# Experimental Demonstration of a Multichannel Elastic Wave Filter in a Phononic Crystal Slab
## 3 Results and Discussions
To illustrate the transmitting wave inside the waveguides, we calculate the out-of-plane displacement at the limit of the first Brillouin zone around \(X\) point in Figure 4c, three consecutive frequencies \(f_{A}=288\) kHz, \(f_{B}=303\) kHz and \(f_{C}=320\) kHz, which respectively, present the waveguide \(W_{1}\) (bottom), \(W_{2}\) (middle) and \(W_{3}\) (top) are calculated. The out-of-plane displacement clearly shows a spatial confinement of the energy inside the defect for three waveguides cases and only interacts weakly with the neighboring pillars. The theoretical results present important characteristics for the possibility of tunable elastic waves filter in slab structure by changing the defect distance between two neighboring pillars. In addition, the strong attenuation in lateral direction (along \(y\) axis) shows the vanishment of the Lamb wave magnitude after three periods. Consequently, a minimum  
Figure 4: **(a)** Supercell technique to measure three waveguide dispersion curves. \(W_{g}\) is the defect distance with \(W_{1}=1.52\) mm, \(W_{2}=1.22\), \(W_{3}=0.88\) mm. **(b)** Dispersion curve for \(W_{1}=1.52\) mm (red), \(W_{2}=1.22\) (orange), \(W_{3}=0.88\) mm (blue). The grey area is the band gap region that ranges from 274 kHz to 350 kHz. **(c)** The out-of plane displacement profile for waveguides, \(W_{1}\) (bottom), \(W_{2}\) (middle) and \(W_{3}\) (top). The dotted-circle indicate the pillar structures. of seven periods between two waveguide lines ensures there is no interaction in the lateral side of the waveguide lines. To verify experimentally the feasibility of our waveguides as a filter, an actual device that consists of three line-defect waveguides as described by Section 2, is manufactured. To measure the transmission coefficient experimentally, a similar setup to the one reported in the Section 2 with \(MSA500\) is used where the spot of laser interferometer is focused at the end of each waveguide line. The scanning area consists of a square surface of \(4\times 4\,\mathrm{mm}^{2}\) divided into 16 measurement nodes. For each frequency, and for each waveguide, an average of the 16 nodes of the out-of-plane displacement measurement is calculated and reported in Figure 5. As predicted, the experimental transmission coefficient reveals three bands inside the band gap. The device acts as a frequency filter in three frequency ranges of \(f_{W_{1}}=274-287\,\mathrm{kHz}\) (red curve), \(f_{W_{2}}=285-300\,\mathrm{kHz}\) (orange curve) and \(f_{W_{3}}=301-315\,\mathrm{kHz}\) (blue curve). It is also noticed that the bands \(W_{1}\) and \(W_{2}\) are completely separated, i.e., without crossing over. For instance, the transmission magnitude of \(W_{2}\) drops immediately at frequency \(f=302\,\mathrm{kHz}\) (orange curve) while at the same time, a notable transmission peak occurs at frequency \(f=305\,\mathrm{kHz}\) which represents \(W_{3}\) channel defects (Blue curve). On the other hand, the bands \(W_{3}\) and \(W_{2}\) have a common narrow band that extends from 280 to 287 kHz. The theoretical transmission shown in Figure 5b has a similar noticeable peak and dip transmission comparable to the experimental data. Finally, an input signal that covers the bandwith of the band gap will be transmitted separately along \(W_{1}\), \(W_{2}\) and \(W_{3}\) according to frequency range of each waveguide. This shows the capability of the device to act as a wave demultiplexer. To demonstrate the separation of the Lamb wave through the filter device, a displacement field measurement based on scanning laser Doppler vibrometer (SLDV) is realized as shown in Figure 6. The measurement procedure is similar to the method described in Section 2. Figure 6  
Figure 5: (**a**) Experimental and (**b**) simulated transmission spectra for three waveguide channels \(W_{1}\), \(W_{2}\),\(W_{3}\). The grey area marks the region for band gap. corresponds to the theoretical and experimental data of the out-of-plane displacement for the wave propagating through the demultiplexing structure.