Characteristic study of anomalous vortex beam through a paraxial optical system

Characteristics of an anomalous vortex beam (AVB) through the paraxial optical systems are studied. The analytical formulas for the AVBs through the aligned and misaligned optical systems are derived by using the Collins integral equation. As the numerical examples, the intensity distributions for the AVBs in the free space and the convergent optical system are given. It is shown that the far fields for AVBs will be determined by the main parameters, such as width of beam waist w0, n, m (n is the order of the AVB, m is the topological charge). Furthermore, the analytical formula of M2 factor of AVB is derived by using the second-order moments associated with the intensity distributions. The study results can provide references for the application of AVBs.