Spatio-temporal trapping of light in parametric three-wave mixing processes

Non-degenerate three-wave mixing processes in χ₂ nonlinear crystals can lead to the generation of trapped states in which the parametrically amplified signal and idler pulses co-propagate with the pump pulse despite the group-velocity mismatch. The existing theory of the trapping effect considers only time-domain dynamics, neglecting diffraction effects. This shortcoming is addressed in this work. For the pump with a Laguerre-Gaussian profile in the transverse plane, we show that the signal and idler trapping generally occur in a series of spatiotemporal localized states. The overall hierarchy of the trapped states is similar to that of the localized states in a 3D quantum well, with time playing the role of a coordinate. We compare the maximal parametric gain across different families of trapped states for different pump configurations and reveal regimes in which simultaneous generation of several trapped states with different orbital angular momenta can be observed.