1. Introduction

Potassium tantalate niobate KTa_1-xNb_xO₃ (x: KTN) single crystals are promising electrooptic (EO) materials explored for a range of phase modulation, photorefractive, electro-holography and beam deflection applications [1–3]. Their dielectric properties, quadratic electrooptic effects, and room temperature linear electrooptic effects (in tetragonal crystals) have been studied as well [4–6]. With the addition of a small amount of lithium to facilitate the crystal growth, a series of high-optical-quality Nb-rich K_0.95Li_0.05Ta_1-xNb_xO₃ (x: KLTN) single crystals have been grown successfully and a systematic study of their optical and electrooptic properties is being conducted by the present authors. The Nb-rich tetragonal ferroelectric K_0.95Li_0.05Ta_0.40Nb_0.60O₃ (0.60: KLTN) single crystals have been found highly attractive, with high Curie temperature (440K) and large piezoelectric properties (d₃₃~400 pC/N and d₃₁~-140 pC/N) at room temperature [7]. The linear electrooptic coefficients r₃₃, r₁₃( = r₂₃), and r_c ($=r_{33}-r_{13}\cdot n_o^3/n_e^3$.) of the 0.60: KLTN single crystals have also been investigated and reported, along with their dependency on frequencies of electric field applied [8].

For electrooptic applications, high magnitudes, broad frequency bandwidth, and low temperature dependence of the linear electrooptic coefficients, especially that of the transverse r₅₁ (or r₄₂) coefficients are highly desirable [9]. However, measured or sometimes derived values of r₅₁ have been scarce in literature due to the fact that the standard EO measuring techniques do not apply to the measurement of r₅₁ [10]. In this paper, we report the measurement of ultra-high linear electrooptic coefficient r₅₁ of the 0.60: KLTN crystals using an AC modulating method based on [11], modified by using coherent light source and via dynamic phase locked signal detection. The frequency dependency of the r₅₁ on modulating electric field is also evaluated in 100 Hz – 100 kHz at room temperature.

2. Experimental and theoretical background

Linear EO effect is defined as, in condensed Voigt notation:

Upon application of an electric field to the crystal, assuming only linear EO effect is appreciable in its ferroelectric phase, the general equation of the indicatrix for the tetragonal KLTN crystal can be written as

In order to determine r₅₁ a light beam with its wave-normal parallel to the y-axis and an electric field along the x-axis are applied, so that the Eq. (3) for E_x≠0(E_y = E_z = 0) reduces to:

The optically uniaxial crystal becomes optically bi-axial upon the application of E_x with the induced optical axis aligned in a direction away from the z-axis by a rotation angle ± β in the x-z plane such that the cross term xz vanishes when$\tan (2\beta )=\frac{2r_{51}{E}_x}{1/n_e^2-1/n_o^2}$.

The room temperature values of n_o( = 2.267) and n_e( = 2.237) of the 0.60:KLTN single crystal were determined by minimum deviation technique at the wavelength of 633nm [8]. Based on the results one has$(1/n_e^2-1/n_o^2)=0.00532$; therefore the value of $2r_{51}{E}_x/(1/n_e^2-1/n_o^2)$is small enough and tan(2β)≈2β is satisfied as long as the electric field applied is moderate (<20 V/mm in this experiment). The EO coefficient r₅₁ can be thus determined by measuring the field-induced rotation angle β of the optical axis:

The configuration of the experiment is shown in Fig. 1 , following a similar design reported by Johnston and Weingart, where the crystal sample is positioned on a precision rotatable positioner (with a resolution of 0.2-degrees) between a pair of crossed polarizer and analyzer [12]. This experiment however uses coherent optics with He-Ne 633 nm laser as light source and adapts dynamic AC measurement technique with phase-locked modulated light intensity detection. White light was used in most relevant reports to avoid possible interference effects [11,12]. While it may be reasonable to assume electrooptic (EO) r coefficients are wavelength insensitive over a certain wavelength range, the white light method nevertheless suffers from high noise floor thus has limited measurement resolution. The phase locked intensity detection method is immune to white noises and enables precision measurements. In addition it allows for small signal measurement so that depolarization of the sample can be avoided.

 

Fig. 1 Arrangement of AC measurement method modified from [12].

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The incident light field of optical frequency ω is polarized in <001> directions of the crystal in its initial position,$F_P=F_z=F_0\sin (\omega t)$, and propagates through the sample’s width in the [010] direction. Since the analyzer is crossed with the polarizer, the configuration is set initially at an optical extinction position thus the light field emerging from the analyzer is F_A = 0.

Upon rotation of the sample an angle α about its <010> axis from the polarizer ([001]) direction toward the analyzer ([100]) direction, an elliptically polarized wave propagates in the crystal, which in general can be viewed as a superposition of two linearly polarized waves,

Considering the magnitude of the linearly polarized light intensity incident upon the sample is$I_P=\frac{1}{T}{\displaystyle {\int }_0^T{(F_P)}^2}dt=\frac{1}{2}{F}_0{}^2$, the magnitude of the light intensity emerging from the analyzer will be$I_A=\frac{1}{T}{\displaystyle {\int }_0^T{(F_{z\text{'}}^{}\sin \alpha +F_{x\text{'}}^{}\cos \alpha )}^2}dt=\frac{1}{2}{F}_0^2{\sin }^2(2\alpha ){\sin }^2(\frac{\Gamma }{2})$. The relative intensity of the light measured by the photodetector thus has sinusoidal dependency on the transformation term attributed to the rotation angle αmodified by an interference term of constant phase retardation due to the sample’s natural birefringence:

Upon the application of a modulating electric field,$E_x=E_0\sin (\Omega t)$ along the <100> axis of the tetragonal crystal, the uniaxial crystal will become optically biaxial. The orientation of the two optical axes can be defined using EO rotation angle ± β where β>0 if the induced rotation is from z’-axis to x’-axis. Taking positive β as an example, the relative intensity of the light in Eq. (7) will have the following form:

Since both the induced β(Ε) rotation and the induced birefringence $\delta n(E)$terms are linearly dependent of the electric field applied, a measurement method that is capable of resolving either β(Ε) or $\delta n(E)$could be effective in rendering the value of r₅₁. In this work, using configuration shown in Fig. 1, we have adapted dynamic method to determine β(Ε). A superimposed modulating electric field E_x(Ω) along the <100> axis induces small optical axis rotation that changes the light intensity output from its reference condition. By precision adjustment of the rotation angle of the crystal, one can optically bias the crystal back to its reference state at E_x = 0, therefore obtaining value β. Coefficient r₅₁ is derived from Eq. (5), $\beta =\frac{r_{51}}{(1/n_e^2-1/n_o^2)}\frac{V}{b}$, where V is the peak value of the AC modulating voltage applied on the sample and b is the electric length of the sample. The most convenient reference state at E_x = 0 is at$\alpha =\frac{\pi }{4}(2m+1)$, m is an integer, where the modulator’s output light intensity is at its maximum. Figure 2 depicts the condition that the modulator is measured around the point where light intensity is at the peak near point “C”. A modulating V_appsin(Ωt) signal leads to a modulated output beam V_outsin(2Ωt) at twice of the signal frequency, offering convenient and sensitive detecting point for β measurement [13].

 

Fig. 2 Illustration of dynamic AC measurement utilizing phase locked modulated intensity at maximum transmission point C where modulated intensity is at twice the modulating frequency.

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The K_0.95Li_0.05Ta_1-xNb_xO₃ crystals used in these investigations were grown by the top-seeded melt growth technique as described in [7]. The Curie temperature of the crystals used in this study was determined as T_C~440 K, by the maxima of their dielectric responses as a function of temperature, upon heating, in c-axis un-poled crystals. The Curie temperature corresponds to a nominal composition of K_0.95Li_0.05Ta_0.40Nb_0.60O₃ or x = 0.60, obtained according to the interpolation relation of Rytz and Chatelain (applicable for x>0.35 in KTa_1-xNb_xO₃) [14]:

The dimensions (length × width × height) of the sample used for EO test are 2.10 mm × 1.36 mm × 1.68 mm, and the height is parallel to the [001] direction. The (010)/(0$\overline{1}$0) faces were polished for light transmission and the (001)/(00$\overline{1}$) faces electroded using air-dry silver paint for single domain treatment, which was subsequently removed with acetone after poling. The sample was poled for ~30 min by applying an electric field of 350 V/mm at 500 K, some 60 K above the Curie temperature. The sample was then slowly cooled to room temperature under electric field. The effectiveness of the poling is verified after poling by measuring the piezoelectric constant d₃₃ and by monitoring its stability after handling and testing, using a d₃₃ meter. The value of d₃₃ = 440 ± 20 pC/N was obtained shortly after poling which remained stable after several months of handling and testing of the sample. Upon removal of the poling electrodes, fresh electrodes were applied onto the (100)/($\overline{1}$00) faces for r₅₁ measurement. The crystallographic orientation relationship of the sample tested is illustrated in Fig. 3.

 

Fig. 3 Illustration of the sample orientation in relation to the direction of the poling field and the direction of the applied AC field for EO r₅₁ measurement. Optical wave propagation direction is vertical to the page, parallel to the [010] direction.

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3. Results and discussion

After recording the angular positions of the crystal positioner corresponding to its first maximum (α₀ = π/4) light intensity position at E_x = 0, the rotation angle α_i is carefully adjusted to α₀ + β_i(Ε_ι) for each given modulating AC field E_xi, so that the total intensity returns to maximum. The intensity maximum point is determined conveniently by monitoring the largest second harmonic component of the modulated light output using the lock-in amplifier set at harmonic frequency = 2Ω. A sequence of modulating electric field is chosen and a family of independent measurement of β is taken to render the value of r₅₁ from the slope of the β vs. E_x(Ω). The measurement results of the 0.60: KLTN sample are shown in Fig. 4 and the linear dependence of β on E_x(Ω) is evident within the range of electric field applied. The EO r₅₁ obtained from the linear fitting of the experimental data is r₅₁ = (1.10 ± 0.14) × 10⁴ pm/V, measured at room temperature, modulated at 1 kHz, and for λ = 633 nm.The r₅₁ reported here should be viewed as an effective EO coefficient$r_{51}^X$($r_{ijk}^X=r_{ijk}^x+r_{ijk}^a=r_{ijk}^x+p_{ijlm}{d}_{lmk}$) that contains both the primary $r_{ijk}^x(\to r_{51}^x,$at constant strain) and the secondary $r_{ijk}^a(\to p_{55}{d}_{51})$ contributions; the latter is a combined elastooptic (p_ijlm) and piezoelectric (d_ijk) effect, arising from the presence of the shear strain x₅ upon the application of E_x on the stress-free piezoelectric tetragonal crystal. Since it is known that p_max≈0.2 for most oxides and halides [15], the secondary contribution of the acoustooptic effect to the effective EO r₅₁ measured in this case is limited and yet to be separately quantified.

 

Fig. 4 Rotation of the optic axis as a function of the modulating electric field in the 0.60: KLTN single crystal.

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To determine the sign of r₅₁, a positive modulating electric field, $E_x=\frac{E_0}{2}[1-\cos (\Omega t)]$ is applied to ensure that only one rotation direction is induced by the field. For such an electric field applied the optic axis rotated from the [001] direction to the [100] direction; the sign of r₅₁ is determined to be positive knowing n_o>n_e (n_o = 2.267 and n_e = 2.237). The uncertainty of repeating measurements of the induced angle β is typically ± 0.04°, which is equivalent to an error band of ± 13% for measured value of r₅₁.

The exceptionally large r₅₁ of 0.60: KLTN is extremely attractive and it is one of the highest r₅₁ values reported to the authors’ knowledge. In Table 1, r₅₁ values of several important EO materials are summarized for comparison purposes [12,16–19]. Note the r₅₁ measured in this work in fact extends the trend that r₅₁ increases with Nb concentration, following the report in [16]: r₅₁ = 7850 ± 1550pm/V in the KTa_0.48Nb_0.52O₃ single crystal and r₅₁ = 5770 ± 1150 pm/V in the KTa_0.53Nb_0.47O₃ single crystal. Günter’s works however, revealed that this trend is observed only in solid solution KLTN system but not in pure KNbO₃ single crystals; its r₅₁ coefficient is only around 105 pm/V [17].

Table 1. Electrooptic coefficients (r₅₁) of some electrooptic single crystals

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The values of r₃₃, r₁₃ and r_C of 0.60: KLTN at room temperature and 1kHz modulating frequency have been reported to be 216.7, −21.2 and 242.9 pm/V, respectively [8]. Table 2 summarizes the dielectric constants in the c- and a-directions and the EO coefficients (at room temperature and 1 kHz frequency) for the 0.60: KLTN crystals measured. Several other important materials, the perovskite KNbO₃ (mm2) and BaTiO₃ (4mm), and tungsten bronze SBN (4mm) crystals are included in Table 2 for comparison purposes [17,19,20].

Table 2. Dielectric constants and electrooptic coefficients of the 0.60: KLTN, BaTiO₃ and SBN single crystals (at room temperature and 1 kHz).

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The frequency of the applied AC modulating field was varied from 100 Hz to 100 kHz to study frequency dependence of the EO r₅₁. The peak level of the applied field was kept below 20 V/mm to avoid disordering the polarization or influences from higher harmonic terms. The results revealed that the r₅₁ of the 0.60: KLTN has insignificant frequency dependence as the values (10,800 ~11,600 pm/V) obtained over the 100 Hz to 100 kHz frequency range were within the measurement resolutions. In contrast, the r₃₃, r₁₃, and r_C of the 0.60: KLTN single crystals of the same boule were found to be strongly dependent on the modulation frequencies, decreasing sharply with the increasing frequency.

It is known that the linear electrooptic effect in a ferroelectric crystal (at T<T_C) of a centric prototype symmetry (perovskite m3m in this case), is fundamentally a quadratic effect biased by spontaneous polarization and can be described in terms of the quadratic polarization optic coefficients g_ijkl of the prototype as [21]

Using the extended relations developed for oxygen octahedral structures we have in the tetragonal phase (of the condensed notation) [22]:

Substituting the experimental values of 0.60:KLTN listed in Table 2 including the linear EO coefficients ([8]), the measured value of spontaneous polarization P_s ~16 μC/cm² ([7]), as well as the values of dielectric constant ε₃₃ ≈790 and ε₁₁ ≈13200 measured in this work for poled samples used in this work, the quadratic coefficients of the 0.60: KLTN crystal are found to be g₁₁~0.10 m⁴/C², g₁₂~–0.01 m⁴/C², and g₄₄~0.147m⁴/C². Compared with the g values of most perovskites, i.e., g₁₁ = 0.14 ± 0.01 m⁴/C², g₁₂ = 0.04 ± 0.01 m⁴/C² and g₄₄ = 0.14 ± 0.02 m⁴/C² summarized by Wemple and DiDomenico the g₄₄ value is in reasonable agreement while the g₁₁ and g₁₂ values are notably lower [23]. The discrepancies may be indicative that in perovskite solid solution systems, the distortions of the BO₆ octahedra, the biaxial components of the refractive index, and the corresponding EO tensor components, may not be sufficiently small to be neglected in the g-coefficient formulism.

Since the most of the perovskite oxides have similar g-coefficients, it is immediately evident by examining the relation of Eq. (10) that a large EO coefficient must be associated with the spontaneous polarization and the permittivity in the transversal direction of the wave-normal [24]. In a poled crystal, dielectric constant (ε₃₃) parallel to the spontaneous polarization (P₃) is in fact related to the polarization through Devonshire theory. Exceptionally large linear EO effect, in this case r₅₁, is possible by a combination of a moderately-large room temperature polarization (P₃) and a very large transverse dielectric permittivity (ε₁₁). Similar observations have been made for tungsten bronze lead barium niobate Pb_1-xBa_xNb₂O₆ [PBN(1-x)%] solid solution single crystals. In its ferroelectric tetragonal compositions, EO r₅₁ is large (~1524 pm/V, PBN57) and in its ferroelectric orthorhombic compositions r_C is large (~216 pm/V, PBN65), both were attributed to the respective large transverse dielectric constants [24].

The sharp contrast in frequency dependence (100 Hz to 100 kHz) of EO coefficients between longitudinal, E(Ω)//P_r, and transversal, E(Ω)⊥P_r, EO effects is likely a direct reflection of the hysteresis behavior of polarization vs. electric field in response to the modulating frequency. In the case of r₅₁, the dominant contribution (since E(Ω)⊥P_r) to its frequency dependence is from permittivity ε₁₁, which has a weak and negative frequency response at room temperature (that is well below its ferroelectric phase transition). In the case of r₃₃, r₁₃, or r_C, the frequency dependence is dominated by the dynamic polarization process in a free crystal, coupled with alternating lattice strains and associated with permittivity variations due to periodic changes of bond charge surface densities. Large frequency dependence is a known behavior for longitudinal EO effect that is often viewed as a limiting factor in EO device applications.

4. Conclusions

The potential applications utilizing the exceptionally large transverse EO r₅₁ coefficient values, as discussed about the case of the KTa_0.48Nb_0.52O₃ single crystal [25], include light modulation and deflection at room temperature and in the frequency ranges (100 – 100 kHz) studied. Compared to the KTa_0.48Nb_0.52O₃ crystals for which the large r₅₁ value is obtained at a temperature near the orthorhombic phase transition (that is slightly above the room temperature), the outstanding EO properties of the 0.60:KLTN single crystal is available at room temperature, well below its ferroelectric phase transition temperature at ~440 K. The combination of the large room temperature electrooptic properties and good optical quality of the as grown crystals opens a range of application opportunities for the KLTN crystals.

This paper reports experimental determinations of the electrooptic transverse coefficient r₅₁, of the solid solution 0.60:KLTN single crystal, by a dynamic AC modulating method coupled with coherent optics and phase sensitive lock-in detection. The electric field induced rotation of the optical axis was measured directly with good precision. Exceptionally large r₅₁ coefficient 11,000 ± 1400 pm/V is found for this crystal, which is consistent with the expected trend that EO r₅₁ coefficient increases with the increasing Nb concentrations in the KLTN solid solutions. The large r₅₁ is attributed to the combination of moderately high spontaneous polarization P₃ and a very large dielectric constant ε₁₁. In addition, the room temperature r₅₁ is found to be frequency insensitive in the range of 100 Hz to 100 kHz, likely due to the disassociation with hysteretic polarization process which otherwise results in steep decrease of longitudinal EO (r₃₃ and r₁₃) coefficients with frequencies.