## 1 Introduction
We will see in the following that the kind of acoustic modes that can be detected in a surf-BLS experiment, and the consequent amount of information that can be achieved, depends on the thickness of the film, its transparency and the presence of a substrate where the acoustic velocity is larger or smaller than in the film. It follows that the above conditions must be carefully considered if one targets a complete elastic characterization of thin films in terms of the whole set of independent elastic constants. Note that even in the case of opaque media, where the penetration depth of light can be as small as ten nanometers, the gained information concerns the whole coherence depth of the acoustic waves, which is of the order of the light wavelength, i.e., hundreds of nanometers. Therefore, for submicrometric film thickness the influence of substrate on the measured frequencies is substantial and must be properly modeled if one wants to extract information about the films. This is different from the case of spin waves in ferromagnetic films, because in that case the presence of a non-magnetic substrate is completely non-influent [21]. Figure 2: Back-scattering geometry for conventional surface Brillouin light scattering (surf-BLS) experiments. The wave vectors of the incident, reflected, transmitted and scattered light are indicated in green, while those of acoustic waves are in red (surface waves) or in blue (bulk waves). All are contained in the plane of incidence. Transparent Films with Micrometric Thickness: Guided Modes and Selective Determination of the Elastic Constants  
The most favorable situation occurs when one deals with a single transparent film of micrometric thickness, supported by a reflecting substrate (such as optically polished Si or GaAs wafers). In this case, it is possible to obtain, in a relatively straightforward way, the whole set of independent effective elastic constants. These are only two for acoustically isotropic films, such as amorphous glasses and soft dielectric films used in electronics, while may increase to three independent constants in the case of epitaxial cubic films, or to five for films of hexagonal (cylindrical) symmetry, such as piezoelectric films with a textured polycrystalline structure. Let us now consider the latter (more complex) case that has been considered in previous investigations devoted, for instance, to ZnO [22] or AlN [23; 24] films. Using different angles of incidence and a careful polarization analysis of the scattered light, one can detect and identify several guided acoustic modes [25], such as the surface Rayleigh wave (RW), the shear horizontal mode (SHM), the longitudinal mode (LM) and the longitudinal bulk (LB) mode, whose propagation direction and polarization are summarized in Figure 3. (Please note that in some papers the LM is labeled as LGM, longitudinal guided mode, and the LB is labeled as LA, longitudinal acoustic wave). Their detection provides a sufficiently large amount of information to estimate the whole set of independent elastic constants, because as summarized in Figure 3, measuring the frequency of the Brillouin peaks associated to the above mentioned modes provides access to the values of the five independent constants, namely \(C_{11}\), \(C_{13}\), \(C_{33}\), \(C_{44}\) and \(C_{66}\). In particular, one has the following selective relationship between the measured velocities and the elastic constants: \(V_{LM}=\sqrt{\frac{C_{11}}{\rho}}\), \(V_{SHM}=\sqrt{\frac{C_{44}}{\rho}}\) and \(V_{RM}=\beta\sqrt{\frac{C_{44}}{\rho}}\) with \(\beta\), a constant that depends on the other elastic constants \(C_{11}\), \(C_{13}\), \(C_{33}\). It must be evaluated numerically, using the model of [25] and usually takes values in the range between about 0.9 and 0.95.