Abstract

Diffraction and reflection of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit are considered in the context of ray theory. A closed-form solution for both diffraction and reflection is obtained, and the polarization character of the diffracted plus reflected electric field is obtained. It is found that the magnitude of the reflected electric field is multivalued for forward scattering. This is interpreted in terms of the variation of the spheroid's Gaussian curvature at the points where grazing ray incidence occurs.

© 1996 Optical Society of America

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), 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, New York, 1983), Chap. 4.
  4. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  5. Ref. 3, appendix A and Sec. 4.8.
  6. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975);Appl. Opt.15, 2028(E) (1976).
    [PubMed]
  7. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  8. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [CrossRef] [PubMed]
  9. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  10. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3.
  11. J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
    [CrossRef]
  12. G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
    [CrossRef] [PubMed]
  13. J.-C. Ravey, P. Mazerone, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
    [CrossRef]
  14. J. C. Ravey, P. Mazerone, “Light scattering by large spheroids in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
    [CrossRef]
  15. J. C. Ravey, “The first extrema in the radiation pattern of the light scattered by micrometer-sized spheres and spheroids,” J. Colloid Interface Sci. 105, 435–446 (1985).
    [CrossRef]
  16. T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28, 4096–4102 (1989).
    [CrossRef] [PubMed]
  17. T. W. Chen, W. S. Smith, “Large-angle light scattering at large size parameters,” Appl. Opt. 30, 6558–6560 (1992).
    [CrossRef]
  18. T. W. Chen, “Simple formula for light scattering by a large spherical particle,” Appl. Opt. 32, 7568–7571 (1993).
    [CrossRef] [PubMed]
  19. B. R. Levy, J. B. Keller, “Diffraction by a spheroid,” Can. J. Phys. 38, 128–144 (1960).
    [CrossRef]
  20. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
    [CrossRef]
  21. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 3.
  22. R. T. Wang, H. C. van de Hulst, “Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  23. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
    [CrossRef]
  24. P. L. Marston, “Transverse cusp diffraction catastrophe: some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am. 81, 226–232 (1987).
    [CrossRef]
  25. C. E. Dean, P. L. Marston, “Opening rate of the cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991).
    [CrossRef] [PubMed]
  26. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  27. A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  28. E. A. Hovenac, “Calculation of far-field scattering from non-spherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991).
    [CrossRef] [PubMed]
  29. J. A. Lock, T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [CrossRef]
  30. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
    [CrossRef]
  31. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  32. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
    [CrossRef]
  33. P. L. Marston, 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] [PubMed]
  34. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  35. G. P. Können, “Appearance of supernumeraries of the second rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [CrossRef]
  36. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid: II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
    [CrossRef] [PubMed]
  37. J. B. Keller, H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. 40, 48–52 (1950).
    [CrossRef]
  38. M. Born, E. Wolf, Principles of Optics, 6th corrected ed. (Pergamon, New York, 1980), pp. 398–399.
  39. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), equation (4.13).
  40. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
  41. Ref. 1, pp. 206–207.
  42. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), Sec. 2.8.
  43. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sect. 4.3.2.
  44. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  45. Ref. 1, Sec. 12.32.
  46. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (NY) 7, 259–286 (1959).
    [CrossRef]
  47. Ref. 43, figure 10.9.
  48. H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
    [CrossRef]
  49. H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (NY) 34, 23–95 (1965).
    [CrossRef]
  50. H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
    [CrossRef] [PubMed]
  51. H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
    [CrossRef] [PubMed]
  52. Ref. 3, Sec. 8.4.7.
  53. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), Chap. 14.
  54. Ref. 1, Sec. 15.22.
  55. Ref. 1, Sec. 16.22.

1996

1995

1994

1993

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

T. W. Chen, “Simple formula for light scattering by a large spherical particle,” Appl. Opt. 32, 7568–7571 (1993).
[CrossRef] [PubMed]

1992

T. W. Chen, W. S. Smith, “Large-angle light scattering at large size parameters,” Appl. Opt. 30, 6558–6560 (1992).
[CrossRef]

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), Sec. 2.8.

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

1991

1989

1988

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
[CrossRef]

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

1987

G. P. Können, “Appearance of supernumeraries of the second rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
[CrossRef]

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

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

1985

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

J. C. Ravey, “The first extrema in the radiation pattern of the light scattered by micrometer-sized spheres and spheroids,” J. Colloid Interface Sci. 105, 435–446 (1985).
[CrossRef]

1984

P. L. Marston, 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]

1983

J. C. Ravey, P. Mazerone, “Light scattering by large spheroids in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

1982

J.-C. Ravey, P. Mazerone, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

1981

1980

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

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

1979

1975

1965

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

1960

B. R. Levy, J. B. Keller, “Diffraction by a spheroid,” Can. J. Phys. 38, 128–144 (1960).
[CrossRef]

1959

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

1950

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

Asano, S.

Barber, P.

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3.

Barton, J. P.

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

Berry, M. V.

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

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th corrected ed. (Pergamon, New York, 1980), pp. 398–399.

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), Chap. 14.

Chen, S.-H.

Chen, T. W.

Dean, C. E.

Evans, B. T. N.

Ford, K. W.

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

Fournier, G. R.

Glantschnig, W. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), equation (4.13).

Gouesbet, G.

Gréhan, G.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sect. 4.3.2.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3.

Hovenac, E. A.

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

E. A. Hovenac, “Calculation of far-field scattering from non-spherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991).
[CrossRef] [PubMed]

Huffman, D. R.

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

Humphreys, W. J.

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

Kaduchak, G.

Keller, H. B.

Keller, J. B.

Kerker, M.

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

Können, G. P.

Levy, B. R.

B. R. Levy, J. B. Keller, “Diffraction by a spheroid,” Can. J. Phys. 38, 128–144 (1960).
[CrossRef]

Lock, J. A.

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid: II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
[CrossRef] [PubMed]

J. A. Lock, T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
[CrossRef]

Marston, P. L.

P. L. Marston, 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] [PubMed]

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), Sec. 2.8.

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

P. L. Marston, “Transverse cusp diffraction catastrophe: 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] [PubMed]

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

Mazerone, P.

J. C. Ravey, P. Mazerone, “Light scattering by large spheroids in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

J.-C. Ravey, P. Mazerone, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

McCollum, T. A.

J. A. Lock, T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

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

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water drops,” 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]

Ravey, J. C.

J. C. Ravey, “The first extrema in the radiation pattern of the light scattered by micrometer-sized spheres and spheroids,” J. Colloid Interface Sci. 105, 435–446 (1985).
[CrossRef]

J. C. Ravey, P. Mazerone, “Light scattering by large spheroids in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

Ravey, J.-C.

J.-C. Ravey, P. Mazerone, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

Sassen, K.

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), Chap. 14.

Smith, W. S.

T. W. Chen, W. S. Smith, “Large-angle light scattering at large size parameters,” Appl. Opt. 30, 6558–6560 (1992).
[CrossRef]

Trinh, E. H.

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

Ungut, A.

Upstill, C.

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

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), Chap. 14.

van de Hulst, H. C.

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9.

Wang, R. T.

Wheeler, J. A.

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

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th corrected ed. (Pergamon, New York, 1980), pp. 398–399.

Yamamoto, G.

Yeh, C.

Am. J. Phys.

J. A. Lock, T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

Ann. Phys. (NY)

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

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

Appl. Opt.

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

W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

E. A. Hovenac, “Calculation of far-field scattering from non-spherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991).
[CrossRef] [PubMed]

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

P. L. Marston, 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] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid: II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975);Appl. Opt.15, 2028(E) (1976).
[PubMed]

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

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
[CrossRef] [PubMed]

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
[CrossRef] [PubMed]

T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28, 4096–4102 (1989).
[CrossRef] [PubMed]

T. W. Chen, W. S. Smith, “Large-angle light scattering at large size parameters,” Appl. Opt. 30, 6558–6560 (1992).
[CrossRef]

T. W. Chen, “Simple formula for light scattering by a large spherical particle,” Appl. Opt. 32, 7568–7571 (1993).
[CrossRef] [PubMed]

Can. J. Phys.

B. R. Levy, J. B. Keller, “Diffraction by a spheroid,” Can. J. Phys. 38, 128–144 (1960).
[CrossRef]

J. Acoust. Soc. Am.

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

J. Appl. Phys.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

J. Colloid Interface Sci.

J. C. Ravey, “The first extrema in the radiation pattern of the light scattered by micrometer-sized spheres and spheroids,” J. Colloid Interface Sci. 105, 435–446 (1985).
[CrossRef]

J. Opt. (Paris)

J.-C. Ravey, P. Mazerone, “Light scattering in the physical optics approximation: application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

J. C. Ravey, P. Mazerone, “Light scattering by large spheroids in the physical optics approximation: numerical comparison with other approximate and exact results,” J. Opt. (Paris) 14, 29–41 (1983).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

Nature

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Figures (9)

Fig. 1
Fig. 1

Laboratory coordinate system xyz and the scattering angles θ, Φ.

Fig. 2
Fig. 2

Coordinate system x″y″z″ attached to the spheroid. The spheroid has the radius a in the x″ and y″ directions and the radius b in the z″ direction.

Fig. 3
Fig. 3

Arbitrarily oriented spheroid with respect to the original and the rotated lab coordinate systems.

Fig. 4
Fig. 4

Outward unit normals m ̂ and n ̂ on the lit side and the shadowed side of the spheroid, respectively.

Fig. 5
Fig. 5

Incident and outgoing flux tubes. The incident flux tube is centered about the point r′, ξ′ and has the cross-sectional area ABr′dr′dξ′. The outgoing flux tube is centered about the scattering angles θ, Φ and has the cross-sectional area R 2Δ sin θdr′dξ′, where Δ is given in Eq. (17).

Fig. 6
Fig. 6

Geometry of the reflected ray. The normal m ̂ to the surface at the point of incidence is in the π − Ψ, η direction with respect to the x′y′z′ rotated lab coordinate system. The incident and reflected unit wave vectors are k ̂ inc and k ̂ ref , respectively.

Fig. 7
Fig. 7

Trajectories of the reflected ray and the reference ray. The spheroid entrance plane is UU′, and the spheroid exit plane is VV′.

Fig. 8
Fig. 8

Scattered intensity in Lorenz–Mie theory and ray theory for (a) the TE polarization and (b) the TM polarization as a function of the scattering angle θ for a conducting sphere with the size parameter 2πa/λ = 50. The diffraction–reflection interference structure in the Lorenz–Mie intensity damps out much faster than in the ray-theory intensity. In this figure the incident intensity is given by I 0 = E 0 2/2μ0 c.

Fig. 9
Fig. 9

Scattered intensity in ray theory for a plane wave with λ = 0.6328 μm and χ = 0° incident upon a spheroid with a = 5 μm as a function of the scattering angle θ in the Φ = 0° plane. The spheroid parameters are (a) b/a = 10, θ = 0°, ϕ = 0°; (b) b/a = 10, θ = 45°, ϕ = 0°; (c) b/a = 10, θ = 90°, ϕ = 0° for a prolate spheroid being tilted from end-on incidence to side-on incidence; (d) b/a = 1, θ = 0°, ϕ = 0° for a sphere; and (e) b/a = 0.1, θ = 0°, ϕ = 0°; (f) b/a = 0.1, θ = 45°, ϕ = 0°; (g) b/a = 0.1, θ = 90°, ϕ = 0° for an oblate spheroid being tilted from end-on incidence to side-on incidence. In this figure the incident intensity is given by I 0 = E 0 2/2μ0 c.

Equations (93)

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E inc = E 0 ( cos χ u ̂ x + sin χ u ̂ g ) exp ( i k z i ω t ) ,
k = 2 π λ .
E scattered ( θ , Φ ) = i E 0 k R exp ( i k R i ω t ) S ( θ , Φ ) × exp [ i δ ( θ , Φ ) ] ( θ , Φ ) ,
x 2 a 2 + y 2 a 2 + z 2 b 2 = 1 ,
z 2 ( sin 2 θ a 2 + cos 2 θ b 2 ) + 2 z ( 1 b 2 1 a 2 ) ( x cos ϕ + y sin ϕ ) × sin θ cos θ + x 2 a 2 ( cos 2 θ cos 2 ϕ + sin 2 ϕ ) + x 2 b 2 sin 2 θ cos 2 ϕ + y 2 a 2 ( cos 2 θ sin 2 ϕ + cos 2 ϕ ) + y 2 b 2 sin 2 θ sin 2 ϕ + 2 x y ( 1 b 2 1 a 2 ) sin 2 θ sin ϕ cos ϕ = 1 ,
x = x cos ϕ + y sin ϕ , y = x sin ϕ + y cos ϕ , z = z .
z upper z lower } = w x ± a b A ( 1 x 2 A 2 y 2 B 2 ) 1 / 2 ,
w = sin θ cos θ ( b 2 a 2 A 2 ) .
A = ( b 2 sin 2 θ + a 2 cos 2 θ ) 1 / 2 ,
B = a
x 2 A 2 + y 2 B 2 = 1 .
x = A r cos ξ , y = B r sin ξ ,
z upper z lower } = w A r cos ξ ± a b A ( 1 r 2 ) 1 / 2 .
m ̂ = ( z lower x ) u ̂ x + ( z lower y ) u ̂ y u ̂ z [ ( z lower x ) 2 + ( z lower y ) 2 + 1 ] 1 / 2
n ̂ = ( z upper x ) u ̂ x ( z upper y ) u ̂ y + u ̂ z [ ( z upper x ) 2 + ( z upper y ) 2 + 1 ] 1 / 2
P inc = E 0 2 2 μ 0 c A B r d r d ξ ,
Δ = | θ r θ ξ Φ r Φ ξ | .
S deflected ( θ , Φ ) = ( k 2 A B r Δ sin θ ) 1 / 2 .
k ̂ inc = u ̂ z ,
k ̂ ref = k ̂ inc 2 m ̂ ( k ̂ inc m ̂ ) = sin θ cos ( Φ ϕ ) u ̂ x + sin θ sin ( Φ ϕ ) u ̂ y + cos θ u ̂ z .
q = r ( 1 r 2 ) 1 / 2
r = q ( 1 + q 2 ) 1 / 2 .
tan η = ( z lower y ) / ( z lower x ) = a b A B q sin ξ a b A 2 q cos ξ + w ,
tan 2 ( π Ψ ) = ( z lower x ) 2 + ( z lower y ) 2 = ( a b A 2 q cos ξ + w ) 2 + ( a b A B q sin ξ ) 2 .
tan ξ = B A tan Ψ sin η tan Ψ cos η w ,
q = A 2 a b [ ( B A tan Ψ sin η ) 2 + ( tan Ψ cos η w ) 2 ] 1 / 2 .
Φ ϕ = η .
cos θ = 1 2 ( m ̂ u ̂ z ) 2
θ = π 2 Ψ .
r 2 = cos 2 θ 2 [ 1 + ( B 2 A 2 1 ) sin 2 ( Φ ϕ ) ] w sin θ cos ( Φ ϕ ) + w 2 sin 2 θ 2 1 + ( B 2 A 2 1 ) [ sin 2 ( Φ ϕ ) + sin 2 θ 2 cos 2 ( Φ ϕ ) ] + ( B 4 A 4 b 2 a 2 B 2 A 2 + w 2 ) sin 2 θ 2 w sin θ cos ( Φ ϕ ) , tan ξ = B A cos θ 2 sin ( Φ ϕ ) cos θ 2 cos ( Φ ϕ ) w sin θ 2 .
θ = π 2 arctan [ ( a b A 2 q cos ξ + w ) 2 + ( a b A B q sin ξ ) 2 ] 1 / 2 , Φ = ϕ + arctan ( a b A B q sin ξ a b A 2 q cos ξ + w 2 ) .
S ref ( θ , Φ ) = k A 2 ( A B a b ) ( 1 r 2 ) sin 2 θ 2 .
S ref ( θ , Φ ) = k b 2 ( B 2 A 2 ) / F ( θ , Φ ) ,
F ( θ , Φ ) = 1 + ( B 2 A 2 1 ) × [ sin 2 ( Φ ϕ ) + sin 2 θ 2 cos 2 ( Φ ϕ ) ] + ( B 4 A 4 b 2 a 2 B 2 A 2 + w 2 ) × sin 2 θ 2 w sin θ cos ( Φ ϕ ) .
S ref ( θ , Φ ) = k b / 2 1 + ( b 2 a 2 1 ) [ cos θ 2 cos ( Φ ϕ ) sin θ sin θ 2 cos θ ] 2 .
E diff ( θ , Φ ) = i k E 0 2 π R exp ( i k R i ω t ) ellipse d x d y × exp [ i k ( x sin θ cos Φ + y sin θ sin Φ ) ] .
x = x , y = A B y
S diff ( θ , Φ ) = ( k A ) 2 B A × J 1 { k A θ [ cos 2 ( Φ ϕ ) + B 2 A 2 sin 2 ( Φ ϕ ) ] 1 / 2 } k A θ [ cos 2 ( Φ ϕ ) + B 2 A 2 sin 2 ( Φ ϕ ) ] 1 / 2 .
α = α R + z lower ( r , ξ ) .
x = β R sin θ cos ( Φ ϕ ) , y = β R sin θ sin ( Φ ϕ ) , z = β R cos θ ,
β R = { A 2 a 2 b 2 [ cos θ w sin θ cos ( Φ ϕ ) ] 2 + sin 2 θ × [ cos 2 ( Φ ϕ ) A 2 + sin 2 ( Φ ϕ ) B 2 ] } 1 / 2 .
x = A r cos ξ + β sin θ cos ( Φ ϕ ) , y = B r sin ξ + β sin θ sin ( Φ ϕ ) , z = z lower ( r , ξ ) + β cos θ ,
β = β R A r cos ξ sin θ cos ( Φ ϕ ) B r sin ξ sin θ sin ( Φ ϕ ) z lower ( r , ξ ) cos θ .
δ ref ( θ , Φ ) = 2 π λ ( α + β α R β R ) π 2 ,
δ ref ( θ , Φ ) = 2 π λ [ z lower A r cos ξ sin θ cos ( Φ ϕ ) B r sin ξ sin θ sin ( Φ ϕ ) z lower cos θ ] π 2 = 4 π α λ sin θ 2 { 1 + ( b 2 a 2 1 ) × [ cos θ 2 cos ( Φ ϕ ) sin θ sin θ 2 cos θ ] 2 } 1 / 2 π 2 .
δ diff = 0 .
E inc = E 0 exp ( i k z i ω t ) ̂ inc ,
̂ inc = cos χ u ̂ x + sin χ u ̂ y = cos ( χ ϕ ) u ̂ x + sin ( χ ϕ ) u ̂ y .
TE ̂ inc = m ̂ × k ̂ inc sin θ inc = sin η u ̂ x cos η u ̂ y ,
TM ̂ inc = k ̂ inc × ( m ̂ × k ̂ inc ) sin θ inc = cos η u ̂ x + sin η u ̂ y ,
k ̂ inc m ̂ = cos θ inc = cos Ψ .
̂ inc = cos γ TE ̂ inc + sin γ TM ̂ inc ,
γ = χ Φ 3 π 2 .
TE ̂ ref = m ̂ × k ̂ ref sin θ ref = sin η u ̂ x cos η u ̂ y ,
TM ̂ ref = k ̂ ref × ( m ̂ × k ̂ ref ) sin θ ref = cos θ cos η u ̂ x + cos θ sin η u ̂ y sin θ η u ̂ z ,
θ inc = θ ref .
̂ ref = ( cos γ ) r TE TE ̂ ref + ( sin γ ) r TE TM ̂ ref ,
m ̂ d sphere = cos ξ d u ̂ x + sin ξ d u ̂ y .
TE ̂ i d sphere = m ̂ d sphere × k ̂ inc = sin ξ d u ̂ x cos ξ d u ̂ y ,
TM ̂ i d sphere = k ̂ inc × ( m ̂ d sphere × k ̂ inc ) = cos ξ d u ̂ x + sin ξ d u ̂ y .
̂ inc sphere = cos χ u ̂ x + sin χ u ̂ y = cos γ d sphere TE ̂ i d sphere + sin γ d sphere TM ̂ i d sphere ,
γ d sphere = χ ξ d 3 π 2 .
a l diff = b l diff = ½ .
π l ( θ ) τ l ( θ ) = l ( l + 1 ) 2 J 0 [ ( l + 1 2 ) θ ] ,
E diff sphere ( θ , Φ ) = i E 0 k R exp ( i k R i ω t ) S diff sphere × ( θ , Φ ) ̂ diff sphere ,
S diff sphere ( θ , Φ ) = ( k a ) 2 J 1 ( k a θ ) k a θ ,
̂ diff sphere = cos ( Φ χ ) u ̂ θ sin ( Φ χ ) u ̂ Φ .
TE ̂ f d sphere = m ̂ d sphere × k ̂ diff sin θ = u ̂ Φ ,
TM ̂ f d sphere = k ̂ diff × ( m ̂ d sphere × k ̂ diff ) sin θ = u ̂ θ ,
̂ diff sphere = cos γ d sphere TE ̂ f d sphere + sin γ d sphere TM ̂ f d sphere .
m ̂ d = cos η d u ̂ x + sin η d u ̂ y ,
tan η d = A B tan ξ d .
TE ̂ i d = m ̂ d × k ̂ inc = sin η d u ̂ x cos η d u ̂ y ,
TM ̂ i d = k ̂ inc × ( m ̂ d × k ̂ inc ) = cos η d u ̂ x + sin η d u ̂ y ,
̂ inc = cos γ d TE ̂ i d + sin γ d TM ̂ i d ,
γ d = χ ϕ η d 3 π 2 .
Φ ϕ = η d .
TE ̂ f d = m ̂ d × k ̂ diff sin θ = sin η d u ̂ x cos η d u ̂ y ,
TM ̂ f d = k ̂ diff × ( m ̂ d × k ̂ diff ) sin θ = cos θ cos η d u ̂ x + cos θ sin η d u ̂ y sin θ u ̂ z ,
̂ diff = cos γ d TE ̂ f d + sin γ d TM ̂ f d ,
E diff + ref = i E 0 k R exp ( i k R i ω t ) { [ S ref exp ( i δ ref ) cos γ r TE + S diff cos γ d ] TE ̂ ref + [ S ref exp ( i δ ref ) sin γ r TE + S diff sin γ d ] TE ̂ ref } .
S ref ( θ = 180 ° , Φ ) = k b / 2 1 + ( b 2 a 2 1 ) cos 2 θ ,
S ref ( θ = 0 ° , Φ ) = k b / 2 1 + ( b 2 a 2 1 ) cos 2 ( Φ ϕ ) sin 2 θ ,
P ref = E 0 2 2 μ 0 c θ = 0 π Φ = 0 2 π S ref 2 ( θ , Φ ) k 2 sin θ d θ d Φ ,
ρ 1 = a 2 b 2 ( a 2 sin 2 Σ + b 2 cos 2 Σ ) 3 / 2 ,
ρ 2 = a 2 ( a 2 sin 2 Σ + b 2 cos 2 Σ ) 1 / 2 ,
K g = ( ρ 1 ρ 2 ) 1 / 2 = 1 + ( b 2 a 2 1 ) cos 2 Σ b .
u ̂ z = sin θ u ̂ x + cos θ u ̂ z , m ̂ = cos θ 2 cos ( Φ ϕ ) u ̂ x + cos θ 2 sin ( Φ ϕ ) × u ̂ z sin θ 2 u ̂ z ,
cos Σ = u ̂ z m ̂ = cos θ 2 cos ( Φ ϕ ) sin θ sin θ 2 cos θ .
S ref = k 2 K g .
S ref ( θ , Φ ) = k b / 2 1 + ( b 2 a 2 1 ) sin 2 θ 2 ,
δ ref ( θ , Φ ) = 4 π a λ sin θ 2 × [ 1 + ( b 2 a 2 1 ) sin 2 θ 2 ] 1 / 2 π 2 ,
S diff ( θ , Φ ) = ( k a ) 2 J 1 ( k a θ ) k a θ .

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