Counterpropagating nondiffracting vortex beams with linear and angular momenta

Closed-form analytical expressions for the electric and magnetic (EM) field components of a high-order Bessel vortex beam of fractional type {α} (HOBVB-F{α}) are derived based on the axial polarization of the vector potentials [denoted by (z,-z)]. The fields' components correspond to the most generalized case of quasi-standing-waves that reduce to perfect (i.e., equiamplitude) standing waves or progressive waves with appropriate choice of the quasi-standing-wave coefficient 0≤Υ≤1. The beam components are exact solutions of the vector wave equation (Helmholtz equation) and Maxwell's equations. Moreover, computations of the EM components and comparison with the results obtained from a transverse polarization scheme [denoted by (x,{-}y)] illustrate the analysis. In addition, the time-averaged angular momentum density and flux are evaluated with particular emphasis on the axial and transverse polarization states of the vector potentials. It is found that the quasi-standing-wave effect does not influence the spatial distributions of the EM components in the transverse polarization scheme (x,{-}y); however, it affects the transverse EM field components in the axial polarization (z,{-}z). Moreover, both the linear and angular momenta density fluxes are shown to reverse sign for a particular value of the half-cone angle {β}. These phenomena anticipate the production of a ``tractor'' beam where particulate matter may be pulled back toward the source, and a spinning reversal effect in which particulate matter may rotate with opposite handedness to a HOBVB-F{α}. The results are particularly useful in applications involving optical laser tweezers, tractor beams, optical spanners, arbitrary scattering, radiation force, angular momentum, and torque.