Optical implementations of the fractional Fourier transform using lenses

The optical structures for implementation of the fractional Fourier transform were analyzed based on the theory of impulse response functions of the lens combination optical system, and the meaning of the standard focal length was discussed. Several general optical setups were provided. By using these setups, the standard focal length can be adjusted easily. It will be significant for designing a spatially, variant filtering system with multiple cascaded optical fractional Fourier transformers. The experimental results performed by a single lens or double lenses are also given.