## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### TDBS Experiments at Two Angles of Probe Incidence and Depth Profiling  
The TDBS experiments were conducted with a Ti:Sapphire laser providing ps laser pulses. A detailed description of the experimental setup is given in Ref. [35]. Here, as is common for experiments with BFO [28; 29; 30; 31; 32; 33; 34], the fundamental radiation at \(\lambda_{0}=808\) nm wavelength was applied for probing the sample, while the frequency-doubled light at 404 nm wavelength was applied for the generation of CAPs via its interband absorption in BFO [28; 29; 30; 35; 36]. The experiments were conducted at different angles of probe incidence on the sample surface (\(\psi=0^{\circ}\) and \(\psi=52^{\circ}\)) with the goal of disentangling the acoustic and optical anomalies in BPFMO films via the TDBS measurements only. The powers of the pump and probe laser radiations were 1.1 mW and 0.6 mW, respectively. Both pump and probe lasers were focused onto the free surface of the BPFMO films using 50X objective lenses. By scanning the pump beam focus relatively to the probe beam focus and measuring the amplitude of the transient thermoreflectance signal as a function of the position, the spatial correlation function of the pump and probe beams was evaluated. Its width of \(\approx\)2.5 \(\upmu\)m at full width at half maximum (FWHM) is a measure of the lateral spatial resolution which is achievable in the TDBS experiments reported here. All the raw measurements of the transient probe reflectivity in the BPFMO films deposited on Si are presented in Figure 1: (a) for probe incidence at \(\psi=0^{\circ}\) and (b) for probe incidence at \(\psi=52^{\circ}\). Each percentage 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. The measured signals are typical for TDBS experiments. At time \(t=0\) ns, the reflectivity changes abruptly due to the pump photon inter-band absorption in the BFO thin films, which causes photo-excitation of charge carriers and subsequent temperature rise near the film/air interface. Note that the penetration depth of the 404 nm pump optical radiation in BFO is about 35-40 nm [28; 29; 30; 35; 36]. These abrupt changes in the probe reflectivity are followed by a long-time relaxation due to motion and recombination of photo-excited carriers and also due to heat diffusion. On top of the relaxation, the reflectivity signals oscillate, which represents the BOs due to the probe photon-CAP interaction. In Figure 2, we present the BOs revealed after the subtraction of the slowly varying contribution to reflectivity of non-acoustic origins: (a) for TDBS measurements at normal incidence (0\({}^{\circ}\)) and (b) for those at oblique incidence (52\({}^{\circ}\)) (set with \(\pm\) 7\({}^{\circ}\) accuracy). Note that each percentage number denotes the Pr substitution concentration associated with the curve with the same ordinate and that the total duration of the BOs corresponds approximately to the time of the CAP arrival on the interface between the film and the substrate. From the data presented in Figure 2, it is evident without any spectral analysis that the frequencies of the BOs, i.e., BFs, at 10% and 15% Pr substitutions are lower than at 5% and 20% substitutions for both probe angles of incidence. This is a signature of the structural phase transition. For quantitative analysis of the change of the BFs as a function of the Pr substitution concentration and of the probe angle of incidence, it is proposed to perform a time-frequency analysis of the acoustic part of each signal. To do so, the synchronous detection technique is used to extract the BF as a function of the time delay between pump and probe pulses.