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Abstract
In this report, we successfully implement a unique cross-field beam deflector by exploiting the modulation of a one-dimensional refractive index in a copper-doped potassium tantalite niobite crystal. A theoretical model is established based on an electrostrictive effect regulated by the dynamic polarized nano-domains to explicate the mechanism of the abnormal beam deflection which is perpendicular to the applied electric field. Experimental results agree well with our theoretical deduction while validating the interactions between the dynamic polarized nano-domains and the applied electric field. Our findings will break the limitation of conventional electro-optic deflectors, paving the way to develop promising optical functional devices with a large field-of-view scanning angle and ultra-low driving voltage.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The localized clusters with randomly oriented polarization in potassium tantalate niobate (KTa1-xNbxO3, KTN) single crystal are formed by the off-center deviation of niobium ions and play a crucial role in governing the dielectric and structural properties of the crystal [1–8]. As a remarkable perovskite-type relaxor ferroelectric, the KTN crystal exhibits several fascinating features especially approaching its paraelectric–ferroelectric phase transition, such as giant quadratic electro-optic effect, abnormal electrostrictive effect, and pyroelectric effect, induced by the polarized nanometric regions (PNRs) and polarized nanometric domains (PNDs) embedded in the bulk material [9–11]. It should be noticed that the PND is a subdivided definition since PNDs are essentially PNRs with larger domain sizes and the existence of time-average spontaneous polarization (PS). Based on these properties, various functional devices such as high-speed electro-optical scanners and electro-optic modulators have been designed and fabricated [12–15]. Up to now, there have been several reports that declare the control of macroscopic behaviors of the crystal by modulating the properties of PNRs and PNDs. For instance, when the sample is rapidly cooled down to its critical state in the vicinity of the phase transition temperature, the PNDs exhibit an anomalously large static susceptibility which can activate the scale-free optical propagation [16,17]. On the other hand, when the sample has a temperature gradient, the ΔT-induced pyroelectric space-charge field reorients the PNR polarization, enabling the decrease of light scattering and thus an enhanced electro-optic effect [18]. Similarly, an external electric field can change the distribution of PNDs, which contributes to the phase transition [19,20]. In recent decades, researches on PNRs and PNDs have drawn great attention because of the relevant enhanced properties of relaxor ferroelectrics and the feasibility of wide applications.
The high-speed optical deflector based on the electro-optic effect is one successful device implementation using the KTN, which has significant applications in optical communication, sensing, and display [21,22]. However, there is a fundamental issue that the applied electric field is essentially parallel to the plane of beam deflection, which prevents a large field of view of optical scanning due to the beam confliction with the two lateral electrodes. Increasing the spacing of the two electrodes can increase the deflection angle in some degree but causes a dramatic increase of the applied voltage. In this paper, we present a unique implementation of cross-field beam deflection with an applied electric field perpendicular to the optical scanning plane. In other words, the optical beam is scanned within a plane sandwiched between the two parallel electrodes. With our new scheme, there is no deflection angle limitation caused by the electrodes. Also, the spacing between the two electrodes can be as thin as sub-millimeters or even micrometers that may be slightly larger than the beam diameter of a focused laser, enabling the feasibility of using ultralow driving voltage. We have also established a theoretical model based on the modulation of an electrostrictive effect in Cu:KTN and found the model fits the experimental results very well. Our results will have a great impact on conventional electro-optic devices especially for the development of beam-controllable functional devices with large field-of-view deflection angle and ultralow driving voltage.
2. Theory of cross-field beam deflection
The mechanism of the interactions between the PNRs/ PNDs and the external electric field has been a question of great interest to researchers for decades. The density distribution of niobium ions along the growth direction of the KTN sample dominates the density distribution of PNDs in the vicinity of Curie temperature. Thus, the dielectric properties, the photoelastic effect, and the electrostrictive behaviors, which are explicitly affected by PNRs and PNDs, also follow the related distribution patterns. The nonlinear relations among several parameters related to the deflection mechanism can be determined by the following equations [23]:
The electrostrictive strain of a cubic perovskite-type (m3m) crystal can be expressed as
where the polarization intensity P1 is modified by PNDs. Due to the photoelastic effect [24], the pattern of strain will further induce the refractive index fluctuation:Here, we integrate the nonlinear relations: H(ρ) = Ψ(ρ)·M(ρ)·Φ(ρ). As such, the relation between the refractive index and the density of dynamic PNDs can be established.
We define that the niobium ions density fluctuates along the y-direction, which means that the density of PNDs varies along this direction (i.e., ρ = ρ(y)). Accordingly, beam deflection angle can be expressed as
Note that Δn0 has a slight impact on the deflection and thus can be ignored. According to Eq. (4) and Eq. (5), the beam deflection phenomenon can be uniquely regulated by the dynamic PNDs. These deducted equations are of relatively complicated manner, but they can be simplified under controlled-variable conditions. Besides, for a certain sample at a determined temperature, Eq. (4) can be fitted to a direct relationship between the beam deflection angle θ and the applied electric field E without other variables.
On the whole, the beam deflection mechanism can be summarized as follows: the reorientation of the PNDs under an electric field generates strong electrostrictive strain inside the sample. Therewith, the strain is transformed to the refractive index change through the photoelastic effect. The density gradient of PNDs causes a refractive index gradient, which eventually results in the beam deflection.
3. Experimental details
To verify our theoretical model and further investigate the beam deflection, we have performed several experiments to measure the temperature-dependent parameters using the Cu:KTN crystal. The sample is zero-cut with a size of 2.0 (x) mm × 2.2 (y) mm × 12.5 (z) mm and the x, y, z axes correspond to the [001], [100], [010] crystallographic directions, respectively. In the experiments, the sample temperature is set with an accuracy of ±0.01°C using a thermoelectric Peltier cooler controlled by a TEC Source (5305, Arroyo Instruments). Figure 1(a) shows the temperature-dependent relative permittivity of the sample measured by utilizing an LCR meter (TH2830, Tonghui). The Curie temperature TC represents the paraelectric-ferroelectric phase transition and is determined to be 21 °C. Meanwhile, the intermediate temperature (T*) is determined to be 31 °C, which represents the formation of PNDs and can be calculated through the linear fitting of the relative permittivity from the Curie-Weiss law. Figure 1(b) shows the electric hysteresis loops measured at different temperatures above the Tc. The P-E curves manifest an increasing double-S-shaped trend which indicates the existence of microscopic ferroelectric structures (i.e., PNRs/PNDs) in the paraelectric state. Also, the size of the PNRs/PNDs varies as temperature changes, resulting in the temperature dependence of the dielectric properties of the sample. It is worth noting that the utilization of TEC will certainly introduce a temperature gradient into the sample along the normal direction of the TEC surface. But in this experiment, the sample is relatively thin, and the deflection direction is perpendicular to the temperature gradient direction. Therefore, this temperature gradient does not affect the experiments involved in this paper.
Fig. 1. Primary ferroelectric properties of the Cu:KTN sample. (a) Temperature dependence of the relative permittivity ɛr. The solid purple line conforms to the Curie-Weiss law. (b) The P-E curves at different temperatures.
The experimental setup of beam deflection is illustrated in Fig. 2(a). A 532-nm beam from a continuous-wave solid-state laser (Verdi G-Series, Coherent Inc.) propagates along the z-direction and is focused into the Cu:KTN sample after being vertically polarized (i.e., along the x-direction). Subsequently, the positions of the output beam are recorded by a high-speed CCD camera (WCD-UCD12-1310, DataRay). As exhibited in Fig. 2(b), there exists an initial deflection angle θ0 along the y-direction in the absence of an external electric field. The refractive index gradient can be roughly calculated by θ0 = L·Δn0 [25]. In our experiment, θ0 is 62.5 mrad and the length of the sample L is 12.5 mm, thus Δn0 = 5×10−3 mm−1 can be obtained. It should be noticed that the density distribution of the niobium ions fluctuates along the same direction of the refractive index gradient. When an external electric field is applied along the x-direction, the beam deflection angle can increase by the angle of θ, as shown in Figs. 2(c) and 2(d) at 23 and 24 °C, respectively. The laser beam is deflected in a plane perpendicular to the applied electric field which has the same direction as the polarization of the propagating laser beam to maximize the electro-optic coupling. It should be noticed that the beam deflection angle θ is further increased along the initial deflection angle θ0 after an external electric field is applied. This phenomenon is contrary to the previous report about the cross-field KTN beam deflector [13]. It is also worth mentioning that the spot of the deflected beam is compressed along the y-direction because of the nonlinear refraction index change along this direction. As a result, the spot presents ellipticity while still maintains a Gaussian-like intensity distribution. But since this ellipticity is kept at a low level even under a relatively large deflection angle, its influence on the beam quality can become insignificant.
Fig. 2. (a) Schematic of the cross-field beam deflection experiment setup. (b) The optical paths under zero-field (black solid line) and nonzero-field (red solid line) conditions. (c)(d) CCD-recorded positions of the deflected output beam under different experimental temperatures.
4. Results and discussion
4.1 Electrostrictive cross-field beam deflection
We design several single-variable experiments to determine the impact factors on deflected angle θ and also to validate our theoretical model. Controllable variables include crystal temperature T, applied electric field E, and the distribution pattern of niobium ions.
The local TC associated with the density of niobium ions along the y-direction is roughly determined by recording in real time the output beam intensity distribution. It is known that the KTN crystals in the paraelectric phase are optically transparent while those in the ferroelectric phase strongly scatter light due to the existence of domain walls. Figure 3(a) shows the results of temperature-dependent transmittances at two relative locations separated by 1 mm along the y-direction. We can see the Tc at these locations are different with an average ΔTC around 2.3 °C. Thus, the Ta/Nb ratio can be estimated by Perry’s linear relation TC = 682x + 33.2, where x is the mole ratio of Nb/(Ta + Nb) [1]. For our sample, the density fluctuation of niobium ions is extremely weak (Δx = 0.003), thus the density variation of PNDs formed around niobium ions is correspondingly weak. For this reason, H(ρ) and Φ(ρ) can be regarded as constants. Meanwhile, since the sample size is on the scale of millimeters, ρ′ = dρ/dy (in which dy = 1 mm) is a relatively appreciable value, namely, ρ′${\gg}$H′(ρ) and ρ′${\gg}$Φ′(ρ). Consequently, θ${\propto}$ρ′ can be concluded.
Fig. 3. (a) CCD images of the transmitted light intensity distribution at different incident positions of the sample under different temperatures. (b) θ dependence on incident position along the y-direction under the condition of T = 23 °C and E = 100 V/mm. (c) θ dependence on temperature under the electric field E = 100 V/mm and −100 V/mm. (d) θ dependence on the electric field under different temperatures.
We first investigate the deflected angle θ dependence on the density of niobium ions, in the case that the temperature and the electric field are both fixed. Under this condition, Eq. (4) can be rewritten as
In the subsequent experiments, beam deflection angles are measured at different incident positions of the sample with 0.1 mm intervals along the y-direction at T = 23 °C and E = 100 V/mm, respectively. The corresponding results are exhibited in Fig. 3(b). The location-dependent deflection phenomenon demonstrates that θ is indeed related to the distribution pattern of niobium ions, and also validate the relation θ${\propto}$ρ′. Meanwhile, the density of niobium ions exhibits a nonlinear decrease, which suggests that ρ′ = ρ′(y).
In the second case, when the electric field and the beam incident position are fixed, namely, Δρ(y)→0, T will be the only variable according to Eq. (4). Under this condition, the variable ρ and the ρ-related parameters, H(ρ) and Ф(ρ), increase as the temperature decreases. One the other hand, ρ′, H′(ρ) and Ф′(ρ) can be regarded as constants due to that Δρ(y)→0. As such, both h and j tend to increase as the temperature decreases. Moreover, the parameters, p11, Q11, and ɛr will increase abnormally in the vicinity of TC owing to the presence of PNDs. Consequently, we infer that θ will present a sharp increase in approaching the ferroelectric phase transition. The experimental results shown in Fig. 3(c) demonstrate that θ is increased with a considerable increment of 20 mrad at TC + 3 °C, which is consistent with the aforesaid analysis. In the experiment, the electric field E is switched from 100 V/mm to −100 V/mm, and the beam incident location is 0.45 mm from the coordinate origin. In addition, we have found that the beam deflection is independent of the direction of the applied electric field, which further confirms that the beam deflection is indeed induced by the electrostrictive strain rather than the localized ferroelectric effect.
In the third case, the beam incident location and the experimental temperature are both fixed. The temperature-sensitive parameters (i.e., ρ, p11, Q11, and ɛr) and the ρ-relevant parameters (h and j) are constants, thus θ will depend solely on E. Equation (4) can be expressed as
and the constants are defined byHere, the beam incident position is at 0.65 mm and the temperature is set separately to 23 °C, 24 °C, and 25 °C. The results illustrated in Fig. 3(d) reveal the nonlinear relations between E and θ. Using Eq. (8), the measured data displayed in red and blue curves can be well fitted to the following equations:
It should be noted that the first term of Eq. (8) takes an increasing proportion with the size growth of ferroelectric structures under the low-field condition. This can be observed in the enhanced linear trend of the θ-E curves in Fig. 3(d) as the temperature approaches TC.
4.2 Electric-field dependent deflection
To further understand the mechanism of the dynamic PNDs modulated by the electric field, the experiments shown in Fig. 3(d) are repeated under different experimental conditions as illustrated in Figs. 4(a)–4(d). The experiments exhibited in Figs. 4(b) and 4(d) are set as references to exclude any possible effect from the temperature gradient. The polarization direction of the incident beam is kept parallel to the electric field during the investigations. Surprisingly, this unique beam deflection phenomenon will disappear when the electric field is parallel to the direction of niobium ions inhomogeneity.
Fig. 4. Dependence of beam deflection on the direction of applied electric field perpendicular (a, b) or parallel (c, d) to the density distribution gradient of Nb ions (blue arrows). The beam deflection direction is indicated by red arrows. (e) Schematic of polar rotation angle θ1 induced by the electrostrictive effect. (f) PNDs’ density and reorientation pattern.
We conclude that the pattern of niobium ions not only induces the PNDs’ density distribution but also changes their disordered reorientation state. Also, the response of PNDs to the electric field is orientation-dependent. This means, in the vicinity of T*, the dynamic PNRs will couple each other and merge into larger ones of PS. In the paraelectric phase, the PS of PNDs prefers to orientate along the density distribution direction of the niobium ions (i.e., the y-direction), thus the crystal maintains non-polarized macroscopically. On the other hand, there exists a slight deformation in the local lattices of the polarized regions along the direction of Ps. Considering the polarization reorientation effect, a giant electrostrictive strain can be generated [26,27]. And it is reasonable to deduce that the polarization reorientation of PNDs results in the electrostrictive strain, and the magnitude of strain is related to the density of PNDs as well as the reorientation angle θ1, as exhibited in Fig. 4(e). Under the experimental conditions shown in Figs. 4(a) and 4(c), the electric field E = 100 V/mm which is large enough to reorient all the PNDs since their size is merely 20–100 nm [18]. As illustrated in Fig. 4(a), the direction of PS is approximately perpendicular to the external electric field. The reorientation of these PNDs can generate huge strain due to the rotation of the polar axes. While under the experimental condition shown in Fig. 4(c), the direction of PS for most PNDs is parallel to the external electric field. The reversion of 180° polarization does not change the deformation direction of the lattices. Therefore, no appreciable strain can be generated. In this case, the resultant strain gradient is prominent according to Eq. (2)–(4), but not enough to support an observable phenomenon of the beam deflection.
5. Conclusion
In summary, one-dimensional refractive index modulation can be induced by the interaction of the external electric field with the reorientation and density distribution of the dynamic PNDs. Based on this model, a unique cross-field beam deflector is successfully implemented using a Cu:KTN sample. The deflection angle can reach 50 mrad under a merely 100-V/mm electric field. In principle, this new beam-deflection scheme has no deflection angle limitation caused by the electrodes, unlike the conventional electro-optical scanning devices. It is feasible to develop high-speed miniature optical scanning devices with large field-of-view deflection and low driving power by simply reducing the crystal thickness or increasing the light propagation length in the crystal. As the crystal growth technique advances, more sophisticated controls on the dynamic PNRs/PNDs, such as three-dimensional refractive index modulations, can be achieved by using various KTNs or other alternatives of ferroelectric relaxors, to find more broad applications in electro-optics and optoelectronics.
Funding
National Natural Science Foundation of China (51975192, 61575097); Science and Technology Research Project of Hubei Provincial Department of Education (Q20202601).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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