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Abstract
Liquid crystal (LC) devices for terahertz phase shifters inevitably use a thick cell gap for the required retardation, severely delaying the LC response. To improve the response, we virtually demonstrate novel LC switching between in-plane and out-of-plane for reversible switching between three orthogonal orientation states, broadening the range of continuous phase shifts. This LC switching is realized using a pair of substrates, each with two pairs of orthogonal finger-type electrodes and one grating-type electrode for in- and out-of-plane switching. An applied voltage generates an electric field that drives each switching process between the three distinct orientation states, enabling a rapid response.
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1. Introduction
Terahertz (THz) technology requires further advances for next-generation wireless communication systems beyond 5G [1,2]. Accordingly, THz modulation technology has become indispensable in various applications [3,4]. Liquid crystals (LCs) exhibit refractive index anisotropy even in the THz band [5,6] and can be applied in various devices that control THz waves as quasi-optical components, e.g., phase shifters [7–16] and gratings [17–19], polarizers [20–22], and filters [23,24]. Tunability is key to such device components, and in that regard, LCs are promising as they can be tuned using external fields at room temperature. Indeed, some LC phase shifters have been successfully demonstrated in the THz frequency range [14–16,25–29].
However, unlike displays in the optical frequency region, a large cell gap of hundreds of micrometers or more is required to satisfy the retardation necessary for sufficient phase modulation at THz frequencies. Therefore, an extremely slow response is inevitable for this type of unusually thick LC device. Switching under an external field shows relatively fast responses on timescales of seconds or fractions of a second, but switching without an external field can require tens or hundreds of seconds or more, which is a serious drawback for LC-based THz devices. Recent research aimed at addressing the issue of this very slow response has demonstrated that applying an electric field to each switching process can effectively solve this problem [30–33].
Our previous study [32] unveiled the dimensional effects of “THz in-plane and THz out-of-plane (TIP-TOP)” switching [30,31], wherein the electrodes mirroring each other on a pair of parallel substrates were composed of finger-type electrodes that produced in-plane (i.e., parallel to the substrate) electric fields, whereas the grating-type ones produced an out-of-plane (i.e., vertical to the substrate) electric fields. Regarding the dimensional effects, varying the dimensions of these electrodes was found to influence the LC in-plane states with the corresponding phase shifts. Interestingly, we found that these significant dimensional effects of the in-plane electrode structures statically and dynamically influence the phase shift and response time in LC switching. These effects become even more remarkable when the electric field from the lateral bus-line electrodes is large. While identifying the mechanism underlying the dimensional effects, we suggested the possibility of novel LC switching by exploiting these dimensional effects and manipulating the continuous reversible switching between the three LC orientation states [32,34].
In this article, we propose novel LC switching between in-plane and out-of-plane for THz phase shifters, which enables continuous reversible switching between three states with a rapid response [32,34]. This is because all three stable states are governed by an electric field. We statically and dynamically demonstrate this new type of switching by showing changes in the phase shifts and dynamic responses for each type of switching. In principle, the ability to reorient LC directors utilizes the three orthogonal orientation states in space, i.e., the azimuthal and polar angles of 90°, which potentially enables exhibiting a theoretical maximum range of continuous phase shifts of an LC.
We initially named this new type of switching “hexadirectional tristable switching,” but the words “hexadirectional” and “tristable” may imply six stable states and three quasi-states at a certain voltage, respectively. “Triangular” or “trigonal switching” was also a possibility owing to the three-pointed shape of the entire switching scheme, particularly as some other types of three-state switching schemes involve an intermediate state between two other states and thus do not enable bidirectional switching between any three states. However, to simplify the name of this type of switching using commonly recognized terms such that most people will easily understand the switching scheme, we refer to it simply as “three-state switching,” “three-state reversible switching,” or “reversible switching between three states.” To be clear, this type of switching consists of bidirectional switching between three pairs of states, and the fact that this switching is reversible in all directions enables a total of six transitions between three independent states under the influence of electric fields.
2. Electrode layout
Figure 1(a) shows the layout of the electrodes on the surface of a substrate in a unit cell, which provides the periodic boundary conditions for the $x$- and $y$-axis directions. The unit cell consists of four pixels, each of which includes two orthogonally arranged pairs of finger-type electrodes, with corresponding pairs colored red and blue as a visual aid in the figure, and one grating-type electrode at the center of each pixel. The former enables in-plane switching [35] along the two orthogonal $x$- and $y$-axis directions, while the latter forms pairs of electrodes with each opposing grating-type electrode on the counter substrate. Each pair effectuates out-of-plane switching. Therefore, electric fields can be independently applied in the $x$-, $y$-, and $z$-axis directions, and accordingly, LC directors can reorient along these three directions, suggesting that the reorientation of LC directors can fully utilize space in three dimensions. Practically, the electrodes that have the same function can be connected throughout all the pixels by placing each pair of finger-type electrodes for in-plane switching and the grating-type electrodes for out-of-plane switching in different layers through an insulation layer [34].
Fig. 1. Schematics showing the geometry of a unit cell used in our calculations. (a) Layout of electrodes on each substrate in a periodic unit structure. (b) Unit cell structure.
As the geometry of an LC cell, two substrates with identical electrode layouts are superimposed on the inner surfaces of the top and bottom substrates and separated by a gap filled with LCs, as shown in Figure 1(b). The gap d can be designed for the required phase shift $\Delta \phi $, which is expressed as $\Delta \phi = 2\pi \cdot \Delta n\cdot d/\lambda $, where $\Delta n$ denotes the birefringence of an LC, and $\lambda $ is the wavelength of a THz wave. For THz waves, $\lambda $ is approximately 50‒500 times larger than that for the waves in the optical region. Inevitably, the retardation $\Delta n\cdot d$ should be increased to modulate THz waves; assuming $\Delta n\sim 0.2$ at 2 THz and $d\sim 100$ μm, we obtain $\Delta \phi \sim 48^\circ $. The larger the expected modulation for THz waves is, the thicker the required d should be, although d varies with $\Delta n$. The biggest problem in this case is that the relaxation time ${\tau _{\textrm{off}}}$ of LCs when removing electric fields becomes extremely long because ${\tau _{\textrm{off}}} \propto {d^2}$ [36].
3. Reversible switching between three states
To remarkably improve upon this issue with LCs for use in THz phase shifters, we innovate electrode structures within a cell configuration by reversibly switching between all three states using electric fields. Figure 2 shows how voltages are applied to the electrodes and how the induced electric field reorients the average of the LC directors in the LC layer. In turn, each LC orientation produces a corresponding phase and is combined with a pair of crossed polarizers that sandwich the LC cell, with one polarization axis being $45^\circ $ to the x-axis to attain the maximum modulation. Initially, in the absence of any electric fields, the LC directors are aligned along the $x$-axis or 45° from the $x$-axis. Alternately applying electric fields in three orthogonal directions enables three-state switching the LCs between in-plane and out-of-plane orientations, which can be operated by applying two in-plane electric fields mutually perpendicular to each other and one out-of-plane counterpart.
Fig. 2. Scheme of reversible switching between three states: (a) in-plane x (IN(X)), (b) in-plane y (IN(Y)), and (c) out-of-plane (OUT) of LC orientations. The side and top views are vertical and horizontal cross-sections taken along the dashed and dashed-dotted lines, respectively, in each illustration of the LC pixel. The IN(X) and IN(Y) states are activated by pairs of parallel electrodes (colored blue and red) for activating in-plane states oriented along the x- and y-axes, respectively. OUT is the out-of-plane state activated by the grating-type electrode (yellow). The pairs of electrodes are hereafter referred to by the state that they produce, e.g., the OUT electrodes.
Figure 2 schematically illustrates how an applied voltage can reversibly orient the LCs in each pixel in each of the three states. In Figure 2(a), an external voltage source is connected to two pairs of finger-type electrodes, which vertically mirror each other, allowing in-plane switching in the same direction on the upper and lower substrates. The in-plane electric fields generated on the upper and lower substrates reorient the average LC directors parallel to the substrate along the $x$-axis, except for some deformed structures caused by nonuniformity in the electric fields around the electrodes. Similarly, in Figure 2(b), the in-plane electric fields generated along the $y$-axis on the upper and lower substrates enable aligning the LC directors in the same average direction. In Figure 2(c), however, an external voltage is supplied to the upper and lower grating-type electrodes. The potential difference between these electrodes generates an out-of-plane electric field largely perpendicular to the substrates. The generated out-of-plane electric field reorients the LCs along the z-axis. In principle, these three states can be selectively induced by applying an electric field along each axis direction using each pair of electrodes, which enables reversible switching between the three states.
4. Static behavior
To virtually demonstrate the static and dynamic behavior of this three-state reversible LC switching, changes in the phase shifts and LC reorientations were calculated using modeling and evaluation software for liquid crystal display (LCD) designers, LCDMaster 3D [37]. The calculation principles and procedures are described in detail in our previous article [32]. The cell gap for this three-state reversible switching was set to 120 µm and assumed to be filled with an LC (RDP-94990, Dainippon Ink & Chemicals Corporation, Tokyo, Japan). This material has refractive indices of ${n_\parallel } = 1.77,\; {n_ \bot } = 1.57$ at $550$ nm and has demonstrated a relatively high birefringence of $\mathrm{\Delta }{n_{THz}} = 0.2$ at THz frequencies [38]. Its other physical properties are described in our previous report [32].
Figure 3 shows potentials in the x–z (vertical) and x–y (horizontal) cross-sections of the LC cell when an external voltage of ±100 V is alternately applied to a group of electrode pairs, along with the resulting new orientations of the LC directors, statically demonstrating the three-state reversible switching. Without an electric field, the initial alignment is 45° from the $x$-axis. The geometry of the cell including the $x$-, $y$-, and $z$-axes is the same as that in Figure 1. When the voltage is applied to the out-of-plane electrodes OUT, the potential difference between the electrodes on the two substrates induces vertical electric fields, which reorient the LCs along these fields, as shown in Figure 3(a) and (b). Pairs of electrodes IN(X) are used for in-plane switching, and the potential is applied to those electrodes. Accordingly, the LC directors switch to reorient along the electric fields induced by the potential difference, which can be observed from the x–z and x–y cross-sections in Figure 3(c) and (d). Further, the in-plane fields orthogonal to the previous ones are induced by pairs of electrodes IN(Y), thereby enabling switching the LC directors in-plane, as shown in Figure 3(e) and (f). In these two types of in-plane switching, the potential applied to the mirrored electrodes between the superimposed substrates has the same sign, and therefore, the LC directors exhibit distorted alignment and inhomogeneity between the electrodes with the same potential on the substrates. Transitioning between these three states can be reversed, thereby enabling a total of six transitions for three-state reversible switching between IN(X), IN(Y), and OUT states.
Fig. 3. Potentials (color scale) and LC directors (white bars and dots) in each of the three states (attained by alternately applying an external voltage of ±100 V to pairs of electrodes: (a)(b) OUT, (c)(d) IN(X), and (e)(f) IN(Y). (a)(c)(e) show the x–z (vertical) cross-section at $y = 50$ μm of the LC unit cell, and (b)(d)(f) show the x–y (horizontal) cross-section. The x–z and x–y cross-sections correspond to AA′BB′ and CC′DD′ in Figure 1(b), respectively.
Combining the LC cell with a pair of crossed polarizers with one polarization axis being 45° from the $x$-axis produces three different states, each of which provides a different phase at 2 THz. Thus, transitioning from one state to another yields a phase shift. Table 1 summarizes the phases for each state calculated throughout the entire space and along the x–z horizontal cross-section at $y = 50$ μm. The difference in the cross-sectional and spatial means indicates the spatial inhomogeneity within the cell. The phase of the initial state is zero because the LC alignment corresponds to the polarization axis, and thus, the transition from the initial state to OUT does not yield any phase shift. The maximum possible phase for the case of $d = 120$ μm and hence $\Delta n\cdot d = 24$ μm at 2 THz should be $\Delta \phi \sim 57.6^\circ $; however, all the absolute values in Table 1 are smaller than this value. The maximum mean phase shift throughout the entire space extracted from the three-state reversible switching is ${\sim} 80^\circ $, which can further be enlarged using larger d and $\Delta n$.
Table 1. Phases of each state at 2 THz.
5. Dynamic behavior
The dynamic responses of the three-state switching are also analyzed by probing changes in the phase of each state when reversibly switching the electrode pairs to which the potential is applied. Figure 4 compares the dynamic changes in the phase at 2 THz from one state to another during the reversible switching. Specifically, a voltage of 100 V was applied between a certain group of electrode pairs for 1 s, followed by a voltage of 100 V between another group of electrode pairs; such transitions are referred to as OUT-to-IN(X), IN(X)-to-OUT, IN(X)-to-IN(Y), and so on. The equilibrated orientations of the LC directors were calculated every 20 µs, from which the phase through the LC cell was deduced. The other calculation conditions were the same as those for the static cases.
Fig. 4. Dynamic phase changes calculated at 2 THz when reversibly switching between three states by applying 100 V to various pairs of electrodes: transitions (a) between out-of-plane and in-plane x (OUT-to-IN(X) and vice versa), (b) between in-plane x and y (IN(X)-to-IN(Y) and vice versa), and (c) between out-of-plane and in-plane y (OUT-to-IN(Y) and vice versa). Each graph includes the mean phase changes in the $x$–$z$ cross-section at $y = 50\;\mathrm{\mu} \textrm{m}$ and those of the entire unit cell space. For the first 1000 ms, 100 V was applied to a certain group of electrode pairs, and then, 100 V was applied to a different group for 1000 ms.
Significantly, all the responses are rapid, with times approaching those required for practical use. This finding is expected because all the switching processes are driven under the influence of electric fields, and theoretically, the response time to an electric field ${\tau _{\textrm{on}}}$ is inversely proportional to the square of the electric field E, i.e., ${\tau _{\textrm{on}}} \propto 1/{E^2}$ [36]. When each voltage signal is applied to induce a transition, every response profile almost levels off within at most 300 ms. More precisely, each response time, defined as the time required to change the phase by 90%, is written along the arrows indicating the transitions in Figure 4. The response time of the transitions to the out-of-plane state generally falls with $15\; \textrm{ms}$, whereas those to the two in-plane states are ∼$160\; \textrm{ms}$, which is approximately ten to eleven times longer than the former. Considering the cell gap is several tens to hundreds of micrometers, which is inevitable for THz phase shifters, we believe that the most effective switching method is indeed to drive all the processes to LC reorientations under the influence of electric fields.
These theoretical calculations statically and dynamically verify LC three-state reversible switching, which is capable of rapid responses and a broad range of phase shifts. Comparing some phases of the states attained by the static and dynamic analyses, however, we note that some resulting phases calculated when reversibly switching from one state to another deviate slightly from those statically calculated. In other words, the leveled-off phases at IN(X) and IN(Y) in the dynamic analysis indicate significantly different values from those obtained in the static analysis. The deviations in the resulting phase values between the static and dynamic analyses are found to be due to the different initial alignment conditions. The static analysis employs the initial alignment states, wherein the pre-tilt and pre-twist angles of the upper and lower interfaces are linearly interpolated throughout the bulk. By contrast, in the dynamic analysis, the initial alignment states correspond to the state before switching. From a practical perspective, the phase shifts calculated by the dynamic analysis may be more plausible, because the initial alignment states are close to those in real switching.
6. Conclusion
We have successfully demonstrated the basic principles of a novel type of switching using LCs for THz phase shifters, which enables a potentially wider range of phase shifts with rapid responses. This new type of LC switching is achieved by one group of electrode pairs for out-of-plane switching and two groups of electrode pairs for in-plane switching, allowing reversible transitions between three LC orientation states, all of which are driven by applying electric fields. In principle, making full use of LC orientational changes in three dimensions, together with optimizing the cell gap and the birefringence of the LC, further broadens the range of phase shifts while maintaining rapid responses. Although this advantageous basic performance is observed virtually, some other factors should be more carefully investigated. For example, the operating voltage, the value of which was simply selected based on previous studies [31,32] was not optimized in this investigation. A more systematic analysis would identify physical mechanisms and influential factors for manipulating the continuous reversible switching between the three LC orientation states: two in-plane states and one out-of-plane state. Nevertheless, further efforts should be devoted to demonstrating the feasibility of this type of switching and to realizing an actual device; to that end, a prototype is currently being prepared.
Funding
National Science and Technology Council (NSTC), formerly known as the Ministry of Science and Technology (MOST), Taiwan (108-2622-M-007-006-CC1, 109-2622-M-007-007-CC1, 110-2221-E-007-092, 111-2221-E-007-023).
Acknowledgments
This study was also funded by Profound Material Technology Co., Ltd., Taiwan via Grants 108-2622-M-007-006-CC1 and 109-2622-M-007-007-CC1 listed above for industrial–academic collaborations with the NSTC. We acknowledge the technical support of LCDMaster 3D by Shintech, Inc., in Japan.
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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