# Phononic Coupled-Resonator Waveguide Micro-Cavities
## 2 Results
A total of five guided waves are apparent, each with their specific modal shape. The two pairs of guided waves in ranges A and C have orthogonal polarizations and hence their dispersion curves can cross without interference. Their respective actual excitations in the experiment depend on the symmetry of the excitation source. Ultrasonic waves in the experimental sample of Figure 1 are excited thanks to a piezoelectric vibrator in contact with the bottom surface in the source region. Coupling from the source region to the resonators occurs mainly with defect 1. Out-of-plane vibrations at the top surface are measured using a scanning laser vibrometer (Polytec Microsystem Analyzer MSA-500, equipped with analog displacement decoder model DD-300; 0.03-24 MHz frequency response with 50 nm/V sensitivity). The source waveform sent to the piezoelectric vibrator is a periodic chirp covering the frequency range of interest. The amplitude of out-of-plane displacement measured at three different locations is plotted in Figure 3a. Measurements averaged over defect 4 are considered typical of CREW cavity C1, whereas measurements for CREW cavities C2 and C3 are averaged over the unit cells just right of defect 8 and defect 11, respectively. Displacements larger than 0.3 pm  can be recorded with this setting. The measurements reveal a sequence of very sharp resonance peaks, in agreement with previous results on a sample with a single chain of seventeen coupled resonators [15]. Compared with that sample, however, the resonance peaks are distributed over a larger span of frequencies, especially in the range from 2.1 to 2.3 MHz. Measurements can be compared directly with a finite element model based on a stochastic force distribution applied in the source region [21]. To this end, a three-dimensional mesh is created to represent the full sample shown in Figure 1, with a perfectly matched layer (PML) surrounding the finite size crystal. Free boundary conditions are still applied on the bottom and top surfaces. Material losses in fused silica are added to the model. A fair correspondence is observed between experiment and numerical simulation in Figure 3. In particular, the distribution of resonance peaks and their central frequencies are rather well-reproduced numerically, but the response levels are less reliable. Compared to the experiment, indeed, a fully spatially random body force distribution is considered numerically, whereas the actual piezoelectric excitation has a more symmetrical and homogeneous distribution. Furthermore, the resonance peaks are very narrow and are resolved experimentally with only a few sampling points. In addition to the frequency response functions shown in Figure 3a, the scanning laser vibrometer can form images of the spatial distribution of the out-of-plane displacement at the surface of the sample at a fixed frequency. We show in Figure 4 the measured displacement fields at three selected frequencies for which maxima of vibration are observed experimentally in CREW cavity C1. Each of these measurements is compared with the finite element numerical simulation obtained at the same frequency. Figure 2: Dispersion of the phononic crystal slab of cross holes. (**a**) The primitive unit cell is defined by four parameters: the lattice constant \(a\), the slab thickness \(h\), the cross length \(b\), and the cross width \(c\). (**b**) A coupled-resonator elastic waveguide (CREW) is formed from defects separated by two lattice constants and created by omitting one hole. The depicted supercell has seven rows and two columns. (**c**) The band structure of the crystal is shown for out-of-plane modes only and for \(b/a=0.9\), \(c/a=0.25\), and \(h/a=0.7\). The complete bandgap appears between the gray regions. The color scale of the bands is proportional to the out-of-plane component of displacement. (**d**) The band structure of the CREW contains a series of flat (small dispersion) bands, each holding a guided Bloch wave. (**e**) Close-up views of three selected frequency ranges are shown.