Analysis of the axial distribution formed by a fracxicon in the paraxial and nonparaxial cases

We analytically and numerically examine the action of the new diffractive optical element with a fractional power dependence of the phase of the radius - "fracxicon" which is a generalization of the classical elements as the axicon and the lens. Analytical expressions for axial distribution formed by a fracxicon are derived based on the method of stationary phase in the paraxial approximation. Also, the numerical simulation of the fracxicon is performed in paraxial and nonparaxial cases with the use of the plane waves expansion method. Analysis of the results showed that the analytical expressions provide high precision calculations for fractional degrees close to unity even at high numerical aperture.