# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 3 Spin and Energy Flow in the Focal Plane for an Optical Vortex with Cylindrical Polarization  
In this section, we show that a similar redistribution of energy and spin fluxes at the focus also occurs for another initial light field, namely, an \(n\)-order cylindrical vector beam, and with an embedded vortex beam with a topological charge \(m\). The Jones vector in the initial plane for a cylindrical vector beam with an optical vortex has the following form:  
\[\mathbf{F}_{m,n}(\theta,\varphi)=B(\theta)\exp(im\varphi)\left( \begin{array}[] {c}\cos n\varphi\\ \sin n\varphi\end{array} \right), \tag{21}\]  
where angles \(\theta\) and \(\varphi\) are defined after (1). Projections of the energy flow vector in the focal plane can be written as follows [38]:  
\[ \begin{array}{l}P_{r}=0,\\ P_{\varphi}=\Omega(r),\\ P_{z}=\frac{1}{2}\left(l_{0,m+n}^{2}+j_{0,m-n}^{2}-j_{2,m+n-2}^{2}-j_{2,m-n+2 }^{2}\right),\\ \Omega(r)\mathbf{=}\int_{1,m+n-1}(J_{0,m+n}+J_{2,m+n-2})\\ \quad+I_{1,m-n+1}(J_{0,m-n}+J_{2,m-n+2}).\end{array}  \tag{22}\]  
It is clear from (22) that there is an azimuthal energy flow in the focal plane of the initial field (21). Moreover, the energy rotates counterclockwise. The expression for the longitudinal component of the energy flow in (22) has terms with a minus sign, that is, at certain radii, the energy flow can be directed in the backward direction (in the negative direction along the \(z\) axis). The expression for the SAM vector longitudinal projection in the focal plane can also be found in [38]:  
\[ \begin{array}{l}S_{z}\mathbf{=}\frac{1}{2}\Big{[}l_{0,m-n}^{2}-l_{0,m+n}^{2} +l_{2,m+n-2}^{2}-l_{2,m-n+2}^{2}\\ \quad+2(-1)^{n}\cos(2(n-1)\varphi)\left(I_{0,m-n}I_{2,m+n-2}-I_{0,m+n}I_{2,m-n+ 2}\right)].\end{array}  \tag{23}\]  
It is seen from (23) that, although the field (21) does not have a longitudinal projection of the SAM in the initial plane (it is equal to zero), the longitudinal projection of the SAM in the focal plane is not zero, and changes its sign when the azimuthal angle \(\varphi\) changes. That is, the polarization at each point in the initial plane is linear, although non-uniform. When going around a circle which is centered on the coordinate center, the direction of the vector of polarization makes \(n\) full turns. And, in the focal plane, areas with different signs of the longitudinal SAM projection are formed (23). That is, the spin Hall effect takes place. When going along a certain circle which is centered on the optical axis in the focal plane, the longitudinal SAM sign (the spin sign) changes \(4(n-1)\) times. The distribution of intensity in the focal plane for the initial beam (21) is as follows:  
\[ \begin{array}{l}I=\frac{1}{2}\Big{(}l_{0,m+n}^{2}+l_{0,m-n}^{2}+l_{2,m+n-2}^{ 2}+l_{2,m-n+2}^{2}\Big{)}+l_{1,m+n-1}^{2}+l_{1,m-n+1}^{2}+\\ (-1)^{n+1}\cos(2(n-1)\varphi)(I_{0,m+n}I_{2,m-n+2}+l_{0,m-n}I_{2,m+n-2}+2I_{1, m+n-1}I_{1,m-n+1}).\end{array}  \tag{24}\]  
The intensity in the focal plane is seen from (24) not to have circular symmetry, except for in the case of \(n=1\). Moreover, the intensity distribution (24) has \(2(n-1)\) side lobes lying on a circle of a certain radius with a center on the optical axis. Next, using (22)-(24), we find the averaged values of the energy (or power) of the entire beam, the total longitudinal energy flux \(\overline{P}_{z}\) and the total longitudinal SAM \(\overline{S}_{z}\):\[ \begin{split}\overline{I}=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} trdrd\varphi=&\frac{1}{2}\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} \left(\dot{J}_{0,m+n}^{2}+\dot{J}_{0,m-n}^{2}+\dot{J}_{2,m+n-2}^{2}+\dot{J}_{0,m -n+2}^{2}\right)rdrd\varphi+\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} \left(\dot{I}_{1,m+n-1}^{2}+\dot{J}_{1,m-n+1}^{2}\right)rdrd\varphi\\ +&(-1)^{n+1}\int\limits_{0}^{\infty}\int\limits_{0}^{2 \pi}\cos(2(n-1)\varphi)(J_{0,m+n}\dot{I}_{2,m-n+2}+\dot{J}_{0,m-n}\dot{I}_{2,m+ n-2}-2\dot{I}_{1,m+n-1})rdrd\varphi=W_{0}+W_{2}+2W_{1}=W.\end{split}  \tag{25}\]  
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