Fig. 1: Concept of OAM operator enabled holography.

a Schematic of the optical OAM multiplication operator \({\mathscr{M}}\), mapping FOAM modes to IOAM modes for holographic reconstruction. b Non-unique operator pathways. A FOAM mode (TC = 1/2) can map to an IOAM mode (TC = 1) via distinct \({\mathscr{M}}\)-pathways, while an IOAM mode (TC = 1) can also be generated from multiple FOAM modes (e.g., TC = 1/2, 1/3 and 1/4). c Operator parametric space representation. Each \({\mathscr{M}}\)-pathway (e.g., TC = 9/4, n = 4, d = 1.6 cm) maps to a unique helical trajectory, where TC is the topological charge of the incident fractional OAM beam, n is the multiplication scaling factor and d is the propagation distance, respectively in the method of coordinate transformation