Abstract

The contributions of complex rays and the secondary radiation shed by surface waves to scattering by a dielectric sphere are calculated in the context of the Debye-series expansion of the Mie scattering amplitudes. Also, the contributions of geometrical rays are reviewed and compared with those of the Debye series. Interference effects among surface waves, complex rays, and geometrical rays are calculated, and the possibility of observing these interference effects is discussed. Experimental data supporting the observation of a surface-wave–geometrical-ray-interference pattern are presented.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 9.
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 4.
  4. B. Van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).
  5. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,”J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  6. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. Theory of the rainbow and the glory,”J. Math. Phys. 10, 125–176 (1969).
    [CrossRef]
  7. V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1976).
  8. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  9. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [CrossRef]
  10. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  11. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  12. J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
    [CrossRef]
  13. J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications: Proceedings of the Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27–52.
    [CrossRef]
  14. V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [CrossRef]
  15. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,”J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  16. B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
    [CrossRef]
  17. S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (N.Y.) 14, 305–332 (1961).
    [CrossRef]
  18. H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y) 34, 23–95 (1965).
    [CrossRef]
  19. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  20. V. Khare, H. M. Nussenzveig, “Theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1976), pp. 723–764.
  21. M. V. Berry, J. F. Nye, F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979), App. C.
    [CrossRef]
  22. F. J. Wright, “The Stokes set of the cusp diffraction catastrophe,” J. Phys. A 13, 2913–2928 (1980).
    [CrossRef]
  23. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sec. 3.3.ii.
    [CrossRef]
  24. M. V. Berry, C. J. Howls, “Stokes surfaces of diffraction catastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).
    [CrossRef]
  25. C. K. Frederickson, P. L. Marston, “Transverse cusp diffraction catastrophes produced by reflecting ultrasonic tone bursts from curved surfaces,” in Proceedings of the IUTAM Symposium of Elastic Wave Propagation and Ultrasonic Nondestructive Evaluation, S. K. Daha, J. D. Achenbach, Y. D. S. Rajapakse, eds. (Elsevier, Amsterdam, 1989).
  26. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y) 7, 259–286 (1959).
    [CrossRef]
  27. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.2.
  28. Ref. 1, Sec. 12.22.
  29. Ref. 1, Sec. 12.34.
  30. H. C. van de Hulst, “A theory of the anti-corona,”J. Opt. Soc. Am. 37, 16–22 (1947).
    [CrossRef]
  31. M. V. Berry, “Uniform approximation for glory scattering and diffraction peaks,”J. Phys. B 2, 381–392 (1969).
    [CrossRef]
  32. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  33. H. M. Nussenzveig, W. Wiscombe, “Forward optical glory,” Opt. Lett. 5, 455–457 (1980).
    [CrossRef] [PubMed]
  34. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 3.
  35. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  36. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Sec. 10.4.
  37. C. W. Querfeld, “Mie atmospheric optics,”J. Opt. Soc. Am. 55, 105–106 (1965).
    [CrossRef]
  38. K. Sassen, “Angular scattering and rainbow formation in pendant drops,”J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  39. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  40. Ref. 1, Sec. 12.31.
  41. Ref. 1, Sec. 17.31.
  42. K. L. Williams, P. L. Marston, “Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation,”J. Acoust. Soc. Am. 78, 1093–1102 (1985).
    [CrossRef]
  43. E. DiSalvo, G. A. Viano, “Surface waves in pion-proton elastic scattering,” Nuovo Cimento A 59, 11–37 (1980).
    [CrossRef]
  44. E. Heyman, L. B. Felson, “High frequency fields in the presence of a curved dielectric interface,”IEEE Trans. Antennas Propag. AP32, 969–978 (1984).
    [CrossRef]
  45. Ref. 6, p. 142.
  46. H. M. Nussenzveig, “Applications of Regge poles to short-wavelength scattering,” in J. E. Bowcock, ed., Methods and Problems of Theoretical Physics (North-Holland, Amsterdam, 1970), p. 225.
  47. E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA CR-182293 DOT/FAA/CD-89/13 (NASA Lewis Research Center, Cleveland, Ohio, 1989).
  48. Ref. 1, Sec. 12.33.
  49. Ref. 1, Sec. 12.32.
  50. J. A. Lock, L. Yang, “Interference between diffraction and transmission in the Mie extinction efficiency,” J. Opt. Soc. Am. A 8, 1132–1134 (1991).
    [CrossRef]

1991 (3)

1990 (1)

M. V. Berry, C. J. Howls, “Stokes surfaces of diffraction catastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).
[CrossRef]

1988 (1)

1985 (1)

K. L. Williams, P. L. Marston, “Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation,”J. Acoust. Soc. Am. 78, 1093–1102 (1985).
[CrossRef]

1984 (1)

E. Heyman, L. B. Felson, “High frequency fields in the presence of a curved dielectric interface,”IEEE Trans. Antennas Propag. AP32, 969–978 (1984).
[CrossRef]

1981 (1)

1980 (5)

E. DiSalvo, G. A. Viano, “Surface waves in pion-proton elastic scattering,” Nuovo Cimento A 59, 11–37 (1980).
[CrossRef]

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

F. J. Wright, “The Stokes set of the cusp diffraction catastrophe,” J. Phys. A 13, 2913–2928 (1980).
[CrossRef]

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sec. 3.3.ii.
[CrossRef]

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

1979 (4)

1977 (1)

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

1976 (2)

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

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1974 (1)

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

1969 (3)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,”J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. Theory of the rainbow and the glory,”J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

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

1965 (2)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y) 34, 23–95 (1965).
[CrossRef]

C. W. Querfeld, “Mie atmospheric optics,”J. Opt. Soc. Am. 55, 105–106 (1965).
[CrossRef]

1961 (1)

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (N.Y.) 14, 305–332 (1961).
[CrossRef]

1959 (2)

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y) 7, 259–286 (1959).
[CrossRef]

1947 (1)

1937 (1)

B. Van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Berry, M. V.

M. V. Berry, C. J. Howls, “Stokes surfaces of diffraction catastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).
[CrossRef]

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sec. 3.3.ii.
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979), App. C.
[CrossRef]

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

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

Bohren, C. F.

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

Bremmer, H.

B. Van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.2.

de Boer, J. H.

DiSalvo, E.

E. DiSalvo, G. A. Viano, “Surface waves in pion-proton elastic scattering,” Nuovo Cimento A 59, 11–37 (1980).
[CrossRef]

Felson, L. B.

E. Heyman, L. B. Felson, “High frequency fields in the presence of a curved dielectric interface,”IEEE Trans. Antennas Propag. AP32, 969–978 (1984).
[CrossRef]

Ford, K. W.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y) 7, 259–286 (1959).
[CrossRef]

Frederickson, C. K.

C. K. Frederickson, P. L. Marston, “Transverse cusp diffraction catastrophes produced by reflecting ultrasonic tone bursts from curved surfaces,” in Proceedings of the IUTAM Symposium of Elastic Wave Propagation and Ultrasonic Nondestructive Evaluation, S. K. Daha, J. D. Achenbach, Y. D. S. Rajapakse, eds. (Elsevier, Amsterdam, 1989).

Gouesbet, G.

Grehan, G.

Heyman, E.

E. Heyman, L. B. Felson, “High frequency fields in the presence of a curved dielectric interface,”IEEE Trans. Antennas Propag. AP32, 969–978 (1984).
[CrossRef]

Hovenac, E. A.

J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
[CrossRef]

E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA CR-182293 DOT/FAA/CD-89/13 (NASA Lewis Research Center, Cleveland, Ohio, 1989).

Howls, C. J.

M. V. Berry, C. J. Howls, “Stokes surfaces of diffraction catastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).
[CrossRef]

Huffman, D. R.

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

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 3.

Keller, J. B.

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications: Proceedings of the Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27–52.
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

Khare, V.

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

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

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1976).

V. Khare, H. M. Nussenzveig, “Theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1976), pp. 723–764.

Können, G. P.

Levy, B. R.

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

Lock, J. A.

Marston, P. L.

K. L. Williams, P. L. Marston, “Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation,”J. Acoust. Soc. Am. 78, 1093–1102 (1985).
[CrossRef]

C. K. Frederickson, P. L. Marston, “Transverse cusp diffraction catastrophes produced by reflecting ultrasonic tone bursts from curved surfaces,” in Proceedings of the IUTAM Symposium of Elastic Wave Propagation and Ultrasonic Nondestructive Evaluation, S. K. Daha, J. D. Achenbach, Y. D. S. Rajapakse, eds. (Elsevier, Amsterdam, 1989).

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.2.

Nussenzveig, H. M.

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

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,”J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

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

V. Khare, 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. I. Direct reflection and transmission,”J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. Theory of the rainbow and the glory,”J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y) 34, 23–95 (1965).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1976), pp. 723–764.

H. M. Nussenzveig, “Applications of Regge poles to short-wavelength scattering,” in J. E. Bowcock, ed., Methods and Problems of Theoretical Physics (North-Holland, Amsterdam, 1970), p. 225.

Nye, J. F.

M. V. Berry, J. F. Nye, F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979), App. C.
[CrossRef]

Querfeld, C. W.

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.2.

Rubinow, S. I.

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (N.Y.) 14, 305–332 (1961).
[CrossRef]

Sassen, K.

Ungut, A.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sec. 3.3.ii.
[CrossRef]

van de Hulst, H. C.

Van der Pol, B.

B. Van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Viano, G. A.

E. DiSalvo, G. A. Viano, “Surface waves in pion-proton elastic scattering,” Nuovo Cimento A 59, 11–37 (1980).
[CrossRef]

Walker, J. D.

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Wheeler, J. A.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y) 7, 259–286 (1959).
[CrossRef]

Williams, K. L.

K. L. Williams, P. L. Marston, “Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation,”J. Acoust. Soc. Am. 78, 1093–1102 (1985).
[CrossRef]

Wiscombe, W.

Wright, F. J.

F. J. Wright, “The Stokes set of the cusp diffraction catastrophe,” J. Phys. A 13, 2913–2928 (1980).
[CrossRef]

M. V. Berry, J. F. Nye, F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979), App. C.
[CrossRef]

Yang, L.

Adv. Phys. (1)

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

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Ann. Phys. (N.Y) (2)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y) 34, 23–95 (1965).
[CrossRef]

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y) 7, 259–286 (1959).
[CrossRef]

Ann. Phys. (N.Y.) (1)

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (N.Y.) 14, 305–332 (1961).
[CrossRef]

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. Heyman, L. B. Felson, “High frequency fields in the presence of a curved dielectric interface,”IEEE Trans. Antennas Propag. AP32, 969–978 (1984).
[CrossRef]

J. Acoust. Soc. Am. (1)

K. L. Williams, P. L. Marston, “Backscattering from an elastic sphere: Sommerfeld–Watson transformation and experimental confirmation,”J. Acoust. Soc. Am. 78, 1093–1102 (1985).
[CrossRef]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,”J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. Theory of the rainbow and the glory,”J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

F. J. Wright, “The Stokes set of the cusp diffraction catastrophe,” J. Phys. A 13, 2913–2928 (1980).
[CrossRef]

J. Phys. B (1)

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

Nonlinearity (1)

M. V. Berry, C. J. Howls, “Stokes surfaces of diffraction catastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).
[CrossRef]

Nuovo Cimento A (1)

E. DiSalvo, G. A. Viano, “Surface waves in pion-proton elastic scattering,” Nuovo Cimento A 59, 11–37 (1980).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

B. Van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Philos. Trans. R. Soc. London (1)

M. V. Berry, J. F. Nye, F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London 291, 453–484 (1979), App. C.
[CrossRef]

Phys. Rev. Lett. (2)

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

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

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sec. 3.3.ii.
[CrossRef]

Sci. Am. (1)

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

Other (19)

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications: Proceedings of the Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27–52.
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum, New York, 1976), pp. 723–764.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1976).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 9.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

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

C. K. Frederickson, P. L. Marston, “Transverse cusp diffraction catastrophes produced by reflecting ultrasonic tone bursts from curved surfaces,” in Proceedings of the IUTAM Symposium of Elastic Wave Propagation and Ultrasonic Nondestructive Evaluation, S. K. Daha, J. D. Achenbach, Y. D. S. Rajapakse, eds. (Elsevier, Amsterdam, 1989).

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 18.2.

Ref. 1, Sec. 12.22.

Ref. 1, Sec. 12.34.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 3.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Sec. 10.4.

Ref. 1, Sec. 12.31.

Ref. 1, Sec. 17.31.

Ref. 6, p. 142.

H. M. Nussenzveig, “Applications of Regge poles to short-wavelength scattering,” in J. E. Bowcock, ed., Methods and Problems of Theoretical Physics (North-Holland, Amsterdam, 1970), p. 225.

E. A. Hovenac, “Droplet sizing instrumentation used for icing research: operation, calibration, and accuracy,” NASA CR-182293 DOT/FAA/CD-89/13 (NASA Lewis Research Center, Cleveland, Ohio, 1989).

Ref. 1, Sec. 12.33.

Ref. 1, Sec. 12.32.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (19)

Fig. 1
Fig. 1

Deflection of a geometrical light ray through the angle Θ by a dielectric sphere of radius a and refractive index n.

Fig. 2
Fig. 2

Comparison of the p-term Debye-series-component intensity I1 with the prediction of geometrical optics, Eq. (6), for all the contributing TE polarized ray trajectories for x = 100 and n = 1.333. (a) Debye p = 0 plus diffraction compared with the reflected geometrical rays; (b) Debye p = 1 compared with the transmitted geometrical rays; (c) Debye p = 2 compared with the single-internal-reflection geometrical rays; (d) Debye p = 3 compared with the two internal reflection geometrical rays.

Fig. 3
Fig. 3

Comparison of the p = 2 Debye-component intensity I1 with the Airy theory approximation of Eq. (13) for the TE polarization state, x = 1000, and n = 1.333.

Fig. 4
Fig. 4

Comparison of the p = 3 Debye-component intensity I1 with the Airy theory approximation of Eq. (13) for the TE polarization state, x = 1000, and n = 1.333.

Fig. 5
Fig. 5

Comparison of the p = 4 Debye-component intensity I1 with the Airy theory approximation of Eq. (13) for the TE polarization state, x = 1000, and n = 1.333.

Fig. 6
Fig. 6

Comparison of the p = 5 Debye-component intensity I1 with the Airy theory approximation of Eq. (13) for the TE polarization state, x = 1000, and n = 1.333.

Fig. 7
Fig. 7

Comparison of the p = 6 Debye-component intensity I1 with the Airy theory approximation of Eq. (13) for the TE polarization state, x = 1000, and n = 1.333.

Fig. 8
Fig. 8

Secondary radiation shed into the far field by a p = 2 surface wave that travels the angular distance ξ = ξ1 + ξ2 along the circumference of the sphere. The path segments marked T denote the shortcuts made by the surface wave through the sphere, and the thick arc segments denote propagation along the sphere surface.

Fig. 9
Fig. 9

Comparison of the p = 1 Debye-component intensity with the transmitted geometrical-ray and surface-wave contributions of Eqs. (6) and (21) for x = 1000 and n = 1.333. (a) The I1 Debye scattered intensity, the TE polarized ray, and the TE polarized surface wave; (b) the I2 Debye scattered intensity, the TM polarized ray, and the TM polarized surface wave.

Fig. 10
Fig. 10

A p = 3 geometrical ray (solid line) and a p = 3 surface wave (dashed line) that interfere in the far field.

Fig. 11
Fig. 11

Comparison of the p = 3 Debye-component intensity with the approximation of Eq. (26) for x = 1000 and n = 1.333. The approximation of Eq. 26) has been offset by a factor of 100 for clarity. (a) The I1 Debye scattered intensity, the TE polarized ray, and the TE polaraized surface wave; (b) the I2 Debye scattered amplitude, the TM polarized ray, and the TM polarized surface wave.

Fig. 12
Fig. 12

A p = 4 geometrical ray (solid line) and the p = 4 complex ray (short-dashed line) that interfere in the far field. The long-dashed line marked R3 is the third-order rainbow ray. The complex ray occurs for scattering angles larger than that of the rainbow ray.

Fig. 13
Fig. 13

Comparison of the p = 4 Debye-component intensity I1 with the approximation of Eq. (27) for the TE polarization state and for x = 600 and n = 1.333. The approximation of Eq. (27) has been offset by a factor of 100 for clarity.

Fig. 14
Fig. 14

A p = 2 surface wave (solid line) and the p = 2 complex ray (short-dashed line) that interfere in the far field. The long- dashed line marked R1 is the first-order rainbow ray. The complex ray occurs for scattering angles smaller than that of the rainbow ray.

Fig. 15
Fig. 15

Comparison of the p = 2 Debye-component intensity I1 with the approximation of Eq. (28) for the TE polarized surface wave and complex ray for x = 100 and n = 1.333. The approximation of Eq. (28) has been offset by a factor of 100 for clarity.

Fig. 16
Fig. 16

Laser beam with a small diameter illuminating a water droplet close to one edge. This focused laser beam causes the p = 3 rainbow and its supernumeraries to be more intense than other rainbows and confines the reflected rays to the forward-scattering hemisphere.

Fig. 17
Fig. 17

Experimental apparatus. Light scattered by the droplets in the angular range 30° lsim; θ lsim; 150° is observed on the viewing screen.

Fig. 18
Fig. 18

Scattered light intensity for a 40-μm-diameter laser beam incident near the edge of an 86.6-μm-diameter water droplet. The scattering angle decreases from left to right. The broad illumination on the right-hand side of the photograph is reflection from the droplet. The interference pattern on the left-hand side is the second-order rainbow and its supernumeraries.

Fig. 19
Fig. 19

Comparison between the Debye-component intensity and experiment. (a) Detector response (normalized) as a function of position on the film for the digitization of the photograph of Fig. 18. The four oscillations between −12 and −6 mm on the film (θ < θC) are the geometrical-ray–surface-wave interference. The rise in the detector response at film positions greater than −6 mm (θ lsim; 100°) is due to reflection by the droplet surface. (b) p = 3 Debye intensity I1 for x = 528.8 and n = 1.333 as a function of the scattering angle.

Equations (65)

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

x = 2 π a λ .
Θ = ( p - 1 ) π + 2 θ i - 2 p θ r ,
sin θ i = n sin θ r .
θ = { Θ - 2 π N if 2 π N Θ 2 π ( N + 1 / 2 ) 2 π ( N + 1 ) - Θ if 2 π ( N + 1 / 2 ) < Θ 2 π ( N + 1 ) ,
L = 2 a ( p n cos θ r - cos θ i ) + 2 a .
E geometrical ray p ( θ ) = E 0 a R [ sin θ i cos θ i 2 sin θ | 1 - p cos θ i n cos θ r | ] 1 / 2 T 21 ( θ i ) × [ R 11 ( θ i ) ] p - 1 T 12 ( θ i ) exp ( i k R ) exp ( 2 π i L / λ ) exp ( i ζ )
Θ R = ( p - 1 ) π + 2 θ i R - 2 p θ r R ,
cos 2 θ i R = n 2 - 1 p 2 - 1 .
E Airy p ( θ ) = E 0 a exp ( i k R ) R ( 2 π sin θ i R sin θ R ) 1 / 2 x 1 / 6 h 1 / 3 T 21 ( θ i R ) × [ R 11 ( θ i R ) ] p - 1 T 12 ( θ i R ) × A i ( - x 2 / 3 Δ h 1 / 3 ) × exp ( 2 π i L R / λ ) exp [ ( i x Δ ) ( p 2 - n 2 p 2 - 1 ) 1 / 2 ] ,
Δ = θ - θ R ,
h = ( p 2 - 1 ) 2 p 2 ( p 2 - n 2 ) 1 / 2 ( n 2 - 1 ) 3 / 2 ,
A i ( u ) 1 2 π 1 / 2 ( 2 3 u 3 / 2 ) - 1 / 4 exp ( - 2 3 u 3 / 2 )
I Airy p ( θ ) = E Airy p ( θ ) 2 ,
sin θ i l ave x .
T = 2 a n ( n 2 - 1 ) 1 / 2
sin θ r C = 1 n .
k shortcut = n k ,
k surface j = k + X j 2 a ( x 2 ) 1 / 3 + i [ 3 1 / 2 X j 2 a ( x 2 ) 1 / 3 - κ a ( n 2 - 1 ) 1 / 2 ] ,
A i ( - X j ) = 0
κ = { 1 for the S 1 scattering amplitude n 2 for the S 2 scattering amplitude .
E surface wave p , j ( θ ) = E 0 a exp ( i k R ) R ( sin θ ) 1 / 2 exp ( i π / 12 ) 2 π 1 / 2 a j 2 ( 2 x ) 1 / 6 × exp ( i p k shortcut T ) exp ( i k surface j a ξ ) exp ( 2 i k a ) × m = 1 p ( p - 1 ) ! ( m - 1 ) ! ( p - m ) ! [ 2 κ ( n 2 - 1 ) 1 / 2 ] m ξ m m ! ,
A i ( - X j ) = a j
Θ = p π - 2 p θ r C + ξ .
I surface wave p ( θ ) = | j = 1 E surface wave p , j ( θ ) | 2 .
Θ C = p π - 2 p θ r C .
I approximate p = 3 ( θ ) = E geometrical ray p = 3 ( θ ) + E surface wave p = 3 , j = 1 ( θ ) 2 ,
I approximate p = 4 ( θ ) = E third geometrical ray p = 4 ( θ ) + E Airy p = 4 ( θ ) 2
I approximate p = 2 ( θ ) = E surface wave p = 2 , j = 1 ( θ ) + E Airy p = 2 ( θ ) 2
2 ψ + n 2 k 2 ψ = 0 ,
k = ω c .
E = - r × ψ , B = - i ω × E
E = i ω c 2 n 2 × B , B = n c r × ψ
ψ ( r , θ , ϕ ) = l m A l m { j l ( n k r ) n l ( n k r ) } P l m ( cos θ ) { cos m ϕ sin m ϕ } ,
j 0 ( n k r ) = sin ( n k r ) n k r , n 0 ( n k r ) = - cos ( n k r ) n k r .
j l ( n k r ) n l ( n k r ) - j l ( n k r ) n l ( n k r ) = ( n k r ) - 2 ,
h l ( 1 ) ( n k r ) = j l ( n k r ) + i n l ( n k r ) , h l ( 2 ) ( n k r ) = j l ( n k r ) - i n l ( n k r ) ,
Ψ ( x ) n 2 = Ψ ( y ) n 1 ,
Ψ ( x ) = Ψ ( y ) ,
x = n 2 k a , y = n 1 k a ,
Ψ ( n k r ) n k r ψ ( n k r ) .
Ψ ( x ) = Ψ ( y ) ,
Ψ ( x ) n 2 = Ψ ( y ) n 1 .
Ψ = H l ( 2 ) ( n 2 k r ) P l m ( cos θ ) { cos m ϕ sin m ϕ } ,
H l ( 2 ) ( n 2 k r ) n 2 k r h l ( 2 ) ( n k r ) .
Ψ 1 = T l 21 H l ( 2 ) ( n 1 k r ) P l m ( cos θ ) { cos m ϕ sin m ϕ }             for r a , Ψ 2 = [ H l ( 2 ) ( n 2 k r ) + R l 22 H l ( 1 ) ( n 2 k r ) ] × P l m ( cos θ ) { cos m ϕ sin m ϕ }             for r a .
T l 21 = - ( n 1 n 2 ) 2 i D ,
R l 22 = [ α H l ( 2 ) ( x ) H l ( 2 ) ( y ) - β H l ( 2 ) ( x ) H l ( 2 ) ( y ) ] D ,
α = { 1 for TE spherical multipole waves n 1 / n 2 for TM spherical multipole waves ,
β = { n 1 / n 2 for TE spherical multipole waves 1 for TM spherical multipole waves ,
D = - α H l ( 1 ) ( x ) H l ( 2 ) ( y ) + β H l ( 1 ) ( x ) H l ( 2 ) ( y ) ,
Ψ = H l ( 1 ) ( n 1 k r ) P l m ( cos θ ) { cos m ϕ sin m ϕ } .
Ψ 1 = [ H l ( 1 ) ( n 1 k r ) + R l 11 H l ( 2 ) ( n 1 k r ) ] P l m ( cos θ ) { cos m ϕ sin m ϕ } for r a , Ψ 2 = T l 12 H l ( 1 ) ( n 2 k r ) P l m ( cos θ ) { cos m ϕ sin m ϕ } for r a .
T l 12 = - 2 i / D ,
R l 11 = [ α H l ( 1 ) ( x ) H l ( 1 ) ( y ) - β H l ( 1 ) ( x ) H l ( 1 ) ( y ) ] D .
( 1 - R l 11 ) ( 1 - R l 22 ) - T l 21 T l 12 = 4 [ - α J l ( x ) + J l ( y ) + β J l ( x ) J l ( y ) ] D ,
J l ( n k r ) n k r j l ( n k r ) .
- α J l ( x ) J l ( y ) + β J l ( x ) J l ( y ) - α H l ( 1 ) ( x ) J l ( y ) + β H l ( 1 ) ( x ) J l ( y ) = 1 2 [ 1 - R l 22 - T l 21 T l 12 1 - R l 11 ] = 1 2 [ 1 - R l 22 - p = 1 T l 21 ( R l 11 ) p - 1 T l 12 ] .
E ( R , θ , ϕ ) = E 0 a R x exp ( i k R ) × [ - i S 2 ( θ ) cos ϕ u ^ θ + i S 1 ( θ ) sin ϕ u ^ ϕ ]
S 1 ( θ ) = l = 1 l max 2 l + 1 l ( l + 1 ) [ a l π l ( θ ) + b l τ l ( θ ) ] ,
S 2 ( θ ) = l = 1 l max 2 l + 1 l ( l + 1 ) [ a l τ l ( θ ) + b l π l ( θ ) ] ,
π l ( θ ) = 1 sin θ P l 1 ( θ ) ,
τ l ( θ ) = d d θ P l 1 ( θ ) ,
l max = x + 4.05 x 1 / 3 + 2.
I 1 ( θ ) = S 1 ( θ ) 2 ,
I 2 ( θ ) = S 2 ( θ ) 2 .

Metrics