  Ince-Gaussian (IG) mode, a recently discovered type of structured Gaussian
beam, corresponds to eigenfunctions of the paraxial wave equation in elliptical
coordinates. This propagation-invariant mode is of significance in various
domains, and in particular, its nonlinear transformation; however, there have
been few relevant studies to date. In this work, we report the parametric
upconversion of IG modes and associated full-field selection rule for the first
time. We demonstrate that IG signals can be perfectly upconverted by a
flattop-beam pump; in contrast, significant mode distortion occurred when using
the most common Gaussian pump. Particular attention was given to the origin of
the distortion, i.e., radial-mode degeneration induced by the sum-frequency
generation excited by Gaussian pump. This proof-of-principle demonstration has
great significance in relevant areas, such as high-dimension quantum frequency
interfacing and upconversion imaging.
