Acoustic waves at any frequency first couples into the

Acoustic waves at any frequency first couples into the corresponding surface mode of the lower surface, Fig. 3. 22a delay of the onset of the upper PnC ensures that component of the incident wave reflected off the lower PnC at an angle θ with the horizontal axis cannot couple the upper PnC surface mode, while it is reflected back into air by the vertical surface of the upper PnC. Thus, any wave component propagating along the upper PnC surface can only be excited by coupling from the lower surface so that the wave initially propagating along the lower surface starts coupling into the upper surface, and then back into the lower surface, Fig. 3.
For a better understanding of mode coupling, acoustic pressure field is probed on the horizontal dashed lines marked as and in Fig. 3(a), which are a/5 away from the respective surfaces. The plots below the FEM simulation results in Fig. 3 present variation of and along the surfaces, which are normalized to a common value in each plot for simplicity. These plots reveal that the peak of appears at higher x as f increases, implying that increases with f, as expected. is visually calculated from the variations of and in Figs. 3 and 4 as the distance between the first minimum of the envelope of (where acoustic atm inhibitor is maximally transferred to the upper surface) and the following maximum (where the energy is maximally coupled back into the lower surface). Visually calculated values in Fig. 3 are 36.3, 38.4, and 42.3 at 1.500MHz, 1.520MHz and 1.540MHz, respectively. Although they follow the general trend of increasing with f, these values are somewhat different from the above-mentioned values calculated from the BS.
The plots of and in Fig. 3 show that the dash-dotted envelope curves do not follow a smooth variation. Instead, each envelope demonstrates a fluctuation, apart from the variation reflecting coupling. The fluctuations are due to the beating behavior of the surface modes on single surfaces [25,27]. The beating occurs due to the fact that each mode at a specific is associated with a counter-propagating mode displaced by a reciprocal lattice vector corresponding to surface periodicity [25]. As the beating and mode coupling superpose, it is difficult to tell where the minima and maxima of the envelope curves occur. Thus, is loosely calculated from Fig. 3 by measuring the distance between a minimum and subsequent maximum of within the coupling region.
The acoustic wave leaves the interacting PnCs mainly from the upper surface at f=1.500MHz, as the device length is such that almost a complete cycle of coupling back and forth is achieved and a new cycle begins, Fig. 3(a). As the frequency reaches 1.520MHz, wave output from the lower PnC is more dominant, Fig. 3(b). Finally, the wave leaves the device almost totally from the lower PnC at 1.540MHz, Fig. 3(c). If one probes the acoustic output from each surface and compares them, variation of coupling length with frequency can be calculated.
FEM simulation results for W=3.5a corresponding to the frequency range where drops with f is presented in Fig. 4 at frequencies of 1.565MHz, 1.570MHz and 1.575MHz. The corresponding values calculated from BS are 47.45, 46.53 and 43.99, respectively. In contrast, measurement of distances in Fig. 4 reveals =42.0, 41.6 and 38.4, respectively. These values are calculated by again measuring the distance between a minimum and the subsequent maximum of the envelope curve for . They are considerably smaller than the expected values. The reason is again the beating behavior that interferes with mode coupling. Thus, it is difficult to tell where maxima and minima occur.
Inspection of Fig. 4(a) shows that there is comparable wave output from the lower and upper surfaces at 1.565MHz. In contrast, waves leave the system mainly from the lower PnC at 1.570MHz and 1.575MHz, Fig. 4(b) and (c). Although a general conclusion cannot be drawn by inspecting Fig. 4 only, it is evident that the ratio of output acoustic power from the two surfaces vary rapidly with frequency. This will be clarified later in the discussion of Fig. 7.