Vectorial structure characteristics of vortex Lorentz-Gauss beam in far-field

Based on the method of vector angular spectrum representation and the method of stationary phase, the analytical expressions for the electromagnetic fields of the vectorial structure, namely the transverse electric (TE) and the transverse magnetic (TM) terms of a vortex Lorentz-Gauss beam, are derived in the far-field. Moreover, the analytical expressions of the energy flux of the TE and TM terms are also presented in the far-field. The influence of the topological charge on the energy flux distributions of the vortex Lorentz-Gauss beam and its vectorial structure are analyzed by numerical examples. The TE term is composed of two or three lobes which are located at the y-axis. The TM term can be viewed as the TE term rotating 90 degrees. When the topological charge is small, the vortex Lorentz-Gauss beam takes on dark hollow shape, and the brightly outer ring is uniform. With increasing the topological charge, the energy flux in the brightly outer ring of the vortex Lorentz-Gauss beam fluctuates, and the change in the central region is relative complicated. With increasing the topological charge, the beam sizes of the vortex Lorentz-Gauss beam and its vectorial structure also augment. However, the beam sizes of the vortex Lorentz-Gauss beam and its vectorial structure will saturate. In the practical application of the vortex Lorentz-Gauss beam, therefore, the topological charge should not be too large.