# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 3 Imaging Technology
### Nonlinear Ultrasonic Array Imaging
#### 3.3.2 Acoustic Modulation Imaging  
The acoustic modulation imaging technique uses low frequency (LF) and high frequency (HF) to modulate each other at the crack to generate a para-flop nonlinear signal for performing nonlinear imaging techniques. The schematic of the acoustic modulation is shown in Figure 17. Excite the high-frequency signal and the low-frequency signal into the test piece separately, and the echo signal in the sample without cracks satisfies the linear superposition of the two. On the contrary, the echo signal in the sample with cracks will produce sidelobe signals, which is the modulation signal [111]. Figure 16: Schematic of the amplitude modulation technique, which consists of a sequence of three acquisitions at the very same focal points with alternative activation of full, odd, or even elements. Signal processing consists of band-pass filtering and time-shifting signals before summation and then subtraction. The modulated signal can be expressed in the following form:  
\[x_{a}(t)=A(t)\sin(2\pi f_{c}t+\varphi_{c}) \tag{17}\]  
\(A(t)\) can be expressed by the following equation:  
\[A(t)=A[1+m_{a}(t)] \tag{18}\]  
where \(m_{a}(t)\) is the modulation function equation and \(A\) is the amplitude parameter. The amplitude modulation intensity or depth can be defined by the amplitude modulation index as follows:  
\[M_{a}=\frac{m_{\max}}{A} \tag{19}\]  
where \(m_{\max}\) is the peak value of the signal \(m_{a}(t)\). The modulated signal \(X_{a}(t)\) can be expressed as follows:  
\[x_{a}(t)=A\cos(2\pi f_{c}t)+\frac{MA}{2}\cos[2\pi(f_{c}-f_{a})t+\varphi]+\frac {MA}{2}\cos[2\pi(f_{c}+f_{a})t+\varphi] \tag{20}\]  
where \(f_{c}\) is the main frequency, \(f_{c}-f_{a}\) is the low-side frequency, and \(f_{c}+f_{a}\) is the high-side frequency. In [112; 113], it was demonstrated that, for nonlinear acoustic detection of cracks, the modulating effect of the crack-induced vibrations on ultrasonic waves can be used. [PERSON] et al. [114] used low-frequency vibration-modulated pulsed surface waves to detect locally closed fatigue cracks in a specimen and studied the effect of crack size changes on the modulated nonlinear parameters. It was found that the modulated response significantly depended on the ratio of the length of the fatigue crack to the size of the plastic yield zone. Thereafter, an acoustic modulation model was developed [115; 116; 117], which can explain the main features observed, to roughly predict the frequency band level obtained by the experiment and select the most suitable acoustic modulation intensity to obtain the best nonlinear modulation signal according to the crack size and the opening and closing states. [PERSON] et al. [118; 119] developed a novel imaging method, subharmonic ultrasonic arrays for crack evaluation (SPACE) based on subharmonic waves and an ultrasonic arrays algorithm, to measure closed-crack depth in the thickness direction. Although the performance of the SPACE method in closed fatigue and stress corrosion cracks is demonstrated [120], because short-burst input waves might cause filter leakage, objects other than cracks such as coarse grains appear in subharmonic images which might degrade the performance of SPACE for identifying closed cracks. To improve the selectivity of closed cracks for objects other than cracks, a nonlinear ultrasonic imaging method, load difference ultrasonic arrays (LDPA), based on the subtraction of responses at different loads and ultrasonic array techniques, was proposed [121]. A schematic illustration of the  
Figure 17: Schematic diagram of sound modulation. proposed method is shown in Figure 18. By applying external static or dynamic loads to closed cracks, the cracks undergo opening and closing state changes. In contrast, the responses at objects other than crack weld defects were independent of external load. Therefore, the nonlinear signal of the crack can be extracted by subtracting the responses at L1 and L2. The study finds that the LDPA method can perform highly selective imaging of closed cracks in the medium.