Abstract

Parhelic circles due to plate-oriented crystals (hence, with main axes vertical) and 120° parhelia change in position when viewed through a rotating polarizer. The parhelic circle moves vertically; its largest shift is found at an azimuthal distance between 90° and 120° from the Sun. The 120° parhelia move both vertically and horizontally. The magnitudes of the shifts are between 0.1° and 0.3°, depending on solar elevation. The mechanism is polarization-sensitive internal reflection by prism faces of the ice crystals. We outline the theory and present three visual and one instrumental observation of the displacements of these halos in polarized light.

© 1998 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, J. Tinbergen, “Polarimetry of a 22° halo,” Appl. Opt. 30, 3382–3400 (1992).
    [CrossRef]
  3. G. P. Können, S. H. Muller, J. Tinbergen, “Halo polarization profiles and the interfacial angles of ice crystals,” Appl. Opt. 33, 4569–4579 (1994).
    [CrossRef] [PubMed]
  4. G. P. Können, “Identification of odd-radius halo arcs and of 44°/46° parhelia by their inner-edge polarization,” Appl. Opt. 37, 1450–1456 (1998).
    [CrossRef]
  5. W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).
  6. G. Szivessy, “Kristaloptik,” in Handbuch der Physik, H. Konen, ed. (Springer, Berlin1928), Vol. 20, p. 702 and pp. 715–718.
  7. H. E. Merwin, “Refractivity of birefringent crystals,” in International Critical Tables, E. W. Washburn, ed. (McGraw-Hill, New York, 1930), Vol. 7, pp. 16–33.
  8. W. Snel van Royen (Leiden, Netherlands1580–1626), latinized name Snellius, is often incorrectly retranslated as Snell. See also Ref. 9.
  9. W. D. Bruton, G. W. Kattawar, “Unique temperature profiles for the atmosphere below an observer from sunset images,” Appl. Opt. 36, 6957–6961 (1997); see authors’ note in Ref. 5.
    [CrossRef]
  10. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).
  11. P. Drude, “Rotationspolarisation,” in Handbuch der Physik, A. Winkelman, ed. (Barth, Leipzig, 1906), Vol. 4, pp. 1347 and 1353.

1998 (1)

1997 (1)

1994 (1)

1992 (1)

1977 (1)

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

Bruton, W. D.

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

Drude, P.

P. Drude, “Rotationspolarisation,” in Handbuch der Physik, A. Winkelman, ed. (Barth, Leipzig, 1906), Vol. 4, pp. 1347 and 1353.

Kattawar, G. W.

Können, G. P.

Merwin, H. E.

H. E. Merwin, “Refractivity of birefringent crystals,” in International Critical Tables, E. W. Washburn, ed. (McGraw-Hill, New York, 1930), Vol. 7, pp. 16–33.

Muller, S. H.

Szivessy, G.

G. Szivessy, “Kristaloptik,” in Handbuch der Physik, H. Konen, ed. (Springer, Berlin1928), Vol. 20, p. 702 and pp. 715–718.

Tape, W.

W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).

Tinbergen, J.

Appl. Opt. (4)

Weather (1)

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

Other (6)

W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).

G. Szivessy, “Kristaloptik,” in Handbuch der Physik, H. Konen, ed. (Springer, Berlin1928), Vol. 20, p. 702 and pp. 715–718.

H. E. Merwin, “Refractivity of birefringent crystals,” in International Critical Tables, E. W. Washburn, ed. (McGraw-Hill, New York, 1930), Vol. 7, pp. 16–33.

W. Snel van Royen (Leiden, Netherlands1580–1626), latinized name Snellius, is often incorrectly retranslated as Snell. See also Ref. 9.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

P. Drude, “Rotationspolarisation,” in Handbuch der Physik, A. Winkelman, ed. (Barth, Leipzig, 1906), Vol. 4, pp. 1347 and 1353.

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Figures (7)

Fig. 1
Fig. 1

Internal reflection of (the wave normal of) the e ray at a prism face (the ray enters and leaves the crystal through basal faces). In addition to the plane of incidence and the basal plane (the plane through the point of reflection and parallel to basal faces of the crystal), the right-hand diagram shows the planes containing the optic axis and, respectively, the incident ray and the reflected ray. The index of refraction for an e ray is determined by the angle γ and its polarization (electric-field vibration) is always in the plane defined by the optic axis and the ray. The angle η defines the polarization angle of the incident e ray with respect to the plane of incidence. The polarization angle of the reflected e ray is -η, which differs from that expected by the Fresnel laws of reflection. As a result, a second ray (an ordinary one, not shown) of opposite polarization is created at reflection. Its wave normal is also in the plane of incidence, but its angle of reflection differs from the angle of incidence i. Reflection of o rays is analogous to that of e rays. The angle i p is the projection of i onto the basal plane.

Fig. 2
Fig. 2

Fine structure of a 120° parhelion due to birefringence. The sequences of ordinary and extraordinary rays in the light paths inside the crystal are indicated: e.g., oeo means ordinary–extraordinary–ordinary ray. The polarization is indicated for each spot. The virtually unpolarized central spot corresponds in position to that of the 120° parhelion in isotropic crystals. The azimuthal displacements of the six satellite spots are independent of solar elevation h sun; their vertical displacement h - h sun in the figure corresponds to a solar elevation of 25°. The figure displays the situation in which the path of the ooo ray is symmetrical with respect to the crystal hexagon (projected angles of incidence of 60° for both prism faces). Rotation of the crystal about its vertical axis causes a change in azimuthal position of the six satellites; their directions are indicated (dashed arrows). The inset depicts the effect of solar-disk smearing of the spots.

Fig. 3
Fig. 3

Relative radiance of the three parhelic circle components as a function of azimuth. The upper component is completely horizontally polarized, the lower component is completely vertically polarized, and the middle component is virtually unpolarized. At azimuths between 90° and 120°, only the two polarized components have significant radiance. The plot is for solar elevation 35°, but the curves depend only weakly on solar elevation.

Fig. 4
Fig. 4

Relative radiance of the seven 120° parhelion components as a function of projected angle of incidence i p (abscissa) of the first reflection at a prism face. The graphs are arranged according to Fig. 2. The relative positions of the spots depend only weakly on i p . The middle spot is the unpolarized component; its radiance is made up of two contributions (ooo and eee). The six other spots are all completely polarized (Fig. 2); their radiances consist of one contribution only (i.e., eeo, eoo, etc.). Note that the relative radiance patterns of diagonal pairs of spots are equal. At i p = 60° (dashed vertical lines) the ray path (neglecting splitting) is symmetrical with respect to the crystal hexagon. The plots are for a solar elevation of 35°, but for other elevations they are similar.

Fig. 5
Fig. 5

Displacement δ of the parhelic circle and of the 120° parhelion when they are viewed through a rotating polarizer as functions of solar elevation. The parhelic circle displacement is vertical, and its value refers to the maximum displacement, which occurs at an azimuth between 90° and 120° from the Sun. The 120° parhelion shifts in both the horizontal and the vertical directions. The visibility Vis (right axis) of the displacements is defined as the displacement in units of that of the 22° halo.

Fig. 6
Fig. 6

Constituent spots of a 120° parhelion created by a nonrotating quartz crystal polished in the shape of an ice plate crystal. The picture was obtained by direct projection into a camera body without lens. The separation of the seven solar images is five times larger than that in ice (compare Fig. 2).

Fig. 7
Fig. 7

Observed radiance of the 120° parhelion for horizontal and vertical polarization (arbitrary units). The background has been subtracted; see Subsection 3.C for further details. The 120° parhelion in horizontal polarization is 0.2° higher in the sky and 0.1° closer to the Sun (concentrate on the inner three contours, where both the radiance and its gradient are maximum). The lobe to the left results from a weakly developed parhelic circle. The observation took place on 6 January 1990 at the U.S. Amundsen–Scott South Pole station. The solar elevation was 22.5°.

Equations (15)

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1 n eff 2 = sin 2   γ n o 2 + cos 2   γ n e 2 ,
n eff   sin   i = n o sin   i     e o ,
n o sin   i = n eff   sin   i     o e .
1 o 3 o 2 oo ,     1 e 3 o 2 eo , 1 o 3 e 2 oe ,     1 e 3 e 2 ee .
h p = h sun oo ,   ee   paths , n o cos   h p = n e cos   h sun oe   path , n e cos   h p = n o cos   h sun eo   path .
n e sin   i p = n o sin   i p     e o , n o sin   i p = n e sin   i p     o e ,
tan   i p = tan   η   cos   γ .
R o e = R e o = sin 2 2 i p cos 2   γ 1 - 1 + 1 / n 2 cos 2   i p sin 2   γ , R o o = R e e = 1 - R o e ,
cos   h sun = n   sin   γ .
Az = 180 ° - 2 i p .
T e basal   face = 1 - sin 2 90 ° - h sun - γ sin 2 90 ° - h sun + γ , T o basal   face = 1 - tan 2 90 ° - h sun - γ tan 2 90 ° - h sun + γ .
T e R e o T o + T e R e e T e + T o R o o T o + T o R o e T e = T e 2 + T o 2 - T e - T o 2 R o e T e 2 + T o 2 .
upper   and   lower   components eo ,   oe   paths = T e T o T e 2 + T o 2   R o e 1 / 2 R o e ,   middle   component ee + oo   path 1 - R o e .
δ = 2 R o e h p oe - h sun ,
F i p ,   γ = cos | i p - 60 ° | + 60 ° × cos | i p - 60 ° | - 60 ° sin   γ   tan   γ .

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