  We present a new family of diffractive lenses whose structure is based on the
combination of two concepts: photon sieve and fractal zone plates with variable
lacunarity. The focusing properties of different members of this family are
examined. It is shown that the sieves provide a smoothing effect on the higher
order foci of a conventional lacunar fractal zone plate. However, the
characteristic self-similar axial response of the fractal zone plates is always
preserved.
