## 1 Introduction  
The key element for acoustic energy emission is the ultrasonic transducer. Different shapes of ultrasonic transducers are designed to create different sound fields and are used in various applications, such as ultrasonic ranging [1, 2] and ultrasound imaging [3]. The directivity of the acoustic field, the width of the beam and other indicators directly affect the accuracy of the instrument which is significant in ultrasound medicine [4] and acousto-optical devices [5, 6, 7]. Acoustic field simulation results of ultrasonic transducers assist researchers in adjusting designs to achieve improved performance. So, accurate calculation of the acoustic field is a key aspect in the design and optimization of ultrasonic transducers, and scholars have carried out a lot of work in this area [8, 9]. The calculation of the acoustic field of an ultrasonic transducer is often equated to the calculation of the acoustic field for embedded sound sources on an infinitely large rigid plane. The problem of planar radiating surfaces has been studied by many scholars [10, 11]. Solving the acoustic field in three-dimensional space is generally described by the fixed solution equation formed by the Helmholtz equation under the boundary conditions [12, 13]. The multi-Gaussian beam superposition model calculates the acoustic field of a piston-type source using multiple Gaussian beams [14]. The calculation is fast but limited by the near-axis condition. The nonparaxial multi-Gaussian model is proposed to address errors arising from non-proximal axes [15]. The Rayleigh integral model is based on the derivation of [PERSON] principle, proposed by the British physicist [PERSON] in 1877, which can be used to obtain accurate solutions and is suitable for many types of acoustic field radiation [16]. Due to the complexity of the boundary conditions in the acoustic field region and the arbitrary shape of the sound source, the calculation process takes a long time and it is difficult to find an analytical solution. The point source method is a numerical calculation of the Rayleigh integral for piston type sources, which greatly simplifies the complexity of the calculation [17; 18]. Compared with the acoustic calculation methods such as finite element [19] and boundary element [20] proposed later, the point source method saves the trouble of associating a larger number of systems of equations. However, due to a large number of sampling points, the point source method is time-consuming. Various methods of numerical deceleration have been proposed to address this problem. [PERSON] et al. propose a method for accelerating the numerical calculation of Rayleigh integrals through the FFT. However, the algorithm is limited by distance and is only suitable for near-field calculations [21]. The angular spectrum method is equivalent in principle to the Rayleigh integral, which is the spatial time-domain representation of the [PERSON] principle, and the angular spectrum method (ASM) is the spatial frequency domain representation of the Huygens principle. In numerical calculations, the ASM uses the two-dimensional fast Fourier transform (FFT), which greatly reduces the computation time and is suitable for applications such as phased arrays that require a large number of acoustic field calculations [22; 23; 24]. However, the calculations may incur a serious aliasing error which seriously affects the computation accuracy. To reduce errors in the propagation, aperture size and propagation distance are limited [25; 26]. [PERSON] et al. proposed a GPU-based solution to accelerate the acoustic field simulations [27]. The look-up table method is often used as an accelerated algorithm in the calculation of holograms of three-dimensional point sets, which has a good reference for accelerated calculation of the acoustic field of piston-type sources [28; 29]. In this paper, an acoustic field look-up table method is presented for fast calculation of the acoustic field.