Abstract

Oblate drops of water illuminated perpendicular to their symmetry axis generate a hyperbolic-umbilic diffraction catastrophe near the primary rainbow [ P. L. Marston and E. H. Trinh, Nature London 312, 529– 531 ( 1984)]. Observations were made of this diffraction catastrophe generated by white-light illumination of acoustically levitated drops of water in air. The observations suggest what generalized rainbows would look like if they were produced in nature when sunlight illuminates large raindrops. Unlike the usual rainbow arc, the transverse cusp of the unfolded catastrophe is not distinctly colored. The hyperbolic-umbilic focal section is distinctly colored as is another diffraction catastrophe generated in the rainbow region when the drop is highly oblate.

© 1991 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, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
    [CrossRef]
  4. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).
  5. J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
    [CrossRef]
  6. J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
    [CrossRef]
  7. C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Part 2 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989); C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered from oblate spheroidal drops,” Appl. Opt. 30, 3441–3449 (1991).
    [CrossRef]
  8. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  9. P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987).
    [CrossRef]
  10. P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: The wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
    [CrossRef]
  11. P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
    [CrossRef]
  12. H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
    [CrossRef]
  13. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  14. J. A. Kneisly, “Local curvature of wavefronts in an optical system,” J. Opt. Soc. Am. 54, 229–235 (1964).
    [CrossRef]
  15. P. L. Marston, in preparation from results given in summary form in Refs. 3 and 10.
  16. G. P. Konnen, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13. For the refractive index of water, see also G. M. Hale, M. R. Querry, “Optical constants of water in the 200-μm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  18. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?,” J. Opt. Soc. Am. 73, 1626–1629 (1983).
    [CrossRef]
  19. G. P. Konnen, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [CrossRef]
  20. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  21. J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984); see discussion of Fig. 4.
    [CrossRef]
  22. 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]
  23. C. V. Raman, S. Ramaseshan, “Diffraction of light by transparent spheres and spheroids: the Fresnel patterns,” Proc. Ind. Acad. Sci. Sect. A 30, 277–283 (1949).

1989 (2)

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). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[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]

1987 (2)

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

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

1986 (2)

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: The wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
[CrossRef]

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

1985 (1)

1984 (3)

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, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984); see discussion of Fig. 4.
[CrossRef]

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

1983 (1)

1981 (1)

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

1979 (1)

1978 (1)

J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

1976 (1)

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

1975 (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

1970 (1)

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

1964 (1)

1949 (1)

C. V. Raman, S. Ramaseshan, “Diffraction of light by transparent spheres and spheroids: the Fresnel patterns,” Proc. Ind. Acad. Sci. Sect. A 30, 277–283 (1949).

Arnott, W. P.

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]

Beard, K. V.

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

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

de Boer, J. H.

Dean, C. E.

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). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Part 2 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989); C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered from oblate spheroidal drops,” Appl. Opt. 30, 3441–3449 (1991).
[CrossRef]

Fraser, A. B.

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

Hannay, J. H.

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984); see discussion of Fig. 4.
[CrossRef]

Kneisly, J. A.

Konnen, G. P.

LoPorto-Arione, S. E.

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

Marston, P. L.

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). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[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]

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

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: The wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (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 (London) 312, 529–531 (1984).
[CrossRef]

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

Nye, J. F.

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984); see discussion of Fig. 4.
[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, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

Pruppacher, H. R.

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Pullen, G. L.

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

Raman, C. V.

C. V. Raman, S. Ramaseshan, “Diffraction of light by transparent spheres and spheroids: the Fresnel patterns,” Proc. Ind. Acad. Sci. Sect. A 30, 277–283 (1949).

Ramaseshan, S.

C. V. Raman, S. Ramaseshan, “Diffraction of light by transparent spheres and spheroids: the Fresnel patterns,” Proc. Ind. Acad. Sci. Sect. A 30, 277–283 (1949).

Simpson, H. J.

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). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

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]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13. For the refractive index of water, see also G. M. Hale, M. R. Querry, “Optical constants of water in the 200-μm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

Adv. Phys. (1)

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

AIP Conf. Proc. (1)

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). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (3)

P. L. Marston, S. E. LoPorto-Arione, G. L. Pullen, “Quadrupole projection of the radiation pressure on a compressible sphere,” J. Acoust. Soc. Am. 69, 1499–1501 (1981); E. H. Trinh, C. J. Hsu, “Equilibrium shapes of acoustically levitated drops,” J. Acoust. Soc. Am. 79, 1335–1338 (1986); E. H. Trinh, “Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity,” Rev. Sci. Instrum. 56, 2059–2065 (1985).
[CrossRef]

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987).
[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. Acoust. Soc. Am. Suppl. (1)

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: The wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
[CrossRef]

J. Appl. Meteorol. (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

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. Acta (1)

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984); see discussion of Fig. 4.
[CrossRef]

Opt. Lett. (1)

Proc. Ind. Acad. Sci. Sect. A (1)

C. V. Raman, S. Ramaseshan, “Diffraction of light by transparent spheres and spheroids: the Fresnel patterns,” Proc. Ind. Acad. Sci. Sect. A 30, 277–283 (1949).

Proc. R. Soc. London Ser. A (2)

J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[CrossRef]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Q. J. R. Meteorol. Soc. (1)

H. R. Pruppacher, K. V. Beard, “A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Q. J. R. Meteorol. Soc. 96, 247–256 (1970).
[CrossRef]

Other (3)

P. L. Marston, in preparation from results given in summary form in Refs. 3 and 10.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13. For the refractive index of water, see also G. M. Hale, M. R. Querry, “Optical constants of water in the 200-μm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Part 2 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989); C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered from oblate spheroidal drops,” Appl. Opt. 30, 3441–3449 (1991).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Locations of once-reflected (twice-refracted) rays leaving an oblate drop to an observer in the four-ray region. (b) Rainbow and cusp caustics that partition the scattering pattern typical of an acoustically levitated oblate drop. The horizontal scattering angle θ increases from left to right where the leftmost point of the rainbow arc is at the usual Descartes ray angle of a sphere, θR ≈ 138°.

Fig. 2
Fig. 2

Rays associated with the primary rainbow in the equatorial plane of a spheroidal drop. The numbers correspond to the rays labeled in Fig. 1(a). The diagram is also applicable to a spherical drop.

Fig. 3
Fig. 3

Anticipated shape of a hyperbolic rainbow in the sky. Large oblate raindrops are horizontally illuminated from the right. The pattern is symmetric with respect to a vertical line through the antisolar point. The vertical spread of the rainbow arc is noticeably reduced relative to the horizontal spread that has the usual half-width of ~42°. The angle between the horizontal cusp points and the rainbow depends on the drop aspect ratio D/H. Consequently the cusps would be well defined only in ideal conditions where the oblate drops do not significantly vary in shape. The lower portion of the horizontal cusps may be obstructed by the shadow of the hill on which the observer is assumed to be standing. The drawing assumes that D/H is between 1.07 and 1.31. Nye’s discussion of the global caustic shape (Fig. 2 of Ref. 8) suggests that the cusp curves from each half meet at a vertical cusp above the antisolar point for this range of D/H. The vertical cusp does not necessarily lie below the rainbow arc as depicted here.

Equations (2)

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( D / H ) 4 = { 3 n 2 / [ 4 ( n 2 - 1 ) ] } 1 / 2 1.311 ,
ψ = 2 arctan [ n / ( 12 ) 1 / 2 ] 42.1 °

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