  Abruptly autofocusing waves and associated ring-Airy (RA) beams are
attracting increasing interest owing to their fascinating properties such as
their ability of abruptly autofocusing to small F-number. Optical frequency
conversion via nonlinear interactions can further expand their applications to
new area, yet are rarely studied. In this work, we report the frequency
conversion of RA beams via sum-frequency generation using perfect flattop and
common Gauss beams as the pump beams. The nonlinear transformation of the
spatial complex amplitude of the signal and associated influences on
autofocusing behavior, under different conditions of interaction location
(i.e., original, autofocusing, and Fourier planes) and pump structure, were
systematically studied and experimentally investigated. This proof-of principle
demonstration provides a general guideline to build the frequency interface for
abruptly autofocusing waves and a reference for relevant studies involving
nonlinear transformation of abruptly autofocusing waves.
