Photoelastic Properties of Trigonal Crystals

Abstract

All possible experimental geometries of the piezo-optic effect in crystals of trigonal symmetry are studied in detail through the interferometric technique, and the corresponding expressions for the calculation of piezo-optic coefficients (POCs) π_im and some sums of π_im based on experimental data obtained from the samples of direct and X/45°-cuts are given. The reliability of the values of POCs is proven by the convergence of π_im obtained from different experimental geometries as well as by the convergence of some sums of POCs. Because both the signs and the absolute values of POCs π₁₄ and π₄₁ are defined by the choice of the right crystal-physics coordinate system, we here use the system whereby the condition S₁₄ > 0 is fulfilled (S₁₄ is an elastic compliance coefficient). The absolute value and the sign of S₁₄ are determined by piezo-optic interferometric method from two experimental geometries. The errors of POCs are calculated as mean square values of the errors of the half-wave stresses and the elastic term. All components of the matrix of elasto-optic coefficients p_in are calculated based on POCs and elastic stiffness coefficients. The technique is tested on LiTaO₃ crystal. The obtained results are compared with the corresponding data for trigonal LiNbO₃ and Ca₃TaGa₃Si₂O₁₄ crystals.

1. Introduction

According to the terminology adopted here, low-symmetry elasto-optic materials are the ones which, in addition to principal components of the tensor of elasto-optic coefficients (ELOCs) p_in (indices i, n = 1, 2, 3), have non-principal non-diagonal components, expressed as p_in, where i, n = 4, 5, 6. In particular, calcium tungstate CaWO₄ (crystal class 4/m) has 10 independent components p_in [1,2], including the non-principal non-diagonal components p₁₆, p₆₁, p₄₅, lithium niobate LiNbO₃ (class 3m) and lithium tantalate LiTaO₃ (class 3m) investigated in this paper, as well as independent components of calcium-tantalum gallosilicate Ca₃TaGa₃Si₂O₁₄ (class 32)–8, including two non-principal problematic coefficients p₁₄ and p₄₁, etc. The mentioned non-principal coefficients are identified as ‘problematic’ because of the ambiguity involved in their determination. To address this ambiguity, in terms of both signs and absolute values, the positive directions of axes of the right crystal-physics coordinate system should be explicitly chosen (see, e.g., [3,4] for details).

Let us point out the reasons for the complexity of the determination of all matrix components p_in and their signs. As it is known, the acousto-optic figure-of-merit M₂ can be found using the Dixon–Cohen method [5,6] based on the expression:

$$M_2=\frac{n_i^6{p}_{in}^2}{\mathsf{\rho }{V}_n^3},$$

(1)

where n_i is a refractive index of a crystal, ρ is its density, and V_n is an acoustic wave velocity.

However, the coefficient p_in appears in Equation (1) in the second power, so the determination of the sign of p_in is impossible. Moreover, for the complex geometry of the experiment, when not one p_in coefficient, but rather the sum of p_in coefficients, is included in Equation (1) (among them, the ones with the indices i, n equal to 4, 5, 6), it is also difficult to determine the absolute values of p_in. It should be mentioned that the expressions for the determination of particular coefficients p_in are absent in [5,6], but the correctness of this method is not doubted, because the values of ELOCs p_in defined in these papers for two dozen of acousto-optic crystals were further confirmed by other authors and, in particular, by other methods. However, the signs of ELOCs were found only for the simplest case of high-symmetry cubic crystals.

The acousto-optic methods of Bergmann-Fues [7] and Pettersen [8] make it possible to determine only the p₁₂/p₁₁ relation or the complex combinations of p_in coefficients, such as (p₁₁ + p₁₂ + 2p₄₄)/(p₁₁ + p₁₂ − 2p₄₄), for cubic crystals. The analogous expressions for the crystals of lower symmetry were not given. The diffraction method of Narasimhamurty [2,9] is useful only for the determination of the combinations of ELOCs, such as $p_{33}{n}_3^3/p_{13}{n}_1^3$, in trigonal crystals (crystal classes 32, 3m). The Brillouin scattering method makes it possible to define the p_in coefficients for cubic crystals [10,11], or only the principal components of the p_in matrix (i, n = 1, 2, 3), for crystals of lower symmetry (the results for three principal ELOCs of tetragonal crystals are given in [12]). All p_in coefficients (including their signs) for LiNbO₃ (symmetry class 3m) and CaMoO₄ (class 4/m) were determined using this method in [13,14]. However, because of the absence of the expressions for ELOCs in these papers, their results cannot be used for the complete study of the elasto-optic effect in low-symmetry crystals. It should be emphasized that the values of the p_in coefficients in the modeling of lithium niobate crystal, as determined by acousto-optic methods, are essentially different according to data provided by different authors: by 1.3–1.5 times for elasto-optic coefficients p₁₂, p₃₁, p₄₁ and by 1.6–2.1 times for p₁₁, p₁₃, p₁₄, p₃₃, p₄₄ [13,15]. Thus, the errors in the determination of ELOCs are high. The acousto-optic techniques used for the determination of different combinations of ELOCs are described in [2,16].

Therefore, only the use of acousto-optic methods prevents the determination of the values and signs of all components of the ELOC matrix with high precision, especially for the crystals in p_in matrices that contain such problematic coefficients as p₁₄, p₄₁, p₂₅, p₅₂, p₁₆, p₆₁, p₆₄, p₄₅, etc. In particular, the absolute values and signs of ELOCs for orthorhombic crystals characterized by the rather simple p_in matrix (without problematic coefficients) were completely determined in [17,18] using three methods (acousto-optic, interferometric and polarization-optical methods) including the application, in addition to uniaxial pressure, of the hydrostatic method to the samples.

Here, the matrix of p_in coefficients for trigonal lithium tantalate crystals LiTaO₃ (symmetry class 3m) was filled based on the experimentally determined matrix of piezo-optic coefficients (POCs) π_im and the known tensor expression

p_in = π_imC_mn,

(2)

where C_mn are elastic stiffness coefficients.

The coefficients π_im were determined by means of the static interferometric method described in papers [2,3,4,19,20,21,22,23]. These papers were focused on the reliability (objectivity) of the obtained results. For this purpose, the maximal number of the experimental geometries was considered and used, the corresponding expressions were written and the specific values of POCs π_im and the sums of POCs, such as π₁₂ + π₁₃, π₂₄ + 2π₄₂ = −(π₁₄ + 2π₄₁), etc., obtained from different experimental geometries, were compared (particularly, the π₁₁ coefficient is determined from six experimental geometries). The main results of the investigations of the photoelasticity (piezo- and elasto-optic effects) of lithium tantalate LiTaO₃ were compared with the results for trigonal lithium niobate LiNbO₃ and catangasite Ca₃TaGa₃Si₂O₁₄ crystals.

It should be noted that the polarization-optical (see, e.g., [16,17,18,24,25,26,27]) and conoscopic [28,29,30,31,32,33] methods are sometimes used for investigations of crystal photoelasticity. However, in this paper, the photoelasticity of lithium tantalate is solely studied using the interferometric technique.

2. Theory: Main Results

Lithium tantalate (LiTaO₃) crystals belong to 3m symmetry class [15] and their matrices of piezo- and elasto-optic coefficients (π_im and p_ik) include eight independent non-zero components [1,2] (Figure 1).

Five independent principal piezo-optic coefficients (POCs) π₁₁, π₁₂, π₁₃, π₃₁ and π₃₃ are determined on the sample of direct cuts (Figure 2a) using simple formulae for their calculation on the basis of experimental data (see below). The main difficulties in filling of POC and ELOC matrices relate to the determination of three non-principal POCs—π₁₄, π₄₁ and π₄₄. The piezo-optic effect (POE) should, to this end, be investigated on the sample of the X/45° cut. In addition to non-principal POCs, the principal POC π₁₁ as well as the sums of principal POCs such as π₂₁ + π₃₁, π₁₂ + π₁₃, π₂₂ + π₂₃, etc. can be determined on this sample. Comparison of these sums of POC with the analogous ones for principal coefficients π_im determined on the direct cut sample makes it possible to confirm the reliability of the POC values as well as piezo-optical identity of the samples cut from the different parts of crystal boule or from different boules.

For the determination of POC π_im using the interferometric method, the change of the optical path must be calculated [3] as:

$$\delta {\Delta }_k=\delta (n_i\cdot d_k)=\delta n_i\cdot d_k+\delta d_k(n_i-1),$$

(3)

where n_i is a refractive index of the crystal and d_k is the crystal thickness in the direction of light propagation.

After substituting the known expressions for the change of the refractive index δn_i under the influence of the mechanical stress σ_m and for the deformation of the sample δd_k along the direction of light propagation, it is easy to obtain the expression for δΔ_k, which contains POCs, represented by π_im, and elastic compliance coefficients, expressed as S_km (see, e.g., [3,20]):

$$\delta {\Delta }_k=-\frac{1}{2}{\pi }_{im}{\sigma }_m{d}_k{n}_i^3+S_{km}{\sigma }_m{d}_k(n_i-1).$$

(4)

For the known method of half-wave stresses (when δΔ_k = λ/2 and σ_m = σ_im is a half-wave mechanical stress), Expression (4) transforms to:

$${\pi }_{im}=-\frac{\lambda }{n_i^3{\sigma }_{im}{d}_k}+2S_{km}\frac{(n_i-1)}{n_i^3}=-\frac{\lambda }{n_i^3{\sigma }_{im}^o}+2S_{km}\frac{(n_i-1)}{n_i^3};$$

(5)

where ${\sigma }_{im}^o={\sigma }_{im}\cdot d_k$ is so called control stress, which is a characteristic of the material; despite the half-wave stress σ_im, which is a characteristic of a sample depended on its dimensions, the indices k, i and m correspondingly designate the directions of light propagation, polarization and the application of uniaxial pressure σ_m, and λ is the light wavelength.

Let us consider the examples of the specific expressions for the determination of the principal POCs based on Formula (2). In particular, under the experimental conditions k = 3 (the direction of light propagation), i = 1 (the direction of light polarization), m = 1 or m = 2 (directions of the application of uniaxial pressure σ₁ or σ₂), for the principal POCs π₁₁ and π₁₂, we obtain the following expressions for the experimental conditions (5):

$${\pi }_{11}=-\frac{\lambda }{n_1^3{\sigma }_{11}^o}+\frac{2S_{13}}{n_1^3}(n_1-1);{\pi }_{12}=-\frac{\lambda }{n_1^3{\sigma }_{12}^o}+\frac{2S_{23}}{n_1^3}(n_1-1),$$

(6)

where it is taken into account that the matrix of elastic compliance coefficients is symmetrical (S_km = S_mk), so S₃₁ = S₁₃, S₃₂ = S₂₃.

Expression (5) is valid for the determination of principal POCs π_im (i, m = 1, 2, 3) on the direct cut sample (Figure 2a), the edges of which are perpendicular to the crystal-physics axes X, Y, Z (the axes of the optical indicatrix). Such expressions are significantly complicated in the case of the investigation of POE on the sample of X/45°-cut (Figure 2b). For example, for the experimental conditions m = 4, k = $\overline{4}$, i = 1 (Figure 2b), the following expression is written for the determination of POC π₁₄ in [3]:

$$\delta {\Delta }_{\overline{4}}=-\frac{{\pi }_{12}+{\pi }_{13}+{\pi }_{14}}{4}\sigma d_{\overline{4}}{n}_1^3+\frac{1}{4}(S_{11}+2S_{13}+S_{33}-S_{44})\sigma d_{\overline{4}}(n_1-1),$$

(7)

and the analogous one for the symmetrical experimental conditions (m = $\overline{4}$, k = 4, i = 1):

$$\delta {\Delta }_4=-\frac{{\pi }_{12}+{\pi }_{13}-{\pi }_{14}}{4}\sigma d_4{n}_1^3+\frac{1}{4}(S_{11}+2S_{13}+S_{33}-S_{44})\sigma d_4(n_1-1),$$

(8)

where σ is a value of the uniaxial pressure.

In the case of the half-wave method (δΔ₄ = $\delta {\Delta }_{\overline{4}}$ = λ/2, σ = σ_im = σ₁₄ and σ = σ_im = ${\sigma }_{1\overline{4}}$), these expressions make it possible to obtain two equations for the determination of POC π₁₄:

$${\pi }_{12}+{\pi }_{13}+{\pi }_{14}=-\frac{2\lambda }{n_1^3{\sigma }_{14}{d}_{\overline{4}}}+(S_{11}+2S_{13}+S_{33}-S_{44})\frac{(n_1-1)}{n_1^3},$$

(9)

$${\pi }_{12}+{\pi }_{13}-{\pi }_{14}=-\frac{2\lambda }{n_1^3{\sigma }_{1\overline{4}}{d}_4}+(S_{11}+2S_{13}+S_{33}-S_{44})\frac{(n_1-1)}{n_1^3}.$$

(10)

These two equations are reduced to one (T. 6) in

Table 1. The expressions for the determination of POCs π_im on the sample of X/45°-cut for crystal classes 32, 3m and $\overline{3}$ m given for the method of half-wave (control) stresses.

Experimental Conditions	Expressions
Sample of X/45°-Cut
m = 1 k = 4($\overline{4}$) i = 1	${\pi }_{11}=-\frac{\lambda }{n_1^3{\sigma }_{11}^o{}_{|k=4(\overline{4})}}+(S_{12}+S_{13}\pm S_{14})\frac{n_1-1}{n_1^3}$	T. 1
	${\pi }_{11}=-\frac{\lambda }{2n_1^3}\left(\frac{1}{{\sigma }_{11}^o{}_{|k=4}}+\frac{1}{{\sigma }_{11}^o{}_{|k=\overline{4}}}\right)+(S_{12}+S_{13})\frac{n_1-1}{n_1^3}$	T. 2
m = 1 k = 4($\overline{4}$) i = $\overline{4}$(4)	${\pi }_{21}+{\pi }_{31}\mp 2{\pi }_{41}=-\frac{2\lambda }{n_4^3{\sigma }_{\overline{4}1(41)}^o}+2(S_{12}+S_{13}\pm S_{14})\frac{n_4-1}{n_4^3}$	T. 3
	${\pi }_{41}=-\frac{\lambda }{2n_4^3}\left(\frac{1}{{\sigma }_{41}^o}-\frac{1}{{\sigma }_{\overline{4}1}^o}\right)-S_{14}\frac{n_4-1}{n_4^3}$	T. 4
	${\pi }_{21}+{\pi }_{31}=-\frac{\lambda }{n_4^3}\left(\frac{1}{{\sigma }_{41}^o}+\frac{1}{{\sigma }_{\overline{4}1}^o}\right)+2(S_{12}+S_{13})\frac{n_4-1}{n_4^3}$	T. 5
m = 4($\overline{4}$) k = $\overline{4}$(4) i = 1	${\pi }_{12}+{\pi }_{13}\pm {\pi }_{14}=-\frac{2\lambda }{n_1^3{\sigma }_{14(1\overline{4})}^o}+(S_{11}+2S_{13}+S_{33}-S_{44})\frac{n_1-1}{n_1^3}$	T. 6
	${\pi }_{14}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{14}^o}-\frac{1}{{\sigma }_{1\overline{4}}^o}\right)$	T. 7
	${\pi }_{12}+{\pi }_{13}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{14}^o}+\frac{1}{{\sigma }_{1\overline{4}}^o}\right)+(S_{11}+2S_{13}+S_{33}-S_{44})\frac{n_1-1}{n_1^3}$	T. 8
m = 4($\overline{4}$) k = $\overline{4}$(4) i = 4($\overline{4}$)	$\begin{array}{l}{\pi }_{22}+{\pi }_{23}\pm {\pi }_{24}+{\pi }_{32}+{\pi }_{33}\pm 2{\pi }_{42}+2{\pi }_{44}=\\ =-\frac{4\lambda }{n_4^3{\sigma }_{44(\overline{4}\overline{4})}^o}+2(S_{11}+2S_{13}+S_{33}-S_{44})\frac{n_4-1}{n_4^3}\end{array}$	T. 9
	$\begin{array}{l}{\pi }_{22}+{\pi }_{23}+{\pi }_{32}+{\pi }_{33}+2{\pi }_{44}=\\ =-\frac{2\lambda }{n_4^3}\left(\frac{1}{{\sigma }_{44}^o}+\frac{1}{{\sigma }_{\overline{4}\overline{4}}^o}\right)+2(S_{11}+2S_{13}+S_{33}-S_{44})\frac{n_4-1}{n_4^3}\end{array}$	T. 10
	${\pi }_{24}+2{\pi }_{42}=-\frac{2\lambda }{n_4^3}\left(\frac{1}{{\sigma }_{44}^o}-\frac{1}{{\sigma }_{\overline{4}\overline{4}}^o}\right)$	T. 11
m = 4($\overline{4}$) k = 1, i = 2	${\pi }_{22}+{\pi }_{23}\pm {\pi }_{24}=-\frac{2\lambda }{n_1^3{\sigma }_{24(2\overline{4})}^o}+2(S_{12}+S_{13}\pm S_{14})\frac{n_1-1}{n_1^3}$	T. 12
	${\pi }_{24}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{24}^o}-\frac{1}{{\sigma }_{2\overline{4}}^o}\right)+2S_{14}\frac{n_1-1}{n_1^3}$	T. 13
	${\pi }_{22}+{\pi }_{23}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{24}^o}+\frac{1}{{\sigma }_{2\overline{4}}^o}\right)+2(S_{12}+S_{13})\frac{n_1-1}{n_1^3}$	T. 14
m = 4($\overline{4}$) k = 1 i = 3	${\pi }_{32}+{\pi }_{33}=-\frac{2\lambda }{n_3^3{\sigma }_{34(3\overline{4})}^o}+2(S_{12}+S_{13}\pm S_{14})\frac{n_3-1}{n_3^3}$	T. 15
	${\pi }_{32}+{\pi }_{33}=-\frac{\lambda }{n_3^3}\left(\frac{1}{{\sigma }_{34}^o}+\frac{1}{{\sigma }_{3\overline{4}}^o}\right)+2(S_{12}+S_{13})\frac{n_3-1}{n_3^3}$	T. 16

Note: the symmetry conditions of the experiment are indicated in brackets, e.g., in Equation (T. 6), the control stresses are written in the form of ${\sigma }_{14\left(1\overline{4}\right)}^o$, i.e., ${\sigma }_{14}^o$ corresponds to the sign (+) at π₁₄, ${\sigma }_{1\overline{4}}^o$ –to the sign (−), etc.

, where the products ${\sigma }_{14}{d}_{\overline{4}}$ and ${\sigma }_{1\overline{4}}{d}_4$ are substituted by the symbols ${\sigma }_{14}^o$ and ${\sigma }_{1\overline{4}}^o$, thus designating the control stresses for the above-mentioned direct and symmetrical experimental conditions.

Because the signs before the coefficient π₁₄ are opposite in Equations (9) and (10), after their subtraction, one can dispose of the principal POCs π₁₂ and π₁₃, as well as the elastic item:

$${\pi }_{14}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{14}^o}-\frac{1}{{\sigma }_{1\overline{4}}^o}\right),$$

(11)

whereas, after the summation of (9) and (10), one can obtain the expression for the calculation of the sum π₁₂ + π₁₃ based on the same control stresses:

$${\pi }_{12}+{\pi }_{13}=-\frac{\lambda }{n_1^3}\left(\frac{1}{{\sigma }_{14}^o}+\frac{1}{{\sigma }_{1\overline{4}}^o}\right)+(S_{11}+2S_{13}+S_{33}-S_{44})\frac{n_1-1}{n_1^3}$$

(12)

This sum of POCs will make it possible to confirm the reliability of the values of principal POCs π₁₂ and π₁₃, the objectivity of the experimentally determined control stresses${\sigma }_{14}^o$, ${\sigma }_{1\overline{4}}^o$ and, correspondingly, the reliability of the value of POC π₁₄.

All other relationships for the calculation of the non-principal POCs and different variants of sums of principal POCs based on the experimental values of the control stresses ${\sigma }_{im}^o$ are given in Table 1. The experimental geometries for the determination of the principal POC π₁₁ on the sample of X/45°-cut are also considered, see formulae (T. 1), (T. 2) in Table 1. The expression for the determination of POC π₄₄ (T. 9) is particularly complex; it includes the complex sums of POCs and elastic compliance coefficients S_km. However, after the summation or subtraction of expressions that differ in terms of sign at some non-principal POCs, simpler expressions can be obtained for direct and symmetrical experimental conditions. In particular, the problematic POCs π₂₄ = − π₁₄ and 2π₄₂ = −2π₄₁ are absent in the expression (T. 10) and all principal POCs and POC π₄₄ as well as all elastic compliance coefficients S_km are absent in (T. 11). Accordingly, the absolute errors of the determination of corresponding non-principal POCs will be significantly lower.

3. The Experimental Technique

The investigated LiTaO₃ crystals were grown at the SRC ‘Electron-Carat’ (Lviv, Ukraine) by means of the Czochralski technique from the congruent melt. The monodomainization of the crystals was carried out by heating them to a temperature that was between 30 and 40 degrees higher than the Curie point, subsequent connection to the dc voltage source (10–15 V), and slow cooling to room temperature.

As mentioned above, lithium tantalate crystals belong to the symmetry class 3m. The matrix of POCs correspondingly includes 8 independent components π_im. For their determination, the samples of the direct cut (Figure 2a) and X/45°-cut (Figure 2b), with the dimensions of about 7 mm × 7 mm × 7 mm, were investigated. The samples withstood high mechanical stress of about 300 kg/cm². The interferometric technique was used for the determination of absolute POCs π_im. The experimental set-up was based on the one-pass Mach–Zehnder laser interferometer, and the sample was placed in one of the interferometer shoulders (see, e.g., [3,21]). For the determination of sign of the optical path change δΔ_k, which was induced by uniaxial pressure, a thin plate of fused silica was placed in the measuring shoulder of the interferometer after the sample. The rotation of the plate from the direction perpendicular to the direction of light beam propagation increased the optical path of the light beam and, correspondingly, led to the displacement of the interferometric band in a certain direction. If under the influence of the uniaxial pressure σ_m, the bands shifted to the same direction, and the sign of δΔ_k was positive, whereas in the opposite direction, it was negative. This sign was placed before λ in the formulae for the calculation of POCs π_im on the basis of control stresses ${\sigma }_{im}^o$ and, moreover, it had to be taken into account that the stresses of compression were attributed to the “minus” sign. A detailed description of the experimental set-up and the procedure of POCs determination was given in paper [21].

Let us recall that in the case of complex POC matrix containing such non-principal components as π₁₄, π₄₁, etc., the ambiguity of the determination of these POCs both on signs and absolute values (see, e.g., [3]) exists depending on the choice of the positive signs of the right coordinate system X, Y, Z. Usually, the signs of the axes are chosen on the basis of piezo-electric effect [3,4,27]. However, the piezo-electric coefficients d_lm of lithium tantalate are relatively low (2–3 times lower than the ones of lithium niobate [15]). Correspondingly, it was impossible to specify the signs of the right coordinate system axes X, Y and Z based on the d_lm coefficients. Therefore, in this case, the positive signs of the axes Y, Z and, correspondingly, the directions 4 and $\overline{4}$, were chosen on the basis of the positive value of the elastic compliance coefficient S₁₄.

The elastic compliance coefficient S₁₄ of the crystals of 32, 3m and $\overline{3}$m classes could be determined via the piezo-optic technique. Namely, if two equations (T. 1 in Table 1) (they differed according to the signs «+» or «–» before the S₁₄ coefficient and the control stresses ${\sigma }_{11}^o{}_{|k=4}$ and ${\sigma }_{11}^o{}_{|k=\overline{4}}$) were added, the expression (T. 2) was obtained for the determination of π₁₁ on the sample of X/45°-cut. If these Equations were subtracted, one obtained the expression for the determination of the S₁₄ coefficient:

$$S_{14}=\frac{\lambda }{2(n_1-1)}\left(\frac{1}{{\sigma }_{11}^o{}_{|k=4}}-\frac{1}{{\sigma }_{11}^o{}_{|k=\overline{4}}}\right).$$

(13)

It should be noted that the values of control stresses ${\sigma }_{11}^o{}_{|k=4}$ i ${\sigma }_{11}^o{}_{|k=\overline{4}}$ traded places depending on the choice of coordinate system on the sample of the X/45°-cut (Figure 3). The elastic compliance coefficient S₁₄ had the sign «+» or «–».

If the uniaxial pressure was applied along the direction 1 (m = 1), we chose the direction of light polarization in the same direction (i = 1) and the direction of light propagation was ensured along 4 (Figure 3a), and thus, we obtained the control stress ${\sigma }_{11}^o{}_{|k=4}$. If the other coordinate system was chosen on the same sample, such as the one in Figure 3b, the control stress had the designation ${\sigma }_{11}^o{}_{|k=\overline{4}}$. Thus, the experimental value of the control stress needed to be placed in the position of the first item in brackets in expression (13) rather than in the position of the second item. Therefore, the S₁₄ coefficient could be determined only to within the sign.

In this investigation, we chose the coordinate system (one of the systems shown in Figure 3) that corresponded to the condition S₁₄ > 0. This condition was used because, in three papers [34,35,36], the coefficients S₁₄ were commensurate with the values and were positive (see

Table 2. Elastic stiffness coefficients C_mk (in 10¹¹ N/m²) and elastic compliance coefficients S_km (in 10⁻¹² m²/N) of LiTaO₃ crystals.

Cmk	C11	C33	C12	C13	C14	C44	Refer.
1.	2.39	2.84	0.41	0.80	−0.22	1.13	[34]
2.	2.38	2.82	0.21	0.73	−0.27	1.17	[35]
3.	2.421	2.752	0.375	0.827	−0.237	1.139	[36]
Skm	S11	S33	S12	S13	S14	S44	Refer.
4.	4.76	4.19	−0.5	−1.20	1.02	9.3	[34]
5.	4.68	4.14	−0.16	−1.17	1.10	9.00	[35]
* 6.	4.72	4.42	−0.37	−1.31	0.98	9.16	* Calculated

Note: the asterisk (*) designates the row where the values of S_km were calculated on the basis of the matrix of elastic shiftiness coefficients C_mk [36] using the inverse matrix method.

). This coordinate system was used for all studies of POE in LiTaO₃ crystals.

It should be emphasized that the value of the S₁₄ coefficient could be defined by the piezo-optic interferometric technique for other experimental conditions, and this was used for the determination of the sum π₃₂ + π₃₃, see formula (T.15) in Table 1. Through the summation of two Equations (T.15), one could obtain the expression (T. 16) for sum π₃₂ + π₃₃, which could be used to prove the reliability of POC values π₃₁ = π₃₂ and π₃₃. By means of the subtraction of Equations (T.15), which differ in terms of the sign before the S₁₄ coefficient, we obtained one more equation for the determination of S₁₄ (under other experimental conditions, on the sample of the X/45°-cut: m = 4 or $\overline{4}$, k = 1, i = 3) based on the control stresses ${\sigma }_{34}^o$ and ${\sigma }_{3\overline{4}}^o$:

$$S_{14}=\frac{\lambda }{2(n_3-1)}\left(\frac{1}{{\sigma }_{34}^o}-\frac{1}{{\sigma }_{3\overline{4}}^o}\right).$$

(14)

The experimental values of the control stresses were ${\sigma }_{11}^o{}_{|k=4}$ = −103 kG/cm and ${\sigma }_{11}^o{}_{|k=\overline{4}}$ = −185 kG/cm; thus, it followed from (13) that S₁₄ = + 1.18 ± 0.31 (in units, 10⁻¹² m²/N). From this point onwards, it was taken into account that the stresses of compressions had the sign ‘−‘. On the other hand, if the experimental values of the control stresses ${\sigma }_{34}^o$ = 450 kG/cm and ${\sigma }_{3\overline{4}}^o$ = 170 kG/cm were substituted in (14), one could obtain S₁₄ = + 1.00 ± 0.17 (in units, 10⁻¹² m²/N). Thus, both values of S₁₄ were commensurate with the limits of experimental errors. However, we used the second one (1.00 ± 0.17), because the error of this S₁₄ coefficient was significantly lower. It should be noted that both values of the S₁₄ coefficient coincided to a significant extent (within the limits of errors) with the values of S₁₄ determined in papers [34,35,36] (see Table 2).

4. Results of Investigations of POE in LiTaO₃ Crystals and Their Analysis

For the calculation of absolute POCs of lithium tantalate, the experimentally defined control stresses, represented as ${\sigma }_{im}^{\mathrm{o}}$ (

Table 3. Results of investigations of POE in LiTaO₃ crystals (λ = 632.8 nm, T = 20 °C, 1 Br = 1 Brewster = 10⁻¹² m²/N).

No.	Experimental Conditions			${\mathit{\sigma }}_{\mathit{i}\mathit{m}}^{\mathbf{o}},\mathbf{kG}/\mathbf{cm}$	πim, Br
	m	k	i
Direct cut samples, Figure 2a
1	1	2	1	σ°11 = −104	π11 = −0.64 ± 0.06
2			3	σ°31 = 104	π31 = 0.56 ± 0.06
3	1	3	1	σ°11 = −175	π11 = −0.63 ± 0.05
4			2	σ°21 = 86	π21 = 0.46 ± 0.07
5	2	1	2	σ°22 = −110	π22 = −0.61 ± 0.06
6			3	σ°32 = 112	π32 = 0.52 ± 0.06
7	2	3	2	σ°22 = −180	π22 = −0.62 ± 0.04
8			1	σ°12 = 84	π12 = 0.48 ± 0.08
9	3	1	3	σ°33 = ∞	π33 = −0.27 ± 0.01
10			2	σ°23 = 65	π23 = 0.70 ± 0.10
11	3	2	3	σ°33 = ∞	π33 = −0.27 ± 0.01
12			1	σ°13 = 62.5	π13 = 0.74 ± 0.10
Sample of X/45°-cut, Figure 2b
13	1	4	1	σ°11 = −103	π11 = −0.63 ± 0.04
14			$\overline{4}$	${\sigma }_{\overline{4}1}^o$ = 70	π41 = −0.29 ± 0.05
15	1	$\overline{4}$	1	σ°11 = −185	π11 = −0.63 ± 0.04
16			4	σ°41 = 200	π12 + π31 = 0.90 ± 0.09
17	4	1	2	σ°24 = 200	π24 = 0.43 ± 0.04
18			3	σ°34 = 450	π31 + π33 = 0.20 ± 0.04
19	4	$\overline{4}$	1	σ°14 = 130	π14 = −0.37 ± 0.10
20			4	σ°44 = 140	π44 = −0.06 ± 0.09
21	$\overline{4}$	1	2	${\sigma }_{2\overline{4}}^o$ = 470	π11 + π13 = 0.14 ± 0.04
22			3	${\sigma }_{3\overline{4}}^o$ = 170	π31 + π33 = 0.20 ± 0.04
23	$\overline{4}$	4	1	${\sigma }_{1\overline{4}}^o$ = 74	π12 + π13 = 1.04 ± 0.12
24			$\overline{4}$	${\sigma }_{\overline{4}\overline{4}}^o$= −1250	π24 + 2π42 = 0.99 ± 0.09

Note: the sign ‘minus’ before the control stresses ${\sigma }_{11}^o$, ${\sigma }_{22}^o$, ${\sigma }_{\overline{4}\overline{4}}^o$ (rows 1, 3, 5, 24, etc.) indicates the shortening of the optical pass under the influence of the uniaxial pressure; when the values of π_im are calculated, this sign is placed before λ; at this point, it is taken into account that the stresses of compression have the sign ‘minus’.

), and the refractive indices indicated in [15] (n₁ = n_o = 2.175, n₃ = n_e = 2.180 for T_room and the wavelength of λ = 632.8 nm) were used. The refractive indices along the diagonal directions 4 and $\overline{4}$ were determined in accordance with the known expression

$$n_4=n_{\overline{4}}=\frac{\sqrt{2}}{\sqrt{a_2+a_3}}=\sqrt{2}/\sqrt{\frac{1}{n_2^2}+\frac{1}{n_3^2}}=\frac{\sqrt{2}{n}_2{n}_3}{\sqrt{n_2^2+n_3^2}},$$

(15)

where a_i = 1/ n_i² is polarization constants.

The elastic compliance coefficients S_km included in the expressions for the determination of POCs π_im are given in Table 2. The best convergence of π_im coefficient calculations (i.e., the closest values of specific POC π_im determined for different experimental geometries, or the same sums of POCs within the limits of the precision of their calculation found from direct measurements or formed from independent POCs π_im) was ensured when the elastic compliance coefficients from [35] was used, see row 5 in Table 2. The elastic shiftiness coefficients C_mn (row 2 in Table 2), for the calculations of ELOCs p_in (see Section 5), were taken from the same paper.

The calculations of the errors of POCs π_im were carried out on the basis of the experimental errors of control stresses ${\sigma }_{im}^{\mathrm{o}}$, which were about 10% of the value of ${\sigma }_{im}^{\mathrm{o}}$, and the errors of elastic compliance coefficients S_km, which were about 5% of the value of S_km, as was also the case in our other papers.

Comments to the Results of POE Investigations in LiTaO₃ Crystals

• As can be seen in Table 3, the π₁₁ coefficient was determined from four experimental geometries (taking into account that π₂₂ = π₁₁) on the direct cut sample (see rows 1, 3, 5, 7) and from two experimental geometries on the sample of the X/45°-cut (rows 13 and 15). The other principal POCs π_im (i, m = 1, 2, 3), namely, π₁₂ = π₂₁, π₁₃ = π₂₃, π₃₁ = π₃₂ and π₃₃, were determined from two experimental geometries. These POCs, defined from different experimental geometries, were the same in terms of the limits of the errors of their determination, thus proving the reliability of the obtained values. The mean-square values of POCs π_im calculated for different experimental geometries as well as their mean-square errors are indicated in the table showing the final results of the investigation of POE in lithium tantalate crystals (

  Table 4. All independent piezo-optic coefficients π_im of LiTaO₃, LiNbO₃ and CTGS crystals (λ = 632.8 nm, T = 20 °C).

  πim, Br	π11	π12	π13	π31	π33	π14	π41	π44
  LiTaO3 (This Work)	−0.63 ± 0.05	0.47 ± 0.08	0.72 ± 0.10	0.54 ± 0.06	−0.27 ± 0.01	−0.43 ± 0.04	−0.29 ± 0.05	−0.06 ± 0.09
  LiTaO3 [37]	−0.62 ± 0.02	0.34 ± 0.03	0.64 ± 0.05	0.43 ± 0.02	−0.07 ± 0.01	0.40 ± 0.07	0.07 ± 0.03	0.41 ± 0.08
  LiNbO3 [27]	−0.38	+0.09	+0.80	+0.50	+0.20	−0.81	−0.88	+2.25
  CTGS [4]	−0.19 ± 0.06	0.22 ± 0.09	0.53 ± 0.12	1.40 ± 0.19	−1.20 ± 0.09	0.72 ± 0.11	0.32 ± 0.11	−0.81 ± 0.40

  Note: the value of POC π₄₄ is lower than the error of its determination.

  ). For comparison, the data on the π_im coefficients for trigonal lithium niobate LiNbO₃ [27], calcium-tantalum gallosilicate Ca₃TaGa₃Si₂O₁₄ (CTGS) [4] and for lithium tantalate LiTaO₃, from the paper [37], are also shown in Table 4.
• The non-principal POCs π₁₄ and π₂₄ were also determined from two independent experimental conditions (taking into account the equalities π₂₄ = −π₁₄, π₄₂ = −π₄₁, which are valid for the crystal class 3m). Correspondingly, the mean-square values of these POCs and their mean-square errors are included in main results (Table 4).
• The errors of the determination of POCs π_im were calculated as mean-square values of the errors of two summands that were included in the expressions for the determination of POCs; specifically, the piezo-optic summand containing the control stresses ${\sigma }_{im}^o$ and elastic summand formed by elastic compliance coefficient S_km or the combination of these coefficients. If the elastic component is formed by the sum of S_km coefficients (see, e.g., Equations (T. 2), (T. 5), (T. 8), etc.), the mean-square error of this sum should be determined. As can be seen in Table 1, the Equation (T. 10) for the determination of POC π₄₄ includes sum of other POCs π_im. Therefore, before the calculation of the π₄₄ coefficient, the mean-square error of the sum of the other POCs included in corresponding equation should be determined.
• The comparison of sums of coefficients determined on the sample of the X/45°-cut (see, e.g., rows 16, 18, 21, etc. in Table 3) with the same sums of independent principal POCs defined on the direct cut sample shows that the results closely coincide with the limits of the error of their determination (

  Table 5. Sums of POCs obtained on the sample of the X/45°-cut (LiTaO₃ crystal).

  No.	Experimental Conditions			Σπim (Based on Direct Measurements), Br	Σπim (Sum of Independent POCs), Br
  	m	k	i
  1	1	4($\overline{4}$)	$\overline{4}$(4)	π21 + π31 = 0.90 ± 0.09	1.01 ± 0.10
  2	4($\overline{4}$)	1	2	π22 + π23 = 0.14 ± 0.04	0.09 ± 0.11
  3	4($\overline{4}$)	1	3	π32 + π33 = 0.20 ± 0.04	0.27 ± 0.06
  4	4($\overline{4}$)	$\overline{4}$(4)	1	π12 + π13 = 1.04 ± 0.12	1.19 ± 0.13
  5	4($\overline{4}$)	$\overline{4}$(4)	4($\overline{4}$)	π24 + 2π42 = 0.99 ± 0.09	1.01 ± 0.09

  ). In particular, the value of the sum of POCs π₃₂ + π₃₃ = 0.20 ± 0.04 Br. The same sum formed from independent POCs π₃₁ = π₃₂ and π₃₃ (see Table 5) is equal to 0.27 ± 0.06 Br. Therefore, there is excellent agreement among the results. The analogous convergence of sums of POCs is also distinctive for other experimental geometries (see Table 5). On the one hand, this points to the reliability of the values of the principal POCs π_im and, on the other hand, to the objectivity of the results obtained on the sample of the X/45°-cut. Note that the sum of POCs π₁₂ + π₁₃ obtained on the basis of the control stresses ${\sigma }_{14}^o$ and ${\sigma }_{1\overline{4}}^o$ coincides with the same sum of independent POCs. Based on the same control stresses, the π₁₄ coefficient was determined, see formula (T. 7) in Table 1. Therefore, the value of π₁₄ coefficient is also reliable. In the same way, the reliability of POC π₄₁ can be proved. For this purpose, the formulae (T. 4), (T. 5) from Table 1 and the control stresses ${\sigma }_{41}^o$, ${\sigma }_{\overline{4}1}^o$ (Table 3) have to be used.
• Note the peculiarity of POE in the LiTaO₃ crystal. That is, the control stress ${\sigma }_{33}^o$ → ∞ (Table 3) for the experimental geometry m = 3, k = 1 (or k = 2), i = 3. This means that when applying uniaxial pressure along the Z = X₃ axis, the optical path does not change, i.e., the piezo-optic and elastic contributions to the optical path change δΔ_k are equal to the absolute values but opposite in terms of their signs. This can be shown on the basis of Expression (4), which describes the optical path change; under the experimental conditions m = 3, k = 1, i = 3. This expression can be written in the form

  $$\frac{\delta {\Delta }_k}{{\sigma }_m{d}_k}=-\frac{1}{2}{\pi }_{im}{n}_i^3+S_{km}(n_i-1)$$

  (16)

  or, for the mentioned experimental conditions, one can obtain (in units, 10⁻¹² m²/N = 1 Br):

  $$\begin{array}{ll}\frac{\delta {\Delta }_1}{{\sigma }_3{d}_1}& =-\frac{1}{2}{\pi }_{33}{n}_3^3+S_{13}(n_3-1)=-\frac{1}{2}(-0.27)\cdot 10.36+(-1.17)\cdot (1.18)=\\ & =+1.400-1.381=0.02\pm 0.16,\end{array}$$

  (17)

  where the error ± 0.16 is calculated as the mean-square value of the sum of errors of the first (piezo-optic) item of (17), i.e., 10% of the value of 1.400, and the second (elastic) item, i.e., 5% of the value of 1.381.

Therefore, the optical path change is an order lower that the error of its determination and two orders lower for each of the items in (17), i.e., δΔ₁/(σ₃d₁) → 0.

6.

For comparison, the POCs π_im for lithium niobate LiNbO₃ [27] and calcium-tantalum gallosilicate CTGS [4] are given in Table 4, which shows the main results for LiTaO₃. As can be seen in Table 4, the principal POCs of these crystals have the same signs (except for π₃₃ coefficient of LiNbO₃). All principal POCs of lithium niobate and lithium tantalate crystals are relatively low (no higher than 1 Br). On the contrary, two coefficients (π₃₁ and π₃₃) of the CTGS crystal are essentially higher than 1 Br.

From the comparison of the obtained values of POCs of π_im of LiTaO₃ crystal (first row in Table 4) with the data of [37] (second row), it follows that the essential discrepancies take place in the values of POCs π₃₃, π₄₁, π₄₄ as well as in the signs of the non-principal POCs π₁₄, π₄₁ and π₄₄. There are two factors that are clearly the reasons for these discrepancies: (1) the slight non-parallelism of the optical faces of a sample was not taken into account in [37], and this can be the source of considerable errors [19,27]; (2) the authors of [37] did not indicate how the positive directions of X, Y and Z axes of right coordinate system were chosen and, correspondingly, how the directions 4 and $\overline{4}$ were assigned on the sample of the X/45°-cut. Moreover, there were no data on the reliability of POC π_im in [37], whereas, in this paper, the reliability was proven by several results for different experimental geometries (see parts 1, 2 and 4 above). Because of considerable attention devoted to the reliability of the results in the present investigation, it can be argued that the obtained results of the study of POE in lithium tantalate are objective.

5. Elasto-Optic Coefficients of LiTaO₃ Crystals

ELOCs p_in were calculated on the basis of the tensor Expression (2), which is detailed in the following form for the 3m crystal class:

$$\begin{gathered} p_{11}={\pi }_{11}{C}_{11}+{\pi }_{12}{C}_{12}+{\pi }_{13}{C}_{13}+{\pi }_{14}{C}_{14},p_{12}={\pi }_{11}{C}_{12}+{\pi }_{12}{C}_{11}+{\pi }_{13}{C}_{13}-{\pi }_{14}{C}_{14}, \\ {p}_{13}=({\pi }_{11}+{\pi }_{12})C_{13}+{\pi }_{13}{C}_{33},p_{31}={\pi }_{31}(C_{11}+C_{12})+{\pi }_{33}{C}_{13},p_{33}=2{\pi }_{31}{C}_{13}+{\pi }_{33}{C}_{33}, \\ {p}_{14}=({\pi }_{11}-{\pi }_{12})C_{14}+{\pi }_{14}{C}_{44},p_{41}={\pi }_{41}(C_{11}-C_{12})+{\pi }_{44}{C}_{14},p_{44}=2{\pi }_{41}{C}_{14}+{\pi }_{44}{C}_{44}. \end{gathered}$$

(18)

POCs π_im from Table 4 (the first row) and elastic shiftiness coefficients C_mk from paper [35] (see Table 2, row 2) were used for the calculations. Their results are given in

Table 6. All independent elasto-optic p_in coefficients of LiTaO₃, LiNbO₃ and CTGS crystals (λ = 632.8 nm, T = 20 °C).

pim	p11	p12	p13	p31	p33	p14	p41	p44
LiTaO3 (This Work)	−0.076 ± 0.016	0.140 ± 0.021	0.191 ± 0.031	0.120 ± 0.016	0.003 * ± 0.008	−0.021 ± 0.006	−0.061 ± 0.013	0.009 * ± 0.011
LiTaO3 [38]	−0.081 ± 0.003	0.081 ± 0.003	0.093 ± 0.02	0.089 ± 0.004	−0.044 ± 0.004	−0.026 ± 0.002	−0.085 ± 0.006	−0.028 ± 0.002
LiNbO3 [39]	−0.021 ± 0.010	0.060 ± 0.012	0.172 ± 0.028	0.142 ± 0.018	0.118 ± 0.017	−0.052 ± 0.007	−0.109 ± 0.018	0.121 ± 0.033
LiNbO3 [13]	−0.026	0.090	0.133	0.179	0.071	−0.075	−0.151	0.146
CTGS [4]	0.008 * ± 0.010	0.044 ± 0.013	0.096 ± 0.022	0.165 ± 0.031	−0.088 ± 0.031	0.029 ± 0.005	0.029 ± 0.015	0.033 ± 0.017

Note: the values of coefficients marked by asterisks (*) are lower than the errors of their determination.

. In addition to the p_in values, the errors of their determination δp_in are also indicated in the Table. They were calculated as mean-square errors of the product π_im⋅C_mk based on the known expression δ(π_im⋅C_mk) = [(δπ_im⋅C_mk)² + (π_im⋅δC_mk)²]^1/2, which was used for each item in (18). For example, the following expression is valid for the determination of the error of the p₁₃ coefficient:

δp₁₃ = [(δπ₁₁⋅C₁₃)² + (δπ₁₁⋅δC₁₃)² + (δπ₁₂⋅C₁₃)² +(π₁₂⋅δC₁₃)² + (δπ₁₃⋅C₃₃)² + (π₁₃⋅δC₃₃)²]^1/2.

(19)

The errors δπ_im are indicated for the values of π_im (Table 4), and the errors of the elastic shiftiness coefficients, represented by δC_mk, are calculated as 5% of the values of C_mk (in addition to the errors of the S_km coefficients). The important conclusion from the analysis of the contributions of items with errors δπ_im and δC_mk to the values of δp_in is that the main contributions are caused by errors of the piezo-optic coefficients, represented by δπ_im. Namely, the contribution of δπ_im to δp₁₁ is 87%, the contribution to δp₁₂ is 95%, the contribution to δp₁₃ is 94%, etc. Thus, a further conclusion that can be drawn is that a significant decrease in the ELOC error, represented by δp_in, requires a significant decrease in the POC error π_im. For this purpose, the precision of the experimental determination of the control stresses, σ^o_im, should be increased from 10% to (3–5%). This is a central problem that can be solved as needed.

It follows from the analysis of the results shown in Table 6 that the results for the p_in coefficients of the LiNbO₃ and LiTaO₃ crystals obtained by Avakyants and co-authors [13,38] (the Brillouin scattering method) are in good agreement with our data on lithium niobate [39]. In particular, the deviation of the values of the p₁₁ coefficient obtained in [13] and [39] from their average value is small, and this deviation is equal to 10.5%. Thus, we found good convergence of the results obtained by means of the static interferometric technique [39] and the dynamic (acousto-optic) method of Brillouin scattering [13]. Unfortunately, such a convergence was not observed in the case of lithium tantalate [38]. It can be assumed that the convergence of our results with the ones of [38] would be better if all parameters (π_im, p_in, S_km, C_mk) are measured on the same samples and, moreover, if the thermodynamic (electrical) conditions of these measurements (with a constant electric field or induction, E = const or D = const) are taken into consideration.

As can also be seen in Table 6, the trigonal LiTaO₃, LiNbO₃ and CTGS crystals reveal a slight overall elasto-optic effect. The maximal values of the ELOCs p_in of these crystals are commensurate and are not higher than 0.191 for LiTaO₃, 0.179 for LiNbO₃ and 0.165 for CTGS. However, the positive characteristics of these crystals—namely high mechanical strength, high optical quality and resistance to aggressive environments—as well as the existence of effective technologies for the growth of large single crystals make these crystals the most important objects in applications related to photo-elastic or acousto-optic devices for the purpose of optical beam control. Maximal elasto-optic or acousto-optic efficiencies of these crystals can be determined from the analysis of surfaces that are indicative of the above-mentioned effect to varying degrees, see, e.g., papers [40,41,42,43,44,45] and others.

6. Conclusions

The piezo-optic effect in trigonal LiTaO₃ crystals was investigated through the use of the interferometric method. The experiment was carried out for the maximal number of possible experimental geometries in order to demonstrate the reliability (objectivity) of the results. In particular, the principal piezo-optic coefficient (POC) π₁₁ was determined from six experimental geometries, and all other principal POCs π_im (i, m = 1, 2, 3) as well as non-principal POCs π₁₄, π₄₁, π₄₄ were determined from two experimental geometries. The convergence of sums of independent POCs, such as π₁₂ + π₁₃ and others, obtained for the samples of the X/45° cut with the ones obtained for the direct cut samples was shown. Our approach makes it possible to prove the reliability of the obtained data and the piezo-optical identity of the samples cut from the different parts of crystal boule, or those from different boules.

The main results of the work are the following:

• All relationships for complex experimental geometries were obtained for the crystals of the 32, 3m and $\overline{3}$m symmetry classes.
• The ambiguity of the determination of problematic POCs π₁₄, π₄₁ was eliminated, in terms of both the sign and the absolute value, through the unambiguous selection of the correct coordinate system. The condition S₁₄ > 0 (S₁₄ is the elastic compliance coefficient) was used for this purpose.
• The effect non-typical for LiTaO₃ crystals was revealed: with the application of uniaxial pressure along the Z axis, the change of the optical path of a light beam δΔ_k → 0.
• All independent components of the matrix of elasto-optic coefficients, represented by p_in, were calculated based on the components of the POCs matrix, π_im, which were determined by means of the interferometric technique. The comparative analysis of the p_in coefficients of LiTaO₃ and LiNbO₃, which were determined through the static interferometric method and the dynamic (acousto-optic) Brillouin scattering method, was carried out.
• The LiTaO₃ crystals, as well as the LiNbO₃ [27,39] and Ca₃TaGa₃Si₂O₁₄ [4] crystals that were previously investigated by us, revealed relatively low elasto-optic effect (the maximal value of ELOC was p_in ≈ 0.2). However, their high mechanical strength, high optical quality and resistance to aggressive environments, as well as the availability of effective technologies for the growth of large single crystals, were found to be excellent baselines for applications that make use of these crystals.