## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### TDBS Experiments at Two Angles of Probe Incidence  
The TDBS experiments were then conducted with the same experimental setup on the BPFMO thin films deposited on LAO substrate, at two angles of probe incidence on the sample surface (\(\psi\) = 0\({}^{\circ}\) and \(\psi\) = 52\({}^{\circ}\)), with the same goal of disentangling the acoustic and optical anomalies in the films via the TDBS measurements only. The obtained results at the two angles were again too close to allow the disentanglement. On top of that, due to the smaller thickness (500 nm) and the inhomogeneity of the BPFMO films, the uncertainties in the extracted refractive indices and sound velocities were too high to consider their values to be reliable. For the evaluation of the CAP velocities in our films, it was therefore decided to use the independent measurement of the optical refractive index (see Section 4.2) together with the Brillouin frequency extracted from the results collected at the normal incidence of the probe beam. The raw measurements of the transient probe reflectivity in the BPFMO films deposited on LAO at normal incidence are shown in Figure 7a. The results after the subtraction of the slowly varying contribution to reflectivity of non-acoustic origins are shown in Figure 7b. Figure 7: **(a)** 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 LaAlO\({}_{3}\) (LAO) for probe incidence at \(\psi\) = 0\({}^{\circ}\). **(b)** The acoustic part of the signals obtained in the (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on LAO after the subtraction of the low-frequency contribution of non-acoustic origins to reflectivity. Each percentage denotes the Pr substitution concentration associated with the curves just below it, and each curve at a given substitution concentration corresponds to a different probed location on the sample. Due to the difficulty in completely removing the non-acoustic contributions without losing the few BOs available on each signal, it was chosen in this part of the study not to use the synchronous detection technique for performing a time-frequency analysis of the signals. For the synchronous detection technique to give confident results, it is mandatory for the treated signal to be a pure sine function. In place of the synchronous detection technique, the segment of the experimental background-free reflectivity signal with the BOs was manually selected on each signal, its envelope removed using Hilbert transform and the resulting normalized oscillating signal fitted with a sine function assuming a constant frequency. For each Pr substitution concentration, the average value of the BFs obtained by the previously described processing of signals registered in different lateral positions was calculated and reported in the third column of Table 3. The uncertainty was chosen to be the maximal difference between the average value and the BFs used in its calculation. Since at 10% and 20% Pr substitution only one signal was treatable, the uncertainties were chosen in this case to be the confidence interval of the used estimator. The proposed uncertainty in the latter case is the only quantitative estimate we have access to, yet it may be undervalued due to the lateral inhomogeneity of the samples. Characterization of Films by Optical Methods, Atomic Force Microscopy, and Scanning Electron Microscopy  
Similar to the case of the BPFMO films deposited on Si substrate, we failed to reliably measure the optical refractive index of the films by purely optical measurements. The reason for this could be partially tentatively attributed to inhomogeneities of the film surfaces revealed by standard and Differential Interference Contrast (DIC) microscopy (Appendix C), optical profilometry (Appendix C), and AFM topography (Appendix D). Some of these inhomogeneities could be consequences of the domain structure of LAO substrate surface [47; 48; 49] (Appendix C).