Backward flow of energy for an optical vortex with arbitrary integer topological charge

We analyze the sharp focusing of an arbitrary optical vortex with the integer topological charge m and circular polarization in an aplanatic optical system. Explicit formulas to describe all projections of the electric and magnetic fields near the focal spot are derived. Expressions for the near-focus intensity (energy density) and energy flow (projections of the Pointing vector) are also derived. The expressions derived suggest that for a left-hand circularly polarized optical vortex with $m > 2$, the on-axis backward flow is equal to zero, growing in the absolute value as a power $2(m-2)$ of the radial coordinate. These relations also show that upon the negative propagation, the energy flow rotates around the optical axis.