Simultaneous measurement of the radial and azimuthal mode indices of a higher-order partially coherent vortex beam based on phase detection.

We explore the phase distribution of the cross-spectral density (CSD) function with an off-axis reference point of a focused higher-order partially coherent vortex (PCV) beam in the focal plane. Our results reveal that the numbers of ring dislocations and screw dislocations in the CSD function equal to the magnitudes of the azimuthal mode index and radial mode index of the incident beam, respectively. In addition, the sign of the azimuthal mode index determines the direction of the phase change of the screw dislocation. This phenomenon can be used to measure the magnitude and sign of the mode indices of a higher-order PCV beam simultaneously. Our results will be useful for quantum information processing, optical storage, and communications.