## 2 Theoretical Analysis of the near Field of an Ultra-Large-Scale Metasurface
### Calculation of Noisy Near-Field Metasurface
#### 2.2.2 Observation Surface Distance of 1.2 m  
When the observation surface is positioned at a distance of 1.2 m from the metasurface, the noise amplitude is set as \(amp=25\). The sampling surface dimensions in Figure 5a,b are \(1.407\) m \(\times\)\(1.407\) m, while those in Figure 5c-g are \(1.005\) m \(\times\)\(1.005\) m. In contrast to Figure 4, it is slightly closer to the metasurface and has a larger noise amplitude setting, whereas Figure 5 has a smaller noise amplitude setting. As depicted in Figure 5a,b, both the 3D and 2D amplitude diagrams reveal that the small metasurface is overwhelmed by noise. The phase diagram also indicates the excitation of the vortex wave with \(l=-5\) mode; however, due to significant noise interference, it becomes challenging to obtain an effective phase for this vortex wave on the submerged sampling surface area. The large metasurfaces shown in Figure 5c-e, with \(R=40\times p\), \(R=60\times p\), and \(R=80\times p\), respectively, exhibit good isolation of the electric field amplitudes from noise disturbances, as evident from both the 3D and 2D amplitude diagrams. The phase diagram further demonstrates the excitation of the vortex wave with \(l=-5\) mode for these large metasurfaces; moreover, as the metasurface size increases, the area of the vortex phase on the sampling surface gradually expands toward obtaining a vortex phase within a central quarter area (\(3.14\times R^{2}/4\)). The ultra-large-scale metasurfaces illustrated in Figure 5f,g generate multi-circle vortex wave amplitudes while effectively isolating them from any interfering noises. The phase diagram reveals the successful acquisition of vortex phases without any interference across all sampling surfaces for the ultra-large-scale metasurfaces shown in Figure 5f,g. Finally, in Figure 5, the OAM spectral mode exhibits high purity unaffected by Gaussian noise.  
Figure 4: Near-field metasurface simulation with noise interference and observation distance of 0.9 m. (**a**) \(R=20\times p\), \(R=0.2\) m, sampling interval = 0.007 m, and metasurface size of \(6.66\times\)\(6.66\lambda^{2}\); (**b**) \(R=28\times p\), \(R=0.28\) m, sampling interval = 0.007 m, and metasurface size of \(9.33\times\)\(9.33\lambda^{2}\); (**c**) \(R=40\times p\), \(R=0.4\) m, sampling interval = 0.005 m, and metasurface size of \(13.33\times 13.33\lambda^{2}\); (**d**) \(R=60\times p\), \(R=0.6\) m, sampling interval = 0.005 m, and metasurface size of \(20\times\)\(20\lambda^{2}\); (**e**) \(R=80\times p\), \(R=0.8\) m, sampling interval = 0.005 m, and metasurface size of \(26.66\times\)\(26.66\lambda^{2}\); (**f**) \(R=100\times p\), \(R=1\) m, sampling interval = 0.005 m, and metasurface size of \(33.33\times\)\(33.33\lambda^{2}\); (**g**) \(R=120\times p\), \(R=1.2\) m, sampling interval = 0.005 m, and metasurface size of \(40\times\)\(40\lambda_{0}{}^{2}\).  
Figure 5: Near-field metasurface simulation with noise interference and an observation distance of 1.2 m. (a) \(R=20\times p\), \(R=0.2\) m, sampling interval = 0.007 m, and metasurface size of \(6.66\times\)\(6.66\lambda^{2}\); (b) \(R=28\times p\), \(R=0.28\) m, sampling interval = 0.007 m, and metasurface size of \(9.33\times\)\(9.33\lambda^{2}\); (c) \(R=40\times p\), \(R=0.4\) m,  
sampling interval = 0.005 m, and super surface size of 13.33 \(\times\) 13.3330\({}^{2}\); (**d**) \(R\) = 60 \(\times\)\(p\), \(R\) = 0.6 m, sampling interval = 0.005 m, and metasurface size of 20 \(\times\) 20x0\({}^{2}\); (**e**) \(R\) = 80 \(\times\)\(p\), \(R\) = 0.8 m, sampling interval = 0.005 m, and metasurface size of 26.66 \(\times\) 26.66x0\({}^{2}\); (**f**) \(R\) = 100 \(\times\)\(p\), \(R\) = 1 m, sampling interval = 0.005 m, and metasurface size of 33.33 \(\times\) 33.330\({}^{2}\); (**g**) \(R\) = 120 \(\times\)\(p\), \(R\) = 1.2 m, sampling interval = 0.005 m, and metasurface size of 40 \(\times\) 40x0\({}^{2}\).