Abstract

In 1652 Huygens derived a formula specifying the rainbow angle for the primary bow (k=1) in terms of the refractive index only. A generalization of this result for any k1 is outlined, along with an alternative representation. The details of the derivation can be found in (Adam, Mathematics Magazine, 2008, under review), but the results as stated may be of interest to the atmospheric optics community.

© 2008 Optical Society of America

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References

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  1. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University Press, 1992).
    [CrossRef]
  2. J. A. Adam, “Noah's Arc: Asine in the Sky,” Mathematics Magazine (under review) (2008).
  3. See http://en.wikiversity.org/wiki/Waves_in_composites_and_metamaterials/Rainbows
  4. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106-117(1991).
    [CrossRef] [PubMed]
  5. M. Vollmer, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger, (Elsevier, Spektrum Akademischer Verlag, 2006), Chap. 5, pp. 121-122.
  6. W. E. Weisstein, “Multiple-angle formulas,” from MathWorld--a Wolfram Web Resource . http://mathworld.wolfram.com/Multiple-AngleFormulas.html

2008 (1)

J. A. Adam, “Noah's Arc: Asine in the Sky,” Mathematics Magazine (under review) (2008).

1991 (1)

Adam, J. A.

J. A. Adam, “Noah's Arc: Asine in the Sky,” Mathematics Magazine (under review) (2008).

Nussenzveig, H. M.

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

van de Hulst, H. C.

Vollmer, M.

M. Vollmer, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger, (Elsevier, Spektrum Akademischer Verlag, 2006), Chap. 5, pp. 121-122.

Wang, R. T.

Weisstein, W. E.

W. E. Weisstein, “Multiple-angle formulas,” from MathWorld--a Wolfram Web Resource . http://mathworld.wolfram.com/Multiple-AngleFormulas.html

Appl. Opt. (1)

from MathWorld--a Wolfram Web Resource (1)

W. E. Weisstein, “Multiple-angle formulas,” from MathWorld--a Wolfram Web Resource . http://mathworld.wolfram.com/Multiple-AngleFormulas.html

Mathematics Magazine (under review) (1)

J. A. Adam, “Noah's Arc: Asine in the Sky,” Mathematics Magazine (under review) (2008).

Other (3)

See http://en.wikiversity.org/wiki/Waves_in_composites_and_metamaterials/Rainbows

M. Vollmer, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger, (Elsevier, Spektrum Akademischer Verlag, 2006), Chap. 5, pp. 121-122.

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

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Equations (19)

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D k ( i ) = k π + 2 i 2 ( k + 1 ) arcsin ( sin i n ) ,
i c = arccos [ n 2 1 k ( k + 2 ) ] 1 / 2 .
D 1 ( i c ) = 2 arccos [ 1 n 2 ( 4 n 2 3 ) 3 / 2 ] .
1 2 ( D 1 ( i c ) π ) = arccos ( n 2 1 3 ) 1 / 2 2 arcsin ( 4 n 2 3 n 2 ) 1 / 2 A 2 B .
sin A = [ ( k + 1 ) 2 n 2 k ( k + 2 ) ] 1 / 2 , cos A = [ n 2 1 k ( k + 2 ) ] 1 / 2 , sin B = [ ( k + 1 ) 2 n 2 n 2 k ( k + 2 ) ] 1 / 2 , cos B = [ ( k + 1 ) 2 ( n 2 1 ) n 2 k ( k + 2 ) ] 1 / 2 .
D 2 ( i c ) = 2 arcsin [ ( n 2 1 ) 1 / 2 ( ( 9 n 2 ) 1 / 2 2 n ) 3 ] ,
D 3 ( i c ) = 2 arccos [ 1 5 ( 15 ) 3 / 2 ( 16 n 2 ) 3 / 2 n 4 ( 27 n 2 32 ) ] .
D 4 ( i c ) = 2 arcsin [ ( n 2 1 ) 1 / 2 ( 25 n 2 ) 3 / 2 ( 25 16 n 2 ) 6 3 n 5 ] ,
D 5 ( i c ) = 2 arccos [ ( 36 n 2 ) 3 / 2 ( 9792 n 2 3125 n 4 6912 ) 7 7 / 2 5 5 / 2 n 6 ] .
D k ( i c ) = k π + 2 arccos ( n 2 1 k ( k + 2 ) ) 1 / 2 2 ( k + 1 ) arcsin ( 1 n 2 ( n 2 1 k ( k + 2 ) n 2 ) ) 1 / 2 .
D 1 ( i c ) = π + 2 arccos ( n 2 1 3 ) 1 / 2 4 arcsin ( 1 n 2 ( n 2 1 3 n 2 ) ) 1 / 2 ,
D 2 ( i c ) = 2 π + 2 arccos ( n 2 1 8 ) 1 / 2 6 arcsin ( 1 n 2 ( n 2 1 8 n 2 ) ) 1 / 2 ,
D 3 ( i c ) = 3 π + 2 arccos ( n 2 1 15 ) 1 / 2 8 arcsin ( 1 n 2 ( n 2 1 15 n 2 ) ) 1 / 2 ,
D 4 ( i c ) = 4 π + 2 arccos ( n 2 1 24 ) 1 / 2 10 arcsin ( 1 n 2 ( n 2 1 24 n 2 ) ) 1 / 2 ,
D 5 ( i c ) = 5 π + 2 arccos ( n 2 1 35 ) 1 / 2 12 arcsin ( 1 n 2 ( n 2 1 35 n 2 ) ) 1 / 2 .
( 1 ) k / 2 sin D k ( i c ) 2 = [ ( k + 1 ) 2 n 2 k ( k + 2 ) ] cos [ ( k + 1 ) ( arccos { ( k + 1 ) 2 ( n 2 1 ) n 2 k ( k + 2 ) } 1 / 2 ) ] [ n 2 1 k ( k + 2 ) ] sin [ ( k + 1 ) ( arccos { ( k + 1 ) 2 ( n 2 1 ) n 2 k ( k + 2 ) } 1 / 2 ) ]
| D k ( i c ) | = 2 ( 1 ) k / 2 arcsin [ Φ ( k , n ) ]
| D k ( i c ) | = 2 arccos [ Φ ( k , n ) ] .
α = arccos { ( k + 1 ) 2 ( n 2 1 ) n 2 k ( k + 2 ) } 1 / 2

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