We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with <i>N</i>×<i>N</i> pixels, in an area <i>d</i>×<i>d</i>, it is necessary to evaluate the Fresnel diffraction pattern at distances <i>z</i>≥<i>d</i><sup>2</sup>/λ<i>N</i>. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ε. If the phase matrix is evaluated at ε=1, we find the discrete Fourier transform matrix.
© 1999 Optical Society of America
Sara Bradburn Tucker, Jorge Ojeda-Castañeda, and W. Thomas Cathey, "Matrix description of near-field diffraction and the fractional Fourier transform," J. Opt. Soc. Am. A 16, 316-322 (1999)