Generation of Light Curves: Demonstration of the Relationship Between the Intensity Distribution and the Characteristics of the Curve’s Evolute

An analytical approach is developed for calculating a phase diffraction axicon with angularly variable spatial frequency that generates a prescribed light curve without iteratively solving an inverse design problem: the axicon frequency function is obtained directly from the polar equation of the target curve. When the axicon is combined with a lens, the intended curve is formed in the focal plane (Fraunhofer region), whereas in the Fresnel zone the intensity pattern is closely related to the curve’s evolute. We systematize this correspondence for rings, ellipses, spirals and polygonal curves, and derive analytical parameter bounds associated with the appearance of inflection points that qualitatively reshape both the evolute and the diffracted field (including hybrid curve–evolute patterns and “rose”-type contours). The qualitative outcome depends on the curve class/type. For a convex curve without inflections the expected curve is formed in in the far field (or in the focal plane), whereas the Fresnel intensity envelope follows the evolute. If the curve has peculiarities, especially inflection points with unbounded evolute branches, hybrid curve–evolute structure appears, and the focal pattern may itself resemble the evolute. For ellipses and polygonal families we analytically identify parameter intervals that mark this transition. The approach is verified experimentally for polygonal axicons, with attention to fabrication tolerances and to matching camera planes with the simulated propagation distances. Measured intensity distribution patterns agree with the corresponding simulations for both convex contours and curves that contain inflection points. The results are of fundamental interest and of practical value for optical trapping and laser surface structuring, where a compact phase element should deliver a controlled bright contour at a chosen working distance.