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We report fast electro-optic response independent of cell thickness in cholesteric liquid crystals (ChLCs). Usually, an electric field normal to the helix axis of ChLCs induces two fast and one slow response components: helical elongation (slow one), helical deformation (fast one), and flexoelectric effect (fast one). In this study, we found that a planarly aligned ChLC applied with an in-plane electric field exhibited only fast response components because the glass substrates suppressed the motion of the helical elongation. Furthermore, we demonstrated complete separation of the remaining fast components by using dielectric measurement system. Consequently, we were able to analyze dynamics of the helical deformation in detail, in which response times exhibited quadratic dependence on the helical pitch and no dependence on the cell thickness.
© 2017 Optical Society of America
1. Introduction
Liquid crystals (LCs) exhibits high responsiveness to external stimuli such as heat, light, electricity, and stress because of the nature of soft materials both with flowability and molecular order, and thus have been used as electro-optical materials that demonstrate a large shift of refractive index and low-voltage driving. The intrinsic characteristics is advantageous over display application requiring thin and low-power devices [1]. On the other hand, when considering device applications requiring thick LC layer, LC devices have a serious drawback in response speed.
Recently, thick nematic LC (NLC) devices with a cell gap greater than 100 μm have been studied for electrically controlling a phase of electromagnetic wave such as millimeter or terahertz waves [2–5]. However, scaling up of the NLC devices produces a serious problem in response times of the electro-optic effect because the decay response time of NLCs is described as [6]
where γ is a viscosity coefficient, d is a cell thickness, and K is an effective elastic coefficient. For example [3], a nematic LC (NLC) device with a thickness of 570 μm demonstrated a phase shift of 90° for electromagnetic wave with a frequency of 1 Thz, whereas the switching time showed 190 s, which is disadvantageous in the practical use. To overcome the problem, LC devices with dynamics independent of cell thickness are greatly desired. Here, we focus on dynamics of cholesteric LCs (ChLCs).ChLCs are known to self-organize into one-dimensional helical structures with periodicities ranging from 100 nm to 10 μm order by controlling chirality [7]. When ChLCs with positive dielectric anisotropy are applied with an electric field normal to the helical axis, whose geometry can be obtained in uniform lying helix (ULH) [8] shown in Fig. 1(a), the ChLCs exhibit multiple modes on deformation of the helical structure [9–11]. We can describe the free energy of the LC as follows [10]:
where K11, K22, and K33 are the elastic constants of splay, twist, and bend deformation, Δε is the dielectric anisotropy, and e1 and e3 are the splay and bend flexoelectric coefficients, respectively. The first three terms are the elastic free energies related to splay, twist, and bend deformation, respectively, the fourth term is the dielectric energy, and the fifth term is the flexoelectric energy. In previous study [11], we investigated an electro-optic effect in ChLCs with the ULH alignment, in which the helix axis lies in the cell-plane direction. The ULH exhibited one slow and two fast response components by applying the electric field [Fig. 1(b)]: (i) flexoelectric effect (FE), (ii) helical elongation (HE) and (iii) helical deformation (HD). The FE component tilts the director around an axis parallel to the electric field, leading to the rotation of the optical axis, in which the rotational direction depends on a polar of the field [10,12,13]. The benefit of the FE component is its short response time, which has a quadratic dependence on the helical pitch, reaching the order of tens of microseconds at a helical pitch of 270 nm [11]. Due to the fast response, considerable attention has been paid to the research of the FE component [14–16]. Both HE and HD components are caused by torque originating from dielectric energy [the fourth term of Eq. (2)]. The HE component involves the movement of the entire helical structure for increasing helical pitch, and thus exhibits a slow response. On the other hand, the HD component exhibits a fast response because reorientation of LC director occurs in the local region equivalent to the scale of the helical pitch. For example [11], in the short-pitch ChLC with the helical pitch of 270 nm, the HE component showed response times of second order, while the HD component showed response times on the order of tens of microseconds comparable to the FE component. Therefore, the HD component is advantageous over device applications in terms of switching speed, and should be investigated in detail as much as the FE component. In this study, we analyzed dynamics of the HD component by demonstrating complete separation of the HD component from the other components. First, we removed the HE component by using planar alignment of ChLCs shown in Fig. 2, in which the glass substrates acted as a barricade for suppressing the motion of the helical elongation. At this time, what is important is application of a homogeneous electric field in the depth direction, and thus bulk electrode was used in our experiment. Furthermore, we separated the HD component from the FE component by using dielectric measurement system. Consequently, we were able to analyze dynamics of the HD, in which response times exhibited quadratic dependence on the helical pitch and no dependence on the cell thickness. This characteristics is believed to be useful for controlling propagation of electromagnetic wave at millimeter or terahertz frequencies.
Fig. 1 (a) Alignment of uniform lying helix and (b) three response components caused by application of an electric field normal to the helix.
2. Optical measurement
The sample was prepared by mixing a nematic LC mixture (MLC-2143, Merck) and a chiral dopant (ZLI-4571, Merck) at a weight ratio of MLC-2143: ZLI-4571 = 84: 16. MLC-2143 has high positive dielectric anisotropy (Δε = 28.3 at 50 kHz), and thus dielectric energy are much greater than flexoelectric energy according to Eq. (2), enhancing the HE and HD components. Also, MLC-2143 has birefringence of 0.16, and extraordinary and ordinary refractive indices are 1.66 and 1.50 from a data sheet provided by Merck, respectively. The concentration of the chiral dopant was selected to make selective reflection (SR) band of the ChLC appear in the ultraviolet range (338-370 nm), and then a helical pitch of the sample was 225 nm. A critical electric field Ec required for unwinding of the helix of ChLCs is known to be described as [1]
where p0 is the helical pitch at zero field. Ec of the sample was 8.1 V/μm, which was examined in the ChLC with ULH alignment shown in Fig. 1(a). A sandwitched cell was fabricated by using bulk aluminum electrodes as a spacer, which generated a homogeneous electric field in the cell-depth direction (Fig. 3). The cell gap and the interval between electrodes were controlled to be 29 μm and 24 μm. The surface of the glass substrates was coated with a polyimide (AL1254, JSR) and rubbed parallelly. Figure 4 shows electro-optic effects in the planarly aligned ChLC cell under crossed polarizers. At zero field, polarization optical microscope (POM) image showed almost dark state, but transmission spectrum exhibited a peak at 400 nm. This inconsistency between the photograph and spectrum are caused by the spectral sensitivity of the camera that has cut-off wavelength in the blue region. The light transmission around 400 nm is attributed to optical rotatory power of ChLCs. In general, light incident parallelly to the helix axis undergoes optical rotatory power Φ given by [7]where n|| and n⊥ are the refractive indices parallel and perpendicular to the director, λ is wavelength, and d is cell thickness.Fig. 3 A planarly aligned ChLC device with bulk electrodes to apply a homogenous electric field in the depth direction.
Fig. 4 The electro-optic effect of the planarly aligned ChLC device: (a) optical micrographs under crossed polarizers at zero field and E = 0.8 Ec, (b) transmission spectra at different fields, and (c) a response curve of a He-Ne laser when applying fields of E = 0.8Ec for 1 ms and 10 s.
Therefore, transmittance of the cell under crossed polarizers can be described as 100sin2Φ. If n|| = 1.63, n⊥ = 1.50, p = 225 nm, and d = 29 μm is assigned, the calculation agrees to the experimental result in the blue region. On the other hand, there was slight mismatch in the wavelength range longer than 500 nm. This is attributed to a leak of light on the boundary between the electrodes and the LC as shown in the photograph of Fig. 4(a). The decrease of n|| originates from the high concentration dope of the chiral dopant. The calculation also indicates that optical rotatory power do not almost affect the transmittance in the wavelength range longer than 600 nm. When a square electric field with a frequency of 50 kHz was applied, transmittance increased, resulting in a bright state at E = 0.8Ec. The switching speed was measured by plotting the transmission of a He-Ne laser (632.8 nm) from the LC cell between the crossed polarizers when applying the square field of E = 0.8Ec for 1 ms and 10 s [Fig. 4(c)]. The laser spot on the cell was approximately 20 μm in the center between electrodes. Their response curves reveal that the ChLC cell exhibits fast response with response times of several tens of microseconds, but no slow response component on the order of seconds. From these results, we attribute the increase in transmittance to electro-induced birefringence based on the HD and FE components. Figure 5 shows index ellipsoids changed by three components of FE, HE, and HD. ChLCs with a short helical pitch equivalent to optical wavelength or below is known to exhibit an optical uniaxial with negative refractive index anisotropy, in which the optical axis is parallel to the helix axis. Refractive indices parallel and perpendicular to helix axis can be described as nh|| = n|| and nh⊥ = (n|| + n⊥)/2, respectively. If one looks at the index ellipsoid from the top of the planarly aligned ChLC (i.e., from the direction parallel to the incident light), one can see a circle of radius nh⊥, thus resulting in the dark state in the POM photograph except the transmission of light originating from optical rotatory power. When applying the electric field normal to the helix axis (parallelly to x-axis), three components induces birefringence. The FE component rotates the optical axis at an angle φ without deforming of the index ellipsoid, and hence induces birefringence ΔnFE = nh⊥(1 – cosφ) on the top view. On the other hand, both HE and HD components deform the index ellipsoid because of increasing of molecules along the electric field, in which the index ellipsoid elongates in the direction along the field (x-axis) and shrinks in the direction normal to the field (y-axis), inducing birefringence from the top view. Since the birefringence induced by their components exhibits the maximum refractive index in the direction along the field, separation in the optical measurement is difficult. However, we can conclude that the HE component was not included in the electro-optic effect from the response curve shown in Fig. 4(c) because the HE component exhibits a long response time on the order of seconds [11]. This means that the glass substrates placed in the direction of the helix axis acted as a barricade to suppress the motion of the helical elongation.
Birefringence induced by the HD and FE components can be estimated by using the following Eq. (5),
where I is transmittance. Figure 6(a) shows birefringence calculated at a wavelength of 633 nm, at which optical rotatory power does not affect the transmittance. Birefringence increased with increasing the electric field, attaining approximately 0.012 at the field intensity of 0.8Ec. The dynamical property had no dependence on the electric field in both rise and decay times [Fig. 6(b)]. Usually, decay response keeps a constant regardless of the field intensity because driving force is elastic restoring force. On the other hand, driving force of rise response is electric force, and hence rise response times in typical NLC cells is inversely proportional to the square of the intensity of the electric field [6]. However, our ChLC cell had no dependence on the field of the electric field. This is because electro-induced torque balances with the elastic force in the ChLC. NLC cells exhibit a small threshold voltage required for driving the molecules, and most molecules align along the electric field at ~10 V. In such a saturated state, in which electro-induced torque is much greater than the elastic force, additional torque enhances the speed of the rotation motion of the molecules. On the other hand, the response in our ChLC cell was not still saturated as shown in Fig. 6(a), in which the electro-induced torque and the elastic force is in balance. In this situation, additional torque contributes to amplitude of the deformation of the FE and HD components, not to the switching speed.Fig. 6 The field-intensity dependence of the electro-optic effect: (a) birefringence and (b) response times.
3. Dielectric measurement
For the next step of analysis of the HD component, separation of the FE and HD components is performed. Here, we use the difference of the change in physics parameter in the direction parallel to the applying electric field (x-axis) as shown in Fig. 5. Refractive index along x-axis in the HD component increases, while that in the FE component keep a constant at nh⊥. The argument holds in dielectric constants, and therefore we can investigate characteristics of only the HD component by measuring a capacitance of the ChLC cell, which is affected by the dielectric constant in the direction parallel to the applying electric field. Since the measurement also is free from optical rotatory power, dynamics of the HD component can be exactly investigated. To analyze the helical-pitch dependence of the HD component, ChLC materials with various pitches were prepared by doping the chiral dopant (ZLI-4571) into the NLC (MLC-2143) at concentrations of 0.3, 1, 2, and 3 wt%. Relationship between the concentrations, the helical pitches, and Ec described as Eq. (3) is shown in Table 1.
Table 1. Material properties used in the experiment.
Figure 7 shows measurement system. Two sinusoidal waves of Vdrive and Vsignal at frequencies of 50 and 100 kHz were applied to the ChLC cell for driving LC molecules and measuring a capacitance C of the ChLC cell, respectively. In the system, Vsignal was selected to be sufficiently less than Vdrive so as not to affect the alignment of LC molecules. The ChLC cell was connected to both a resistance of R = 50 Ω and a lock-in amplifier with time resolution of 100 μs (Stanford Research Systems, SR844). A current i induced by Vsignal has a phase difference θ owing to capacity component C of the ChLC cell and gives rise to a voltage of VR = iR on the resistance. When the lock-in amplifier is synchronized with the frequency of f = 100 kHz, VR and θ are output. Using the VR and θ, an admittance YLC of the ChLC cell can be described as,
The real and imaginary part are given bywhere ω is an angular frequency. As a result, the measurement system enable us to analyze a change in capacitance C when applying Vdrive. A cell used in the measurement system had an interval of 50 μm between electrodes and a cell gap of 30 μm as shown in Fig. 7(b). In such a cell, electric fields gets across both the ChLC and the glass substrates, and thus C is not pure capacitance comprising only the LC component. This means that by using the measurement system, we can investigate dynamical characteristics of the ChLC, but cannot directly investigate the amplitude of the dielectric constant along x-axis. Figures 8 shows response curves of C when the ChLC with p = 1.0 μm was applied with an electric field of 0.8Ec. The change of C resulted from a change of the dielectric constant of the ChLC along the x-axis because a dielectric constant of the glass substrate does not change by applying Vdrive. Rise and decay times of the HD component were 782 and 815 μs, respectively. Next, response times in the ChLCs with various pitches were examined by the measurement system (Fig. 9). Rise and decay times kept a constant regardless of the intensity of the electric field, while had a strong dependence on the helical pitch. The response time of the short-pitch ChLC used in optical measurement was too fast to be examined owing to the limitation of the time resolution of the lock-in amplifier. The response times were proportional to the square of the helical pitch as shown in Fig. 9(c). Furthermore, we investigated the thickness dependence of response times using the ChLC with p = 1.0 μm [Figs. 10(a) and 10(b)]. The measured response times showed errors of several tens microseconds in some points. This is caused by a spike noise produced in the calculation of the lock-in amplifier. Since the spike noise is generated at random and with low probability, true value can be obtained by increasing the number of points. Response times kept a constant in the ChLC cell with cell gaps of dC = 17, 30, and 49 μm, demonstrating submillisecond switching even in the thick cell with the gap of 49 μm. Namely, the HD component exhibits thickness-independent dynamics, which is clearly different from dynamics of the NLC materials depending on the thickness. For a comparison, we investigated the decay time of the planar NLC cell applied with an electric field in the depth direction by using the measurement system [Fig. 10(c)]. The decay response time in the NLC had a strong dependence on the cell gap dN, and increasing the cell gap increased the decay response time, resulting in the order of seconds at the cell gap of dN = 19 μm.Fig. 8 (a) Rise response and (b) decay response curves of the capacitance calculated by measuring VR and θ.
Fig. 9 The helical pitch dependence of response times: (a) rise time, (b) decay time, and (c) response times versus helical pitches.
Fig. 10 The thickness dependence of response times: (a) rise time, (b) decay time, and (c) a comparison between the NLC and the HD component of the ChLC.
The difference of key parameters between NLCs and ChLCs can be described by considering director distribution of LC molecules when applying the electric field (Fig. 11). LC molecules exhibits the property as continuum, and hence spatial change of LC director gives rise to elastic restoring force. In the case of the NLC cell, LC director is sinusoidally modulated in the depth direction by the electric field, and an angle α formed by the LC director and the easy axis takes the maximum αMax at the center of the cell (z = dN/2). In a thin cell, the LC director rapidly changes, while in a thick cell, the LC director gradually changes, indicating that increasing the cell thickness decreases elastic restoring force. On the other hand, the director distribution induced by the HD component is retained in both thin and thick ChLC cells, although the number of the helix increases, resulting in the same decay time regardless of the cell thickness. However, if the helical pitch increases, the LC director gradually changes, decreasing the elastic restoring force. This means that the key parameter that affects the decay response time of the HD component is the helical pitch. This logic supports the experimental results. The thickness-independent dynamics is believed to be useful for device applications in millimeter or terahertz waves requiring large retardation. In future works, we will demonstrate a terahertz phase shifter with improved response times using the planarly aligned ChLC device.
Fig. 11 Director distribution of LC molecules when applying the electric field into NLC and ChLC cells.
Finally, we mention the importance of the use of the bulk electrodes. If the bulk electrodes are replaced by common interdigitate electrodes, an inhomogeneous field is generated, resulting in the HE component. The electro-optic effect including the HE component is greatly different from that in this study, and strongly affects reflection characteristics of ChLCs. We will report the difference in detail in another paper.
4. Conclusion
We investigated an electro-optic effect of planarly aligned ChLCs applied with an electric field normal to the helix axis. In the device, the motion of helical elongation with long response times was suppressed by glass substrates, and fast switching based on two components of helical deformation and flexoelectric effect was demonstrated. Furthermore, we analyzed the dynamical properties of the helical deformation by using dielectric measurement system, which demonstrates a complete separation from the flexoelectric effect. Consequently, the helical deformation exhibited quadratic dependence on helical pitch and no dependence on cell thickness, indicating that the ChLC device potentially can be used as fast tunable devices for millimeter or terahertz waves.
Funding
JSPS KAKENHI Grant Numbers 26820112 and 26420290.
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