Two-dimensional fractional Fourier transform and its optical implementation

Abstract

The IVactional Fourier transform of order a is defined in a manner sucli that tlu' common Fourier transform is a special case with order a = 1. Tlie definition is easil}^ extended to two dimensions just repeating the transibrm in x and y directions independently. The properties of the separable two dimensional fractional Fourier transform defined in this manner are derived and several oj)- tical implementations are given. However, this definition, ibr certain purposes, motivatcxi us to look for a new, non-separabhi definition. We ])resent sucli a d('iinition of the two dimensional fractional Fourier transform with its optical implementation. The usefulness of the new definition is justified with a noise filtering example.