## 4 Local Acoustic Dispersion Curve of GESA Modified Sample
The extracted signals contain wavelength components corresponding to \(\lambda_{k}=120/\)k with \(k=1,2,3,\ldots\)They were obtained after high-pass filtering the thermoelastic signal. (**b**) Corresponding spectra of (**a**), revealing spectral peaks corresponding with SAW components at different wavelengths. Figure 7: (**a**) Normalized \((x,t)\)-dependence of the high-pass filtered laser beam deflection signal at the coating surface of the GESA modified sample, excited by a transient grating with a fringe of 120 \(\upmu\)m. The diffraction grating was scanned (\(\mathrm{d}x=0.5\)\(\upmu\)m) in the direction of wave propagation. To guide the eye, Rayleigh waves with a wavelength of 120 \(\upmu\)m are indicated. (**b**) Normalized \((k,t)\)-signal of the logarithmically scaled signal obtained after performing a one-dimensional Fourier transform along the \(x\)-direction of the signal in (**a**). By virtue of the non-sinusoidal glass etch relief of the diffraction grating, multiple wavenumber components were excited, emerging as relative bright horizontal lines in the Figure. The experimental dispersion curve was fitted with the thickness \(d\), the density \(\rho_{1}\) and the Rayleigh velocity \(c_{R_{1}}\) of the coating material as fitting parameters, while keeping the substrate's density and Rayleigh velocity fixed at known literature values of \(\rho_{2}=7650\) kg/m\({}^{3}\) and \(c_{R_{2}}=3024\) m/s [42]. The resulting fitted dispersion curve is shown, together with the experimental data, in Figure 9. In Table 2, the best fitting parameters, together with their least-squares uncertainties, are listed. Via the relations \(c_{T}=c_{R}(1+\
u)/(0.862+1.14\
u)\) and \(E=2\rho c_{T}^{2}(1+\
u)\)[43, 44], the Young's modulus \(E_{1}\) and \(E_{2}\) of the coating and substrate materials, respectively, were derived. For Poisson's ratio, have used _ad hoc_ a value of 0.3, which is roughly equal to the value of T91 steel [1, 42] and of FeCrAlY [45]). The relatively large uncertainty on the calculated Young's modulus of the coating layer can primarily be attributed to fitting covariance with the coating density, which unfortunately is inherent to this inverse problem. Taking into account the (relatively large) margin of uncertainty, the extracted value of \(22\pm 4\) m for the coating thickness is consistent with the thickness of 19 \(\pm\) 4 m as estimated from the SEM images in Figure 1.  
The value of density of the coating material (5.8 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\)) is smaller than the one of the substrate (7.65 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\)). This can be explained by the inherent lower density of the alloy components, as depicted in Table 1.  
To investigate the validity of these results, the Young's modulus of the specimen was independently measured by a (destructive) nano-indentation measurement using a Berkovich tip and small loads (\(\sim\)100 mN) [46, 47]. At 12 points, selected arbitrarily along the surface of the sample, \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(c_{R_{1}}\) (m/s) & \(\rho_{1}\) (kg/m\({}^{3}\)) & \(d\) (\(\mu\)m) & \(c_{R_{1}}\) (m/s) & \(\rho_{2}\) (kg/m\({}^{3}\)) \\ \hline
**fitted** & (3.31 \(\pm\) 0.05) \(\times\) 10\({}^{3}\) & (5.8 \(\pm\) 1.3) \(\times\) 10\({}^{3}\) & 22 \(\pm\) 4 & \\
**assumed** & & & & 3024 & 7650 \\ \hline \hline  & \(E_{1}\) (GPa) & \(\
u_{1}\) & \(E_{2}\) (GPa) & \(\
u_{2}\) \\ \hline
**calculated** & 190 \(\pm\) 50 & 0.3 & 202 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Obtained best fitted parameters for the GESA modified coating and their least-squared uncertainties together with the assumed T91 steel substrate’s properties (top table). From these fitted parameters, Young’s moduli of the coating and substrate materials were calculated (bottom Table). Figure 9: Experimental data obtained from the signal traces in Figure 8 and best parameter fitted coating-substrate dispersion curve (solid gray line), as a function of wavelength (**top**) and frequency (**bottom**).