## 1 Introduction
### Transparent Films with Deeply Submicrometric Thickness: Dispersion Curves and Interference Effects
Such a behavior (that was present in the case of Si substrate but not for a fused quartz one) was interpreted in terms of a reduction of the effective elastic constants of the film in a layer near the interface, due to the lattice misfit between the film and the Si substrate. In the above examples of Figures 6 and 7, a best fit procedure of the measured dispersion curves to the calculated ones permitted to achieve a determination of four of the five independent elastic constants. As for thick films discussed in the previous section, the fifth elastic constant \(C_{66}\) can be determined from measurement of the phase velocity of shear horizontal modes (Love modes) in a depolarized scattering experiment, as shown in Figure 8 for an InSe film [41]. Use of an opaque substrate (typically crystalline silicon) is recommended in order to take advantage of the presence of a reflecting interface which enhances the Brillouin cross section, even if long acquisition times (several hours) are usually necessary. Figure 6: **(Left)** Brillouin spectra relative to the three GaSe films of different thickness \(h\), for an angle of incidence \(\theta=70^{\circ}\). The peaks corresponding to the Rayleigh (R) and Sezawa (S) guided acoustic modes are clearly seen. In the case of the thicker film, h = 990 nm, the discrete Sezawa modes are not resolved anymore and the peak due to bulk longitudinal acoustic waves (LA) is observed; (Right) Experimental values of the phase velocity of the Rayleigh and Sezawa acoustic modes (points with error bars) as a function of the ratio between the film thickness and the acoustic wavelength. The dispersion curves (solid lines) have been obtained from a best fit procedure assuming the elastic constants of the film material as free parameters. Adapted with permission from [32], IOP Publishing, 1999. To conclude this Section, let us note that in the case of a soft film on a hard substrate [25], instead, one observes only the RW, which evolves into a leaky wave as soon as the film thickness exceeds a few nanometers, radiating energy into the bulk (as for AlN on Si [45], CaF2 on GaAs [46] or TiN over high-speed steel [47]). As a consequence, the RW peak broadens and finally disappears from the spectra. In these cases, only a relatively limited information about the shear elastic constant C\({}_{44}\) can be extracted from the measurement. Figure 8: Brillouin spectra taken at an angle of incidence \(\theta\) = 65\({}^{\circ}\). From an InSe, 79 nm thick, film. The peaks corresponding to the Rayleigh wave (R) and to the Love mode (L) are clearly seen in the p-p spectrum and the p-s spectrum, respectively. Note that the ghost of the Rayleigh peak is also present in the p-s spectrum, because of the finite extinction ratio of the polarization analyzer. Reprinted with permission from [41], IOP Publishing, 1999. Figure 7: **(Left) Brillouin spectra from ZnO films of different thickness on (001)-cut Si substrate for fixed backscattering angle \(\theta\) = 45\({}^{\circ}\); (Right) Experimental data of phase velocity of first two Rayleigh modes for the ZnO/Si structure and theoretical dispersion curves, plotted as a function of the product between the acoustic wavevector and the film thickness, evaluated by use of elastic constants of bulk ZnO (dashed lines), of effective constants of the film (dash-dotted lines) and of modified constants of the film (solid line). Adapted with permission from [39], IEEE, 1991. **