Tight focusing cylindrical vector beams with fractional order

By simulating tight focusing of vector beams with azimuthal polarization of fractional order 0 < m < 1 ( m = 1 is the azimuthal polarization; m = 0 is the linear polarization), it is shown that the shape of the intensity distribution in the focal spot changes from elliptical ( m = 0 ) to round ( m = 0.5 ) and ends with an annular ring ( m = 1 ). The shape of the distribution of the longitudinal component of the Poynting vector (energy flux) in the focal spot changes in a different way: from circular ( m = 0 ) to elliptical ( m = 0.5 ), and it ends in an annular ring ( m = 1 ). The diameter of the focal spot at full width at half-maximum for a beam with azimuthal polarization ( m = 1 ) with an optical vortex of the first order for a numerical aperture N A = 0.95 is 0.46 of the wavelength, and the diameter of the axial energy flux for linearly polarized light ( m = 0 ) is 0.45 of the wavelength. Therefore, the answers to the questions of whether the focal spot is round or elliptical and whether the focal spot is minimal with azimuthal polarization with a vortex or with linear polarization without a vortex depend on whether we are considering the intensity at the focus or the energy flow. In the second simulation, the effect of the deviation of the beam order from m = 2 (i.e., the case when the backflow is observed at the center of the focal spot) was investigated. It was shown that for integer values of the beam order, the transverse components of the Poynting vector are equal to zero, but for fractional values, they are not.