Abstract

By using Taylor expansions, simple expressions are obtained for the deflection of light by ice crystals. With these simplified formulas, the intensity distributions of halos as a function of scattering angle are calculated analytically near the halo angle. It is found that the intensity distributions of halos depend on the number of degrees of freedom of the generating set of crystals. The differences in the purity of the colors of various types of halo are explained subsequently on the basis of their intensity distributions. An analytical description of the shape of the halo or of the halocaustic near the halo angle is obtained also. On the basis of the obtained intensity distributions, the polarization of refraction halos as a function of scattering angle is calculated, in which both contributions (birefringence of ice and polarization by refraction) are taken into account. It is found that the polarization of parhelia and tangent arcs shows a strong maximum near the inner edge of the halo over an angular range of 0.1°, followed by a similar maximum of reversed polarization at 0.5° from the first one. The 22° halo shows a less strong maximum near its edge over an angular range of 0.5°. Halos at 46° from the sun also show a strong polarization near their inner edges, but the direction of the polarization is perpendicular to the polarization of the 22° halo edges. The possibility for detecting ice crystals on Venus by polarimetry near the halo angle is discussed briefly.

© 1983 Optical Society of America

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References

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  1. G. P. Können, “Polarization of haloes and double refraction,” Weather 32, 467–468 (1977).
    [CrossRef]
  2. G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).
  3. R. White, “Intensity plots of the parhelia,” Q. J. R. Meteorol. Soc. 103, 169–175 (1977).
    [CrossRef]
  4. A. B. Fraser and G. J. Thompson, “Analytic sun pillar model,” J. Opt. Soc. Am. 70, 1145–1148 (1980).
    [CrossRef]
  5. R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).
  6. R. S. McDowell, “The formation of parhelia at higher solar elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
    [CrossRef]
  7. R. White, “Deviation produced by anisotropic prisms,” J. Opt. Soc. Am. 70, 281–287 (1980).
    [CrossRef]
  8. G. P. Können and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  9. W. J. Humphreys, Physics of the Air (Dover, New York, 1964).
  10. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).
  11. W. Tape, “Analytic foundations of halo theory,” J. Opt. Soc. Am. 70, 1175–1192 (1980).
    [CrossRef]
  12. W. Tape, “Folds, pleats and halos,” Am. Sci. 70, 467–474 (1982).
  13. For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.
  14. P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.
  15. H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.
  16. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  17. A. B. Fraser, “What size of ice crystals causes the halo?” J. Opt. Soc. Am. 69, 1112–1118 (1979).
    [CrossRef]
  18. J. Veverka, “A polarimetric search for a Venus halo during the 1969 inferior conjunction,” Icarus 14, 282–283 (1971).
    [CrossRef]

1983 (1)

1982 (1)

W. Tape, “Folds, pleats and halos,” Am. Sci. 70, 467–474 (1982).

1980 (3)

1979 (2)

1977 (2)

G. P. Können, “Polarization of haloes and double refraction,” Weather 32, 467–468 (1977).
[CrossRef]

R. White, “Intensity plots of the parhelia,” Q. J. R. Meteorol. Soc. 103, 169–175 (1977).
[CrossRef]

1974 (1)

R. S. McDowell, “The formation of parhelia at higher solar elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[CrossRef]

1971 (1)

J. Veverka, “A polarimetric search for a Venus halo during the 1969 inferior conjunction,” Icarus 14, 282–283 (1971).
[CrossRef]

de Boer, J. H.

Fraser, A. B.

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

Hobbs, P. V.

P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

Können, G. P.

G. P. Können and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[CrossRef] [PubMed]

G. P. Können, “Polarization of haloes and double refraction,” Weather 32, 467–468 (1977).
[CrossRef]

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).

McDowell, R. S.

R. S. McDowell, “The formation of parhelia at higher solar elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[CrossRef]

Messiah, A.

For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.

Tape, W.

W. Tape, “Folds, pleats and halos,” Am. Sci. 70, 467–474 (1982).

W. Tape, “Analytic foundations of halo theory,” J. Opt. Soc. Am. 70, 1175–1192 (1980).
[CrossRef]

Thompson, G. J.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).

van de Hulst, H. C.

H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.

Veverka, J.

J. Veverka, “A polarimetric search for a Venus halo during the 1969 inferior conjunction,” Icarus 14, 282–283 (1971).
[CrossRef]

White, R.

R. White, “Deviation produced by anisotropic prisms,” J. Opt. Soc. Am. 70, 281–287 (1980).
[CrossRef]

R. White, “Intensity plots of the parhelia,” Q. J. R. Meteorol. Soc. 103, 169–175 (1977).
[CrossRef]

Am. Sci. (1)

W. Tape, “Folds, pleats and halos,” Am. Sci. 70, 467–474 (1982).

Appl. Opt. (1)

Icarus (1)

J. Veverka, “A polarimetric search for a Venus halo during the 1969 inferior conjunction,” Icarus 14, 282–283 (1971).
[CrossRef]

J. Atmos. Sci. (1)

R. S. McDowell, “The formation of parhelia at higher solar elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[CrossRef]

J. Opt. Soc. Am. (5)

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

R. White, “Intensity plots of the parhelia,” Q. J. R. Meteorol. Soc. 103, 169–175 (1977).
[CrossRef]

Weather (1)

G. P. Können, “Polarization of haloes and double refraction,” Weather 32, 467–468 (1977).
[CrossRef]

Other (7)

G. P. Können, Polarized Light in Nature (Cambridge U. Press, Cambridge, to be published); (Dutch ed., Thieme, Zutphen, the Netherlands, 1980).

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

For the properties of the Dirac delta function and the Heaviside step function see, e.g., A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I.

P. V. Hobbs, Ice Physics (Clarendon, Oxford, 1974), p. 202.

H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 245.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970).

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Figures (10)

Fig. 1
Fig. 1

Definition of vectors, planes, and angles. If the position of the axial vector P ^ of a crystal is fixed in space, the relevant angles can be defined; d refers to the normal plane of a crystal and shows the projection of some vectors in this plane. See Appendix A for the notation.

Fig. 2
Fig. 2

Intensity distribution of the circumzenithal arc as a function of scattering angle θ in the solar vertical for a finite sun. θ(α0, 0) denotes the position of this halo for a point-shaped sun.

Fig. 3
Fig. 3

Intensity distribution of a parhelion as a function of scattering angle θ for a point source and for a finite sun. θh denotes the halo angle. The figure refers to zero solar elevation, but for another solar elevation the intensity distribution is the same.

Fig. 4
Fig. 4

Intensity distribution of a tangent arc as a function of scattering angle θ in the solar vertical for a point source and for a finite sun. θh denotes the halo angle.

Fig. 5
Fig. 5

Intensity distribution of the 22° halo as a function of scattering angle θ for a point source and for a finite sun. θh denotes the halo angle.

Fig. 6
Fig. 6

Quantity of polarized light Ipol = I1I2 for the circumzenithal arc as a function of scattering angle θ in the solar vertical for a finite sun. θedge denotes the inner edge of the halo. Near θedge only ordinary refracted rays contribute to the light of the halo. For comparison, at the intensity maximum of the halo, Ipol is also given for Fresnel refraction alone. If Ipol > 0, the polarization is in the plane of scattering.

Fig. 7
Fig. 7

Quantity of polarized light Ipol = I1I2 for the parhelion as a function of scattering angle θ for a finite sun. θedge denotes the inner edge of the halo. Near θedge only ordinary refracted rays contribute to the light of the halo. For comparison, at the intensity maximum of the halo, Ipol is also given for Fresnel refraction alone. If Ipol > 0, the polarization is in the plane of scattering.

Fig. 8
Fig. 8

Quantity of polarized light Ipol = I1I2 for a tangent arc as a function of scattering angle θ in the solar vertical for a finite sun. θedge denotes the inner edge of the halo. Near θedge only ordinary refracted rays contribute to the light of the halo. For comparison, at the intensity maximum of the halo, Ipol is also given for Fresnel refraction alone. If Ipol > 0, the polarization is in the plane of scattering.

Fig. 9
Fig. 9

Quantity of polarized light Ipol = I1I2 for the 22° halo as a function of scattering angle θ for a finite sun. θedge denotes the inner edge of the halo. Near θedge, only ordinary refracted rays contribute to the light of the halo. For θθedge > s + 0.11°, only Fresnel refraction contributes to the polarization of the 22° halo. For any θ, the polarization is in the plane of scattering.

Plate II
Plate II

(G. P. Kônnen, p. 1629). Birefringence of ice crystals caused a remarkable polarization of the parhelion. Rotating a polarizer before the eye changes its position with respect to the sun by 0.11° (photographed by A. Tramper).

Tables (3)

Tables Icon

Table 1 Properties of Halos

Tables Icon

Table 2 Properties of Degenerate Halos

Tables Icon

Table 3 Dispersion of Halos and the Rainbow

Equations (64)

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D = i + i - A ,
sin i = n sin r ,             sin i = n sin r , n = ( n 2 - sin 2 h cos 2 h ) 1 / 2 ,
sin θ / 2 = sin D / 2 cos h ,
ϕ = q + ϕ ,
cos ϕ = cos h sin D sin θ
sin D m + A 2 = n sin A 2 .
I d N / d ω .
θ = θ ( i , h ) ,             D = D ( i , h ) , ϕ = ϕ ( i , h , q ) q + ϕ ( i , h ) .
α = i - i h ,
( D ( α , h ) α ) h = 0 α = 0 = 0.
D ( α , h ) h = D n d n d h = sin A cos r cos i ( n 2 - 1 ) sin h n cos 3 h ,
D ( α , h ) D ( 0 , 0 ) + 1 / 2 ( 2 D ( α , 0 ) α 2 ) α = 0 α 2 + 1 / 2 ( 2 D ( 0 , h ) h 2 ) h = 0 h 2
D ( 0 , 0 ) + C 1 α 2 + C 2 h 2 .
C 1 = 1 / 2 ( 2 D α 2 ) h = 0 α = 0 1 / 2 ( 2 D i 2 ) h = 0 i = i h ,
C 1 = n cos 2 r h sin i h - cos 2 i h sin r h n cos i h cos 2 r h 1 cos 2 r h ( 1 - 1 n 2 ) tan i h .
( 2 D h 2 ) h = 0 α = 0 = ( d 2 n d h 2 ) h = 0 ( sin A cos i cos r ) h = 0 α = 0 = n 2 - 1 n sin 2 r h cos i h cos r h = n 2 - 1 n 2 sin r h cos i h = 2 n 2 - 1 n 2 tan i h ,
C 2 = ( 1 - 1 n 2 ) tan i h .
D ( α , h ) = D ( 0 , 0 ) + C 1 α 2 + C 2 h 2 .
sin [ θ ( α , h ) / 2 ] = sin [ D ( α , h ) / 2 ] cos h ,
( 2 θ ( 0 , h ) h 2 ) h = 0 = - 2 tan [ θ ( 0 , 0 ) / 2 ] + ( 2 D ( 0 , h ) h 2 ) h = 0 .
θ ( α , h ) = θ ( 0 , 0 ) + C 1 α 2 + C 3 h 2 ,
C 3 = ( 1 - 1 n 2 ) tan i h - tan [ θ ( 0 , 0 ) / 2 ] C 2 - tan [ θ ( 0 , 0 ) / 2 ] .
cos ( α + i h ) = sin Σ / cos h .
α α 0 - sin Σ 2 sin ( α 0 + i h ) h 2 ,
θ ( h ) = θ ( 0 , 0 ) + C 1 α 0 2 + [ C 3 - α 0 sin sin ( α 0 + i h ) ] h 2 θ ( 0 , 0 ) + C 1 α 0 2 + C 3 h 2 .
θ ( h ) = θ ( α 0 , 0 ) + C 3 h 2 ,
cos ϕ ( α , h ) = cos h sin D ( α , h ) sin θ ( α , h ) ,
1 - ϕ 2 ( α , h ) 2 ( 1 - h 2 2 ) [ sin D ( 0 , 0 ) + cos D ( 0 , 0 ) d D sin θ ( 0 , 0 ) + cos θ ( 0 , 0 ) d θ ] ( 1 - h 2 2 ) [ 1 + cotan θ ( 0 , 0 ) d θ ] × [ 1 - cotan θ ( 0 , 0 ) d θ ] .
1 - ϕ 2 ( α , h ) 2 1 - [ 1 / 2 - ( C 2 - C 3 ) cotan θ ( 0 , 0 ) ] h 2 ,
ϕ ( h ) = tan [ θ ( 0 , 0 ) / 2 ] h ,
ϕ ( h , q ) = q + tan [ θ ( 0 , 0 ) / 2 ] h .
ϕ ( h ) = { tan [ θ ( 0 , 0 ) / 2 ] ± tan Σ } h C 4 h ,
ϕ ( h ) = { tan [ θ ( α 0 , 0 ) / 2 ] ± tan Σ } h C 4 h ,
I ( θ ) = d N / d ω ,
Ī ( θ ) = 2 π s 2 max ( θ h , θ - s ) θ + s I ( y ) g ( y - θ ) d y ,
Ī ( θ ) = 1 2 s max ( θ h , θ - s ) θ + s I ( y ) d y .
I ( θ ) = d h / d l .
θ - θ ( α 0 , 0 ) = C 3 C 2 4 ϕ 2 C 5 ϕ 2 ,
d l = [ sin 2 θ + ( d θ d ϕ ) 2 ] 1 / 2 d ϕ [ sin 2 θ ( α 0 , 0 ) + ( d θ d ϕ ) 2 ] 1 / 2 d ϕ = [ sin 2 θ ( α 0 , 0 ) + 4 C 2 5 ϕ 2 ] 1 / 2 d ϕ ,
d h d l = d h d ϕ d ϕ d l = 1 C 4 1 [ sin 2 θ ( α 0 , 0 ) + 4 C 2 5 ϕ 2 ] 1 / 2 .
I ( θ ) = δ [ θ - θ ( α 0 , 0 ) - C 5 ϕ 2 ] C 4 [ sin 2 θ ( α 0 , 0 ) + 4 C 2 5 ϕ 2 ] 1 / 2 ,
I ( θ ) = δ [ θ - θ ( α 0 , 0 ) ] ,
Ī ( θ ) = 2 π s 2 δ [ y - θ ( α 0 , 0 ) ] [ s 2 - ( y - θ ) 2 ] 1 / 2 d y = 2 π s 2 { s 2 - [ θ - θ ( α 0 , 0 ) ] 2 } 1 / 2 .
I = d α / d D .
I ( θ ) = 1 ( θ - θ h ) 1 / 2 ,
Ī ( θ ) = 1 s ( θ - θ h + s ) 1 / 2 ,             - s < θ - θ h < s , Ī ( θ ) = 1 s [ ( θ - θ h + s ) 1 / 2 - ( θ - θ h - s ) 1 / 2 ] ,             θ - θ h > s ,
I ( θ ) = d h d ϕ d α d θ = 1 C 4 d α d θ
θ ( α , h ) = θ h + C 1 α 2 + C 3 C 4 2 ϕ 2 θ h + C 1 α 2 + C 5 ϕ 2 .
I ( θ ) = 1 ( θ - θ h - C 5 ϕ 2 ) 1 / 2 ,
θ - θ h = C 5 ϕ 2 C 3 C 4 2 ϕ 2 .
Ī ( θ ) = 2 π s 2 max ( θ h , θ - s ) θ + s [ s 2 - ( y - θ ) 2 y - θ h ] 1 / 2 d y .
( θ - θ h ) ϕ 2 / 3 ,
θ ( α , h ) = θ h + C 1 α 2 + C 3 h 2
N π θ - θ h ( C 1 C 3 ) 1 / 2 θ - θ h             if θ - θ h > 0 , N = 0             if θ - θ h < 0.
I ( θ ) = d N / d θ ,
I ( θ ) = H ( θ - θ h ) ,
H ( x ) = 1 , x > 0 , H ( x ) = 0 , x < 0.
Ī ( θ ) = 1 / 2 + 1 π arc sin θ - θ h s + θ - θ h π s 2 [ s 2 - ( θ - θ h ) 2 ] 1 / 2 ,             - s < θ - θ h < s , Ī ( θ ) = 1 ,             θ - θ h > s .
P = - I pol I 1 + I 2 + I B ,
I 1 / I 2 = [ 1 - tan 2 ( i h - r h ) tan 2 ( i h + r h ) ] 2 [ 1 - sin 2 ( i h - r h ) sin 2 ( i h + r h ) ] - 2 .
I 1 / I 2 = cos - 4 ( θ h / 2 ) .
n eff = n o + sin 2 γ ( n e - n o ) n o + sin 2 γ δ n .
cos θ h + A 2 d θ h = sin ( A / 2 ) ( n eff - n o ) .
I pol = Ī ( θ - θ edge ) - cos 4 11 ° Ī ( θ - θ edge - 0.11 ° ) , I pol = Ī ( θ - θ edge - 0.15 ° ) - cos 4 23 ° Ī ( θ - θ edge )