Free-space diffraction and the fractional Fourier transform

An explicit expression is derived for free-space propagation in terms of the fractional Fourier transform. The expression contains two arbitrary parameters, which can be, for example, an original phase curvature together with a scaling parameter or, alternatively, the order of fractional transformation. Appropriate choice of these parameters is discussed. This approach is then applied to the free-space propagation of Gaussian beams.