Fast and exact diffraction integral calculus: A comparison with fresnel approximation

In Lens-less Digital Holographic Microscopy1, 2, 3, amplitude and phase of an object can be recovered from the irradiance of a diffraction pattern. To do this, an iterative algorithm with constrictions is applied, which is based on the calculation of diffraction patterns between two fixed planes. In one dimension and for N pixels on each straight line segment, we must calculate N2 values of the impulse response function, h, in order to evaluate the optical field on N pixels along another straight line segment. By using translation and permutation symmetries of h, and without any approximation, in this paper we will show that to calculate the diffraction pattern over N pixels we need only N values of h. Adding to this, if iterations are applied between two pixel lines, the N values of h are calculated only one time, and then we achieved a significative reduction in calculation time. Finally, our exact calculations were compared with Fresnel diffraction patterns reported by Goodman4, and we found that when distance between object and diffraction planes diminishes then the diffraction pattern irradiance differences are increased up to 53 % with respect to Fresnel approximation.