# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 2 Modelling Method
### Angular Spectrum Method  
ASM was introduced from the field of optics to the field of acoustics in the 1970s [4]. It is a standard method to extrapolate the ultrasonic field from the point measurement data set of the pressure field in a single plane. First, the propagation and attenuation of numerous plane waves emitted by the probe in the medium are calculated. Then, the propagation of the sound field is simulated by dividing all the superimposed plane waves by a certain weight function. Similar to the Rayleigh integral, ASM is the spatial frequency-domain representation of Huygens' Principle [5]. It has been widely used in sensor modeling and analysis, and more and more attention is paid to it. ASM can also be used to deal with complex situations, such as the distribution of sound fields in anisotropic media, and the transmission and reflection of sound beams across curved interfaces. The principle of the traditional angular spectrum method is shown in Figure 1.  
A measurement technique for analyzing the velocity distribution on ultrasonic surfaces was proposed in 1989, which was based on the angular spectrum method of wave [6]. In 2008, [PERSON] et al. [7] developed a model for studying transient ultrasonic waves radiated by a transducer in a liquid and transmitted through a planar interface to a solid. This extends the transient situation of the ASM developed for the monochrome situation. In 2012, the ASM was extended to non-uniform tissues. The hybrid angular spectrum (HAS) method was proposed by [PERSON] et al. [8; 9]. The approach proposes an inhomogeneous medium modeled using voxels, and its principle is shown in Figure 2. Each voxel has a unique sound velocity, absorption coefficient, and density. To find the pressure diagram at a distance from the input pressure diagram Z, the calculation of each layer alternates between the spatial domain and the frequency domain. In the spatial domain, the result of multiplying the pressure field at the entrance of the layer by the transfer function is subjected to a Fourier transform and then multiplied by the linear propagation term representing the average phase shift of the interlayer distance. The initial pressure field is transformed by FFT into a wave spectrum \(A_{1}^{*}(a,\beta)\) into plane 1, which is then transformed by FFT after multiplying by a linear propagation term representing the distance between the layers into a spatial domain pressure field, where it becomes the pressure field at the entrance to plane2. We can model complex and irregular inhomogeneous tissue geometries and quickly and accurately calculate the complex pressure distribution in a non-uniform three-dimensional model through controlling the size and number of voxels. Then, in 2017, [PERSON] et al. [10] calculated the pressure field of the cylindrical bending transducer array and compared the numerical results with those obtained through Field II. Meanwhile, they verified the accuracy of the method proposed by [PERSON] et al. When using the ASM to calculate the field radiated by a sound source of limited size, how to avoid or eliminate the aliasing phenomenon in the frequency domain and the distortion caused by the phase shift will be the focus of research.  
Figure 1: Traditional angular spectrum method.