Identifying orbital angular momentum of vectorial vortices with Pancharatnam phase and Stokes parameters

In this work, an explicit formula is deduced for identifying the orbital angular moment (OAM) of vectorial vortex with space-variant state of polarization (SOP). Different to scalar vortex, the OAM of vectorial vortex can be attributed to two parts: the azimuthal gradient of Pancharatnam phase and the product of the azimuthal gradient of orientation angle of SOP and relevant solid angle on the Poincar\'{e} sphere. With our formula, a geometrical description for OAM of light beams can be achieved under the framework of the traditional Poincar\'{e} sphere. Numerical simulations for two types of vectorial vortices have been carried on to confirm our presented formula and demonstrate the geometrical description of OAM. Furthermore, the finding will pave the way for precise characterization of OAM charge of vectorial vortices.