## A general setting for halo theory

Applied Optics, Vol. 38, Issue 9, pp. 1552-1625 (1999)

http://dx.doi.org/10.1364/AO.38.001552

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### Abstract

We describe a general framework for systematically treating halos that are due to refraction in preferentially oriented ice wedges, and we construct an atlas of such halos. Initially we are constrained neither by the interfacial angles nor the orientations of real ice crystals. Instead we consider “all possible” refraction halos. We therefore make no assumption regarding the wedge angle, and only a weak assumption regarding the allowable wedge orientations. The atlas is thus a very general collection of refraction halos that includes known halos as a small fraction. Each halo in the atlas is characterized by three parameters: the wedge angle, the zenith angle of the spin vector, and the spin vector expressed in the wedge frame. Together with the sun elevation, the three parameter values for a halo not only permit calculation of the halo shape, they also give much information about the halo without extensive calculation, so that often a crude estimate of the halo’s appearance is possible merely from inspection of its parameters. As a result, the theory reveals order in what seems initially to be a staggering variety of halo shapes, and in particular it explains why halos look the way they do. Having constructed and studied the atlas, we then see where real or conceivable refraction halos, arising in specific crystal shapes and crystal orientations, fit into the atlas. Although our main goal is to understand halos arising in pyramidal crystals, the results also clarify and unify the classical halos arising in hexagonal prismatic crystals.

© 1999 Optical Society of America

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

**History**

Original Manuscript: May 18, 1998

Revised Manuscript: August 6, 1998

Published: March 20, 1999

**Citation**

Walter Tape and Günther P. Können, "A general setting for halo theory," Appl. Opt. **38**, 1552-1625 (1999)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-9-1552

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### References

- F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A 1, 520–526 (1984). [CrossRef]
- R. A. R. Tricker, “Arcs associated with halos of unusual radii,” J. Opt. Soc. Am. 69,1093–1100 and 1195 (1979). [CrossRef]
- Half of SO(3) and the partition can be visualized as follows. The orientations in SO(3) whose (one-point) zenith loci are on the upper hemisphere are pictured as the upper hemisphere with a line segment 0 ≤ ϕ < 360 attached at each point and pointing radially outward—a boy’s head with a brush haircut. The segment—strand of hair—attached to the sphere at Y is the coset Zu = {zrot(ϕ) · u}, where u, corresponding to ϕ = 0 and thought of as a point on the sphere, is the rotation that takes Y to k and that has horizontal rotation axis. The halo-making sets are the sets consisting of entire segments—no hairs are split. The zenith locus of a halo-making set is the portion of scalp from which its hair is growing. (The brush cut picture can be extended to most of SO(3) by extending the hemisphere downward toward the South Pole -k. But the South Pole cannot be added to complete the picture, since there the above description of u is not enough.)
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- The group W of orthogonal transformations w satisfying wk = ±k is a subgroup of the group O(3) of all orthogonal transformations, and Z is a normal subgroup of W. The quotient group W/Z consists of the four cosets Z · e, Z · xrot, Z · yref, and Z · zref, which correspond to the four lines of Table 1. The homomorphism of W onto {e, -e, yref, -yref} that is implicit in the theorem can be expressed as w = zrot(ϕ) · yrefi · zrefj → Z · yrefi · zrefj → (-1)jyrefi+j, the first mapping being the quotient map onto W/Z. The quotient group formally captures our intuition that we often do not worry about zrot(ϕ).
- A subtle and logically important point that can nevertheless be ignored on a first reading is that whether a given rotation u is a plate (Parry, alternate Parry, etc.,) orientation depends on the wedge under consideration. Technically, this is because the vector N1(u) depends on the wedge as well as on u, since u starts with the crystal oriented so that the specified wedge is in standard orientation. For example, the plate orientations for wedge 1 6 have the form u = zrot(ϕ)yrot(-45), whereas the plate orientations for wedge 3 1 have the form u = zrot(ϕ)yrot(45). They are not the same.
- J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).
- Thus Par ij = Pu, where P = N3 and where the matrix u gives the orientation, or frame, of wedge i j. In Section 1 we said that since the spin vector P is a wedge vector, then Pu is independent of u. That, however, assumed there was only one wedge under consideration (and one spin vector). Now Pu depends on u. The technical explanation is that P(u) [e.g., Eq. (24)] depends not only on u but on the wedge under consideration. But it is probably best to think less technically: At any moment there is the vector P, and there are the wedge frame vectors A, B, C for the wedge under consideration. Then Pu is given by Eq. (21) as always. Equation (23) is also correct, with A, B, C being the columns of the matrix u. Of course Pu depends on the choice of spin vector as well as on the frame.
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- W. Tape, Atmospheric Halos (American Geophysical Union, Washington, D.C., 1994). [CrossRef]
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- O. R. Norton, Science Graphics, Bend, Oregon 97708 (personal communication, 1996).
- T. Kobayashi, “Vapour growth of ice crystals between -40 and -90 C,” J. Meteorol. Soc. Jpn. 43, 359–367 (1965).
- T. Kobayashi, K. Higuchi, “On the pyramidal faces of ice crystals,” Contrib. Inst. Low Temp. Sci. Hokkaido Univ. Ser. A, No. 12, 43–54 and 13 plates (1957).
- If there are at least three linearly independent vectors among Pv1, … , Pvk, then there is at most one induced pole symmetry w* for the given w, P, and V. Otherwise there may be more than one w*.

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