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

We use four-wave mixing in a heated rubidium vapour to couple the orthogonal azimuthal and radial mode numbers of Laguerre-Gaussian beams. The selection rules for the angular mode number are governed by angular momentum conservation, while the role of the radial mode number is more esoteric. One would not normally expect these orthogonal components to interact but we demonstrate experimentally that a clean Laguerre-Gauss mode LG^l_p = LG⁰₁ can be generated by converting LG¹₀ and LG^-1₀ near-infrared pump beams -- but only if the length of the atomic medium exceeds the Rayleigh range. This leads to strikingly different conversion behaviour in thick- and thin-medium regimes; with the Gouy phase, and it's relative importance in each regime, the key to understanding this conversion. Our experimental investigation of the transition between these regimes bridges the gap between previous experiments in atomic thick media and work in nonlinear crystals. This sets a clear starting point to explore efficient radial-to-azimuthal and radial-to-radial mode conversion in the thick-medium regime.