Design lenses forming paraxial longitudinal distribution according to their spatial spectra

We consider radially-symmetric diffractive optical elements forming along optical axis a set of local foci or zero values by the specific law. The axial distribution is determined by the spatial spectrum of the radial function of the optical element that allows us to call these elements longitudinally-spectral lenses. The theoretical explanation of the effect is based on the reduction of the on-axis Fresnel-Hankel transform to the one-dimensional Fourier transform. We investigate different lenses, including that forming the longitudinal-modal distribution proportional to the Airy and Gauss-Hermite functions.