# Strongly Focused Circularly Polarized Optical Vortices Regulated by a Fractal Conical Lens
## 3 Numerical Simulations and Analysis  
In terms of the analytical expression derived above, we graphically illustrate the tight focusing characteristics of circularly polarized optical vortex in cases with and without a FCL. Figure 2a-c, respectively, plot the intensity distributions of \(x\)-polarized component, \(y\)-polarized component, and \(z\)-polarized component \(|\text{E}_{i}|^{2}\) (\(i=x,y,z\)) of a strongly focused RHC polarized Gaussian beam (\(\sigma_{x}=+1\), \(m=0\)) without a FCL. For a RHC polarized Gaussian beam (\(\sigma_{x}=+1\), \(m=0\)), it can be seen from Figure 2a-c that in the focal plane, both \(x\)-polarized component and \(y\)-polarized component present peak-centered intensity distributions, whereas the \(z\)-polarized component forms a donut-shaped configuration. Thus, the total intensity distribution forms an on-axis-maximum profile (Figure 2d). These intensity distributions are also clearly verified from the transverse line scans through the focus (see Figure 2e). It is of great interest to find from Figure 2f that the phase of the \(z\)-polarized component presents a helical structure, which indicates that through strongly focused optical system, the targeted light beam has become an optical vortex even though the incident light beam does not possess a vortex. This phenomenon occurs because, through a highly focused optical system, partial SAM of the incident circularly polarized light field could be transferred to OAM, which leads to a helical phase of the longitudinal component of the focal electric field. Figure 3 illustrates the tight focusing of a RHC polarized Gaussian beam (\(\sigma_{x}=+1\), m = 0) with a FCL of stage \(S=3\). Comparing Figure 3a,b with Figure 2a,b, one can obviously note that the transverse components have the same intensity distributions in cases with or without a FCL. Moreover, only a slight difference can be found in the magnitude of the longitudinal component \(|\text{E}_{z}|^{2}\) between Figures 2c and 3c. Thus, one may have an impression that the modulation effect of FCL on a tightly focused Gaussian beam is small and mainly reflected on the longitudinal component. However, the  
Figure 2: Calculated 2-D distributions \(|\text{E}|^{2}/|\text{E}^{2}_{\text{max}}\) of a strongly focused RHC polarized (\(\sigma_{x}=+1\)) optical vortex with charge \(m=0\), i.e., Gaussian beam, without a FCL: (**a**) \(x\)-polarized component, (**b**) \(y\)-polarized component, (**c**) \(z\)-polarized component, (**d**) total field, (**e**) line scan of focus, and (**f**) the helical phase structure of the \(z\)-polarized component of the focal electric field. following simulations show that a FCL of different stage \(S\) could significantly modulate the tight focusing intensity of circularly polarized optical vortex. Figure 4 depicts the strongly focused RHC polarized optical vortex (\(\sigma_{x}=+1\), \(m=1\)) with a FCL of stage \(S=3\). In this case, the \(x\)-polarized component, \(y\)-polarized component and \(z\)-polarized component all present donut-shaped intensity distributions. As a result, the total intensity demonstrates a dark-centered distribution. In addition, the magnitude of \(x\)-polarized component and \(y\)-polarized component are identical, and meanwhile the \(z\)-polarized component's magnitude becomes much larger when compared with Figure 3. This phenomena can be explained as follows: for the RHC polarized vortex (\(\sigma_{x}=+1\), \(m=1\)), in the focal plane, the double helical phase appears (where \(\exp[(m+1)i\varphi]=\exp(2i\varphi)\) in Equation (6)) when the directions of the helical phase and circular polarization are the same, and it can be verified from the phase helical phase distribution of \(z\)-polarized electric field in Figure 4f. In addition, the dark hollow radius of the longitudinal polarized component depends on the topological charge, therefore, the dark area in Figure 4c becomes much wider than that in Figure 3c.