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Rich F. Coleman
Kuo-Nan Liou


A complete and traceable geometric ray-tracing solution for finite hexagonal columns and plates arbitrarily oriented in space has been developed by means of analytic geometry. In addition, an analytic expression for the cross-sectional area for arbitrarily oriented hexagons also has been derived based on which Fraunhofer diffraction and extinction and scattering cross sections in the limit of geometric optics can be computed exactly. The program involving geometrical reflection and refraction and Fraunhofer diffraction was used to compute the scattered intensities corresponding to two components of polarization for randomly oriented columns and plates in a horizontal plane and three-dimensional space. The scattered intensities were subsequently normalized to yield the nondimensional phase function commonly used in radiative transfer analyses. Numerical computations have been performed to study the effects of size, shape, orientation, and absorption on the scattering phase function and linear polarization. We show that the scattering phase functions for columns and plates, having approximately the same value, are quite similar except columns have a broader 22° halo pattern. The polarization patterns for these two shapes as well as for spheres and circular cylinders, however, are distinctly different, especially between 30 and 60° and between 130 and 140° scattering angles. We also show that hexagonal columns and plates randomly oriented in a horizontal plane do not generate a full scattering pattern of significant magnitude for an obliquely incident beam, and that the scattering patterns vary significantly with the oblique angle of the incident beam. At the 10.6 μm infrared wavelength, because of the strong absorption of ice, the scattering pattern is basically attributed to diffraction and external reflection but with a noticeable 7° halo maximum due to two refractions. Comparisons with experimental results for plates having a mode radius of 20 μm reveal a general agreement in regions from about 30–160° scattering angles.

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