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.
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