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Abstract
Pulsed-biased higher speed (> 10 MHz @ 30 V, 100 ns pulse width) and DC-biased wider angle (105 mrads @ 410 V/mm) linear electro-optic (EO) deflection is reported in a thermally-controlled domain engineered (DE) ferroelectric (FE) potassium tantalate niobate [KTa1−xNbxO3, KTN] crystal. DE-FE KTN crystals can not only provide a higher transmittance and larger linear EO coefficient, but also enable higher speed (10X) and wider angle (2X) deflection than that of its paraelectric equivalent. The physical mechanism behind the optimization of injected space charge on high deflection angles at high speeds is also addressed. This significantly improves its use in megahertz EO applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Many applications such as sensing, printing, varifocal lens, laser material processing and optical coherence tomography using a swept light source (SS-OCT) require high speed and/or wide angle deflection [1–3]. For example, Higher OCT scanning speed within a SS-OCT system allows larger scan areas and faster acquisition times which lead to a reduction in motion artifacts that are caused by retinal, organ or bodily movement [2]. Motion artifacts often decrease the scan quality of in-vivo OCT images in ophthalmic or cardiologic applications. Higher speed deflection and lensing enables access to higher spatial resolution and reductions in total processing times in laser manufacturing applications [1]. EO KTN deflectors, due to their large deflection angle and high speed response, within an SS-OCT system have promoted the creation of high quality, high speed image rendering and data acquisition at a speeds greater than 100kHz [3–9]. Most endeavors have been focused on utilizing and improving their paraelectric (PE) characteristics since these characteristics are a result of their EO Kerr, permittivity and space-charge controlled characteristics [10–18]. However, these properties often coexist with negative paraelectric attributes such as: (1) the electric field-induced phase transition, which transitions the crystal into a multi-domain ferroelectric state and potentially distorts the beam quality; (2) the frequency and electric field dependent dielectric loss, both of which could limit the deflection angle and speed at high frequencies; and (3) the aperture size, which can affect the complexity of KTN-based systems.
Space charge plays a crucial role in the deflection capabilities of KTN deflectors. However, it has not been explored in ferroelectric (FE) KTN crystals due to its multi-domain ferroelectric state. Although a high transmittance and large linear EO coefficient were recently reported in KTN crystals via a thermally controlled domain engineered (DE) process by our team [19], there was a lack of discussion on how this thermally controlled DE process affects the deflection angle and speed of ferroelectric KTN single crystals. In this paper, a thorough investigation is conducted on the influence of the thermally controlled domain engineering process on the linear EO deflection angle and speed of KTN crystals. It is found that a higher speed (10X), wider angle (2X) deflection can be achieved by optimizing the domain engineering process, which also optimizes the injected space charge and polar linear EO coefficient. This presents a significant advancement in the capabilities of KTN deflector that could further increase the scanning speed of SS-OCT and other applications towards megahertz functionality and find use within new megahertz required applications [3].
2. Theoretical influence of space charge density in KTN deflectors
Space charge characteristics are accessible to all ferroelectrics whose side surfaces are coated with electrodes [4–8]. If the electrodes have a work function near the electron affinity of the ferroelectric, it creates an ohmic interface that causes electrons or space charge to gather at this interface. An applied electric field increases the space charge density near this interface which enables deflection in ferroelectric crystals with band structures similar to KTN [20]. If the electric field distribution is modeled in consideration of the electrons that are already stored within electron traps,
This equation shows that the deflection for a DE-FE KTN crystal is dependent on the linear EO coefficient, space charge density and the permittivity, but the extent that each parameter impacts the deflection characteristics is somewhat unclear given the complexity of this equation. A comparison between the ferroelectric and paraelectric band structure of KTN shows that: the space charge density of the ferroelectric phase should change in comparison to its paraelectric phase since the ferroelectric bandgap is different from its paraelectric equivalent [20,24]. Later sections use our experimental analysis and calculations to better clarify the importance of each parameter and expose other potential optimization methods that may further enhance EO linear deflection in DE-FE KTN deflectors.
3. Experiment procedure
Two rectangularly shaped KTN crystal samples (KTN1 and KTN2) were cut from a high quality KTN crystal ingot that had a composition of KTa${_{1-x}}$Nb${_{x}}$O${_{3}}$: $x$ = 0.389 and a Curie temperature ($T_C$) of 22 $^\circ$C. KTN1 and KTN2 had a dimension 8$\times$3$\times$5.5 mm$^3$ and 13.2$\times$5.4$\times$0.85 mm$^3$, respectively, which describe their dimensions across the a$_1{\times }$a$_2{\times }$c axis. Ti/Au electrodes were coated onto the onto the 8$\times$3 mm$^2$ and 13.2$\times$5.4 mm$^2$ side surfaces. The growth axis was assumed to be the c-axis and was along the z-direction. All the side surfaces were optically polished for both samples. KTN2 was used to measure the permittivity vs. temperature (15 $^\circ$C to 80 $^\circ$C) at frequencies 100 kHz to 50 MHz; the electric field induced permittivity and dielectric loss at 18 $^\circ$C, 16 $^\circ$C, 14 $^\circ$C, 27 $^\circ$C and 28 $^\circ$C; the deflection angle at 18 $^\circ$C to 4 $^\circ$C and deflection speed at 18 $^\circ$C.
For all the following experiments, the rapidly cooled DE-FE recipe is initially applied to the crystal. The recipe proceeds as follows: (1) heat the KTN crystal to $T_C$ + 58 $^\circ$C to initiate the reorientation of polar nanoregions (PNRs), (2) rapidly cool the crystal at a cooling rate of 1.67 $^\circ$C/s to $T_C$ + 8 $^\circ$C to quench the number and size of PNRs and slow cool the crystal at a rate of 0.0115 $^\circ$C/s to $T_C$ - 4 $^\circ$C to promote the growth of critical nucleation of these PNRs to form large-scale domains [19]. KTN1 was used for the XRD confirmation of the DE-FE transition and electric field induced Raman curve. The temperature dependent XRD spectra of KTN1 was collected by attaching the temperature controlling stage and KTN1 onto the MRD Cradle Sample Stage and using the PIXcel 3D and Xcelerator detector within the Malvern Panalytical XPert3 MRD System. Three LCR meters were used for these measurements: the Keysight U1733C (KUC), Keysight E4980A (KEA) and Agilent 4294A (A4A) with a frequency range of 100 Hz – 100 kHz, 10 Hz – 2 MHz, and 40 Hz – 110 MHz, respectively. The Agilent system was calibrated to a 50 pF capacitor and 50 $\Omega$ resistor. Electric field induced permittivity and dielectric loss were collected using a high voltage amplifier, DC blocking circuit and the KEA LCR meter. The beam image in response to an electric field was recorded with a camcorder and the electric induced Raman curve was collected with a Raman setup similarly described in Ref [19].
The deflection angle and speed experiment setup is shown in Figure S1. For the deflection angle measurements, a linearly polarized Helium Neon 632 nm laser propagates through and hits the crystal at a beam diameter of approximately 0.5 mm and hits a wall that is approximately 1333.5 mm away from the KTN2. A DC voltage is applied to the crystal across the a${_1}$-axis of KTN2. Using these experimental results and Eq. (5), the PE charge density is estimated and graphed as a function of the applied voltage. These results are used to estimate x which is the location of the propagation path of the laser with respect to the cathode of the KTN deflector, and x is used with the DE-FE deflection results to estimate the DE-FE charge density versus the voltage graph. For the deflection speed measurements, a collimated frequency doubled Nd:YAG 532 nm laser propagates through and hits the crystal at a beam diameter of approximately 0.5 mm and gets focused into a photodetector that outputs this detected signal to a TDS6604 oscilloscope. An Agilent 33120A function generator creates a 10 Vpp continuous sinusoidal wave that is applied to the KTN crystal at frequencies ranging from 10 Hz to 15 MHz. the pulsed response was measured by applying a 30 V, 100 ns pulse width signal with a 600 ps rise time from an Avtech AVL-3A-C pulse generator and recording its response using a HCA-S-200M-Si photodetector. The propagation path of the laser in all experiments were parallel to the growth axis ( c-axis) of the KTN sample. Both KTN samples in all experiments were covered with thermal tape and attached to a Peltier temperature controlling stage with a thermal temperature sensor.
4. Results & discussion
4.1 Temperature & frequency dependence of the dielectric loss & relative permittivity in KTN crystals
Figure 1(a) approximates the $T_C$ to be $22 \,^{\circ }\textrm {C}$ by observing the peak permittivity in the frequency dependent permittivity versus temperature curve. The permittivity stays at approximately the same value throughout the 100 – 50 MHz frequency range except within $10 \,^{\circ }\textrm {C}$ above the peak permittivity. At the peak value, the permittivity decreases from 23,000 at 10 MHz to 20,000 at 50 MHz for example. Figure 1(b) shows the frequency dependent dielectric loss versus temperature which could also be used to estimate the $T_C$ of the sample. The dielectric loss stays below 0.02 at all frequencies below 4 MHz. At and above 4 MHz, it starts to peak at $T_C$ followed by a quadratic decrease with increasing temperatures, a rapid linear decrease from $T_C$ to 19 $^{\circ }\textrm {C}$ and a slow linear decrease starting from 19 $^{\circ }\textrm {C}$ which is similar to the permittivity (Fig. 1(a)). The peak value as well as the values between $T_C$ to 30 $^{\circ }\textrm {C}$ continue to increase quadratically with frequency, reach 0.36 at 50 MHz, and seemingly start to stabilize towards 0.40. Above 30 $^{\circ }\textrm {C}$, both the dielectric loss and permittivity at all frequencies normalize to their respective values at 100 kHz. For example, the dielectric loss is less than 0.05 and the relative permittivity is 2,000 at 30 $^{\circ }\textrm {C}$ or $T_C\, + \,18 \,{\rm {^\circ C}}$. The frequency response of the dielectric loss is the same below 19 $^{\circ }\textrm {C}$ except it seems to stabilize towards 0.1.
Fig. 1. (a) Permittivity versus temperature (15 $^{\circ }\textrm {C}$ to 80 $^{\circ }\textrm {C}$) and (b) dielectric loss versus temperature (15 $^{\circ }\textrm {C}$ to 40 $^{\circ }\textrm {C}$) from 100 Hz to 50 MHz for KTN1.
A comparison between the ferroelectric (T ${\leq }$ 18 $^{\circ }\textrm {C}$) and paraelectric (T ${\geq }$ $T_C$) dielectric characteristics of the crystal reveals that although the paraelectric permittivity was always greater than the ferroelectric permittivity at all frequencies, the paraelectric dielectric loss displays characteristics that impact its deflection capabilities. Later sections also reveal that the ferroelectric deflection characteristics is less reliance on permittivity than their paraelectric equivalent. Below 2 MHz, the paraelectric dielectric loss was always less than the ferroelectric dielectric loss. Above 4 MHz, a paraelectric temperature range that has a greater dielectric loss than its ferroelectric equivalent is created and this temperature range increases as a function of frequency. For example, the paraelectric dielectric loss between 22 $^{\circ }\textrm {C}$ and 28 $^{\circ }\textrm {C}$ is greater than the ferroelectric dielectric loss below 18 $^{\circ }\textrm {C}$ at 20 MHz. These frequency characteristics limit the permittivity which also can limit the deflection angle.
4.2 Temperature analysis of the XRD spectra of a transparent DE-FE KTN single crystals
Figure 2 shows an illustration of the XRD analysis of the sample and the measured XRD spectra as a function of temperature for KTN1. During the XRD analysis, it was discovered that the c-axis of the crystal is not perfectly parallel to its growth axis. In response, the MRD Cradle sample required a chi rotation of 0.011 degrees for the x-ray beam to be parallel to the c-axis. During the deflection test experiment and beam profile images, it was also found that a similar rotation is required to reveal the high-quality beam profile of the DE-FE KTN phase. A $T_C$ of 22 $^{\circ }\textrm {C}$ can also be approximated with Fig. 2(b) by observing a change and shift between the (002) and (200) peaks at the 2$\theta$ values of 45.2 to 45.6 respectively. These peaks are often used for the XRD study of the domain orientation of a tetragonal structure [25–28]. The c-domains and a-domains are known to be responsible for the (002) and (200) peaks in the XRD spectra, respectively. The intensities of these peaks can also be correlated with the populations of the c- and a-domains of KTN1 [26–28].
From Fig. 2(b), the (002) peak decreases as the temperature decreases due to the thermally controlled domain engineered recipe. From T = 23 $^{\circ }\textrm {C}$ to T = 18 $^{\circ }\textrm {C}$, (200) peak starts to increase while the (002) continues to decrease until the (200) peak intensity becomes greater than the (002) peak intensity. In other words, the a-domain peak becomes larger than the c-domain peak at 18 $^{\circ }\textrm {C}$. This shows that the transparent DE-FE KTN crystal may be in an a-domain dominant ferroelectric state. The XRD plot of Fig. 2(c) shows the tetragonal phase transition results with and without the thermally controlled domain engineered recipe at $T_C\, - \,4 \,{\rm {^\circ C}}$ (18 $^{\circ }\textrm {C}$) represented by 18 $^{\circ }\textrm {C}\_$non-DE and 18 $^{\circ }\textrm {C}\_$DE, respectively. A comparison of these two tetragonal phased results shows that the non-DE state was in a c-domain dominate state while the DE state was in a-domain dominant state.
Fig. 2. (a) An Illustration of the XRD setup, (b) XRD Spectra of DE KTN1 versus temperature (ranging from 18 $^{\circ }\textrm {C}$ to 70 $^{\circ }\textrm {C}$ ) (c) XRD of KTN1 at a multidomain (18 $^{\circ }\textrm {C}\_$non-DE) and single (18 $^{\circ }\textrm {C}\_$DE) domain state for KTN1.
4.3 Electric field induced properties of transparent DE-FE KTN vs. PE KTN
Figure 3 shows the electric field induced permittivity of KTN2 in its paraelectric state ( 27 $^{\circ }\textrm {C}$ and 28 $^{\circ }\textrm {C}$ and transparent ferroelectric state (18 $^{\circ }\textrm {C}$, 16 $^{\circ }\textrm {C}$ and 14 $^{\circ }\textrm {C}$). These temperatures are chosen because the maximum deflection properties are often found within this temperature range. The permittivity of KTN2 in its paraelectric state (Fig. 3(b)) shows the typical 3 stages of electric field response: (1) a small and slight increase, (2) a rapid increase and (3) an inflection point with a rapid decrease [18,29,30]. The dielectric loss versus electric field curve (Fig. 3(d)) also follows a similar pattern as the permittivity versus electric field curve (Fig. 3(b)). It should also be noted that the dielectric loss of the paraelectric state of KTN2 always stays below 4%. However, if the combination of its electric field and high frequency response are considered, this can further limit the paraelectric deflection characteristics of KTN crystals. As the temperature approaches $T_C$, the initial permittivity ($\epsilon _{r0}$), the inflection point ($\epsilon _{r-IP}$) and electric field-dependent rate of change ($\pm \epsilon _{r}/E$) increase and the electric field of the inflection point ($E_{IP}$) decrease. Figure 3(b) and 3(d) also confirm that an electric field-induced phase transition occurs at the inflection point of both curves [18,29,30].
Fig. 3. (a,c) Permittivity and (b,d) dielectric loss versus electric field of KTN2 at ferroelectric phase (18 $^{\circ }\textrm {C}$, 16 $^{\circ }\textrm {C}$ and 14 $^{\circ }\textrm {C}$) and paraelectric phase (27 $^{\circ }\textrm {C}$ and 26 $^{\circ }\textrm {C}$).
The ferroelectric electric field dependent permittivity (Fig. 3(a)) and dielectric loss (Fig. 3(c)) displays no signs of a phase transition which was further supported by the Raman analysis in Fig. 4. The ferroelectric response displays a small but linear decrease in the electric field induced permittivity that starts to become purely linear as we decrease the temperature to 14 $^{\circ }\textrm {C}$ or $T_C\, - \,8 \,{\rm {^\circ C}}$ (Fig. 3(a)). The ferroelectric electric field dependent dielectric loss rapidly peaks at 100 V/mm but decreases and saturate to a dielectric loss lower than the peak (Fig. 3(c)). This saturation level, seen within Fig. 3(c) at 350 V/mm, seem to increase as a function of temperature. However, since the ferroelectric deflection angle may be less dependent on the dielectric response, this may not be a great concern.
Considering the electric field, temperature and frequency response of permittivity, we can observe the paraelectric deflection characteristics of KTN crystals to be greatly hindered at frequencies greater than 4 MHz due to the increased dielectric loss as a function of frequency, temperature and electric field and the electric field-induced phase transition. Besides the electric field-induced phase transition, a large permittivity also inhibits the deflection speed of the crystal due to its large initial value (10,000+) and this value being further enhanced by its electric field induced response at specific temperatures. Figure 4 further confirms the transparent DE-FE transition and the elimination of electric field-induced phase transition by observing the beam images and electric field induced Raman curves. The peak position and structure of the Raman spectra reveals that the DE-FE phase of KTN2 is in a ferroelectric state (Fig. 4). As the applied electric field increases from 0 to 470 V/mm, the peak intensity and position of the Raman spectra does not change significantly [18,31–34]. This implies that the applied electric field and its induced strain may not affect the tetragonal crystal structure of KTN2.
4.4 Deflection beam image, angle and speed of transparent DE-FE KTN vs PE KTN
The beam profile images of KTN2 shows the typical beam deformation with increasing applied electric fields (Fig. 5). The DE-FE beam maintains their profile from 0 to 106 V/mm, starts slightly deforming at 118 to 165 V/mm then starts to fade and deform above 176 V/mm. The DE-FE beam image maintain its profile from 0 to 341 V/mm, slightly deform at 353 to 400 V/mm and gets stretched at 412 V/mm (Fig. 5(a)). A comparison of the PE and DE-FE beam images shows that the DE-FE beam profile is larger than the PE beam and maintains its shape until 412 V/mm. The PE beam shows the typical beam deformation that gets stretched horizontally with the applied electric field starting at 176 V/mm (Fig. 5(b)).
Fig. 5. Beam profile images verse electric field of the (a) DE-FE transparent KTN2 at 18 $^{\circ }\textrm {C}$ and the (b) PE KTN2 at 27 $^{\circ }\textrm {C}$.
The experimentally measured deflection angle and its respective calculated charge density were shown in Fig. 6. A maximum deflection angle of 105 mrads at an applied electric field of 410 V/mm is seen at 18 $^{\circ }\textrm {C}$ or $T_C\, - \,4 \,{\rm {^\circ C}}$ whereas a maximum deflection angle of 50 mrads at an applied electric field of 245 V/mm was seen at 27 $^{\circ }\textrm {C}$ or $T_C\, + \,5 \,{^\circ \rm {C}}$ (Fig. 6(a) and 6(c)). Figure 6(c) shows that due to the electric field-induced phase transition of the PE KTN2, the maximum deflection angle is limited to 50 mrads. The maximum deflection angle of the transparent DE-FE phase of KTN2 at 18 $^{\circ }\textrm {C}$ was two times greater than that of its PE phase equivalent. PE deflection outperforms the deflection range if the DE-FE deflector operates at 14 $^{\circ }\textrm {C}$ and below.
Fig. 6. Experimentally measured deflection angle versus electric field and its respective calculated charge density versus voltage curve under an assumed distance x from the cathode in the (a,b) transparent DE-FE phase and (c,d) PE phase.
The PE charge density curve shows that although we aligned the beam to be as close to the cathode as possible, the calculated charge density most closely resembles those calculated in previous reports at a distance of x = 0.1d where d is the gap between electrodes of the KTN deflector as shown in Figure S1 [35–37]. For example, the charge density at 100 V is approximately −75 C/m${^{3}}$ and it has been reported to be as high as −50 C/m${^{3}}$ at the same voltage. It can also be seen that the beam image profile gets distorted at Voltages above 150 V (176 V/mm) which is a space charge density of −140 C/m${^{3}}$. Due to this, the charge density of the DE-FE KTN deflector was calculated with respect to a x = 0.1d. Figure 6(d) shows that the maximum charge density for x = 0.1d is approximately −170 C/m$^3$ at 200 V for the PE deflector. Figure 6(b) shows that the charge density decreases with temperature from 18 $^{\circ }\textrm {C}$ to 12 $^{\circ }\textrm {C}$ but starts to increase as the temperature decreases from 12 $^{\circ }\textrm {C}$ to 4 $^{\circ }\textrm {C}$ for the DE-FE deflector. This transition point will be further discussed in a later section.
Figure 7 shows the frequency response of transparent DE-FE KTN2 deflector. The DE-FE KTN2 can response to a continuously applied sinusodial waveform throughout the 10 Hz to 15 MHz range. The deflection angle range response seems to fluctuate between the 10 Hz to 2 MHz which could be due to the frequency dependant efficiency of EO modulation property of KTN crystals [38]. However, a decreased deflection range is initiated at 4 MHz and continues to decrease towards 15 MHz. This could be due to the decreased influence of space charge polarization mechanism as a function of frequency. Thus, the frequency response above 4 MHz may be more dependent on other polarization mechanisms like dipolar (orientational) polarization since this polarization mechanism can appear at frequencies less than 10 GHz. The implications of these polarization mechanisms will also be further discussed in Section 4.5. In response to a 30 V, 100 ns width pulsed signal with a 600 ps rise time, DE-FE KTN2 displays a maximum rise time of 20 MHz due to the RC time constant characteristics of the pulse generator-KTN system (Fig. 7(b)). The capacitance of the transparent DE-FE KTN2 was approximately 1.5 nF and the impedance of the pulse generator was 50 ${\Omega }$. This results in a rise time of 75 ns which is equivalent to 13 MHz. If the electric field response and 600 ps rise time of the pulse generator were considered, the dielectric loss may approach 0.1 which would decrease the permittivity and capacitance; therefore, it may cause a maximum response time of 20 MHz (50 ns) from the transparent DE-FE KTN2 sample at an applied voltage of 30 V without a significant loss in the deflection range that is often seen in PE KTN deflectors. The results suggest that the DE-FE deflector can always surpass the PE deflector in deflection range; and better maintain its deflection characteristics at high frequencies as adjustments were made to the operating temperature of the DE-FE deflector. Since the permittivity of PE KTN2 was approximately 10 times greater than transparent DE-FE KTN2, paraelectric continuous deflection speed is often limited to 100 kHz to 2 MHz due to its large dielectric loss [39–42]. Transparent DE-FE KTN2 can reach a continuous deflection speed of 15 MHz since it was less reliant on space charge polarization mechanisms than its paraelectric equivalent (Fig. 7(a)).
Fig. 7. (a) The photodetected deflection waveform of KTN2 at a temperature of 18 $^{\circ }\textrm {C}$ at frequencies of 10 Hz to 15 MHz and (b) its 100 ns pulsed response.
4.5 Space-charged-optimization and linear EO analysis within the transparent DE-FE KTN deflectors
When an electric field is applied to a ferroelectric crystal, the total polarization generated is the sum of four sources of polarization mechanisms: electronic, ionic, dipole( orientational) and interfacial( space charge). Interfacial ( space charge ) polarization is created due to the accumulation of charges within the crystal at the metal-crystal interface and typically occurs at all frequencies less than 10 kHz. Dipolar (orientational) polarization emerges when dipole moments within a crystal rotate themselves upon an applied electric field. These only appear at frequencies less than 10 GHz. And the remaining polarization mechanisms ( electronic and ionic) occur at frequencies far above 10 GHz and therefore may always be existent within our frequency response test (Fig. 6). The permittivity always decreases with increased frequencies since certain polarization mechanisms become less effective at higher frequencies [43,44].
Within the DC to 50 MHz frequency scale, the frequency response of KTN deflectors seem to depend mostly on space charge and dipolar polarization mechanisms since an applied electric field results in a space charge density and a reorientation of dipole moments. Both mechanisms are shown by the various space charge density measurements; and the change in permittivity versus temperature and applied electric field [15,18,23,35–37,39–41]. A PE KTN deflector near $T_C$ is shown to be quadratically and linearly dependent on its space charge density and permittivity, respectively; and their values are on the order of 200 C/m$^3$ and 18,000 respectively. Its space charge characteristics may be due to its band structure characteristics while its permittivity characteristics may greatly depend on its randomly oriented dipoles which are known to get enhanced at varying cooling rates [20,23,45–48]. Although its electric field-induced phase transition mechanism limits the PE deflection range of KTN crystals within the DC to 50 MHz frequency scale, it may be that both its quadratic dependence on space charge and large dielectric losses above 2 MHz creates a large decrease in deflection capabilities since their space charge polarization mechanisms become less effective at high frequencies [42,43].
Creating a DE-FE KTN crystal near $T_C$ orients its dipole moments along a single direction which can be shown by a large decrease in permittivity (from approximately 18,000 to 3,000) and a linear dependence on space charge density as shown by Eq. (5). This decreased dependance on space charge density within DE-FE KTN crystals seem to enable stronger deflection responses at high frequencies (20 MHz in this paper) than its paraelectric equivalent since dipolar polarization mechanism are allowed to be more dominant in this DE-FE state. This can be further optimized by harnessing its resonant frequency characteristics; selecting a temperature that has the lowest space charge density, but the largest deflection; and/or modifying the band structure characteristics which may enable access to higher frequencies [38]. It should be noted that these characteristics are also applicable when comparing the deflection angle of the DE-FE KTN to its PE equivalent. PE relies on not only the EO Kerr effect but also the space charge density, permittivity and any associated characteristics of those properties such as the electric field induced phase transition which will limit the maximum applicable electric field. The relationship between the DE-FE KTN and its properties are more complex.
To observe this more clearly using Eq. (5), if we assume that $-\rho /{{\varepsilon }_{0}}{{\varepsilon }_{r}}\left ( x-d/2 \right )-V/d=-\rho /{{\varepsilon }_{0}}{{\varepsilon }_{r}}$; ${\left ( 1/n_{e}^{2}-1/n_{o}^{2} \right )}^{2}$ and $1/n_{o}^{2}+\,1/n_{e}^{2}$ are negligible;
This equation shows that the charge density enables deflection to exist but decreases the deflection angle as it increases. The linear electro optic coefficient causes the defection angle to increase if $\tau _{51}$ is between 0 < $\tau _{51}$ < 1 and approaches zero. Once $\tau _{51}$ > 1, it starts to cause a decrease in the deflection angle while the permittivity always enables an increased deflection angle. Considering this and the ability for DE-FE KTN cryrstals to have a higher maximum applicable electric field allows DE-FE KTN deflectors to display a larger deflection angle than PE KTN deflectors. Equation (6) shows that although DE-FE deflection requires space charge density, a small space charge density accompanied with a small linear electro optic effect will enable a large deflection angle as long as the permittivity is greater than or equal to 10$^3$. In consideration of this and the dielectric dispersion studies of PE KTN crystals, it may be possible that the DE-FE KTN crystals can reach the GHz range [49].
5. Conclusion
In addition to a high transmittance and large EO coefficient, a domain engineered ferroelectric transparent KTN crystal at 18 $^{\circ }\textrm {C}$ enables higher (10X) speed (15 MHz continuous with a 10 Vpp Sine signal and 20 MHz with a 30 Volt 100 ns pulsed signal), wider (2X) angle (105 mrads at a DC-Biased signal of 410 V/mm ) linear EO beam deflection than its paraelectric equivalent. Since space charge can also be trapped in domain engineered ferroelectric transparent KTN crystals, these crystals show that the injected space charge and polar linear EO coefficient can be optimized using the thermally controlled domain engineering process to enable access to higher deflection speeds and wider deflection angles. This also concludes that the combined effects of space charge density, polar electro optic coefficient and permittivity play a significant role in improving electro-optic KTN based deflection applications.
Funding
Office of Naval Research (N00014-17-1-2571).
Acknowledgments
This research was sponsored and partially supported by the Office of Naval Research (ONR) under Grant Number N00014-17-1-2571. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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