  We investigate the propagation properties of the optical vortices emitted
from a collectively polarized electric dipole array as a simple model of the
several emitters. Using an analytical approach based on the Jacobi-Anger
expansion, we derive a relationship between the topological charge of the
optical vortices and the source conditions of the emitter, and clarify and
report our new finding; there exists an intrinsic split of the singular points
in the electric field due to the spin-orbit interaction of the dipole fields.
