# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 4 Conclusions  
In summary, we have investigated the propagation of elastic waves along the parallel and perpendicular directions of epoxy nanoridges with high aspect ratios deposited at the surface of a glass substrate. With the help of the finite element method (FEM), we have been able to discriminate the modes belonging to the 1D grating from the full modes of the whole nanostructure. To do so, we have introduced a band-sorting method and separated the modes as a function of their localization, polarization, and symmetry. In the direction parallel to the nanoridges, we found that, due to the high aspect ratio, the surface acoustic waves coincided with fundamental and higher order Lamb modes of a single plate. Due to the finite size of the nanoridge, we also found harmonic vibrations of the fundamental Lamb modes and two edges' eigenmodes localized at the free corners of the nanoridge.  
In the direction perpendicular to the nanoridges, we have shown the emergence of band gaps for SAW due to the interaction of flat branches with the Rayleigh wave near the sound line of the SiO\({}_{2}\) substrate. We also described the evolution of the band gaps as a function of the geometrical parameters of the nanoridges. In particular, we have shown that a polarization mode conversion occurs by changing the height of the ridges. The results presented here can be used as a support in perspective of Brillouin light scattering measurements in this high aspect ratio nanoridge structure [24].  
We also investigated the modes of vibration when the grooves between the solid nanoridges (epoxy) are filled with a viscous liquid (Cargille). The liquid layers enable the constitution of a finite 1D multilayer array at the origin of wave propagation on top of the semi-infinite substrate. In the direction parallel to the nanoridges, the modes follow the dispersion curves of a 1D infinite multilayer array with harmonic modes, whereas in the direction perpendicular to the nanoridges, propagative modes with group velocities different from zero have been introduced. These dispersive modes are the consequence of the coupling between the eigenmodes of the nanoridges through the liquid layers. The manipulation and the control of the elastic properties of such devices, and moreover the behaviors emerging from the periodic arrangement of surfaces, could enable developing a new generation of SAW devices. Since some of the features in the band diagram can be very sensitive to the physical  
Figure 9: Dispersion curves calculated in the direction perpendicular to the nanoridges for different heights of the liquid (Cargille) infiltrating the grooves of height \(h=540\) nm.  
and geometrical parameters of the infiltrating fluid, such a device can be useful for sensing purposes, as both theoretically and experimentally proposed recently, in mixed solid/liquid phononic crystal structures [29; 30; 31].  
[PERSON] designed and performed the simulations; [PERSON] supervised the research; [PERSON] and [PERSON] prepared the manuscript; [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] discussed the results and reviewed the manuscript.  
The work was partially supported by ERC SmartPhon No. 694977.  
The authors declare no conflicts of interest