## 7 Simulation  
The simulation is carried out utilizing a Richards-Wolf transformation (RW) [34]. The numerical aperture of the spherical lens is assumed to be NA = 0.95, that is, the field is limited by an aperture with a radius of 4 \(\upmu\)m, and focal length is\(f=1.31\upmu\)m. The wavelength \(\lambda\) is 532 nm, the incident wave is a Gaussian beam with the radius \(\sigma=1.33\upmu\)m, the size of the initial and output fields is \(8\times 8\upmu\)m, \(400\times 400\) points, and \(\alpha=0\). The simulation is obtained at a distance from \  
ZP, but are not calculated inside it. From Figure 1a, the SAM before the lens is seen to be equal to 1, while the OAM is equal to 0, since \(n=0\). After the lens, the SAM decreases and becomes equal to about 0.76, and the OAM increases up to 0.24. The sum of the SAM and OAM remains unchanged and is equal to 1. Such behavior of the SAM and OAM occurs in accordance with Equations (10), (11), (19) and (21) at \(\alpha=0\). If the topological charge of the initial field (1) is 1, then both the SAM and OAM before the lens are equal to 1 (Figure 1b). Just after the diffractive lens, due to the spin-orbit conversion, the SAM decreases to 0.76, while the OAM increases and becomes equal to 1.24. The sum of them before and after the lens remains unchanged and is equal to 2. This result confirms the correctness of Equation (20). Figure 2 shows the change in the normalized total longitudinal SAM and OAM after the lens depending on \(\alpha\) for the initial light field (1) at \(n=0\) (a) and \(n=1\) (b). From Figure 2a, at \(\alpha=0\) (circular polarization) the SAM and OAM after the lens is seen to be the same as in Figure 1a. As \(\alpha\) increases, the initial circular polarization becomes elliptical and, therefore, the initial SAM decreases (curve 1) and the part of the SAM that converts into the OAM decreases (curve 2). Additionally, when \(\alpha=1\) (linear polarization) both the SAM and OAM after the lens (and before the lens) are equal to zero. The behavior of the SAM and OAM after the lens for an optical vortex (1) with \(n=1\) is shown in Figure 2b. From it the maximum SAM for the initial field (1) with circular polarization (\(\alpha=0\)) is seen to be equal to 0.76. Furthermore, the minimal value of the SAM is zero, for the field with linear polarization (\(\  
whereas the initial SAM is equal to 0.6. That is, because of the spin-orbit conversion, only a small part of the SAM (about 0.1) transferred into the OAM. From Figure 3, the minimum value of the SAM after the lens is seen to be approximately 0.35 for a numerical aperture close to 1. Therefore, the maximum value of 0.25 is converted from the SAM into the OAM. As the numerical aperture decreases, curve 1 in Figure 3 tends to a value of 0.6, i.e., to the initial SAM value before the lens. That is, it follows from Figure 3 that the effect of the spin-orbit conversion in a spherical lens with a numerical aperture less than 0.7 can be neglected. Shown in Figure 4 are the intensity \(I\) (first and third rows) and the longitudinal SAM \(S_{z}\) (second and fourth rows) for different values of the parameter \(\alpha\): 0 (first column), 0.5 (second column), 0.75 (third column), 0.9 (fourth column), and 1 (fifth column). The topological charges are \(n=3\) (first and second rows), \(n=5\) (third and fourth rows). It is seen from Figure 4 that, according to the theory (6)-(8), the distribution of the longitudinal SAM is radially symmetric (red ring in the first and second columns) in the focus of the field (1) for a small parameter \(\alpha\). As the parameter \(\alpha\) increases, the red ring in Figure 4 becomes an ellipse (second and third column), and when \(\alpha\) is close to 1, one part  
Figure 4: Intensity distributions \(I\) (**a–e,k–o**) and longitudinal SAM \(S_{z}\) (**f–j,p–t**) calculated at the beam focus (1) for different \(\alpha\): 0 (**a,f,k,p**), 0.5 (**b,g,l,q**), 0.75 (**c,h,m,r**), 0.9 (**d,i,n,s**), and 1 (**e,j,o,t**). The topological charges are \(n=3\) (**a–j**), \(n=5\) (**k–t**).