## Abstract

We developed a novel technique to evaluate the elasto-optic coefficients of electrooptic (EO) single crystals. Notably, this method uses the deformation of the crystal generated by the space charge formed by the electrons injected into the crystal. For the first time, to our knowledge, the coefficient *p*_{12} was quantified separately from *p*_{11} for KTa_{1-x}Nb* _{x}*O

_{3}(KTN) with this method. Both the coefficients exhibit significant temperature dependence caused by polarization fluctuations. Genuine EO coefficients

*g*

^{g}_{11}and

*g*

^{g}_{12}were calculated by excluding photoelastic contributions from the nominal EO coefficients.

*g*

^{g}_{12}was negligibly small compared to the nominal coefficient before the exclusion. This indicates that the conventional nominal coefficient

*g*

^{n}_{12}is actually composed of strain-induced components but does not reflect the pure effect.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The electrooptic (EO) effect is a phenomenon in which the refractive index of a substance changes according to the voltage applied to it. Various kinds of optical modulators have been developed by using this effect. Most of these devices are intensity modulators or phase modulators. This is because the optical phase modulations produced by the EO material is small. Recently, however, different EO devices than these modulators have been developed by using a crystal material KTa_{1-x}Nb* _{x}*O

_{3}(KTN). One of them is an optical beam deflector that scans a beam with a voltage [1] and the other is a varifocal lens [2]. Both of them are capable of bending light rays by utilizing a remarkably large second-order EO effect (Kerr effect) of KTN [3,4]. Their advantage is that they respond faster than other devices due to the response characteristic of oxide EO materials. There is a literature that reports 700-kHz operation of a KTN deflector [5] whereas galvanometer mirror deflectors typically respond up to several kHz. The deflection angle of this type of device is not as large as galvanometer mirrors, but full angles exceeding 160 mrad have been obtained [6]. Wavelength-swept light sources for cross-sectional profiling systems have been developed by using this beam deflector as a key component [7].

When designing these devices, it is essential to simulate spatial distributions of refractive indices. Recently, it was found that the photoelastic effect plays an important role in the index modulation of KTN [2,8]. Therefore, a complete simulation of such a device requires coefficients *p*_{11} and *p*_{12} of the effect. There are reported values for *p*_{11} or (*p*_{11} - *p*_{12}) [9,10]. However, because |*p*_{12}| is small, this value has been unknown for KTN. Moreover, the conventional EO coefficients *g*_{11} and *g*_{12} of KTN contain photoelastic influence and pure coefficients that exclude the photoelastic effect are also unheard of.

This paper aims to evaluate *p*_{11} and *p*_{12} separately for KTN and exclude the effects of these parameters from the EO coefficients. To do this, we developed a novel method for measuring *p*_{11} and *p*_{12} with the help of a space charge in the crystal. When we inject electrons into a KTN crystal and form a space charge inside, a generated electric field warps the crystal via the electrostrictive effect [11,8]. We measured modulations of the refractive indices of the crystal caused by the EO effect and the distortion via the photoelastic effect. Both *p*_{11} and *p*_{12} were successfully evaluated from these data with other material parameters such as elastic stiffness coefficients. Although this is the first evaluation of both *p*_{11} and *p*_{12}, to our knowledge, the results were consistent with conventionally reported results. From *p*_{11} and *p*_{12}, we also calculated the genuine EO coefficients *g ^{g}*

_{11}and

*g*

^{g}_{12}excluding the effect of photoelasticity and found that

*g*

^{g}_{12}is negligibly weak compared to the conventional nominal

*g*

_{12}. Therefore, we concluded that the conventional

*g*

_{12}basically stems from the electrostrictive effect and the photoelastic effect. This method can be used not only for KTN but also for other EO materials to analyze the EO properties.

## 2. EO effect, photoelastic effect and measurement theory

This section describes modulations of refractive indices for an EO crystal with a space charge. We deal with crystals without the first order effects such as KTN crystals in the paraelectric phase. Thus, the crystals do not exhibit the Pockels effect and the piezoelectric effect but exhibit the Kerr effect and the electrostrictive effect. First, we compare a conventional EO approach with a new approach incorporating the photoelastic effect and then introduce a theory for evaluating the photoelastic effect.

In the conventional theory of the Kerr effect [3,12], field-induced modulations of the refractive indices are written as follows.

*n*

^{g}_{x}is for a light polarization parallel to

*x*-direction and Δ

*n*

^{g}_{y}for

*y*-direction.

*n*

_{0}is the original refractive index and

*g*coefficients are the electro-optic coefficients. We assumed that

*P*is the only non-zero component of the low-frequency polarization in the crystal block (Fig. 1).

_{x}*P*is approximated by

_{x}*D*=

_{x}*εE*for materials with huge permittivities such as KTN and other ferroelectric materials. Furthermore, in the block with such a permittivity and a pair of electrodes covering the entire area of two opposing surfaces, it is possible to assume that the polarization is substantially uniform.

_{x}However, a spatial distribution of *P _{x}* appears when a space charge is formed in the block. In fact, by applying a high voltage with appropriate electrodes, we can inject electrons into the crystal block, where they are trapped by localized states and form a space charge. The space charge is stable and is conserved even after the voltage has been removed. Here we assume that the charge density

*ρ*is constant throughout the block and the following equation holds according to Gauss’s law [13].

*ε*is the permittivity,

*V*is the voltage applied between the electrodes and

*d*is the electrode gap or the block thickness. The origin of the coordinate

*x*is placed at the center of the block. With this

*P*, Eq. (1) can be rewritten as

_{x}*g*

^{g}_{11}is positive, the distribution works as a convex lens for

*x*-polarized light. The index distribution has a peak at

*εV*/(

*ρd*) and moves with

*V*, which explains the beam deflection function of this crystal block [14].

On the other hand, such a material exhibits the electrostrictive effect, that is, the material is strained by voltage application [11]. It is a kind of quadratic effect and the driving force is proportional to the square of the polarization induced by the voltage. However, when the polarization is not spatially uniform, the strains are affected by the elasticity and are not necessarily proportional to the square of the polarization [2]. Recently we reported that a KTN crystal is spherically distorted when there is a space charge in the crystal. Then the faces with the electrodes in Fig. 1 are warped. Assuming that *P _{x}* varies spatially according to Eq. (2), the strain that accompanies this distortion is given by the following equations [8].

*Q*

_{11}and

*Q*

_{12}are the electrostrictive coefficients.

*c*

_{11}and

*c*

_{12}are the elastic stiffness components.

*x*

_{c}is the position where the strain component

*e*is zero and the distance between this position and the center of the distortion sphere is

_{yy}*r*. These two parameters are obtained by the following equations.

_{a}*p*

_{11}and

*p*

_{12}as elasto-optic coefficients according to Refs. [9] and [15]. The optical path length of the light transmitting through the block is modulated by these index modulations, together with those of the genuine electrooptic effect in Eq. (3). Also, the deformation itself changes the optical path length. Thus, the total optical path length change is expressed as follows.

*L*is the block length. Using Eqs. (3), (4), (5), and (6), we obtain

*g*coefficients are defined as follows.

*g*

^{n}_{11}and

*g*

^{n}_{12}are the nominal electrooptic coefficients that are obtained from the common experimental phase shift data with

*ρ*= 0. Indeed, the following common formulae are obtained by putting

*ρ*= 0 in Eq. (8) [12].

*g*

^{n}_{11}and

*g*

^{n}_{12}that consist of the genuine EO coefficients, the photoelastic components, and the contribution from the electrostrictive deformation. We can obtain the last components by evaluating the deformation caused by voltage application [11, 16]. However, it has not been clear how much fractions of the EO coefficients are the photoelastic contribution. We did not know if the genuine effect really existed or if the EO effect was entirely due to the electrostrictive deformation.

We were successfully able to evaluate the elasto-optic coefficients *p*_{11} and *p*_{12} by utilizing Eq. (8). The equation can be simplified as follows when *V* = 0.

*g*coefficients defined by Eqs. (9) consist of the elasto-optic coefficients, the elastic coefficients and the electrostrictive coefficients. Therefore, if

*g*

^{n}_{11},

*g*

^{n}_{12},

*c*

_{11},

*c*

_{12},

*Q*

_{11}, and

*Q*

_{12}are known, we can obtain the values of the elasto-optic coefficients

*p*

_{11}and

*p*

_{12}by measuring the profile shown in Eq. (11) and evaluating the coefficients shown in Eq. (12). The following is the equation used for the evaluation.

## 3. Experimental

The sample crystals were cut from a Li-doped KTN single crystal boule grown with the top-seeded solution growth method [17,18]. The size was 4 mm x 3.2 mm x 1.2 mm. All the faces were parallel to the crystallographic (100) planes. The direction of light passage was parallel to the 4-mm edge. The transition temperature between the cubic and tetragonal phases was 31 °C, which indicates the approximate composition as K_{0.95}Li_{0.05}Ta_{0.73}Nb_{0.27}O_{3}. Film electrodes were formed on the two 4 mm x 3.2 mm faces. When measuring *g*_{11}* ^{n}* and

*g*

_{12}

*, we used platinum film electrodes that form Schottky junctions with the crystal and prevent electron injection into the crystal accompanying voltage application [1,16]. On the other hand, when evaluating profile coefficients such as*

^{n}*α*, we used titanium electrodes to form a space charge by applying voltages to the crystal blocks [1]. To realize a uniform charge distribution (constant

_{x}*ρ*), we injected electrons from both sides of the block by inverting the voltage sign and regulating the voltage level. We also used a 405 nm light-emitting diode (LED) for controlling the charge density. Uniformly irradiating this violet light reduces the charge density; that is, it excites trapped electrons to discharge them [19]. The permittivity of a KTN crystal changes with the temperature according to Curie-Weiss’ law [18,20]. Therefore, during all measurements, we kept the permittivity constant by controlling the temperature with a thermoelectric cooler (TEC, Peltier device). The temperature was always controlled above the phase transition temperature so that the crystal was in the paraelectric phase.

In this evaluation method, it is necessary to observe the spatial profiles of the optical path length shown in Eq. (11). This can be done by using the phase shift method with a Mach Zehnder interferometer. We monitored the interference between the light output from the crystal block and a plane wave by using a two-dimensional image detector (a charge coupled device) and calculated the spatial profile of the optical path length from the obtained images. Detailed measurement methods are explained in Refs. [10] and [13]. The wavelength was 685 nm. The nominal EO coefficients *g ^{n}*

_{11}and

*g*

^{n}_{12}can also be measured by using the same experimental setup with Eq. (10). Figure 2 shows the results. The original refractive index

*n*

_{0}of this crystal was determined to be 2.237 at this wavelength by the prism coupling method. It is noteworthy in Fig. 2 that

*g*

^{n}_{12}strongly depended on the temperature and approached zero at elevated temperatures, which has not been reported in the literatures that we know.

The electrostrictive coefficients *Q*_{11} and *Q*_{12} can be evaluated simply by applying a voltage to a crystal block and measuring the amount of deformation. In this study, we used the values thus obtained, *Q*_{11}: 0.070 m^{4}/C^{2} and *Q*_{12}: −0.026 m^{4}/C^{2} [16]. As regards the elastic stiffness, we can use acoustic waves. We measured velocities of transversal and longitudinal waves with propagation directions along crystallographic <100> and <110> axes for obtaining both *c*_{11} and *c*_{12} [10]. We used an Olympus thickness gauge 38Dl PLUS with a frequency of 5 MHz for this measurement. Figure 3 shows the elastic stiffness coefficients as functions of the temperature. The horizontal axis shows the sample temperature as the difference from the phase transition temperature *T _{c}*. The softening, or the reduction of

*c*

_{11}on cooling, is said to be caused by fluctuations of polar nano regions (PNR) [21,22].

## 4. Results and discussions

Figure 4 shows examples of the optical path length distributions along the *x*-axis for the two optical polarizations. The origin of the *x*-axis was at the center of the sample. As described, the thickness of the crystal block was 1.2 mm. The paired electrodes were on the left and right ends of this graph and the left electrode is connected to the ground. The temperature was 35.7 °C. We applied −247 V and then 400 V to the right electrode in order to inject electrons into the crystal and to achieve a uniform charge density. The figure shows the crystal condition immediately after the charging process and voltage removal. The experimental plots are close to the ideal parabolic curves of Eq. (11) shown as the solid lines with the extrema at *x* = 0. This indicates good *ρ* uniformity because Eq. (11) was derived by assuming a constant *ρ*. The data deviated from the parabola in the case of poor injection conditions and poor uniformity. On the other hand, the extrema in Fig. 4 themselves are not zero. These non-zero extrema are not included in Eq. (3) which is based on the conventional EO theory but support the validity of Eqs. (8) and (11), namely *β _{x}* and

*β*. Due to the periodic nature of interference, the offsets

_{y}*β*and

_{x}*β*are not automatically deduced from data acquired by the Mach- Zehnder interferometer. However, we gradually reduced |

_{y}*ρ*| by discharging the crystal using a 405-nm LED, tracked changes in the curves as in Fig. 4 and corrected periodic jumps in the collected data. We were able to observe that the spatial variation coefficients

*α*and

_{x}*α*gradually vanished with the offset coefficients

_{y}*β*and

_{x}*β*and that both Δ

_{y}*s*and Δ

_{x}*s*eventually became zero in the entire crystal, which means that

_{y}*ρ*= 0.

Figure 5 shows the values of *α _{x}*,

*β*,

_{x}*α*and

_{y}*β*, thus obtained at 35.7 °C. The horizontal axis shows -

_{y}*α*and the vertical axis indicates values of the other parameters. The rightmost data were first acquired immediately after the charging process at the maximum

_{x}*ρ*

^{2}and the leftmost data were obtained last at the lowest

*ρ*

^{2}. The figure shows good linearity among the four parameters, consistent with Eq. (12). The ratios among them should be determined by coefficients such as

*g*

^{n}_{11},

*g*

^{s}_{11}and

*g*

^{e}_{11}. Conversely, by using Eq. (13), we can determine such electrooptic

*g*coefficients and elasto-optic coefficients

*p*

_{11}and

*p*

_{22}from the data in this figure.

Before introducing the values of the elasto-optic coefficients, we discuss *g ^{n}*

_{12}/

*g*

^{n}_{11}. As described, the nominal EO coefficients

*g*

^{n}_{11}and

*g*

^{n}_{12}as shown in Fig. 2 can be evaluated by measuring Δ

*s*and Δ

_{x}*s*at

_{y}*V*≠ 0 and

*ρ*= 0. On the other hand,

*α*,

_{x}*β*,

_{x}*α*and

_{y}*β*as in Fig. 5 were evaluated at

_{y}*V*= 0 and

*ρ ≠*0. Therefore these experiments are much different from each other. However there is a relationship between the results of these two experiments according to our theory, that is

*ρ*= 0 with the right side of Eq. (14) evaluated at

*V*= 0. The horizontal axis shows the sample temperature. The circles show the left side of Eq. (14) and the squares show the right side. The values obtained from the two different experiments have the same values at all temperatures in this region, confirming our theory.

Figure 7 shows the elasto-optic coefficients calculated from the acquired data with Eq. (13). For this plot, we drew lines as shown in Fig. 5 at each temperature and calculated the ratios of *β _{x}* and

*β*to

_{y}*α*. Also, we used the data in Figs. 2 and 3. To our knowledge, this is the first paper that presents both

_{x}*p*

_{11}and

*p*

_{12}separately for KTN crystals. However, there are a few reports dealing with the photoelastic effect of the crystals. Wemple et al. reported measurement results of (

*p*

_{11}-

*p*

_{12}) with a theory of their temperature dependence [9]. According to the authors, KTN crystals have two types of photoelastic effects. One of them is related to interband transitions and is not considered to have significant temperature dependence. The other photoelastic component is caused by lattice-polarization fluctuations. This component strongly depends on the temperature because the permittivity of a KTN crystal drastically changes with temperature at around the phase transition temperature and the polarization fluctuation increases as the permittivity increases. The data shown in Fig. 7 exhibit such a temperature dependence with a tendency to converge to fixed values in the high-temperature region corresponding to the interband transition components. The data is in good agreement with the experimental data presented in their report. Both their experimental (

*p*

_{11}-

*p*

_{12}) and our

*p*

_{11}do not show the drastic increase as expected from Curie-Weiss’ law of dielectrics, which Wemple used in their theory. The permittivity of KTN crystals is said to deviate from Curie-Weiss law because of random fields associated with PNRs. In particular, the permittivity is significantly reduced with Li-doped KTNs [23]. This is the reason why the temperature dependence is not as great as Wemple predicted. With respect to Fig. 7, it should be additionally noted that

*p*

_{12}approaches zero with rising temperature and that the contribution of the interband transitions to this component is negligibly small compared to

*p*

_{11}.

Figure 8 shows *g ^{g}*

_{11}and

*g*

^{g}_{12}obtained using this method together with the elasto-optic coefficients. The scale of the vertical axis is the same as in Fig. 2. We used these EO coefficients to evaluate the pure EO effect excluding the influence of strains. They may be called clamped coefficients but it should be noted that our crystal was not actually mechanically clamped, that is, it was thermodynamically unclamped. Notable in this figure is that

*g*

^{g}_{12}is negligibly weak compared to the nominal EO coefficient

*g*

^{n}_{12}. This indicates that

*g*

^{n}_{12}is basically composed of strain-induced components and that the significant temperature dependence of this coefficient is caused by the above-mentioned polarization fluctuations. At this stage, the actual value of

*g*

^{g}_{12}is not clear because it is much weaker than the conventional values. It is unknown whether

*g*

^{g}_{12}has a finite non-zero value or not. This value should have an origin from the interband transitions [24]. Therefore, it is noteworthy that, in relation to the value of

*g*

^{g}_{12}, the interband component of

*p*

_{12}is negligibly weak, as shown in Fig. 7. This is one of the next issues for KTN and other related EO crystals along with the origin of the slight temperature dependence of

*g*

^{g}_{11}appearing in Fig. 8.

## 5. Summary

In this report, we introduced a novel method to evaluate elasto-optic coefficients of electrooptic single crystals with the help of space charges and elasticity. For KTN blocks, we measured index modulations that were induced by space charges via the electrooptic effect and the photoelastic effect. From these data with experimental elastic stiffness and electrostrictive data, we evaluated the coefficients *p*_{11} and *p*_{12} separately, which is, to our knowledge, the first quantification example of *p*_{12} for KTN. As it is for *p*_{11}, *p*_{12} also exhibited a strong temperature dependence related to polarization fluctuations. The genuine EO coefficient *g ^{g}*

_{12}derived from these data was negligibly small compared to the nominal EO coefficient

*g*

^{n}_{12}, which is the conventional coefficient. This indicates that the conventional

*g*

^{n}_{12}basically consists of strain originated components. We believe that the method presented can also be applied to other EO materials and will help analyze their EO properties.

## Disclosures

The authors declare no conflicts of interest.

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