Tight focusing of light with mth-order circular-azimuthal polarization was investigated. This is a new type of inhomogeneous hybrid polarization that combines the properties of mth order cylindrical polarization and circular polarization. Using the Richards-Wolf formalism, we obtained analytical expressions in the focal spot for the projections of the electric and magnetic field, the intensity distribution, the projections of the Poynting vector, and the spin angular momentum. It was shown theoretically and numerically that at the focus, the intensity has 2(m+1) local maxima located on a circle centered on an on-axis intensity null. It was shown that 4m vortices of a transverse energy flow were produced at the focus, with their centers located between the local intensity maxima. It was also shown that in the focal plane, the transverse energy flow changes the direction of rotation 2(2m+1) times around the optical axis. It is interesting that the longitudinal projection of the spin angular momentum at the focus changes sign 4m times. In those areas of the focal plane where the transverse energy flow rotates counterclockwise, the longitudinal projection of the spin angular momentum is positive, and the polarization vector rotates counterclockwise in the focal plane. Conversely, if the energy flow rotates clockwise, the polarization vector rotates clockwise, and the longitudinal projection of the spin angular momentum is negative. Numerical simulations are in agreement with the theoretical investigation.
Spin angular momentum density and transverse energy flow of tightly focused kaleidoscope-structured vector optical fields