Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations

The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms. This function has interesting properties and Fourier transform relations. For example, the Fourier transform of the sampled chirp is also a sampled chirp for some sampling rates. These properties are essential in interpreting the aliasing and its effects as a consequence of sampling of the quadratic phase function, and lead to interesting and efficient algorithms to simulate Fresnel diffraction. For example, it is possible to construct discrete Fourier transform (DFT)-based algorithms to compute exact continuous Fresnel diffraction patterns of continuous, not necessarily, periodic masks at some specific distances. © 2004 Society of Photo-Optical Instrumentation Engineers