In beam shaping applications, the minimization of the number of necessaryoptical elements for the beam shaping process can benefit the compactness ofthe optical system and reduce its cost. The single freeform surface design forinput wavefronts, which are neither planar nor spherical, is therefore ofinterest. In this work, the design of single freeform surfaces for a givenzero-\'etendue source and complex target irradiances is investigated. Hence,not only collimated input beams or point sources are assumed. Instead, apredefined input ray direction vector field and irradiance distribution on asource plane, which has to be redistributed by a single freeform surface togive the predefined target irradiance, is considered. To solve this designproblem, a partial differential equation (PDE) or PDE system, respectively, forthe unknown surface and its corresponding ray mapping is derived from energyconservation and the ray-tracing equations. In contrast to former PDEformulations of the single freeform design problem, the derived PDE ofMonge-Amp\`ere type is formulated for general zero-\'etendue sources incartesian coordinates. The PDE system is discretized with finite differencesand the resulting nonlinear equation system solved by a root-finding algorithm.The basis of the efficient solution of the PDE system builds the introductionof an initial iterate constuction approach for a given input direction vectorfield, which uses optimal mass transport with a quadratic cost function. Aftera detailed description of the numerical algorithm, the efficiency of the designmethod is demonstrated by applying it to several design examples. This includesthe redistribution of a collimated input beam beyond the paraxialapproximation, the shaping of point source radiation and the shaping of anastigmatic input wavefront into a complex target irradiance distribution.
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