Fractional Fourier transform properties of lenses or other elements or optical environments are used to introduce one or more positive-definite optical transfer functions outside the Fourier plane so as to realize or closely approximate arbitrary non-positive-definite transfer functions. Designs can be straightforwardly obtained by methods of approximation. The invention provides for single and multiple channel controllable optical processors which may be used for image or other types of computation, including computation with complex-valued arithmetic, visual color, and wide-spectrum optical signals. The invention also provides for synthesis of Fractional Fourier transforms, extensions to non-Hermite basis functions, phase-shifting filter elements and non-quadratic graded-index materials (which do not naturally invoke the Fractional Fourier transform operation). Applications include integrated optics, optical computing systems, particle beam systems, radiation accelerators, astronomical observation methods, and controllable lens systems.
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