## 2 Theory and Methods  
The OSM is an optimized dielectric nanostructure of catenary [39; 40]. Compared with the DM, the OSM can achieve continuous wavefront sampling in 0-2\(\pi\), with improved diffraction efficiency and continuous phase distribution in a wider spectral band [38]. Here, based on the photonic spin hall effect [41] and catenary optics [22], an OSM structure was designed. It can transform different vortices in a large mode space into focusing modes of different azimuth coordinates in the transverse plane by photon momentum transformation (PMT), thus, OAMs can be determined through a single-shot measurement. PMT is an optical transformation that transforms the received OAMs' mode to the focusing modes on the focal plane with different angular positions [35; 37], which can be expressed as:  
\[M(r,\phi)=g(r)\exp(l_{0}\phi^{2}/2) \tag{1}\]  
where:  
\[ \begin{cases}g(r)=1&r_{i}\leq r\leq r_{o}\\ g(r)=0&\text{otherwise}\end{cases}  \tag{2}\]\[\Phi(\phi)=\frac{l_{0}}{2}\phi^{2} \tag{3}\]  
where \(M\) is the phase mask, \(l_{0}\) is the quadratic phase coefficient, \(\phi\) is the azimuthal coordinate defined as \(\tan^{-1}\) (y/x), and \(r_{i}\) and \(r_{o}\) represent the inner and outer radius of the ring phase mask. By flattening the polar coordinates into Cartesian coordinates, the azimuthal-quadratic phase contour can be converted into a cylindrical quadratic phase contour. \(\phi\) can also be expressed as:  
\[\phi=\frac{s}{s_{0}} \tag{4}\]  
where \(s_{0}=\pi r_{0}\) and \(r_{0}=(r_{i}+r_{o})/2\), respectively, and \(s\) is the focusing position of the detection position from the origin position [37]. Then, after the vortex light with topological charge \(l\) acts on the OSM, the phase distribution of the outgoing light is as follows:  
\[\Phi(\phi)+l\phi=\frac{l_{0}}{2}(\frac{s}{s_{0}})^{2}+l\frac{s}{s_{0}}=\frac{l_ {0}}{2s_{0}^{2}}(s+\frac{ls_{0}}{l_{0}})^{2}-\frac{l^{2}}{2l_{0}}=\Phi(\phi^{ \prime})-\frac{l^{2}}{2l_{0}} \tag{5}\]  
where \(\phi^{\prime}\) denotes a new azimuthal coordinate. Neglecting the azimuth-independent term on the right-hand side of the equation, we find that there is only a mode-dependent rotation through the PMT:  
\[s=-\frac{ls_{0}}{l_{0}} \tag{6}\]  
According to Equation (4), the theoretical focus \(\phi_{l}\) can be expressed as:  
\[\phi_{l}=-\frac{l}{l_{0}},\;\phi_{l}\in[-\pi,\;\pi] \tag{7}\]  
Here, \(\phi_{l}\) is the radian corresponding to the focus and the origin (the focus position when the topological charge is zero) of the vortex light with the topological charge \(l\) passing through the OSM. Then, the focusing spot is symmetric around the azimuth coordinate \(\phi=0\). As the topological charge of the vortex increases from zero to the positive and negative ends, the focusing mode will take the zero point as the origin and rotate to both ends with an azimuth angle spaced \(1/l_{0}\) apart. By recording the azimuth coordinates of the focusing mode, the received OAM mode can be determined as follows. \[l=-l_{0}\phi_{l},\;\phi_{l}\in[-\pi,\;\pi] \tag{8}\]  
According to equation \(f=\frac{k_{0}r_{0}^{2}}{l_{0}}\)[37], where \(k_{0}\) is the wavenumber in free space, \(r_{0}=(r_{i}+r_{o})/2\), and the focal length is independent of the topological charge of the incident OAM mode. Therefore, different OAM modes can be detected in the same transverse plane of the focal length. According to Equation (4), the detection of vortex optical topological charge is correlated with \(l_{0}\), and \(l_{0}\) is positively correlated with the phase gradient that the designed metasurface can accommodate. As shown in Figure 1a, the \(2\pi\) periodic phase distribution within the metasurface changes rapidly from right to left, reducing the structural space that accommodates \(2\pi\) phase adjustment. To ensure the phase control ability of DM, it is necessary for the metasurface retention space to accommodate sufficient nanopillars for \(2\pi\) phase adjustment, which limits the maximum \(l_{0}\) and the maximum detectable OAMs.