## 2 Theoretical Analysis of the near Field of an Ultra-Large-Scale Metasurface
### Calculation of a Noiseless Near-Field Metasurface  
The calculation program for determining the near-field of the array was designed according to Figure 1. In this interim design, vortex wave modes with \(l=-5\) were generated at a calculated frequency of 10 GHz and a wavelength of 0.03 m (\(\lambda_{0}\)). The observation distance was set at both 0.9 m and 1.2 m. To verify the mode purity, the OAM spectrum can be obtained by expanding the vortex wave mode on the electric field of the observed surface according to Equation (1). \[A_{l}=\frac{1}{2\pi}\int\limits_{0}^{2\pi}E(\varphi)e^{-jl\varphi}d\varphi \tag{1}\]  
The normalized amplitude, phase, and OAM spectrum of vortex waves observed on the observation surface with different sizes at an observation distance of 0.9 m are depicted in Figure 2. A total of 201 \(\times\) 201 points were sampled on the observation surface. In Figure 2a, the sampling interval is 0.007 m and the period of the unit cell is 10 mm, resulting in a sampling surface size of 1.407 m \(\times\) 1.407 m. On the other hand, in Figure 2b-d, the sampling interval is reduced to 0.005 m, leading to a smaller sampling surface size of 1.005 m \(\times\) 1.005 m due to the better centering capabilities of larger arrays compared to smaller arrays. For example, when \(R\) = 20 \(\times\)\(p\), it requires a larger sampling surface for displaying generated vortex waves. The small metasurface \(R\) = 20 \(\times\)\(p\) in Figure 2a exhibits periodic fluctuations in both the three-dimensional (3D) and two-dimensional (2D) OAM amplitude graphs. The phase diagram reveals a vortex phase with multiple turns, resulting in the generation of sidelobes for the vortex beam. The OAM spectrum depicted in Figure 2a demonstrates the generation of a high-mode purity vortex wave with \(l=-5\). The OAM amplitude graphs in Figure 2b-d demonstrate that as the metasurface size increases (\(R\) = 40 \(\times\)\(p\), \(R\) = 80 \(\times\)\(p\), \(R\) = 120 \(\times\)\(p\)), the edges of both the 3D and 2D patterns exhibit ideal circular shapes. Moreover, the energy distribution converges toward the center of the metasurface while gradually reducing the radius of the resulting OAM beam. Simultaneously, there is an increase in the normalized amplitude of vortex waves. Additionally, enlarging the metasurface size leads to a decrease in phase cycles and consequently reduces the sidelobes. Furthermore, Figure 2b-d illustrate that with increasing metasurface size, there is a gradual improvement in purity within the calculated OAM spectral patterns, and specifically for Figure 2c,d models, where the purity approaches close to 98%. The normalized amplitude, phase, and OAM spectrum of the vortex waves observed on the observation surface with different metasurface sizes are depicted in Figure 3 for an observation distance of 1.2 m. A total of 201 \(\times\) 201 points are sampled on the observation surface. In Figure 3a, the sampling interval is 0.007 m, resulting in a sampling surface size of 1.407 m \(\times\) 1.407 m. For Figure 3b-d, the sampling interval is 0.005 m, leading to a sampling surface size of 1.005 m \(\times\) 1.005 m. In Figure 3a, where \(R\) = 20 \(\times\)\(p\) represents a small metasurface, it can be observed that both the 3D and 2D normalized amplitude graphs exhibit periodic fluctuations in the amplitude margin. The phase diagram demonstrates multiple turns for vortex phases, which generate more sidelobes for the vortex beamformation. The OAM spectrum in Figure 3a reveals the generation of an \(l=-5\) vortex wave with high mode purity. From Figure 3b-d, as the metasurface size increases, ideal circular edges without fluctuations appear in both the 3D and 2D OAM amplitude graphs. The energy converges toward the center of the metasurface, resulting in a gradually smaller OAM beam radius and increased normalization amplitude for the vortex waves. As the metasurface size increases, the number of phase cycles decreases, indicating a reduction in sidelobes.