Berry's phase of guided vortices in an optical fiber

It is shown that the difference in structure and propagation constants of guided vortices of an optical fiber is connected with a topological phase of the given vortices. The field state of CV and IV vortices is uniquely presented by a pair of numbers: a topological phase I and polarization helicity sigma(z). Symmetry relationships, which are characteristic for the topological phases, extend to this pair of numbers. Thus, a simultaneous change of the sign l and sigma(z) leaves the vortex state unchanged. The sign change of only one of the numbers transforms the vortex from the state CV ( or IV) into another state IV ( or CV). These symmetry relationships reflect the result of spin-orbital interaction in vortices and determine the eligibility of quantum numbers: topological charge and helicity.