Abstract

The scattering enhancement known as the glory was observed in forward scattering from bubbles in liquids. A physical-optics model of the forward glory is detailed, based on transmitted waves reflected within the bubble. Some aspects of the model are compared with the Mie theory and with features in the cross-polarized light from single bubbles. Clouds of small bubbles rising in water show an angular structure in the forward glory light that is useful for estimating the bubble size.

© 1991 Optical Society of America

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References

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  1. H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
    [CrossRef]
  2. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  3. V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.
  4. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  5. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  6. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  7. D. S. Langley, P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981).
    [CrossRef]
  8. W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
    [CrossRef]
  9. 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).
    [CrossRef]
  10. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 105–114.
  11. H. M. Nussenzveig, W. J. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
    [CrossRef] [PubMed]
  12. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.
  14. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  15. G. E. Davis, “Scattering of light by an air bubble in water,” J. Opt. Soc. Am. 45, 572–581 (1955).
    [CrossRef]
  16. J. McK. Ellison, C. V. Peetz, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London Ser. B 74, 105–123 (1959).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  18. P. L. Marston, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
    [CrossRef]
  19. 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).
    [CrossRef]
  20. 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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the computations in Figs. 8 and 9 are done correctly.]
    [CrossRef]

1988

1987

1984

1983

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

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).
[CrossRef]

1982

1981

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

1980

1977

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

1974

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

1959

J. McK. Ellison, C. V. Peetz, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London Ser. B 74, 105–123 (1959).

1955

1908

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Arnott, W. P.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Bryant, H. C.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

Davis, G. E.

Ellison, J. McK.

J. McK. Ellison, C. V. Peetz, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London Ser. B 74, 105–123 (1959).

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 105–114.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Jarmie, N.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.

Langley, D. S.

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).
[CrossRef]

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).
[CrossRef]

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

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the computations in Figs. 8 and 9 are done correctly.]
[CrossRef]

Lock, J. A.

Marston, P. L.

W. P. Arnott, P. L. Marston, “Optical glory of small 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, “Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories,” J. Opt. Soc. Am. 73, 1816–1818 (1983).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the computations in Figs. 8 and 9 are done correctly.]
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
[CrossRef] [PubMed]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Peetz, C. V.

J. McK. Ellison, C. V. Peetz, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London Ser. B 74, 105–123 (1959).

Probert-Jones, J. R.

van de Hulst, H. C.

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

Wiscombe, W. J.

Ann. Phys.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt.

J. Acoust. Soc. Am.

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).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. Lett.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

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

Proc. Phys. Soc. London Ser. B

J. McK. Ellison, C. V. Peetz, “The forward scattering of light by spheres according to geometrical optics,” Proc. Phys. Soc. London Ser. B 74, 105–123 (1959).

Sci. Am.

H. C. Bryant, N. Jarmie, “The glory,” Sci. Am. 231, 60–71 (1974).
[CrossRef]

Other

V. Khare, “Surface waves and rainbow effects in the optical glory,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 417–464.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 105–114.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

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

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). [Note: the numerical factor in Eq. (21b) should be 0.0235, not 0.042, but the computations in Figs. 8 and 9 are done correctly.]
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Diagram of ray paths having P = 2 and P = 3 chords inside a bubble. The resulting glory waves appear to come from ringlike sources that cut the plane at F2 and F3, respectively. The incidence and refraction angles for a ray are θ and ρ, and b represents the radius of its ringlike source.

Fig. 2
Fig. 2

Comparison of the physical-optics model and the exact Mie theory for near-forward, cross-polarized scattering from a sphere of size parameter x = 4000. The locations of maxima and minima are accurately modeled by the (2, 0) wave alone. The irradiance approximation is improved when the (3, 0) wave is included.

Fig. 3
Fig. 3

Diagram of apparatus for observing forward glory effects from bubbles.

Fig. 4
Fig. 4

Photographs of (a) copolarized and (b) cross-polarized virtual sources for forward glory waves. The bubble has a radius of 1.88 mm and is nearly motionless in a silicon oil of refractive index 1.403. Several source rings are discernible in the original negatives.

Fig. 5
Fig. 5

Measured radii of ringlike sources inside bubbles. The horizontal lines show the sequence of radii expected for ray paths with P chords. The top two lines and data sets have been shifted upward to facilitate comparison of the data with the appropriate ring in the sequence.

Fig. 6
Fig. 6

Photograph of the far-zone cross-polarized scattering from a bubble of radius 0.197 mm in a viscous silicon oil. The incident light polarization was along the vertical direction.

Fig. 7
Fig. 7

Measured angular spacing of the far-zone ring structure from 11 photographs such as Fig. 6 and from Mie computations for 8 bubble sizes. The line shows the spacing predicted by condition (14) for the (2, 0) glory wave.

Fig. 8
Fig. 8

Photograph of cross-polarized, near-forward scattering from small bubbles rising in water. The ring structure indicates an average bubble radius of ≈90 μm.

Tables (2)

Tables Icon

Table I Glory Waves and Physical Optics Model Results for a Bubble with Size Parameter x = 4000 In Liquids Giving m = 1/1.403 and m = 0.75

Tables Icon

Table II Analytic Expressions for the Incidence Angles of Several (P, L) Forward Glory Rays

Equations (14)

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

sin θ = m sin ρ .
ϕ = 2 θ - 2 P ρ + ( P - 1 ) π ,
η ( s ) = η 0 + k ( s - b ) 2 / 2 α ,
α = a [ 1 + sin θ / 2 ( P tan ρ - tan θ ) ]
E j ( s , ψ ) = E i F j ( ψ ) q - 1 / 2 exp ( i η + i μ ) ,
F 1 ( ψ ) = c 1 sin 2 ψ + c 2 cos 2 ψ ,             F 2 ( ψ ) = ½ ( c 2 - c 1 ) sin 2 ψ .
c j = r j P - 1 ( 1 - r j 2 ) ( - 1 ) ( P - 1 ) ( j - 1 ) ,             j = 1 , 2 ,
E j ( R , ϕ , ξ ) k E i exp ( i k R + i μ ) 2 π i R q 1 / 2 D j ( ϕ , ξ ) ,
D j ( ϕ , ξ ) = 0 s W j exp ( i η ) d s ,
W 1 ( ϕ , ξ , s ) = π ( c 1 + c 2 ) J 0 ( k s sin ϕ ) + π ( c 1 - c 2 ) J 2 ( k s sin ϕ ) cos 2 ξ ,
W 2 ( ϕ , ξ , s ) = π ( c 1 - c 2 ) J 2 ( k s sin ϕ ) sin 2 ξ
D j b λ 0 α 1 / 2 W j ( ϕ , ξ , b ) exp [ i k η 0 + i π / 4 - i k α ( 1 - cos ϕ ) ] .
I j = E j 2 2 b 2 ( α - a ) π a 3 x I R W j 2 ,
Δ ϕ λ 0 / 2 b ,

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