# Experimental Demonstration of a Multichannel Elastic Wave Filter in a Phononic Crystal Slab
## 3 Results and Discussions  
Before analyzing band diagrams of defect modes, it is necessary to calculate the mode dispersion of the perfect crystal which is considered as a reference for the band structure of defect modes. By changing the value \(k\) in the first irreducible Brillouin zone, FEM allows us to solve the eigen-value problem and calculate the band diagram of the phononic crystal. As illustrated in Figure 2a, the band diagram of the perfect crystal shows a complete band gap as well as two partial band gaps for the geometrical parameters listed in the previous section. The two partial band gaps occur at frequency ranges 382 to 405 kHz and 485 to 497 kHz while the complete band gap exists in the range 272 to 347 kHz. This latter appears to be a Bragg band gap, which is due to destructive interferences of the scattering waves from the pillar array. To compare these simulations with experimental results, a transmission through a finite crystal that consists of 7 pillars periods in the direction of incoming wave is shown as a function of frequency in Figure 2b. Numerically, the out-of-plane polarized excitation source is set as the input signal and perfectly matched layers (PML) are incorporated in the direction of outgoing waves to avoid any reflection from the boundaries. On the experimental side, the spatial scanning area of \(1.6\times 1.6\,\mathrm{mm}^{2}\) (\(5\times 5\) points spatial grid) on the perfect crystal, i.e., without defect lines, is considered using MSA-500 setting as described earlier. The measurements are represented in the form of Reduced Displacement (RD) according to Equation (3) [31]. \[u_{\mathrm{dB}n}=20\cdot log_{10}\left(\frac{1}{N}\sum_{k=1}^{N}\frac{|u_{k}|} {1\,\mathrm{nm}}\right) \tag{3}\]  
where \(N\) is the number of measurements made during an integration time of 3.2 ms. The index \(\mathrm{dB}n\) indicates that the quantity \(u\) is expressed in normalized dB scale with reference to one nanometer of displacement (nm). A good agreement between experimental and simulation results shows a pronounced dip in the frequency span between 262 to 347 kHz kHz suggesting the existence of a band gap region as predicted by the dispersion curve. A small discrepancy in the simulated curve showing low frequency shift is justified by the mismatch between the theoretical values of the physical parameters used in the simulation and the experimental values of the printed material that are sensitive to the fabrication conditions. To show the omnidirectionality of the band gap, we conduct a surface topography measurement with laser Doppler vibrometer (SLDV) \(PSV\)-5003\(DHV\)by \(Polytec\) as depicted in Figure 3a. The instrument has the capability to scan larger surface areas of sizes ranging from a few mm\({}^{2}\) to several cm\({}^{2}\). This capability is sufficient to illustrate a potential situation for the actual wave travelling in devices of larger dimensions. The scanning covered a surface area of \(113\times 56.8\,\mathrm{mm}^{2}\) (proportionate to \(97\times 49\) points spatial grid) and took place on the backside surface with no phononic pillar-based structure, Figure 3b. The laser vibrometer is perpendicularly positioned at about 0.8 m distance from the top surface of the device. During the measurement we used only one SLDV head with no noise filter to detect the out-of-plane vibration of the plate surface. A similar procedure is repeated by using a piezoelectric ceramic with an out-of-plane polarized excitation. The same chirped signal  
Figure 2: (**a**) Band diagram of unit cell (**b**) simulated (black) and experimental (red) transmission for the band gap between 260 kHz to 347 kHz. was applied to the piezoelectric transducer ensuring broadband mechanical excitation ranging from 250 to 400 kHz.