Three-dimensional array foci of generalized Fibonacci photon sieves

We present a new kind of photon sieves on the basis of the generalized Fibonacci sequences. The required numbers and locations of axial foci can be designed by generalized Fibonacci photon sieves (GFiPS). Furthermore, the three-dimensional array foci can be controllable and adjustable by the optical path difference scaling factor (OPDSF) when the amplitude modulation is replaced with the phase modulation. Multi-focal technologies can be applied to nano-imaging, THZ, laser communications, direct laser writing, optical tweezers or atom trapping, etc.