Abstract

Light scattered from spherical bubbles in water manifests an enhancement in the backward direction analogous to the well-known optical glory of a drop. Unlike the glory for water drops, in which the rays travel partially on the drop’s surface, the glory for bubbles is due to rays that are refracted and multiply reflected within the scatterer and is an example of a transmitted wave glory. Photographs of glory scattering for freely rising air bubbles in water are displayed for bubbles having diameters of less than 300 μm. A physical-optics model for backscattering is developed for spherical bubbles. Computed glory patterns from both Mie-series calculations and the physical-optics model agree with the observed patterns. The patterns of freely rising air bubbles having a diameter of ≲300 μm are essentially those predicted for a spherical scatterer. The interference of different classes of glory rays is more clearly seen for bubbles in water than for the previously studied case of bubbles in oil.

© 1988 Optical Society of America

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References

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  1. H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc Am. 37, 16–22 (1947);Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.
  2. H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–69 (July1974).
  3. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys Rev. Lett. 38, 1279–1282 (1977);H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory” J Opt. Soc. Am. 69, 1068–1079 (1979).
  4. D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
  5. P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
  6. P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
  7. P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt Instrum. Eng.489, 130–141 (1984).
  8. D. S. Langley, P. L. Marston, “Critical angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
  9. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).
  10. W. P. Arnott, P. L. Marston, “Backscattering from a slightly spheroidal air bubble in water: a novel unfolding of the optical glory,” J. Opt. Soc. Am. 3 (13), P117 (1986).
  11. P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983);erratum, 78, 1128 (1985).
  12. J. W. Goodman, Introduction to Fourier Optks (McGraw-HillNew York, 1968).
  13. Some notational differences with the previous work in Refs. 4–7 should be noted. The index n here was previously denoted by ℓ.The azimuth φ of the observation direction was previously denoted by ρ. In Refs. 4 and 5 the angle of refraction ν of a ray within the bubble is denoted by ρ. The copolarized amplitudes were often referred to as polarized amplitudes. The function denoted as W’ in Ref. 4 is denoted here as Q[see Eq. (13)].
  14. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
  15. M. V. Berry, “Waves and Thorn’s theorem,” Adv. Phys. 25, 1–26 (1976).
  16. G. E. Davis, “Scattering of light by an air bubble in water,” J. Opt. Soc. Am. 45, 572–581 (1955).
  17. P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).
  18. M. V. Berry, “Uniform approximations for glory scattering and diffraction peaks,” J. Phys. B 2, 381–392 (1969).
  19. This merging of azimuthally adjacent rays is also present for water drops and addresses the “physical essence of the explanation of the glory” requested in R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1980), p. 146.
  20. P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816—1818 (1983).
  21. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
  22. H. M. Nussenzvieg, W. J. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
  23. P. L. Marston, D. S. Langley, “Forward optical glory from bubbles (and clouds of bubbles) in liquids and other novel directional caustics,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. V. Varadan, V. K. Varadan, eds. (Pennsylvania State U. Press, University Park, Pa., 1987), pp. 419–429.
  24. D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984), Chap. 3.
  25. J. J. Stephens, P. S. Ray, T. W. Kitterman, “Far-field impulse response verification of selected high-frequency optics backscattering analogs,” Appl. Opt. 14, 2169–2176 (1975).
  26. T. R. Zhang, C. DeWitt-Morette, “WKB cross section for polarized glories of massless waves in curved space-times,” Phys. Rev. Lett. 52, 2313–2316 (1984); Eq. (13) therein.
  27. R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

1986 (1)

W. P. Arnott, P. L. Marston, “Backscattering from a slightly spheroidal air bubble in water: a novel unfolding of the optical glory,” J. Opt. Soc. Am. 3 (13), P117 (1986).

1985 (1)

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

1984 (3)

D. S. Langley, P. L. Marston, “Critical angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).

T. R. Zhang, C. DeWitt-Morette, “WKB cross section for polarized glories of massless waves in curved space-times,” Phys. Rev. Lett. 52, 2313–2316 (1984); Eq. (13) therein.

P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).

1983 (2)

P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816—1818 (1983).

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983);erratum, 78, 1128 (1985).

1982 (2)

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).

1981 (1)

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).

1980 (3)

1977 (1)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys Rev. Lett. 38, 1279–1282 (1977);H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory” J Opt. Soc. Am. 69, 1068–1079 (1979).

1976 (1)

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

1975 (1)

1974 (1)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–69 (July1974).

1969 (1)

M. V. Berry, “Uniform approximations for glory scattering and diffraction peaks,” J. Phys. B 2, 381–392 (1969).

1955 (1)

1947 (1)

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc Am. 37, 16–22 (1947);Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.

Arnott, W. P.

W. P. Arnott, P. L. Marston, “Backscattering from a slightly spheroidal air bubble in water: a novel unfolding of the optical glory,” J. Opt. Soc. Am. 3 (13), P117 (1986).

Berry, M. V.

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

M. V. Berry, “Uniform approximations for glory scattering and diffraction peaks,” J. Phys. B 2, 381–392 (1969).

Boyd, R. W.

Bryant, H. C.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–69 (July1974).

Clift, R.

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

Davis, G. E.

DeWitte-Morette, C.

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

DeWitt-Morette, C.

T. R. Zhang, C. DeWitt-Morette, “WKB cross section for polarized glories of massless waves in curved space-times,” Phys. Rev. Lett. 52, 2313–2316 (1984); Eq. (13) therein.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optks (McGraw-HillNew York, 1968).

Grace, J. R.

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

Greenler, R.

This merging of azimuthally adjacent rays is also present for water drops and addresses the “physical essence of the explanation of the glory” requested in R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1980), p. 146.

Hanson, T. J. B.

P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).

Jarmie, N.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–69 (July1974).

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys Rev. Lett. 38, 1279–1282 (1977);H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory” J Opt. Soc. Am. 69, 1068–1079 (1979).

Kingsbury, D. L.

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).

Kitterman, T. W.

Langley, D. S.

D. S. Langley, P. L. Marston, “Critical angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983);erratum, 78, 1128 (1985).

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).

P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt Instrum. Eng.489, 130–141 (1984).

P. L. Marston, D. S. Langley, “Forward optical glory from bubbles (and clouds of bubbles) in liquids and other novel directional caustics,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. V. Varadan, V. K. Varadan, eds. (Pennsylvania State U. Press, University Park, Pa., 1987), pp. 419–429.

D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984), Chap. 3.

Marston, P. L.

W. P. Arnott, P. L. Marston, “Backscattering from a slightly spheroidal air bubble in water: a novel unfolding of the optical glory,” J. Opt. Soc. Am. 3 (13), P117 (1986).

P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).

D. S. Langley, P. L. Marston, “Critical angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).

P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816—1818 (1983).

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983);erratum, 78, 1128 (1985).

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).

P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt Instrum. Eng.489, 130–141 (1984).

P. L. Marston, D. S. Langley, “Forward optical glory from bubbles (and clouds of bubbles) in liquids and other novel directional caustics,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. V. Varadan, V. K. Varadan, eds. (Pennsylvania State U. Press, University Park, Pa., 1987), pp. 419–429.

Matzner, R. A.

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

Nelson, B.

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys Rev. Lett. 38, 1279–1282 (1977);H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory” J Opt. Soc. Am. 69, 1068–1079 (1979).

Nussenzvieg, H. M.

Ray, P. S.

Stephens, J. J.

van de Hulst, H. C.

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc Am. 37, 16–22 (1947);Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.

Weber, M. E.

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

Williams, K. L.

P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).

Wiscombe, W. J.

Zhang, T. R.

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

T. R. Zhang, C. DeWitt-Morette, “WKB cross section for polarized glories of massless waves in curved space-times,” Phys. Rev. Lett. 52, 2313–2316 (1984); Eq. (13) therein.

Adv. Phys. (1)

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

Appl. Opt. (3)

Appl. Sci. Res. (1)

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).

J. Acoust. Soc. Am. (2)

P. L. Marston, K. L. Williams, T. J. B. Hanson, “Observation of the acoustic glory: high-frequency backscattering from an elastic sphere,” J. Acoust. Soc. Am. 74, 1555–1563 (1984).

P. L. Marston, D. S. Langley, “Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: models for diffracted axial focusing,” J. Acoust. Soc. Am. 73, 1464–1475 (1983);erratum, 78, 1128 (1985).

J. Opt. Soc Am. (1)

H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc Am. 37, 16–22 (1947);Light Scattering by Small Particles (Wiley, New York, 1957), p. 207.

J. Opt. Soc. Am. (5)

J. Phys. B (1)

M. V. Berry, “Uniform approximations for glory scattering and diffraction peaks,” J. Phys. B 2, 381–392 (1969).

Opt. Lett. (1)

Phys Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys Rev. Lett. 38, 1279–1282 (1977);H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory” J Opt. Soc. Am. 69, 1068–1079 (1979).

Phys. Rev. D (1)

R. A. Matzner, C. DeWitte-Morette, B. Nelson, T. R. Zhang, “Glory scattering by black holes,” Phys. Rev. D 31, 1869–1878 (1985); Fig. 3 therein.

Phys. Rev. Lett. (2)

T. R. Zhang, C. DeWitt-Morette, “WKB cross section for polarized glories of massless waves in curved space-times,” Phys. Rev. Lett. 52, 2313–2316 (1984); Eq. (13) therein.

D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).

Sci. Am. (1)

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–69 (July1974).

Other (7)

P. L. Marston, D. S. Langley, “Strong backscattering and cross polarization from bubbles and glass spheres in water,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt Instrum. Eng.489, 130–141 (1984).

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

This merging of azimuthally adjacent rays is also present for water drops and addresses the “physical essence of the explanation of the glory” requested in R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1980), p. 146.

J. W. Goodman, Introduction to Fourier Optks (McGraw-HillNew York, 1968).

Some notational differences with the previous work in Refs. 4–7 should be noted. The index n here was previously denoted by ℓ.The azimuth φ of the observation direction was previously denoted by ρ. In Refs. 4 and 5 the angle of refraction ν of a ray within the bubble is denoted by ρ. The copolarized amplitudes were often referred to as polarized amplitudes. The function denoted as W’ in Ref. 4 is denoted here as Q[see Eq. (13)].

P. L. Marston, D. S. Langley, “Forward optical glory from bubbles (and clouds of bubbles) in liquids and other novel directional caustics,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. V. Varadan, V. K. Varadan, eds. (Pennsylvania State U. Press, University Park, Pa., 1987), pp. 419–429.

D. S. Langley, “Light scattering from bubbles in liquids,” doctoral dissertation (Washington State University, Pullman, Wash., 1984), Chap. 3.

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Figures (11)

Fig. 1
Fig. 1

Backward-directed rays for a bubble in water with m = 0.7496. The numbers on the right-hand side refer to the number of chords in the bubble.

Fig. 2
Fig. 2

Coordinate system for the calculation of the fields in the exit and observation planes.

Fig. 3
Fig. 3

General wave-front shape in the exit plane for each class of glory ray. For clarity, the quadratic approximation given by Eq. (4) has been extended beyond its useful domain.

Fig. 4
Fig. 4

Physical-optics and Mie-theory comparisons of the copolarized and cross-polarized backscattering patterns for (a) ka = 100 and (b) ka = 1000. The solid lines (physical optics) and short-dashed lines (Mie series) represent computations for the copolarized component. The long-dashed lines (physical optics) and the dashed-dotted line (Mie series) represent computations for the cross-polarized component. The scale on the left-hand side is for the copolarized component. These calculations correspond to cases in which the azimuthal angle φ has the value ±45 or ±135 deg. For other values of φ, the cross-polarized irradiance for both the Mie-theory model and the physical-optics model may be inferred, since each is proportional to sin2 (2 φ).

Fig. 5
Fig. 5

Comparison of Mie-theory and model results of cross-polarized irradiance for a spherical bubble in water with ka = 1988 relevant to Fig. 8(b). The angular scale is typical of the observations in Figs. 7 and 8 below.

Fig. 6
Fig. 6

Top view of the experimental apparatus for observing the cross-polarized backscattering. A measuring microscope (not shown) is mounted above the scattering cell for bubble-diameter measurements. The bubble size relative to the other apparatus and the tilt of the cell windows have been exaggerated.

Fig. 7
Fig. 7

Photograph of cross-polarized near backscattering for a bubble of radius 122.0 μm with ka = 1988, showing the fourfold symmetric azimuthal dependence characteristic of spherical scattered.This is the experimental result used in Fig. 8(b).

Fig 8
Fig 8

Photographs of cross-polarized near backscattering by a freely rising bubble in water. The first and third quadrants of each composite pattern are computer-generated results from physical optics and Mie theory, respectively. The bubble radii and ka values for (a)–(d) are given in Table 2. In (a), the experimental photo has extraneous background scattering near the origin. White lines are borders of the synthesized patterns.

Fig. 9
Fig. 9

Distances and angles needed to describe scattering caused by the reflected axial ray. θ is the incident angle, and γ is the backscattering angle.

Fig. 10
Fig. 10

Distances and angles needed to describe scattering caused by the single-bounce axial ray. θ and γ are the incident and backscattered angles, respectively. The refraction angle v is illustrated for m = 0.6.

Fig. 11
Fig. 11

Model cross-polarized irradiance at a series of fixed angles measured from an experimental photograph for various bubble radii to ascribe a best-fit radius to experimental data. Ring numbers and angular locations for curves a-h are given in Table 3. Lines are correctly ordered in magnitude for a bubble radius of ≈130 μmas indicated by an arrow on the axis. This result is pertinent to Fig. 8(c).

Tables (3)

Tables Icon

Table 1 Focal Circle Parameters for m = 0.7496

Tables Icon

Table 2 Experimental Data for Fig. 8

Tables Icon

Table 3 Angular Locations of Bright and Dark Rings in Fig. 8(c)

Equations (32)

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b p = a sin θ p ,
α p = a ( 1 + cos θ p 2 ( p τ p 1 ) ) ,
q p = | α p α p a | ,
E n , p g ( s , ψ ) = E i q 1 / 2 F n ( ψ ) exp [ i η ¯ + i k ( s b ) 2 2 α ] , n = 1 , 2
F 1 ( ψ ) = C 1 sin 2 ψ + C 2 cos 2 ψ ,
F 2 ( ψ ) = ( C 1 C 2 ) sin 2 ψ 2 ,
C j = ( 1 ) p ( 1 j ) r j ( p 1 ) ( 1 r j 2 ) , j = 1 , 2 ,
r 1 = sin ( θ p ν p ) sin ( θ p + ν p ) ,
r 2 = tan ( θ p ν p ) tan ( θ p + ν p ) .
η ¯ = μ + 2 k a ( 1 cos θ p + p m cos ν p ) .
E n , p g ( γ , φ ) = k E i 2 π i q 1 / 2 exp ( ikr + i η ¯ ) r D n ( γ , φ ) ,
D n = { F n ( ψ ) exp [ i k ( s b ) 2 / 2 α ] } × exp [ i k ( x x + y y ) / r ] d x d y
D n ( γ , φ ) = 0 2 π F n ( ψ ) Q ( γ , φ ; ψ ) d ψ ,
Q ( γ , φ ; ψ ) = 0 s exp i k ( s b ) 2 2 α exp [ i u cos ( ψ φ ) ] d s ,
F = exp ( i k α p cos γ ) F p .
Q p ( γ , φ ; ψ ) = B p b exp [ i u p cos ( φ ψ ) ] ,
Q ( 0 , φ ; ψ ) ( 2 π | α | k ) 1 / 2 exp ( i π / 4 ) , k α 1 , k b 1 .
Q ( γ , φ ; ψ ) b ( 2 π | α | k ) 1 / 2 exp ( i π / 4 ) × exp [ i k α ( 1 cos γ ) ] exp [ i u p cos ( ψ φ ) ] .
D n ( γ , φ ) = b ( 2 π | α | k ) 1 / 2 × exp ( i π / 4 ) exp [ i k α ( 1 cos γ ) ] V n ( γ , φ ) ,
V n ( γ , φ ) = π { [ c 1 c 2 ] J 2 ( u p ) G n ( φ ) + δ 1 , n [ c 1 + c 2 ] J 0 ( u p ) } ,
E n s = p E n , p g + δ 1 , n p E n , p a ,
I n = I R 4 r 2 a 2 | E n s E i | 2 ,
J 2 ( u 3 ) ( 2 π u 3 ) 1 / 2 cos ( u 3 5 π 4 ) , cos ( u 3 5 π 4 ) = exp ( i 5 π 4 ) exp ( i u 3 i π 2 ) + exp ( i u 3 ) 2 ,
k b 3 Δ γ = π .
I 1 = I R | S S + cos 2 φ | 2 ( k a ) 2 ,
I 2 = I R | S + sin 2 φ | 2 ( k a ) 2 ,
E 1 , p a ( γ ) = E i h p c p a r exp ( ikr + i ζ p + μ ) ,
h p 2 = ( d σ d Ω ) p = b p sin γ | d b p d γ | ,
ζ 0 = 2 k a [ 1 cos ( γ / 2 ) ] k a ( 1 cos γ ) .
ζ 2 = 2 k a [ ( 1 cos θ ) + 2 m cos ν ] k a ( 1 cos γ ) ,
θ m γ 2 ( 2 m ) .
( d σ d Ω ) 2 = a 2 4 sin ( m γ 2 m ) sin γ m 2 m a 2 4 ( m 2 m ) 2

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