## 3 Results  
### Analysis and Simulation  
The wavefront sampling of DM phase modulation is insufficient, and the detection capability of the OAMs is low, especially for higher-order OAMs. Here, we give explanations  
Figure 1: Comparison of the OSM and DM for detection and separation of the different OAMs’ modes at 633 nm. (**a**). PMT diagram of LCP vortex light passing through a phase plate when \(l_{0}\) = 200, \(r_{i}\) = 50 \(\mu\)m, and \(r_{o}\) = 80 \(\mu\)m. (**b**). OSM (\(l_{0}\) = 200) simulation results of the OAMs at \(l\) = \(-\)620 to \(l\) = 620, \(\Delta l\) = 10. (**c**). Simulation results of DM (\(l_{0}\) = 200) with period \(p\) = 600 nm and under the same conditions as (**b**). According to the square result, the detection fails when \(\phi_{l}\) exceeds \(\pm\)1.36 rad. (**d**). Intensity curves along the white dotted circle line in (**b**,**c**).  
from theoretical derivation. If the phase coverage range of a single rectangular nanocolumn structure is \(\frac{2\pi}{n}\) and the period is \(p\), the difference can be expressed as:  
\[|\Phi\left(s2\right)-\Phi\left(s1\right)|=\frac{l_{0}}{2s_{0}^{2}}\Big{|}s_{2}^{ 2}-s_{1}^{2}\Big{|}=\frac{2\pi}{n},s_{2}-s_{1}>p \tag{10}\]  
From Equation (10), it can be deduced that:  
\[|s_{1}|<\frac{2\pi s_{0}^{2}}{nl_{0}p}-\frac{p}{2} \tag{11}\]  
According to Formulas (4) and (11), it can be expressed as:  
\[|\phi_{\max}^{\prime}|<\frac{2\pi s_{0}}{nl_{0}p}-\frac{p}{2s_{0}} \tag{12}\]  
When \(|\phi|>|\phi_{\max}^{\prime}|\), the topology detection cannot be realized by the structure.  
As shown in Figure 1a, the vortex passes through the metasurface with ideal phase distribution and form focusing spots at corresponding positions in the focal plane. The focus spots are distributed in a circular direction, and the course rotates clockwise. Then, it can be inferred that the whole optical system is relatively simple, and the differentiation of the different OAM modes is also relatively simple and convenient. The topological charge of incident light can be inferred only by comparing the location of the focusing light spot.  
Here, the differences between the OSM and DM in sorting OAMs are further studied. The separation of LCP vortex light was calculated through the OSM and DM phase plates with \(l_{0}=200\). Then, for a better comparison, a numerical simulation of the OSM and DM at \(l=-620\) to \(l=620\), \(\Delta l=10\) was conducted here, and the simulation results were statistically sorted out to form Figure 1b,c. As can be seen in Figure 1b,c, the number of the OAM modes detected by the OSM is much larger than that detected by DM structure. Figure 1c indicates the simulation results are consistent with Equation (12), i.e., according to the square result, when \(\phi_{l}\) exceeds \(\pm 1.36\) radians, OAMs detection fails using the DM structure. However, due to the advantages of its structure, the OAMs' mode can be effectively detected by the OSM structure when DM detection fails, which significantly improves the detection range of the OAMs. Furthermore, in the same detection range, the OSM can also generate stronger focusing spots than DM, as shown in Figure 1d (normalization is performed here for the convenience of comparison). According to the above information, the OSM has higher efficiency than DM for PTM, which makes it have a larger mode resolution range for OAMs, and improves the practical application value in related fields.