Spatial Anisotropy of Photoelasticity Determined by Path Difference in Ba₃TaGa₃Si₂O₁₄ Crystals

Abstract

The elastic and photoelastic coefficients of Ba₃TaGa₃Si₂O₁₄ (BTGS) crystals were determined by the quantum–mechanical calculation technique. Based on these data, extreme piezo-optic surfaces π′°_km were constructed, which describe the change in the path difference in light beams in the crystal under the influence of mechanical stress. The results for BTGS crystals are compared with the ones for other crystals of the langasite group (La₃Ga₅SiO₁₄, Ca₃Ga₂Ge₄O₁₄, Ca₃TaGa₃Si₂O₁₄ and Ca₃NbGa₃Si₂O₁₄). The global maxima of the π′°_km surfaces for BTGS crystals significantly exceed the ones for the other crystals mentioned above and, accordingly, BTGS crystals can be suitable for use in polarization-optic light modulators and devices based on them. The acousto-optic efficiency of BTGS crystals was evaluated. The correlations between the magnitude of the piezo- and elasto-optic coefficients and the parameters of the unit cell of the studied crystals were determined.

1. Introduction

Sensitive elements of optical modulating devices are, as a rule, bulk crystalline materials. The number of photoelastic materials that have been used in practice includes few crystals, so the task of finding new effective materials for controlling laser radiation is relevant.

Interest in crystals of the langasite group [1,2,3,4,5,6] is due to the fact that they reveal a unique combination of various physical properties. For instance, the temperature stability of physical characteristics, in particular elastic [7,8], as well as a large piezoelectric effect and a high coefficient of electromechanical coupling (e.g., the value of the piezoelectric coefficient d₁₁ of langasite group crystals is in the range of 4…7 pC/N [9,10,11], that is 2…3 times higher than in quartz crystals [9])—makes them a popular material for development of acoustoelectronic devices [6]. In particular, they are used in pressure and detonation sensors, resonators in tunable generators, substrates of thermostable slices for acoustoelectronic filters on surface and bulk acoustic waves, Q-switches, as well as temperature-stable broadband monolithic filters used in mobile communication systems [12,13,14]. Because of a high optical damage threshold of ~1 GW/cm² [15], these crystals can be used as sensitive elements in acousto-optical converters. The optical properties of crystals of the LGS group have also been widely studied, see, e.g., works [1,4,9,13,16,17]. However, the photoelastic properties of these crystals were not investigated until our first papers [18,19,20]. In our previous works, the piezo-, elasto- and acousto-optical properties of several representatives of this group were experimentally studied: crystals of langasite La₃Ga₅SiO₁₄ (LGS) [18] and calcium halogermanate Ca₃Ga₂Ge₄O₁₄ (CGG) [20], which have a disordered structure (see [21]), and crystals of catangasite Ca₃TaGa₃Si₂O₁₄ (CTGS) [19] with an ordered structure (see [22]). However, for most crystals of the LGS group (for a list of these crystals, see, e.g., [21,22]), the piezo-optic effect (POE), the elasto-optic effect (ELOE), and the acousto-optic efficiency have not been studied. It is these effects that are interesting from the point of view of the application of crystals of the LGS group, because of the stability of physical characteristics, high optical quality of the crystals, their high mechanical strength [19,20], and transparency in the deep ultraviolet region (up to ~250 nm [1,9,20]).

Since it is problematic to experimentally study a large number of crystals by the laborious interferometric method [19,20,23], in paper [24] the photoelastic (piezo- and elasto-optic) coefficients of ordered CTGS and Ca₃NbGa₃Si₂O₁₄ (CNGS) crystals were calculated by the quantum–mechanical technique. The calculation results for CTGS crystals [24] agree well with the experimental data [19]. Therefore, in this paper, the elastic, piezo- and elasto-optic coefficients of ordered batangasite Ba₃TaGa₃Si₂O₁₄ (BTGS) crystals were calculated in a similar way. Based on these results, the spatial anisotropy of photoelasticity effects can be studied by the methods of indicative or extreme surfaces, see, e.g., [25,26,27,28]. Here the extreme surfaces describing the change in the path difference of the orthogonally polarized light beams

δΔ_k = δ(Δn_k·d_k)

(1)

under the influence of mechanical stress σ_m is constructed and the maximum values of this effect are found (here Δn_k is the birefringence of the crystal, d_k is the thickness of the sample in the direction of light propagation). The results obtained for BTGS crystal are compared with similar ones for LGS, CTGS, CNGS and CGG.

2. Material and Method for Determining Photoelastic Coefficients

BTGS crystals were chosen for calculating the piezo-optic coefficients (POCs) π_im and elasto-optic coefficients (ELOCs) p_ik because their photoelastic properties have not been elucidated yet. In addition, because of the high refractive indices (n_i ≈ 2 [29]), which are decisive for the value of the acousto-optic figure-of-merit M₂, since M₂ ~ $n_i^6$ [30,31,32], for these crystals, one can expect a higher AO efficiency than for other crystals of the LGS group, as well as a higher efficiency in terms of path difference, which can be used in polarization-optical light modulators and, accordingly, pressure sensors based on them [33,34,35,36].

In papers [24,37,38] quantum–mechanical calculation of POCs and ELOCs for a number of crystals using the CRYSTAL program [39] ensured good agreement with experimental data. Here, the calculation of photoelastic constants of BTGS crystals was also carried out using version CRYSTAL17 of this program [40]. The program implements automated algorithms for determination of elastic and piezoelectric [41], piezo-optic and elasto-optic [42] tensors for crystals of any symmetry. The hybrid PBE0 functional of the density functional theory (DFT) is used [43], which is known to provide accurate strain-related properties of solids [44,45,46]. The POB-TZVP-rev2 atom-centered Gaussian-type function basis set has been adopted for all elements [47,48]. The integration was performed on a 5 × 5 × 5 k-grid chosen by the Monkhrost-Pack method. Convergence of the self-consistent field step of the calculations is determined by a threshold on the energy set to 10⁻⁸ Hartree.

3. Results and Discussion

3.1. Elastic and Photoelastic Coefficients

BTGS crystals belong to the point group 32 [2] and, accordingly, have seven independent coefficients of elastic stiffness C_mk and elastic compliance S_km, as well as eight independent elasto-optic p_ik and piezo-optic π_im coefficients [49,50]. The calculated values of elastic C_mk, S_km and photoelastic p_ik, π_im coefficients for these crystals are indicated in

Table 1. Elastic coefficients C_mk (10⁹ N/m²) and S_km (10⁻¹² m₂/N) of BTGS crystals calculated by quantum–mechanical technique (in comparison with literature data).

Cmk	C11	C12	C13	C33	C14	C44
[22]	166.0	–	–	–	6.3	–
[51] **	160.5	77.4	85.5	170.9	3.03	63.3
Our data	164.9	72.0	83.6	192.0	–6.38	66.3
Skm	S11	S12	S13	S33	S14	S44
[51] *	9.32	−2.75	−3.28	9.14	0.58	15.85
Our data	8.50	−2.41	−2.65	7.50	1.05	15.28

* The symbol (*) denotes experimentally obtained results; the symbol (**) denotes C_mk coefficients calculated based on the known relation C_mk = S_km⁻¹ and experimentally determined S_km.

and

Table 2. Piezo-optic π_im (in units of 10⁻¹² m²/N = Br = Brewster) and elasto-optic p_ik coefficients of BTGS crystals calculated by the quantum–mechanical technique.

π11	π12	π13	π31	π33	π14	π41	π44
−0.18	0.65	0.78	1.31	−1.48	0.26	0.11	−0.46
p11	p12	p13	p31	p33	p14	p41	p44
0.08	0.162	0.189	0.185	−0.067	0.023	0.013	−0.032

.

As seen from Table 1, the agreement between the calculated (our data) and experimental [22,51] values of the elastic coefficients C_mk and S_km is mostly good. Further, these coefficients were used for calculation of the AO efficiency and constructing extreme surfaces of the change in the optical path difference δΔ_k between orthogonally polarized beams induced by the uniaxial pressure σ_m.

The highest value of POCs for BTGS crystals (Table 2) corresponds to the coefficient π₃₃, as for other studied crystals of the LGS group [18,19,20]. For ELOCs, we see that the highest coefficients are p₁₃ and p₃₁, so the transverse effect prevails, as in CTGS [19] and CGG [20]. Let us pay attention to an interesting observation: if the values of the POE and ELOE in crystals of the LGS group are related to the crystal lattice parameters, then larger photoelastic effects are generally observed in CTGS [12,19], LGS [18] and BTGS crystals with a larger unit cell volume (V = 283.80 ų [52], 294.28 ų [53] and 326.03 ų [2], respectively) compared to the small value of V = 279.96 ų [54] for CGG crystals and, accordingly, the predominantly smaller values of POCs and ELOCs for this crystal [20]. It was also found that the maximum values of POE or ELOE are sensitive to the ratio of the cell parameters c/a. Namely, in the LGS crystal with a higher value of c/a = 0.6233 [55], longitudinal effects corresponding to the coefficients π₃₃ and p₃₃ [18] predominate, while in BTGS, CTGS and CGG crystals with smaller values of the c/a ratio (0.6097, 0.6148 and 0.6158, respectively [2,52,54]), transverse effects with the coefficients π_i31, p₁₃ and p₃₁ [18,19,20] are more significant.

Note that crystals of the LGS group are often divided into ordered and disordered, see, e.g., papers [10,22,54,55]. If, according to the piezoelectric coefficients, there is a large difference between ordered crystals (e.g., CTGS, CNGS) and disordered ones (e.g., LGS, LGT) [10,11], then no clear patterns have been found for the photoelastic coefficients π_im, p_ik. We can only note that for crystals with an ordered structure (CTGS, CNGS, BTGS) transverse elasto-optic effects prevail (i.e., the coefficients p₁₃, p₃₁ are larger), whereas for crystals with a disordered structure (LGS, CGG), similar patterns have not been revealed for either ELOE or POE.

3.2. Evaluation of Acousto-Optic Efficiency

Let us estimate the value of the AO figure-of-merit of BTGS crystal based on the known [30,31,32] expression

$$M_2={\displaystyle \frac{n_i^6{p}_{ik}^2}{\rho V_k^3},}$$

(2)

where ρ is the crystal density and V_k is the acoustic wave velocity.

For the conditions of AO interaction with a longitudinal acoustic wave, which correspond to the maximum ELOC p₃₁ = 0.185 (Table 2), expression (2) should be substituted with this value of p₃₁. At refractive index n₃ = 1.8776 (for the wavelength of 632.8 nm [29]), the velocity of the longitudinal acoustic wave V₃ = 5908 m/s, calculated by the Christoffel method [50] based on the elasticity coefficients C_mk (Table 1), and the density of BTGS crystal ρ = 5514 kg/m³ [2] were used. The result is M₂ = 1.66·10⁻¹⁵ s³/kg. In terms of this M₂ value, BTGS crystal is approximately two times superior to the strontium borate (SrB₄O₇) crystal, suitable for AO modulation of light in the ultraviolet range [56] and has a value comparable to the maximum of M₂ of the CTGS crystal [19]. However, to find the maximum AO efficiency of BTGS crystal, it is necessary to construct AO figure-of-merit (M₂) surfaces and analyze the anisotropy of these surfaces using the technique described, e.g., in [28].

3.3. Extreme Piezo-Optic Surfaces of the Path Difference

The path difference POE δΔ_k according to expression (1) consists of changing the birefringence Δn_k and the sample thickness d_k under the influence of mechanical stress σ_m. Such POE finds its application in polarization-optical light modulators, particularly, in pressure gauges based on them [33,34,35,36]. We call such surfaces extreme, because each point of the surface is obtained not simply as a result of calculation using a known expression, but as a consequence of optimization (searching for the maximum path difference in accordance with the direction of uniaxial pressure applying or light wave vector). This optimization is based on the well-known classical Levenberg–Marquardt algorithm [57]. To avoid the uncertainty caused by reaching a local maximum, the search was performed several times for each point of the surface, with 10 different initial values for each optimization parameter (i.e., the polar or azimuthal angle of the spherical coordinate system). It should also be noted that the calculations for each point of the surface were carried out independently, and reaching a local (non-global) maximum in such a case is usually reflected in the shape of the extreme surface and violates its symmetry caused by the symmetry of the crystal, or forms ‘dips’ on the surface. Such situations are easily traced; however, they were not encountered in the calculations of this work. The construction of extreme surfaces of the path difference is detailed, particularly, in [27].

Here the objective function of optimization is the absolute value of ${{\pi }^{\prime }}_{km}^o$ corresponding to the definition of POE by path difference [58]:

$${{\pi }^{\prime }}_{km}^o=2{\displaystyle \frac{\delta (\Delta n_k\cdot d_k)}{{\sigma }_m{d}_k}.}$$

(3)

Based on this expression, we obtain the equation of the ${{\pi }^{\prime }}_{km}^o$ surface, which describes the change in the path difference for all directions $\overrightarrow{m}$ of the applied uniaxial pressure and all light propagation directions $\overrightarrow{k}$, namely

$${{\pi }^{\prime }}_{km}^o={{\pi }^{\prime }}_{im}{({n^{\prime }}_i)}^3-{{\pi }^{\prime }}_{jm}{({n^{\prime }}_j)}^3-2\Delta {n^{\prime }}_k\cdot {S^{\prime }}_{km},$$

(4)

where the dashes indicate the spatial distribution of all effects: ${{\pi }^{\prime }}_{km}^o$ is POE in terms of the path difference; ${{\pi }^{\prime }}_{im}$ and ${{\pi }^{\prime }}_{jm}$ are POEs consisting refractive indices of orthogonally polarized waves ${n^{\prime }}_i$ and ${n^{\prime }}_j$ changes; $\Delta {n^{\prime }}_k$ is the birefringence; is the effective elastic compliance coefficient. It should be emphasized that all possible geometries of piezo-optic interaction were considered, and not only the conditions of orthogonality of $\overrightarrow{k}$ and $\overrightarrow{m}$ directions.

The calculation by the method of extreme surfaces (see, e.g., [27,28]) was carried out in two equivalent variants. In the first variant of the calculation, for each possible direction $\overrightarrow{k}$ of light propagation, such a direction $\overrightarrow{m}$ of uniaxial pressure σ_m was determined, which would provide the maximum of the path difference ${{\pi }^{\prime }}_{km}^o$. The dependence of the obtained maximum values ${{\pi }^{\prime }}_{km}^o$ on the direction $\overrightarrow{k}$ of the light wave can be represented in the form of a surface, which we call the wave vector extreme surface. The distances of the points of this surface from the origin of coordinates correspond to the absolute values of the maximum ${{\pi }^{\prime }}_{km}^o$ for each direction of the wave vector $\overrightarrow{k}$. Among all ${{\pi }^{\prime }}_{km}^o$ maxima obtained as a result of the calculation, the global maxima of this value were determined, which correspond to the maximum achievable POE of the path difference. In the second variant of the calculation, for each possible direction $\overrightarrow{m}$ of uniaxial pressure, such a direction of wave vector $\overrightarrow{k}$ was determined, which would also provide the maximum of ${{\pi }^{\prime }}_{km}^o$ surface. The corresponding extreme surface is called the surface of mechanical stress. Both types of surfaces have all the symmetry elements of the point group of the crystal and the center of inversion (i.e., they correspond to the Laue group of the crystal [59]). The symmetry reveals in the existence of few equivalent global maxima. Although the wave vector and mechanical stress surfaces differ in form, the maximum achievable global maxima of the POE must be the same for both of them.

Parameters of LGS, CGG, CTGS, CNGS and BTGS crystals (π_im, S_km, n_i) used for construction of extreme surfaces ${{\pi }^{\prime }}_{km}^o$ were taken from [18,19,20,24,29] and Table 1 and Table 2.

The examples of the extreme surfaces of the POE by the path difference, constructed in accordance with (4), are shown in Figure 1 and Figure 2. The results of the optimization for the experimentally studied (by interferometric method) crystals CGG [20], CTGS [19] and LGS [18] and the theoretically studied (by quantum–mechanical method) crystals CTGS [24], CNGS [24] and BTGS (this paper) are indicated in

Table 3. Maximal values and angle parameters for the extreme surfaces of POE ${{\pi }^{\prime }}_{km}^o$ in CGG, CTGS and LGS crystals; the optimization is based on the experimental data.

Crystal	Light Wave				Direction of Uniaxial Pressure Applying		α, deg.	The Global Maximum, Br
	θk, deg.	φk, deg.	θi, deg.	φi, deg.	θm, deg.	φm, deg.
CGG	104	90	90 (o), 14 (e)	0 (o), 90 (e)	90	0	90	6.8
CTGS	101.3	90	90 (o), 11.3 (e)	0 (o), 90 (e)	90	0	90	10.4
LGS	91	90	90 (o), 1 (e)	0 (o), 90 (e)	−4.5	90	95.5	10.8

Hereinafter θ_k, φ_k, θ_i, φ_i, θ_m and φ_m are the angles of spherical coordinate system for the directions of wave $\overrightarrow{k}$, light polarization $\overrightarrow{i}$ and the direction of uniaxial pressure applying $\overrightarrow{m}$. This spherical coordinate system is connected with the crystal-physics Cartesian coordinate system X, Y, Z; the designations (o) and (e) correspond to the polarization directions of the ordinary and extraordinary beams, respectively, α is the angle between $\overrightarrow{k}$ and $\overrightarrow{m}$ vectors.

and

Table 4. Maximal values and angle parameters for the extreme surfaces of POE ${{\pi }^{\prime }}_{km}^o$ in CTGS, CNGS and BTGS crystals; the optimization is based on the data obtained by quantum–mechanical method.

Crystal	Light Wave				Direction of Uniaxial Pressure Applying		α, deg.	The Global Maximum, Br
	θk, deg.	φk, deg.	θi, deg.	φi, deg.	θm, deg.	φm, deg.
CTGS	124	90	90 (o), 35 (e)	0 (o), 90 (e)	90	0	90	11.0
	155	90	90 (o), 65 (e)	0 (o), 90 (e)	60	90	95	11.0
CNGS	146.4	90	90 (o), 56.4 (e)	0 (o), 90 (e)	52	90	94.4	11.6
BTGS	90	90	90 (o), 0 (e)	0 (o), 0 (e)	2.4	90	87.6	14.5
	90	90	90 (o), 0 (e)	0 (o), 0 (e)	90	0	90	14.5

. These tables provide data on the position of one of the global maxima, the positions of other ones can be determined using the symmetry elements of the point group 32 supplemented by inversion in accordance with the symmetry of the expressions used for calculations (specifically, the use of the absolute value of ${{\pi }^{\prime }}_{km}^o$ as the objective function).

As seen from Table 3, for CGG and CTGS crystals, the maximum values of the ${{\pi }^{\prime }}_{km}^o$ surfaces were obtained for the case of orthogonal conditions of piezo-optic interaction (α = 90°), when the direction $\overrightarrow{m}$ of pressure applying coincides with the main crystallographic axis X, and light propagates in the main crystallographic plane YZ in directions determined by the angles θ_k = 104° and 101.3° to the optical axis Z of the considered crystals. For LGS crystal, the most effective geometry of piezo-optic interaction corresponds to the case when the light propagates and the uniaxial pressure is applied in the crystallographic plane YZ. The optimal geometry of interaction is not strictly orthogonal, the angle α is equal to 95.5°. The smallest global value of POE by the path difference ${{\pi }^{\prime }}_{km}^o$ is obtained for CGG crystal (Table 3). This result is obviously caused by the values of the refractive indices n_i, birefringence Δn_k and piezo-optic coefficients π_im of this crystal, which are lower compared to other representatives of the langasite group [20].

It should be emphasized that Figure 1 and Figure 2 give a qualitative representation of the anisotropy of the POE in terms of the path difference, whereas Table 3 and Table 4 present quantitative results of the maximum (global) values of the piezo-optic extreme surfaces of the path difference.

The most important results presented in Table 3 and Table 4 include the following:

(1)

The directions of light propagation $\overrightarrow{k}$ indicated in these tables do not correspond to the main crystallographic axes (an exception exists only for BTGS crystal, see rows 4 and 5 in Table 4); the directions of light propagation $\overrightarrow{k}$ and uniaxial pressure applying $\overrightarrow{m}$ are either strictly orthogonal (α = 90°, see rows 1 and 2 in Table 3 and rows 1 and 5 in Table 4), or deviate from orthogonality by small angles from 2.4° to 5.5° (see row 3 in Table 3 and rows 2–4 in Table 4);

(2)

Among the experimentally and theoretically studied crystals, the maximal value of ${{\pi }^{\prime }}_{km}^o$ was obtained for BTGS crystal (14.5 Br);

(3)

CGG crystal reveals the lowest global maximum of ${{\pi }^{\prime }}_{km}^o$ (6.8 Br); for comparison, the maximum value of the POE (in terms of path difference) for quartz found by the extreme surface method is equal to 7.4 Br, which is also significantly (by a factor of 2) lower than that for BTGS crystal (Table 4). Therefore, it is BTGS crystals that should be preferred when using them as sensitive elements of polarization-optic light modulators and devices based on them.

4. Conclusions

All elastic and photoelastic (piezo-optic and elasto-optic) coefficients of BTGS crystals were calculated using the quantum–mechanical method realized in the CRYSTAL17 program. Based on the calculated elastic compliance coefficients S_km and piezo-optic coefficients π_im, as well as the known values of the refractive indices n_i for the light wavelength λ = 632.8 nm, extreme surfaces of the POE ${{\pi }^{\prime }}_{km}^o$ were constructed, which describe the spatial distribution of the change in the path difference δΔ_k caused by the influence of the mechanical stress σ_m. The maximum (global) values of the piezo-optic extreme surfaces of the path difference were found by the Levenberg–Marquardt method. The results obtained for BTGS crystals are compared with the ones for other crystals of the langasite group: LGS, CGG, CTGS and CNGS. The main result of the research is that the BTGS crystal reveals the highest value of the POE ${{\pi }^{\prime }}_{km}^o$ among all studied crystals of langasite group (14.5 Br). This value is significantly (twice) higher than the one for quartz (7.4 Br). Therefore, BTGS crystals should be preferred when used as sensitive elements of polarization-optic light modulators and devices based on them.

The value of the AO figure-of-merit M₂ of BTGS crystal was estimated based on the highest elasto-optic coefficient p₃₁. According to the obtained value of M₂ = 1.66·10⁻¹⁵ s³/kg, BTGS crystal is twice as much as the strontium borate crystal (SrB₄O₇), which is suitable for acousto-optic modulation of light in the ultraviolet range.

It should be emphasized that for the studied crystals, a correlation was found between the value of the piezo-optic π_im and elasto-optic p_ik coefficients on the one hand and the volume of the crystal lattice on the other. Moreover, the values of the π_im and p_ik coefficients correlate with the value of the ratio of the unit cell parameters c/a.

It was also found that for crystals with an ordered structure (CTGS, CNGS, BTGS), the transverse elasto-optic effects (corresponding to the coefficients p₁₃, p₃₁) prevail, and for crystals with a disordered structure (LGS, CGG), similar regularities were not determined for either ELOE or POE.

Based on the results obtained, it can be stated that by changing the parameters of the unit cell by incorporating the appropriate cations in the crystal structure, the photoelastic properties of crystals of the langasite group can be controlled. For example, replacing the Ca²⁺ cation with Ba²⁺ in crystals with an ordered structure (CTGS, BTGS) leads to a significant increase in the maximum POE value by the path difference ${{\pi }^{\prime }}_{km}^o$ (by about 50%), namely, from 10.4 Br for CTGS to 14.5 Br for BTGS.

It is clear that before developing recommendations for the practical use of an effective (based on quantum–mechanical calculation) photoelastic material, a thorough experimental verification (by interferometry methods) of the calculated values of POCs and ELOCs should be carried out. In other words, the method of quantum–mechanical calculation should be considered as an effective express technique for the initial assessment of the photoelastic characteristics of considered crystals.