Fractional Fourier transform in optical setups

The fractional order Fourier transform was introduced into optics on the basis of light propagation through a piece of graded-index fiber of proper length. Lohmann reinvented the fractional Fourier transform operation based on the Wigner distribution function that can be applied not only for wave propagation in free space, but for passage through a lens system, too. In this paper, it is shown that optical implementation of the fractional Fourier transform of different orders requires optical systems of different (Fourier planes) configurations.