## 1 Introduction
The exact value of \(\beta\) can be found according to the following implicit expression derived by [PERSON] and [PERSON] [26]:  
\[\Delta\varepsilon_{\alpha\beta}=k_{\alpha\beta\gamma\delta}\;\mathrm{S}_{ \gamma\delta} \tag{1}\]0  
\[\Delta\varepsilon_{\alpha\beta}=k_{\alpha\beta\gamma\delta}\;\mathrm{S}_{ \gamma\delta} \tag{1}\]1 Schematic view of the four kind of acoustic modes that can be detected in Brillouin spectra taken from transparent films of micrometric thickness deposited on a reflecting substrate. The Rayleigh wave (RW), the longitudinal mode (LM) and the longitudinal bulk (LB) waves are polarized in the sagittal plane, while the shear horizontal mode (SHM) is polarized perpendicular to the sagittal plane, so that the latter gives origin to depolarized light scattering. To illustrate the above considerations, in Figure 4, one can see typical spectra relative to an AlN film of 1.7 \(\upmu\)m thickness measured on a wide frequency range, up to 100 GHz, where several peaks are present. Some of them, corresponding to the Rayleigh wave (RW) and the longitudinal mode (LM), change their frequency position with \(\theta\), according to Equation (5). The peak corresponding to the longitudinal bulk wave (LB) instead, stays almost fixed with changing \(\theta\), because it corresponds to the interaction mechanism of bulk phonons, according to Equation (6). A deeper insight into the characteristics of \"surface\" waves show that the peak at lower frequency is actually a doublet, corresponding to scattering from either the Rayleigh wave (polarized scattering, p-p) or the shear horizontal mode (SHM) that is depolarized (p-s) [22]. Measuring the frequency shift f of each of these two peaks, one can determine the phase velocity of the corresponding acoustic modes according to Equation (5). For the LB peak, instead, the corresponding phase velocity is obtained by Equation (6). Measuring the LB wave at normal incidence permits to obtain \(C_{33}=\rho\times V_{LB}^{2}\). Finally, the value of \(C_{13}\) can be estimated, even if less precisely, from the dependence of \(V_{\text{LB}}\) on the angle \(\alpha\), performing measurements at different incidence angles. Note however that if one uses the above procedure, disregarding the contribution of the piezoelectric constants, one obtains the values of the so called \"piezoelectrically-stiffened\" effective elastic constants. If one instead wants to estimate the \"true\" elastic constants values, one should also consider the contribution of the piezoelectric terms in the numerical model and this implies that the values of \(C_{11}\), \(C_{13}\), \(C_{33}\) are slightly modified [24]. Please note that if the studied film has only two [27; 28], or three [29] independent elastic constants, it is not necessary to look for the depolarized scattering from the SHM (that is much less intense than the other modes [30]), because the information extracted from the RW and either LM or LB is sufficient to determine the two independent constants \(C_{11}\) and \(C_{44}\). \[\Delta\varepsilon_{\alpha\beta}=k_{\alpha\beta\gamma\delta}\;\mathrm{S}_{ \gamma\delta} \tag{1}\]2 **(Left)** Comparison between two Brillouin spectra, relative to a 1.7 \(\upmu\)m thick AlN film, supported by Si, extending to a large frequency range, collected at different angles of incidence. The labels LM and LB refer to the longitudinal mode and the longitudinal bulk acoustic waves, respectively. The incoming light was to p-polarized and no polarization analysis was made on the scattered light; **(Right)** Comparison between two Brillouin spectra, corresponding to p-polarized incoming light and p- or s-polarized scattered light (labeled p-p and p-s, respectively). It is seen that in the former case one can detect the peaks corresponding to the surface Rayleigh wave (RW) and to the longitudinal mode (LM), while in the latter case, only the peak relative to shear horizontal mode (SHM) is detected. The angle of incidence of light was \(\vartheta=60^{\circ}\). Adapted with permission, from [24] IEEE, 2017.