Backflow in vector Gaussian beams

The phenomenon of Poynting backflow in a single vector Gaussian beam is examined. The paraxial Maxwell equations and their exact solutions containing terms proportional to the small parameter $\varepsilon=\frac{\lambda}{2\pi w_0}$, or to its square, where $w_0$ is the beam waist, are made use of. Explicit analytical calculations show that just these additional expressions are responsible for the occurrence of the reversed Poynting-vector longitudinal component for selected polarizations. Concrete results for the time-averaged vector for several Gaussian beams with $n=1$ ($n$ being the topological index of the beam) are presented. Depending on the choice of polarization, the backflow area is located around the beam's axis, has an annular character or is absent. In the case of $n=2$, backflow areas were found as well. The magnitude of the backflow proved to be proportional to $\varepsilon^4$.