## 1 Introduction  
Plate-like structures are widely used in various kinds of equipment, and fatigue crack is one of their main damage forms. Fatigue cracks will cause stress concentrations in the structure. If corrective measures are not taken in time, a crack will accelerate its expansion under the action of load and eventually lead to structural fracture. Therefore, the use of non-destructive evaluation (NDE) for the detection or identification of cracks at an incipient stage is essential for maintaining the integrity and safety of a structure. Lamb waves, with their characteristics of long propagation distance and slow energy attenuation, have been considered as one of the most potential NDE monitoring technologies [1; 2; 3; 4; 5; 6]. Existing linear Lamb wave monitoring technique uses changes in time-domainsignal features (e.g., acoustic velocity variation [7], mode conversion [8], time of flight (TOF) [9], and transmission and reflection coefficients [10]) to detect a small number of visible fatigue cracks, macroscopic holes, and notches. However, the technique based on the temporal signal features is insensitive to the embryonic stage of mechanical damage (much smaller than the probing wavelength) because small damage artifacts are not expected to induce evident changes in the linear features to be extracted from the ultrasonic wave. To overcome the aforementioned deficiencies, nonlinear Lamb wave monitoring techniques based on the frequency domain characteristics of signals have attracted increasing attention in research because they can detect micro-cracks effectively. Existing research results show that Lamb wave nonlinearity comes from three main sources: the elastic nonlinear effect of structures or materials [11], the geometric nonlinearity of structures [12], and the boundary nonlinearity of contact [13]. These nonlinear components carry damage information and have been used intensively for identifying micro-cracks. Material nonlinearity is the high-order, sub-harmonic variation generated in the structural response signal under fundamental frequency excitation, which is mainly manifested in the second-order harmonic signals. Geometric nonlinearity is the harmonic spectrum that occurs symmetrically near the basic signal frequency under the excitation of the frequency modulation signal. Boundary nonlinearity is a type of local nonlinearity that can generate higher harmonic components. According to the detection principle, there are four methods for nonlinear ultrasonic detection: vibration acoustic modulation technique [14], nonlinear resonance technique [15], harmonic wave technique [16], and wave-mixing technique [17]. Vibration acoustic modulation technique has advantages in detecting interface contact state and closed cracks, but this method requires the imposition of additional low-frequency vibrations on the tested specimen and is vulnerable to boundary nonlinearity, which makes the detection system difficult to operate. The nonlinear resonance method has stringent requirements for instrument and experimental operation. The harmonic method is the most widely used method for detecting micro-cracks, but the main difficulty with harmonic generation methods is that they are greatly affected by the nonlinearity of signal transmitting/receiving equipment and inherent material nonlinearity. [PERSON] et al. [18] used the finite element method to research the interaction between nonlinear single S\({}_{0}\) mode Lamb wave and micro-cracks embedded in a thin plate. [PERSON] et al. [19] extended the use of temporal signal processing to Lamb wave second harmonic components and located fatigue cracks using the TOF time-domain feature of the second harmonic. [PERSON] et al. [20] decomposed the time reversal operator (TRO) at the second harmonic frequency and introduced an imaging algorithm based on the total focus method to locate fatigue cracks.