Abstract

The electromagnetic fields scattered when a plane wave is incident on an oblate spheroid in the side-on orientation may be calculated using a generalization of Mie theory, and the results may be decomposed in a Debye series expansion. A number of optical caustics are observed in the computed scattered intensity for the one internal reflection portion of the Debye series for scattering angles in the vicinity of the first-order rainbow, and are analyzed in terms of the rainbow, transverse cusp, and hyperbolic umbilic caustics of catastrophe optics. The specific features of these three caustics are described, as is their assembly into the global structure of the observed caustics for spheroid scattering. It is found that, for a spheroid whose radius is an order of magnitude larger than the wavelength of the incident light, the interference structure accompanying the transverse cusp and hyperbolic umbilic caustics is only partially formed.

© 2010 Optical Society of America

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  1. P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
    [CrossRef]
  2. J. F. Nye, “Rainbow scattering from spheroidal drops--an explanation of the hyperbolic umbilic foci,” Nature 312, 531-532 (1984).
    [CrossRef]
  3. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588-590 (1985).
    [CrossRef]
  4. P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226-232 (1987).
    [CrossRef]
  5. P. L. Marston, Erratum, J. Acoust. Soc. Am. 83, 1976 (1988)
    [CrossRef]
  6. P. L. Marston, C. E. Dean, and H. J. Simpson, “Light scattering from spheroidal drops: Exploring optical catastrophes and generalized rainbows,” in Drops and Bubbles: American Institute of Physics Conference Proceedings 197, T. G. Wang, ed. (American Institute of Physics, 1989), pp 275-285.
  7. C. E. Dean and P. L. Marston, “Opening rate of transverse cusp diffraction catastrophe in light scattering by oblate spheroidal drops,” Appl. Opt. 30, 3443-3451 (1991).
    [CrossRef]
  8. C. E. Dean and P. L. Marston, Errata, Appl. Opt. 32, 2163 (1993). Also as a correction to , please replace d in Eq. (6a) with 1/d.
    [CrossRef]
  9. J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397-417 (1992).
    [CrossRef]
  10. S. Holler, Y. Pan, R. K. Chang, J. R. Bottiger, S. C. Hill, and D. B. Hillis, “Two-dimensional angular optical scattering for the characterization of airborne microparticles,” Opt. Lett. 23, 1489-1491 (1998).
    [CrossRef]
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  12. S. Asano and G. Yamamoto, Erratum, Appl. Opt. 15, 2028 (1976)
    [CrossRef]
  13. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712-723 (1979).
    [CrossRef]
  14. S. Asano and M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962-974 (1980).
    [CrossRef]
  15. M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1-26 (1976).
    [CrossRef]
  16. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.24.
  18. F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A (in press).
  19. Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
    [CrossRef]
  20. G. Kaduchak and P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697-4701 (1994).
    [CrossRef]
  21. P. L. Marston and G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band,” Appl. Opt. 33, 4702-4713 (1994).
    [CrossRef]
  22. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520-1526 (1998).
    [CrossRef]
  23. H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468-3473, 3547 (1991).
    [CrossRef]
  24. G. Kaduchak, P. L. Marston, and H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white light and laser illumination,” Appl. Opt. 33, 4691-4696 (1994).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 5.
  26. W. J. Humphreys, Physics of the Air (Dover, 1964), pp. 483-491.
  27. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002), Sections 1.2, 2.1.
    [CrossRef]
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.23.
  29. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Section 3.2.
  30. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Section 2.3.
  31. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Fig. 7.
  32. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106-117(1991).
    [CrossRef]
  33. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236 (4), 116-127 (1977).
    [CrossRef]
  34. J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A8, 1541-1552 (1991).
    [CrossRef]
  35. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eq. 10.4.32.
  36. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eqs. 10.4.59, 10.4.60.
  37. T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a caustic,” Philos. Mag. 37, 311-317(1946).
  38. J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey's integral,” J. Chem. Phys. 75, 2831-2846(1981).
    [CrossRef]
  39. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
    [CrossRef]
  40. W. Mobius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105-254 (1907-1909).
  41. W. Mobius, “Zur Theorie des Regenbogens und ihrir experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493-1558(1910).
    [CrossRef]
  42. G. P. Konnen and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961-1965 (1979).
    [CrossRef]
  43. W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
    [CrossRef]
  44. W. P. Arnott and P. L. Marston, “Unfolded optical glory of spheroids: Backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429-3442(1991).
    [CrossRef]
  45. A. Thaning, Z. Jaroszewicz, and A. T. Friberg, “Diffractive axicons in oblique illumination: analysis and experiments and comparison with elliptical axicons,” Appl. Opt. 42, 9-17(2003).
    [CrossRef]
  46. 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]
  47. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980), Appendix 2.

2003 (1)

2002 (2)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002), Sections 1.2, 2.1.
[CrossRef]

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

1998 (2)

1994 (3)

1993 (1)

1992 (3)

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397-417 (1992).
[CrossRef]

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Section 2.3.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Fig. 7.

1991 (5)

1989 (1)

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

1988 (1)

P. L. Marston, Erratum, J. Acoust. Soc. Am. 83, 1976 (1988)
[CrossRef]

1987 (1)

P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226-232 (1987).
[CrossRef]

1985 (1)

1984 (2)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops--an explanation of the hyperbolic umbilic foci,” Nature 312, 531-532 (1984).
[CrossRef]

1981 (1)

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey's integral,” J. Chem. Phys. 75, 2831-2846(1981).
[CrossRef]

1980 (3)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980).

S. Asano and M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962-974 (1980).
[CrossRef]

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980), Appendix 2.

1979 (3)

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712-723 (1979).
[CrossRef]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
[CrossRef]

G. P. Konnen and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961-1965 (1979).
[CrossRef]

1977 (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236 (4), 116-127 (1977).
[CrossRef]

1976 (2)

S. Asano and G. Yamamoto, Erratum, Appl. Opt. 15, 2028 (1976)
[CrossRef]

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

1975 (1)

1969 (1)

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]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a caustic,” Philos. Mag. 37, 311-317(1946).

1910 (1)

W. Mobius, “Zur Theorie des Regenbogens und ihrir experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493-1558(1910).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eq. 10.4.32.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eqs. 10.4.59, 10.4.60.

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002), Sections 1.2, 2.1.
[CrossRef]

Arnott, W. P.

W. P. Arnott and P. L. Marston, “Unfolded optical glory of spheroids: Backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429-3442(1991).
[CrossRef]

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

Asano, S.

Berry, M. V.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980).

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980), Appendix 2.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
[CrossRef]

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

Bottiger, J. R.

Chang, R. K.

Connor, J. N. L.

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey's integral,” J. Chem. Phys. 75, 2831-2846(1981).
[CrossRef]

de Boer, J. H.

Dean, C. E.

C. E. Dean and P. L. Marston, Errata, Appl. Opt. 32, 2163 (1993). Also as a correction to , please replace d in Eq. (6a) with 1/d.
[CrossRef]

C. E. Dean and P. L. Marston, “Opening rate of transverse cusp diffraction catastrophe in light scattering by oblate spheroidal drops,” Appl. Opt. 30, 3443-3451 (1991).
[CrossRef]

P. L. Marston, C. E. Dean, and H. J. Simpson, “Light scattering from spheroidal drops: Exploring optical catastrophes and generalized rainbows,” in Drops and Bubbles: American Institute of Physics Conference Proceedings 197, T. G. Wang, ed. (American Institute of Physics, 1989), pp 275-285.

Farrelly, D.

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey's integral,” J. Chem. Phys. 75, 2831-2846(1981).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

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

Gouesbet, G.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Grehan, G.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Han, Y. P.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Hill, S. C.

Hillis, D. B.

Holler, S.

Hovenac, E. A.

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A8, 1541-1552 (1991).
[CrossRef]

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, 1964), pp. 483-491.

Jaroszewicz, Z.

Kaduchak, G.

Konnen, G. P.

Langley, D. S.

Lock, J. A.

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A8, 1541-1552 (1991).
[CrossRef]

F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A (in press).

Marston, P. L.

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520-1526 (1998).
[CrossRef]

G. Kaduchak, P. L. Marston, and H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white light and laser illumination,” Appl. Opt. 33, 4691-4696 (1994).
[CrossRef]

G. Kaduchak and P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697-4701 (1994).
[CrossRef]

P. L. Marston and G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band,” Appl. Opt. 33, 4702-4713 (1994).
[CrossRef]

C. E. Dean and P. L. Marston, Errata, Appl. Opt. 32, 2163 (1993). Also as a correction to , please replace d in Eq. (6a) with 1/d.
[CrossRef]

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Section 2.3.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Fig. 7.

H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468-3473, 3547 (1991).
[CrossRef]

C. E. Dean and P. L. Marston, “Opening rate of transverse cusp diffraction catastrophe in light scattering by oblate spheroidal drops,” Appl. Opt. 30, 3443-3451 (1991).
[CrossRef]

W. P. Arnott and P. L. Marston, “Unfolded optical glory of spheroids: Backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429-3442(1991).
[CrossRef]

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

P. L. Marston, Erratum, J. Acoust. Soc. Am. 83, 1976 (1988)
[CrossRef]

P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226-232 (1987).
[CrossRef]

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588-590 (1985).
[CrossRef]

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

P. L. Marston, C. E. Dean, and H. J. Simpson, “Light scattering from spheroidal drops: Exploring optical catastrophes and generalized rainbows,” in Drops and Bubbles: American Institute of Physics Conference Proceedings 197, T. G. Wang, ed. (American Institute of Physics, 1989), pp 275-285.

Mees, L.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Mobius, W.

W. Mobius, “Zur Theorie des Regenbogens und ihrir experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493-1558(1910).
[CrossRef]

W. Mobius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105-254 (1907-1909).

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236 (4), 116-127 (1977).
[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, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Section 3.2.

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397-417 (1992).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops--an explanation of the hyperbolic umbilic foci,” Nature 312, 531-532 (1984).
[CrossRef]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
[CrossRef]

Pan, Y.

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a caustic,” Philos. Mag. 37, 311-317(1946).

Ren, K. F.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Sato, M.

Simpson, H. J.

G. Kaduchak, P. L. Marston, and H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white light and laser illumination,” Appl. Opt. 33, 4691-4696 (1994).
[CrossRef]

H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468-3473, 3547 (1991).
[CrossRef]

P. L. Marston, C. E. Dean, and H. J. Simpson, “Light scattering from spheroidal drops: Exploring optical catastrophes and generalized rainbows,” in Drops and Bubbles: American Institute of Physics Conference Proceedings 197, T. G. Wang, ed. (American Institute of Physics, 1989), pp 275-285.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eq. 10.4.32.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eqs. 10.4.59, 10.4.60.

Thaning, A.

Trinh, E. H.

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

Tropea, C.

F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A (in press).

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980).

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980), Appendix 2.

van de Hulst, H. C.

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106-117(1991).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.23.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.24.

Wang, R. T.

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
[CrossRef]

Wu, S. Z.

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Xu, F.

F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A (in press).

Yamamoto, G.

Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. (1)

W. Mobius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105-254 (1907-1909).

Adv. Phys. (1)

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

Ann. Phys. (Leipzig) (1)

W. Mobius, “Zur Theorie des Regenbogens und ihrir experimentallen Prufung,” Ann. Phys. (Leipzig) 33, 1493-1558(1910).
[CrossRef]

Appl. Opt. (15)

G. P. Konnen and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961-1965 (1979).
[CrossRef]

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106-117(1991).
[CrossRef]

C. E. Dean and P. L. Marston, “Opening rate of transverse cusp diffraction catastrophe in light scattering by oblate spheroidal drops,” Appl. Opt. 30, 3443-3451 (1991).
[CrossRef]

C. E. Dean and P. L. Marston, Errata, Appl. Opt. 32, 2163 (1993). Also as a correction to , please replace d in Eq. (6a) with 1/d.
[CrossRef]

G. Kaduchak and P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697-4701 (1994).
[CrossRef]

P. L. Marston and G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band,” Appl. Opt. 33, 4702-4713 (1994).
[CrossRef]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520-1526 (1998).
[CrossRef]

H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468-3473, 3547 (1991).
[CrossRef]

G. Kaduchak, P. L. Marston, and H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white light and laser illumination,” Appl. Opt. 33, 4691-4696 (1994).
[CrossRef]

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29-49 (1975).

S. Asano and G. Yamamoto, Erratum, Appl. Opt. 15, 2028 (1976)
[CrossRef]

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712-723 (1979).
[CrossRef]

S. Asano and M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962-974 (1980).
[CrossRef]

W. P. Arnott and P. L. Marston, “Unfolded optical glory of spheroids: Backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429-3442(1991).
[CrossRef]

A. Thaning, Z. Jaroszewicz, and A. T. Friberg, “Diffractive axicons in oblique illumination: analysis and experiments and comparison with elliptical axicons,” Appl. Opt. 42, 9-17(2003).
[CrossRef]

J. Acoust. Soc. Am. (3)

P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226-232 (1987).
[CrossRef]

P. L. Marston, Erratum, J. Acoust. Soc. Am. 83, 1976 (1988)
[CrossRef]

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

J. Chem. Phys. (1)

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey's integral,” J. Chem. Phys. 75, 2831-2846(1981).
[CrossRef]

J. Math. Phys. (1)

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. (1)

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A8, 1541-1552 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A (in press).

Nature (2)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops--an explanation of the hyperbolic umbilic foci,” Nature 312, 531-532 (1984).
[CrossRef]

Opt. Commun. (1)

Y. P. Han, L. Mees, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Grehan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Opt. Lett. (2)

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

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. London 291, 453-484 (1979), Fig. 13.
[CrossRef]

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a caustic,” Philos. Mag. 37, 311-317(1946).

Phys. Acoust. (2)

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Section 2.3.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Fig. 7.

Phys. Rep. (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002), Sections 1.2, 2.1.
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397-417 (1992).
[CrossRef]

Prog. Opt. (2)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980).

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257-346 (1980), Appendix 2.

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236 (4), 116-127 (1977).
[CrossRef]

Other (8)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.23.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Section 3.2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eq. 10.4.32.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), Eqs. 10.4.59, 10.4.60.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), Section 13.24.

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

W. J. Humphreys, Physics of the Air (Dover, 1964), pp. 483-491.

P. L. Marston, C. E. Dean, and H. J. Simpson, “Light scattering from spheroidal drops: Exploring optical catastrophes and generalized rainbows,” in Drops and Bubbles: American Institute of Physics Conference Proceedings 197, T. G. Wang, ed. (American Institute of Physics, 1989), pp 275-285.

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Figures (7)

Fig. 1
Fig. 1

Geometry for the physical optics model for a plane wave scattered by a sphere or by an oblate spheroid. The central incident ray makes an angle θ i o with the normal to the spheroid surface in the horizontal z x plane. The central ray’s scattering angle is Θ o . The w v plane, with w horizontal and v vertical, is the spheroid’s exit plane. A far-zone viewing screen is normal to the Z axis and has its origin at the point where it intersects the exiting central ray. The angular position of a point on the exit plane with respect to this origin is Δ H in the horizontal direction and Δ V in the vertical direction.

Fig. 2
Fig. 2

The sign of the horizontal and vertical radii of curvature, R and S, respectively, of the wavefront in the exit plane at the position of the central ray as a function of θ i o and a / b . The horizontal radius R of Eq. (15) is infinite on the line V V , and the vertical radius S of Eq. (28) is infinite on the line U U . The point h of Eq. (34), at which both radii of curvature diverge, is the condition for the hufs. The region above the line V V is called region A, and the region below V V is called region B.

Fig. 3
Fig. 3

Locus of zero Gaussian curvature as a function of w and v in the spheroid exit plane (left-hand figures) and its caustic image as a function of Δ H and Δ V on the far-zone viewing screen (right-hand figures). The radius ratio for (a) is a / b = 1 , and a / b progressively increases in (b), (c), and (d), where (c) corresponds to the hufs condition of Eq. (34). The branches of the zero Gaussian curvature locus and their respective caustic images are labeled 1 and 2. At the hufs condition, the part of the zero Gaussian curvature locus that is mapped to the vertical portion of the rainbow detaches from the part of the locus that is mapped into the horizontal portion of the rainbow and attaches to the part of the locus that is mapped into the transverse cusp caustics. The dashed lines on the caustics in (c) and (d) denote regions of greatly reduced intensity in Figs. 4, 5.

Fig. 4
Fig. 4

Polar plot of the intensity as a function of the angles Θ and Φ for a vertically polarized incident plane wave with wavelength λ = 0.5145 μm scattered by an oblate spheroid with refractive index n = 1.334 + i ( 1.2 × 10 9 ) , horizontal radius a = 6.0 μm , and horizontal-to-vertical radius ratio (a)  a / b = 1.0001 , (b)  a / b = 1.05 , (c)  a / b = 1.15 , (d)  a / b = 1.25 , (e)  a / b = 1.30 , and (f)  a / b = 1.36 .

Fig. 5
Fig. 5

Polar plot of the intensity as a function of the angles Θ and Φ for an unpolarized incident plane wave with wavelength λ = 0.5145 μm scattered by an oblate spheroid with refractive index n = 1.334 + i ( 1.2 × 10 9 ) , horizontal radius a = 6.0 μm , and horizontal-to-vertical radius ratio (a)  a / b = 1.0001 , (b)  a / b = 1.05 , (c)  a / b = 1.15 , (d)  a / b = 1.25 , (e)  a / b = 1.30 , and (f)  a / b = 1.36 .

Fig. 6
Fig. 6

Intensity as a function of the angle Θ along the horizontal axis of the exit plane for a vertically polarized incident plane wave with wavelength λ = 0.5145 μm scattered by an oblate spheroid with refractive index n = 1.334 + i ( 1.2 × 10 9 ) , horizontal radius a = 6.0 μm , and radius ratio a / b = 1.30 .

Fig. 7
Fig. 7

Intensity as a function of the angles Θ and Φ or, equivalently, Δ H and Δ V , on the far-zone viewing screen in the vicinity of the Descartes rainbow angle for a vertically polarized incident plane wave with wavelength λ = 0.5145 μm scattered by an oblate spheroid with refractive index n = 1.334 + i ( 1.2 × 10 9 ) , horizontal radius a = 6.0 μm , and radius ratio a / b = 1.30 .

Equations (55)

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Θ 0 = ( p 1 ) π + 2 θ i 0 2 p θ t 0 ,
sin ( θ i 0 ) = n sin ( θ t 0 ) .
θ i = θ i 0 + ε ,
Θ = Θ 0 + Δ ,
Θ = ( p 1 ) π + 2 θ i 2 p θ t ,
sin ( θ i ) = n sin ( θ t ) .
E ( Δ H , Δ V ) = ( i k / 2 π Z ) exp ( i k Z ) t 21 0 ( r 11 0 ) p 1 t 12 0 ( Q H Q V ) 1 / 2 d w d v exp { i [ Φ ( w , v ) + k w Δ H k v Δ V ] } ,
Φ ( w , v ) = Φ ( 0 ) + k w 2 / 2 R a k h w 3 / 3 a 2 + k v 2 / 2 S a + ,
R = cos ( θ i 0 ) / [ 2 ( 1 p A 0 ) ] ,
h = [ sin ( θ i 0 ) / cos 3 ( θ i 0 ) ] [ ( 1 p A 0 3 ) 4 ( 1 p A 0 ) 3 ] ,
S = cos ( Θ 0 ) sin ( θ i 0 ) / sin ( Θ 0 ) ,
Q H = 1 ,
Q V = | 1 / cos ( Θ 0 ) | ,
A 0 = cos ( θ i 0 ) / [ n cos ( θ t 0 ) ] .
cos ( θ i r ) = [ ( n 2 1 ) / ( p 2 1 ) ] 1 / 2 ,
A r = 1 / p ,
R ,
h r = ( p 2 1 ) sin ( θ i r ) / [ p 2 cos 3 ( θ i r ) ] = ( p 2 1 ) 2 ( p 2 n 2 ) 1 / 2 / [ p 2 ( n 2 1 ) 3 / 2 ] .
z r / a = sin ( Θ 0 ) sin ( θ i 0 ) + { cos ( Θ 0 ) cos ( θ i 0 ) / [ 2 ( 1 p A 0 ) ] } , x r / a = cos ( Θ 0 ) sin ( θ i 0 ) { sin ( Θ 0 ) cos ( θ i 0 ) / [ 2 ( 1 p A 0 ) ] } .
z = a sin ( θ i 0 ) / sin ( Θ 0 ) .
E ( Δ H , 0 ) = exp { i [ k Z + Φ ( 0 ) π / 4 ] } { ( k a ) 7 / 6 ( 2 π ) 1 / 2 / [ ( k Z ) h r 1 / 3 ] } [ sin ( θ i r ) / sin ( Θ r ) ] 1 / 2 × t 21 r ( r 11 r ) p 1 t 12 r A i [ ( k a ) 2 / 3 Δ H / h r 1 / 3 ] .
A i ( X ) = ( 3 1 / 3 / 2 π ) d w exp { i [ w 3 ( 3 ) 1 / 3 X w ] } .
A i ( X ) 1 / ( π 1 / 2 X 1 / 4 ) cos ( 2 X 3 / 2 / 3 π / 4 ) for     X > 0 1 / ( 2 π 1 / 2 | X | 1 / 4 ) exp ( 2 | X | 3 / 2 / 3 ) for     X < 0.
P ( X , Y ) = d w exp [ i ( w 4 + X w 2 + Y w ) ] .
( 2 X / 3 ) 3 = Y 2 .
P ( X , 0 ) ( π / X ) 1 / 2 exp ( i π / 4 ) for     X > 0 ( π / | X | ) 1 / 2 exp ( i π / 4 ) [ 1 + i ( 2 ) 1 / 2 exp ( i X 2 / 4 ) ] for     X < 0.
Φ ( w , v ) = k w 2 / 2 R a + k v 2 / 2 S a k g w v 2 / 8 a 2 + ,
S = { cos ( θ t 0 ) / [ n β ( 1 β ) ] } { 2 β [ ( 1 A 0 ) / ( 2 β A 0 ) ] } cos ( θ i 0 ) ,
β = 2 ( a / b ) 2 ( 1 A 0 ) cos 2 ( θ t 0 ) .
g cp = 8 ( a / b ) 2 sin ( θ i cp ) ( 1 A cp ) 2 / cos ( θ i cp ) .
cos 2 ( θ i cp ) = [ n 2 2 ( a / b ) 2 ( n 2 1 ) ] 2 / { 4 ( a / b ) 2 [ n 2 ( a / b ) 2 ( n 2 1 ) ] } .
( a / b ) cp = n / { 2 cos ( θ t cp ) [ n cos ( θ t cp ) cos ( θ i cp ) ] } .
n / [ 2 ( n 2 1 ) ] 1 / 2 a / b { n / [ 2 ( n 1 ) ] } 1 / 2 ,
( a / b ) hufs = ( n / 2 ) [ 3 / ( n 2 1 ) ] 1 / 2
E ( Δ H , Δ V ) exp [ i ( k a R Δ H 2 / 2 + π / 4 ) ] { ( 2 k a R ) 5 / 4 / [ ( k Z ) ( π R 2 g ) 1 / 2 ] } P ( X , Y ) ,
X = ( 2 k a R ) 1 / 2 [ Δ H 4 / ( S R g ) ] ,
Y = 2 [ ( 2 k a R ) 3 / 4 / ( R 2 g ) 1 / 2 ] Δ V .
( 2 R 2 g / 27 ) [ Δ H 4 / ( S R g ) ] 3 = Δ V 2 ,
2 R 2 g / 27 = ( 4 / 27 ) ( a / b ) 2 sin ( θ i cp ) cos ( θ i cp ) [ ( 1 A cp ) / ( 1 2 A cp ) ] 2 .
E ( X , Y , Z ) = d s d t exp [ i ( s 3 + t 3 Z s t X s Z t ) ] .
Φ ( w , v ) = k w 2 / 2 R a k h w 3 / 3 a 2 + k v 2 / 2 S a k g w v 2 / 8 a 2 + .
Δ H = [ 1 / ( 4 R 2 h ) ] + ( 8 h 2 S 2 / g ) Δ V 2 / ( S / R + 8 h / g ) 2
( 2 S 2 g / 27 ) [ Δ H ( 4 / S 2 g ) ( S / R + 4 h / g ) ] 3 / ( S / R + 8 h / g ) 2 = Δ V 2
± ( 8 h / g ) 1 / 2 Δ V = Δ H ,
X = ( k a / 2 ) 2 / 3 ( 3 / h ) 1 / 3 { [ Δ H ( 8 h / g ) 1 / 2 Δ V ] + ( 3 / 16 S 2 h ) ( S / R 8 h / g ) ( S / R + 8 h / 3 g ) } ,
Y = ( k a / 2 ) 2 / 3 ( 3 / h ) 1 / 3 { [ Δ H + ( 8 h / g ) 1 / 2 Δ V ] + ( 3 / 16 S 2 h ) ( S / R 8 h / g ) ( S / R + 8 h / 3 g ) } ,
Z = ( k a / 2 ) 1 / 3 ( 3 / h ) 2 / 3 ( 1 / 2 S ) ( S / R + 8 h / g ) .
Δ cp = 16 h / ( g S ) 2 ,
E ( Δ H , 0 ) [ ( k a / k Z ) ] ( 2 / g ) 1 / 2 ( Δ H Δ cp ) 1 / 4 [ 1 + ( Δ H / Δ cp ) 1 / 2 ] 1 / 2 exp ( 2 i k a Δ H 3 / 2 / 3 h 1 / 2 ) [ i ( k a / k Z ) ] ( 2 / g ) 1 / 2 ( Δ H Δ cp ) 1 / 4 [ 1 ( Δ H / Δ cp ) 1 / 2 ] 1 / 2 exp ( 2 i k a Δ H 3 / 2 / 3 h 1 / 2 ) for     0 < Δ H < Δ cp ,
E ( Δ H , 0 ) [ ( k a / k Z ) ] ( 2 / g Δ H ) 1 / 2 [ 1 + ( Δ cp / Δ H ) 1 / 2 ] 1 / 2 exp ( 2 i k a Δ H 3 / 2 / 3 h 1 / 2 ) [ ( k a / k Z ) ] ( 2 / g Δ H ) 1 / 2 [ ( 1 ( Δ cp / Δ H ) 1 / 2 ] 1 / 2 exp ( 2 i k a Δ H 3 / 2 / 3 h 1 / 2 ) [ 2 i ( k a / k Z ) ] ( 2 / g Δ H ) 1 / 2 ( 1 Δ cp / Δ H ) 1 / 2 exp i k a ( Δ cp / h ) 1 / 2 ( Δ H Δ cp / 3 ) for     Δ cp < Δ H .
E ( Δ H , 0 ) { ( 2 ) 3 / 2 π 1 / 2 / [ ( k Z ) g 1 / 2 h 1 / 12 Δ cp 1 / 4 ] } exp ( i π / 4 ) { ( k a ) 7 / 6 A i [ ( k a ) 2 / 3 Δ H / h 1 / 3 ] i [ h 1 / 6 ( k a ) 5 / 6 / ( 2 Δ cp 1 / 2 ) ] A i [ ( k a ) 2 / 3 Δ H / h 1 / 3 ] } .
E ( Δ H , 0 ) { ( k a ) / [ ( k Z ) ( g Δ cp ) 1 / 2 ] } exp [ 2 i ( k a ) Δ H 3 / 2 / h 1 / 2 ] { 2 ( k a ) 5 / 4 / [ ( k Z ) π 1 / 2 g 1 / 2 ( Δ cp h ) 1 / 8 ] } exp ( i π / 4 ) exp [ 2 i ( k a ) Δ H 3 / 2 / h 1 / 2 ] P ( X , 0 ) ,
X = ( k a ) 1 / 2 ( Δ cp Δ H ) / ( Δ cp h ) 1 / 4 .
E ( Δ H , Δ V ) i { 2 7 / 2 π ( k a / 2 ) 4 / 3 / [ ( k Z ) g 1 / 2 h 1 / 6 ] } A i { ( k a / 2 ) 2 / 3 [ Δ H ( 8 h / g ) 1 / 2 Δ V ] / h 1 / 3 } × A i { ( k a / 2 ) 2 / 3 [ Δ H + ( 8 h / g ) 1 / 2 Δ V ] / h 1 / 3 } .
( 8 h / g ) hufs = lim R , S S / R = 12 / n 2 .

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