## 3 Numerical Simulations and Experiment  
To demonstrate the effectiveness of the proposed method to calculate the acoustic field distribution, we calculated the acoustic field distribution generated by the transducer using the point source method, the ALUT method, and the ASLUT method, respectively. Acousto-optical devices are one of the main areas of application for ultrasonic transducers. We use the structure and parameters of the acousto-optic modulator to calculate the ultrasonic transducer acoustic field [33]. The overall structure of the transducer is shown in Figure 6a, which consists of an ultrasonic transducer, an acoustic-optical crystal, and a perfectly matched layer. In this simulation, a diamond transducer shape is chosen. The driving frequency of the transducer is 80 MHZ, and the wavelength of the sound wave propagating in the crystal is 71.8 \(\upmu\)m. If boundary discretization coarser than \(\lambda/2\) is employed, the resulting acoustic field would be distorted because of spatial aliasing [34; 35; 36]. For simplicity, we set the sampling spacing to 25 \(\upmu\)m. Thus, the rhombic electrode is represented by a two-dimensional matrix that consists of 400 \(\times\) 200 sampling points, as shown in Figure 6b. Any point in the transducer area is assigned a value of 1, indicating that the points on the electrode vibrate synchronously, while the points not in the transducer area do not vibrate. And the initial phase of the vibration at each point on the electrode is set to 0. To more clearly represent the direction and position relationship, the long side of the transducer is set to the Y-axis and the short side is set to the X-axis, and the propagation direction of the acoustic field is the Z-axis. The propagation distance of the acoustic field calculation is 12  
Figure 5: The flow chart of the ASLUT method. cm. The sampling spacing of propagation depth is the same as the sampling spacing of electrode surface grid points; both are 25 \(\upmu\)m.  
In order to demonstrate the acceleration effect of the ALUT and ASLUT methods for point source superposition calculations, the calculation times and acoustic field results of the three methods are presented as shown in Figure 7a. the ASM is also included in the comparison as a well-known method for acoustic field calculations. As shown in Figure 7b, the calculation time of the acoustic field of the point source method is 17,760 s, and the calculation time of the ALUT method is 6803.68 s, which is around one third of the point source method. The calculation time of the ASLUT method is only 949.79 s, which is about one eighteenth of the point source method. The ASM took the least time, only 15.40 s. However, the root-mean-square error of the calculated results of the angular spectrum method and the point source method increased with the propagation distance as shown in Figure 7c, which confirms that the ASM is limited by the propagation distance. The ALUT method and ASLUT method are acceleration algorithms based on point-source method. Moreover, their calculation results are the same as those obtained using the point-source method. To test the applicability of implementing the sub-region lookup table method under different computational volumes, a comparison was made between five different algorithms: point source method (labeled \(N1\)), ALUT method (labeled \(N2\)), ASLUT method which stored a single point source and 3 \(\times\) 3 point source array (labeled \(N3\)), ASLUT method which stores a single point source and 3 \(\times\) 3, 5 \(\times\) 5 point source arrays (labeled \(N4\)), and ASLUT method which stores a single point source and 3 \(\times\) 3, 5 \(\times\) 5, 7 \(\times\) 7 point source arrays (labeled \(N5\)). As the planar radiation source expands in the two-dimensional direction, the increase in computation time increases rapidly with the number of sampling points as shown in Figure 8. The LUT method significantly reduces the computation time, especially with a large number of sampling points.