KTa1-xNbxO3 is known for its huge Kerr effect, which is a second order electrooptic (EO) effect. By utilizing the large refractive index change Δn of this EO effect, a fast optical beam deflector has been realized. However, anomalous spatial distributions of Δn 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 Δn distribution.
© 2015 Optical Society of America
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. KTa1-xNbxO3 (KTN) is a ferroelectric oxide . 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 . 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 . 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 g11, g12 and g44 . With the device structure shown in Fig. 1, Ex is the only non-zero field component. The refractive index modulation with Ex is commonly written as follows in the conventional EO theory.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.9].Eqs. (1) and (3) should have a parabolic profile in the same way as the retardation given by Eqs. (2) and (3).
We measured the spatial distribution of the retardation R and compared it with the above equations. The composition of the sample crystal was approximately K0.95Li0.05Ta0.73Nb0.27O3. Lithium was added to the crystal growth ingredients to facilitate crystal growth . 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 . 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 Δnx, 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.12].
With our model, we regard Py = Pz = 0 and all the elements of [PP] except for the first as being zero. Therefore, the strain elements are expressed as follows with the conventional theory.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 .
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 eyy and ezz are expressed as follows.Eqs. (9) and (10) for Eq. (11), we used Euler’s equation in the calculus of variation:Eq. (3) but Px(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 eyy is coupled to exx. Therefore, a linear function of x is added to exx. The other point is that the coefficient for Px2 is different. As the sign of Q12 is opposite to that of Q11, the coefficient in Eq. (12) is smaller than that of the conventional theory. As Eq. (13) indicates, the radius of the spheric distortion r0 is inversely proportional to the parameter H, which is related to the symmetry of Px(x) with respect to the center of the block. When Px(x) is asymmetric, H becomes large and the distortion curvature also becomes strong. When Px(x) is symmetric, H is zero. Then both r0 and ra diverge and the x-dependence of eyy disappears although a constant term remains in Eq. (9). eyy converges to the conventional value shown in Eq. (8) when Px is uniform. Then exx also converges to the value shown in Eq. (8).
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) . Circles indicate the FEM result for exx, squares are for eyy and diamonds are for ezz. Here we assumed Px(x) with a form of Eq. (3) and with a constant charge density ρ. P0 was zero. Then r0 and ra are expressed as follows by using Eqs. (13) and (14).
With Fig. 4, ρ was −96 C/m3 and the relative permittivity εr was 28000. This means that V was 300 V. We assumed Q11, Q12, c11 and c12 to be 0.11 m4/C2, 0.02 m4/C2, 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, Px is zero at the cathode and becomes stronger with the distance from the cathode. Therefore, exx increases parabolically with x. However, exx 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 y- and z- directions because of the distortion, which means that eyy and ezz are positive there. Then the part contracts along the x-direction and exx 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 ezx were negligible as shown in the figure, which is consistent with our model.
With the help of Eq. (7), we plotted the retardation derived from Eqs. (9) and (12) in Fig. 5. This time we assumed that p11 and p12 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 Δn is, of course, caused by the anomaly in the strain component exx shown in Fig. 4. With a constant ρ, Δnx is written as
Out next step is to determine the values of pij, Qij and cij. We evaluated cij by measuring the velocities of acoustic waves in the crystals [13, 14]. As regards Qij, there is a report dealing with experimental values obtained for KTN crystals . However, the values are not for crystals doped with lithium. Moreover, p12 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 gs11 defined by Eq. (17) equals the conventional EO coefficient g11 observed with a uniform electric field, which is reported in . 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.
In conclusion, we observed and analyzed anomalous electrooptic index modulations for KTa1 - xNbxO3 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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