## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### TDBS Experiments at Two Angles of Probe Incidence
However, the samples observed by spectral reflectance unambiguously show pronounced fringes that ensure homogeneous average thicknesses over large areas, at least with centimeter size. In the optical range of probe light, the absorption can be neglected and the modulation of reflectance is accurately defined similar to the case of the films deposited on Si substrate. The corresponding optical thicknesses \(\left(nH\right)_{\text{opt}}\) at 830 nm wavelength are given in the fifth column in Table 3.  
On the other hand, the absolute reflectance and the contrast of fringes change a lot with the position of the analysis. This is attributed to the fact that, as shown by the optical microscopy results (Appendix C), the light is strongly scattered instead of being fully specularly reflected, as a consequence of the roughness over large scales and/or the fluctuation of refractive index associated with the random orientation of microcrystallites. This prevents the separate measurement of the geometrical thickness (\(H\)) and the refractive index (\(n\)). Then, for the determination of the refractive index, the geometrical thickness of the film should be measured independently. Unfortunately, the films deposited on LAO substrate were too hard for scratching and it was impossible to apply the AFM scratch method for the thickness measurements. Because of this, we had to cleave the samples along the direction normal to the surface and the thickness of the films was measured by scanning electron microscope (SEM) side imaging (Appendix E). According to the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Pr substitution concentration & \(H\) [nm] & \(f_{0}\) [GHz] & \(H_{\text{SEM}}\) [nm] & \(\left(nH\right)_{\text{opt}}\) & \(n=\frac{\left(nH\right)_{\text{opt}}}{H_{\text{SEM}}}\) & \(v_{S}=\frac{\ell_{\text{B}}v_{\text{0}}}{2m}\) \\
5\% & 500 & 25.4 \(\pm\) 1.0 & 500 \(\pm\) 20 & 1310 \(\pm\) 10 & 2.62 \(\pm\) 0.12 & 3917 \(\pm\) 334 \\
10\% & 500 & 25.8 \(\pm\) 0.7 & 500 \(\pm\) 20 & 1300 \(\pm\) 10 & 2.60 \(\pm\) 0.12 & 4009 \(\pm\) 294 \\
12.5\% & 500 & 21.3 \(\pm\) 2 & 500 \(\pm\) 20 & 1320 \(\pm\) 10 & 2.64 \(\pm\) 0.13 & 3260 \(\pm\) 467 \\
15\% & 500 & 24.5 \(\pm\) 1.5 & 500 \(\pm\) 20 & 1310 \(\pm\) 10 & 2.62 \(\pm\) 0.12 & 3778 \(\pm\) 404 \\
20\% & 500 & 26.5 \(\pm\) 0.7 & 500 \(\pm\) 20 & 1290 \(\pm\) 10 & 2.58 \(\pm\) 0.12 & 4102 \(\pm\) 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation of the parameters of (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on LaAlO\({}_{3}\) (LAO) by combining TDBS, optical reflectivity, and scanning electron microscope (SEM) measurements. measurement by SEM presented in the fourth column in Table 3, the thickness of the film is 500 \(\pm\) 20 nm whatever the Pr substitution concentration. Then, with these data and with the earlier measured optical thickness, it is possible to deduce the refractive indices of the films; these are reported in the sixth column in Table 3. It appears that the average refractive indices, although with a similar magnitude as those published for pure BFO [41; 42; 43; 44], are slightly weaker than the lowest reported one (extraordinary refractive index \(n_{e}\ =\) 2.66 at 830 nm [44]). Finally, using the obtained refractive indices and BFs measured at normal incidence (see the third column in Table 3), we revealed the values of the longitudinal acoustic velocities in these films: \(v_{5}\ =\ f_{0}\lambda_{0}/2n\), presented in the last column of Table 3.  
Optical and Acoustic Anomalies Accompanying Structural Transformation of BPFMO Films Deposited on LAO Substrate  
According to our measurements (Table 3), the refractive index of BPFMO films deposited on LAO substrate exhibits a weak maximum at Pr substitution concentration of x = 0.125 correlated with the expected position of the structural transition.