1. Introduction

Next generation of power grids requires new, more accurate and reliable instrumentation. Optical voltage sensors (OVSs) are an essential part of this new instrumentation [1,2]. OVSs employ linear electro-optic effect in a transducer crystal linked to the rest of the instrument by an optical fiber [3]. Since 1970s, significant efforts have been made for OVSs development. However, despite the simplicity of underlying physical principles, development of practical OVSs for electric power industry proved to be a nontrivial problem: to our knowledge, no commercial high-voltage OVSs are available for the moment. In our opinion, this is largely due to the shortcomings of electro-optic materials commonly employed in high-voltage OVSs.

OVSs for electric power lines operate outdoors and should maintain the prescribed accuracy – typically 0.2% or better – in a wide range of environmental conditions: temperature, precipitation, stray electric fields from other phases, etc. This leads to specific requirements to electro-optic (EO) materials for OVSs. To be insensitive to environmental perturbations of the electric field inside EO crystal, an OVS should use longitudinal EO effect when the induced birefringence depends only on the projection of external electric field E onto optical wavevector [4]. Phase difference of eigenwaves imposed by the longitudinal EO effect is

The most popular material for high-voltage OVSs is bismuth germanate Bi₄Ge₃O₁₂ (BGO). It is a cubic crystal allowing longitudinal EO effect with thermal coefficient of about 10^‒4 K^‒1 [5]. Unfortunately, its EO response can be distorted by birefringence imposed by built-in and extrinsic stress, and this is the case for any EO cubic crystal. This can seriously limit the accuracy of OVSs [6–8].

For AC voltage sensors, the problem of stress-induced birefringence can be resolved by using anisotropic EO crystals instead of cubic ones. Natural birefringence much stronger than stress-induced one stabilizes EO response by decoupling it from stress-induced birefringence [9]. Temperature dependence of natural birefringence can be used for measurement of crystal temperature and compensation of temperature dependence of half-wave voltage.

Crystals of langasite family can be thought as suitable anisotropic EO crystals for high-voltage OVSs. Langasites are the group of about 100 compounds isomorphic with langasite La₃Ga₅SiO₁₄. They belong to symmetry group 32 and thus permit longitudinal EO effect for anisotropic propagation directions along 2-fold crystallographic axes [10]. Langasites are usually considered as promising piezoelectric materials and do not draw much interest as electro-optic media because of relative weakness of EO effect. Data on optical and electro-optic properties of langasites are fragmentary and insufficient for estimation of their usability for high-voltage OVSs. Paper [11] reports room-temperature refractive indices and EO coefficients of langasite (LGS), langataite La₃Ga_5.5Ta_0.5O₁₄ (LGT), and langanite La₃Ga_5.5Nb_0.5O₁₄ (LGN), as well as temperature dependence of EO coefficients $r_{11}$, $r_{41}$ of LGS between 40 K and 500 K. Linear fit of experimental data of [11] yields thermal coefficient for $r_{11}$ of LGS close to ‒3 × 10^‒4 K^‒1 between 120 K and 325 K. However, this result needs to be verified because of big spread of experimental data: standard deviation of experimental $r_{11}$ values of [11] is 11% which is much larger than systematic temperature dependence of $r_{11}$. No data on temperature dependence of EO effect in other langasites are available.

In this paper we measure temperature dependencies of unclamped half-wave voltage for longitudinal EO effect for 3 langasites: LGS, LGT, and catangasite Ca₃TaGa₃Si₂O₁₄ (CTGS), together with temperature dependence of their natural linear birefringence. As well, room-temperature ratios of unclamped EO coefficients $r_{41}/r_{11}$ are measured in these crystals, along with the room-temperature refractive indices of CTGS at 1.55 µm. Obtained data form the ground for estimation of applicability of studied crystals for high-voltage OVSs.

2. Samples

Single crystals of LGS, LGT and CTGS are grown by Czochralski method using “Kristall-3m” pullers with RF-heating. The initial charge is prepared from 99.99%-pure La₂O₃, CaCO₃, Ta₂O₅, Ga₂O₃ and SiO₂ powders. Powders are mixed in stoichiometric composition and synthesized at 1200 C. Crystals are grown in an Ar-atmosphere with admixture of oxygen (0.5 ~2 vol. %) using Ir crucible. The growth rate is set to 1 ~3 mm/h. The rotation rate of the grown crystal was within the range of 5 ~25 rpm. The “as grown” single crystals are annealed for 24 hours at the temperature of above 1000 C.

Samples are cut in a shape of parallelepipeds whose sides are parallel to the crystallographic axes: x-side parallel to one of 2-fold axes a, and z-side parallel to optical axis c. The sizes of samples are: LGS 5.10 mm (x) × 10.40 mm (y) × 5.85 mm (z), LGT 4.94 mm (x) × 14.84 mm (y) × 6.03 mm (z), CTGS 5.04 mm (x) × 14.85 mm (y) × 6.02 mm (z). xz faces are optically polished. The accuracy of orientation of parallelepiped sides in respect to crystallographic axes is better than 15ʹ.

3. Methods

Electro-optic properties and natural birefringence are measured with modulation polarization interferometer shown in Fig. 1. It comprises light source S (superluminescent diode Thorlabs SLD1550S–A40, center wavelength 1539.5 nm), fiber polarization controller PC, polarization-insensitive circulator Circ, single-mode fiber F (Corning SMF-28) with angled (8°) tip, collimating lens L, linear polarizer P – Glan prism, photoelastic polarization modulator M (Hinds Instruments PEM-100, modulator head IS-50), sample crystal Q₁ with electrodes EL, compensator crystal Q₂, mirror Mirr, photodetector PD, high-voltage supply V, signal acquisition board ADC (National Instruments PXIe-6366), and computer C. Transmission plane of polarizer P is oriented at 45° in respect to anisotropy axes of modulator M and optical axis c₁ of sample Q₁. Compensator crystal Q₂ is identical to the sample Q₁, optical axes of sample c₁ and compensator c₂ are mutually perpendicular to compensate for the retardation of sample Q₁ which exceeds coherence length of broadband light source S. We use transverse geometry of measurement of EO effect instead of longitudinal one: voltage is applied to electrodes EL on yz faces of sample Q₁, while wavevector of light k is parallel to y-sides of sample, i.e. orthogonal to optical axis c₁ and 2-fold axis a₁. In this geometry, EO phase retardation in a crystal of symmetry group 32 can be written as [10]

 

Fig. 1 Polarization interferometer used for electro-optic and birefringence measurements.

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For measurement of temperature dependencies sample Q₁ is placed into optical cryostat (Janis VPF-100, not shown in Fig. 1). One of yz faces of sample is attached by silver paint (Electrolube SCP) to cryostat sample holder which is 35 × 40 mm flat copper plate acting as the electric ground. High-voltage electrode on another yz face is formed by the same silver paint. Temperature of sample is measured as a mean of readings of Pt100 thermistor (ZIEHL TF101F) placed on free yz sample’s face, and thermal sensor of cryostat (LakeShore DT-670B-CU) mounted next to sample holder at the cold finger. The temperature of sample is altered by steps, measurements are made only at steady temperature. Electrostatic edge effect correction factor a is calculated numerically using CST EM Studio software as

For room temperature measurement of ratio of EO coefficients $r_{41}/r_{11}$ and birefringence $n_e-n_o$ sample Q₁ is put on rotation mount (Standa 7R150, not shown in Fig. 1) with 2ʹ angular resolution; rotation axis coincides with 2-fold crystallographic axis of the sample, i.e. with x₁ axis of Fig. 1. Formulae for EO effect in crystals of point group 32 [10] give $r_{41}/r_{11}$ ratio as

The output light intensity of interferometer Fig. 1 is

For measurement of ordinary refractive index we use confocal reflectometer shown in Fig. 2. Confocal method is applicable for parallelepiped-shaped samples used in our EO measurements. Compared to other confocal schemes for refractive index measurement [14,15], our scheme provides a reasonable compromise between accuracy and simplicity. The coincident notations of Fig. 2 and Fig. 1 denote the same elements (see the explanation for Fig. 1); unlike interferometer Fig. 1, fiber F has right-angle tip. Remaining elements are the platform T established on computer-controlled linear translator (Standa 8MT173-30), lenses L₁ (Thorlabs A260C, f₁ = 15.29 mm) and L₂ (Thorlabs A280C, f₂ = 18.4 mm), lock-in amplifier LIA (Stanford Research SR830), and optical chopper Ch (Stanford Research SR540). Sample Q₁ and mirror Mirr are mounted on translator platform T. The light beam emergent from fiber F is sharply focused by lenses L₁, L₂. The beam is directed along y-axis of Fig. 2, orthogonally to optical axis of sample c₁. Transmission plane of polarizer P is orthogonal to optical axis of sample, so only ordinary wave is excited in the sample. The reflected power at photodetector PD strongly depends on relative position of focused beam waist and surfaces of sample and mirror, reaching maximum when any surface is exactly in the beam waist. Translating platform T with sample Q₁ and mirror Mirr along y-axis produces peaks of output power shown in the inset of Fig. 2. Peaks 1 and 2 are associated with faces of sample, peak 3 – with mirror Mirr in the presence of sample in front of it, and peak 4 – with mirror Mirr in the absence of sample. To observe peak 4, the sample is removed from optical system upon recording of first three peaks. The gap between sample Q₁ and mirror Mirr should be larger than the confocal parameter of the probe beam to have peaks 2 and 3 well separated. In a paraxial approximation, the distances between peaks 1, 2 and 3, 4 are $\Delta y_{21}=L_y{n}_{air}{n}_o^{-1}$, $\Delta y_{43}=L_y(1-n_{air}{n}_o^{-1})$. This yields ordinary index as

 

Fig. 2 Confocal reflectometer for refractive index measurement.

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We were unable to obtain satisfactory results for extraordinary index by confocal method. The reason is astigmatism of a uniaxial medium in respect to an extraordinary beam [17] resulting in splitting of reflection peaks 2 and 3. In weakly birefringent crystals like LGS, pairs of split peaks appear as single merged peaks, whose positions are defined by an ordinary index in compliance with theory of [17]. For crystals with stronger birefringence like CTGS, split peaks are resolved, but their positions substantially deviate from paraxial theory of [17].

4. Results and discussion

Measured temperature dependencies of unclamped longitudinal double-pass half-wave voltage $V_{\pi }$ Eq. (3) for LGS, LGT and CTGS are shown in Fig. 3. Half-wave voltage is derived from observables using Eq. (11). Color of points in Fig. 3, 4 indicates the difference of temperatures of current and previous measurement: red means heating of sample, blue – cooling. Error bars show confidence intervals for statistical error of individual measurements. For CTGS error bars are not shown since they are less than data point size. In LGS and CTGS measurements are made at the same point of sample $(x_0,z_0)$, in LGT – at 3 different points of sample. For all studied crystals, half-wave voltage grows linearly with temperature. Lines are linear least square fits of experimental points.

 

Fig. 3 Longitudinal half-wave voltage versus temperature for LGS, LGT, and CTGS.

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Fig. 4 Measured DC phase retardation versus temperature for LGS, LGT, and CTGS.

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Figure 4 displays temperature-dependent component of measured DC phase retardation versus sample temperature for LGS, LGT and CTGS. Curves are polynomial least square fits of experimental values of ${\psi }_{\text{DC}}(T)$.

Dependence of measured unclamped half-wave voltage $\upsilon$ and DC phase retardation ${\psi }_{\text{DC}}$ on the angle of incidence $\theta$ to xz-face of CTGS crystal are shown in Fig. 5a and Fig. 5b respectively. Lines are fits of experimental points: linear for Fig. 5a, and quadratic for Fig. 5b.

 

Fig. 5 Room-temperature half-wave voltage (a) and DC phase retardation (b) versus angle of incidence to xz-face for CTGS.

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Measured unclamped electro-optic parameters of LGS, LGT, and CTGS are collected in Table 1. The 2nd column of Table 1 contains linear fits of experimental temperature dependencies of half-wave voltage, $T_C$ is the temperature in Celsius degrees, ${\sigma }_{V_{\pi }}$ is a relative standard error of fit $V_{\pi }(T_C)$, ${\eta }_Q$ is a ratio of half-wave voltages of α-quartz and studied crystal at 0 C, $\chi$ is a thermal coefficient for half-wave voltage Eq. (2), $\delta T_{0.1\%}$ is a required accuracy of measurement of temperature of EO crystal for an OVS of accuracy class 0.1%, calculated from Eq. (2).

Table 1. Measured electro-optic parameters of langasites

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Average confidence interval $6\sigma$ for statistical error of half-wave voltage measurement at a given temperature is 0.16% for LGS, 0.27% for LGT, and 0.23% for CTGS. As shown in Table 1, standard error of linear fit of half-wave voltage temperature dependence varies from 0.03% to 0.12%. The latter value can be assumed as an estimation of repeatability of half-wave voltage measurement, since no statistically significant systematic deviation from linear fit is observed in all temperature cycles except cycle 2 for LGS. In the latter cycle, a hysteresis-like temperature dependence of half-wave voltage is obtained with maximum gap between cooling and heating branches of 0.2% and average gap of 0.1%, as can be seen from Fig. 3a. This is somewhat larger than half of the confidence interval $3\sigma$ for individual experimental points but close to estimated measurement repeatability. No temperature hysteresis is detected in cycle 1 for LGS as well as in all other temperature cycles. Therefore, hysteresis in cycle 2 can be treated as an occasional phenomenon not related to the properties of the sample. It can be associated with temperature dependent displacement of probe beam in respect to the sample due to thermal deformation of cryostat. In our LGS sample, the sensitivity to such a displacement is associated mainly with electric field inhomogeneity due to electrostatic edge effect, the contribution of intrinsic inhomogeneity of sample is minor. Mean difference of LGS half-wave voltages for cycles 1 and 2 is 0.18%. Anticipated cause of this difference is a shift of sample as a result of manipulations with cryostat, which is taken off from the setup for evacuation before every temperature cycle. The difference of half-wave voltage thermal coefficients $\chi$ for two LGS cycles is 12%. The change of half-wave voltage of LGS in the entire range of measured temperatures is about 0.4%, which confidently surpasses estimated measurement uncertainty.

Refractive indices and temperature dependencies of natural linear birefringence of LGS, LGT, and CTGS are listed in Table 2. ${\sigma }_{n_o}$ is a standard error of ordinary index measurement. Temperature derivative of birefringence $\partial \Delta n/\partial T$ is found from Eq. (12) using fitting functions ${\psi }_{\text{DC}}(T)$ of Fig. 4 and room-temperature thermal expansion coefficients ${\alpha }_{11}$ [18]. For LGS and LGT, reference values of $n_o$ and $\Delta n$ from [11] are given. Our value of $n_o$ for LGS is in good agreement with reference [11]. The discrepancy with reference birefringence for LGT is 5.5 × 10^‒4, which exceeds declared error of dispersion formula [11]. Possible error sources of our measurement are inaccuracies of the angle $\theta$ and the sample length $L_y$. As follows from Eq. (6), relative error of birefringence measurement $\delta \Delta n=-\delta \text{}{L}_y-2\delta \theta$, where $\delta \text{}{L}_y$ is a relative length error, $\delta \theta =\Delta {\theta }_{meas}/\Delta \theta -1$ is a relative angle error, $\Delta {\theta }_{meas}$, $\Delta \theta$ are measured and actual increment of angle, respectively. With sample length error of 0.025 mm, and angle inaccuracy $\delta \theta =4\times {10}^{-3}$, which corresponds to angle error of 1′ per 4°, the error of measured birefringence can be as large as 2.5 × 10^‒4, which is only a half of discrepancy with data of [11]. Thus, it can be supposed that birefringence of LGT varies from sample to sample within several units of 10^‒4, possibly because of differences in composition and fabrication technology.

Table 2. Ordinary refractive index, birefringence, and its temperature dependence for langasites

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Temperature derivative of birefringence $\partial \Delta n/\partial T$ in LGS turns to zero at 158.3 K. At higher temperatures, this derivative is negative, like in other studied langasites and α-quartz. At lower temperatures it becomes positive with a maximum at 127 K and anticipated second zero near 100 K. Between 115 and 172 K $\left|\partial \Delta n/\partial T\right|$ is less than 3.3 × 10^‒8 K^‒1. In the range of 200 ~310 K, $\partial \Delta n/\partial T$ for LGS can be appropriately fitted by a linear function:

Room-temperature longitudinal half-wave voltage calculated from Eq. (3) for wavelength of 1.54 μm using $r_{11}$ values of [11] is 43.4 kV for LGS and 38.5 kV for LGT against ours 45.2 kV for LGS and 42.7 kV for LGT, i.e. our values are higher by 4% for LGS and by 11% for LGT. This mismatch probably reflects differences between samples and their inhomogeneity. Inside our LGT sample, measured half-wave voltage Eq. (11) varies by at least 3.6%. Even larger variations of EO coefficient $r_{11}$ of LGS of up to 20% on a scale of 10 mm are reported in [19]. According to [19], inhomogeneity of EO coefficients is associated with spatial variations of crystal composition due to instability of growth conditions. It seems reasonable to suggest that the difference of composition of samples can be responsible – at least, partially – for the mismatch of electro-optic coefficients of our samples and samples of [11].

Data of Tables 1, 2, along with piezoelectric coefficient of CTGS $d_{11}=-4.58$ pm/V [12] yield room-temperature unclamped EO coefficients of CTGS at 1.54 μm $r_{11}=\text{0}\text{.537}$ pm/V, $r_{41}=-0.324$ pm/V; $d_{11}$ and $r_{11}$ are assumed to have opposite signs, like in LGS and LGT [11].

Our temperature dependence of EO effect in LGS significantly differs from that of [11]. Thermal coefficient for $r_{11}$ ${\chi }_{r_{11}}\approx -3\times {10}^{-4}$ K^‒1 [11] is 2 orders of magnitude larger in absolute value, than our thermal coefficient for longitudinal half-wave voltage $\chi$. Relation between these coefficients stems from Eq. (3) [10]:

Our study indicates that electro-optic properties of langasites strongly depend on presence of rare-earth atoms in a unit cell. LGS and LGT incorporating lanthanum have 5 times smaller half-wave voltage than CTGS not having rare-earth atoms, while piezoelectric constants of all 3 crystals are close [11,12]. Studied crystals follow the Miller rule [20] linking linear and nonlinear susceptibilities of quadratic media: LGS and LGT with larger static permittivity and refractive indices have larger EO coefficients. More interestingly, rare-earth containing langasites have much lower thermal coefficients of half-wave voltage and natural birefringence, than rare-earth free CTGS. Correlation between temperature dependencies of refractive index and EO coefficients can also be treated as a manifestation of the Miller rule. Especially interesting is temperature dependence of natural birefringence in LGS in which 1st derivative of birefringence on temperature turns to zero at 158 K. Shifting this point to room temperature – for instance, by a modification of chemical composition of crystal – could give new materials for temperature-insensitive polarization optics.

Rare-earth containing LGS and LGT look promising for application in high-voltage optical sensors. They have low thermal coefficient of half-wave voltage, which is essential for sensors for outdoor applications: in LGS-based instrument of accuracy class 0.1% the temperature of EO crystal should be measured with standard error of about 40 K versus less than 3 K for α-quartz and 6.5 K for BGO. LGS seems especially attractive because of more established technology of manufacturing. CTGS with relatively low electro-optic coefficients and strong temperature dependence of EO effect looks much less interesting for electro-optic OVSs.

5. Conclusion

We experimentally investigated electro-optic properties of single crystals of langasite (LGS), langataite (LGT), and catangasite (CTGS) in the temperature range of 170 ~310 K, and natural birefringence of these crystals in the range of 115 ~310 K using modulation polarization interferometer. As well, we measured room temperature refractive indices of CTGS at 1.54 µm with confocal reflectometer. Electro-optic properties of langasites demonstrate prominent dependence on the presence of rare earths in a composition of crystal. Lanthanum-containing LGS and LGT have 5 times less half-wave voltage and much weaker temperature dependence of electro-optic effect, than rare-earth free CTGS. LGS exhibits non-monotonic dependence of natural birefringence $\Delta n$ on temperature T, with $\partial \Delta n/\partial T=0$ at 158 K, and $\left|\partial \Delta n/\partial T\right|<3.3\times {10}^{-8}$ K⁻¹ within 115 ~172 K.

LGT and especially LGS can be considered as promising electro-optic materials for AC high-voltage optical sensors due to permitted longitudinal EO effect for anisotropic propagation directions along 2-fold crystallographic axes, weak temperature dependence of this effect, and reasonably high EO coefficients. Search for EO materials for high-voltage OVSs can be extended to other langasites with rare earths. Another interesting property of these crystals is weak temperature dependence of birefringence in certain temperature intervals which can be useful for development of temperature insensitive polarization optical elements and devices.