# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 3 Results and Discussion
### Propagation Parallel to the 1D Phononic Nanoridge
We called these modes A\({}_{0}\), S\({}_{0}\), SH\({}_{0}\), and SH\({}_{1}\) '-like' modes in reference to the antisymmetric and symmetric propagation modes of the plate that are commonly called A\({}_{\mathrm{n}}\) and S\({}_{\mathrm{n}}\) for the modes with \(xy\) polarization, and SH\({}_{\mathrm{n}}\) for the modes with \(z\) polarization. The calculation of the intensity of the displacement fields at the wave vector \(q=0.0135\) nm\({}^{-1}\), which is indicated by a red arrow on the dispersion curves that are represented in Figure 3b,e, clearly shows the localization of the field inside the nanoridge together with their components and symmetries. This confinement is in perfect agreement with the corresponding modes of an infinite plate (not shown here). However, and contrary to an infinite plate, each of the fundamental plate 'like' modes is followed by a set of higher harmonics corresponding to stationary waves along the height of the nanoridge, and displaying more and more oscillations in the vertical direction. An illustration of the second and third harmonics is presented in Figure 3c,f. Another intriguing feature, which is actually not present in the dispersion curves of an infinite plate, concerns the modes lying in the region below the fundamental A\({}_{0}\)-like mode of the nanoridge. To give a better representation of the full set of modes calculated in Figure 2, we show in Figure 4a the phase velocity diagram versus the wave vector \(q\). The diagram is represented with a band-sorting calculation following Equations (1) and (2). The red (resp. blue) color corresponds to modes that have a high value of \(P_{xy}\) (resp. \(P_{z}\)), meaning a strong \(xy\) (resp. \(z\)) polarization. As we can see, at a high-wave vector (\(q=0.05\) nm\({}^{-1}\)), the pseudo modes 'A\({}_{0}\)-like' and 'S\({}_{0}\)-like' of \(xy\) polarization, and their harmonics, below the transverse velocity of epoxy (1350 m/s), move toward a new limit represented by a blue solid line. This line corresponds to the velocity of the well-known Raleigh wave of epoxy. It means that, at a high \(q\) vector, the 'A\({}_{0}\)-like' and 'S\({}_{0}\)-like' modes move toward a localized mode at the surface of the nanoridge without fully penetrating inside the epoxy plate. In addition, two modes appear well below the Rayleigh wave velocity of epoxy and move towards another limit, which is represented by a pink solid line. As shown in Figure 4b, these modes are localized in the corners of the nanoridges. Moreover, one of the two modes is \(xy\) polarized (mode 1), while the other is \(z\) polarized (mode 2). This conclusion is supported both by the color of the modes '1' and '2' in Figure 4a and the components of their elastic fields, as represented Figure 4b. It is well known that such wedge waves, which are confined at solid edges, exhibit a phase velocity below those of surface and bulk waves [26]. These modes exist because of the finite size of the nanoridge. Figure 3: (**a**) Dispersion curves of the modes obtained by applying the relation \(P_{xy}\) (Equation (1)) on the full dispersion curves of the nanoridge in Figure 2. (**b**) Displacement field intensities of both \(S_{0}\)-like and A\({}_{0}\)-like modes and their corresponding \(u_{x}\) and \(u_{y}\) components at the wave vector \(q=0.0135\) nm\({}^{-1}\), which is indicated by a red arrow in (**a**). (**c**) Representation of the second and third harmonics of the \(S_{0}\)-like and \(A_{0}\)-like modes. (**d**) Same as (**a**) by applying the relation \(P_{z}\) (Equation (2)). (**e**) Same as (**b**) for the SH\({}_{0}\)-like and SH\({}_{1}\)-like modes and (**f**) second and third harmonics of the SH\({}_{0}\)-like and SH\({}_{1}\)-like modes. The red dotted lines in (**a**) and (**d**) give the dispersion curves of the Lamb and SH modes, respectively, in an infinite plate with a thickness of \(w=250\) nm.