Image rotation, Wigner rotation, and the fractional Fourier transform

TLDR

The fractional Fourier transform, exemplified by degree 1/2 performing an ordinary Fourier transform twice, was introduced into optics through graded‑index fiber that performs a full Fourier transform over a proper length. This study assigns degree p = 1 to the ordinary Fourier transform, explores its algorithmic isomorphism with image rotation and Wigner distribution rotation, and proposes two optical setups to realize fractional Fourier transforms. The mechanism involves cutting a graded‑index fiber into shorter segments to split the ordinary Fourier transform into fractional transforms, revealing an algorithmic isomorphism with image rotation and Wigner distribution rotation, and enabling two optical setups for implementation. Ozaktas and Mendlovic’s work, “Fourier transforms of fractional order and their optical implementation,” appears in Opt.

Abstract

In this study the degree p = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform.