# In situ determination of the optical axis orientation in a single grain using time-domain Brillouin microscopy

M. Lejman, G. Vaudel, V. Juvé, M C Weber, I C Infante, B. Dkhil, V. Gusev, P. Ruello

► To cite this version:

M. Lejman, G. Vaudel, V. Juvé, M C Weber, I C Infante, et al. In situ determination of the optical axis orientation in a single grain using time-domain Brillouin microscopy. Applied Physics Letters, 2025, 126 (1), 10.1063/5.0245340. hal-05173277

HAL Id: hal-05173277

https://cnrs.hal.science/hal-05173277v1

Submitted on 21 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
*In-situ* determination of the optical axis orientation in a single grain using
time-domain Brillouin microscopy

M. Lejman,<fnref n="1" /> G. Vaudel,<fnref n="1" /> V. Juvé,<fnref n="1" /> M. C. Weber,<fnref n="1" /> I. C. Infante,<fnref n="2" /> B. Dkhil,<fnref n="3" /> V. Gusev,<fnref n="4" /> and P. Ruello *<fnref n="1" />

In this paper we develop a method that combines optical birefringence properties and time-domain Brillouin scattering microscopy to determine *in-situ* the optical axis orientation of each single micrometer size grain in a polycrystalline sample. We illustrate the method by investigating the room temperature multiferroic material BiFeO₃ where the optical axis coincides with the ferroelectric polarization direction. We are able to find the grain orientation and also provide the sound velocity (longitudinal and transverse) since the method is based on the Brillouin scattering process. These advances open interesting perspectives for probing the anisotropy of a micrometer size grain with an extension to the evaluation of the ferroelastic domain orientation in a non-contact and non-destructive manner.

PACS numbers:

The distribution of grain orientations in polycrystalline
materials is fundamental in materials science. The grains
spatial distribution and their crystallographic orienta-
tion (anisotropy) are indeed key in understanding ma-
terial properties e.g., in mechanics, transport phenom-
ena or optics [1, 2]. Visualizing grains and grain bound-
aries is standard in metallurgy (metallography combined
with surface etching [3]) and mapping the grain crys-
tallographic orientation is routinely achieved with elec-
tron back scattering measurement (EBSD)[4]. X-ray to-
mography is also a powerful technique to reconstruct the
grain distribution [5]. Optical methods have been devel-
oped for years to probe the crystal orientation using the
birefringence property in crossed-polarized microscopy
scheme [2]. The absolute crystal orientation, i.e. the
angle between the normal of the visualized surface and
the optical axis (OA), is usually indeed accessible with
optical interferometer combined with a careful analysis
of the Fresnel coefficients [6–8]. However, these meth-
ods are applied to large transparent single crystals in
general. They do not work for micrometer size grains
of a polycrystalline materials (or small ferroelastic do-
mains/variants in the case of ferroelectrics for instance)
and usually cannot be suitable to semi-opaque materials.
These aforementioned restrictions exclude many practi-
cal situations in materials science. For small grains in
anisotropic materials, it is necessary to use focused op-
tical beam to get the spatial selectivity. Micro-Raman
scattering method combined with a modeling of the op-

tical phonon dispersion curve enables to reconstruct the
anisotropy of a single grain in polycrystalline sample,
i.e. to find its optical axis direction [9]. But the lat-
ter method requires the values of elastic constant (exper-
imentally measured or numerically determined by first
principle calculation). Micro-Brillouin scattering method
also enables to probe the grain orientation by analysing
the sound velocity anisotropy. By comparing the experi-
mental values of the sound velocities (longitudinal and
transverse waves) with the simulated ones (computed
anisotropy of velocities, i.e. the slowness surface) it is
possible to deduce the crystal orientation [10–13]. But
again, this method unfortunately requires to know the
elastic stiffness tensor.

To circumvent all these limitations, we present a
method that does not require, large single crystalline
samples, absolute optical intensity measurements (Fres-
nel coefficients), or knowledge of elastic constants. Our
approach, which applies to any birefringent material,
is based on the precise measurement of the effec-
tive extraordinary Brillouin frequencies (polarization-
dependent Brillouin scattering) which depend on the di-
rection of the optical axis. We only need to use the ordi-
nary and extraordinary refractive indices values as input
parameters. We validate this method with the ferroelec-
tric material BiFeO₃ that is a highly birefringent uniax-
ial material. Besides the finding of the OA direction, our
method also provides the values of sound velocities since
it relies on the Brillouin process. Finally, our method be-
comes also relevant to determine *in-situ* the orientation
of the ferroelectric polarization direction since it usually
coincides with the OA direction. Such ferroelectric polar-
ization orientation is frequently determined with piezo-

<footnote n="1">¹Institut des Molécules et Matériaux du Mans, UMR CNRS 6283, Le Mans Université, 72085 Le Mans, France.</footnote>

<footnote n="2">²Institut des Nanotechnologies de Lyon, CNRS UMR 5270 ECL INSA UCBL CPE, 69621 Villeurbanne Cedex, France.</footnote>

<footnote n="3">³Université Paris Saclay, CentraleSupelec, UMR CNRS 8580, Laboratoire Structures,
Propriétés et Modélisation des Solides, Gif-Sur-Yvette, France.</footnote>

<footnote n="4">⁴Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR CNRS 6613,
Institut d'Acoustique-Graduate School (IA-GS), Le Mans Université, Le Mans, France</footnote>

<footnote>*Electronic address: pascal.ruello@univ-lemans.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: (color online) (a) Principle of the time-domain optical reflectivity experiment with an inclined grain of BiFeO₃. The orange arrow depicts the probe polarization $\vec{E}_0$ whose orientation is defined by the angle $\alpha$ between the probe polarization and the neutral direction $n_{eff}$. (b) Electron Back Scattering Diffraction (EBSD) image of the sample revealing the variable crystallographic orientation of the polycrystalline BiFeO₃ that can be probed by focused laser beams. Each color corresponds to a defined crystalline orientation in the rhombohedral (R) system. (c) Typical time-resolved signals revealing several oscillating components (Brillouin oscillations) coming from the presence of different longitudinal (LA) and transverse (TA) acoustic modes with additional modulation due to the optical birefringence of the crystal as discussed in the main text. The inset is the acoustic phonon signal only. (d) (adapted from [17]) Description of all the optical Brillouin scattering processes (photon-phonon collision) taking place with an arbitrary incident probe beam ($k_i$ and $k_s$ are the incident and scattered probe photon wavevector, respectively). $q_{ph}^{\alpha-\alpha}(q_{ph}^{\alpha-\beta})$ is the momentum transfer taking place when the incident and the scattered probe electric field propagate along the ordinary "o" (extraordinary "e") channel. $q_{ph}^{\alpha-\alpha}(q_{ph}^{\alpha-\beta})$ is the momentum transfer taking place when the incident and the scattered probe electric field propagate along the ordinary (extraordinary) and extraordinary (ordinary) channels, respectively. The probe beam has an arbitrary angle of incidence relative to the optical axis (OA).</figcaption></figure>

force microcopy that, however, requires a bottom electrode for this electric measurement [14]. Our optical method does not require specific sample preparation, except appropriate polishing to attain optical quality for the surface.

Our method is based on an optical pump-probe method to realize time-domain Brillouin scattering (TDBS) microscopy. This pump-probe technique consists in the optical excitation of the material by a femtosecond laser pump and in the detection of the photoexcited acoustic waves by measuring the transient reflectivity with the time-delayed femtosecond laser probe [15, 19]. A sketch of the incident pump and probe beams relative to the grain surface is shown in Fig. 1(a). This pump excites an arbitrary selected grain of a polycrystalline BiFeO₃ sample whose grain distribution is shown in Fig. 1(b) obtained by Electron Back Scattering Diffraction (EBSD). The colors indicate the crystallographic orientation. The orientation of the grain (i.e. the optical axis direction) is defined by the angle $\theta$ (Fig. 1(a)) be-

tween the vector normal of the BiFeO₃ surface (S) and the optical axis (OA) direction (which corresponds to the trigonal axis (C₃) for the 3m point symmetry group of BiFeO₃). In the particular case of our study, the BFO sample is excited by a UV pump beam with 3.1 eV photon energy. We remind that the direct band gap of BFO is around 2.8 eV [16]. These conditions thus permit above band-gap optical interband transitions and also lead to efficient deposition of energy within a rather thin skin depth that is estimated at about 50 – 60 nm at a pump energy of 3.1 eV [16]. As a result of the photoexcitation, acoustic waves are emitted from the surface as discussed previously [10, 17]. Note that photogenerated acoustic front is well described as a plane wave front [18–20] since the pump beam diameter of 15 µm is much larger than the pump optical penetration depth. The pump beam diameter of 15 µm remains smaller or equivalent to a grain diameter resulting in the possibility to excite a single grain. The pump-generated acoustic phonons create a strain field that propagates in the
grain and modulates the refractive index which in turn, through the photoelastic effect, leads to the scattering of the delayed probe laser pulse by the moving acoustic phonon front. The photogenerated coherent acoustic phonons are detected by the probe beam of 2.1 eV energy (590 nm) i.e. below band-gap detection scheme. Note that a single achromatic objective is used to focus both the pump and the probe beams in the normal incidence geometry. A dichroic mirror is used to combine both the pump and the probe beams before the objective. Due to momentum conservation during the interaction between the probe light and the moving acoustic front, only the acoustic phonon component with the Brillouin frequency ($f_{ph}$) is detected [18–20]. In the back-scattered configuration of our experiments, and for an incident probe beam normal to the surface, the momentum conservation is achieved for $q_{ph} = 2k_{probe}$ where $q_{ph}$ is the momentum of acoustic phonon and $k_{probe}$ the photon momentum of the probe beam. In this latter equation, we have $k_{probe} = 2\pi n/\lambda_{probe}$ and $\lambda_{probe}$ and $n$, the probe wavelength in vacuum and the refractive index of BFO at the wavelength $\lambda_{probe}$ (for below-band gap detection process, i.e. probing a transparent medium, the refractive index $\tilde{n}$ is reduced to its real part $n$), respectively. The time-domain Brillouin signal leads to sinusoidal modulation of the optical reflectivity at the frequency $f_{ph}$ ($\frac{\Delta R}{R}(2\pi f_{ph})$). Its origin can also be understood as the result of optical interferences between the beam reflected by the free surface and the beam scattered by the moving acoustic front [15, 19, 20]. The time-domain Brillouin signal can also be viewed as a heterodyne process where the contribution of phonon to the transient reflectivity is $\frac{\Delta R}{R}(2\pi f_{ph}) \propto E_r(2\pi\nu_{probe}) \times E_s^*(2\pi\nu_{probe} \pm 2\pi f_{ph})$ with $E_r$, $E_s$ and $\nu_{probe}$, the probe electric field reflected at the free surface, the scattered one (conjugated) by phonons and the probe light frequency respectively [15, 19, 20]. Typical Brillouin oscillations obtained in different grains are shown in Fig. 1(c) with their time derivatives in inset. The existence of multiple modulations in the oscillating signal comes from the existence of different Brillouin modes (longitudinal-LA and transverse-TA acoustic modes) whose frequencies can be controlled by the grain orientation as discussed previously [10, 21]. As we will show in this paper, the Brillouin frequencies drastically depend on the probe polarization since the probe beam is decomposed into two beams when entering the anisotropic medium: the first one having a probe electric field $E$ perpendicular to the plane containing the OA and the surface vector $S$ (ordinary light) and a second one having a probe electric field within that plane (extraordinary light). For example, if the incident probe electric field is aligned with the direction noted $n_o$ ($n_{eff}$) sketched with a blue (red) arrow in the Fig. 1(a), the light will propagate along the ordinary (effective extraordinary) channel. This effect of birefringence is known for years in standard Brillouin spectroscopy

but restricted to experiments performed with large single crystals [22]. The different phonon-photon scattering processes are summarized in Fig. 1(d) where we describe the Brillouin processes for a probe beam with an oblique incidence for the clarity of the presentation [17]. For a probe beam in normal incidence having an arbitrary orientation of its electric field $E$ defined by the angle $\alpha$ (see Fig. 1(a)), relative to the neutral axes "$n_o$" and "$n_{eff}$", up to three Brillouin frequency $f_{ph}$ components are then detected when a single acoustic mode propagating at the sound velocity $V_S$ is probed [17]:

$$f_o = \frac{2V_s n_o}{\lambda_{probe}}, f_{eff} = \frac{2V_s n_{eff}}{\lambda_{probe}}, f_{eo} = \frac{V_s(n_o + n_{eff})}{\lambda_{probe}}, \quad (1)$$

The frequency $f_o$ and $f_{eff}$ are the ordinary and effective extraordinary Brillouin frequencies while $f_{eo} (= f_{oe})$ are known as the acousto-optic mode conversion Brillouin frequencies [17]. These formulas (Eq. 1) work both for the transverse TA and LA modes ($V_S = V_{LA}, V_{TA}$). Importantly, $n_{eff}$ depends on the angle $\theta$ (i.e. on the optical axis direction) with [2]:

$$n_{eff}^2 = \frac{n_o^2 n_e^2}{n_o^2 \sin^2(\theta) + n_e^2 \cos^2(\theta)} \quad (2)$$

We see that $n_{eff} = n_e$ (pure extraordinary index) when $\theta = 90^\circ$ (probe wavevector propagation perpendicularly to the OA) and $n_{eff} = n_o$ when $\theta = 0^\circ$ (probe wavevector propagation along the OA).

The dependence of $n_{eff}$ on $\theta$ angle presented in Eq. 2 will be exploited in our method to find the optical axis orientation. From a practical point of view, when we arbitrary select a grain, we can straightforwardly determine when the electric field of the probe $E$ is along one of the two neutral axis: there are indeed two particular angles $\alpha$ where the probe light electric field is parallel to the neutral axis $n_o$ (probing with the ordinary index $n_o$, $\alpha = \pi/2$) or parallel to the second neutral axis $n_{eff}$ (probing with the extraordinary index $n_{eff}$, $\alpha = 0$, see Fig. 1(a)). For these two particular situations, we have only one Brillouin process, i.e. either the "o-o" or "e-e" process and no mode conversion (e-o/o-e) [17]. Even if we cannot know how these neutral axes are orientated on the surface of a given grain in the laboratory frame when we perform a time-domain Brillouin experiment, we can localize them by analysing the value of the Brillouin frequencies. By tuning $\alpha$ with the probe half plate, we register time-domain Brillouin signals until we can detect the minimum and maximum Brillouin frequencies for both the LA or TA modes. According to the theory shown in Eq. 1, and for a negative uniaxial material with $n_e < n_{eff} < n_o$ (case of BiFeO$_3$), these minimum and maximum Brillouin frequencies coincide with $f_{eff}$ ($\alpha = 0^\circ$) and $f_o$ ($\alpha = 90^\circ$) respectively. So, using Eq.
<figure><img src="image_3.png" /><figcaption>Figure 2: (color online) Brillouin oscillations and the corresponding fast Fourier transform (FFT) obtained with time-domain Brillouin experiments performed in different arbitrary selected grains named 1 (a), 2 (b) and 3 (c), respectively. (d) The determination of the effective refractive index $n_{eff}$ from the analysis of the Brillouin frequencies enable (Eq. 4) to find the optical axis direction $\theta$ (i.e. the ferroelectric polarization direction) of different grains. (e) Determination of the optical axis orientation of a (001)$_{pc}$ single ferroelectric domain BiFeO$_3$ crystal from data shown in Ref. [23]. The inset shows the direction of the optical axis relative to the surface normal, i.e. to the direction [001] in the cubic frame.</figcaption></figure>

1, once we have found these extrema ($f_{eff}$ and $f_o$) for either the TA or LA modes, we can then deduce the effective index $n_{eff}$ providing we know the ordinary axis $n_o$ with:

$$n_{eff} = n_o f_{eff}^{TA} / f_o^{TA} = n_o f_{eff}^{LA} / f_o^{LA} \quad (3)$$

With Eq. 2, the ordinary $n_o$ and extraordinary $n_e$ indices as input parameters and with the experimentally determined $n_{eff}$, we can then deduce the angle $\theta$ which defines the optical axis direction (i.e., the orientation of the ferroelectric polarization) relative to the normal vector of the irradiated surface (S), with [2]:

$$\sin \theta = \sqrt{\frac{\frac{n_0^2}{n_{eff}^2} - 1}{\frac{n_0^2}{n_e^2} - 1}} \quad (4)$$

In Fig. 2 we show different time-domain Brillouin signals for different grains of the BiFeO$_3$ polycrystalline sample and for variable probe polarization angle ($\alpha$). The fast Fourier transform (FFT) reveals the acoustic modes

(LA, TA) whose frequencies and amplitude vary depending on the considered grain. We first observe that for grain 1, only the LA mode is observed, for grain 2 we detect one TA mode and one LA mode while for grain 3 two TA modes (TA$_1$, TA$_2$) and the LA mode are identified. The number of Brillouin modes is governed by the symmetry of the grain and the involved physical mechanism of generation, as already discussed [10]. Note for example that observing only one TA mode in the grain 2 tells that the symmetry plane $m$ of the 3$m$ point group is necessarily perpendicular to the photoexcited surface. This means that the optical axis is rotated around the $Ox_1$ axis by an amount $\theta$ that we will determine in the following (Note that $x_1$, $x_2$ and $x_3$ are the eigenvectors basis of the rhombohedral lattice in the hexagonal frame when $x_3$ coincides with the OA). As for the grain 3, the observation of 3 acoustic modes indicates that the propagation direction of the acoustic waves (along S) is not a high symmetry direction (no mode degeneracy). We remind that with optical birefringence we have only access to the angle $\theta$ and we cannot extract the full lattice orientation for this general case represented by the grain 3. It is worth to note that whatever the grain, we observe (see zooms on the FFT signals) that these LA
<table><caption>Table I: Experimental Brillouin frequencies, sound velocities and optical axis orientation (θ) measured in three grains of a BiFeO₃ polycrystalline sample and in a (001)pc single ferroelectric domain BiFeO₃ crystal.</caption><thead><tr><th>Brillouin frequency GHz</th><th>f<sub>eff</sub>(TA<sub>1</sub>)</th><th>f<sub>o</sub>(TA<sub>1</sub>)</th><th>f<sub>eff</sub>(TA<sub>2</sub>)</th><th>f<sub>o</sub>(TA<sub>2</sub>)</th><th>f<sub>eff</sub>(LA)</th><th>f<sub>o</sub>(LA)</th><th>Angle θ (°)</th></tr></thead><tbody><tr><td colspan="8">Polycrystalline sample (λ<sub>probe</sub>=590 nm)</td></tr><tr><td>Grain 1</td><td></td><td></td><td></td><td></td><td>46.9</td><td>50.9</td><td>78</td></tr><tr><td>Grain 2</td><td>23.15</td><td>23.5</td><td></td><td></td><td>50.8</td><td>51.6</td><td>24.5 ± 0.5</td></tr><tr><td>Grain 3</td><td>21.7</td><td>22.5</td><td>29.2</td><td>30.2</td><td>48.5</td><td>50.5</td><td>39.6 ± 2</td></tr><tr><td>Single crystal BiFeO<sub>3</sub> (001)<sub>pc</sub> [23] (λ<sub>probe</sub>=800 nm)</td><td>19.85</td><td>20.8</td><td></td><td></td><td>32.55</td><td>34.15</td><td>52.5 ± 0.5</td></tr><tr><td colspan="8">Velocity (m/s)</td></tr><tr><td></td><td>V<sub>TA<sub>1</sub></sub></td><td>V<sub>TA<sub>1</sub></sub></td><td>V<sub>TA<sub>2</sub></sub></td><td>V<sub>TA<sub>2</sub></sub></td><td>V<sub>LA</sub></td><td>V<sub>LA</sub></td><td></td></tr><tr><td colspan="8">Polycrystalline sample (λ<sub>probe</sub>=590 nm)</td></tr><tr><td>Grain 1</td><td></td><td></td><td></td><td></td><td>4924</td><td>4923</td><td></td></tr><tr><td>Grain 2</td><td>2261</td><td>2272</td><td></td><td></td><td>4962</td><td>4990</td><td></td></tr><tr><td>Grain 3</td><td>2170</td><td>2176</td><td>2921</td><td>2920</td><td>4884</td><td>4884</td><td></td></tr><tr><td>Single crystal BiFeO<sub>3</sub> (001)<sub>pc</sub> [23] (λ<sub>probe</sub>=800 nm)</td><td>2973</td><td>2971</td><td></td><td></td><td>4876</td><td>4878</td><td></td></tr></tbody></table>

and TA Brillouin frequencies exhibit a significant shift in frequency (few percents) when the probe polarization is rotated from α = 0° to α = 90° dependence. This is the direct consequence of the large optical birefringence of BiFeO₃ (see Eq. 1). Note that such effect of the probe polarization on the Brillouin spectrum was not observed in our previous work [10] since at that time the probe polarization was circular. As explained before, we can identify the minimum (f<sub>eff</sub>) and the maximum (f<sub>o</sub>) Brillouin frequencies for each grain (shown in insets of FFT curves in Fig. 2(a-c)) and for each set of mode (TA₁, TA₂ and LA modes). The data are given in Table 1. With Eq. 3 we can deduce the values of the effective extraordinary refractive index n<sub>eff</sub> and with Eq. 4 we deduce the θ angle (i.e. the optical axis orientation) with the transverse and longitudinal Brillouin frequency. An average value of θ angle is then summarized in Table 1 and are shown in Fig. 2(d) as diamond symbols for each grain. We have also plotted the theoretical θ angle as a function of n<sub>eff</sub> in Fig. 2(d) (purple curve). With this analysis, we find that the grain 1 has an angle θ ~ 78°, very close to orthogonality between S and the OA. This result is consistent with our analysis of the physical mechanism of photo-generation of acoustic waves discussed in Ref. [10]. The grain 2 has an angle θ ~ 24.5 ± 0.5° and for grain 3, we find θ ~ 39.6 ± 2° with a lower resolution due, tentatively, to the quality of the signal presumably. To illustrate the robustness of the method, we have also applied the same procedure to the Brillouin signals obtained in a (001)<sub>pc</sub> single crystal of BiFeO₃ (pc means pseudo-cubic) whose signal is shown in Ref. [23]. The values of the Brillouin frequencies are given in the table 1. We find an angle θ ~ 52.5 ± 0.5° which is in accordance with the value we extract when we assume a perfect cubic unit cell, i.e. θ = 54.7° (see inset of Fig. 2(e)), where the ferroelectric polarisation (i.e. the OA axis) is along the [111] direction. To complete this analysis, with the values of n<sub>o</sub> and n<sub>eff</sub> (Note that for the BFO single crystal, for the probe wavelength of λ<sub>probe</sub>=800 nm, we took n<sub>o</sub> = 2.8 and n<sub>e</sub> = 2.6 from Ref. [16]), according to Eq. 1, we

deduce the corresponding sound velocities and values are given in the Table 1. The departure between estimates of the sound velocity, obtained with either n<sub>o</sub> or n<sub>eff</sub> and for the different Brillouin frequencies (LA, TA), provides the typical error bar.

As a summary, we demonstrate that it is possible to probe in-situ the optical axis direction in micrometer size grains of a uniaxial polycrystalline material with the time-domain Brillouin microscopy. One advantage of the time-domain Brillouin experiment is based on the optical generation of a quite intense acoustic pulse in a selected grain, compared to the weaker thermally activated acoustic mode signal detected in standard Brillouin spectroscopy. Such large acoustic signal favors a significant Brillouin signal. As a consequence, we do not need a tandem Fabry-Perot interferometer for detecting Brillouin signal. The method we present here works providing the knowledge of the ordinary and extraordinary indices of the material. This method also enables to access the values of the sound velocities for a sound propagation perpendicular to the grain surface. And it can be extended to any grain even from slightly opaque materials, as soon as the probe beam penetrates enough (few micrometers with a probe beam energy below the band gap) to get a sufficiently long-lived time-domain Brillouin signal. We applied this method to the ferroelectric material BiFeO₃ for which the optical axis coincides with the ferroelectric polarisation direction. In the context of ferroelectric based photonic applications, this method should work to determine in-situ the orientation of micrometer size ferroelastic domains for instance. Finally, with the most recent development of ASOPS-based pump-probe scheme (asynchronous optical sampling system [24, 25]), our method could be envisioned in a fast scanning process of a polycrystalline surface within a few minutes, which could be profitable for large scale imaging.

**Acknowledgments** The authors thank the French National Research Agency for support with the project THz-MUFINS project (Grant No. ANR-21-CE42-0030) and the support of Région Pays de la Loire for the project
[1] Materials Science, 1st Edition - January 1, 1970, S. W. Jones, ISBN: 9780408701105

[2] E. Hecht, Optics, 4th Edition, published by Pearson Education, Inc, publishing as Benjamin Cummings, Copyright 2002

[3] Metallography and Microstructures, Vol. 9, ASM Handbook, ASM International, Materials Park, OH, 2005.

[4] J. I. Goldstein, D. E. Newbury, J. R. Michael, N. W. M. Ritchie, J. H. Scott, D. C. Joy, "Backscattered Electrons", Scanning Electron Microscopy and X-Ray Microanalysis, Springer New York, pp. 15-28 (2018).

[5] S. A. McDonald, P. Reischig, C. Holzner, et al., "Non-destructive mapping of grain orientations in 3D by laboratory X-ray microscopy," Sci. Rep. 5, 14665 (2015).

[6] R. M. A. Azzam, "Determination of the optic axis and optical properties of absorbing uniaxial crystals by reflection perpendicular incidence ellipsometry on wedge samples," Appl. Opt. 19(18) 3092-3095 (1980).

[7] J. Wang and B. He, "Optical axis determination method," J. of the Opt. Soc. of Am. B. 40, 1102-1112 (2023).

[8] D-C. Su and C-C. Hsu, "Method for determining the optical axis and (n,e,n)o of a birefringent crystal," Appl. Opt. 41(19), 3936-3940 (2002).

[9] J. Hlinka, J. Pokorny, S. Karimi et al., "Angular dispersion of oblique phonon modes in BiFeO3 from micro-Raman scattering," Phys. Rev. B. 83, 020101(R) (2011).

[10] M. Lejman, G. Vaudel, I. C. Infante, et al., "Giant Ultrafast Photoinduced Shear Strain in Ferroelectric BiFeO3," Nat. Comm. 5, 4301 (2014).

[11] S. Sandeep, T. Thréard, E. De Lima Savi, et al., "3D characterization of individual grains of coexisting high-pressure H2O ice phases by time-domain Brillouin scattering," J. Appl. Phys. 130, 053104 (2021).

[12] Y. Wang, D. H. Hurley, Z. Hua, et al., "Nondestructive characterization of polycrystalline 3D microstructure with time-domain Brillouin scattering," Scripta Mater. 166, 34-38 (2019).

[13] Y. Wang, D. H. Hurley, Z. Hua, et al., "Energy material with optically generated coherent acoustic phonons," Nat. Comm. 11 1597 (2020).

[14] S.V. Kalinin, B.J. Rodriguez, A.L. Kholkin, "Piezoresponse Force Microscopy and Spectroscopy". In: Bhushan, B. (eds) Encyclopedia of Nanotechnology. Springer, Dordrecht (2012).

[15] V. E. Gusev and P. Ruello, "Advances in applications of time-domain Brillouin scattering for nanoscale imaging", Appl. Phys. Rev. 5, 031101 (2018).

[16] S. G. Choi, H. T. Yi, S.-W. Cheong, "Optical anisotropy and charge-transfer transition energies in BiFeO3 from 1.0 to 5.5 eV," Phys. Rev. B 83, 100101(R) (2011).

[17] M. Lejman, G. Vaudel, I. C. Infante, "Ultrafast acoustooptic mode conversion in optically birefringent ferroelectrics," Nature Comm. 7, 12345 (2016).

[18] V. Gusev and A. Karabutov, *Laser Optoacoustics*, AIP, New York (1993).

[19] C. Thomsen, H. T. Grahn, H. J. Maris, et al., "Surface generation and detection of phonons by picosecond light

pulses," Phys. Rev. B 34, 4129 (1986).

[20] P. Ruello, V. Gusev, "Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action," Ultrasonics, 56, 21-35 (2015).

[21] P. Ruello, T. Pezeril, S. Avanesyan, et al., "Photoexcitation of gigahertz longitudinal and shear acoustic waves in BiFeO3 multiferroic single crystal," Appl. Phys. Lett. 100 (2012) 212906.

[22] T. C. Chey and S. Le, "Determination of Birefringence by Brillouin Spectroscopy," J. of the Opt. Soc. of Kor. 2(2), 45-49 (1998).

[23] V. Juvé, R. Gu, S. Gable, et al., "Ultrafast light-induced shear strain probed by time-resolved x-ray diffraction: Multiferroic BiFeO3 as a case study," Phys. Rev. B 102, 220303 (2020).

[24] A. Bartels, R. Cerna, C. Kistner, et al., "Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling," Rev. Sci. Instr. 78, 035107 (2007).

[25] R. Gebs, G. Klatt, C. Janke, et al., "High-speed asynchronous optical sampling with sub-50fs time resolution," Opt. Express 18, 5974 (2010).