# Spin-Orbital Transformation in a Tight Focus of an Optical Vortex with Circular Polarization
## 7 Simulation Results  
This section presents the results of numerical simulation of light field (1) focusing on different topological charges of incident beams. The simulation was performed by using the solution of the Richards-Wolf integral [35]. Focusing of light with a wavelength of 633 nm by aplanatic objectives was considered. This wavelength was chosen because it can be implemented in practice using a conventional helium-neon laser. As an example of the calculated patterns of intensity and SAM, the results of focusing for vortices with topological charges of \(m=0\) (no vortex) and \(m=1\) are presented. Figure 2 shows the intensity pattern and its individual components for \(m=0\). Patterns of the SAM vector components for the same topological charge (\(m=0\)) are presented in Figure 3. Figures 4 and 5 show the distribution of intensity and SAM for \(m=1\). Figure 6 shows the ratio \(\hat{S}_{z}/\hat{L}_{z}\) depending on the numerical aperture of the focusing lens. All figures are given for the numerical aperture of the focusing lens equal to NA = 0.95. Microobjectives with 100\(\times\) magnification have such a numerical aperture. A large numerical aperture was chosen for modeling because the effect of SOC is noticeable starting from a numerical aperture of 0.8 and reaches a maximum at a numerical aperture of 1. Figure 2: The distribution of intensity \(I\) (**a**) and its components \(I_{x}\) (**b**), \(I_{y}\) (**c**), \(I_{z}\) (**d**) in the focus of the beam with circular polarization (without vortex) for NA = 0.95. Figure 4: The distribution of intensity \(I\) (**a**) and its components \(I_{x}\) (**b**), \(I_{y}\) (**c**), \(I_{z}\) (**d**) in the focus of the optical vortex with unit topological charge and with circular polarization for NA = 0.95. Figure 3: The patterns of the SAM components \(S_{x}\) (**a**), \(S_{y}\) (**b**), \(S_{z}\) (**c**) in the focus of the beam with circular polarization (without vortex) for NA = 0.95. The comparison of Figures 2a and 3c shows that at \(m=0\), the intensity \(I\) and the lengthwise SAM projection \(S_{z}\) in the focal plane are almost the same in shape (bright round spot) and in magnitude \(I_{\max}\simeq S_{z,\max}\simeq 16\). This follows from Equations (3) and (4), since \(I\simeq S_{z}\simeq\gamma_{0,0}^{2}\), because it follows from (9) that \(\gamma_{0,0}^{2}\gg\gamma_{2,2}^{2}\), \(\gamma_{0,0}^{2}\gg 2\gamma_{1,1}^{2}\). The comparison of Figures 4a and 5c shows that at \(m=1\), the intensity \(I\) and the lengthwise SAM projection \(S_{z}\) in the focal plane have the form of a ring with approximately the same radius and size, although \(S_{z,\max}\) - 4 is slightly less than \(I_{\max}\) - 5. This follows from Equations (3) and (4): \(I=\gamma_{0,1}^{2}+\gamma_{2,3}^{2}+2\gamma_{1,2}^{2}\), \(S_{z}=\gamma_{0,1}^{2}-\gamma_{2,3}^{2}\), taking into account (9): \(\gamma_{0,1}^{2}\gg\gamma_{2,3}^{2}\), \(\gamma_{0,1}^{2}\gg 2\gamma_{1,2}^{2}\). Figure 5: The patterns of the SAM components \(S_{x}\) (**a**), \(S_{y}\) (**b**), \(S_{z}\) (**c**) in the focus of optical vortex with unit topological charge and with circular polarization for NA = 0.95. Figure 6: The ratio \(\hat{S}_{z}/\hat{L}_{z}\) in the focus of light field (1) at different numerical apertures of the focusing lens for beams (1) with different topological charges: \(m=0\) (without vortex, blue line 1), \(m=1\) (red line 2), \(m=-1\) (yellow line 3). The vertical dotted line drawn through the NA = 0.87. Figure 6 shows the ratio of SAM to OAM \(\hat{S}_{z}/\hat{L}_{z}\) in the focus of light field (1) at different numerical apertures of a spherical lens and for different topological charges of incident light: \(m=0\) (line 1), \(m=1\) (line 2), \(m=-1\) (line 3). In Section 4, specific values \(\hat{S}_{z}\) and \(\hat{L}_{z}\) were also obtained for the numerical aperture NA = 0.87. It is obtained that \(\hat{S}_{z}=W_{0}-W_{2}\approx 3W/4\), \(\hat{L}_{z}\approx W/4\) for \(m=0\), and \(\hat{S}_{z}\approx 3W/4\), \(\hat{L}_{z}=W+W/4\approx 5W/4\) for \(m=1\).