## 3 Modeling, Simulation and Analysis
### Modelling and Simulation of the Metasurface
After the optimization, the threshold is selected as 288\({}^{\circ}\) (i.e., the normalized value is 0.8 relative to 360\({}^{\circ}\)), which is optimal for our proposal. Due to the final slot-type distribution pattern, a model of the metasurface generating a CP quasi-non-diffractive vortex wave carrying \(l=-1\) mode OAM can be built accurately. Figure 3: Scheme of the calculation process of the final slot-type distribution pattern. Phase distribution patterns of the (**a**,**e**) object waves, (**b**,**f**) reference waves, and (**c**,**g**) infergrams with the y- and x-polarization, respectively. Slot-type distribution patterns of the (**d**) y-, (**h**) x- and (**i**) left-handed circularly polarized quasi-non-vortex waves, respectively. The threshold is important for designing metasurfaces, so it is necessary to analyze how it affects the performance of the metasurface. According to the holographic principle, the designed metasurface is equivalent to a hologram containing the interference information of a reference wave and an objective wave, and can also be regarded as the absolute value of the phase difference between the two waves, which is consistent with the principle of impedance modulation. Therefore, the threshold represents the deviation between the sampled reference wave phase and the desired objective wave phase. If the deviation is too small, only less energy can be collected from the slots, resulting in a poor beamforming at the far-field. On the contrary, when the deviation is too large, many redundant slots are set in many unnecessary places, and the radiated energy interferes with the objective beam and deteriorates the wavefront phase. The selection range and sensitivity of the threshold are also related to the type and characteristics of the objective wave. For example, when the objective wave is a plane wave, the optional range of the threshold is narrow, usually in the range of 60-110\({}^{\circ}\). Moreover, the sensitivity of the hypersurface to the threshold change is about 10\({}^{\circ}\). When the expected wave becomes a quasi-non-diffractive vortex wave, the optimal variation range is extended to a range from 60-300\({}^{\circ}\) and the sensitivity is changed to 36\({}^{\circ}\). This is because the vortex wave is a hollow beam and possesses a spiral phase profile. The complex radiation characteristics increase the demand for energy. An expected metasurface is created in the ANSYS Electromagnetics Suite according to a one-to-one mapping relationship between the numerical values and the types of slots. In Figure 5a, a top view of the proposed metasurface is presented, which is composed of 440 inhomogeneous slot units and a monopole as the central feeding source. Full-wave simulation is performed, far-field radiation patterns and scattering parameters of the proposed metasurface are obtained and illustrated in Figure 5b-d. It can be seen from Figure 5c that, for the most part, the reflection coefficient (\(|S11|\)) remains below \(-10\) dB from 19.5 GHz to 20.5 GHz, especially at \(-26\) dB at the central frequency of 20 GHz, exhibiting a good impedance matching. The 3D far-field radiation pattern of the quasi-non-diffractive wave is a hollow beam with a small diverging radius, as depicted in Figure 5b. In addition, its normalized power levels on the planes of \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\) are illustrated in Figure 5d. There is a null existing in the main lobe of the 2D radiation pattern, demonstrating that a vortex wave with OAM is produced in the normal direction. Moreover, because it is not in the non-diffracting superposition zone, the side lobes are relatively high, which is in accordance with the theory of quasi-non-diffraction. The simulation results demonstrate that the proposed holographic generator of a quasi-non-diffractive vortex wave possesses a good performance in the far-field radiation. Figure 4: Calculation flow chart for determining the threshold.