Matrix description of near-field diffraction and the fractional Fourier transform

We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional in put. We show that for an input with N X N pixels, in an area d X d, it is n ecessary to evaluate the Fresnel diffraction pattern at distances z greater than or equal to d(2)/lambda N. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special ph ase matrix with fractional parameter epsilon. If the phase matrix is evalua ted at epsilon = 1, we find the discrete Fourier transform matrix. (C) 1999 Optical Society of America [S0740-3232(99)02302-9].