## 4 Application of the SLaTCoW Method: Estimation of Ultrasonic Attenuation
Several points should be mentioned here that could explain the difference between the values estimated using the proposed method and those reported in the literature:  
* The samples used in the literature are single crystals (attenuation is mainly due to coupling to electrons mediated by deformation potential), while the sample used in this work is a polycrystalline plate (attenuation could be due to grain boundary scattering, interaction with dislocations, etc.). * The purity of the samples used in experiments are not exactly the same. For example, in [50], the purity of aluminum is 99.9999% or 99.999%, but the aluminum sample used in this work is with purity of 99.0%. * In [50], the authors measured the ultrasonic attenuation along specific and different propagation directions (namely, [100], [110], and [111] of aluminum single crystal), whereas the attenuation obtained in this work is measured in a polycrystalline alu  
\[\overline{u}(s,\omega)=\int_{0}^{L}\widetilde{u}(x,\omega)e^{-sx}\mathrm{d}x, \tag{4}\]0 Illustration of the minimization process: (**a**) map in the plane \((k_{i},f)\) of the change in the position of the \(S_{2}\) and \(S_{1b}\) theoretical branches with changing the ultrasonic attenuation \(\alpha\) from 0 (bright yellow) to 1 dB/cm/MHz (dark blue), the experimental results shown in Figure 4b are reported; (**b**) curves showing the evolution of the root-mean-square deviation (RMSD) vs. \(\alpha\), calculated with the \(S_{2}\) mode (red circles), the \(S_{1b}\) mode (blue crosses), and both the \(S_{2}\) and \(S_{1b}\) modes (black triangles). minum sample and potentially over multiple grains, which would tend to increase the attenuation. * The low SNR due to the attenuation, where the estimation is done close to the ZGV point, is the main source of measurement uncertainty in our experiments and could thus be an additional source for the discrepancy. Although the sample used in this study and those measured in the literature are not the same (as explained in the first point above), the results of ultrasonic attenuation (\(\alpha\)) obtained in this work (see Table 2) are still inspiring. Hence, the comparison with \(\alpha\) measured by other means on the same tested sample could strengthen the confidence in our results in order to fully demonstrate the quantitative measurement ability of the proposed method. Furthermore, the all-optical technique used in this paper, based on slowly propagating modes in the vicinity of the ZGV point, could also give access to the local mechanical properties and thickness of the sample by using the value of the ZGV resonance frequency. Therefore, the method shows good ability for determining, at the same time, the mechanical properties and the ultrasonic attenuation of samples with plate-like structures. The local attenuation, namely the ultrasonic attenuation measured at one local point of the scanning distance on the aluminum plate, is estimated by using the approach reported in [24], and the attenuation coefficient (denoted by \(\alpha_{loc}\)) is defined by the following formula [24]:  
\[\overline{u}(s,\omega)=\int_{0}^{L}\widetilde{u}(x,\omega)e^{-sx}\mathrm{d}x, \tag{4}\]1  
where \(V_{ph}=\omega/k_{S_{1}S_{2}}\) is the phase velocity at the \(S_{1}S_{2}\)-ZGV point (see Figure 1 in Section 2) and \(\tau=2Q_{AI}/\omega\) is the time-decay constant (see Section 3). At the \(S_{1}S_{2}\)-ZGV point, we get \(V_{ph}=11.54\) km/s and \(\tau=0.48\) us. Using the values of \(V_{ph}\) and \(\tau\) in Equation (10), we get \(\alpha_{loc}=15.61\) dB/cm, which is equal to 0.39 dB/cm/MHz if divided by the ZGV resonance frequency at 40.28 MHz. The value of the attenuation coefficient calculated by Equation (10) is lower than, but quite similar to, that estimated by the presented approach in this work (see Table 2). The difference between the values of the attenuation coefficients calculated by the two different methods can be mainly explained by the following two reasons.