Simultaneous detection of the sign and magnitude of the Poincaré Hopf index of the electromagnetic Gaussian Schell model polarization singular vector beams

We report a method to determine the sign as well as the magnitude of the Poincaré Hopf index (PHI) of electromagnetic Gaussian Schell model (EGSM) polarization singular vector beams. This method is based on the analysis of the orientation angle (OA) of polarization. OA distribution is found to form petal like structure around the singularity. The sign and magnitude of PHI are related to the number and the orientation direction of petals around the singularity.