# 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
Let us also write an intensity distribution formula in the focal plane for the initial field (1) [36]:  
\[I(r,\varphi,z=0)=\left|F_{x}\right|^{2}+\left|F_{y}\right|^{2}+\left|F_{z} \right|^{2}=J_{0,m}^{2}+J_{2,m+2}^{2}+2J_{1,m+1}^{2}. \tag{6}\]  
Now let us find the average characteristics of the light field in the focal plane: the energy (or power) of the entire beam \(\overline{I}=W\), full longitudinal energy flow \(\overline{P}_{z}\), full longitudinal SAM \(\overline{S}_{z}\) and full longitudinal OAM \(\overline{L}_{z}\). \[ \begin{array}{l}\overline{I} =\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}Irrd\varphi= \int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\left(J_{0,m}^{2}+J_{2,m+2}^{2} +2J_{1,m+1}^{2}\right)rdrd\varphi\\ =W_{0}+W_{2}+2W_{1}.\end{array}  \tag{7}\] \[ \begin{array}{l}\overline{P}_{z} =\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}P_{z}rdrd\varphi= \frac{1}{2}\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}rdrd\varphi\left(J_{0, m}^{2}-J_{2,m+2}^{2}\right)\\ =W_{0}-W_{2}=\overline{I}-2W_{2}-2W_{1}.\end{array} \] (8) \[ \begin{array}{l}\overline{S}_{z} =\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}S_{z}rdrd\varphi= \frac{1}{2}\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}rdrd\varphi\left(J_{0, m}^{2}-J_{2,m+2}^{2}\right)\\ =W_{0}-W_{2}=\overline{I}-2W_{2}-2W_{1}.\end{array} \] (9) \[ \begin{array}{l}\overline{L}_{z} =\int\limits_{0}^{\infty}rdr\int\limits_{0}^{2\pi}d\varphi L_{z}(r, \varphi)\\ =\int\limits_{0}^{\infty}rdr\int\limits_{0}^{2\pi}d\varphi\left(mJ_{0,m}^{2}+( m+2)J_{2,m+2}^{2}+2(m+1)J_{1,m+1}^{2}\right)=mW_{0}+(m+2)W_{2}+2(m+1)W_{1}.\end{array}  \tag{10}\]Expressions (7)-(10) are obtained based on the following integral from [39]:  
\[ \begin{array}{ll}W_{a}&=2\pi\int\limits_{0}^{\infty}\big{|}J_{\alpha,\beta}(r) \big{|}^{2}rdr\\ &=4\pi f^{2}\int\limits_{0}^{\theta_{0}}\sin^{2\alpha+1}\left(\frac{\theta}{2} \right)\cos^{5-2\alpha}\left(\frac{\theta}{2}\right)|B(\theta)|^{2}d\theta.\end{array}  \tag{11}\]  
It follows from (9) and (10) that the sum of SAM and OAM (angular momentum) is equal to the total energy of the beam \(\overline{I}\) multiplied by the topological charge \(m\) plus 1:  
\[\overline{S}_{z}+\overline{L}_{z}=(m+1)\overline{I} \tag{12}\]  
In order to understand how the average characteristics of the initial field (1) change at the focus, let us find these characteristics in the initial plane:  
\[\overline{S}_{z,0}=\ \overline{I},\ \overline{P}_{z,0}=\overline{I},\ \overline{L}_{z,0}=m\overline{I}. \tag{13}\]  
From a comparison of (12) and (13), it is seen that the sum of the longitudinal SAM and OAM for the wave (1) is preserved during focusing. The longitudinal SAM at the focus decreases by the amount \(\overline{S}_{z,0}-\overline{S}_{z}=2(W_{2}+W_{1})\) relative to the initial value. This decrease is clear; it is due to the spin-orbit conversion effect. Indeed, from a comparison of (10) and (13), it is seen that the magnitude of the longitudinal OAM at the focus increases by exactly the same value: \(\overline{L}_{z}-\overline{L}_{z,0}=2(W_{2}+W_{1})\). It is not yet clear why the longitudinal energy flow decreases by the same number of times, as follows from comparison of (8) and (13): \(\overline{P}_{z,0}-\overline{P}_{z}=2(W_{2}+W_{1})\). On the one hand, a part of the beam energy being in the initial plane at all, directed along the optical axis in the focal plane, is clear to propagate not only along it, but also in the transverse direction. However, if we average the crosscut projections of the energy flow vector (2) over the entire focal plane, we obtain zeros:  
\[ \begin{array}{ll}\overline{P}_{x}&=\int\limits_{0}^{\infty}\int\limits_{0}^ {2\pi}P_{x}rdrd\varphi=-\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\Omega( r)\sin\varphi drd\varphi=0,\\ \overline{P}_{y}&=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}P_{y}rdrd \varphi=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\Omega(r)\cos\varphi drd \varphi=0.\end{array}