# 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
 \tag{14}\]  
Integrals in (14) are equal to zero since integrals of \(\sin\varphi\) and \(\cos\varphi\) over the azimuthal angle \(\varphi\) over an integer period are equal to zero. That is, there is a transverse rotation of the energy flow in the focal plane (2), but, on average, the energy flow along the transverse axes is zero. But, although there is no averaged crosscut energy flow in the focal plane, because the energy still rotates along a circular path, the averaged value of the azimuthal rotating energy flow is nonzero and should be equal to the following expression:  
\[\overline{P}_{\varphi}=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}P_{ \varphi}rdrd\varphi=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\Omega(r)rdrd\varphi. \tag{15}\]  
It is interesting that, when \(m=0\) (there is no optical vortex), the azimuthal energy flux will remain unchanged and will be equal to \(P_{\varphi,m=0}=\Omega(r)=J_{1,1}(J_{0,0}+J_{2,2})\), and the full azimuthal energy flow (15) will not be changed since it is formed by the spin-orbit conversion. The azimuthal energy flow in the focal plane also has a crosscut SAM; the azimuthal projection of the spin density vector (4) is directed tangentially to any circle in the focal plane with the center lying on the optical axis:  
\[\overline{S}_{\varphi}=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}S_{\varphi }rdrd\varphi=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\Omega(r)rdrd\varphi. \tag{16}\]  
That is, for light that rotates in a circle in the focal plane, the polarization vector also rotates in a plane perpendicular to the vector \(\mathbf{S}_{\varphi}\). If \(\Omega(r)>0\) in (4), then the polarization vector rotates counterclockwise. Now we see that the initial longitudinal energy flow at the focus is preserved, but redistributed; a part of it goes to the rotating transverse energy flow:  
\[\overline{P}_{z,0}=\overline{P}_{z}\ +2(W_{2}+W_{1})=\overline{P}_{z}+\overline{P}_ {\varphi} \tag{17}\]  
Note that the total energy flow, which propagates along the optical axis (8), consists of two terms, \(\overline{P}_{z}=W_{0}-W_{2}\). Both terms are positive, and the second is always less than the first [39], but the second term is included in the expression with a minus sign. The physical meaning of this term is that this part of the energy flow propagates along the \(z\) axis, but in the opposite direction. That is, the term \(W_{2}\) describes the total reverse energy flow in the focal plane. Thus, in (17), we can distinguish not two, but three terms that have a certain physical meaning:  
\[\overline{P}_{z,0}=W=\overline{P}_{z+}+\overline{P}_{z-}+\overline{P}_{\varphi }=W_{0}-W_{2}+2(W_{2}+W_{1}). \tag{18}\]  
In (18), \(\overline{P}_{z+}=W_{0},\ \overline{P}_{z-}=-W_{2}\). It was previously shown for the case of an optical vortex with linear polarization that, with a numerical aperture NA = 0.95, only a half of the initial beam energy intersects the focal plane in the positive direction of the optical axis per unit time [40]. Similarly, the initial longitudinal spin decreases, but does not disappear, and is redistributed so that a part of it changes sign, and a part goes to the azimuthal spin of the transverse rotating energy flow:  
\[\overline{S}_{z,0}=W=W_{0}-W_{2}+2(W_{2}+W_{1})=\overline{S}_{z+}+\overline{S }_{z-}+\overline{S}_{\varphi} \tag{19}\]  
A part of the light flow in the focal plane has changed its spin sign from positive to negative, \(\overline{S}_{z-}=-W_{2}\), because circularly polarized light with right-handed polarization, which propagates in the opposite direction to the optical axis, appears to an observer as left-handedly circularly polarized light. And, along the positive direction of the \(z\) axis, only a part of the light with right-handed circular polarization and SAM equal to \(\overline{S}_{z+}=W_{0}\) propagates. These last three equations, Equations (17)-(19), are the main result of this work. The general picture can be presented as follows.