  Studying the conversion between transverse light modes via four-wave mixing
in a heated rubidium vapour, we demonstrate and explain a transfer between
azimuthal and radial mode numbers. These relate to orthogonal modal dimensions,
which one would not normally expect to interact. While angular momentum
conservation in this nonlinear process dictates the selection rules for the
angular mode number, the role of the radial mode number is more esoteric. We
demonstrate systematically that the Gouy phase is the key to understanding this
conversion, leading to strikingly different conversion behaviour in the thick
and thin medium regime. Our experimental investigation of the transition
between these regimes bridges the gap between previous experiments in atomic
thick media and work in nonlinear crystals. Our work sets a clear starting
point to explore new territory in the thick medium regime, allowing efficient
radial-to-azimuthal and radial-to-radial mode conversion.
