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Abstract: Cantor rings are rotational symmetric pupils that are generated from a Cantor set of a given level of growth. These pupils have certain fractal properties. For example, it is known that when illuminated by a general spherical wavefront they provide self-similar patterns at transverse planes in the Fraunhofer region. In this paper, we study the response of Cantor rings when illuminated by a Bessel light beam conforming what we call fractal axicons. It is shown that, with this kind of illumination a close replica of the radial profile of the pupil is obtained along the optical axis, i.e., we show that the axial behaviour of these pupils has self-similarity properties that can be correlated to those of the diffracting aperture. The influence of several construction parameters is numerically investigated. [Copyright &y& Elsevier]