# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 3 Imaging Technology
### Nonlinear Ultrasonic Array Imaging
#### 3.3.1 Harmonic Imaging  
The concept of harmonic imaging was first proposed by [PERSON] et al. in 1997 [97]. When a continuous pulse train is used to excite large-amplitude ultrasonic waves, it can cause forced vibrations at discontinuities inside the material. As the sound waves propagate in the discontinuous area, the area \"closes\" or \"opens\" with the sound wave cycle, intensifying the time-domain waveform distortion, and the frequency domain is manifested as the appearance of harmonics. Relevant studies have shown that harmonic imaging technology can effectively reduce the width of the lateral and axial main lobe of the imaging system and reduce the sidelobe level, thereby significantly improving the spatial resolution and contrast resolution of the system, and can reduce the artifacts caused by sidelobes [98; 99]. A 1-D plane longitudinal wave propagating along the \(\alpha\)-axis can be expressed as a nonlinear wave [100]:  
\[u(x,t)=u_{1}(x,t)+u_{2}(x,t)+u_{3}(x,t)+\dots, \tag{15}\]  
where: \(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t)\) denotes fundamental, second harmonic, and third harmonics, respectively. Letting the solution of the fundamental wave be \(b\), then:  
\[ \begin{split} u_{2}(x,t)&=\frac{c_{2}}{8c_{1}}k^{2 }A_{1}^{2}ax\cos(2 kx-2\omega t),\\ u_{3}(x,t)&=\frac{1}{2t}k^{3}\gamma A_{1}^{3}\bigg{(} \frac{c_{2}^{2}}{c_{1}^{2}}-\frac{c_{3}}{c_{1}}\bigg{)}\beta x\sin(3 kx-3\omega t )+\frac{c_{2}^{2}}{32c_{1}^{2}}k^{4}A_{1}^{3}\gamma x^{2}\cos(3 kx-3\omega t), \end{split}  \tag{16}\]  
where \(c_{1}\) is the second-order elastic constant; \(c_{2}\) and \(c_{3}\) are the higher-order elastic constants; \(A_{1}\) is the fundamental wave amplitude; and \(\alpha\), \(\beta\), and \(\gamma\) are the contact nonlinearity factors to represent the effect of contact nonlinearity effects of defects within the solid on nonlinear higher harmonics. There are three main methods of harmonic separation in harmonic imaging technology: RF filtering techniques, pulse inversion techniques, and pulse amplitude modulation techniques [101]. The RF filtering method uses a filter to remove the fundamental components and only retain the harmonic components. When the frequency band of the emitted fundamental signal is controlled narrowly, the resulting second harmonic and fundamental components are separated into their frequency bands. Since ultrasonic array imaging is a broadband system, the nonlinearity of the detection system or the spectral leakage of the emitted signal will be mixed with the harmonic components, making it difficult to effectively separate the nonlinear signal from the overlapping interference. [PERSON] et al. [102] first proposed subharmonic ultrasonic arrays for crack evaluation (SPACE). His team developed a SPACE imaging device, as shown in Figure 11. To produce images, propagation times along the paths such as A-B-C-D in Figure 11 were calculated and subtracted before the summation of signals received from each element of the array sensor. The device was capable of acquiring sub-harmonic signals, which were used for sub-harmonic phased-array nonlinear imaging of closed cracks. However, a single-array SPACE could image only the vicinity of a transmission focal point (TFP) when the TFP was fixed. [PERSON] et al. [103] developed a confocal SPACE that defines multiple TFPs for imaging closed cracks over a wide area. [PERSON] et al. [104] proposed a multi-signal classification method (MUSIC) based on subharmonic ultrasonic phased-array nonlinear imaging. A numerical simulation was used to compare the proposed method with a conventional one, and MUSIC give a narrower beam width. Experiments also demonstrate that MUSIC can improve the resolution of sub-harmonic imaging. Pulsed inversion techniques are an effective method to solve the decrease in system spatial resolution and contrast resolution due to the problems of ultrasonic array broadband systems.