Complex vortices formed by coaxial superposition of circular and planar vortices

Complex vortices distributed axial symmetrically on different radii were achieved theoretically by axially superimposing circular and planar vortices at their cores. The simulation results show that the intensity distributions of these vortices are axial symmetrical. A graphical solution method for determining positions of complex vortices is proposed. The positions of the formed vortices can be obtained exactly by applied the graphical method to the above superposition of vortices with arbitrary parameters. The graphical results are consistent with that of the numerical simulation.