# High-Precision Noncontact Guided Wave Tomographic Imaging of Plate Structures Using a DHB Algorithm
## 3 Experimental Set-Up and Specim
\[g(t)\ =\ \sum_{n\ =\ 1}^{N}f(t-n\Delta t) \tag{5}\]  
In the above equation, \(\Delta t\ =\ \Delta s/c\) and \(f(t)\) is the out-of-plane displacement of the guided wave generated by a single line laser source. It is expressed as a function of time in the thermoelastic regime. The displacement is proportional to the derivative of the absorbed laser energy function. The energy distribution of the line laser beams was regarded as a square pulse, and \(f(t)\) can be approximated by  
Figure 10: Schematic of the line laser sources of 1-D and Gaussian model: (**a**) One-dimensional theory; (**b**) Gaussian model. Figure 9: Improved transmission using a laser source. Values shown are approximate fractions of the original energy. the square pulse derivative. The relationship between the Lamb wave frequency dispersion curve and the corresponding parameters is given in Figure 11.  
The signal is generated on the specimen surface when the illuminated laser beam transmits through the slit mask. The wavelength corresponds to the element gap of the slit array. The relation between the wavelength, phase velocity, and wave frequency is shown in Equation (6), where \(d\) is the specimen thickness, \(f\) is frequency, and \(\lambda\) is wavelength. The principle of generating the desired mode is shown in Figure 11.  
\[C_{ph}\ =\ f\cdot\lambda\ =\ f\cdot\Delta\mathrm{s}=\ f\cdot d\left(\frac{ \Delta\mathrm{s}}{d}\right). \tag{6}\]  
As shown in Figure 11, a diagonal line with a slope of \(\Delta\mathrm{s}/\mathrm{d}\) is described in the dispersion curves of the Lamb wave. The modes of the curves that have cross points with the diagonal line can be generated. Therefore, various modes with different frequencies can be generated simultaneously. The experimental setup is shown in Figure 9.0 A Nd:YAG pulsed laser with a wavelength of 1064 nm and a pulse duration of 6 ns was used as the energy source. A Quantel Brilliant B pulsed YAG laser was used with principal radiation at 1064 nm, max energy/pulse 850 mJ, max average power 7.5 W, and pulse duration 6 ns. This laser beam was expanded and illuminated on a slit mask. The slit mask is located close to the specimen surface to suppress the diffraction phenomena. Figure 9.1 The frequency dispersion curve of a Lamb wave and the corresponding parameters: (**a**) Phased velocity; (**b**) Group velocity. Figure 9.2 Schematic diagram of the experimental setup. An unfocused planar air-coupled transducer with a circular aperture was used as the receiver. It was placed on an automatic scanning robot arm. The diameter of its aperture area is 10 mm. The air-coupled transducer used in the experiment is a transducer capable of transmitting and receiving broadcast frequencies. It can detect an ultrasonic wave in air with a frequency range of 0.2 to 0.5 MHz. The lift-off distance of this air-coupled transducer from the specimen surface is 7 mm. The air-coupled transducer has a small amplitude because it receives ultrasonic waves through the air noncontacting the specimen. Therefore, an amplifier is used to amplify the signal. There are two key points that contribute to the imaging quality of the tomography. One is the algorithm, and another one is the high-accuracy acquisition of signals. By using the automatic scanning robot arm, the air-coupled transducer can be positioned automatically with the right receiving angle. The details of the automatic position will be illustrated. The wave propagation direction and the receiving direction of the air-coupled transducer should be kept in the same straight line in practical application. Although the line source is based on a one-dimensional model, the pulse laser has a Gaussian model [36]. This makes the accurate positioning of the air-coupled transducer very important. Thus, an automatic positioning robot arm was developed in this work. Figure 13 shows the characteristics of the Gaussian beam. In the wave traveling toward the focus \((z\ =\ 0)\), the beam width is continuously reduced, so that the beam width is minimized at the focus.