# Spin-Orbital Transformation in a Tight Focus of an Optical Vortex with Circular Polarization
## 6 Separate Measurement of the OAM and SAM in the Focus  
In reference [37], formulas are given that relate the force \(\overset{\rightarrow}{F}\) and the moment of the force \(\overset{\rightarrow}{T}\), which influence on a Rayleigh (smaller than wavelength) microparticle with a complex permittivity \(\varepsilon\), with the intensity \(I\), the canonical vector of the energy flow \(\overset{\rightarrow}{P}_{0}\), and the SAM vector \(\overset{\rightarrow}{S}\):  
\[\overset{\rightarrow}{F} =\frac{\mathrm{Re}(\varepsilon)}{4}\
abla I+\frac{\mathrm{Im}( \varepsilon)}{2}\overset{\rightarrow}{P}_{\emptyset}, \tag{20}\] \[\overset{\rightarrow}{T} =\frac{\mathrm{Im}(\varepsilon)}{2}\overset{\rightarrow}{S}.\]  
In (20), we use next notation:  
\[ \begin{array}{l}I=\left|E_{x}\right|^{2}+\left|E_{y}\right|^{2}+\left|E_{z} \right|^{2},\\ \overset{\rightarrow}{P}=\overset{\rightarrow}{P}_{0}+\overset{\rightarrow }{P}_{s}=\frac{\mathrm{Re}}{8\pi c}\left(\overset{\rightarrow}{E}^{*} \times\overset{\rightarrow}{H}\right)=\frac{\mathrm{Im}}{16\pi c}\left( \overset{\rightarrow}{E}^{*}\cdot(\
abla)\overset{\rightarrow}{E}\right)+ \frac{1}{2}\
abla\times\overset{\rightarrow}{S},\\ \overset{\rightarrow}{S}=\frac{\mathrm{Im}}{16\pi c}\left(\overset{ \rightarrow}{E}^{*}\times\overset{\rightarrow}{E}\right),\end{array}  \tag{21}\]  
where \(c\) is the speed of light in vacuum and \(\overset{\rightarrow}{P}_{s}\) is the spin flow. Therefore, the method of separate measurement of the SAM and OAM in the focal plane of the field (1) is as follows. It follows from (3) that at \(m=0\), there is an intensity maximum \(I(\rho=0)=\gamma_{0,0}^{2}\) in the focus center. The microparticle in focus will move to the center, where the intensity is maximal and the gradient is equal to zero. Due to the absorption of the particle (\(\mathrm{Im}\varepsilon\
eq 0\)), the moment of the force (20) effecting on the particle will rotate it around its center of mass and around the optical axis. The value of this moment of forces will be proportional to the lengthwise projection of the SAM vector: \(T_{z}\sim S_{z}\). At a large numerical aperture (close to unity), the focal spot is smaller than the wavelength and the particle captured by such a focal spot will rotate under the influence of the total longitudinal component of the SAM equal to \(\hat{S}_{z}=W_{0}-W_{2}\approx W/2\). Therefore, by measuring the speed of the particle rotation, it is possible to estimate the force moment, which will be proportional to the SAM \(\hat{S}_{z}\). To measure the transverse energy flux \(P_{\varphi}\), it is necessary to form a ring of light in the focus. It follows from (3) that at \(m=1\) the intensity on the optical axis will be zero in the focus, since \(I(\rho=0)=\gamma_{0,1}^{2}+\gamma_{2,3}^{2}+2\gamma_{1,2}^{2}=0\)This means that a light ring will form in the focus, because of the presence of an optical vortex with a topological charge \(m=1\). The particle in focus will shift to the ring due to the minimum intensity gradient on the ring, and the force \(F_{\theta}\sim P_{\theta}\) will shift this particle along the light ring. This force will be proportional to the transverse energy flow \(P_{\theta}\), which, in turn, is proportional to the longitudinal component of the angular momentum and the lengthwise projection of the OAM: \(J_{z}=L_{z}=\rho P_{\theta}\). It was shown in [36] that the distribution of the transverse flux in the focal plane of the field (1) is described by the expression \(P_{\theta}=\gamma_{1,m+1}(\gamma_{0,m}+\gamma_{2,m+2})\). The speed of the particle moving along a ring of small radius will be influenced not by the full longitudinal OAM, but by a partial one, since the particle is captured not by the entire circular trajectory, but only by its part. By experimental estimation of the value of the light ring, which falls on the particle (\(\gamma<1\)), it is possible to estimate the part of the total longitudinal OAM that is transferred to the particle \(\gamma\hat{L}_{z}=\gamma(2W-W_{0}+W_{2})\approx 3\gamma W/2\).