Gouy phase-matched angular and radial mode conversion in four-wave mixing

Studying the conversion between transverse light modes via four-wave mixing in a heated rubidium vapor, we demonstrate and explain a transfer between azimuthal and radial mode numbers. They 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 behavior in the thick- and thin-medium regimes. 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 the thick-medium regime, allowing efficient radial-to-azimuthal and radial-to-radial mode conversion.