Tailoring the focal region of abruptly autofocusing and autodefocusing ring-Airy beams

We theoretically study the properties of the focal region of abruptly autofocusing and, their variant, autodefocusing ring-Airy beams, under the action of a conical phase gradient. By expanding the analysis of 1D Airy beams to cylindrically symmetric Airy beams, we derive analytic formulas for the position and dimensions of the focus. Our analysis covers in a unified way both beam types, and numerical simulations over a broad parameters range are in excellent agreement with our theoretical predictions. Our results allow the tailoring of the focal region both in size and in position by tuning the initial parameters.