2 2 2
1 2 cos tL 2 cos tS
L L S
A A B
c c c , (4.13)
where A1, A2 and B2 are the amplitudes of the input longitudinal wave, transmitted longitudinal wave, and the transmitted shear wave, respectively. The corresponding power and amplitude graphs are shown in the right side of Fig. 4.18(a). From the equation, it can be clearly inferred that as the refracted angle increases, the amplitude of the transmitted shear wave also increases while the amplitude of the transmitted longitudinal wave stays unity due to total transmission condition. To confirm the theoretical analysis, simulations of various transmitted angles are presented in Fig. 4.18(b). The results show good accordance with the theoretical results by depicting increased shear amplitudes as the refracted angles become larger.
the plane-stress condition, which is applied throughout this work. Accordingly, the lowest symmetric Lamb wave (S0), which is a good approximation [36, 39] for in- plane longitudinal wave in thin plates, is implemented for the experiment.
The experimental set-up for visualizing the wave propagation is shown in Fig.
4.19(b). The host plate is fixed along a metallic wall by using magnet blocks.
Specifically, four magnets are placed between the plate and the wall near the four corners, and additional magnets are attached on the other side of the plate to fix itself along the wall. In this way, we can realize the traction-free as well as plane- stress condition by freeing the plate from the wall. It should be noted that the magnet blocks are placed far enough from the source transducer to avoid undesirable interference on the magnetostriction phenomenon of the employed transducer.
On the other hand, to experimentally realize the widely-uniform S0 wave, as employed in the simulation in Fig. 4.16, we implemented a giant MPT, as shown in Fig. 4.19(b). This lab-made ultrasonic transducer consists of permanent magnets, a solenoid copper coil array, and an acryl housing to configure the components in their positions. Most importantly, a magnetostrictive material (here, we utilize a nickel patch of dimension: 300 mm × 25mm × 0.1 mm) must be bonded onto the plate to induce strain, which is responsible for the elastic wave generation. In this work, we utilize a shear couplant to couple the patch with the plate. When the giant MPT is installed onto the plate, the dynamic magnetic field induced into the coil of the MPT generates a strain in the patch, which in turn induces a strain in the plate.
In this particular structural assembly, our MPT predominantly generates the longitudinal mode due to the orthogonally arranged copper coil array and static magnetic field [106]. Additionally, the MPT was tuned to our target frequency (100 kHz) by setting the interval to half wavelength. In a thin plate, this induces the lowest-symmetric Lamb (S0) mode, as explained above. For more information on the detailed working principles of such MPTs, see Refs [66, 103, 106].
The source signal for the experiment was initially generated using a function generator (Agilent 33220A), then amplified by a power amplifier (AE TECHRON 7224), and then sent to the MPT. As for the input signal, a Gabor pulse (or modulated Gaussian pulse) of 100 kHz, which is shown in Fig. 4.19(b), was used to ensure adequate frequency localization. A digital oscilloscope was used to conveniently handle the source and received signals. The transmitted field is picked up by a laser scanning vibrometer (Polytec PSV-400), which is aimed perpendicular to the plate surface in order to scan the displacement fields. The received data were automatically averaged 800 times to remove unwanted signal noises. Additionally, a band-pass filter embedded in the laser vibrometer software was employed to eliminate unwanted frequency components.
As shown in Fig. 4.20, we scanned the transmitted regions, which are indicated by the purple squares in Figs. 4.16(a) and 4.16(c), for the refraction and focusing cases, respectively. The zoomed-in views at specific times (t) from the transient wave simulations are shown in Figs. 4.20(a) and (b), along with the coordinate information with respect to the x-y axes to specifically denote the region of interest.
We set the scanning region to be 0.2 m × 0.15 m for the refraction case as shown in Fig. 4.20(a). The mesh grid was set to 14.3 mm × 15 mm in order to divide the entire region into 15 × 11 points to ensure that the neighboring measurements are within the subwavelength scale. The sequential images with about 4 μs of time interval match well with the simulation, which also reveal the actual propagation phenomenon in the region of interest.
On the other hand, for the focusing case, which is shown in Fig. 4.20(b), we plot the fields with a scale that is 1.75 times greater than that for the refraction case (shown in Fig. 4.20(a)) in order to properly display the enhanced amplitudes resulting from the concentration. For these two cases and the others throughout this work, the input power is fixed to 1 (in units of normalized divergence) for consistency. We utilized the same mesh dimension as in the former case; however, here 11 × 9 points were fit into the region of interest. The experimental results shown in Fig. 4.20(b) are also in good agreement with the simulation result. The numerical and experimental results, which match well with the theoretical prediction, demonstrate the excellent performance of the designed substructured metasurfaces in beam pattern manipulation.
Remarks on the experimental results
For the experimental results, the sequential images are captured from an imaging program embedded in the laser scanning vibrometer (Polytec PSV-400). In other words, no signal post-processing was performed and the propagating wave fields
are drawn from the raw data only. Although band-pass filter and averaging method were applied beforehand, we did not perform any signal processing with the raw data to observe the actual results in real environment. Therefore, unlike the results from simulations, the experimental results contain some noises from other frequency components due to high sensitivity of the laser vibrometer. Nevertheless, the results confirm that the beams are successfully refracted and focused by the subwavelength metasurface layers.
To thoroughly investigate the experimental results, we additionally analyzed the raw data. We performed Fast-Fourier-Transforms (FFT) to efficiently derive the frequency components. As can be seen in Fig. 4.21, the raw data (although containing other noise components) has predominantly 100 kHz component which was in fact the frequency at the input source.