Fracxicon as hybrid element between the parabolic lens and the linear axicon

In previous studies we shown on the basis of the geometrical-optical analysis that the fracxicon (generalized optical element of the parabolic lens and the linear axicon) formally produces an infinitely large value of the intensity on the optical axis. Within frame of the paraxial wave model this caustic effect was not found. In this paper, we consider plane wave diffraction on the fracxicon in nonparaxial wave model and we show that at high numerical aperture the fracxicon with an exponent close to 3/2, is actually analogue of the hyperbolic lens optimally focusing incident beam.