Abstract

Oblate drops of water can produce caustics where, unlike a simple Airy caustic, more than two rays merge. We extend previous treatments of generalized primary rainbows based on catastrophe optics [Opt. Lett. 10, 588 (1985); Proc. R. Soc. (London) A 438, 397 (1992)] to rays having (p − 1) = 2 to 5 internal reflections. The analysis is for a horizontally illuminated ellipsoid with a vertical symmetry axis. Aspect ratios causing a vanishing of the vertical curvature at the equator for the outgoing wave front are found from generalized ray tracing. In response to infinitesimal deformation, the axial caustic of real glory rays unfolds producing cusps. Laboratory observations with laser illumination demonstrate that cusps resulting from rays with five internal reflections extend into Alexander's dark band when the drop's aspect ratio is near 1.08. The evolution of this p = 6 scattering pattern as cusps meet the quinary rainbow is suggestive of an E6 catastrophe. For ellipsoids of varying aspect ratio and refractive index N, there is an organizing singularity associated with an exceptionally flat outgoing wave front from spheres with N = p.

© 1994 Optical Society of America

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References

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  1. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  2. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590(1985).
    [CrossRef] [PubMed]
  3. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 1–234.
  4. 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 (1991).
    [CrossRef] [PubMed]
  5. H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M. S. degree project rep. (Department of Physics, Washington State University, Pullman, Wash., 1988);P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
    [CrossRef]
  6. G. Kaduchak, “Observation of the E6 diffraction catastrophe by optical scattering from levitated drops in rainbow region,” M. S. degree project rep. (Washington State University, Seattle, Wash., 1992);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 (1994).
    [CrossRef] [PubMed]
  7. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  8. J. F. Nye, “Rainbow from ellipsoidal water drops,” Proc. R. Soc. London 438, 397–417 (1992).
    [CrossRef]
  9. C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered from oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991);Erratum 32, 2163 (1993).
    [CrossRef] [PubMed]
  10. M. V. Berry, “Waves and Thom's theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  11. W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wavefronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
    [CrossRef]
  12. W. P. Arnott, 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] [PubMed]
  13. W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964) pp. 475–495;see alsoJ. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 228–232.
  15. 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]
  16. 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);Erratum 83, 1976 (1988).
    [CrossRef]
  17. W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506(1988).
    [CrossRef]
  18. D. S. Langley, P. L. Marston, “Forward glory scattering from bubbles,” Appl. Opt. 30, 3452–3458 (1991).
    [CrossRef] [PubMed]
  19. D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
    [CrossRef] [PubMed]
  20. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
    [CrossRef]
  21. S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. 14, 305–332 (1961).
    [CrossRef]
  22. P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
    [CrossRef]
  23. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  24. P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.
  25. J. F. Nye, Department of Physics, University of Bristol, Bristol BS8 1TL, UK (personal communication, June1993).
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    [CrossRef]
  27. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179.
  28. S. D. H. Andreasson, S. E. Gustafsson, N. O. Hailing, “Measurment of the refractive index of transparent solids and fluids,” J. Opt. Soc. Am. 61, 595 (1971).
    [CrossRef]

1994 (1)

1992 (1)

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

1991 (6)

1989 (1)

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

1988 (1)

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);Erratum 83, 1976 (1988).
[CrossRef]

1985 (1)

1984 (2)

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

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

1982 (1)

1976 (1)

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

1971 (1)

1964 (1)

1961 (1)

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

Andreasson, S. D. H.

Arnott, W. P.

Berry, M. V.

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

Dean, C. E.

Gustafsson, S. E.

Hailing, N. O.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964) pp. 475–495;see alsoJ. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Kaduchak, G.

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, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.

G. Kaduchak, “Observation of the E6 diffraction catastrophe by optical scattering from levitated drops in rainbow region,” M. S. degree project rep. (Washington State University, Seattle, Wash., 1992);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 (1994).
[CrossRef] [PubMed]

Kneisly, J. A.

Langley, D. S.

Marston, P. L.

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]

W. P. Arnott, 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] [PubMed]

D. S. Langley, P. L. Marston, “Forward glory scattering from bubbles,” Appl. Opt. 30, 3452–3458 (1991).
[CrossRef] [PubMed]

C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered from oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991);Erratum 32, 2163 (1993).
[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 (1991).
[CrossRef] [PubMed]

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

W. P. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506(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);Erratum 83, 1976 (1988).
[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 (London) 312, 529–531 (1984).
[CrossRef]

P. L. Marston, D. S. Langley, “Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops,” J. Opt. Soc. Am. 72, 456–459 (1982).
[CrossRef]

P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 1–234.

Morrell, M. J.

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

Nye, J. F.

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

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

J. F. Nye, Department of Physics, University of Bristol, Bristol BS8 1TL, UK (personal communication, June1993).

Rubinow, S. I.

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

Simpson, H. J.

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 (1991).
[CrossRef] [PubMed]

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M. S. degree project rep. (Department of Physics, Washington State University, Pullman, Wash., 1988);P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
[CrossRef]

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179.

Trinh, E. H.

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

van de Hulst, H. C.

R. T. Wang, H. C. van de Hulst, “Rainbows: 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), pp. 228–232.

Wang, R. T.

Adv. Phys. (1)

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

Ann. Phys. (1)

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

Appl. Opt. (7)

J. Acoust. Soc. Am. (2)

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);Erratum 83, 1976 (1988).
[CrossRef]

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

J. Opt. Soc. Am. (3)

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

Nature (London) (2)

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

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

Opt. Lett. (1)

Proc. R. Soc. London (1)

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

Other (9)

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

P. L. Marston, G. Kaduchak, “Secondary and higher-order generalized rainbows and unfolded glories of oblate drops: analysis and laboratory observations,” in Light and Color in the Open Air, Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 12–15.

J. F. Nye, Department of Physics, University of Bristol, Bristol BS8 1TL, UK (personal communication, June1993).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, New York, 1992), Vol. 21, pp. 1–234.

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M. S. degree project rep. (Department of Physics, Washington State University, Pullman, Wash., 1988);P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989).
[CrossRef]

G. Kaduchak, “Observation of the E6 diffraction catastrophe by optical scattering from levitated drops in rainbow region,” M. S. degree project rep. (Washington State University, Seattle, Wash., 1992);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 (1994).
[CrossRef] [PubMed]

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964) pp. 475–495;see alsoJ. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 228–232.

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Figures (10)

Fig. 1
Fig. 1

(a) Profile of an ellipsoidal drop, (b) Ray diagram for the equatorial plane of an oblate water drop showing the Descartes ray of the secondary rainbow (solid line) and a representative general ray (dashed line) having an angle of incidence i and refraction r. The vertices j = 1, 2, 3, and 4 are labeled. The midway point for the ray is denoted by S. (c) Similar to (b) but for five internal reflections, and only the Descartes-ray vertices 1, 2, and 7 are labeled. The dashed ray leaves the drop directed backward and has i = 58.5°.

Fig. 2
Fig. 2

Vertical focusing of rays infinitesimally displaced from the equatorial plane of a drop. Symmetry conditions for the wave front leaving the drop to have a vanishing vertical curvature in the equatorial plane are designated by A and B. (a) p = 3 where the vertical double arrows represent refraction or reflection at the jth vertex with 1 and 4 being refraction and 2 and 3 being internal reflection. The diagram is symmetric about the midpoint S [compare Fig. 1(b)]. (b), (c), and (d) are corresponding diagrams for p = 4, 5, and 6, respectively, except the rays are shown only up to the midpoint.

Fig. 3
Fig. 3

Diagram (following van de Hulst14) of the range of the horizontal scattering angle θ swept by letting the angle of incidence i in the equatorial plane increase from 0° to 90° for a drop of water. The diagram applies to a sphere as well as to the equatorial plane of an ellipsoidal drop. R denotes a rainbow or Airy caustic and T designates a transition (or aperture) event where i = 90°. Forward-glory and backward-glory rays occur for i ≠ 0 and are designated by FG and BG, respectively.

Fig. 4
Fig. 4

Locus of the aspect ratio, Eq. (3a), giving vertical far-field focusing at the indicated horizontal scattering angle θ as the angle of incidence i ranges from 0° to 90° in 1° steps. The locus here is for the twice-reflected (p = 3) ray and is similar in form to a previous analytical result2 descriptive of the D4+ diffraction catastrophe of the p = 2 ray. The concentration of points on the right occurs for i close to the rainbow angle of incidence iR. The upper and lower trajectories, ordinarily representing cusp-point loci, meet at a D4+ focal section where θ = 129.0° and D/H = 1.425 in agreement with observations (Ref. 15). D/H ranges from 1.309 to 1.5, where a lips event is on the left.

Fig. 5
Fig. 5

Locus from Eq. (3b) analogous to Fig. 4, except that now the upper and lower cusp trajectories meet at an E6 focal section of the secondary rainbow in general agreement with observations (Ref. 15). The corresponding analysis for the primary rainbow was given by Nye.8 D/H ranges from 1.511 to 1.550.

Fig. 6
Fig. 6

Locus of the aspect ratio for the p = 6 ray giving far-field focusing at the indicated horizontal backscattering angle Γ = 180° − θ. The calculation indicates that small deformations of a drop are sufficient to generate cusps in the far-field scattering. The locus is from Eq. (10), where i increases from the backward glory value ibg = 58.5° to 90° in 0.5° steps. The caustic trajectory initially describes a single cusp point that unfolds the glory (or axial caustic) at Γ = 0 as D/H increases from unity. At D/H = 1.069 the cusp locus is joined by a second cusp having virtually the same scattering angle (Fig. 7), and the cusp-point trajectories end at the quinary Airy caustic where D/H = 1.084. The cusp loci cut across Alexander's dark band, which is bounded by the p = 2 and 3 Airy caustics at Γ = 42.0° and 51.0°, respectively, for the refractive index N = 4/3 used in the calculation.

Fig. 7
Fig. 7

(a) Expanded view of the locus as in Fig. 6, with i ranging from 72° to 90°. A slight angular offset between upper and lower cusp loci is visible near the transition event at D/H = 1.069, but the offset decreases with increasing D/H. (b) As in (a) but with higher resolution close to the focal section near D/H = 1.084. Distinct cusp-point loci are not resolved. Similarities of (a) with the region close to the E6 focal section in Fig. 5 suggest that the cusp-point loci here also terminate at an E6 focal section on the right.

Fig. 8
Fig. 8

a, Ultrasonically levitated 1.78-mm-diameter drop of water with an aspect ratio D/H = 1.09 corresponding to b, the far-field scattering pattern, where a bright cusplike feature and the Airy pattern of the secondary arc are visible. The backscattering direction is out of view on the right and the primary rainbow is obscured (see text), c and d show scattering patterns for slightly rounder drops than in a. In d, the Airy caustics of both the secondary (left) and primary (right) rainbows are visible along with cusps that extend into Alexander's dark band. In c, light incident upon one side of the drop is blocked to remove the secondary arc and any p = 7 rainbow contribution whereas the p = 2 and 6 contributions remain. In b, c, and d, the bright patch to the left of the cusp point is related to the Airy caustic of the p = 6 ray. The displayed angular widths (based on the focal length and magnification) are 30°, 19°, and 25° for b, c, and d, respectively.

Fig. 9
Fig. 9

Sequence as in Fig. 8c but for increasing aspect ratios close to and above D/H = 1.09. The bright scattering contribution from the primary rainbow saturated the film. This sequence shows that caustics originally at the cusp boundaries shift vertically with increasing aspect ratio as is most clearly evident in c and d (see text). With additional flattening the nearly horizontal caustics shift outside the vertical field of view. Those caustics appear to terminate at the quinary Airy caustic, which is concavo from the left as a consequence of the drop's oblateness. The displayed angular width is 19°.

Fig. 10
Fig. 10

Caustic interpretation of scattering patterns corresponding to the focal section D/H of 1.084. Unlike Figs. 8 and 9, θ increases from right to left. The number of (p = 6) rays are shown encircled in the different regions bounded by caustics. Rays that do not participate in the singularity (such as the p = 2 rays to the left of the secondary arc) are not included. The number decreases by four in crossing the cusp curve because the curve is interpreted to be a superposition of two cusps. Photographs (e.g., Figs. 8a, 8b, 8d, and 9a) indicate that the intensity of the usual quinary Airy caustic decreases rapidly with increasing magnitude of the vertical scattering angle. That decrease may result from an aperture restriction within the drop that limits the vertical extent of the two-ray region and the quinary Airy caustic.

Equations (15)

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

sin i = N sin r ,
y = cos r N 1 cos i .
q 2 = ( 1 + 2 y 1 cos r ) / ( 4 cos 2 r ) ,
q 2 = { ( 3 y + 2 ρ ) + [ ( 3 y + 2 ρ ) 2 16 y ρ ] 1 / 2 } / 8 y ρ 2 ,
θ = 180 I ± 2 ( i p r ) , 0 θ 180 ,
( q L 1 ) 2 = { [ 2 N / ( N 1 ) ] + 1 } / 4 .
q T = [ 3 N 2 / 4 ( N 2 1 ) ] 1 / 2 ,
q 2 = [ ( y + ρ ) ± ( y 2 + ρ 2 ) 1 / 2 ] / 4 y ρ 2 ,
( q L 1 ) 2 = [ ( η + 1 ) + ( η 2 + 1 ) 1 / 2 ] / 4 ,
q 2 = { α ± [ α 2 ( 4 + 16 ρ / y ) ] 1 / 2 } / 8 ρ 2 ,
β 3 + β 2 ( γ + 1 ) + β ( γ 2 ) γ = 0 ,
β = 1 4 ( q cos r ) 2 ,
γ = ( 2 cos r ) / y ,
L 1 = y L E ,
L j = L ij + 2 L E cos r , 1 < j p ,

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