## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### TDBS Experiments at Two Angles of Probe Incidence and Depth Profiling
It also follows from the comparison of the two parts of Figure 3 that the quality of the signals detected with 52\({}^{\circ}\) probe incidence is generally worse than that at normal incidence of the probe, as could have been expected. In fact, we attempted but failed to accumulate additional reliable data at \(\psi=60^{\circ}\). The reliable extracted BFs are used to calculate the weighted average value of the BF in each sample for the two different probe angles of incidence, \(\psi=0^{\circ}\) and \(\psi=52^{\circ}\). The weight of each BF in the average is the reliability weight \(R_{W}\), given by \(R_{W}=1-\)NMRSE, which is applied in order to allow for the most  
Figure 3: Brillouin frequency (BF) images as a function of the time delay between pump and probe pulses and of the Pr substitution concentration from TDBS measurements conducted (**a**) at normal incidence (0\({}^{\circ}\)) and (**b**) at oblique incidence (52\({}^{\circ}\)). reliable BF values to contribute more than others to the average BF value. To evaluate the uncertainty of the reported BF values, the distributions of the reliably extracted BFs are plotted in Figure 4 for the two probe angles of incidence, \(\psi=0^{\circ}\) (blue points) and \(\psi=52^{\circ}\) (orange triangles), and for the different Pr substitution concentrations: (a) 5%, (b) 10%, (c) 15%, and (d) 20%. These distributions are fitted by Gaussian distributions with the positions of their maxima fixed at the average BF values (see Figure 4). The uncertainty is chosen for each Pr substitution concentration to be the FWHM of the Gaussian distribution, as depicted by the horizontal lines in Figure 4. Note that the Gaussian fit is not intended to be accurate, but rather here is merely used to quantify the uncertainty, which increases where the distribution deviates strongly from a Gaussian (due to in-depth inhomogeneity). The calculated average BF values are reported in Figure 5 as a function of the Pr substitution concentration for TDBS measurements at normal incidence (\(\psi=0^{\circ}\)) with filled blue points and for those at oblique incidence (\(\psi=52^{\circ}\)) with filled red triangles. The error bars in all cases are associated with the FWHM of the distributions shown in Figure 4. Despite this uncertainty, it can be clearly seen that the frequencies of the BOs at 10% and 15% Pr substitution are substantially lower than at 5% and 20% substitutions for both probe angles of incidence. Unfortunately, the measurements conducted at \(\psi=0^{\circ}\) and \(\psi=52^{\circ}\) only appeared to be insufficient for the reliable disentanglement of \(n\) and \(\
u_{S}\). The data summarized in Table 1 demonstrate that, although the BFs measured at normal and oblique incidence are distinct, their bare values are too close,  
Figure 4: Distributions of the reliably extracted BFs for the two probe angles of incidence, \(\psi=0^{\circ}\) (blue) and \(\psi=52^{\circ}\) (orange), and for the different Pr substitution concentrations: (**a**) 5%, (**b**) 10%, (**c**) 15%, and (**d**) 20%. Gaussian fits of the distributions are also shown. Figure 5: Average BF values as a function of the Pr substitution concentration for TDBS measurements at normal incidence (\(\psi=0^{\circ}\)) with filled blue points and for those at oblique incidence (\(\psi=52^{\circ}\)) with filled red triangles. The error bars in all cases are associated with the full width at half maximum (FWHM) of the distributions shown in Figure 4.  
leading to high uncertainty in the evaluation of the refractive index \(n_{\text{TDBS}}\) and of the acoustic velocity \(\
u_{\text{TDBS}}\). We note that optical anisotropy of the films is assumed to be negligible. Due to the expected absence of any strong texture in the orientations of BPFMO grains when the films were deposited on the Si substrate, the films can be characterized by one refractive index. The absence of strong texture is confirmed by the XRD measurements in the same films reported in Ref. [13] and in Appendix B.