## 2 Method
### Acoustic Field Look-Up Table Method  
Due to a large number of sampling points in the acoustic field, the calculation of the numerical solution based on the Rayleigh integral takes a long time. The look-up table (LUT) method can be used to solve this problem. The LUT method is an algorithm that accelerates the computational process by pre-computing part of the data. In hologram calculations, a three-dimensional object is discretized as a collection of points and the complex amplitude information of the hologram is obtained by superimposing the spherical waves of the discretized points. The look-up table method saves time in the computation by pre-storing the repetition information in the spherical wave superposition calculation. The process of calculating planar sound sources through the Rayleigh integral model can also be accelerated by pre-storing information. Unlike hologram calculations, where the depth information of a 3D object is compressed into a 2D hologram, the acoustic field is calculated from a 2D source surface to the acoustic field in 3D space. In this subsection, we elaborate on the reasons for the long time taken to calculate the acoustic field. And an acoustic field look-up table method is proposed to reduce the time. Figure 2a,b shows an outline of the process of calculating the acoustic field in three dimensions by the point source method. As shown in Figure 2a, the Source layer is the acoustic source plane where the electrode is located, and slice 1 to slice \(M\) is the spatial location planes of different depths. To facilitate the calculation, we make the source point spacing and the acoustic field distribution sampling point spacing the same, both denoted by \(\Delta p\), with \(N\)\(\times\)\(N\) elements in each layer. Figure 2b shows the location relationships between the acoustic source point and sound field sampling points in a one-dimensional manner. There are \(N\) vibration points in the source plane. From slice 1 to slice \(M\), there are \(N\) sound field sampling points in each layer. According to Equation (2), the distance  
Figure 1: Procedure for computing the numerical solution of the Rayleigh integral. relationship between any source point in the source plane and the sampling point in space can be expressed as Equation (3):  
\[r=\sqrt{\left(x-x_{n}\right)^{2}+\left(y-y_{n}\right)^{2}+z} \tag{3}\]  
where \(r\) is the distance between the sampling point (\(x_{n}\), \(y_{n}\), \(z\)) in space and the acoustic source point (\(x\), \(y\), 0). In order to obtain the three-dimensional sound field distribution in space, it is necessary to perform the \(M\times N^{2}\) calculation as in Equation (3), which is undoubtedly a tedious task. The ALUT method was proposed to solve this problem. When calculating the acoustic field distribution of any layer, the distance relations between source points and the sampled points in the spatial plane have overlapping parts. As shown in Figure 2c, based on the distance of different layers from the source layer, the corresponding \(M\) tables are calculated according to Equation (3). The corresponding table for each layer has \(2N-1\) pixels, and the pixel spacing is the same as the spacing of the sound field sampling points as \(\Delta p\). The distance correspondences of the nth vibration point in the slice m can be extracted from the \((N+1-n)\)th sampling point to the \((2N-n)\)th sampling point of the nth table. Thus, the distance term between the distribution of acoustic source points and the sampling points at slice \(m\) can be obtained by table m. As a result, the flow of the sound field calculation is simplified from Equations (2)-(4):  
\[P=\sum\limits_{n=1}^{N}\frac{ipcu}{\lambda}\cdot T_{n} \tag{4}\]  
\[T_{n}=\frac{\epsilon^{-\delta R}}{R} \tag{5}\]  
where \(T_{n}\) represents the distance term of the acoustic field radiated by the nth source point into three dimensions. The distance term \(T_{\rm n}\) is calculated in advance and stored in the table.