## 3 Results
### Model Establishment and Simulation  
Then, an \(l_{0}=200\) OSM was designed to ensure the ease of production, and the left circular polarization (LCP) OAM was simulated and sequenced at wavelength 633 nm. Figure 2a illustrates the three-dimensional structure of the OSM. The \(\Lambda\) of each column of the catenary in the OSM decreases gradually from right to left. Considering the detection ability of the OSM and the feasibility of the device fabrication, the \(l_{0}\) was set to 200, and the corresponding minimum \(\Lambda\) value of the metasurface was about 300 nm. The OSM was designed as a streamlined array with H = 600 nm and interval \(d_{1}\) = 700 nm, where \(w_{1}=100\) nm was the optimized streamlined catenary. To ensure the feasibility of manufacturing, the streamline was truncated by \(d_{2}=0.75\Lambda\), and vertical bars of width \(w_{2}=100\) nm were added to the truncated region to increase phase continuity and amplitude uniformity. Furthermore, a ring OSM with \(r_{1}=50\) um and \(r_{0}=80\) um was selected as an example of a proof-of-concept demonstration. The opening of the streamline catenary in the upper semicircle of the metasurface was outward, while the opening of the lower semicircle was inward. Then, the streamline catenary \(\Lambda\) gradually decreases with the circumference from right to left, and the \(\Lambda\) position reaches the minimum at the left radius of the inner circle. In addition, TiO\({}_{2}\) was selected as the constituent material of the OSM due to its high refractive index and low loss at visible frequency, and it realized this larger aspect ratio processing in metalenses at visible wavelengths [42], meta-optics [43], and spin-decoupled metasurfaces [37]. Then, Figure 2b shows the ordering of the OAMs by PTM. The designed OSM transforms vortex beams with different topological charges into circular focusing modes in the transverse focal plane by Pancharatnam-Berry phase regulation. Thus, OAMs can be determined through a single measurement, giving them simple and fast sorting capabilities that may find potential applications in quantum communication. Then, FDTD simulation was used to simulate the designed OSM, and the simulation results are shown in Figure 3a (the LCP is filtered, and the RCP is retained to obtain a sharper focused image). Here, the FDTD simulation results of the different OAMs with topological charges between \(-600\) and \(550\) through the PMT were shown, where the adjacent interval \(\Delta l\) was \(50\). According to the simulation results, it can be seen that all the focusing modes under the same focal length are located on the same circumference, and the spot position rotates clockwise as the topological charge increases from negative to positive. Moreover, when the topological charge is in the range of \(1=\pm 2.5l_{0}\), the focus spot is double-arc cross-focusing, and the intensity of the focus spot is stronger. Then, when the topological charge changes in the range of \(\pm 2.5l_{0}\) \(\rightarrow\pm 3.14l_{0}\), the focus gradually becomes a single arc, and the intensity gradually decreases. However, the position of the focal spot  
Figure 2: The design of the OSM and PTM detection diagram. (**a**). Schematic diagram of the OSM with SiO\({}_{2}\) substrate. The OSM graphic structure consists of TiO\({}_{2}\) streamline arrays with H = 600 nm, \(d_{1}\) = 700 nm, \(w_{1}\) = 100 nm, \(r_{1}\) = 50 \(\upmu\)m, and \(r_{0}\) = 80 \(\upmu\)m. The streamlines are truncated (\(d_{2}\) = 0.75\(\upLambda\)) to guarantee the feasibility of fabrication. The vertical structure with width \(w_{2}\) = 100 nm was added in the truncated region to improve the phase continuity and amplitude uniformity. (**b**). Schematic diagram of the OSM detection, OAMs sequencing by spin-controlled PMT. Vortex beams with different spins are converted into circular focusing modes on the screen in the transverse focal plane with topological charge-dependent azimuth rotation.