# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 5 Conclusions  
With sharp focusing of an optical vortex with topological charge \(m\) and right-handed circular polarization, because of the spin-orbit conversion, a part of the total longitudinal spin \(S_{z}\) equal to the energy \(2(W_{1}+W_{2})\), as seen from analytical expressions (7)-(10) obtained in the work, goes into the longitudinal OAM \(L_{z}\). In this case, the sum of the total longitudinal SAM and OAM is conserved and equal to \((m+1)W\), where \(W\) is the total energy of the beam. And exactly the same part of the energy \(2(W_{1}+W_{2})\) transforms from the total energy flow to the azimuthal energy flow (transverse Poynting vector) at the focus. Although there is some contradiction here, it is believed that the spin of light does not transfer energy [41]. Then how does circularly polarized light (even without an optical vortex) form a focal spot with an azimuthal energy flow in it? In [41], the authors compare the canonical energy flux and the Poynting vector. The Poynting vector can be represented as the sum of two vectors: the orbital flux (canonical energy flux) and the spin flux. The spin flux is a rotor from the spin angular momentum. Since the divergence from the rotor is zero, the spin flux does not participate in the energy transfer, although it does participate in the formation of the electromagnetic field distribution. For example, the presence of a backward energy flow (negative Poynting vector) is simply explained by the fact that the spin flow is negative and greater in magnitude than the orbital flow. When, using a numerical solution (FDTD method) of [PERSON]'s equations, we calculate the amplitudes of the magnetic and electric fields in the reverse flow region, the vectors \(\mathbf{F}\) (electric), \(\mathbf{H}\) (magnetic) and \(\mathbf{k}\) (wave vector) form the right-handed triple of vectors if \(k_{z}<0\). The Poynting vector at a sharp focus is much simpler to calculate than the canonical energy flux. And, therefore, many vector problems at a sharp focus are easier to solve using the Poynting vector. But the main thing is that both energy flows (canonical flow and Poynting vector) satisfy the differential and integral laws of energy conservation. Therefore, the results obtained on their basis are equal if we consider the average characteristics of energy flows when the energy flow crosses closed surfaces. In this work, we considered just such an averaged energy flow intersecting the initial and the focal planes. And it is shown that the total energy flow in the incident plane (longitudinal only) is equal to the total energy flow in the focal plane, if we take into account the reverse energy flow and transverse flow. Note that, without taking into account the reverse flow near the focus, the energy balance cannot be obtained. Let us also note one feature associated with the direction of the energy flow around the optical axis in the focal plane when focusing the optical vortex. It is known that, in scalar vortex beams near the center of the optical vortex, the azimuthal energy flux tends towards infinity as the radial variable tends towards zero (towards the center of the optical vortex) [41]. But, with sharp focusing of a circularly polarized vortex beam (due to the presence of a longitudinal component of the field), the azimuthal energy flux tends towards zero as it approaches the center of the optical vortex. This follows from the comparison of expressions (2) and (6). The azimuthal energy flux in the center (for \(kr<<1\)is proportional to \(\Omega(r)\sim(kr)^{2m+1}\), and the intensity is proportional to \(I(r)\sim(kr)^{2m}\), so their ratio linearly tends towards zero: \(\Omega(r)/I(r)\sim(kr)\). It is interesting to note that an optical vortex with a topological charge \((m+2)\), which is present in (20), was experimentally discovered in [4] in the sharp focus of the optical vortex as a result of an increase in the radius of the circular trajectory when the microparticle rotates.