Abstract

The intensity of light scattered by an air bubble in water is predicted by the geometric-optics calculation of Davis (1955) to have a divergent angular derivative as the critical scattering angle ϕc is approached. Effects of diffraction in the angular region near ϕc are described here. The Fraunhofer diffraction for scattering angles ϕϕc is estimated using a simplified physical-optics approximation. A ringing and decay of the far-field intensity is predicted that is formally similar to the near-field diffraction of a straight edge. Observation of millimeter radius bubbles in water with collimated monochromatic illumination confirm the existence of this ringing which has a quasi period ≃ 25 mrad. The diffraction calculation gives an approximate description of the relative ϕ of the observed maxima and minima. Fringes with a lower contrast and spacing ≃ 0.3 mrad were also observed; they appear to be caused by the interference of rays with distinct paths. Implications for the critical angle scattering of white light are discussed.

© 1979 Optical Society of America

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References

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  1. G. E. Davis, “Scattering of light by an air bubble in water,” J. Opt. Soc. Am. 45, 572–581 (1955).
    [Crossref]
  2. R. J. Withrington, “Light scattering by bubbles in a bubble chamber,” Appl. Opt. 7, 175–181 (1968).
    [Crossref] [PubMed]
  3. G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Phil. Soc. 6, 379–402 (1838).
  4. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  5. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am.,  236 No. 4, 116–127 (1977).
    [Crossref]
  6. V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [Crossref]
  7. D. A. Glaser, in Handbuch der Physik edited by S. Flugge, Vol. 45, (Springer-Verlag, Berlin, 1958).
  8. G. R. Fowles, Introduction to Modern Optics2nd ed. (Holt, New York, 1975, Sec. 2.7–2.10.
  9. L. A. Segel, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977) Appendix 9.1.
  10. See for example, ref. 8, Fig. 5.25.
  11. D. C. Blanchard and L. D. Syzdek, “Production of air bubbles of a specified size,” Chem. Engng Sci. 32, 1109–1112 (1977).
    [Crossref]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, San Francisco, 1968) Chap. 5.
  13. Ref. 12, Sec. 7.2.
  14. M. E. Mascart, Traite D’Optique Vol. I (Gairfhier-Villars et Fils, Paris, 1889), Sec. 195.
  15. The polarization dependence of this phenomenon was not determined in these observations.
  16. P. L. Marston and R. E. Apfel, “Acoustically forced shape oscillations of hydrocarbon drops levitated in water,” J. Colloid Interface Sci. 68, 280–286 (1979).
    [Crossref]
  17. P. L. Marston and R. E. Apfel, “Quadrupole resonance of drops driven by modulated acoustic radiation pressure-experimental properties,” (unpublished).
  18. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere,” J. Math. Phys. 10, 82–124 (1969).
    [Crossref]
  19. D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
    [Crossref]

1979 (1)

P. L. Marston and R. E. Apfel, “Acoustically forced shape oscillations of hydrocarbon drops levitated in water,” J. Colloid Interface Sci. 68, 280–286 (1979).
[Crossref]

1977 (2)

D. C. Blanchard and L. D. Syzdek, “Production of air bubbles of a specified size,” Chem. Engng Sci. 32, 1109–1112 (1977).
[Crossref]

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am.,  236 No. 4, 116–127 (1977).
[Crossref]

1974 (1)

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

1970 (1)

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

1968 (1)

1955 (1)

1838 (1)

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Phil. Soc. 6, 379–402 (1838).

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Phil. Soc. 6, 379–402 (1838).

Apfel, R. E.

P. L. Marston and R. E. Apfel, “Acoustically forced shape oscillations of hydrocarbon drops levitated in water,” J. Colloid Interface Sci. 68, 280–286 (1979).
[Crossref]

P. L. Marston and R. E. Apfel, “Quadrupole resonance of drops driven by modulated acoustic radiation pressure-experimental properties,” (unpublished).

Blanchard, D. C.

D. C. Blanchard and L. D. Syzdek, “Production of air bubbles of a specified size,” Chem. Engng Sci. 32, 1109–1112 (1977).
[Crossref]

Davis, G. E.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics2nd ed. (Holt, New York, 1975, Sec. 2.7–2.10.

Glaser, D. A.

D. A. Glaser, in Handbuch der Physik edited by S. Flugge, Vol. 45, (Springer-Verlag, Berlin, 1958).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, San Francisco, 1968) Chap. 5.

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Ludwig, D.

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

Marston, P. L.

P. L. Marston and R. E. Apfel, “Acoustically forced shape oscillations of hydrocarbon drops levitated in water,” J. Colloid Interface Sci. 68, 280–286 (1979).
[Crossref]

P. L. Marston and R. E. Apfel, “Quadrupole resonance of drops driven by modulated acoustic radiation pressure-experimental properties,” (unpublished).

Mascart, M. E.

M. E. Mascart, Traite D’Optique Vol. I (Gairfhier-Villars et Fils, Paris, 1889), Sec. 195.

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am.,  236 No. 4, 116–127 (1977).
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

Segel, L. A.

L. A. Segel, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977) Appendix 9.1.

Syzdek, L. D.

D. C. Blanchard and L. D. Syzdek, “Production of air bubbles of a specified size,” Chem. Engng Sci. 32, 1109–1112 (1977).
[Crossref]

Van de Hulst, H. C.

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

Withrington, R. J.

Appl. Opt. (1)

Chem. Engng Sci. (1)

D. C. Blanchard and L. D. Syzdek, “Production of air bubbles of a specified size,” Chem. Engng Sci. 32, 1109–1112 (1977).
[Crossref]

J. Colloid Interface Sci. (1)

P. L. Marston and R. E. Apfel, “Acoustically forced shape oscillations of hydrocarbon drops levitated in water,” J. Colloid Interface Sci. 68, 280–286 (1979).
[Crossref]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. Lett. (1)

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am.,  236 No. 4, 116–127 (1977).
[Crossref]

Trans. Cambridge Phil. Soc. (1)

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Phil. Soc. 6, 379–402 (1838).

Other (10)

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

P. L. Marston and R. E. Apfel, “Quadrupole resonance of drops driven by modulated acoustic radiation pressure-experimental properties,” (unpublished).

D. A. Glaser, in Handbuch der Physik edited by S. Flugge, Vol. 45, (Springer-Verlag, Berlin, 1958).

G. R. Fowles, Introduction to Modern Optics2nd ed. (Holt, New York, 1975, Sec. 2.7–2.10.

L. A. Segel, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977) Appendix 9.1.

See for example, ref. 8, Fig. 5.25.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, San Francisco, 1968) Chap. 5.

Ref. 12, Sec. 7.2.

M. E. Mascart, Traite D’Optique Vol. I (Gairfhier-Villars et Fils, Paris, 1889), Sec. 195.

The polarization dependence of this phenomenon was not determined in these observations.

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Figures (5)

FIG. 1
FIG. 1

Total reflection in the plane of scattering of a ray with an impact parameter b larger than the critical impact.parameter bc. Only one quadrant of the cross section of the bubble is shown.

FIG. 2
FIG. 2

Illustration of the geometric procedure for locating the coordinates (x,y) of a point Q on the curved virtual wave front. The upper line of length h is parallel to the x axis; h is also the distance between its point of contact with the bubble’s surface [labeled (h,s)] and the point Q. The rotated coordinate axes u,v are introduced to facilitate a Taylor expansion of the wave front.

FIG. 3
FIG. 3

View of the apparatus in the plane of scattering which was horizontal in the laboratory. The bottom of the bubble was in contact with a needle that prevented the bubble from floating out of the scattering plane. The top of the bubble (which is the side displayed here) was viewed by a microscope. The wavelength in air of the illumination was 632.8 nm.

FIG. 4
FIG. 4

Photograph of the critical angle far-field diffraction from a 2.15-mm-diam bubble. The scattering angle ϕ decreases from left to right. The vertical bands have a characteristic separation of approximately 25 mrads; they manifest the ringing of the intensity caused by diffraction. The incident beam was polarized with its electric field parallel to the scattering plane.

FIG. 5
FIG. 5

The dashed curve is the smoothed relative intensity inferred from a microdensitometer scan of the photographic negative. The bubble diameter was 1.65 mm. The solid curve is given by Eq. (30) which is the result of the physical optics approximation. Two parameters were adjusted to facilitate this comparison.

Equations (33)

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ϕ c = π 2 θ c ,
sin θ c = n 1 ,
x = h [ 1 ( 1 + m 2 ) 1 / 2 ] ,
y = s m h ( 1 + m 2 ) 1 / 2 ,
m = tan ϕ = tan [ 2 cos 1 ( b / a ) ] .
u = x sin 2 θ c + y cos 2 θ c ,
υ = x cos 2 θ c + y sin 2 θ c .
υ = α u 2 ,
α = 1 / a cos θ c .
U ( u , t ) = r j e i ( k α u 2 ω t ) ,
r 1 = cos θ ( n 2 sin 2 θ ) 1 / 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 ,
r 2 = n 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 n 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 .
r 1 1 ( 8 / n cos θ c ) 1 / 2 ,
r 2 1 + n 2 ( 8 / n cos θ c ) 1 / 2 ,
tan ( δ 1 / 2 ) = ( sin 2 θ n 2 ) 1 / 2 / cos θ ,
tan ( δ 2 / 2 ) = n 2 tan ( δ 1 / 2 ) .
δ 1 ( 8 / n cos θ c ) 1 / 2 ,
δ 2 n 2 ( 8 / n cos θ c ) 1 / 2 .
β 1 = ( 8 / n a ) 1 / 2 / cos θ c ,
β 2 = n 2 β 1 .
f j ( η ) = r j ( u ) e i k ( α u 2 η u ) d u ,
f j = ± 0 e i [ k ( α u 2 + η u ) + β j u 1 / 2 ] d u ,
w = η [ ( a / λ ) cos θ c ] 1 / 2 .
g = e i [ ψ j ( w , 0 ) + k η 2 / 4 α ] w e i ψ j ( z , w ) e i ( π / 2 ) z 2 d z ,
ψ j ( z , w ) = β j 2 1 / 2 [ ( w + z ) 1 / 2 w 1 / 2 ] ( λ a cos θ c ) 1 / 4 .
g = e i [ ψ j ( w , 0 ) + k η 2 / 4 α ] [ F ( w ) F ( ) ] ,
F ( w ) = 0 w e i ( π / 2 ) z 2 d z = C ( w ) + i S ( w ) ,
C ( w ) = 0 w cos ( π 2 z 2 ) d z ,
S ( w ) = 0 w sin ( π 2 z 2 ) d z ,
g g * = [ C ( w ) + ½ ] 2 + [ S ( w ) + ½ ] 2 .
I = I 0 ( a / R ) 2 g g * / 8 ,
I t = I 0 ( a / R ) 2 [ g g * / 8 + G ( ϕ ) ] ,
I e = K ( τ 1 / γ τ 0 1 / γ )