Phase Matching of Orbital Angular Momentum in Rare Earth Ion Doped Solid State Systems
In this work, we demonstrate the generation of stimulated photon echos carrying unique topological charge that results from the temporal phase matching conditions of three independent beams spatially and spectrally overlapped in a cryogenically cooled rare earth ion doped solid state system. A sample of $Tm^{3+}:YAG$ was used to generalize the momentum phase matching condition to include the orbital angular momentum of the input fields. The input fields and corresponding photon echo were characterized via astigmatic transform, with results mapping directly to the expected behavior of traditional stimulated photon echos. These results demonstrate that rare earth ion doped systems are capable of spatial multiplexing, spatial filtering, and the generation of structured light, opening pathways towards real time optical multi-modal signal processing.