Complex-valued Fresnel-transform sampling

VanderLugt [Appl. Opt. 29, 3352 (1990)] presented sampling rates for the amplitude of Fresnel diffraction patterns. These apply to any plane in a coherent optical system. Although these sampling rates represent the amplitude of diffraction patterns accurately, they are not adequate for the retention of complete information in complex-valued Fresnel diffraction patterns. I show this by considering the ability to reconstruct the original input image through backward diffraction of the forward diffraction pattern of such an image. I then extend the VanderLugt sampling techniques such that reliable sampling of the phase of these Fresnel diffraction patterns can also be achieved. The analysis is restricted to lensless optical systems. The new sampling rates are tested with numerical computations of Fresnel diffraction patterns and rigorous scalar diffraction patterns in both forward and backward directions.