  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.
