New analytical diffraction expressions for the Fresnel–Fraunhofer transition regime

We propose a method to obtain improved expressions for the diffraction of circular apertures in the boundary between near and far field. The method is based on an analysis of the center value of the diffraction pattern derived from the Fresnel expression. The method is shown to be applied to both amplitude, phase, and complex apertures, and is compared to Fresnel propagation in simulations. We show how these new expressions allow for the use of the Fraunhofer approximation at Fresnel numbers at least twice as large as the conventional expressions. This facilitates analytical Fraunhofer propagation to be used for much larger apertures, at shorter propagation distances, and with more complex aperture transmission functions.