Fresnel transform as a projection onto a Nijboer-Zernike basis set.

The Fresnel transform is widely used in optics to calculate the free-space propagation of paraxial fields. Generally, there is no analytical solution for the Fresnel transform; therefore, the numerical methods are used often. In this Letter, we propose a new semi-analytical method to calculate the Fresnel transform, which is based on an extended Nijboer–Zernike theory. We calculate two examples to investigate how the sampling rate and maximal number of Zernike polynomialsaffect the accuracy of our results, and then use this method to calculate the reconstruction of two different kinds of holograms. At the end, we discuss the advantages and disadvantages of our method.