# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 1 Introduction  
Calculation of the intensity distribution, Poynting vector (energy flow), spin and orbital angular momentum vectors in a sharp focus of laser light is an actual and interesting problem. At the strong focus, all six projections of the electric and magnetic vectors of the electromagnetic beam strength have approximately the same weight and must all be taken into account in the calculations. An adequate analytical description of the behavior of coherent monochromatic light at the focus is provided by the Richards-Wolf theory [1]. Based on this theory, many interesting results have been obtained characterizing the behavior of light in the area of a sharp focus. There are a lot of effects that have been obtained at the focus: spin-orbit conversion [2; 3; 4; 5; 6; 7], optical (photon) wheels [8; 9; 10; 11], polarized Mobius strips [12; 13; 14; 15; 16; 17], spin and orbital Hall effects [18; 19; 20; 21], optical magnetization [22; 23; 24], reverse energy flow [25; 26; 27; 28] and optical skyrmions [29; 30; 31]. To demonstrate different effects at the focus, laser radiation with different polarization states is used. Light with radial, azimuthal and cylindrical polarization in combination with the vortex phase [32; 33] and Poincare beams [34; 35] are used.  
In [36; 37], we analyzed in detail characteristics of laser beams with circular polarization at a sharp focus. Analytical expressions for projections of an energy flux vector (Umov-Poynting vector), a spin angular momentum (SAM) vector and for a longitudinal projection of the orbital angular momentum (OAM) at the strong focus were obtained. Similarexpressions in the strong focus of a vortex beam with cylindrical polarization we obtained in [38]. For different beam types (a circularly polarized optical vortex has a spin, and a vortex with cylindrical polarization has no spin), the same expression for the average energy flow along the optical axis in the focal plane was obtained. This expression shows that only a part of the initial light energy crosses the focal plane in the optical axis direction. It can be assumed that the rest of the energy is directed in the transverse direction. But the energy fluxes averaged over the beam cross section in the focal plane along two transverse Cartesian axes are equal to zero [36; 37; 38]. A question arises. If only a part of the initial light energy of the beam crosses the focal plane in the longitudinal direction, and there is no energy flow in the transverse direction, what happens with the remaining part of the initial beam energy in the focal plane?  
In this work, we show that, in the focal plane, in addition to the forward energy flow, there is a reverse energy flux, or toroidal energy flow, as well as an azimuthal energy flow (energy rotation in a circle), if the initial beam has a vortex (OAM beam), or there is a circular polarization in the initial plane. Calculations are based on the Richards-Wolf formalism and using numerical simulation. If the initial beam with only cylindrical polarization (without a vortex) is considered, then there is no rotating transverse energy flow in the focal plane, but there are two energy flows that compensate each other; one transverse flow propagates towards the optical axis, and the second moves away from the optical axis.