Abstract

A physical-optics approximation is derived for light scattering by dielectric spheres with refractive indices less than their surroundings, and it is applied to air bubbles in water. The approximation gives the coarse structure in the scattering when the scattering angle Ø is near the critical scattering angle Ø<sub>c</sub>, where Ø<sub>c</sub> ࣃ 83° for bubbles in water. Diffraction has been observed to be important in the critical region because of an abrupt change in the amplitude of the reflected wave [P. L. Marston, J. Opt. Soc. Am. 69, 1205–1211 (1980)]. Interference that is due to a refracted wave produces oscillations in the intensity with an angular quasi-period of magnitude (λ/a)<sup>½</sup> rad near Ø<sub>c</sub>, where λ is the wavelength and a is the radius. Unlike diffraction-related oscillations, the interference oscillations increase in magnitude as forward scattering is approached. Optical tunneling through bubbles is also discussed.

© 1981 Optical Society of America

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. H. M. Nussenzveig, "Complex angular momentum theory of the rainbow and the glory," J. Opt. Soc. Am. 69, 1068–1079 (1979).
  3. G. P. Können and J. H. de Boer, "Polarized rainbow," Appl. Opt. 18, 1961–1965 (1979).
  4. P. L. Marston, "Critical angle scattering by a bubble: physicaloptics approximation and observations," J. Opt. Soc. Am. 69, 1205–1211 (1979); 70, 353(E) (1980).
  5. D. L. Kingsbury and P. L. Marston, "Mie scattering near the critical angle of bubbles in water," J. Opt. Soc. Am., to be published (March 1980).
  6. Unlike in Ref. 4, h0,2 and r2 are written here without minus signs in front of their right-hand sides because this convention gives a description of the interference that is convenient for comparison with Mie theory.
  7. G. E. Davis, "Scattering of light by an air bubble in water," J. Opt. Soc. Am. 45, 572–581 (1955).
  8. H. K. V. Lotsch, "Reflection and refraction of a beam of light at a plane interface," J. Opt. Soc. Am. 58, 551–561 (1968).
  9. Ref. 8, footnote 62.
  10. L. A. Segel, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Appendix 9.1.
  11. Equations (18) and (19) differ from Ref. 4 because it is not assumed in these equations that sin η ࣃ η.
  12. W. T. Welford, "Illumination and photography of bubble chambers," in Bubble and Spark Chambers, R. P. Shutt, ed. (Academic, New York, 1967), Vol. 1, pp. 233–313.
  13. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 301–302, Eqs. 7.3.32, 7.3.33, 7.3.9, 7.3.10.
  14. G. Mie, "Beiträge zur Optik triber Medien, speziell kolloidaler Metallosungen," Ann. Phys. (Leipzig) 25, 377–445 (1908).
  15. W. J. Wiscombe, "Improved Mie scattering algorithms," Appl. Opt. 19, 1505–1509 (1980).
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 27–35.
  17. The absolute phases of S1, and S2 are not calculated here because the coordinate system used in Sec. 1 was not centered on the bubble and because wave-front curvature orthogonal to the scattering plane introduces an undetermined phase factor independent of the polarization. Equation (25) is to be used in the evaluation of Eq. (28).
  18. P. L. Marston, "Rainbow phenomena and the detection of nonsphericity in drops," Appl. Opt. 19, 680–685 (1980).
  19. J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.
  20. P. L. Marston, "Critical angle diffraction in high frequency scattering by fluid spheres and cylinders," J. Acoust. Soc. Am. 66, S80(A) (1979); 67, 718(E) (1980).
  21. I. N. Court and F. K. von Willisen, "Frustrated total internal reflection and applications of its principle to laser cavity design," Appl. Opt. 3, 719–726 (1964).
  22. A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface—electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).

1980 (2)

1979 (2)

1975 (1)

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface—electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).

1968 (1)

1964 (1)

1956 (1)

J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.

1955 (1)

1908 (1)

G. Mie, "Beiträge zur Optik triber Medien, speziell kolloidaler Metallosungen," Ann. Phys. (Leipzig) 25, 377–445 (1908).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 301–302, Eqs. 7.3.32, 7.3.33, 7.3.9, 7.3.10.

Court, I. N.

Davis, G. E.

de Boer, J. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 27–35.

Keller, J. B.

J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.

Kingsbury, D. L.

D. L. Kingsbury and P. L. Marston, "Mie scattering near the critical angle of bubbles in water," J. Opt. Soc. Am., to be published (March 1980).

Können, G. P.

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.

Lotsch, H. K. V.

Love, J. D.

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface—electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).

Marston, P. L.

P. L. Marston, "Rainbow phenomena and the detection of nonsphericity in drops," Appl. Opt. 19, 680–685 (1980).

D. L. Kingsbury and P. L. Marston, "Mie scattering near the critical angle of bubbles in water," J. Opt. Soc. Am., to be published (March 1980).

P. L. Marston, "Critical angle scattering by a bubble: physicaloptics approximation and observations," J. Opt. Soc. Am. 69, 1205–1211 (1979); 70, 353(E) (1980).

P. L. Marston, "Critical angle diffraction in high frequency scattering by fluid spheres and cylinders," J. Acoust. Soc. Am. 66, S80(A) (1979); 67, 718(E) (1980).

Mie, G.

G. Mie, "Beiträge zur Optik triber Medien, speziell kolloidaler Metallosungen," Ann. Phys. (Leipzig) 25, 377–445 (1908).

Nussenzveig, H. M.

Seckler, B. D.

J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.

Segel, L. A.

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

Snyder, A. W.

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface—electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 301–302, Eqs. 7.3.32, 7.3.33, 7.3.9, 7.3.10.

van de Hulst, H. C.

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

von Willisen, F. K.

Welford, W. T.

W. T. Welford, "Illumination and photography of bubble chambers," in Bubble and Spark Chambers, R. P. Shutt, ed. (Academic, New York, 1967), Vol. 1, pp. 233–313.

Wiscombe, W. J.

Ann. Phys. Leipzig (1)

G. Mie, "Beiträge zur Optik triber Medien, speziell kolloidaler Metallosungen," Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (4)

Commun. Pure Appl. Math. (1)

J. B. Keller, R. M. Lewis, and B. D. Seckler, "Asymptotic solution of some diffraction problems," Commun. Pure Appl. Math. 9, 207–265 (1956), example 19.

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface—electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).

J. Opt. Soc. Am. (3)

Other (12)

Ref. 8, footnote 62.

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

Equations (18) and (19) differ from Ref. 4 because it is not assumed in these equations that sin η ࣃ η.

W. T. Welford, "Illumination and photography of bubble chambers," in Bubble and Spark Chambers, R. P. Shutt, ed. (Academic, New York, 1967), Vol. 1, pp. 233–313.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 301–302, Eqs. 7.3.32, 7.3.33, 7.3.9, 7.3.10.

P. L. Marston, "Critical angle scattering by a bubble: physicaloptics approximation and observations," J. Opt. Soc. Am. 69, 1205–1211 (1979); 70, 353(E) (1980).

D. L. Kingsbury and P. L. Marston, "Mie scattering near the critical angle of bubbles in water," J. Opt. Soc. Am., to be published (March 1980).

Unlike in Ref. 4, h0,2 and r2 are written here without minus signs in front of their right-hand sides because this convention gives a description of the interference that is convenient for comparison with Mie theory.

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

P. L. Marston, "Critical angle diffraction in high frequency scattering by fluid spheres and cylinders," J. Acoust. Soc. Am. 66, S80(A) (1979); 67, 718(E) (1980).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 27–35.

The absolute phases of S1, and S2 are not calculated here because the coordinate system used in Sec. 1 was not centered on the bubble and because wave-front curvature orthogonal to the scattering plane introduces an undetermined phase factor independent of the polarization. Equation (25) is to be used in the evaluation of Eq. (28).

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