Lenses to form a longitudinal distribution matched with special functions

We study radially symmetric diffractive optical elements to generate an array of local foci or intensity zeros in the paraxial region by a certain law. The axial distribution is defined by the spatial spectrum of the optical element's radial function, enabling the elements to be called longitudinal-spectrum lenses. The theoretical explanation of the effect is based on the reduction of the Fresnel-Hankel transform to the one-dimensional Fourier transform. The various lenses are analyzed include those generating longitudinal distribution proportional to the Airy and Hermite-Gaussian functions.