Toroidal Vortices of Energy in Tightly Focused Second-Order Cylindrical Vector Beams

In this paper, we simulate the focusing of a cylindrical vector beam (CVB) of second order, using the Richards–Wolf formula. Many papers have been published on focusing CVB, but they did not report on forming of the toroidal vortices of energy (TVE) near the focus. TVE are fluxes of light energy in longitudinal planes along closed paths around some critical points at which the flux of energy is zero. In the 3D case, such longitudinal energy fluxes form a toroidal surface, and the critical points around which the energy rotates form a circle lying in the transverse plane. TVE are formed in pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). We show that when light with a wavelength of 532 nm is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 μm (0.85 λ) from the axis (with a period along the z-axis of 0.8 μm (1.5 λ)). The vortices arise in pairs: the vortex nearest to the focal plane is twisted clockwise, and the next vortex is twisted counterclockwise. These vortices are accompanied by saddle points. At higher distances from the z-axis, this pattern of toroidal vortices is repeated, and at a distance of about 0.7 μm (1.3 λ), a region in which toroidal vortices are repeated along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, these toroidal vortices are not observed.

Toroidal polarization vortices in tightly focused beams with singularity

In this paper, we numerically investigated tight focusing of cylindrical vector beams of the second order using Richards-Wolf formulae. It was shown that intensity rings where the Poynting vector was equal to zero appeared not only in the focal plane but also in nearby planes. For example, a lens with numerical aperture NA=0.95 was shown to generate periodical toroidal vortices with a 0.8-m period along the z-axis at a distance of about 0.45 m from the axis. The vortices were generated pairwise, with the closest-to-focus vortex having clockwise helicity and the subsequent being anticlockwise. The vortices were also characterized by saddle points. When focusing an optical beam passed through a narrow annular aperture, no toroidal vortices were observed.

Properties of electrostatic correcting systems with annular apertures

Image formation in electron microscopes with circular hole and annular apertures is studied theoretically. The apertures—the circular hole aperture being negative with respect to the annular aperture—produce an additional electrostatic field that exerts a force on the electrons directed toward the optical axis. The resulting deflection angle decreases with increasing distance from the optical axis. This electrostatic field results in a correcting effect of the unavoidable spherical aberration of round electron lenses; the deflection toward the optical axis increases stronger than linearly with increasing distance from the optical axis. Analytical formulae are given for the correcting effect of circular hole and annular apertures. The expressions are based on the Davisson–Calbick formula, which is used to calculate focal length of a simple electrostatic lens.