## Abstract

KTa_{1-}* _{x}*Nb

*O*

_{x}_{3}is known for its huge Kerr effect, which is a second order electrooptic (EO) effect. By utilizing the large refractive index change Δ

*n*of this EO effect, a fast optical beam deflector has been realized. However, anomalous spatial distributions of Δ

*n*were observed with this beam deflector. This anomaly is ascribed to distortions caused by the electrostrictive effect that occurs when voltage is applied. We assumed a spheric distortion and used a variational method to deduce an analytic solution for the strains that accompany this distortion. The analytic solution coincides with numerical results obtained with the finite element method. In addition, the solution agrees well with the experimentally obtained Δ

*n*distribution.

© 2015 Optical Society of America

## 1. Introduction

The electrooptic (EO) effect is a phenomenon where the refractive indices of a substance are modulated with an externally applied electric field. Because an optical phase can be modulated with an applied voltage, various optical phase modulators and intensity modulators have been realized with interferometers. Of the various kinds of materials exhibiting the EO effect, ferroelectric oxides are noted for their speed although their EO coefficients are not as large as those of liquid crystal materials. KTa_{1-}* _{x}*Nb

*O*

_{x}_{3}(KTN) is a ferroelectric oxide [1]. Although KTN can be ferroelectric, it is usually used in its paraelectric phase above the phase transition temperature because paraelectric KTN has a huge second order EO effect, namely the Kerr effect. This huge Kerr effect allows us to realize large optical beam steering functions in addition to the conventional phase modulating functions. A prism-shaped beam deflector and a space charge controlled (SCC) beam deflector have been developed with paraelectric KTNs [2, 3]. The deflection angle of the SCC deflector exceeds 10 degrees. Moreover, the device is much faster than conventional galvanometer scanners. By using these features, a wavelength-tunable laser with a high scanning speed has been developed and used as the key component of a swept-source type optical coherence tomography system [4, 5].

The SCC deflector is illustrated in Fig. 1, and it is simply a KTN crystal block with a pair of electrodes. When we apply a voltage, electrons are injected into the crystal and form a space charge. The space charge generates a spatial distribution of an electric field in the crystal and thus creates a spatially distributed refractive index [6]. This index distribution bends light rays and works as an optical beam deflector. Therefore, index control is essential for SCC deflector operation. However, anomalous index modulations are often observed in the spatial index profile; the sign of the index change is opposite to that of the conventional theory near the cathode. We discuss this anomaly in KTN crystals in this letter. The anomaly is caused by the strain that accompanies electrostrictive distortion [7]. We propose a spheric distortion model for an SCC KTN and deduce an analytic solution that explains experimental results well.

## 2. Conventional theory and anomalous index modulation

Above the phase transition temperature, a KTN crystal has m3m symmetry. Then the non-zero EO coefficients are *g*_{11}, *g*_{12} and *g*_{44} [8]. With the device structure shown in Fig. 1, *E _{x}* is the only non-zero field component. The refractive index modulation with

*E*is commonly written as follows in the conventional EO theory.

_{x}*n*

_{0}is the original refractive index and

*ε*is the permittivity. Δ

*n*and Δ

_{x}*n*are the refractive index changes for

_{y}*x*- and

*y*-polarized light, respectively. These equations are sometimes written as follows using a low frequency polarization ${P}_{x}\approx \epsilon {E}_{x}$.

*g*

_{11}has a positive value and

*g*

_{12}is negative [8]. When both polarized lights are input into a KTN block, the

*x*-polarized light is retarded in relation to the

*y*-polarized light because of this birefringence. We define the retardation

*R*as follows with the block length

*L*.

*R*should always be negative with this formula because the sign of

*g*

_{11}

*- g*

_{12}is positive. For use as a beam deflector, electrons must be injected through the cathode with an applied voltage. Then

*P*exhibits a spatial variation. If we assume that the injected electrons are uniformly distributed in the KTN block and the constant charge density is

_{x}*ρ*, the spatial function

*P*(

_{x}*x*) distribution can be expressed as follows using Gauss’s law [9].

*d*is the distance between the electrodes and we place the origin of

*x*at the midpoint between the electrodes. The cathode is at –

*d*/2 and the anode at

*d*/2.

*P*

_{0}is a constant determined by the boundary conditions. When we apply a dc voltage, assuming

*P*

_{0}= 0,

*P*and

_{x}*E*are zero at the cathode. The refractive index distribution given by Eqs. (1) and (3) should have a parabolic profile in the same way as the retardation given by Eqs. (2) and (3).

_{x}We measured the spatial distribution of the retardation *R* and compared it with the above equations. The composition of the sample crystal was approximately K_{0.95}Li_{0.05}Ta_{0.73}Nb_{0.27}O_{3}. Lithium was added to the crystal growth ingredients to facilitate crystal growth [10]. The block size was 1.2 mm (*x* direction in Fig. 1) x 3.2 mm (*y*) x 4.0 mm (*z*). We kept the sample temperature about 3 K above the phase transition temperature to control the relative permittivity at 17,500. The retardation measurement method has been described in detail elsewhere [11]. Figure 2 shows the *R* distribution. The horizontal axis shows the *x*-coordinate. The left end corresponds to the cathode and the right end to the midpoint. The anode was set at + 0.6 mm and is not shown in the figure. We acquired these data with a continuously applied voltage of 300 V. With this voltage, a sufficient number of electrons are injected into the KTN crystal. The gradient of the curve is caused by these injected electrons in accordance with Eqs. (3) and (2). The retardation is, in fact, dominated by Δ*n _{x}*, and the gradient of the curve approximates the beam deflection angle for the

*x*-polarized light. As the curve seems to fit a parabolic function, it appears that the theory described above with Eqs. (1) to (3) explains the experimental result well. However, there is a serious problem. As described above by Eq. (2),

*R*should always be negative according to the conventional theory. The curve in Fig. 2 is negative in most places in the KTN crystal. However, it becomes positive in the vicinity of the cathode at the left end. This phenomenon, namely a bidirectional index change, cannot be explained by the conventional theory because the KTN did have an inversion symmetry of m3m.

## 3. Theory of strain and spheric distortion model

The anomaly is explained by the strains induced by the electrostrictive effect. We investigated the influence of strain on the refractive index. Conventionally, the electrostrictive effect is expressed as follows.

We defined [*e*], [

*Q*] and [

*PP*] as follows [12].

*e*] is the strain and [

*Q*] is the electrostrictive coefficient. When strains are induced in the crystal, the refractive index is changed via the elastooptic effect. The effect is expressed as modulations of the optical impermeability tensor [

*β*(

*P*)].

*p*] is the elastooptic coefficient and has the same structure as [

*Q*]. [

*β*(

*P*)] and [Δ

*β*(

*P*)] are functions of the polarization and have the same structure as [

*e*]. As regards lights propagating along z direction, the refractive index changes are expressed as follows.

*p*

_{11}and

*p*

_{12}are the elastooptic coefficients.

With our model, we regard *P _{y}* =

*P*= 0 and all the elements of [

_{z}*PP*] except for the first as being zero. Therefore, the strain elements are expressed as follows with the conventional theory.

*Q*

_{11}is positive and

*Q*

_{12}is negative [12]. Thus, with a low frequency polarization (or an electric field), the crystal expands along the direction of the polarization and shrinks along the direction perpendicular to it. The polarization is weak near the cathode but strong near the anode as explained earlier with Eq. (3). Therefore, the shrinkage is stronger near the anode than the cathode and this shrinkage imbalance warps the crystal as shown in Fig. 3. The simple substitution of Eq. (8) for Eq. (7) gives the same result as Eq. (1) with an appropriate coefficient transformation. However, this is not accurate because of the presence of elasticity. The strain is not determined only by the local polarization following Eq. (8) but is affected by the interaction with the adjacent part of the crystal [7].

We assumed a spherical distortion model to analyze the strain distribution in the KTN block (Fig. 3). In this model, the KTN block is distorted and the two faces of the block become concentric spheres with a fixed center point. Then strain components *e _{yy}* and

*e*are expressed as follows.

_{zz}*r*

_{0}is the distance between the sphere center O

_{o}and the center of the block.

*r*is the distance between O

_{a}_{o}and the point where

*e*is zero. These two values will be fixed later.

_{yy}*e*is a function of

_{xx}*x*and yet to be determined. The other strain components

*e*,

_{yz}*e*and

_{zx}*e*indicate shear strains. These are all zero if the distortion is precisely spherical. We assume the local free energy as follows.

_{xy}*c*] is the elastic stiffness coefficient, which has the same structure as [

*Q*]. We ignored the

*EP*term because it has no meaningful effect. In addition, we define the following total energy.The integration is performed throughout the entire block and uniformity is assumed along the

*y*- and

*z*-directions.

*S*is the area of the electrode. We deduce

*e*as a function of

_{xx}*x*so that it minimizes

*F*. Substituting Eqs. (9) and (10) for Eq. (11), we used Euler’s equation in the calculus of variation:

*y*is a function of

*x*and

*y’*is its derivative. We put

*e*as

_{xx}*y*and obtain

*r*

_{0}and

*r*in

_{a}*e*are also chosen so that

_{xx}*F*is minimized as

*G*and

*H*are calculated from

*P*(

_{x}*x*) by using the following formulae.

*P*(

_{x}*x*) to have the form shown in Eq. (3) but

*P*(

_{x}*x*) can be an arbitrary function of

*x*. Equations (9) and (12) are the obtained analytical solutions for the strain in a KTN crystal block with a space charge. The most important difference between Eq. (12) and the conventional formula Eq. (8) is that

*e*is coupled to

_{yy}*e*. Therefore, a linear function of

_{xx}*x*is added to

*e*. The other point is that the coefficient for

_{xx}*P*

_{x}^{2}is different. As the sign of

*Q*

_{12}is opposite to that of

*Q*

_{11}, the coefficient in Eq. (12) is smaller than that of the conventional theory. As Eq. (13) indicates, the radius of the spheric distortion

*r*

_{0}is inversely proportional to the parameter

*H*, which is related to the symmetry of

*P*(

_{x}*x*) with respect to the center of the block. When

*P*(

_{x}*x*) is asymmetric,

*H*becomes large and the distortion curvature also becomes strong. When

*P*(

_{x}*x*) is symmetric,

*H*is zero. Then both

*r*

_{0}and

*r*diverge and the

_{a}*x*-dependence of

*e*disappears although a constant term remains in Eq. (9).

_{yy}*e*converges to the conventional value shown in Eq. (8) when

_{yy}*P*is uniform. Then

_{x}*e*also converges to the value shown in Eq. (8).

_{xx}## 4. Results and discussions

Figure 4 shows spatial distributions of the strain components. The solid lines are those of Eqs. (9) and (12). We also plot the strain components that we calculated numerically with the finite element method (FEM) [7]. Circles indicate the FEM result for *e _{xx}*, squares are for

*e*and diamonds are for

_{yy}*e*. Here we assumed

_{zz}*P*(

_{x}*x*) with a form of Eq. (3) and with a constant charge density

*ρ*.

*P*

_{0}was zero. Then

*r*

_{0}and

*r*are expressed as follows by using Eqs. (13) and (14).

_{a}With Fig. 4, *ρ* was −96 C/m^{3} and the relative permittivity *ε _{r}* was 28000. This means that

*V*was 300 V. We assumed

*Q*

_{11},

*Q*

_{12},

*c*

_{11}and

*c*

_{12}to be 0.11 m

^{4}/C

^{2}, 0.02 m

^{4}/C

^{2}, 250 GPa and 110 GPa respectively. The thickness

*d*was 1.2 mm. The left end of Fig. 4 is the cathode and the right end is the anode. As described above,

*P*is zero at the cathode and becomes stronger with the distance from the cathode. Therefore,

_{x}*e*increases parabolically with

_{xx}*x*. However,

*e*was not zero at the cathode but had a negative value there. This phenomenon cannot be explained by Eq. (8); it is caused by the warping distortion. The part in the block is stretched along the

_{xx}*y*- and

*z*- directions because of the distortion, which means that

*e*and

_{yy}*e*are positive there. Then the part contracts along the

_{zz}*x*-direction and

*e*becomes negative. As seen in Fig. 4, the analytical solutions (9) and (12) agree well with the numerically obtained strain values of the FEM. In addition, the numerical shear strains such as

_{xx}*e*were negligible as shown in the figure, which is consistent with our model.

_{zx}With the help of Eq. (7), we plotted the retardation derived from Eqs. (9) and (12) in Fig. 5. This time we assumed that *p*_{11} and *p*_{12} were 1.18 and −0.16, respectively. The other parameters were the same as those in Fig. 4. The solid line shows the analytical result. The circles are small fraction of the data taken from those in Fig. 2. The figure indicates that our spherical distortion model was successful in reproducing the experimental result. The change in the retardation sign at around −0.46 mm was also reproduced. This anomalous retardation or Δ*n* is, of course, caused by the anomaly in the strain component *e _{xx}* shown in Fig. 4. With a constant

*ρ*, Δ

*n*is written as

_{x}*g*

^{e}_{11}and

*g*

^{s}_{11}as follows.

*g*

^{s}_{11}>

*g*

^{e}_{11}, the condition for the appearance of the anomalous index modulation is

Out next step is to determine the values of *p _{ij}*,

*Q*and

_{ij}*c*. We evaluated

_{ij}*c*by measuring the velocities of acoustic waves in the crystals [13, 14]. As regards

_{ij}*Q*, there is a report dealing with experimental values obtained for KTN crystals [12]. However, the values are not for crystals doped with lithium. Moreover,

_{ij}*p*

_{12}is so weak that we have not found any reliable experimental data even for undoped KTN [14, 15]. We assumed that the EO effect is totally governed by the electrostrictive and elastooptic effects. Thus we ignored index modulations that were independent of strains and caused directly by electric fields. This gave a condition where

*g*

^{s}_{11}defined by Eq. (17) equals the conventional EO coefficient

*g*

_{11}observed with a uniform electric field, which is reported in [8]. With this condition, we determined the other parameters so that they reproduced the observed retardation profile the best. However, a more detailed study of such parameters should help us obtain a deeper understanding of the anomalous strains and index modulations.

## 5. Conclusions

In conclusion, we observed and analyzed anomalous electrooptic index modulations for KTa_{1 -} * _{x}*Nb

*O*

_{x}_{3}single crystals. The anomaly is caused by anomalous strains that result from the warping of the KTN crystal block with a space charge formed in the crystal. Using a spherical distortion model, we derived an analytical solution for the spatial profiles of strains and refractive index modulations in the crystal and succeeded in fitting the profile to the experimental result.

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