Abstract

A vector ray tracing (VRT) model is developed to simulate optical caustic structures in the primary rainbow region of light scattering from oblate droplets. The changes of the optical caustic structures in response to shape deformation of oblate droplets are investigated. Then the curvature calculated from the simulated rainbow fringes is compared with that from the measured rainbow fringes and good agreement is found. Furthermore, according to the generalized rainbow patterns and the relation between aspect ratio and curvature of the rainbow fringe, non-sphericities in terms of aspect ratio of an oblate water droplet is measured with high measurement accuracy.

© 2013 Optical Society of America

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  1. C. Tropea, “Optical particle characterization in flows,” Annu. Rev. Fluid Mech.43(1), 399–426 (2011).
    [CrossRef]
  2. H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques, (Heidelberg: Springer-Verlag, 2003).
  3. M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
    [CrossRef]
  4. N. Semidetnov, “Investigation of laser Doppler anemometer as instrumentation for two-phase flow measurements,” Doctoral Dissertation, Leningrad Institute for Precision Mechanics and Optics (1985).
  5. N. Damaschke, H. Nobach, N. Semidetnov, and C. Tropea, “Optical particle sizing in backscatter,” Appl. Opt.41(27), 5713–5727 (2002).
    [CrossRef] [PubMed]
  6. N. Roth, K. Anders, and A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal, 1996), pp. 9.21–9.26.
  7. J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett.24(23), 1696–1698 (1999).
    [CrossRef] [PubMed]
  8. J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
    [CrossRef]
  9. H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
    [CrossRef]
  10. P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature312(5994), 529–531 (1984).
    [CrossRef]
  11. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature312(5994), 531–532 (1984).
    [CrossRef]
  12. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett.10(12), 588–590 (1985).
    [CrossRef] [PubMed]
  13. P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am.81(2), 226–232 (1987).
    [CrossRef]
  14. 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.
  15. C. E. Dean and P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt.30(24), 3443–3451 (1991).
    [CrossRef] [PubMed]
  16. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust.21, 1–234 (1992).
    [CrossRef]
  17. P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transf.63(2–6), 341–351 (1999).
    [CrossRef]
  18. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. Lond. A438(1903), 397–417 (1992).
    [CrossRef]
  19. 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(24), 3468–3473 (1991).
    [CrossRef] [PubMed]
  20. 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(21), 4691–4696 (1994).
    [CrossRef] [PubMed]
  21. 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(21), 4697–4701 (1994).
    [CrossRef] [PubMed]
  22. 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(21), 4702–4713 (1994).
    [CrossRef] [PubMed]
  23. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt.37(9), 1520–1526 (1998).
    [CrossRef] [PubMed]
  24. J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A8(10), 1541–1552 (1991).
    [CrossRef]
  25. J. A. Lock and F. Xu, “Optical caustics observed in light scattered by an oblate spheroid,” Appl. Opt.49(8), 1288–1304 (2010).
    [CrossRef] [PubMed]
  26. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
    [CrossRef]
  27. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
    [CrossRef]

2013 (1)

H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
[CrossRef]

2011 (1)

C. Tropea, “Optical particle characterization in flows,” Annu. Rev. Fluid Mech.43(1), 399–426 (2011).
[CrossRef]

2010 (1)

2002 (1)

2001 (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
[CrossRef]

2000 (1)

M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
[CrossRef]

1999 (2)

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transf.63(2–6), 341–351 (1999).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett.24(23), 1696–1698 (1999).
[CrossRef] [PubMed]

1998 (1)

1994 (3)

1992 (2)

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. Lond. A438(1903), 397–417 (1992).
[CrossRef]

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

1991 (3)

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(2), 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,” Nature312(5994), 529–531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature312(5994), 531–532 (1984).
[CrossRef]

1980 (1)

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

1979 (1)

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
[CrossRef]

Anders, K.

N. Roth, K. Anders, and A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal, 1996), pp. 9.21–9.26.

Berry, M. V.

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

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
[CrossRef]

Damaschke, N.

Dean, C. E.

Frohn, A.

N. Roth, K. Anders, and A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal, 1996), pp. 9.21–9.26.

Giannoulis, D.

Hishida, K.

M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
[CrossRef]

Hovenac, E. A.

Kaduchak, G.

Kawaguchi, M.

M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
[CrossRef]

Langley, D. S.

Lock, J. A.

Maeda, M.

M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
[CrossRef]

Marston, P. L.

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transf.63(2–6), 341–351 (1999).
[CrossRef]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt.37(9), 1520–1526 (1998).
[CrossRef] [PubMed]

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(21), 4697–4701 (1994).
[CrossRef] [PubMed]

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(21), 4702–4713 (1994).
[CrossRef] [PubMed]

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(21), 4691–4696 (1994).
[CrossRef] [PubMed]

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

C. E. Dean and P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt.30(24), 3443–3451 (1991).
[CrossRef] [PubMed]

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(24), 3468–3473 (1991).
[CrossRef] [PubMed]

P. L. Marston, “Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field,” J. Acoust. Soc. Am.81(2), 226–232 (1987).
[CrossRef]

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett.10(12), 588–590 (1985).
[CrossRef] [PubMed]

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature312(5994), 529–531 (1984).
[CrossRef]

Nobach, H.

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. Lond. A438(1903), 397–417 (1992).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature312(5994), 531–532 (1984).
[CrossRef]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
[CrossRef]

Riethmuller, M. L.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett.24(23), 1696–1698 (1999).
[CrossRef] [PubMed]

Roth, N.

N. Roth, K. Anders, and A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal, 1996), pp. 9.21–9.26.

Semidetnov, N.

Simpson, H. J.

Trinh, E. H.

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature312(5994), 529–531 (1984).
[CrossRef]

Tropea, C.

H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
[CrossRef]

C. Tropea, “Optical particle characterization in flows,” Annu. Rev. Fluid Mech.43(1), 399–426 (2011).
[CrossRef]

N. Damaschke, H. Nobach, N. Semidetnov, and C. Tropea, “Optical particle sizing in backscatter,” Appl. Opt.41(27), 5713–5727 (2002).
[CrossRef] [PubMed]

Upstill, C.

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

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett.24(23), 1696–1698 (1999).
[CrossRef] [PubMed]

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
[CrossRef]

Xu, F.

H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
[CrossRef]

J. A. Lock and F. Xu, “Optical caustics observed in light scattered by an oblate spheroid,” Appl. Opt.49(8), 1288–1304 (2010).
[CrossRef] [PubMed]

Yu, H. T.

H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
[CrossRef]

Zimmer, L.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett.24(23), 1696–1698 (1999).
[CrossRef] [PubMed]

Annu. Rev. Fluid Mech. (1)

C. Tropea, “Optical particle characterization in flows,” Annu. Rev. Fluid Mech.43(1), 399–426 (2011).
[CrossRef]

Appl. Opt. (8)

C. E. Dean and P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt.30(24), 3443–3451 (1991).
[CrossRef] [PubMed]

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(24), 3468–3473 (1991).
[CrossRef] [PubMed]

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(21), 4691–4696 (1994).
[CrossRef] [PubMed]

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(21), 4697–4701 (1994).
[CrossRef] [PubMed]

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(21), 4702–4713 (1994).
[CrossRef] [PubMed]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt.37(9), 1520–1526 (1998).
[CrossRef] [PubMed]

N. Damaschke, H. Nobach, N. Semidetnov, and C. Tropea, “Optical particle sizing in backscatter,” Appl. Opt.41(27), 5713–5727 (2002).
[CrossRef] [PubMed]

J. A. Lock and F. Xu, “Optical caustics observed in light scattered by an oblate spheroid,” Appl. Opt.49(8), 1288–1304 (2010).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (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(2), 226–232 (1987).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transf. (2)

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transf.63(2–6), 341–351 (1999).
[CrossRef]

H. T. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transf.126, 105–112 (2013).
[CrossRef]

Meas. Sci. Technol. (1)

M. Maeda, M. Kawaguchi, and K. Hishida, “Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows,” Meas. Sci. Technol.11(12), L13–L18 (2000).
[CrossRef]

Nature (2)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature312(5994), 529–531 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature312(5994), 531–532 (1984).
[CrossRef]

Opt. Lett. (2)

Part. Part. Syst. Charact. (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact.18(4), 196–204 (2001).
[CrossRef]

Philos. Trans. R. Soc. Lond. A (1)

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. Lond. A291(1382), 453–484 (1979).
[CrossRef]

Phys. Acoust. (1)

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

Proc. R. Soc. Lond. A (1)

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. Lond. A438(1903), 397–417 (1992).
[CrossRef]

Prog. Opt. (1)

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

Other (4)

N. Roth, K. Anders, and A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal, 1996), pp. 9.21–9.26.

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.

N. Semidetnov, “Investigation of laser Doppler anemometer as instrumentation for two-phase flow measurements,” Doctoral Dissertation, Leningrad Institute for Precision Mechanics and Optics (1985).

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques, (Heidelberg: Springer-Verlag, 2003).

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Figures (12)

Fig. 1
Fig. 1

VRT in an oblate droplet.

Fig. 2
Fig. 2

Definition of the scattering angles.

Fig. 3
Fig. 3

Rays associated with the cusp caustic.

Fig. 4
Fig. 4

Comparison of the cusp location calculated by VRT and that by the analytical solution in the primary rainbow region for oblate water droplets with different aspect ratios.

Fig. 5
Fig. 5

Evolution of the primary rainbow fringe and the HU fringe as the aspect ratio of an oblate water droplet increases.

Fig. 6
Fig. 6

Primary rainbow fringes for oblate droplets with the same refractive index and aspect ratio but different equatorial radii.

Fig. 7
Fig. 7

Primary rainbow fringes for oblate droplets with the same aspect ratio and equatorial radius but different relative refractive indices.

Fig. 8
Fig. 8

Primary rainbow fringes for oblate water droplets with different aspect ratios.

Fig. 9
Fig. 9

Generalized rainbow patterns from an oblate water droplet: (a) a = 0.80 mm and a/c = 1.03, (b) a = 0.84 mm and a/c = 1.23. The horizontal scattering angle θ increases from left to right.

Fig. 10
Fig. 10

Comparison of the curvatures of primary rainbow fringes calculated from VRT simulation and that from generalized rainbow patterns.

Fig. 11
Fig. 11

Comparison of water droplet aspect ratio inverted from the corresponding generalized rainbow patterns to that observed from direct imaging.

Fig. 12
Fig. 12

Relative errors of water droplet aspect ratio inverted from the corresponding generalized rainbow patterns.

Equations (7)

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

L 01 = 1 m [ L 0 ( L 0 n A ) n A ] [ 1 1 m 2 + 1 m 2 ( L 0 n A ) 2 ] 1/2 n A ,
L 12 = L 01 2( L 01 n B ) n B , L 2 =m[ L 12 ( L 12 n C ) n C ]+ [ 1 m 2 + m 2 ( L 12 n C ) 2 ] 1/2 n C ,
θ= cos 1 [ a y / ( a x 2 + a y 2 ) 1/2 ],
ϕ= cos 1 [ ( a x 2 + a y 2 ) 1/2 / ( a x 2 + a y 2 + a z 2 ) 1/2 ].
| sin θ i 0 cos θ i ε 1 / a 2 ε 2 / c 2 1 /a asin2 θ r ε 1 ε 2 acos2 θ r +a |0,
asin θ i ( 2 cos 2 θ r c 2 sin2 θ r cos θ i c 2 sin θ i 1 a 2 ) ε 2 +cos θ i ( 1 c 2 1 a 2 ) ε 1 ε 2 =0.
a/c=m [ 2( m 2 sin 2 θ i )2 ( m 2 sin 2 θ i ) 1/2 ( 1 sin 2 θ i ) 1/2 ] 1 /2 .

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