  Fourier transforms are ubiquitous mathematical tools in basic and applied
sciences. We here report classical and quantum optical realizations of the
discrete fractional Fourier transform, a generalization of the Fourier
transform. In the integrated configuration used in our experiments, the order
of the transform is mapped onto the longitudinal coordinate, thus opening up
the prospect of simultaneously observing all Transformation orders. In the
context of classical optics, we implement discrete fractional Fourier
transforms, both integer and fractional, of exemplary wave functions and
experimentally demonstrate the shift theorem. Moreover, we apply this approach
in the quantum realm to transform separable and highly entangled biphoton wave
functions. The proposed approach is versatile and could find applications in
various fields where Fourier transforms are essential tools, such as quantum
chemistry and biology, physics and mathematics.
