## 1 Introduction
However, one can derive the following simple guiding rule. Longitudinal acoustic waves give origin only to polarized scattering, i.e., the light polarization of the scattered photons is the same of that of incoming photons. In particular, if the incoming and scattered beams are both polarized perpendicular to the plane of incidence, s-s scattering, one has that the cross section is proportional to \(|\mathbf{g}|^{2}=4k_{12}^{2}\), while for p-p scattering one has \(|\mathbf{g}|^{2}=4\big{|}(k_{11}-k_{12})cos^{2}\frac{\psi}{2}+k_{12}cos\psi\big{|} ^{2}\), where \(\psi\) is the scattering angle, formed by \(\mathbf{K}i\) and \(\mathbf{K}s\). Instead, shear horizontal (transverse) waves that are polarized perpendicular to the scattering plane, give origin only to depolarized scattering (p-s or s-p scattering), and the cross section is proportional to \(|\mathbf{g}|^{2}=(k_{11}-k_{12})^{2}\cos\frac{\psi}{2}\). From the brief description above, it follows that the polarization of the light scattered by acoustic waves can be either parallel or perpendicular to that of the incident light and this fact is of great help for the experimentalist in order to select the scattering from specific acoustic modes, as we will show in the following. Moreover, as anticipated in the Introduction, there is another interaction mechanism that contributes to the light scattering and is particularly relevant in opaque samples: the rippling of the free surface, or of the interface of a thin film, induced by the acoustic phonons. It contributes to polarized scattering, only, but the relative importance of the elastooptic and the ripple mechanism depends on the specific medium under investigation, with a prevalence of the former (latter) in the case of transparent (opaque) media, as we will see in the following [6; 7]. From the experimental point of view, the most simple and popular surf-BLS interaction geometry, suitable for thin films and opaque samples, is that represented in of Figure 2, called backscattering geometry, where the scattering angle \(\psi=\pi\) and the same lens (typically a camera objective with f number comprised between 2 and 4 and focal length of 35 or 50 mm) is used for focusing the laser beam on the sample and for collecting the light that is back-scattered within a solid angle. A discussion of the influence of the\(f\) number of the collecting lens on the measurement accuracy can be found in Refs. [19; 20]. Two classes of phonons contribute to the scattering in this geometry. First, through the conservation of the momentum component parallel to the surface, one can reveal acoustic waves propagating parallel to the free surface (\"surface-like\" waves) with an acoustic wavenumber \(Q_{S}=2k_{i}\) sin(\(\theta\)) with \(k_{i}=2\pi/\lambda\), with \(\lambda\) the optical wavelength (typically 514.5 or 532 nm) and \(\theta\) the angle of incidence of light. Moreover, when the film thickness is much larger than the wavelength of light, \"bulk-like\" acoustic waves with wavevector \(Q_{B}=2\) nk\({}_{i}\) enter the scattering process, with n the refractive index of the film. Due to optical refraction, these bulk waves propagate within the film material at an angle \(\alpha=\) sin\({}^{-1}\) (sin \(\theta\)/n), as seen in Figure 2.  
One can therefore directly derive the connection between the frequency shift f of Brillouin peaks and the phase velocity v of the corresponding acoustic mode, as follows:  
\[\mathrm{v_{S}}=2\pi\mathrm{f}/\mathrm{Q_{S}}=\lambda\mathrm{f}/2\mathrm{sin}(\theta) \tag{5}\]  
for surface-like modes, and  
\[\mathrm{v_{B}}=2\pi\mathrm{f}/\mathrm{Q_{B}}=\lambda\mathrm{f}/2\mathrm{n} \tag{6}\]  
for bulk-like modes. **3. Determination of the Elastic Constants of Nanometric or Micrometric Films**  
Let us now consider the case of a film with either micrometric or submicrometric thickness (such as those used in acoustic-wave resonators), whose elastic constants are to be determined.