Focusing properties of phase-only generalized Fibonacci photon sieves

We propose a new algorithm to extend the standard Fibonacci photon sieve to the phase-only generalized Fibonacci photon sieve (GFiPS) and find that the focusing properties of the phase-only GFiPS are only relevant to the characteristic roots of the recursion relation of the generalized Fibonacci sequences. By switching the transparent and opaque zones on the basis of the generalized Fibonacci sequences, we not only realize adjustable bifocal lengths, but also give their corresponding analytic expressions. Besides, we investigate a special phase-only GFiPS, a spiral-phase GFiPS, which can present twin vortices along the axial coordinate. Compared with the single focusing system, bifocal system can be exploited to enhance the processing speed, and offer a broad range of applications, such as direct laser writing, optical tweezers or atom trapping and paralleled fluorescence microscope.