## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### TDBS Experiments at Two Angles of Probe Incidence and Depth Profiling
The BF images presented in Figure 3 were  
Figure 1: Raw measurements of the transient probe reflectivity in the (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on Si: (**a**) for probe incidence at \(\psi=0^{\circ}\) and (**b**) for probe incidence at \(\psi=52^{\circ}\). Each percentage number denotes the Pr substitution concentration associated with the curves just above it, and each curve at a given substitution concentration corresponds to a different probed location on the sample. Figure 2: Acoustic part of the signals obtained in the (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on Si after the subtraction of the low-frequency contribution of non-acoustic origins to reflectivity: (**a**) for time-domain Brillouin scattering (TDBS) measurements at normal incidence (0\({}^{\circ}\)) and (**b**) for those at oblique incidence (52\({}^{\circ}\)). Each percentage number denotes the Pr substitution concentration (x) associated with all curves with the same ordinate, and each curve at a given substitution concentration corresponds to a different probed location on the sample. obtained by this signal processing technique, which had previously been successful in TDBS imaging of polycrystalline aggregates at high pressure [35]. Signals are here processed with a Hann moving window of 62 ps duration at FWHM, limiting the depth resolution at the structural phase transition to about 140 nm. At each temporal position of the moving window, the BF is obtained by means of maximizing the modulus of a complex integral (the integral over the time window of the product between the complex exponential whose time-dependent phase corresponds to that of the measured signals and the complex exponential \(\exp(-2\mathrm{i}\gamma\mathrm{r}_{\mathrm{B}}t)\), where \(f_{\mathrm{B}}\) stands for the BF). The signal-to-noise ratio (SNR) within the window, as well as the normalized root-mean-square error (NRMSE), is also calculated at each time position of the moving window, to serve as goodness criteria. In Figure 3, the time-frequency analysis of all signals is shown: (a) for TDBS measurements at normal incidence (0\({}^{\circ}\)) and (b) for those at oblique incidence (52\({}^{\circ}\)). The color scale is common and depicts the value of the BF. The goodness criteria were here chosen as: (i) SNR should be greater than 3 dB and (ii) NRMSE should be lower than 15%. The white regions in Figure 3 correspond to the regions where at least one of these criteria is not fulfilled. The signals measured at different lateral positions on a sample with given Pr substitution concentration are packed together (signals from each sample are separated by black horizontal lines). This representation allows the conclusion that all samples are relatively laterally homogeneous, as can also be seen in Figure 2, where all signals at a given Pr substitution concentration and probe angle of incidence are superimposed. From the BF images in Figure 3, it is additionally important to notice that the BF appears to vary as a function of time for a given lateral position. Indeed, TDBS is a known modality for depth-profiling and imaging [20; 21; 37; 38; 39; 40]. The distribution of the BFs as a function of the time delay between the probe and pump pulses can be transformed in the depth distributions of the BF. The BF images in Figure 3 reveal variations of the BFs with depth up to 30%, which demonstrates the depth inhomogeneity of the films. This could be caused by inhomogeneity of the texturing of the films, which influences the detected frequencies via the orientation of the elastically anisotropic grains. Note that at normal incidence of the probe, the BF is directly proportional to the product of the acoustic velocity \(v_{S}\) and of the refractive index \(n\).