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Coincidence of the longitudinal components of the Poynting vector and the canonical energy in the sharp focus of an optical vortex with azimuthal polarization
V.V. Kotlyar1,2, S.S. Stafeev 1,2, E.V. Telegina 1,2

1 Samara National Research University, Moskovskoye Shosse 34, Samara, 443086, Russia;
2 Image Processing Systems Institute, NRC “Kurchatov Institute”,
Molodogvardeiskaya Str. 151, Samara, 443001, Russia

  Full text (PDF)

DOI: 10.18287/COJ1855

Article ID: 1855

Language: Russian

Abstract:
This paper theoretically demonstrates that, at the sharp focus of an azimuthally polarized optical vortex, the longitudinal component of the Poynting vector coincides with the longitudinal component of the canonical energy flux vector. This occurs because, during propagation and focusing, an azimuthally polarized optical vortex has no longitudinal component of the electric-field vector and, therefore, no longitudinal component of the spin flux vector. It is numerically demonstrated that, in and near the focal plane, there are subwavelength regions characterized by the negative longitudinal component of the canonical energy flux, suggesting that a reverse energy flow occurs in these regions. In these regions with reverse energy flow, the wave vector is directed oppositely to the direction of light propagation. The reverse energy flow occurs at the rings of zero intensity. Near the zero-intensity rings, the longitudinal projection of the local wave vector reaches large values (more than wavenumber).

Keywords:
tight focusing, Poynting vector, energy flow, canonical flow, azimuthal polarization, optical vortex.

Acknowledgements:
This work was funded by the Russian Science Foundation under project No. 26-12-00117 (theoretical part) and the RF Ministry of Science and Higher Education under a government project of FSRC “Crystallography and Photonics” RAS (numerical simulation).

Citation:
Kotlyar VV, Stafeev SS, Telegina EV. Coincidence of the longitudinal components of the Poynting vector and the canonical energy flow in the sharp focus of an optical vortex with azimuthal polarization. Computer Optics 2026; 50(5): 1855. doi: 10.18287/COJ1855.

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