Frequency suppression of director oscillations in AC-driven liquid-crystal-based terahertz phase shifters

Chenxiang Liu; Yu Wang; Li Li; Peng Tan; Shuai Li; Guanchao Wang; Wenpeng Guo; Zhenghao Li; Xingkai Che; Hao Tian; Hao Tian; Hao Tian

doi:10.1364/OE.504179

1. Introduction

Terahertz phased arrays have demonstrated tremendous potential in wireless communication, radar, and terahertz imaging [1–5]. Each unit of the phased array can be considered as a phase shifter, and the uniformity of the modulated wavefront directly affects the overall performance of the phased arrays [6]. Electro-tunable thick nematic liquid crystals (LCs) with a large modulation depth provide a solution for terahertz phased array units and have been widely proven to be suitable for terahertz sensors and phase modulation devices [7–11]. Compared to the magnetic modulation performed in earlier studies, the electric modulation enables miniaturization of the device and is compatible with microelectronic processing [12–14].

Generally, direct-current (DC) electric field can cause damage to LCs, such as causing electrochemical decomposition or electrode damage [15], which undoubtedly reduces the reliability of such devices. Compared with DC-drive, AC is chemically less harmful to nematic LC materials, and while using AC, the shielding effect of the insulating orientation layers can almost be neglected. In the recent years, electric modulation of LCs has been gradually extended from amplitude modulation to frequency modulation. Novel frequency-dependent LC materials such as dual-frequency LCs have been gradually introduced into devices [16–18], and it is necessary to investigate the LC dynamics under frequency modulation. AC-drive also poses certain problems, such as flexoelectric domains under low-frequency alternating electric fields [19,20] and complex electrohydrodynamic behaviors, such as Williams domains and herringbone domain patterns at high frequencies accompanied by profound nonlinear processes [21–25]. However, the collective oscillations of the AC-driven nematic LCs still introduce instabilities to the transmitted phases while guaranteeing the absence of domains. This behavior is more pronounced in thicker LCs. Thus, the AC-driven characteristics of an LC need to be further investigated to optimize the performance of the modulator.

In this work, we mainly analyze the electrical Fréedericksz transition of AC-driven LCs theoretically and experimentally, and reveal the suppression of frequency-induced oscillations in the low frequency range. The application of the Frank-Leslie equation has been extended to calculate the amplitude of small and large-angle oscillations of LCs under alternating electric fields. Based on the numerical calculations, the equations for DC and AC modulation have been derived by using perturbation method, and the frequency threshold to suppress the modulation instability has been estimated. The thick LC phase shifters for terahertz wave modulation with terahertz transparent electrodes PEDOT:PSS were fabricated to verify the abovementioned theoretical prediction. The Sénarmont compensator method and digital holography were applied to measure the stability and domain state of the phase modulation of an AC-driven LC phase shifter, demonstrating the suppression of director oscillations by increasing the frequency in a non-convective state. The terahertz time-domain spectra (TDS) in our study revealed that the device can provide a phase shift of 360° at 2 THz in agreement with the theoretical results, suggesting that phase shifters with frequencies above the threshold frequency are competent for modulation applications. These results provide basis for the dynamics research and instability characterization of AC-driven LC devices.

2. Fundamental equations

LC materials are widely used in phase-modulated devices because of their continuously tunable and stable properties at optical and even terahertz wavebands. In LCs without flow, the LC molecules only rotate without directional motion, in which case the elastic strain produced due to splay, twist, bend deformations, etc., and the strain produced due to an external field jointly determine the LC state. As shown in Fig. 1(a), consider a simple LC system with a sandwich structure, where an electric field is applied on both sides by a pair of parallel flat electrodes, and LCs close to the pole plate are arranged in parallel. The boundary condition eliminates the effect of distortion deformation for the system, which causes the LC directors to be deflected only in the x-z plane. Consequently, the director field can be written in terms of the deflection angle n = (cos θ, 0, sin θ) . In general, the free energy of nematic LC under an electric field can be written as [26]

(1)$$f = \frac{1}{2}{K_{11}}{(\nabla \cdot {\textbf n})^2} + \frac{1}{2}{K_{22}}{({\textbf n} \cdot \nabla \times {\textbf n})^2} + \frac{1}{2}{K_{33}}{({\textbf n} \times \nabla \times {\textbf n})^2} - \frac{1}{2}\Delta \varepsilon {({\textbf n} \cdot {\textbf E})^2}$$

Here, K₁₁, K₂₂, and K₃₃ represent the splay, twist, and bend elastic constants. $\Delta \varepsilon = {\varepsilon _\parallel } - {\varepsilon _ \bot }$ is the dielectric anisotropy; ${\varepsilon _\parallel }$ and ${\varepsilon _ \bot }$ represent the relative dielectric constants under parallel and perpendicular electric fields, respectively. E represents the electric field.

Fig. 1. (a) Diagram of liquid crystal terahertz phase shifter. (b) Distribution of director along the thickness at 0-10 V.

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The phase-field method can be used to construct the kinetic equations from the free energy to reveal the evolution of the system under the action of the electric field [27,28]. It could trace the phase transition process through phase-field parameters, and is therefore, one of the ideal methods to study Fréedericksz transition. Herein, the deflection angle was chosen as the phase-field parameter, which is a non-conservative quantity whose evolution equation satisfies the Landau-Ginzburg equation [29], expressed as follows:

(2)$$\frac{{\partial \theta }}{{\partial t}} ={-} {L_\theta }\frac{{\delta f}}{{\delta \theta (r,t)}}$$

Here, L_θ represents the mobility of the non-conserved quantity θ. The Landau-Ginzburg equation could be reduced to the LC kinetics equation by substituting the free energy expression. The resulting kinetics equation is referred to as the Frank-Leslie equation [30], which is applied to the DC electric field and describes the dynamics of the Fréedericksz transition.

(3)$$\begin{array}{c} \frac{\partial }{{\partial z}}[({K_{11}}\textrm{co}{\textrm{s}^2}\theta + {K_{33}}\textrm{si}{\textrm{n}^2}\theta )\frac{{\partial \theta }}{{\partial z}}] - ({K_{33}} - {K_{11}})\textrm{sin}\theta \textrm{cos}\theta {(\frac{{\partial \theta }}{{\partial z}})^2}\\ + \textrm{sin}\theta \textrm{cos}\theta {\varepsilon _a}{E^2} = {\gamma _1}\frac{{\partial \theta }}{{\partial t}} \end{array}$$

In the following discussion, the LC parameters for the calculation are given as K₁₁ = 17.1 × 10⁻¹² N, K₂₂ = 26.4 × 10⁻¹² N, and as K₃₃ = 35.6 × 10⁻¹² N, which are the parameters of the nematic LC of the laboratory. γ₁= 0.2 Pa·s is the rotational viscosity [31], which is obtained by fitting the experimental measurements. ${\varepsilon _a} = {\varepsilon _0}\Delta \varepsilon $ represents dielectric anisotropy. Figure 1(b) illustrates the distribution of the deflection angle along the thickness as the result of the calculated stable solution for the abovementioned LC system. The results reveal that the deflection was large in the middle and decreased to both sides, and the maximum value of the deflection angle was obtained in the central layer. With the symmetric Dirichlet's boundary condition of anchoring angle, the positive LC will be driven and deflected along the direction of the electric field, while the deflection of the LC is suppressed by the elastic force originating from the boundary anchoring effect of the two polar plates. It is observed that between the RMS voltages of 2.5 V and 2.8 V, a critical voltage occurred, which determines whether the LC starts to deflect or not. It is the critical point where the electric field drives the LC molecules to break through the elastic force binding, which is also known as the Fréedericksz transformation threshold voltage. However, it has been shown that a DC electric field acting on the space charge induced by orientation distortion would drag the LC to flow. This flow speed increased with the electric field strength, which would inevitably cause damage to the electrode [15].

AC-drive has been applied to some devices modulated by LCs due to the low damage caused to electrodes by AC-drive as compared to DC-drive. However, the AC-drive could not be intuitively equated to DC-drive because of the varying dynamic evolution. In this study, the Frank-Leslie equation was generalized to AC-drive, where the electric field E is extended to alternating electric fields $\widetilde E$, in the following form:

(4)$$\begin{array}{c} \frac{\partial }{{\partial z}}[({K_{11}}\textrm{co}{\textrm{s}^2}\theta + {K_{33}}\textrm{si}{\textrm{n}^2}\theta )\frac{{\partial \theta }}{{\partial z}}] - ({K_{33}} - {K_{11}})\textrm{sin}\theta \textrm{cos}\theta {(\frac{{\partial \theta }}{{\partial z}})^2}\\ + \textrm{sin}\theta \textrm{cos}\theta {\varepsilon _a}{{\tilde{E}}^2} = {\gamma _1}\frac{{\partial \theta }}{{\partial t}} \end{array}$$

Because of the bijective mapping between the extreme deflection angle and the director distribution under the same anchoring conditions, we considered the middle layer of the liquid crystals, i.e., ${\left. {\frac{{\partial \theta }}{{\partial z}}} \right|_{{\theta _m}}} = 0$, and defined the function $\eta ({\theta _m}) = {\left. {\frac{{{\partial^2}\theta }}{{\partial {z^2}}}} \right|_{{\theta _m}}}$. Subsequently, the equation for the maximum value θ_m was obtained as follows:

(5)$$({K_{11}}\textrm{co}{\textrm{s}^2}{\theta _m} + {K_{33}}\textrm{si}{\textrm{n}^2}{\theta _m})\eta ({\theta _m}) + \textrm{sin}{\theta _m}\textrm{cos}{\theta _m}{\varepsilon _a}{\tilde{E}^2} = {\gamma _1}\frac{{\partial {\theta _m}}}{{\partial t}}$$

The extreme value equation is a constant differential equation, and the function describes the degree of distortion of the LC. In the following section, Eq. (5) will be discussed from the small-angle case and the large-angle case.

3. AC-drive calculation

When the deflection angle is small, the problem can be properly simplified. After choosing a sinusoidal functional basis, the solution can be expressed as a superposition of sinusoidal functions. Especially when the deflection angle is small, based on small signal approximation, the degeneracy of the nonlinear term will make the angular distribution of the directors along the thickness approximated as a sinusoidal spatial pattern [32]:

(6)$$\theta = C(t )\sin \left( {\frac{{\pi z}}{d}} \right)$$

Where d is thickness of LC. With $\eta ({\theta _m}) ={-} {(\pi /d)^2}{\theta _m}$, Eq. (5) becomes:

(7)$$- {(\frac{\pi }{d})^2}({K_{11}}\textrm{co}{\textrm{s}^2}{\theta _m} + {K_{33}}\textrm{si}{\textrm{n}^2}{\theta _m}){\theta _m} + \textrm{sin}{\theta _m}\textrm{cos}{\theta _m}{\varepsilon _a}{\tilde{E}^2} = {\gamma _1}\frac{{\partial {\theta _m}}}{{\partial t}}$$

The numerical results of the maximum value oscillations under conditions of DC drive and AC drive are illustrated in Fig. 2(a), and the time is expressed by the intrinsic relaxation time τ₀ = γd²/(π²K)/(V²/V_th²−1) [33]. Herein, a voltage of V_rms= 2.75 V was used by the calculation, which is slightly larger than the Fréedericksz transition threshold voltage V_th of 2.71 V exhibited by the LC of the laboratory. This was done to satisfy the condition of small deflection angle. The characteristics of the solution are presented as a stable distribution superimposed on symmetrical alternating oscillations. Figure 2(c) shows and linearly fits the inverse relationship of the oscillation amplitude with frequency using logarithmic coordinates. It is observed that as the frequency increased, the oscillation amplitude decreased inversely with frequency, limited by the relaxation rate of the LC molecules. The high-frequency solution converged to the DC steady-state solution, demonstrating the suppression effect of the frequency on the oscillation. More detailed calculations reveal that the deflection angle obtained using the small angle approximation is about 82% of the value obtained without approximation.

Fig. 2. Calculated results of maximum values for DC and different frequency conditions (d = 100 µm). (a) Calculated results for small angle (V_rms = 2.75 V). (b) Calculated results for large angle (V_rms = 10 V). (c) Variation of small angle amplitude with frequency and inverse fit results. (d) Variation of large angle amplitude with frequency and inverse fit results.

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Because of the low voltage leading to excessively long relaxation times in the small angle case, the case of the large angle deflection is more relevant. In the large angle case, the condition of small signal approximation is no longer satisfied due to increased elastic deformation of LCs. By assuming that the relaxation process of an LC can be decomposed into a quasi-static evolutionary process, the relaxation kinetic process of the LC is calculated with the help of steady-state distribution, and Eq. (2) is approximated as follows:

(8)$$({K_{11}}\textrm{co}{\textrm{s}^2}{\theta _m} + {K_{33}}\textrm{si}{\textrm{n}^2}{\theta _m})\eta ({\theta _m})(\frac{{{\theta _m}}}{{{\theta _\infty }}}) + \textrm{sin}{\theta _m}\textrm{cos}{\theta _m}{\varepsilon _a}{\tilde{E}^2} = {\gamma _1}\frac{{\partial {\theta _m}}}{{\partial t}}$$

Furthermore, the numerically calculated oscillations of a large-angle AC-driven LC are shown in Fig. 2(b). In the AC-driven case, the behavior of the solution in the large angle case is basically the same as that in the small angle case. However, the asymmetry of the oscillating waveform is more pronounced, indicating that the large-angle elastic moment has a stronger torsional effect on the LC molecules. Since the electric field in the equation is a squared term, the oscillation of the LC demonstrates the quadratic response of the electric field. The amplitude of oscillation versus frequency is shown in Fig. 2(d). It is observed that with an increase in frequency, the amplitude gradually decreased, showing the overall suppression of the oscillation by frequency. A logarithmic function of the deflection angle yielded highly linear results, indicating that this relationship follows an inverse relationship.

4. Calculation of stabilization frequency

The suppression of instability by frequency can be calculated by using the perturbation method. It is assumed that the director distribution at high frequency is obtained from the steady state after perturbation, i.e., the slow field corresponds to the time average of the deflection angle. On the other hand, the fast field corresponds to the oscillation of directors caused by the alternating electric field. According to Eq. (5), the slow and fast fields, respectively, satisfy the following equation:

(9)$$\begin{array}{c} ({K_{11}}\textrm{co}{\textrm{s}^2}\left\langle {{\theta_m}} \right\rangle + {K_{33}}\textrm{si}{\textrm{n}^2}\left\langle {{\theta_m}} \right\rangle )\eta (\left\langle {{\theta_m}} \right\rangle )\\ + \frac{1}{2}\textrm{sin}\left\langle {{\theta_m}} \right\rangle \textrm{cos}\left\langle {{\theta_m}} \right\rangle {\varepsilon _a}E_0^2 = 0 \end{array}$$

(10)$$({\kappa _1} + {\kappa _2}E_0^2[1 + \textrm{cos}(2\omega t + {\phi _0})])\psi + \beta E_0^2\textrm{cos}(2\omega t + {\phi _0}) = {\gamma _1}\frac{{\partial \psi }}{{\partial t}}$$

Here,

(11)$$\begin{array}{c} {\kappa _1}(\left\langle {{\theta_m}} \right\rangle ) = ({K_{11}}\textrm{co}{\textrm{s}^2}\left\langle {{\theta_m}} \right\rangle + {K_{33}}\textrm{si}{\textrm{n}^2}\left\langle {{\theta_m}} \right\rangle )\eta ^{\prime}(\left\langle {{\theta_m}} \right\rangle )\\ + 2({K_{33}} - {K_{11}})\textrm{sin}\left\langle {{\theta_m}} \right\rangle \textrm{cos}\left\langle {{\theta_m}} \right\rangle \eta (\left\langle {{\theta_m}} \right\rangle ) \end{array}$$

(12)$${\kappa _2}(\left\langle {{\theta_m}} \right\rangle ) = \frac{1}{2}\textrm{cos}(2\left\langle {{\theta_m}} \right\rangle ){\varepsilon _a}$$

(13)$$\beta (\left\langle {{\theta_m}} \right\rangle ) = \frac{1}{2}\textrm{sin}\left\langle {{\theta_m}} \right\rangle \textrm{cos}\left\langle {{\theta_m}} \right\rangle {\varepsilon _a}$$

Considering $\eta ({\theta _m}) = {\left. {\frac{{{\partial^2}\theta }}{{\partial {z^2}}}} \right|_{{\theta _m}}}$, the slow field ${\theta _m}$ is the steady state distribution caused by RMS voltage. From the equation of the fast field, it can be observed that the response frequency of the fast field is twice the frequency of the applied electric field, and the solution can be expressed as

(14)$$\psi = \Psi \textrm{cos}(2\omega t + {\psi _0})$$

where,

(15)$$\Psi = \frac{{\beta (\left\langle {{\theta_m}} \right\rangle )E_0^2}}{{\sqrt {\kappa _1^2(\left\langle {{\theta_m}} \right\rangle ) + 4\gamma _1^2{\omega ^2}} }}$$

The critical frequency is represented by ${\omega _0} = \beta {E_0}^2/({2{\gamma_1}|\Psi |} )$. When the condition of ω > ω₀ is satisfied, the instability can be suppressed. Through numerical calculation, it is revealed that this value is inversely proportional to d. When d = 100 µm and 2Ψ = 1°, its value is approximately 1.31 Hz under a driving RMS voltage of 5 V, coinciding with the results of Fig. 2(d).

5. Experimental section

The LC phase shifters having a classical sandwich structure were fabricated to test the abovementioned theory. Each device consisted of a pair of 500-µm-thick quartz substrates, a pair of thin film electrodes, a pair of polyimide orientation layers, and a thick LC layer having a thickness of hundreds of microns. Its parallel-plate electrodes were uniform, and they used the novel organic and transparent film poly (3, 4-ethylenedioxythiophene): poly (styrene sulfonate) (PEDOT:PSS). It is a highly conductive organic and transparent thin-film electrode with a high terahertz band transparency [34,35]. The LC molecules were initially arranged in parallel, with directors lying in the plane of the electrode. Sec. 2 and Sec. 3 of Supplement 1 show more details about the PEDOT:PSS electrode and the preparation of LC phase shifters.

The electro-optical effect is very suitable for studying the state of LC systems owing to its high sensitivity and accuracy [36]. Because of the small change in transmittance caused by deflection angle oscillations, small oscillations in LCs cannot be gauged by measuring transmittance. To address this problem, the electro-optical test was carried out by using the Sénarmont compensator method, as is shown in Fig. 3(a). A 632.8 nm laser beam becomes circularly polarized light after passing through a polarizer and a quarter-wave plate with the main axis at 45° to it, and the circularly polarized light is modulated by a fabricated liquid crystal phase shifter and then passes through a polarizer after receiving the light intensity using a CCD. The liquid crystal phase shifter is oriented in the same direction as the quarter waveplate, while the line polarizer is 45° away from them. Due to the designed deflection of the positive liquid crystal, the whole system can be approximated as an electrically adjustable waveplate, and the polarization state of the outgoing circularly polarized light will be converted among line polarization, elliptical polarization and circular polarization with the change of phase shift of the phase shifter. When the line polarized light is exactly ± 45° with the orientation direction, after a line polarizer will be received in the photoelectric detector with the phase of the phase shifter is a sinusoidal function of the light intensity, as shown in Eq. (16).

(16)$$I = {I_0}\frac{{1 + \textrm{sin}(\delta \phi )}}{2}$$

The phase modulator of a thick LC of 300 µm was measured, and the amount of phase modulation obtained by applying different frequencies with time is shown in Fig. 3(b). The instability was determined by Eq. (14), which indicates that the oscillation amplitude of the AC-driven LC deflection angle decreases inversely with increasing frequency and is in agreement with the experiment. The oscillation amplitude at V_rms= 10 V is shown in Fig. 3(c), and after further fitting, it is observed that the oscillation amplitude decreased inversely with increasing frequency, showing the suppression effect of frequency on the oscillation. Therefore, the important factor that proves that the effect of AC-drive is close to that of DC-drive is the drive frequency. When the drive frequency is higher than the critical frequency ω₀, the oscillation caused by AC-drive will be suppressed. At this time, it is reasonable to equate the AC-drive to DC-drive.

Fig. 3. Measurement results of the Sénarmont compensator method. (a) Diagram of the optical path of the Sénarmont compensator method used for electro-optical testing. (b) Phase modulated signal versus time. (c) Inverse of oscillation amplitude of phase modulated signal versus frequency and fitting result.

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Based on the abovementioned conclusions, the terahertz modulation performance of a 500-µm-thick AC-driven terahertz phase shifter was measured under the condition of satisfying the threshold frequency (f = 500 Hz) using the terahertz time-domain spectroscopy method [37,38]. Sec. 1 of Supplement 1 shows more details of the TDS for experiments. The measured time-domain spectra are shown in Fig. 4(a), while the frequency-domain spectra are shown in Fig. 4(b). The sample is homogeneous along the thickness direction, and the refractive index properties of the LC layer can be extracted by the multilayer reflection model, the calculation formula is shown in Eq. (17).

(17)$${n_s}(\omega ) = \phi (\omega )\frac{c}{{\omega d}} + 1$$

where, $\phi (\omega )$ is the phase angle of complex transmittance.

Fig. 4. Phase shift characteristics of AC-driven terahertz LC phase shifters. (a) Terahertz time-domain spectra of 0-15 V RMS. (b) Terahertz power spectra of 0-15 V RMS. (c) Effective refractive index of 0.3-2.0 THz. (d) Phase shift of 0.3-2.0 THz.

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Based on the results of the TDS, the effective refractive index (n_eff) of the LC layer can be obtained by subtracting the contributions from the quartz substrate and the film. It is found that the n_eff started to increase after reaching the threshold voltage (about 2.7 V) and tended to be constant when the voltage continued to increase, as is shown in Fig. 4(c), which corresponded to the deflection characteristic of the director. The measured phase-shift characteristics are shown in Fig. 4(d), which were in good accordance with the results of the electronically controlled phase-shift characteristics obtained from the director calculation [39].

(18)$$\delta = \frac{{2\pi }}{\lambda }\int_0^d {(\frac{{{n_e}{n_o}}}{{\sqrt {n_e^2\textrm{co}{\textrm{s}^2}\theta + n_o^2\textrm{si}{\textrm{n}^2}\theta } }} - {n_o})} dz$$

It is worth noting that the phase shift increased almost linearly with the RMS voltage from 2.2 V to 4 V under AC-drive, and this anomaly cannot be explained by solving the Frank-Leslie equation. This may be caused by the long relaxation time of the LCs at smaller angles, where the intrinsic relaxation time may be on the order of hours, well beyond the experimentally measured time interval. The use of the overvoltage-drive method as well as the process of continued operation outside this range ensures the accuracy and speed of the electro-tunable LC device.

However, the increased deflection of the director enhanced the gradient of the refractive index along the thickness direction, which in turn decreased the transmittance. Figure 5(a) shows the transmittance of AC-driven device, where the device maintained high performance with a transmittance that was well stabilized at 0.4-1.4 THz and was higher than 0.45 at AC drive voltages below 15 V RMS. Figure 5(b) shows the typical extracted transmittance characteristics of the device at 1 THz, where the transmittance decreases with voltage to converge to a limiting value of about 0.47, which means that the insertion loss of the device is less than 3.3 dB at drive voltages below 15 V (and less than 3 dB below 8 V drive voltage). The overall curve satisfies the exponential decline, but a retracted region of decreasing and then increasing is experimentally observed at 3 V RMS, which may imply a chaotic distribution of LC molecules at weak alternating voltages.

Fig. 5. Transmittance characteristics of AC-driven terahertz LC phase shifter. (a) Transmittance of phase shifter. (b) The relationship of phase shifter electronically controlled transmittance.

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The LC oscillation above the critical frequency had been well suppressed, but the domain structure caused by the AC might exist and affect the uniformity of the phase modulation of the device. To ensure the applicability of the theory, the digital holography technique was applied to confirm whether the domain structure is generated, and the optical path is shown in Fig. 6(a). A laser beam with a wavelength of 632.8 nm passes through a spatial optical filter to filter out the higher order modes. Then, the wavefront was formed as a plane wave by lens. The beam is then split into two beams by a beam splitter, forming a typical Mach-Zehnder interference system. One beam passes through the sample called the object light and the other beam passes through the attenuator, called the reference light. The two beams are combined into a single beam at a small angle to form an interference pattern that is received by the CCD. The off-axis digital holography algorithm, adaptive focusing algorithm, phase compensation and FFT-based least squares unwrapping algorithm [40] are used to obtain the phase distribution and finally the measurement results of the refractive index distribution of the liquid crystal.

Fig. 6. Digital holographic measurement results (a) The schematic diagram of digital holographic measurement. (b) RMSE of refractive indices obtained from digital holography measurements. (c) 1.0 V phase plane distribution measurement results. (d) 3 V phase plane distribution measurement results. (e) 6 V phase plane distribution measurement results.

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Figure 6(b) reveals that the root-mean-square-error (RMSE) of the refractive index measured was in the range of 0.5 - 6 V RMS, and the frequency of 500 Hz was within 2 × 10⁻⁵. This result might contain the phase measurement error caused by the inhomogeneity of the spin coating and frictional orientation during the preparation of the transparent electrode film of PEDOT:PSS. There was no obvious trend of refractive index RMSE observed, and no obvious patterns or other changes in the surface distribution of the refractive index at RMS voltage of 1 V, 3 V and 6 V were observed as shown in Fig. 6(c), (d) and (e). This indicates that there is no obvious electrostatic instability in the LC driven by the AC voltage, and its transmission wavefront does not exhibit additional distortion due to voltage change. Overall, the AC-driven LC phase shifter device exhibited good performance, and this phase shifter driven at frequencies above the threshold frequency can be competent for modulation applications.

6. Conclusion

In summary, by generalizing the Frank-Leslie equation to alternating electric fields, we revealed the dynamic response of AC-driven LCs for small and large angle deflections. Based on the elasticity theory of nematic LC and electro-optical experiments, the suppression of instability by frequency at low-frequency electric fields was verified and elucidated. The fabricated 500-µm-thick LC phase shifter could achieve a phase shift of 360° at 2 THz, and the trend of electric field modulation was in accordance with the theoretical prediction. In addition, the stability of thick AC-driven LC phase shifters was investigated using digital holography. The phase shifter with a 300-µm-thick LC layer driven at a frequency of 500 Hz to satisfy the threshold frequency was found to have excellent wavefront modulation characteristics, with the RMSE of the refractive index being within 2 × 10⁻⁵. This work provides an experimental and theoretical reference for the study of AC-driven LC devices, as well as a basis for the study of frequency-modulated LC devices.

Funding

National Natural Science Foundation of China (12004085, 12074092); Natural Science Foundation of Heilongjiang Province (TQ2023A006, YQ2022A010); Fundamental Research Funds for the Central Universities (2023FRFK06002); China National Postdoctoral Program for Innovative Talents (BX20200111); Heilongjiang Provincial Postdoctoral Science Foundation (LBH-Z22121).

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.

Supplemental document

See Supplement 1 for supporting content.

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Frequency suppression of director oscillations in AC-driven liquid-crystal-based terahertz phase shifters

Abstract

1. Introduction

2. Fundamental equations

3. AC-drive calculation

4. Calculation of stabilization frequency

5. Experimental section

6. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (6)

Equations (18)

Optics Express