We obtain analytical expressions for the complex amplitudes of optical vortices deformed by astigmatic transforms, i.e., passed either through a cylindrical lens or through an inclined spherical lens. We also obtain similar analytical expressions describing propagation of an optical vortex generated when a Gaussian beam illuminates an inclined spiral phase plate (SPP) or when an elliptic Gaussian beam illuminates a SPP (not inclined). All these optical vortices with a topological charge (TC) $n$ are described by the $n$-th order Hermite polynomial with a complex argument. It is shown that the argument is real only on a straight line in the transverse plane of the laser beam. There are $n$ intensity nulls on this line. The treated here astigmatic transforms are used to determine the integer TC of optical vortices. We conduct a comparative experimental study of different astigmatic transforms and we show that the transform with a cylindrical lens is the best for determining the TC. Unlike other similar works, in this study we achieve transformation of $n$-degenerate intensity null of an optical vortex with the TC $n=100$ into $n$ isolated first-order intensity nulls.

We propose a new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM). The aB-modes are described by the $n$th-order Bessel function of the first kind with complex argument. The asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter $c\ge 0$: when $c=0$, the aB-mode is identical to a conventional radially symmetric Bessel mode; with increasing $c$, the aB-mode starts to acquire a crescent form, getting stretched along the vertical axis and shifted along the horizontal axis for $c\gg 1$. On the horizontal axis, the aB-modes have a denumerable number of isolated intensity zeros that generate optical vortices with a unit topological charge of opposite sign on opposite sides of 0. At different values of the parameter $c$, the intensity zeros change their location on the horizontal axis, thus changing the beam’s OAM. An isolated intensity zero on the optical axis generates an optical vortex with topological charge $n$. The OAM per photon of an aB-mode depends near-linearly on $c$, being equal to $\hslash (n+c{I}_1(2c)/I_0(2c))$, where $\hslash$ is the Planck constant and $I_n(x)$ is a modified Bessel function.

We show that if a nonzero $E$-field intensity is found in the focus of a circularly polarized optical vortex with phase singularity or a higher-order azimuthally polarized beam with polarization singularity, an energy backflow will occur at the center of the focal plane. Using the Richards-Wolf formulas, which describe the electromagnetic field near the tight focus of an aplanatic system containing an ideal spherical lens, we prove that the negative on-axis propagation of light occurs when focusing (i) a left-hand circularly polarized optical vortex with the topological charge $m=2$ or (ii) a conventional second-order azimuthally polarized radially symmetric laser beam. Notably, in the former case, the negative energy flow rotates on a spiral about the optical axis, while in the latter case, the Poynting vector is perpendicular to the focal plane at any point. The theoretical conclusions are confirmed by numerical simulation.

Using the Richards–Wolf formulas for an arbitrary circularly polarized optical vortex with an integer topological charge $m$, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expressions for the intensity (energy density) and for the energy flux (components of the Poynting vector) in the focal plane of an aplanatic optical system. For $m=2$, from the obtained expressions it follows that the energy flux near the optical axis propagates in the reversed direction, rotating along a spiral around the optical axis. On the optical axis itself, the reversed flux is maximal and decays rapidly with the distance from the axis. For $m=3$, in contrast, the reversed energy flux in the focal plane is minimal (zero) on the optical axis and increases (until the first ring of the light intensity) as a squared distance from the axis.

We considered a generalization of the Laguerre–Gaussian (LG) laser beam family by using a complex shift of the beam complex amplitude in Cartesian coordinates. In this case, LG-beams lose their axial symmetry. The normalized orbital angular momentum is the sum of the beam topological charge and the term which is in square dependence on the asymmetry parameter. By optical trapping and moving the polystyrene microspheres in the focus of the asymmetric LG-beam, it is proven that the velocity of the microspheres increases with increasing the asymmetry parameter and constant topological charge.

We theoretically show that optical vortices conserve the integer topological charge (TC) when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with same-sign different TC, the resulting TC of the beam is shown to equal the sum of all constituent TCs. If the beam is composed of an on-axis superposition of Laguerre-Gauss modes (n, 0), the resulting TC equals that of the mode with the highest TC. If the highest positive and negative TCs of the constituent modes are equal in magnitude, the “winning” TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. In the case of equal weight coefficients of both optical vortices, TC of the entire beam equals the greatest TC by absolute value. We have given this effect the name “topological competition of optical vortices”.