Abstract

Transmission of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit is considered in the context of ray theory. The transmitted electric field is added to the diffracted plus reflected ray-theory electric field that was previously derived to obtain an approximation to the far-zone scattered intensity in the forward hemisphere. Two different types of cross-polarization effects are found. These are (a) a rotation of the polarization state of the transmitted rays from when they are referenced with respect to their entrance into the spheroid to when they are referenced with respect to their exit from it and (b) a rotation of the polarization state of the transmitted rays when they are referenced with respect to the polarization state of the diffracted plus reflected rays.

© 1996 Optical Society of America

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References

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  1. W. D. Bachalo, “Method for measuring the size and velocity of spheres by dual-beam light-scatter interferometry,” Appl. Opt. 19, 363–370 (1980).
    [Crossref] [PubMed]
  2. W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
  3. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [Crossref] [PubMed]
  4. A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [Crossref] [PubMed]
  5. S. V. Sanker, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
    [Crossref]
  6. M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
    [Crossref] [PubMed]
  7. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975);Appl. Opt.15, 2028(E) (1976).
    [PubMed]
  8. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [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. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [Crossref]
  14. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [Crossref]
  15. G. P. Können, “Appearance of supernumeraries of the second rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [Crossref]
  16. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 316, 529–531 (1984).
    [Crossref]
  17. J. P. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 316, 531–532 (1984).
    [Crossref]
  18. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [Crossref] [PubMed]
  19. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
    [Crossref]
  20. M. Bottlinger, H. Umhauer, “Modeling of light scattering by irregularly shaped particles using a ray-tracing method,” Appl. Opt. 30, 4732–4738 (1991).
    [Crossref] [PubMed]
  21. 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]
  22. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection,” Appl. Opt. 35, 500–514 (1996).
    [Crossref] [PubMed]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34–35, 46, 57.
  24. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 498.
  25. C. P. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 383, 402–403.
  26. Ref. 24, Chap. 10.
  27. Ref. 25, Sect. 5.3–5.7 and 8.2.
  28. Ref. 25, Sect. 8.3.
  29. Ref. 25, Sect. 8.4.
  30. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [Crossref] [PubMed]
  31. J. B. Keller, H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. 40, 48–52 (1950).
    [Crossref]
  32. D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [Crossref] [PubMed]
  33. A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
    [Crossref] [PubMed]
  34. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [Crossref]
  35. Ref. 23, Secs. 12.22 and 12.23.
  36. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sect. 5.2.1.
  37. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
    [Crossref]
  38. J. A. Lock, T. A. McCollum, “Further throughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [Crossref]
  39. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), figure 2.5.
    [Crossref]
  40. M. V. Berry, “Waves and Thorm's theorem,” Adv. Phys. 25, 1–26 (1976).
    [Crossref]
  41. D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
    [Crossref] [PubMed]
  42. Ref. 23, Sec. 4.41.
  43. Ref. 25, Sec. 3.2.
  44. Ref. 38, figure 4.20–4.22.
  45. T. W. Chen, “Simple formula for light scattering by a large spherical dielectric,” Appl. Opt. 32, 7568–7571 (1993).
    [Crossref] [PubMed]
  46. Figure 8 of Ref. 21 also contains the contributions of 2, 4, and 6 internal reflections. When these are removed, our result agrees exactly [E. A. Hovenac, NASA Lewis Research Center, Cleveland, Oh. 44115 (personal communication, August1994)].
  47. T. W. Chen, L. Yang, New Mexico State University, Las Cruces, N.M. 88003 (personal communication, December1994).
  48. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [Crossref]
  49. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [Crossref] [PubMed]
  50. H. J. Simpson, 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] [PubMed]
  51. G. Kaduchak, P. L. Marston, 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, 4961 (1994).
    [Crossref] [PubMed]
  52. G. Kaduchak, 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] [PubMed]
  53. 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]
  54. The only published photograph to my knowledge is in A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980), figure 18b.
    [Crossref] [PubMed]

1996 (1)

1995 (1)

1994 (5)

1993 (1)

1992 (1)

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

1991 (7)

1989 (1)

1987 (1)

1985 (1)

1984 (3)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 316, 529–531 (1984).
[Crossref]

J. P. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 316, 531–532 (1984).
[Crossref]

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

1983 (1)

1981 (3)

1980 (5)

1979 (2)

1976 (1)

M. V. Berry, “Waves and Thorm's theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

1975 (2)

1973 (1)

1969 (1)

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

1950 (1)

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.

Ashkin, A.

Bachalo, W. D.

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), figure 2.5.
[Crossref]

M. V. Berry, “Waves and Thorm's theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Bohren, C. P.

C. P. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 383, 402–403.

Bottlinger, M.

Boyd, R. W.

Burkhard, D. G.

Chen, S.-H.

Chen, T. W.

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

T. W. Chen, L. Yang, New Mexico State University, Las Cruces, N.M. 88003 (personal communication, December1994).

Dziedzic, J. M.

Evans, B. T. N.

Fournier, G. R.

Fraser, A. B.

Glantschnig, W. J.

Gouesbet, G.

Gréhan, G.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sect. 5.2.1.

Hill, S. C.

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

Hirleman, E. D.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

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

Figure 8 of Ref. 21 also contains the contributions of 2, 4, and 6 internal reflections. When these are removed, our result agrees exactly [E. A. Hovenac, NASA Lewis Research Center, Cleveland, Oh. 44115 (personal communication, August1994)].

Huffman, D. R.

C. P. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 383, 402–403.

Kaduchak, G.

Kassim, A. M.

Keller, H. B.

Keller, J. B.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 498.

Können, G. P.

Lock, J. A.

Marston, P. L.

McCollum, T. A.

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

Nussenzveig, H. M.

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

Nye, J. F.

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

Nye, J. P.

J. P. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 316, 531–532 (1984).
[Crossref]

Sanker, S. V.

Sassen, K.

Sato, M.

Schneider, M.

Shealy, D. L.

Simpson, H. J.

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 316, 529–531 (1984).
[Crossref]

Umhauer, H.

Ungut, A.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), figure 2.5.
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34–35, 46, 57.

Yamamoto, G.

Yang, L.

T. W. Chen, L. Yang, New Mexico State University, Las Cruces, N.M. 88003 (personal communication, December1994).

Yeh, C.

Adv. Phys. (1)

M. V. Berry, “Waves and Thorm's theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Am. J. Phys. (1)

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

Appl. Opt. (23)

D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
[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]

H. J. Simpson, 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] [PubMed]

G. Kaduchak, P. L. Marston, 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, 4961 (1994).
[Crossref] [PubMed]

G. Kaduchak, 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] [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]

The only published photograph to my knowledge is in A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980), figure 18b.
[Crossref] [PubMed]

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

S. V. Sanker, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
[Crossref]

M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
[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]

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]

M. Bottlinger, H. Umhauer, “Modeling of light scattering by irregularly shaped particles using a ray-tracing method,” Appl. Opt. 30, 4732–4738 (1991).
[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]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection,” Appl. Opt. 35, 500–514 (1996).
[Crossref] [PubMed]

W. D. Bachalo, “Method for measuring the size and velocity of spheres by dual-beam light-scatter interferometry,” Appl. Opt. 19, 363–370 (1980).
[Crossref] [PubMed]

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

D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[Crossref] [PubMed]

A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[Crossref] [PubMed]

J. Appl. Phys. (1)

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. Math. Phys. (1)

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

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Nature (2)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 316, 529–531 (1984).
[Crossref]

J. P. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 316, 531–532 (1984).
[Crossref]

Opt. Eng. (1)

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

Opt. Lett. (1)

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

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

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), figure 2.5.
[Crossref]

Other (15)

Ref. 23, Sec. 4.41.

Ref. 25, Sec. 3.2.

Ref. 38, figure 4.20–4.22.

Figure 8 of Ref. 21 also contains the contributions of 2, 4, and 6 internal reflections. When these are removed, our result agrees exactly [E. A. Hovenac, NASA Lewis Research Center, Cleveland, Oh. 44115 (personal communication, August1994)].

T. W. Chen, L. Yang, New Mexico State University, Las Cruces, N.M. 88003 (personal communication, December1994).

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

Ref. 23, Secs. 12.22 and 12.23.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sect. 5.2.1.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34–35, 46, 57.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 498.

C. P. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 383, 402–403.

Ref. 24, Chap. 10.

Ref. 25, Sect. 5.3–5.7 and 8.2.

Ref. 25, Sect. 8.3.

Ref. 25, Sect. 8.4.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the transmitted ray. The unit normal m ̂ 0 to the surface at the point of entrance is in the π – ψ0, η0 direction with respect to the x′ y′ z′ rotated lab coordinate system. The portion of the ray inside the spheroid has length s 01 and is in the ψ01, η01 direction. The unit normal n ̂ 1 to the surface at the point of exit on the shadowed side of the spheroid is in the ψ1, η1 direction.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Rays transmitted through a spheroid for n = 1.333. (a) b/a = 1 corresponding to a sphere. The caustics are a cusp of revolution pointing outward and an axial spike caustic. (b) b/a = 1.33. The cusp starts to retract into the spheroid. (c) b/a = 1.5123. All the rays focus at a single point. (d) b/a = 1.67. The caustics are a cusp of revolution pointing inward and an axial spike caustic. (e) b/a = 2.0. Paraxial rays have a scattering angle of θ = 0° corresponding to a forward glory. (f) b/a = 2.25. The paraxial rays form a butterfly of revolution caustic and a second axial spike caustic. (g) Butterfly caustic of (f) magnified by a factor of 11.67. The caustic begins at location 1 and then continues in order to the locations 2, 3, 4, 5. (h) b/a = 2.45. The second-cusp caustic points outward. As a function of r 0′ there are two critical angles for total internal reflection.

Fig. 4
Fig. 4

Phase-space diagram for the caustics produced by the transmitted rays when r 0′ and b/a are varied while the refractive index n is held fixed. In region A there are two caustics, a cusp of revolution pointing outward and an axial spike. Along the line α α ¯ corresponding to Eq. (55) these caustics contract to a point focus. In region B there are two caustics, a cusp of revolution pointing inward and an axial spike. In region C there are three caustics, a cusp of revolution pointing inward and two axial spikes. In regions D, E, F, there are four caustics. In D they are two cusps pointing inward and two axial spikes. In E they are an inward-pointing cusp, a butterfly, and two axial spikes. In F they are an inward-pointing and an outward-pointing cusp, and two axial spikes. The line β β ¯ corresponds to the forward glory given by Eq. (58), and the line β γ ¯ is the transmission rainbow. The cross-hatched regions denote the absence of transmitted rays as a result of total internal reflection as in Eq. (33).

Fig. 5
Fig. 5

TE and TM polarization directions for the reflected ray and the transmitted ray of Eq. (60) and Eqs. (61)(64), respectively.

Fig. 6
Fig. 6

Contributions to the transmitted electric field. The contributions proportional to cos Δ01 are the polarization terms that describe transmission by a sphere or end-on spheroid. The contributions proportional to sin Δ01 are the first type of cross-polarization terms.

Fig. 7
Fig. 7

Contributions to the reflected plus transmitted electric field. The contributions proportional to cos Δ R 1 are the polarization terms that describe scattering by a sphere or end-on spheroid. The contributions proportional to sin Δ R 1 are the second type of cross-polarization terms.

Fig. 8
Fig. 8

Intensity as a function of the scattering angle θ with Φ = 0° for a plane wave with λ = 0.6328 μm and χ = 90° incident upon a sphere with a = 10.071 μm and n = 1.333. The solid curve is the Lorenz–Mie theory result, the dashed curve is Eq. (84), and the dotted curve is the generalized eikonal approximation of Ref. 45.

Fig. 9
Fig. 9

Intensity as a function of the scattering angle θ with Φ = 0° for a plane wave with λ = 0.6328 um and χ = 90° incident upon a spheroid with a = 10.071 μm, n = 1.333, and b/a = 1.5 In (a) the spheroid is in the end-on orientation, with θ = 0° and ϕ = 0°. In (b) it is in the side-on orientation, with θ = 90° and ϕ = 0°. The solid curve is Eq. (84), and the dashed curve is the generalized eikonal approximation of Ref. 47.

Fig. 10
Fig. 10

Intensity as a function of the scattering angle θ with $ = 0° for a plane wave with λ = 0.6328 μm and χ = 90° incident upon a spheroid with a = 3.021 μm, n = 1.333, and b/a = 1.5. In (a) the spheroid is in the end-on orientation, with 0 = 0° and ϕ = 0°. In (b) it is in the side-on orientation, with θ = 90° and ϕ = 0°. The solid curve is Eq. (84), and the dashed curve is the exact solution of Ref. 49.

Equations (88)

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k = 2 π λ ,
E inc = E 0 ( cos χ û x + sin χ û y ) exp ( i k z i ω t ) .
E trans ( θ , Φ ) = i E 0 k R exp ( i k R i ω t ) S trans ( θ , Φ ) × exp [ i δ trans ( θ , Φ ) ] trans ( θ , Φ ) .
x 2 a 2 + y 2 a 2 + z 2 b 2 = 1 ,
z upper z lower } = w A r cos ξ ± a b A ( 1 r 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 = A r cos ξ , y = B r sin ξ ,
k ̂ i 0 = û z .
m ̂ 0 = sin Ψ 0 cos η 0 û x + sin Ψ 0 sin η 0 û y cos Ψ 0 û z .
tan η 0 = a b A B q 0 sin ξ 0 a b A 2 q 0 cos ξ 0 + w ,
tan 2 ( π Ψ 0 ) = ( a b A 2 q 0 cos ξ 0 + w ) 2 + ( a b A B q 0 sin ξ 0 ) 2 ,
q 0 = r 0 ( 1 r 0 2 ) 1 / 2 .
k ̂ t 0 = sin Ψ 01 cos η 01 û x + sin Ψ 01 sin η 01 û y + cos Ψ 01 û z .
k ̂ t 0 = 1 n k ̂ i 0 + ( cos θ i 0 n cos θ t 0 ) m ̂ 0 ,
θ i 0 = Ψ 0
n sin θ t 0 = sin Ψ 0 .
Ψ 01 = Ψ 0 θ t 0 η 01 = η 0 + π .
x = x 0 s 01 sin Ψ 01 cos η 0 , y = y 0 s 01 sin Ψ 01 sin η 0 , z = z 0 + s 01 cos Ψ 01 ,
s 01 = 2 a b A ( 1 r 0 ) 1 / 2 [ ( cos Ψ 01 + w sin Ψ 01 + cos η 0 ) + a b A B q 0 sin Ψ 01 ( sin ξ 0 sin η 0 + B A cos ξ 0 cos η 0 ) ( cos Ψ 01 + w sin Ψ 01 + cos η 0 ) 2 + a 2 b 2 A 2 B 2 sin 2 Ψ 01 ( sin 2 η 0 + B 2 A 2 cos 2 η 0 ) ] .
x 1 = x 0 s 01 sin Ψ 01 cos η 0 = A r 1 cos ξ 1 , y 1 = y 0 s 01 sin Ψ 01 sin η 0 = B r 1 sin ξ 1 , z 1 = z 0 + s 01 cos Ψ 01 ,
r 1 2 = r 0 2 2 r 0 B s 01 sin Ψ 01 × ( sin ξ 0 sin η 0 + B A cos ξ 0 cos η 0 ) + s 01 2 B 2 sin 2 Ψ 01 ( sin 2 η 0 + B 2 A 2 cos 2 η 0 ) ,
tan ξ 1 = r 0 sin ξ 0 s 01 B sin Ψ 01 sin η 0 r 0 cos ξ 0 s 01 A sin Ψ 01 cos η 0 .
s 01 ( cos Ψ 01 + w sin Ψ 01 cos η 0 ) a b A ( 1 r 0 2 ) 1 / 2 = a b A ( 1 r 1 2 ) 1 / 2 .
s 01 ( cos Ψ 01 + w sin Ψ 01 cos η 0 ) a b A ( 1 r 0 2 ) 1 / 2 = a b A ( 1 r 1 2 ) 1 / 2 .
n ̂ 1 = sin Ψ 1 cos η 1 û x + sin Ψ 1 sin η 1 û y + cos Ψ 1 û z ,
tan η 1 = a b A B q 1 sin ξ 1 a b A 2 q 1 cos ξ 1 w ,
tan 2 Ψ 1 ( a b A 2 q 1 cos ξ 1 w ) 2 + ( a b A B q 1 sin ξ 1 ) 2 ,
q 1 = r 1 ( 1 r 1 2 ) 1 / 2 .
k ̂ i 1 = k ̂ t 0 .
cos θ i 1 = n ̂ 1 k ̂ i 1 = cos Ψ 01 cos Ψ 1 sin Ψ 01 sin Ψ 1 cos ( η 0 η 1 ) .
sin θ t 1 = n sin θ i 1 .
k ̂ t 1 = sin θ cos ( Φ ϕ ) û x + sin θ sin ( Φ ϕ ) û y + cos θ û z ,
k ̂ t 1 = n k ̂ i 1 + ( cos θ t 1 n cos θ i 1 ) n ̂ 1 .
cos θ = n cos Ψ 01 + ( cos θ t 1 n cos θ i 1 ) cos Ψ 1 ,
tan Φ = ( cos θ t 1 n cos θ i 1 ) sin Ψ 1 sin η 1 n sin Ψ 01 sin η 0 ( cos θ t 1 n cos θ i 1 ) sin Ψ 1 cos η 1 n sin Ψ 01 cos η 0 .
ξ 0 = η 0 = ξ 1 = η 1 = Φ ϕ π .
tan Ψ 0 = b a q 0 ,
s 01 = 2 b ( 1 r 0 2 ) 1 / 2 { cos Ψ 01 + b a q 0 sin Ψ 01 cos 2 Ψ 01 + b 2 a 2 sin 2 Ψ 01 } ,
r 1 = [ r 0 2 2 r 0 s 01 a sin Ψ 01 + s 01 2 a 2 sin 2 Ψ 01 ] 1 / 2 = | r 0 s 01 a sin Ψ 01 | ,
tan Ψ 1 = b a q 1 ,
θ i 1 = Ψ 01 + Ψ 1 ,
θ = θ t 1 Ψ 1 .
S trans ( θ , Φ ) = ( k 2 A B r 0 Δ sin θ ) 1 / 2 ,
Δ = | θ r 0 θ ξ 0 Φ r 0 Φ ξ 0 | .
α R = z min ,
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 .
α 0 = α R + z 0 ( r 0 , ξ 0 ) .
x = β R sin θ cos ( Φ ϕ ) , y = β R sin θ sin ( Φ ϕ ) , z = β R cos θ
x = A r 1 cos ξ 1 + β 1 sin θ cos ( Φ ϕ ) , y = B r 1 sin ξ 1 + β 1 sin θ sin ( Φ ϕ ) , z = Z 1 ( r 1 , ξ 1 ) + β 1 cos θ
β 1 = β R A r 1 cos ξ 1 sin θ cos ( Φ ϕ ) B r 1 sin ξ 1 sin θ sin ( Φ ϕ ) z 1 cos θ .
L trans = α 0 + n s 01 + β 1 α R β R = z 0 + n s 01 A r 1 cos ξ 1 sin θ cos ( Φ ϕ ) B r 1 × sin ξ 1 sin θ sin ( Φ ϕ ) z 1 cos θ ,
δ trans = 2 π λ L trans 3 π 2
b 2 a 2 = n 2 n 2 1 ,
z = ( b 2 a 2 ) 1 / 2 ,
b 2 a 2 = n n 1 ,
q 0 2 = n 2 ( b 2 a 2 1 ) 2 b 2 / a 2 b 2 a 2 .
̂ inc = cos ( χ ϕ ) û x + sin ( χ ϕ ) û y .
TE ̂ inc = m ̂ × k ̂ inc sin θ inc , TE ̂ inc = k ̂ inc × ( m ̂ × k ̂ inc ) sin θ inc .
TE ̂ i 0 = k ̂ i 0 × m ̂ 0 sin θ i 0 , TM ̂ i 0 = k ̂ i 0 × ( k ̂ i 0 × m ̂ 0 ) sin θ i 0 ,
TE ̂ t 0 = k ̂ t 0 × m ̂ 0 sin θ t 0 , TM ̂ t 0 = k ̂ t 0 × ( k ̂ t 0 × m ̂ 0 ) sin θ t 0 ,
TE ̂ i 1 = k ̂ i 1 × n ̂ 1 sin θ i 1 , TM ̂ i 1 = k ̂ i 1 × ( k ̂ i 1 × n ̂ 1 ) sin θ i 1 ,
TE ̂ t 1 = k ̂ t 1 × n ̂ 1 sin θ t 1 , TM ̂ t 1 = k ̂ t 1 × ( k ̂ t 1 × n ̂ 1 ) sin θ t 1 ,
̂ inc = cos γ 0 TE ̂ i 0 + sin γ 0 TM ̂ i 0 ,
γ 0 = χ ϕ η 0 π 2 .
t 0 = t TE 0 cos γ 0 TE ̂ t 0 + t TM 0 sin γ 0 TE ̂ t 0
TE ̂ t 0 = cos Δ 01 TE ̂ i 1 + sin Δ 01 TM ̂ i 1 , TM ̂ t 0 = sin Δ 01 TE ̂ i 1 + cos Δ 01 TM ̂ i 1 ,
tan Δ 01 = sin Ψ 1 sin ( η 0 η 1 ) sin Ψ 10 cos Ψ 1 + cos Ψ 10 sin Ψ 1 cos ( η 0 η 1 )
i 1 = ( t TE 0 cos γ 0 cos Δ 01 t TM 0 sin γ 0 sin Δ 01 ) TE ̂ i 1 + ( t TE 0 cos γ 0 sin Δ 01 + t TM 0 sin γ 0 cos Δ 01 ) TM ̂ i 1 .
t 1 = ( t TE 0 t TE 1 cos γ 0 cos Δ 01 t TM 0 t TE 1 sin γ 0 sin Δ 01 ) TE ̂ t 1 + ( t TE 0 t TM 1 cos γ 0 sin Δ 01 + t TM 0 t TM 1 sin γ 0 cos Δ 01 ) TM ̂ t 1 ,
TE ̂ t 1 = cos Δ R 1 TE ̂ ref + sin Δ R 1 TM ̂ ref , TM ̂ t 1 = sin Δ R 1 TE ̂ ref + cos Δ R 1 TM ̂ ref ,
tan Δ R 1 = sin Ψ 1 sin ( Φ ϕ η 1 ) sin θ cos Ψ 1 cos θ sin Ψ 1 cos ( Φ ϕ η 1 ) ,
[ S 2 S 3 S 4 S 1 ]
E diff + ref + trans = i E 0 k R exp ( i k R i ω t ) [ S 2 S 3 S 4 S 1 ] [ sin γ cos γ ] .
[ sin γ cos γ ]
γ = χ ϕ η 3 π 2 = χ Φ 3 π 2 .
[ S 2 sin γ + S 3 cos γ S 4 sin γ + S 1 cos γ ]
TE ̂ i 0 = cos Ω TE ̂ inc sin Ω TM ̂ inc , TM ̂ i 0 = sin Ω TE ̂ inc + cos Ω TM ̂ inc ,
Ω = γ 0 γ = π + η η 0 = π + Φ ϕ ζ 0 ,
[ S 2 S 3 S 4 S 1 ] = S diff [ 1 0 0 1 ] + S ref exp ( i δ ref ) [ r TM 0 0 r TE ] + S trans exp ( i δ trans ) [ cos Δ R 1 sin Δ R 1 sin Δ R 1 cos Δ R 1 ] × [ t TM 1 0 0 t TE 1 ] [ cos Δ 01 sin Δ 01 sin Δ 01 cos Δ 01 ] × [ t TM 0 0 0 t TE 0 ] [ cos Ω sin Ω sin Ω cos Ω ] .
S 1 ( θ , Φ ) = S diff + S ref exp ( i δ ref ) r TE + S trans × exp ( i δ trans ) ( t TE 0 t TE 1 cos Δ R 1 cos Δ 01 cos Ω t TE 0 t TM 1 sin Δ R 1 sin Δ 01 cos Ω t TM 0 t TE 1 × cos Δ R 1 sin Δ 01 sin Ω t TM 0 t TM 1 sin Δ R 1 × cos Δ 01 sin Ω ) ,
S 2 ( θ , Φ ) = S diff + S ref exp ( i δ ref ) r TM + S trans × exp ( i δ trans ) ( t TM 0 t TM 1 cos Δ R 1 cos Δ 01 cos Ω t TM 0 t TE 1 sin Δ R 1 sin Δ 01 cos Ω t TE 0 t TM 1 × cos Δ R 1 sin Δ 01 sin Ω t TE 0 t TE 1 sin Δ R 1 × cos Δ 01 sin Ω ) ,
S 3 ( θ , Φ ) = S trans exp ( i δ trans ) ( t TM 0 t TM 1 cos Δ R 1 cos Δ 01 × sin Ω + t TE 0 t TM 1 cos Δ R 1 sin Δ 01 cos Ω + t TE 0 t TE 1 sin Δ R 1 cos Δ 01 cos Ω t TM 0 t TE 1 sin Δ R 1 sin Δ 01 sin Ω ) ,
S 4 ( θ , Φ ) = S trans exp ( i δ trans ) ( t TE 0 t TE 1 cos Δ R 1 cos Δ 01 × sin Ω t TM 0 t TE 1 cos Δ R 1 sin Δ 01 cos Ω t TM 0 t TM 1 sin Δ R 1 cos Δ 01 cos Ω + t TE 0 t TM 1 sin Δ R 1 sin Δ 01 sin Ω ) .
D E = S diff cos γ , D M = S diff sin γ , R E = S ref r TE cos γ , R M = S ref r TM sin γ , T EE = S trans t TE 0 t TE 1 cos γ 0 , T EM = S trans t TE 0 t TM 1 cos γ 0 , T ME = S trans t TM 0 t TE 1 sin γ 0 , T MM = S trans t TM 0 t TM 1 sin γ 0 ,
I diff + ref + trans ( θ , Φ ) = E 0 2 2 μ 0 c 1 k 2 R 2 [ S diff 2 + R E 2 + R M 2 + ( T EE 2 + T MM 2 ) cos 2 Δ 01 + 2 ( R E D E + R M D M ) cos δ ref + 2 ( T EE D E + T MM D M ) cos Δ R 1 cos Δ 01 cos δ trans + 2 ( T EE R E + T MM R M ) cos Δ R 1 × cos Δ 01 cos ( δ trans δ ref ) + ( T ME 2 + T EM 2 ) sin 2 Δ 01 + 2 ( T MM T EM T EE T ME ) cos Δ 01 sin Δ 01 + 2 ( T EE R M T MM R E ) sin Δ R 1 × cos Δ 01 cos ( δ trans δ ref ) + 2 ( T EM R M T ME R E ) cos Δ R 1 × sin Δ 01 cos ( δ trans δ ref ) 2 ( T ME R M T EM R E ) sin Δ R 1 × sin Δ 01 cos ( δ trans δ ref ) + 2 ( T EE D M T MM D E ) sin Δ R 1 cos Δ 01 cos δ trans + 2 ( T EM D M T ME D E ) cos Δ R 1 sin Δ 01 cos δ trans 2 ( T ME D M + T EM D E ) sin Δ R 1 sin Δ 01 cos δ trans ] .
Δ = θ r 0 .

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