Propagation of the radially-limited vortical beam in a near zone. Part I. Calculation algorithms

On an example of plane wave diffraction by a circular aperture in a near zone (the order of several wavelengths) comparison of calculation algorithms such as vectorial Rayleigh-Sommerfeld diffraction integral (RS) and plane wave expansion (PWE) on accuracy and speed of calculations is executed. In a scalar case that corresponds to calculation of a cross-section components of an electric field, results differ only in area very close to the aperture. In a vector case at calculation of the longitudinal component in PWE method there is a singularity in the domain of spectral frequencies which radius is close to unit. Various variants of avoiding of this singularity are offered. On distance of several wavelengths results of two considered algorithms coincide and differ from finite-difference time domain (FDTD) method only in scale (the root-mean-square deviation with account of scale makes less than 2%). Thus, the algorithms considered in given work allow to receive for essentially smaller time structurally true (but a little overestimated on amplitude) a picture of diffraction in a near zone. Such "overestimation" of amplitude can be connected by that considered RS and PWE methods do not mean presence y-components in diffraction picture of initially x-polarized field. In the second part of the paper a modification of PWE method allowing to consider presence of all vector components is considered.