The tight focusing of an optical vortex with an integer topological charge (TC) and linear polarization was considered. We showed that the longitudinal components of the spin angular momentum (SAM) (it was equal to zero) and orbital angular momentum (OAM) (it was equal to the product of the beam power and the TC) vectors averaged over the beam cross-section were separately preserved during the beam propagation. This conservation led to the spin and orbital Hall effects. The spin Hall effect was expressed in the fact that the areas with different signs of the SAM longitudinal component were separated from each other. The orbital Hall effect was marked by the separation of the regions with different rotation directions of the transverse energy flow (clockwise and counterclockwise). There were only four such local regions near the optical axis for any TC. We showed that the total energy flux crossing the focus plane was less than the total beam power since part of the power propagated along the focus surface, while the other part crossed the focus plane in the opposite direction. We also showed that the longitudinal component of the angular momentum (AM) vector was not equal to the sum of the SAM and the OAM. Moreover, there was no summand SAM in the expression for the density of the AM. These quantities were independent of each other. The distributions of the AM and the SAM longitudinal components characterized the orbital and spin Hall effects at the focus, respectively.

In this paper, the tight focusing of high-order cylindrical vector beams (beams with polarization singularity) was investigated. Using the Richards-Wolf formalism, there were obtained expressions for all projections of the electric and magnetic light fields with m-order polarization singularity in the focus of the aplanatic system. Also expressions for the longitudinal component of the Poynting vector were obtained. It was shown that these beams produce in the focal areas with the direction of the Poynting vector opposite to the direction of propagation of the beam. Moreover, the negative values could be comparable in absolute value with positive values; however, this strong inverse energy flow is obtained only while laser light is focused by a lens with high numerical aperture. Of particular interest is the case when the beam order is two (m = 2). In this case, the region where the Poynting vector longitudinal projection is negative is located on the optical axis. If the order of the beam is more than two (m > 2), than the reverse flow occurs near the optical axis and has a shape of a "tube." Moreover, the width of the negative values region (the diameter of the "tube") increases with increasing order of the beam, however, the absolute value of energy backflow decreases. Earlier, we reported on the observation of a spiral negative energy flow from the center of the focal plane of a focused vortex beam. In this paper, the negative propagation of a laminar near-axis energy flow is reported.

In this work, we study three different types of astigmatic (anisotropic) Hermite-Gaussian (aHG) beams whose complex amplitude in the Fresnel diffraction zone is described by the complex argument of a Hermite polynomial of degree (n, 0). The first-type beam is a circularly symmetric optical vortex with topological charge n that has passed through a cylindrical lens. The outgoing optical vortex ‘splits’ into n first-order optical vortices, carrying an orbital angular momentum (OAM) per photon of n. The second-type beam is a Hermite-Gaussian (HG) beam of order (n, 0) generated by passing an elliptic HG beam through a cylindrical lens, which acts by imprinting an OAM into the original HG beam. The OAM of such a beam is a sum of vortex and astigmatic components and can reach large values (tens and hundreds of thousands per photon). We derive an exact formula to describe the OAM of these aHG beams. Under certain conditions, the zero intensity lines of the aHG beam ‘merge’ into an n-fold degenerate intensity null on the optical axis, with the OAM of the beam becoming equal to n. The third type is an elliptical optical vortex with topological charge n that has passed through a cylindrical lens. With a special choice of the ellipticity degree (1:3), the beam retains its structure upon propagation, with the on-axis degenerate intensity null not ‘splitting’ into n optical vortices. The beam is shown to carry a fractional OAM not equal to n. Using intensity distributions of the aHG beam in the foci of two cylindrical lenses, we measure a normalized OAM of the elliptic aHG beam, with the deviation from a theoretical estimate being as low as 7% (experimental OAM-13.62 and theoretical OAM-14.76).