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V.V. Kotlyar

443001, Samara, Russia, Molodogvardeyskaya 151;

**DOI: **10.18287/2412-6179-CO-1165

**Pages: **5-15.

**Full text of article:** Russian language.

**Abstract:**

In this work, the sharp focusing of a laser beam whose initial polarization pattern is formed by superposition of a cylindrical *m*^{th}-order vector beam and a homogeneous linearly polarized beam is considered theoretically and numerically. Although in the source plane of such a beam both the angular spin momentum and the third Stokes parameter are equal to zero, we reveal that given odd *m*, subwavelength local regions are formed in the focal plane, where transverse vortex energy flows occur and the third Stokes parameter (the on-axis component of the angular spin momentum) is non-zero. Thus, at odd *m*, at the focus of such a beam there are – sub-regions with elliptical polarization of light with alternating handedness in the adjacent sub-regions (clockwise and counterclockwise). This phenomenon can be interpreted as a variant of an optical Hall effect. We note that at even *m*, the field at the focus is linearly polarized at every point and no transverse energy flow is observed.

**Keywords**:

linear and circular polarization, sharp focusing, Richards-Wolf formulas, Stokes vector, spin angular momentum.

**Citation**:

Kotlyar VV, Stafeev SS, Zaitsev VD. Sharp focusing of on-axis superposition of a high-order cylindrical vector beam and a beam with linear polarization. Computer Optics 2023; 47(1): 5-15. DOI: 10.18287/2412-6179-CO-1165.

**Acknowledgements**:

This work was funded by the Russian Science Foundation under project No. 22-22-00265 (Section “Theory”) and the RF Ministry of Science and Higher Education within a government project of the FSRC “Crystallography and Photonics” RAS (Section “Numerical simulation”).

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