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• Freeform optics. • Reflector design/optimization. • Optimal transportation. • Finite source illuminance. • Far field illumination. We address the "freeform optics" inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed far-field distribution, for a finite light source. When the finite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting, the inverse problem is well-posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and numerically. We are not aware of any similar mathematical formulation in the finite source case: i.e. the source has an "étendue" with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflector and the reflection map. Under this parameterization, we can construct a smooth loss/misfit function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study. [ABSTRACT FROM AUTHOR]