Maria Bayard Dühring - Solid Mechanics

6.5 High aspect ratio electrodes 51
SH2 has one of order 2 and SH3 has one of order 3. The same is found for the
VP modes where the mode shapes have increasing order in the x1- and x2-direction.
Thus, more modes can exist for larger aspect ratios as the electrode is allowed to
vibrate with modes of higher order.
The fraction of mechanical energy in the electrode Emech,elec with respect to
the total mechanical energy in the structure Emech,tot is plotted in figure 6.14 (b)
for the six modes. The energy is more confined to the electrode for increasing
aspect ratio. The electrode thus acts as a mechanical resonator, which slows down
the SAW velocity because of mechanical energy storage. For modes of the same
type, the fraction of mechanical energy in the electrode tends to the same value
for increasing aspect ratio. The SH modes tend to a value around 0.97, which is
larger than the limit for the VP modes at 0.93. The fact that the limits do not
reach 1 explains that the wave is still (slightly) propagating, or rather coupled from
one electrode to the other by surface waves. If the energy was fully trapped the
surface wave would not propagate at all. For increasing aspect ratio the mode
shapes tend to clean cantilever vibrations of slender beams, and the only energy left
in the substrate is what connects the cantilever modes to the substrate. Modes of
the same polarization type tend to the same energy ratio because they deflect in the
same direction where the stiffness in the substrate is the same. As the wavelength
gets shorter for increasing mode order for a fixed aspect ratio, the aspect ratio for
the higher order modes must be bigger before they reach the energy limits.
6.5.2 Finite structure and acousto-optical interaction
The problem is now extended to a finite structure with twelve electrode pairs by
applying PMLs at the vertical borders, see figure 6.15. The electrodes are excited
with an alternating electrical potential. The values of the phase velocity and the
Figure 6.15 The geometry of the acousto-optical problem with Ni electrodes on a LiNbO3
substrate. Perfectly matched layers absorb the waves at the boundaries and the thick
square indicates the optical domain.