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Flatau, PJ, Stephens GL, Draine BT.  1990.  Light-Scattering by Rectangular Solids in the Discrete-Dipole Approximation - a New Algorithm Exploiting the Block-Toeplitz Structure. Journal of the Optical Society of America a-Optics Image Science and Vision. 7:593-600.   10.1364/josaa.7.000593   AbstractWebsite

The discrete-dipole approximation is used to study the problem of light scattering by homogeneous rectangular particles. The structure of the discrete-dipole approximation is investigated, and the matrix formed by this approximation is identified to be a symmetric, block-Toeplitz matrix. Special properties of block-Toeplitz arrays are explored, and an efficient algorithm to solve the dipole scattering problem is provided. Timings for conjugate gradient, Linpack, and block-Toeplitz solvers are given; the results indicate the advantages of the block-Toeplitz algorithm. A practical test of the algorithm was performed on a system of 1400 dipoles, which corresponds to direct inversion of an 8400 × 8400 real matrix. A short discussion of the limitations of the discrete-dipole approximation is provided, and some results for cubes and parallelepipeds are given. We briefly consider how the algorithm may be improved further.

Flatau, PJ, Draine BT.  2014.  Light scattering by hexagonal columns in the discrete dipole approximation. Optics Express. 22:21834-21846.   10.1364/oe.22.021834   AbstractWebsite

Scattering by infinite hexagonal ice prisms is calculated using Maxwell's equations in the discrete dipole approximation for size parameters x = pi D/lambda up to x = 400 (D = prism diameter). Birefringence is included in the calculations. Applicability of the geometric optics approximation is investigated. Excellent agreement between wave optics and geometric optics is observed for large size parameter in the outer part of the 22 degree halo feature. For smaller ice crystals halo broadening is predicted, and there is appreciable "spillover" of the halo into shadow scattering angles < 22 degrees. Ways to retrieve ice crystal sizes are suggested based on the full width at half-maximum of the halo, the power at < 22deg, and the halo polarization. (C) 2014 Optical Society of America