Non-paraxial analytical solution for the generation of focal curves

A solution for the transmittance of elements forming planar and nonplanar focal curves in the non-paraxial case is proposed in this paper. The treatment is based on the scalar diffraction theory and makes use of the stationary phase method in evaluation of the diffraction integral. The equation of a stationary point enables one to establish mapping relations between the geometrical regions of the element and the corresponding points of the focal curve. These regions turn out to be described by conical curves. The mapping relations can be manipulated by a phase function imposed onto the focal curve. The amplitude factors of the derived transmittances can be omitted and the desired intensity distribution along the focal curve can then be obtained by an appropriate change in the aperture of pure phase elements. The approach is illustrated by examples of elliptical, hyperbolic, conical and circular zone plates. The focus generations for both non-paraxial and paraxial cases of these elements are compared using numerical simulations.