# High-Precision Noncontact Guided Wave Tomographic Imaging of Plate Structures Using a DHB Algorithm
## 3 Experimental Set-Up and Specim
Wave past focus increases beam width almost linearly. Suppose that the electromagnetic wave is traveling in the +z direction, and also assume that the Gaussian beam changes like a Gaussian function in the \(x\) and \(y\) axis directions perpendicular to the \(z\) axis. Then, Equation (7) can be obtained. Figure 9.3  
Here, \(K\) is the wave number, and the gauss beam moves in the +z direction, so the beam width widens and the phase changes, so if \(q(z)\) in Equation (8) is defined as plural, \(Z_{R}\) can be defined by considering Figure 9.4 Referring to Figure 13, \(W_{o}\) has the smallest beam width \((z\ =\ 0)\), so measurement is possible. That is, since Wo is half the beam width, \(Z_{R}\) is also determined by \(W(z)\). \(Z_{R}\) is called Rayleigh length, and \(W(z)\) is increased by \(\sqrt{2}\) times at \(Z\ =\ Z_{R}\), and the position of \(Z\ =\ Z_{R}\), which is doubled by the area of the beam, becomes Rayleigh length. Figure 9.5  
Even if the Gaussian beam travels in the +z direction, the total power must not change, so \(A(z)\) can be expressed as Equation (8). Substituting 2 into Equation (7), the Gaussian beam formula is  
Figure 9.6 Gaussian beam propagation factor. defined as Equation (9). By installing the lens using the formula, a plane wave is generated through the lens, and when passing through the slit, a desired wave can be created. Figure 9.7  
As shown in Figure 14, the scanning process consisted of four steps, and the initial condition is given in step 1. The air-coupled transducer should firstly revolve at a certain angle \(\alpha\) around the test center when it is necessary to change the detection points. Then, the slit mask is rotated to make the wave propagation direction stay in the same straight line with the new receiving angle. This process can be very complicated with manual computation and operation. Thus, automatic control was adopted in this work. The LED laser line sources and photoresistors were installed on the edge of the slit mask and the rotation center of the air-coupled transducer. They were marked as LED\({}_{1/2}\) and R\({}_{1/2}\), respectively. All the automatic movement in this work is based on a step motor. To rotate the slit mask, LED\({}_{1}\) will be turned on at the same time as the step motor M\({}_{1}\). The illumination direction of LED\({}_{1}\) is the same as that of the wave propagation. The step motor M\({}_{1}\) will stop by itself once the light of LED\({}_{1}\) points to the photoresistor R\({}_{1}\). Then, LED\({}_{2}\) will turn on together with M\({}_{2}\), and the step motor M\({}_{2}\) will also stop automatically once the light of LED\({}_{2}\) points to the photoresistor R\({}_{2}\). The new detection point can be fixed with the right receiving angle automatically based on the process discussed above. The complex calculations and manual operation can be omitted. The design of the automatic scanning is shown in Figure 9.8 The key part here is the photoresistor. Its electrical resistance will decrease immediately if it is illuminated by the light of the laser LED. Then, the corresponding circuit that was used to control the step motor will close. Conversely, the circuit will turn off. In this work, a single chip computer was used to control the step motor. The step motor will be fixed by its magnetic pole once the circuit is controlled by the photoresistor. It can guarantee that the experimental equipment will never be influenced by potential factors from outside. In other words, the movement of the transducer can only be controlled by a program. Then, the detection points can be positioned with very high accuracy with the right receiving angle of the air-coupled transducer. Figure 9.9 Schematic of the automatic scanning robot arm. In a practical environment, the photoresistor can also be influenced by sunlight or the bulbs in the lab.