# Experimental Demonstration of a Multichannel Elastic Wave Filter in a Phononic Crystal Slab
## 3 Results and Discussions
As noted in Figure 3c, sending a wave within the band gap frequencies, \(f=265\) kHz, results in a total reflection along all directions of the crystal inducing multiple interferences which generate standing waves pattern limited by the white line that marks the crystal boundaries of the device. The effect of this omnidirectionality is corroborated by numerical simulation of a harmonic single source vibrating at the same frequency as shown in Figure 3d. We also observe the same standing waves form as a result of the total reflection from the pillar array. In addition, the numerical simulation shows a strong attenuation of the incident wave along a short distance inside the pillar based phononic crystal. Different defects inside the perfect crystal are introduced by inserting a space gap between two rows of pillars in the wave propagating direction. The defect width ensures the presence of multiple waveguide modes inside the band gap. The band diagram of the waveguiding prediction is based on supercell technique. Figure 4a illustrates the \(1\times 7\) supercell with defect width \(W_{g}\) that has been used to calculate dispersion relations and the eigenvectors of the defect modes. This technique assumes a periodic boundary condition, with phase varying, along \(x\) direction and periodic continuity, without phase change, in \(y\) direction. The 8 rows of pillars that separate each of these linear waveguides, along \(y\) axis, are sufficient to avoid any interaction between the waveguide lines in the frequency band gap region. The geometry variables of defect-containing structures, i.e., lattice parameters, \(a\), height, \(h_{p}\) and radius, \(r_{p}\) of pillar are maintained as perfect crystals to ensure a similar frequency gap limit. To run numerical simulation for guiding effect, three defects widths \(W_{1}=1.52\) mm, \(W_{2}=1.2\) mm and \(W_{3}=0.88\) mm are set up. These waveguide apertures are carefully designed to cover a wide range of frequencies within the band gap and to ensure a selective transmission in each path of the waveguides. Figure 3: (**a**) PSV-500 set-up to measure the surface cartography displacement for demultiplexer device, (**b**) scanning area (red region) equivalent to 113 mm \(\times\) 56.8 mm surface dimension, (**c**) experimental surface measurement at the band gap frequency, \(f=265\) kHz, (**d**) simulated displacement at similar band gap frequency, as (**c**). The white lines mark the boundaries of the pillar based phononic crystal in both (**d**) and (**c**). The numerical study summarized in Figure 4b reveals several pass bands that do not exist in native perfect crystal structures due to the defect modes associated with the different waveguide widths. Basically, the three different waveguide thicknesses insert three branches at different cutoff frequencies inside the band gap, as depicted in Figure 4b with marks \(A\), \(B\), \(C\). It covers range of frequencies \(f_{W_{1}}=275-288\) kHz, \(f_{W_{2}}=288-303\) kHz and \(f_{W_{3}}=310-320\) kHz that represents waveguides \(W_{1}\), \(W_{2}\) and \(W_{3}\). The position and frequencies of these pass bands are adjustable as a function of the waveguides' widths. At higher frequencies, beyond the \(C\) band, numerous transmitting bands occur inside the band gap, which are related to new modes or to the folding back of the waveguide bands that appear at low frequencies of the band gap. This folding is due to the pillar edges periodicity of the waveguide line. The transmitting bands close to the band gap upper limit are not used in the demultiplexing operation for two reasons: \((1)\) the pass band has interaction with the bulk modes, i.e., Lamb modes, which induce a radiation in other directions of propagation \((2)\) each pass band crisscrossing with one another and create a weakly interaction between transmitting bands. Thus, these two conditions reduce the energy confinement within the waveguide and therefore have been ignored in our transmission band selection.