  The Fourier transform proves indispensable in the processing of classical
information as well as in the quantum domain, where it finds many applications
ranging from state reconstruction to prime factoring. An implementation scheme
of the $d$-dimensional Fourier transform acting on single photons is known that
uses the path encoding and requires $O(d \log d)$ optical elements. In this
paper we present an alternative design that uses the orbital angular momentum
as a carrier of information and needs only $O(\sqrt{d}\log d)$ elements,
rendering the path-encoded design inefficient. The advantageous scaling and the
fact that our approach uses only conventional optical elements allows for the
implementation of a 256-dimensional Fourier transform with the existing
technology. Improvements to our design, as well as explicit setups for low
dimensions, are also presented.
