# Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces
## 3 Results
### Theoretical Zero-Thickness Holograms Based on Acoustic Sieve Metasurfaces  
To approximate the PS array, an acoustic sieve metasurface (ASM) that is a zero-thickness rigid screen with many small holes of subwavelength size is introduced. A sound field similar to that of a PS can be created by sound waves passing through a small hole. Despite the fact that a small hole in a rigid screen cannot be considered an ideal PS because it must radiate an ideal spherical wave, it is found that the sound field diffracted from a small hole is nearly identical to the sound field diffracted from an ideal PS (see the next paragraph for specific evidence), which is important because the reconstruction of a holographic image is largely dependent on the fields of diffraction units (such as holes or PSs) in the image plane. Furthermore, the transmitted wave from the small hole propagates forward, whereas the reflected wave has no bearing on image reconstruction. As a result, the concept of ASMs is introduced for the realization of holograms. Figure 2: Holographic imaging based on the PS array. The background medium is chosen as air with a sound velocity \(c_{0}=340\) m/s and a density \(\rho_{0}=1.2\) kg/m\({}^{3}\), and the radiation frequency is set at \(f_{0}=34\) kHz, so the wavelength is \(\lambda_{0}=c_{0}/f_{0}=10\) mm. (**a**) Desired image (a four-pointed star). The black parts in the figure represent the presence of sound energy with equal magnitudes (the sound energy at these locations can be considered to be 1), and the white parts represent the absence of sound energy (the sound energy at these locations can be considered to be 0); (**b**) The overall map of the hologram based on the PS array designed by the modified genetic algorithm. Each black dot represents a PS. The distance between grid nodes is \(D=\lambda_{0}\) and the distance of the desired image plane from the hologram is \(z_{0}=120\lambda_{0}\); (**c**) Normalized energy distribution in the image plane. First, as shown in Figure 3a, the lateral profile of the sound field from a PS in the middle of a circular rigid boundary with a diameter of 30\(\lambda_{0}\) is investigated. On the red dashed line in Figure 3a, the lateral (\(x\)-\(y\) plane) energy profiles in the \(z\)-cut plane of \(z=4\lambda_{0}\) are calculated. This distance (4\(\lambda_{0}\)) is beyond the evanescent region while remaining small enough that the simulation model does not consume excessive computational resources or memory space. Then, in the same model, the PS is replaced with holes of various diameters (\(d=0.01\lambda_{0}\), 0.02\(\lambda_{0}\), 0.03\(\lambda_{0}\),..., \(\lambda_{0}\)) to allow plane sound waves to pass through. The lateral energy profiles of the same location (the red dashed line) generated by plane sound waves passing through holes of various diameters are calculated using the FEM-based numerical simulation. The lateral normalized energy profiles \(E_{\text{PS/hole}}(x_{k},0,4\lambda_{0})/E_{\text{PS/hole}}(0,0,4\lambda_{0})\) of the ideal PS and holes of various diameters are shown in Figure 3b. As can be seen, the smaller the hole's diameter, the closer the energy profile approaches to the ideal PS. By calculating their RMSEs (shown by blue circles in Figure 3c), the degree of approximation between PS and holes of various diameters can be quantitatively verified  
\[\text{RMSE}=\sqrt{\sum_{k=1}^{K}\left[\frac{E_{\text{hole}}(x_{k},0,4\lambda_ {0})}{E_{\text{hole}}(0,0,4\lambda_{0})}-\frac{E_{\text{PS}}(x_{k},0,4\lambda _{0})}{E_{\text{PS}}(0,0,4\lambda_{0})}\right]^{2}/(K-1)}\]  
where \(E_{\text{hole}}(x_{k},0,4\lambda_{0})\propto p_{\text{hole}}^{2}(x_{k},0,4 \lambda_{0})\) and \(E_{\text{PS}}(x_{k},0,4\lambda_{0})\propto p_{\text{PS}}^{2}(x_{k},0,4\lambda _{0})\) are the energy profiles of the hole and ideal PS, respectively, and \(K=1500\) is the sampling number in the \(x\)-direction. The calculated RMSE values (blue circles) and relative transmitted energies (red dots) for holes of various diameters are shown in Figure 2c.