# Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces
## 3 Results  
### Hologram Realized by the Array of Identical Point Sources  
First, the holographic principle based on an array of identical point sources (PSs) is discussed. As shown in Figure 1, the PSs (represented by red dots) with source strength \(Q_{0}\) are located at the grid nodes (represented by \(G_{m,n}\)) of an \(N\times N\) array with a distance \(D\) between two neighboring nodes, where \(m\), \(n=1\), \(2\), \(3\), \(\ldots\), \(N\). Every grid node is assigned a value of 1 or 0, with 1 indicating the presence of a PS and 0 indicating the absence of a PS. According to the Rayleigh-Sommerfeld theory, the sound pressure of an array of PSs at any point \((x,y,z)\) in space can be described as a superposition of sound fields from all of these PSs  
\[p(x,y,z)=\frac{\mathrm{i}\rho_{0}c_{0}k_{0}Q_{0}}{2\pi}\sum_{m=1}^{N}\sum_{n=1 }^{N}A_{m,n}\frac{\mathrm{e}^{-\mathrm{i}k_{0}R_{m,n}}}{R_{m,n}} \tag{1}\]  
where \(A_{m,n}=0\) or 1 is the value of \(G_{m,n}\), \(R_{m,n}=\sqrt{\left(x-x_{m,n}\right)^{2}+\left(y-y_{m,n}\right)^{2}+z^{2}}\) is the distance from \(G_{m,n}\) to the point \((x,y,z)\), and \((x_{m,n},y_{m,n},0)\) is the coordinate of \(G_{m,n}\). Figure 1: Schematic diagram of a hologram based on an array of identical PSs. The PSs (represented by red dots) are at the grid nodes (represented by \(G_{m,n}\)) of an \(N\times N\) array with a distance \(d\) between two neighboring nodes, where \(m,n=1\), \(2\), \(3\), \(\ldots\), \(N\). The grid node \(G_{m,n}\) takes the value \(A_{m,n}=0\) or 1, where 1 means a PS exists in this grid node and 0 means no PS. According to the Rayleigh–Sommerfeld theory, the diffraction pattern in the image plane is the superposition of the sound waves generated by these PSs. Combined with Equation (1), a desired real image (e.g., the four-pointed star pattern in this paper, as shown in Figure 2a; the black parts in the figure represent the presence of sound energy with equal magnitudes and the sound energy at these locations can be considered to be \(1\); the white parts represent the absence of sound energy, and the sound energy at these locations can be considered to be \(0\)) in the target plane can be designed by using a modified genetic algorithm that only contains the mutation and evolution operations for a chromosome encoding the value (\(0\) or \(1\)) of every grid node. The specific operation process of the modified genetic algorithm is as follows:  
1. Randomly generate a chromosome \(A_{1}\) (each subsequent generation of the chromosome is named \(A_{2}\), \(A_{3}\),..., \(A_{n}\),... ). \(A_{1}\) contains all \(A_{m,n}\), each \(A_{m,n}\) as a gene, and the value of each gene is only \(0\) or \(1\). Moreover, let \(A=A_{1}\). 2. Calculate the sound field \(p_{1}(x,y,z_{0})\) (each subsequent generation of the sound field is named \(p_{2}\), \(p_{3}\),..., \(p_{n}\),... ) generated by chromosome \(A\) in the desired image plane (the distance from the hologram is \(z_{0}\)) based on Equation (1), and let \(p(x,y,z_{0})=p_{1}(x,y,z_{0})\). 3. Using \(p(x,y,z_{0})\) and the desired sound field \(p_{\text{desired}}(x,y,z_{0})\) in the image plane, the root-mean-square error (RMSE) is calculated as the value of the fitness function \(F_{1}\) (each subsequent generation of the fitness function is represented by \(F_{2}\), \(F_{3}\),..., \(F_{n}\),... ), and let \(F=F_{1}\).