# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 2 Modelling Method
### Multi-Gaussian Beam Method  
In general, many spherical or plane waves need to be superimposed in the calculation of the probe emission acoustic field. However, The sound field of plane probe and focus probe can be simulated by using a small number of Gaussian beam superposition. The boundary condition of multiple single Gaussian acoustic beams superimposed to simulate the acoustic field of the ultrasonic probe is called Multivariate Gaussian Sound Beam (MGB). This model was derived from the Helmholtz wave equation by [PERSON] [11] in 1988 and demonstrated that the acoustic beam radiated into the water by a circular piston probe could be accurately simulated by superimposing only ten Gaussian beams. Due to its high computational efficiency and good simulation of ultrasonic waves, MGB has been widely applied in the modeling process of ultrasonic arrays. When there is no delay and deflection, the total radiation sound field of N array elements can be obtained by the following Equation (1) [12]:  
In a single medium, the Rayleigh integral of the sound field radiated by the \(n\)th rectangular source is:  
\[p_{n}(x,\omega)=\frac{-i\omega p_{1}v_{0}(\omega)}{2\pi}\int_{s}\frac{\exp(ikr )}{r}ds \tag{1}\]  
where \(\omega\) is the angular frequency, \(p_{1}\) is the density of the medium, \(V_{0}(\omega)\) is the normal velocity of the particles on the probe surface, and \(r\) is the distance between any point on the probe surface and the position \(x\) of the medium. The extended multivariate Gaussian sound beam model can calculate the radiated sound field \(P_{n}(x,\omega)\) of the \(n\) array unit through the superposition of 2-D Gaussian sound beams. The expression can be written as:  
\[p_{n}(x,\omega)=v_{0}(\omega)[\sum_{m=1}^{10}A_{m}\xi_{1}(x,z_{\omega})]\cdot[ \sum_{m=1}^{10}A_{m}\xi_{1}(y,z_{\omega})] \tag{2}\]  
where \(A_{m}\) are 10 complex constants. Following [PERSON] principle, the total sound field of an ultrasonic transducer array with \(N\) elements can be calculated by superimposing each array element. The sound pressure of a linear phased-array transducer can be written as:  
\[p(x,\omega)=\sum_{n=0}^{N-1}p_{n}(x,\omega)\exp(j\omega t_{n}) \tag{3}\]  
MGB can not only be used to simulate the emission sound field of the piston and rectangular ultrasonic array transducer, but also be used to simulate the propagation  
Figure 2: Hybrid angular spectrum method. of a sound field in a heterogeneous anisotropic medium and irregular interface and the waveform conversion in the propagation process. In 2003, [PERSON] et al. [13] proposed to calculate the radiated sound field of a rectangular sensor by the superposition of 2-D Gaussian sound beams. This model is called the extended multi-Gaussian beam method (EMGB). However, when EMGB technology is applied to ultrasonic arrays, since it needs to rely on paraxial approximation, modeling errors will occur when the steering angle is greater than about 20 degrees. In 2007, [PERSON] et al. proposed a new multi-Gaussian beam model, called the linear phase multi-Gaussian beam model (LMGB), and proved that it could eliminate these steering angle restrictions. In the 2010s, [PERSON] et al. [14] calculated the radiated sound field distribution of an ultrasonic arrays probe with a wedge shape by using the extended multi-Gaussian beam model and measured the ultrasonic arrays probe with an incident angle greater than \(10^{\circ}\) (refractive angle greater than \(20^{\circ}\)). The results are modified to enlarge the deflection range of the sound field calculation and the application range of the Gaussian superposition method. In 2019, [PERSON] et al. [15] compared the simulation results of beam propagation through austenitic steel using ordinary multi-Gaussian beam (OMGB) and LMGB models at different steering angles and found that, especially in anisotropic media, the LMGB method was superior to the OMGB method when the steering angle of the beam was greater than \(20^{\circ}\).