## 4 Local Acoustic Dispersion Curve of GESA Modified Sample  
In order to determine a local acoustic dispersion curve of the GESA coated sample, the transient grating evolution of the tangential component of the gradient of the normal displacement at the surface of the GESA modified coating-substrate sample was detected by the GILD setup for a set of 10 different diffraction grating spacings ranging between 200 and 18 \(\upmu\)m and which were scanned laterally with a scanning step of \(\mathrm{d}x=0.5\)\(\upmu\)m. As mentioned above, for each grating spacing, multiple wavenumber components were excited, because of the non-sinusoidal glass etch relief of the diffraction mask. As a beneficial consequence, for each fundamental grating spacing, multiple points on the acoustic dispersion curve could be extracted. In Figure 7a the acoustic spatiotemporal evolution of the laser beam deflection signal profile at the coating surface of the GESA modified sample is shown for a grating spacing of 120 \(\upmu\)m. The slowly varying thermal decay was removed by a 1 st order high-pass filter with a cut-off frequency of 2 MHz to enhance the visibility of the acoustic signal contribution. Each horizontal line corresponds to a particular spatial position of the diffraction grating. Since the location of the detection point was not altered during the spatial scan of the diffraction grating, each line appears to be phase shifted with respect to the previous line. As a result, a spatiotemporal rhombus pattern is created due to the crossing of right- and left-traveling surface waves with a wavelength of 120 \(\upmu\)m. SAWs with a smaller wavelength component appear as nearly vertical lines as they propagate with a higher frequency. As a consequence of the asymmetric location of the detection point with respect to the center of the Gaussian spatial profile of the laser beam, a relatively large portion of left-traveling waves were detected. In order to select the acoustic traces corresponding to the discrete wavenumber components, a one-dimensional spatial Fourier transform was calculated. The result of this operation is depicted in Figure 7b. Each bright line accords to a particular wavenumber component of the acoustic field. Especially the discrete wavenumber \(k=8.38\) mm\({}^{-1}\), corresponding to the fundamental wavelength of \(\lambda\sim 120\)\(\upmu\)m is clearly visible. In Figure 8a, the temporal evolution of the normalized signal is plotted for several these bright lines. Each of the lines carries information about the dispersive relation between the phase velocity and frequency/wavelength, which can be extracted after transforming the time signal into Fourier domain, as is shown in Figure 8b. Using a Lorentzian frequency peak fitting procedure, the fundamental frequency of each signal was extracted. This procedure was repeated for all the 10 grating spacings resulting in a complete acoustic dispersion curve. The width of the SAW frequency peak was used as the uncertainty of the extracted frequency. Together with the wavelength of the SAW, which could be exactly calculated using the wavenumber  
Figure 6: Schematic representation of the scanning algorithm for the topographical mapping of the GESA coating’s elastic properties. component of the bright line of local maximal displacement in Figure 7b, a particular phase velocity was associated with each frequency-wavelength couple. The resulting experimental dispersion curve is shown in Figure 9. As error bars, we have taken the widths at half maximum \(f\) of the respective peaks in the signal spectrum, and the propagated uncertainties on the phase velocity \(c\), via \(c=\lambda f\). The relative uncertainty on the grating spacing was much smaller than the one on the frequency, so that it could be considered as negligible. Figure 8: (**a**) Set of acoustic signals selected from Figure 7b with a relatively large wavenumber component.