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The solar optical properties of Saharan mineral dust observed during the Saharan Mineral Dust Experiment (SAMUM) were explored based on measured size-number distributions and chemical composition. The size-resolved complex refractive index of the dust was derived with real parts of 1.51-1.55 and imaginary parts of 0.0008-0.006 at 550 nm wavelength. At this spectral range a single scattering albedo omega(o) and an asymmetry parameter g of about 0.8 were derived. These values were largely determined by the presence of coarse particles. Backscatter coefficients and lidar ratios calculated with Mie theory (spherical particles) were not found to be in agreement with independently measured lidar data. Obviously the measured Saharan mineral dust particles were of non-spherical shape. With the help of these lidar and sun photometer measurements the particle shape as well as the spherical equivalence were estimated. It turned out that volume equivalent oblate spheroids with an effective axis ratio of 1:1.6 matched these data best. This aspect ratio was also confirmed by independent single particle analyses using a scanning electron microscope. In order to perform the non-spherical computations, a database of single particle optical properties was assembled for oblate and prolate spheroidal particles. These data were also the basis for simulating the non-sphericity effects on the dust optical properties:omega(o) is influenced by up to a magnitude of only 1% and g is diminished by up to 4% assuming volume equivalent oblate spheroids with an axis ratio of 1: 1.6 instead of spheres. Changes in the extinction optical depth are within 3.5%. Non-spherical particles affect the downwelling radiative transfer close to the bottom of the atmosphere, however, they significantly enhance the backscattering towards the top of the atmosphere: Compared to Mie theory the particle non-sphericity leads to forced cooling of the Earth-atmosphere system in the solar spectral range for both dust over ocean and desert.