## 1 Introduction
### Applications of the Time-Domain Brillouin Scattering for Evaluation of Bismuth Ferrite Thin Films  
TDBS is a picosecond acoustic experimental technique for the generation and detection of ultrashort acoustic pulses in transparent-to-the-probe-wavelength materials using ultrafast lasers, and was originally referred to as picosecond interferometry [16, 17, 18, 19]. In this technique, a pump laser pulse generates a picosecond coherent acoustic pulse (CAP) of nanometers-length in the material, while a time-delayed probe laser pulse exhibits additional scattering by this photo-generated CAP. Detecting this additionally scattered light via heterodyning [16, 17, 18, 19] provides information on the acoustical (sound velocity), optical (refractive index), and acousto-optical (photoelastic constant) parameters of the material in the current position of the CAP [18, 20, 21]. In materials that are transparent for probe light and are spatially homogeneous, the detected signal of the CAP-inducedprobe reflectivity variation has a sinusoidal form with respect to the increasing time-delay of the probe laser pulse relative to the pump laser pulse. This sinusoid is commonly called the Brillouin oscillation (BO), while its frequency is called the Brillouin frequency (BF). The period \(T\) of this sinusoid \(\left(\frac{1}{T}\,=\,f\,=\,\frac{2\pi\
u_{S}}{\lambda_{0}}\cos\phi\right)\) contains information on both the velocity of the CAP propagation, \(\
u_{S}\), and the index of refraction, \(n\), at the probe wavelength \(\lambda_{0}\), and depends on the angle between the directions of the propagation of the CAP and the probe light, \(\phi\). If the experiments are conducted at a single angle \(\phi\), then to disentangle \(\
u_{S}\) and \(n\), the independent measurements of \(n\) are required. These are commonly achieved by ellipsometry [22; 23; 24; 25]. However, both \(\
u_{S}\) and \(n\) can be measured by the TDBS technique simultaneously if the experiments are conducted at two or several angles \(\phi\)[26; 27]. The TDBS technique has been previously applied for the evaluation of BFO bulk single crystals [28; 29; 30] and thin films [31; 32; 33; 34]. In Ref. [31], the sound velocity in the 360 nm rhombohedral BFO film grown on SrTiO\({}_{3}\) (STO) (110) single-crystal substrates by pulsed laser deposition was estimated as \(\
u_{S}\cong 4760\)-4880 m/s. In Ref. [32], the coherent longitudinal acoustic phonons in La and Nb co-doped polycrystalline BFO (Bi\({}_{0.8}\)La\({}_{0.2}\)Fe\({}_{0.99}\)Nb\({}_{0.01}\)O\({}_{3}\) (BLFNO)) films were monitored by TDBS. BLFNO thin films were deposited on p-type (001)-Si, on yttrium stabilized (001)-ZrO\({}_{2}\) (YSZ), and on (001)-STO at 550 \({}^{\circ}\)C, then cooled down to room temperature following rapid thermal processing. The BLFNO film on (001)-STO was epitaxially grown with tetragonal structure, which was evidenced by X-ray diffraction. The film thickness was about 200 nm. It was revealed that the measured BF importantly depends on the substrate and, in particular, in the tetragonal BFO film on STO it was nearly 30% lower than in the films on Si and YSZ substrates. However, the refractive indices of the grown films were not measured. Thus, in the evaluation of sound velocities and the elastic modulus, the possible inequality in refractive indices of different films was not taken into account. In Ref. [33], thin films of rhombohedral-like BFO, tetragonal BFO, rare-earth-doped Bi\({}_{0.7}\)Dy\({}_{0.3}\)FeO\({}_{3}\), and Bi\({}_{0.6}\)La\({}_{0.1}\)Dy\({}_{0.3}\)FeO\({}_{3}\) were grown on (001)-oriented LaAlO\({}_{3}\) (LAO) and MgO substrates by pulsed laser deposition. The elastic moduli of rhombohedral, tetragonal, and rare-earth-doped BFO were determined from the measured Brillouin frequencies in conjunction with variable-angle spectroscopic ellipsometry for the determination of the complex refractive index of thin films. Film thicknesses measured by ellipsometry were determined to be in the range from 290 to 450 nm.