marston, D. S. Langley, in Near Zero: New Frontiers in Physics, B. S. Deaver, C. W. F. Everitt, Eds. (Freeman, San Francisco, 1984).

1983 (1)

1982 (2)

P. L. Marston, D. S. Langley, D. L. Kingsbury, Appl. Sci. Res. 38, 373 (1982).
[CrossRef]

J.-C. Ravey, P. Mazeron, J. Opt. Paris 13, 273 (1982).
[CrossRef]

1981 (6)

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

D. L. Kingsbury, P. L. Marston, Appl. Opt. 20, 2348 (1981).
[CrossRef] [PubMed]

P. L. Marston, D. L. Kingsbury, J. Acoust. Soc. Am. 70, 1488 (1981).
[CrossRef]

P. L. Marston, D. L. Kingsbury, J. Opt. Soc. Am. 71, 192 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
[CrossRef]

D. L. Kingsbury, P. L. Marston, J. Opt. Soc. Am. 71, 358 (1981).
[CrossRef]

D. S. Langley, P. L. Marston, Phys. Rev. Lett. 47, 913 (1981).
[CrossRef]

1980 (1)

1979 (2)

P. L. Marston, J. Opt. Soc. Am. 69, 1205 (1979); J. Opt. Soc. Am. 70, 353 (E) (1980).
[CrossRef]

B. D. Johnson, R. C. Cooke, J. Geophys. Res. 84, 3761 (1979); F. Avellan, F. Resch, Int. J. Multiphase Flow 9, 649 (1983).
[CrossRef]

1975 (1)

1974 (1)

G. B. Wallis, Int. J. Multiphase Flow 1, 491 (1974).
[CrossRef]

1971 (1)

1964 (1)

1956 (1)

W. L. Haberman, R. K. Morton, Trans. Am. Soc. Civ. Eng. 121, 227 (1956).

1955 (1)

1954 (1)

W. Siemes, Chem. Ing. Tech. 26, 614 (1954).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

1888 (1)

C. Pulfrich, Ann. Phys. Chem. Leipzig 33, 209 (1888).

Abdelmessih, A. H.

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

Andreasson, S. D. H.

Brim, B. L.

Clift, R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), Chap. 7.

Cooke, R. C.

B. D. Johnson, R. C. Cooke, J. Geophys. Res. 84, 3761 (1979); F. Avellan, F. Resch, Int. J. Multiphase Flow 9, 649 (1983).
[CrossRef]

Davis, G. E.

Durst, F.

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

Farnell, G. C.

G. C. Farnell, in The Theory of the Photographic Process, C. E. K. Mees, T. H. James, Eds. (Macmillan, New York, 1966), Chap. 6, p. 72.

Grace, J. R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), Chap. 7.

Gustafsson, S. E.

Haberman, W. L.

W. L. Haberman, R. K. Morton, Trans. Am. Soc. Civ. Eng. 121, 227 (1956).

Halling, N. O.

Hartland, S.

S. Hartland, R. Hartley, Axisymmetric Fluid–Liquid Interfaces (Elsevier, Amsterdam, 1976).

Hartley, R.

S. Hartland, R. Hartley, Axisymmetric Fluid–Liquid Interfaces (Elsevier, Amsterdam, 1976).

Johnson, B. D.

B. D. Johnson, R. C. Cooke, J. Geophys. Res. 84, 3761 (1979); F. Avellan, F. Resch, Int. J. Multiphase Flow 9, 649 (1983).
[CrossRef]

Johnson, J. L.

Kingsbury, D. L.

Langley, D. S.

P. L. Marston, D. S. Langley, D. L. Kingsbury, Appl. Sci. Res. 38, 373 (1982).
[CrossRef]

D. S. Langley, P. L. Marston, Phys. Rev. Lett. 47, 913 (1981).
[CrossRef]

P. L. Marston, D. S. Langley, in Near Zero: New Frontiers in Physics, B. S. Deaver, C. W. F. Everitt, Eds. (Freeman, San Francisco, 1984).

Liska, J. J.

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

Love, S. P.

Marston, P. L.

P. L. Marston, J. L. Johnson, S. P. Love, B. L. Brim, J. Opt. Soc. Am. 73, 1658 (1983).
[CrossRef]

P. L. Marston, D. S. Langley, D. L. Kingsbury, Appl. Sci. Res. 38, 373 (1982).
[CrossRef]

D. L. Kingsbury, P. L. Marston, J. Opt. Soc. Am. 71, 358 (1981).
[CrossRef]

P. L. Marston, D. L. Kingsbury, J. Opt. Soc. Am. 71, 192 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
[CrossRef]

D. L. Kingsbury, P. L. Marston, Appl. Opt. 20, 2348 (1981).
[CrossRef] [PubMed]

P. L. Marston, D. L. Kingsbury, J. Acoust. Soc. Am. 70, 1488 (1981).
[CrossRef]

D. S. Langley, P. L. Marston, Phys. Rev. Lett. 47, 913 (1981).
[CrossRef]

P. L. Marston, J. Opt. Soc. Am. 69, 1205 (1979); J. Opt. Soc. Am. 70, 353 (E) (1980).
[CrossRef]

P. L. Marston, D. S. Langley, in Near Zero: New Frontiers in Physics, B. S. Deaver, C. W. F. Everitt, Eds. (Freeman, San Francisco, 1984).

Martin, W. W.

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

Mazeron, P.

J.-C. Ravey, P. Mazeron, J. Opt. Paris 13, 273 (1982).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

Morton, R. K.

W. L. Haberman, R. K. Morton, Trans. Am. Soc. Civ. Eng. 121, 227 (1956).

Murty, M. V. R. K.

Pogorzelski, R. J.

Pulfrich, C.

C. Pulfrich, Ann. Phys. Chem. Leipzig 33, 209 (1888).

Ravey, J.-C.

J.-C. Ravey, P. Mazeron, J. Opt. Paris 13, 273 (1982).
[CrossRef]

Siemes, W.

W. Siemes, Chem. Ing. Tech. 26, 614 (1954).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 499.

Tsai, W. C.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.

Wallis, G. B.

G. B. Wallis, Int. J. Multiphase Flow 1, 491 (1974).
[CrossRef]

Weber, M. E.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), Chap. 7.

Wiscombe, W. J.

Ann. Phys. Chem. Leipzig (1)

C. Pulfrich, Ann. Phys. Chem. Leipzig 33, 209 (1888).

Ann. Phys. Leipzig (1)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

Appl. Opt. (3)

Appl. Sci. Res. (1)

P. L. Marston, D. S. Langley, D. L. Kingsbury, Appl. Sci. Res. 38, 373 (1982).
[CrossRef]

Chem. Ing. Tech. (1)

W. Siemes, Chem. Ing. Tech. 26, 614 (1954).
[CrossRef]

Int. J. Multiphase Flow (2)

W. W. Martin, A. H. Abdelmessih, J. J. Liska, F. Durst, Int. J. Multiphase Flow 7, 439 (1981).
[CrossRef]

G. B. Wallis, Int. J. Multiphase Flow 1, 491 (1974).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. L. Marston, D. L. Kingsbury, J. Acoust. Soc. Am. 70, 1488 (1981).
[CrossRef]

J. Geophys. Res. (1)

B. D. Johnson, R. C. Cooke, J. Geophys. Res. 84, 3761 (1979); F. Avellan, F. Resch, Int. J. Multiphase Flow 9, 649 (1983).
[CrossRef]

J. Opt. Paris (1)

J.-C. Ravey, P. Mazeron, J. Opt. Paris 13, 273 (1982).
[CrossRef]

J. Opt. Soc. Am. (7)

Phys. Rev. Lett. (1)

D. S. Langley, P. L. Marston, Phys. Rev. Lett. 47, 913 (1981).
[CrossRef]

Trans. Am. Soc. Civ. Eng. (1)

W. L. Haberman, R. K. Morton, Trans. Am. Soc. Civ. Eng. 121, 227 (1956).

Other (7)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 499.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), Chap. 7.

G. C. Farnell, in The Theory of the Photographic Process, C. E. K. Mees, T. H. James, Eds. (Macmillan, New York, 1966), Chap. 6, p. 72.

Reciprocity failure may alter the H-D curve shape at different incident light intensities. Tests indicate this effect was negligible under the conditions of this experiment.

S. Hartland, R. Hartley, Axisymmetric Fluid–Liquid Interfaces (Elsevier, Amsterdam, 1976).

P. L. Marston, D. S. Langley, in Near Zero: New Frontiers in Physics, B. S. Deaver, C. W. F. Everitt, Eds. (Freeman, San Francisco, 1984).

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Figures (11)

Fig. 1
Fig. 1

Normalized scattered intensity from a bubble with ka = 1633 and the electric field parallel to the scattering plane. (a) Three models: the solid curve is the Mie theory, the dashed curve is a physical-optics approximation, and the dotted curve is from simple geometric optics. (b) The solid curve is experimental data taken from the photograph in Fig. 6(a), and the dashed curve is the physical-optics model.

Fig. 2
Fig. 2

Rays in the scattering plane emerging at the scattering angle ϕ = 50°. Rays are specified by the parameters p = the number of chords within the bubble, and l = the number of crossings of the optic axis.

Fig. 3
Fig. 3

Rays used to model the fine-structure interference period. Points F0 and F2 are the virtual source locations, respectively, of the rays labeled 0 and 2 in the limit as the dashed 0′ and 2′ rays approach them, and αp is the distance from Fp to the exit plane.

Fig. 4
Fig. 4

Top view of apparatus set up for angle calibration. To observe bubble scattering the rotatable reflector was removed and bubbles were injected by a needle entering through the bottom of the aluminum block. A microscope above the top opening was used to measure bubbles. The screen (used during angle calibration of the rotatable reflector) is at the optical-transform plane of the lens. The systematic uncertainty in the angle calibration was <0.02°.

Fig. 5
Fig. 5

Measurements and model for the angular frequency of fine-structure lines. Data are displayed for 120 rising and 23 pendant bubbles, whose radii as and ap, respectively, were measured by microscope. Mie results were obtained from high-resolution computations.

Fig. 6
Fig. 6

Photographs of far-zone scattering from rising bubbles aligned with a scale showing the scattering angle ϕ. The bubble radii and incident polarizations are (a) ar = 0.1235 mm, j = 2; (b) 0.3890 mm, j = 2; (c) 0.5808 mm, j = 1; (d) 0.0461 mm, j = 1. The corresponding intensity profiles are shown in Figs. 1(b) and 79.

Fig. 7
Fig. 7

Normalized scattered intensity from a bubble with ka = 5144 and the electric field parallel to the scattering plane (j = 2 scattering). The solid curve is data taken from the photograph in Fig. 6(b), and the dashed line is the physical-optics model.

Fig. 8
Fig. 8

Like Fig. 7 but with ka = 7680, j = 1, and the solid curve corresponding to the photograph in Fig. 6(c).

Fig. 9
Fig. 9

The j = 1 scattered intensity for ka = 612. The solid curve is data from the photograph in Fig. 6(d), the dotted curve is the Mie result, and the dashed curve is the physical-optics model.

Fig. 10
Fig. 10

Normalized scattered intensity for ka of (a) 2748 and (b) 9699. These ka were determined from the respective radii of 0.2078 and 0.7335 mm which were inferred from the measured fine-structure spacing. The solid curves are measurements; in (b) they were smoothed to remove the fine structure. The dashed curve is the physical-optics approximation.

Fig. 11
Fig. 11

Coordinate system and angles used in determining the locations of scattered-wave virtual sources.

Equations (31)

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ϕ = 2 ( - 1 ) l [ p ρ p - θ p + π 2 ( l - p + l ) ] ,
sin θ p = m sin ρ p .
α p = a [ 1 + ½ ( p τ - 1 ) - 1 cos θ p ] ,
η p ( θ p , ρ p ) = 2 k a ( 1 - cos θ p + m p cos ρ p ) ,
tan ( δ j / 2 ) = m 2 ( 1 - j ) ( sin 2 θ 0 - m 2 ) 1 / 2 / cos θ 0 ,
2 π = 2 k a { [ cos θ 2 ( 1 - cos Δ θ 2 ) - sin θ 2 sin Δ θ 2 ] ( 2 J - 1 ) + cos θ 0 ( 1 - cos Δ θ 0 ) - sin θ 0 sin Δ θ 0 } + δ j ( θ 0 ) - δ j ( θ 0 ) ,
δ j ( θ 0 ) m 2 ( 1 - j ) ( 8 tan θ c ) 1 / 2 ( θ 0 - θ c ) 1 / 2
Δ ϕ = λ o / B 2 - 0 [ ( λ o / B 2 ) 3 / 2 ] ,
Δ ϕ λ o / B 2 ,
I j = S j 2 ( 2 / k a ) 2
( Δ ϕ ) M = λ o / ( B 3 - B 2 ) .
S p , j = ( - i k a / 2 ) F p , j exp ( i γ p , j ) .
γ 0 , j = - 2 k a cos θ 0 - H ( ϕ c - ϕ ) δ j ,
F 0 , j = [ F ( w ) - F ( - ) ] 2 - 1 / 2 exp ( - i π / 4 ) ,
F ( w ) = 0 w exp [ i ( π / 2 ) z 2 ] d z ,
w = [ ( a / λ 0 ) cos θ c ] 1 / 2 sin ( ϕ c - ϕ ) ,
γ 1 , j = 2 k a ( m cos ρ 1 - cos θ 1 ) ,
F 1 , j = 2 ( 1 - r j 2 ) D 1 / 2 H ( ϕ c - ϕ ) ,
r 1 = sin ( ρ 1 - θ 1 ) sin ( ρ 1 + θ 1 ) ,             r 2 = tan ( θ 1 - ρ 1 ) tan ( θ 1 + ρ 1 ) ,
D = sin θ 1 cos θ 1 2 1 - ( m - 1 cos θ 1 / cos ρ 1 ) sin ϕ .
tan θ 1 = m sin ( ϕ / 2 ) [ 1 - m cos ( ϕ / 2 ) ] - 1
x cos ( ϕ + π / 2 ) + y sin ( ϕ + π / 2 ) = a ( - 1 ) l sin θ p ,
x = a ( - 1 ) l ( sin θ p cos ϕ - sin θ p cos ϕ ) / sin ( ϕ - ϕ ) , y = a ( - 1 ) l sin θ p sec ϕ + x tan ϕ .
d d θ p = θ p + ( ϕ θ p + ϕ d ρ p d ρ p d θ p ) ϕ
= θ p + 2 ( - 1 ) l ( p τ - 1 ) ϕ ,
x F = lim ϕ ϕ θ p θ p ( d / d θ p ) ( sin θ p cos ϕ - sin θ p cos ϕ ) ( d / d θ p ) [ sin ( ϕ - ϕ ) ]
= - a ( - 1 ) l sin θ p sin ϕ - ½ a ( p τ - 1 ) - 1 cos θ p cos ϕ ,
y F = a ( - 1 ) l sin θ p cos ϕ - ½ a ( p τ - 1 ) - 1 cos θ p sin ϕ .
x cos ϕ + y sin ϕ = a .
x E = a cos ϕ - a ( - 1 ) l sin θ p sin ϕ ,
y E = a sin ϕ - a ( - 1 ) l sin θ p cos ϕ .

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