Quantifiable example of complementarity relation between optical orbital angular momentum and angular position

A light beam with phase singularity (PS) characterized with azimuthally symmetric angular positions (APs) can be constructed by the rotationally symmetric superposition of numerous fractional vortex light beams with identical charges. The profile of this light beam deforms after propagation owing to its orbital angular momentum (OAM) noneigenvalue, and the deformation degree can be evaluated by the degree of phase dislocation associated with the PS. The expectation value of the mean deviation of its OAM from its characteristic charge value and the variation of the degree of phase dislocation obey a proportionality relation. This light beam offers a quantifiable example of a complementarity relation between the observables of optical OAM and AP, which are the OAM noneigenvalue and intensity AP variation, respectively. This complementarity relation is significant for an existing experiment employing the quantum state of this light beam.