## 2 Methods  
The principle of our method is shown in Figure 2 and briefly reviewed as follows: An eddy current is induced by the exciting coil and then an alternative current of sinusoid form is applied. A PM interacts with the eddy current and functions as the probe. Similar to the process in ECT and LET, defects in the test sample would induce a redistribution of the eddy current, thus leading to a change in the Lorentz force on the conductor. The Lorentz force is a vector, with all its components reflecting different features of the eddy current and its reaction force acts on the PM. A laser cantilever magnet (LCM) system, similar to the laser-cantilever-tip system used in Atomic Force Microscopy (AFM) [20], is applied to measure all components of this force via displacement measurements. Just as the traditional ECT, the PM and exciting coil can be on the same side or on both sides, the form with the PM and exciting coil on both sides are used to illustrate the principle in Figure 2. The detailed form of the net force is discussed as follows. By applying an alternating input current \(\mathbf{I}=I_{0}\sin\omega t\) to the exciting coil, the exciting coil generates a magnetic field, which can be described as follows:  
\[\mathbf{B}_{p}=\mathbf{B}(x,y,z)\sin\omega t \tag{1}\]  
As the PM interacts with this primary magnetic field, the exciting coil exerts a magnetostatic force on the PM. This magnetostatic force has the same phase as the primary magnetic field, and its dependence on time can be simply described as follows:  
\[\mathbf{F}_{\rm c}(t)=\mathbf{F}_{c}(x,y,z)\sin\omega t \tag{2}\]  
Figure 2: Principle of the vector-based eddy current testing method. \(\mathbf{F}_{\rm c}\) is the magnetostatic force. (**a**) Testing sample without a defect. \(\mathbf{F}_{L}\) is the reaction of the Lorentz force acting on the PM without a defect; (**b**) Testing sample with a defect. \(\mathbf{F}_{L}^{\prime}\) is the reaction of the Lorentz force acting on the PM with a defect. 1. Exciting coil; 2. PM; 3. Copper plate; 4. Defect. According to [PERSON]'s equations and [PERSON]'s law, this alternating magnetic field induces an eddy current inside the conductor. \(\sigma\) is the electrical conductivity of the sample, and the eddy current satisfies the following:  
\[\
abla\times\mathbf{j}=-\sigma\frac{\partial\mathbf{B}}{\partial t} \tag{3}\]  
Thus, its dependence on time can be written as follows:  
\[j(t)=j\cos\omega t \tag{4}\]  
Note that \(\mathbf{B}_{p}\) and \(\mathbf{j}\) have a \(\pi/2\) phase difference. The interaction between the PM's static magnetic field and the eddy current produces the [PERSON] force acting on the conductor, equally obvious but less widely recognized is the fact that by virtue of [PERSON]'s law, an opposite force (reaction force) acts on the PM [11]. In our study, this force is time-dependent and has the same phase as the eddy current, thus can be described as following form:  
\[\mathbf{F}_{L}=\mathbf{F}_{L}(x,y,z)\cos\omega t \tag{5}\]  
Thus, the total force acting on the PM is described as follows:  
\[\mathbf{F}_{PM}(t)=\mathbf{F}_{c}(x,y,z)\sin\omega t+\mathbf{F}_{L}(x,y,z)\cos\omega t \tag{6}\]  
Note that, although both \(\mathbf{F}_{c}\) and \(\mathbf{F}_{L}\) are dependent on the magnetic field generated by the exciting coil, there is no direct relation between them and they both proportional to the input current \(I\). If the PM and the coil are on the same side, \(\mathbf{F}_{c}\) can be more significant because of the reduced distance between them. For the sake of readily operation, our experiment set them on the different sides with the conductor. For any component \(i\) (\(x\), \(y\) or \(z\)) of this force, we have the following:  
\[F_{PMi}(t)=F_{cl}\sin\omega t+F_{Li}\cos\omega t=\sqrt{F_{ci}^{2}+F_{Li}^{2}} \sin(\omega t+\varphi) \tag{7}\]  
where \(\varphi=\arctan(F_{Li}/F_{ci})\). We can calculate the above relation through the trigonometric functions. Thus, the force acting on the PM also has a sinusoidal form, leading to an oscillation motion for the PM.