## 2 Theory
### Annular Array Transducer with Phase-Shifted Radiation  
When the array transducer shown in Figure 2b is used to generate and receive nonlinear waves, each element of the array transducer can be operated individually. In nonlinear ultrasonic testing, a tone burst with many cycles is typically used to drive the element. Here, we introduce a phase shift technique that improves SHG through phase control of the second harmonic components reflected at the stress-free boundary.  
Here is a brief description of the phase shift technique. Figure 3 shows a sine wave of two cycles with the zero phase. Waveforms with phase shifts of \(\pi/2\) and \(\pi\) are also shown in the same figure. When every element of an array transducer emits waves without a phase shift, the array transducer works as a single-element transducer. However, when there is a phase shift between the elements, the wave fields in some positions will be strongly enhanced.  
A time-delayed signal in the time-domain is equivalent to multiplying the frequency domain signal by the phase term, which is linear in frequency and proportional to the delay. The relationship between the phase angle and the delay time is given by \(\phi=\omega\Delta t\). Thus, the fundamental and second harmonic wave fields generated by an array transducer with the phase shift can be calculated in the frequency domain by introducing an exponential term \(\exp(i\phi^{(m)})\) to each element \(m\) and adding contributions from all elements.  
The single element approach can be extended to calculate the radiated and received beam fields of the four-element array transducer in the pulse-echo mode. Nonlinear interaction that may occur between the elements of the array transducer is ignored. The fundamental velocity fields in the forward propagation region (\(0\leq z\leq z_{0}\)) due to the excitation of the \(j\)th element, \(v_{i}^{(j)}\), and the generated second harmonic velocity fields, \(v_{2i}^{(j)}\), can be expressed in the same forms as Equations (11) and (12), respectively. The sound source prescribed on the \(j\)th element is now given by the following:  
\[v_{1}^{j}(x^{\prime},y^{\prime},z^{\prime}=0)= \left\{\begin{array}{c}v_{0}\mathrm{e}^{\mathrm{i}\phi^{(j)}}, \ 0\leq x^{\prime 2}+y^{\prime 2}\leq b_{j}^{2},\ j=1\ \text{(central disk element)}\\ v_{0}\mathrm{e}^{\mathrm{i}\phi^{(j)}},a_{j}^{2}\leq x^{\prime 2}+y^{\prime 2}\leq b_{j}^ {2},\ j=2,3,4\ \text{(outer ring element)}\end{array}\right. \tag{19}\]  
where \(\phi^{(j)}\) is the transmission phase shift of the \(j\)th element, and \(a\) and \(b\) denote the inner and outer diameters, respectively. The diameter \(b_{1}\) of the center disk element is equal to the diameter \(a_{2}\), so there is no gap between the two adjacent elements.  
The propagated and reflected velocity fields due to the phase-shifted radiation of the \(j\)th element can be found using the integral solutions for the single-element transducer described in the previous section. Next, it is necessary to calculate the received average velocity of the \(j\)th element by summing the contributions from the phase-shifted radiation of all elements of the array transducer. The average velocity received on each element at the source position (\(z=2z_{0}\)) can be written as the following:  
\[\widetilde{v}_{n}^{(j)}(2z_{0})=\frac{1}{S^{(j)}}\int_{S^{(j)}}\left[\sum_{m=1 }^{4}v_{nr}^{(m->j)}(x,y,2z_{0})e^{\mathrm{i}\phi^{(m)}}\right]dS^{(j)},n=1,2 \tag{20}\]  
where \(S^{(j)}\) is the area of the \(j\)th element, and \(v_{nr}^{(m->j)}(x,y,2z_{0})e^{\mathrm{i}\phi^{(m)}}\) is the reflected velocity arriving at the \(j\)th element when the phase-shifted \(m\)th element is radiated. Finally, the received total velocity of the \(n\)th harmonic is given by the following:  
\[\widetilde{v}_{n}(2z_{0})=\sum_{j=1}^{4}\widetilde{v}_{n}^{(j)}(2z_{0}) \tag{21}\]  
and the uncorrected nonlinear parameter, \(\beta^{\prime}\), can be calculated by Equation (18).