Caustic Interpretation of the Abruptly Autofocusing Vortex beams

We propose an effective scheme to interpret the abruptly autofocusing vortex beam. In our scheme, a set of analytical formulae are deduced to well predict not only the global caustic, before and after the focal plane , but also the focusing properties of the abruptly autofocusing vortex beam, including the axial position as well as the diameter of focal ring. Our analytical results are in excellent agreement with both numerical simulation and experimental results. Besides, we apply our analytical technique to the fine manipulation of the focusing properties with a scaling factor. This set of methods would be beneficial to a broad range of applications such as particle trapping and micromachinings.