Phase memory preserving harmonics from abruptly autofocusing beams

We demonstrate both theoretically and experimentally that the harmonics from abruptly auto-focusing ring-Airy beams present a surprising property, they preserve the phase distribution of the fundamental beam. Consequently, this "phase memory" inherits to the harmonics the abrupt autofocusing behavior, while, under certain conditions, their foci coincide in space with the one of the fundamental. Experiments agree well with our theoretical estimates and detailed numerical calculations. Our findings open the way for the use of such beams and their harmonics in strong field science.