M-Y-Γ-X-M-Γ band structures

For the anisotropy of the substrate—90°-rotated ST-cut quartz, the complete band structure is calculated and plotted along the connecting lines between M-Y-Γ-X-M-Γ points on the boundary of the irreducible Brillouin zone.

First we shall introduce the concept of the sound cone. The sound cone is usually defined as the slowest bulk wave velocity of the substrate in the dispersion relation. The constant velocity surface forms a cone in the 2-D dispersion relation plot, but in the band structures that will be discussed later, the cone is sectioned as lines (the sound lines). Surface waves with dispersion relation above the sound lines are considered to be leaky for the reason that the apparent wavenumber on the surface support a certain angle such that the disturbance created by the surface wave can be transformed as the bulk wave and leaks. However if the particle motion of surface wave has very little compo-nent on that of the bulk wave, the propagation loss may be negligible. We plotted the sound lines from not only the slowest but all the three bulk wave velocities in the sub-strate according to the slowness curve Fig 2.8, and along M-Y-Γ-X-M-Γ. It is shown in Fig 2.20.

The band structures of the phononic crystals with r/a = 0.1, 0.2, 0.4 are plotted in

Fig 2.21 (unfiltered) and Fig 2.22 (color marked). Instead of making a dichotomy of the bulk/surface modes; Rayleigh/Love modes, we use graded colors and opacity to indicate the polarization (ROT) and energy depth (DOE/λ), respectively. Where the reds show the polarizations have larger components perpendicular to the sagittal plane and the blues have larger components in the sagittal plane. The dots with higher opacity repre-sent the energy are concentrated near the surface. Also the three sound lines are plotted in the figures for reference. With the graded color we can observe more things from the band structures such as mode coupling phenomenon, and it is also too assertive to clas-sify the modes in a “black or white” way especially in the PCs consist of anisotropic materials.

We shall notice that all modes near the Γ point seem to be surface modes as they have small DOE/λ. But actually it is a false appearance caused by the finite-depth mod-el. Near the Γ point (|k| → 0) the wavelength approaches infinity therefore in a fi-nite-thick plate the DOE/λ of all modes naturally approach zero. Most modes in this re-gion behave like the plate modes and the surface modes have not developed yet. So we may ignore this region.

Some invisible lines influence the dispersion curves significantly. (See Fig 2.21.) Compared with the sound lines we found them identical. Consider the plate modes as bulk waves trapped and reflecting in the quartz plate, their phase velocities are always larger than that of the bulk waves. Thus the plate modes are always located above the corresponding sound lines. For example under the slowest shear wave (quasi-SV) sound line there is no plate modes, and between quasi-SV and quasi-SH sound lines there are plate modes with polarization in the sagittal plane, and above the quasi-SH sound line the plate modes with SH polarization also show. In addition, near the quasi-longitudinal

sound line, mode coupling can be observed.

Mode coupling also exist between the plate modes and the surface modes. A clear example is the quasi-Rayleigh mode and the quasi-in-plane type plate mode shown in the black circle in Fig 2.22 (c). In general with the existence of the PCs that contain complex 3-D boundary conditions, the polarization of the surface modes is unlikely to be uncoupled (i.e., have no component) with the plate modes, just with different propor-tion. For example the quasi-Love modes can be only slightly coupled with qua-si-in-plane type plate modes. Mode coupling happens when the two modes being closed in the band structure, and level repulsion will be observed, which is that the two disper-sion curves avoid from crossing each other, and the two modes becomes like each other.

Away from the coupling region, the two modes revert to their original appearance.

Considering the fact that the dispersion curves of the plate modes depend on the substrate thickness, and the plate modes becomes more and denser in the band structure above the sound line with increasing of the substrate thickness (eigenfunctions along the thickness), we may conclude that the more and more discrete plate modes becomes con-tinuous modes that represent bulk modes with different down-propagating angles, when the substrate’s thickness approaches infinity (again the plate modes can be regarded as reflecting bulk modes). The coupling between the surface modes and the plate modes in the calculation predicts that between the surface modes and bulk modes in the real situ-ation, that is, it indicates the leakiness of the surface modes.

It is also observed that the “folded” surface modes is also coupled with the plate modes that have not experience the folding. An example is shown in the black circle in Fig 2.22 (a). From the group velocities and mode shapes we may think the folded bands are due to the periodicity in k-space and is from another reciprocal lattice so they have wavenumbers 2π/a – k, different from the unfolded bands’ wavenumber k, so there

should not be coupling. But actually there is.

Assume that there are two modes propagating in a fictional homogeneous medium, which the dispersion curves are like Fig 2.23 (a). Then consider a phononic crystal with small modulation, with period a, the dispersion curves will repeat with the period equals to the reciprocal lattice, like Fig 2.23 (b). The crossing point becomes a singular point since the group velocity (propagating direction) is not defined. To fix the unusual phe-nomenon an anti-crossing (i.e. level repulsion) will occur here so the singularity is not exist. As the modulation of the PC increase, the level repulsion or the coupling becomes stronger, as Fig 2.23 (c). The Bragg band gap indicated by the black circle in Fig 2.23 (a) can also be explained by such interpretation, and the coupling of right and left propa-gating waves forms standing waves on the edges of the gap. For a strongly modulated PC, one cannot even classify a mode to a reciprocal lattice, and the waves can only be regarded as a summation of Bloch waves with different G’s so the coupling will gener-ally occur.

So we can conclude that in the PC, a surface mode become leaky as it lies above the sound lines, no matter it is a folded band or not. The degree of leakiness depends on the magnitude of the component of the surface mode on the bulk mode. The degree of leakiness of a “folded” surface modes is also depends on how strong the periodic mod-ulation is.

As for the band gap, we found no complete (all direction) band gap for Love waves in the three PCs. But for the partial band gap, all the three PCs have, and without sur-prise, the gap width grows wider as the hole radius increases. For the same PC structure, it is found that the partial band gap of the Rayleigh wave is much smaller than that of the Love wave. It is due to the fact that SH-type SAWs are more sensitive to the surface loadings.