Fourier transforms of fractional order and their optical interpretation

The original definition of the derivatives of a function makes sense only for integral orders, i.e. we can speak of the first or second derivative and so on. However, it is possible to extend the definition of the derivative to noninteger orders by using an elementary property of Fourier transforms. Bracewell shows how fractional derivatives can be used to characterize the discontinuities of certain functions [ 11. An example from the field of optics is related to the Talbot effect [ 2 1, in which self-images of an input object are observed at the 2D planes z=NzO for integer N (z is the axial coordinate and z0 a characteristic distance). Using a self-transformation technique [ 31, it was shown that N could also take on certain rational values. In this letter we define fractional Fourier transformations in a similar spirit. The 0th Fourier transform of a function S(x, y) will be denoted as .P~(x, y)], or simply S”fwhen there is no room for confusion. We require that our definition satisfy two basic postulates. First, .9iIfshould be the usual first Fourier transform, defined as +cc +oo