# Phononic Coupled-Resonator Waveguide Micro-Cavities
## 2 Results  
The phononic crystal slab sample we consider is shown in Figure 1. The crystal itself is a square lattice of cross-holes [16] with lattice constant \(a=714~{}\upmu\)m. Between each group of four nearest holes, a square-shape membrane resonator is defined and is connected to its four neighbors. Figure 1: (**a**) Optical microscope image of the sample considered in this work. The 500 \(\upmu\)m-thick fused silica plate is processed by femtosecond laser-assisted wet etching (FLAE). Cross-holes are removed from the plate to define a phononic crystal slab structure with 18 rows and 20 columns. Defects, numbered from 1 to 11, define a set of three coupled-resonator waveguide cavities. The source region, placed inside the crystal structure, is in contact from the bottom with a piezoelectric vibrator. Out-of-plane displacements of the top surface are measured by a scanning laser vibrometer. The lattice spacing is \(a=714~{}\upmu\)m. (**b**) Scanning electron microscope view of a cross-hole and zoomed view at a corner showing the side-wall rugosity. The phononic crystal slab has a large complete bandgap that is detailed later on. As a note, cross-holes provide a much wider bandgap width than circular holes, for instance [16]. There are even other unit-cell designs that still allow gaining on the bandgap width [17], but we selected cross-holes in this work because of the simpler footprint for technology. For frequencies within the complete bandgap, there are only evanescent Bloch waves and the coupling between defects decreases exponentially with their separation. A total of eleven identical defects are introduced and are grouped in three coupled-resonator waveguide cavities: defects 1-4 form CREW cavity C1, defects 5-8 form CREW cavity C2, and defects 9-11 form CREW cavity C3). Defect resonators in the cavities are separated by a distance \(2a\). Comparatively, the shortest distance between two cavities is \(4a\). Hence, it is expected that CREW cavities are only weakly coupled. Experimental details regarding sample preparation and characterization are identical with Ref. [15] and are not repeated here. The phononic crystal dispersion properties are summarized in Figure 2. All computations are performed using the finite element method. Phononic band structures are obtained using the unit-cell of Figure 2a for the phononic crystal and using the unit-cell of Figure 2b for the CREW. Periodic boundary conditions are applied on all lateral sides, whereas the bottom and top surfaces are left free; the Bloch wavevector is fixed and frequencies are obtained by solving an eigenvalue problem [13]. The band structures are shown only for out-of-plane polarized Bloch waves, as only that displacement component is accessible in our experiments. Geometrical parameters are chosen to ensure the existence of a wide complete bandgap, extending between 1.7 and 4 MHz  except from a flat bulk band appearing around 3.2 MHz. The smallest length in the structure is \(a-b=a/10=71\)\(\upmu\)m, much larger than the estimated resolution of \(\approx 1\)\(\upmu\)m  of the technological process for the holes. The CREW considered in this paper are formed from defects separated by two lattice constants. Defects are simply created by omitting to etch chosen cross holes. As a result, it is expected that elastic waves can become spatially localized around the defects, for frequencies inside the complete bandgap, decaying exponentially away from the defect center [18; 19; 20]. If defects are placed not too far away from one another, they are in their mutual near-fields and are thus evanescently coupled. Guided elastic waves can then transit from one defect to the next. The dispersion of guided waves in a CREW is very smooth and is dictated by the coupling coefficients between resonant defects [13]. In Figure 2, many guided bands appear within the complete bandgap, each of them originating from a given resonant mode. In relation to experiments, we have selected three frequency ranges around 2.25 MHz (range A), 3 MHz (range B), and 3.55 MHz (range C).