  Optical orbital angular momentum transformation and corresponding
azimuthal-mode selection rules have been studied exhaustively for various
nonlinear optical interactions. However, nonlinear transformation of radial
mode has not been systematically studied since the pioneering work [Phys. Rev.
A 56, 4193, 1997]. In this paper, we theoretically study and experimentally
verify the radial modal transitions of Laguerre-Gauss (LG) modes in parametric
upconversion. Specifically, we provide a general solution that describes the
sum-frequency generation (SFG) field excited by two arbitrary LG modes. Based
on the solution, one can predict the full spatial complex amplitude of SFG
fields upon propagation precisely and readily obtain the associated full-field
selection rule including both azimuthal and radial modes. This work provides a
theoretical basis for quantum and nonlinear optical research involving
parametric upconversion of complex structured light, and paves the way for
future work on full-field transformation of spatial modes in other nonlinear
interactions.
