Diffraction optical devices of modest degrees of freedom (DOF), such as deformable mirrors, have not been exploited as general-purpose transformers of optical fields. Described in this Letter is a method that guides deformable mirrors to optimal surfaces allowed by the DOF for various desirable outcomes. The method is based on a modal optimization procedure with the help of Walsh functions in controlling the variables of the mirrors, i.e. the actuators. It is shown that a deformable mirror of modest DOF can provide field transformations for arbitrary beam-splitting, formation of ring-shaped beams, and coherent beam combining.
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