# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 2 Spin and Energy Flows in the Focal Plane for a Circularly Polarized Optical Vortex
In the initial plane, there is only a longitudinal (axial and forward) energy flow and an axial projection of the SAM (spin). And this longitudinal energy flow and longitudinal spin density are equal to the power (total energy) of the vortex beam with right-handed circular polarization. That is, all photons in the initial plane propagate forward and \"rotate\" around their axis counterclockwise (the spin is +1), and each photon has angular momentum \((m+1)\hbar\) where \(\hbar\) is the Planck's constant. In the focal plane, both the total energy flow and the total SAM of the beam are preserved, but are redistributed so that a part of the energy flux propagates in the opposite direction, a part goes to form transverse rotation of energy in a circle in the beam cross section and a part of the total spin goes to form azimuthal spin of light rotating in the focal plane. Figure 1 schematically shows how the energy flow and spin density are redistributed at the beam focus (1). The deficiency of the previous arguments is that they are based on Equations (15) and (16), which are not derived strictly and are obtained intuitively. But there is another approach to justify the decrease in the longitudinal energy flux in the focal plane. It is based on exact formulas for projections of the electric and magnetic field strength vectors at a sharp focus for the initial field (1):  
\[ \begin{array}{l}F_{x,m}=\frac{\varphi^{-1}}{\sqrt{2}}e^{im\varphi}\left(J_{0,m}+ e^{2i\varphi}J_{2,m+2}\right),\\ F_{y,m}=\frac{\varphi^{2}}{\sqrt{2}}e^{im\varphi}\left(J_{0,m}-e^{2i\varphi}J_{2,m+2}\right),\\ F_{z,m}=-\sqrt{2}im^{\varphi}e^{(m+1)\varphi}J_{1,m+1},\\ H_{x,m}=\frac{-m^{2}}{\sqrt{2}}e^{im\varphi}\left(J_{0,m}+e^{2i\varphi}J_{2,m+ 2}\right),\\ H_{y,m}=\frac{\varphi^{-1}}{\sqrt{2}}e^{im\varphi}\left(J_{0,m}-e^{2i\varphi} J_{2,m+2}\right),\\ H_{z,m}=\sqrt{2}im^{+1}e^{i(m+1)\varphi}J_{1,m+1}.\end{array}  \tag{20}\]  
When the initial optical vortex \(\exp(im\varphi)\) propagates through a spherical lens, two additional optical vortices, \(\exp(i(m+2)\varphi)\) and \(\exp(i(m+1)\varphi)\), are seen from Equation (20) to be formed, where the expressions of (20) describe field components in the focus of the beam (1). That is, a part of the energy goes to the formation of these vortices. The last optical vortex has only longitudinal projections of the electric and magnetic fields \(F_{z,m}\) and \(H_{z,m}\), and therefore propagates along the focal plane and does not cross it. The power of this vortex is \(2W_{1}\), as follows from (11). Another beam with angular harmonic \(\exp(i(m+2)\varphi)\) is described by transverse field projections (20), but we can notice that it is included in the \(F_{x}\) field with a plus sign and in \(H_{y}\) with a minus sign. This optical vortex must propagate along the \(z\) axis so that the triple of vectors \(\mathbf{F_{x}}\), \(\mathbf{H_{y}}\) and \(\mathbf{k_{z}}\) is right. But, since the OAM harmonic \(\exp(i(m+2)\varphi)\) enters \(H_{y}\) with a minus sign, then \(k_{z}<0\) for it, that is, this beam intersects the focal plane in the backward direction. Therefore, the power \(2W_{2}\) in (8) is subtracted from the total beam power \(W\). It turns out that the total longitudinal energy flow in the focal plane equals, as in (8), \(\overline{P}_{2}=W-2W_{2}-2W_{1}\). Or, in other words, the longitudinal energy flux in the focal plane is equal to the direct energy flux, which is determined by the energy optical vortex \(\exp(im\varphi)\), minus the reverse energy flow, which is determined by the energy of the optical vortex \(\exp(i(m+2)\varphi)\): \(\overline{P}_{z}=W_{0}-W_{2}\). Figure 1: The distribution of photons in the incident plane with a spin of +1 (right-handed circular polarization), which all propagate along the optical axis, is schematically shown. In the focal plane, some photons propagate in the opposite direction and have a spin of \(-1\), and some photons rotate along a closed trajectory, creating an azimuthal energy flow and an azimuthal spin flow.