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Abstract
A polarization-independent terahertz (THz) phase shifter was proposed using a liquid crystal (LC) grating with subwavelength periodic alignment. The LC grating was constructed with one-dimensional periodic planar alignment and was designed based on the effective medium theory. The phase of the transmitted wave was theoretically independent of the polarization state and the phase was shifted by transition from a periodic planar alignment to a homeotropic alignment. The LC grating was fabricated using a nematic LC and photoalignment layers. The easy axes of the photoalignment layers were periodically regulated using a grating photomask with a subwavelength pitch. There was minimal dependence of the obtained phase shift on the polarization state, and the results were in agreement with the theoretical calculations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Numerous practical devices and systems using electromagnetic waves within the terahertz (THz) spectral range have been proposed in various fields, including wireless communications, physical chemistry, biological imaging, nondestructive inspection, and security [1−7]. The active control of the propagation of THz waves is important for practical implementation of these applications. Liquid crystals (LCs) are promising materials for active THz devices because they exhibit optical anisotropy in the THz spectral range and respond to a variety of external fields [8−10]. Fundamental LC THz devices have been proposed [11−26]. Chen et al. reported a phase shifter based on magnetically controlled birefringence in a homeotropically aligned nematic LC [11]. Scherger et al. studied an electrically tunable beam steering device using a wedge nematic LC cell [15]. Wu et al. reported an electrically tunable phase shifter using a homogeneously aligned nematic LC cell with transparent graphene electrodes [16]. Most of these LC THz devices have a polarization dependence because the homogeneously aligned nematic LCs are uniaxial media; however, there is also a demand for polarization-independent THz devices [27]. Our previous study showed that randomly aligned nematic LC cells with subwavelength domains were able to be applied for polarization-independent THz phase shifters [17]. In this report, using a 0.1-mm-thick LC layer, a voltage-controlled phase shift of ∼0.1 rad in the THz spectral range was demonstrated [17]. Wang et al. reported polarization-independent THz phase modulation using a chiral nematic LC [18,21]. They indicated that the domain structures in the randomly aligned nematic LC cause non-negligible scattering losses [18]. Additionally, control of the domain sizes, i.e., control of the active frequency, is difficult in the random alignment.
In the visible range, diffractive optical elements using spatial alignment distributions of LC molecules are of interest for polarization imaging, polarimetry, optical signal processing, etc [28−31]. We investigated LC diffractive optical elements fabricated using photoalignable liquid crystalline polymeric films as alignment layers [31]. The polymeric photoalignment films easily realize complex alignment patterning via photolithography, polarization holography, and polarized beam drawing [31]. Our previous work also investigated the subwavelength periodic alignment structures of the liquid crystalline polymers and their optical properties in the visible range [32−34]. These studies showed that subwavelength alignment structures of LCs were useful for enhancement and compensation of birefringence. Recently photopatterned LC based THz devices was also presented [35,36].
In the present study, a polarization-independent THz phase shifter is proposed using an LC cell with a subwavelength periodic alignment distribution. The LC cell had planar alignment in its initial state but the alignment direction (i.e., the director) was modulated one-dimensionally in the cell. The subwavelength alignment distribution was designed to realize a zero-birefringence medium in the THz spectral range based on the effective medium theory and the ordinary and extraordinary refractive indices of the LC. The effective refractive index of the LC grating is controlled via the alignment transition from a planar to a homeotropic state. Therefore, a polarization-independent phase shift can be obtained by applying an electric field vertical to the LC layer. The LC cell was fabricated using polymeric photoalignment films and a low-molar-mass nematic LC. The periodic alignment pattern was formed using a grating photomask with a grating pitch of 50 µm. The complex transmittance of the LC cell was investigated via THz time-domain spectroscopy by changing the polarization state of the incident THz pulses. Using transmissive electrodes, the phase shift between the planar and homeotropic states was also measured. The experimental results were explained based on the theoretical model of the subwavelength alignment structure. This study advances tunable metamaterials and various active devices in the THz spectral range.
2. Methods
Based on the effective medium theory, a zero-birefringence material was designed using a one-dimensional binary grating of nematic LCs with positive uniaxial anisotropy [32,37]. Figure 1 shows a schematic of the subwavelength LC grating. In the xyz-coordinate system, the director ${\boldsymbol n}$ is in the xy-plane in the initial state, the grating vector ${\boldsymbol K}$ is parallel to the x-axis, the grating pitch is $\Lambda $, and the angle between ${\boldsymbol n}$ and ${\boldsymbol K}$ is given by
Fig. 1. Schematic of the one-dimensional binary LC grating. The director ${\boldsymbol n}$ is in the xy-plane and parallel or perpendicular to the x-axis.
Fig. 2. Theoretical anisotropy of the subwavelength LC grating. The red line represents the birefringence ${\mathop{\rm Re}\nolimits} ({{n_\parallel } - {n_ \bot }} )$ and the blue line represents the dichroism ${\mathop{\rm Im}\nolimits} ({{n_\parallel } - {n_ \bot }} )$.
To demonstrate the polarization-independent phase shift, an LC cell with a subwavelength grating structure was fabricated using photoalignment polymeric films (RN3222, Nissan Chemical, Japan). The anchoring energy for the photoalignment film is provided by irradiating it with polarized UV light and subsequent annealing. The alignment easy axis (i.e., the director on the film) is perpendicular to the polarization direction. The photoalignment polymer was spin-coated on two quartz glass substrates (HERALUX, ShinEtsu QUARTZ, Japan). The thickness of the two photoalignment films was 0.1 µm each. The photoalignment films were exposed twice to a UV lamp (KIN-ITSU-KUN 50, Yamashita Denso, Japan) with a 313-nm-bandpass filter, a linearly polarizer, and a grating photomask with a pattern of lines and spaces (CA25E 20LPMM, Edmund Optics). The grating pitch of the photomask was 50 µm and the lines and spaces were each 25 µm wide. For the first exposure, the polarization direction was set parallel to the grating vector of the photomask. In the second exposure, the photomask was shifted 25 µm to the grating vector direction and the polarization direction was set to be perpendicular to the grating vector direction. The intensity of the UV light was 6.3 mW/cm2 and the exposure time was 2.4 s for the respective exposures. The substrates were then annealed at 140°C for 10 min and an empty planar cell was constructed using the two substrates. The two alignment layers were separated using ball spacers with a diameter of 0.3 mm. Using a polarizing optical microscope (ECRIPSE E200, Nikon, Japan) and a micro-stage, the grating vectors of the two alignment layers were adjusted to be mutually parallel. The spatial phase difference of the grating structures in the two alignment films was also adjusted to be zero. The nematic LC (5CB, Tokyo Chemical Industry, Japan) was then injected into the empty cell. Figure 3(a) shows a schematic of the grating LC cell. The alignment state of the LC was observed via polarizing optical microscopy. The transmittance and phase of the fabricated LC cell in the THz spectral range were measured via a THz time-domain spectroscopic system (TAS7500, Advantest, Japan), which emits (detects) linearly polarized THz pulses. All the measurements were conducted at room temperature (i.e., the nematic phase temperature of 5CB).
Fig. 3. Schematic illustrations of the grating LC cell; (a) the off-state and (b) the on-state. Blue rods and circles represent the LC molecules. The arrows in the RN3222 layers represent the alignment easy axes.
3. Results and discussion
Figure 4 shows the polarizing optical microscopy images of the grating LC cell. It is difficult to distinguish the two alignment regions with $\theta = 0$ and $\theta = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$, but this indicates that a planar alignment was formed in the cell except around the boundaries of the two regions. The alignment distribution around the boundaries including the discontinuities will have no small effect on the transmission property in the THz spectral range. However, for simplicity, we assume the binary periodic alignment [Eq. (1)] in the theoretical analysis.
Fig. 4. Polarization optical microscopy images of the fabricated LC grating under (a) the dark condition and (b) the bright condition. The arrows represent the transmission axes of the polarizers.
Figure 5(a) shows the measured transmittance of the LC cell in the THz spectral range. The transmittance spectra for ${\boldsymbol E}\parallel {\boldsymbol K}$ and ${\boldsymbol E} \bot {\boldsymbol K}$ were almost identical, where ${\boldsymbol E}$ is the electric field of the linearly polarized THz pulses. The transmittance spectra were simulated based on the Fresnel equations [20,40]. For these calculations, a three-layered model was constructed in the form of glass substrate, LC layer, glass substrate, and the alignment films were ignored. This was because the thickness of the dielectric alignment layer was substantially smaller than the wavelengths. The refractive index of free space was assumed as 1, the thickness of the substrates was ${d_{\textrm{glass}}} = 0.96\textrm{ mm}$, the refractive index of the substrates was ${n_{\textrm{glass}}} = 1.96 + i0.9 \times {10^{ - 14}}f$, the thickness of the LC layer was $d = 0.34\textrm{ mm}$, the effective refractive index of the LC layer for ${\boldsymbol E}\parallel {\boldsymbol K}$ was ${n_\parallel } = 1.65 + i0.024$, and the effective refractive index of the LC layer for ${\boldsymbol E} \bot {\boldsymbol K}$ was ${n_ \bot } = 1.66 + i0.022$. Here, ${d_{\textrm{glass}}}$ and ${n_{\textrm{glass}}}$ were determined based on the experimentally observed THz transmission spectrum for the quartz glass substrate, d was measured using an optical microgage (C11011, Hamamatsu Photonics, Japan) for the empty cell, and ${n_\parallel }$ and ${n_ \bot }$ were the calculated values using Eqs. (3) and (4), ${n_\textrm{o}} = 1.58 + i0.031$, ${n_\textrm{e}} = 1.74 + i0.014$, and $F = 0.5$. Figure 5(b) shows the calculation results. The calculated transmittance spectra for ${\boldsymbol E}\parallel {\boldsymbol K}$ and ${\boldsymbol E} \bot {\boldsymbol K}$ were also almost identical and were in reasonable agreement with the measured spectra. This indicated that the transmittance of the subwavelength LC grating was able to be well explained based on the effective medium theory and the scattering loss was negligibly small for such a thickness. Figure 6 shows the measured and calculated phase difference between ${\boldsymbol E}\parallel {\boldsymbol K}$ and ${\boldsymbol E} \bot {\boldsymbol K}$. For comparison, the results for homogeneously aligned LC cell structures (i.e., for $F = 1$ and $F = 0$) are also shown in Fig. 6. The homogeneous LC cell was fabricated without the photomask exposure. The thickness of the LC layer was also adjusted to 0.34 mm. The measured phase difference for the homogeneous LC cell agreed well with the calculation results for $F = 1$ or $F = 0$ . This indicated that the values of the calculation parameters, ${n_\textrm{o}}$ and ${n_\textrm{e}}$, were nearly reasonable for the LC used in the experiment. The measured phase difference for the grating LC cell was inconsistent with the calculated data. The birefringence estimated based on the measured phase difference was ${\mathop{\rm Re}\nolimits} ({{n_\parallel } - {n_ \bot }} )={-} 0.03$, while the theoretical birefringence was $- 0.01$. The measured birefringence corresponds to the theoretical value for $F = 0.43 \pm 0.01$ (i.e., $w \sim 21{ \mu}\textrm{ m}$). Based on Fig. 4, we think that a gap on this scale may exist in the fabricated cell. We should precisely adjust F to reduce the absolute value of the effective birefringence. The difference between the measurement and simulation results was also affected by the refractive indices of the LC, their dispersions, and the alignment distribution throughout the cell. It is difficult to characterize these effects quantitatively. However, the experimental and calculation results clearly demonstrate that the effective birefringence can be controlled by the subwavelength alignment distribution.
Fig. 5. Transmission spectra of the subwavelength LC grating: (a) the measured data and (b) the calculated data. The red (blue) open circles represent the measured transmittance for ${\boldsymbol E}\parallel {\boldsymbol K}$ (${\boldsymbol E} \bot {\boldsymbol K}$). The red (blue) lines represent the calculated transmittance for ${\boldsymbol E}\parallel {\boldsymbol K}$ (${\boldsymbol E} \bot {\boldsymbol K}$). The green (orange) circles represent the measured transmittance of the quartz glass substrate (the quartz glass substrate with the NiCr layer).
Fig. 6. Phase differences between the transmitted waves for ${\boldsymbol E}\parallel {\boldsymbol K}$ and ${\boldsymbol E} \bot {\boldsymbol K}$. The circles, triangles, and squares represent the measured data for $F \sim 0.5$ (the fabricated LC grating), $F = 1$, and $F = 0$. The red, blue, and green lines represent the theoretical data for $F = 0.5$, $F = 1$, and $F = 0$.
Figure 7 shows phase shifts induced by an applied voltage. The phase shift was defined as the difference between the phase angle of the complex transmittance of the LC cell with no voltage and that with a sufficiently high voltage. In the experiment, we fabricated the LC cell with the subwavelength grating structure using NiCr-coated substrates. NiCr thin layers, which were coated on the substrates using vacuum vapor deposition (SVC-7PS 80, Sanyu Electron, Japan), were employed as transparent electrodes in the THz spectral range [41,42]. The measured transmittance spectra of the substrate without and with the NiCr layer are shown in Fig. 5(a). This show that the transmission loss of the NiCr layer is relatively small. The sheet resistance, which was measured using a for probe system (SR4-SS, Astellatech, Japan), was ${\sim} {10^3}{ \Omega}\textrm{ /sq}$. The thickness, which was measured using an optical interference microscope (VertScan, Ryoka Systems, Japan), was 0.01 µm. The conductivity was ${\sim} {10^5}\textrm{ S/m}$. A square-wave voltage with a frequency of 1 kHz and an amplitude of 5.0 V, which is a sufficient voltage to switch from the planar alignment to the homeotropic alignment, was applied to the LC via the NiCr layers [Fig. 3(b)]. In the measured data, the phase shift for ${\boldsymbol E}\parallel {\boldsymbol K}$ was nearly the same as that for ${\boldsymbol E} \bot {\boldsymbol K}$ [Fig. 7(a)]. The phase shift amount agreed well with the calculations [Fig. 7(b)]. In the calculations, the refractive index of the LC under the applied voltage was assumed as ${n_\textrm{o}}$, and it had no dependence on the polarization state. The other parameters were the same as mentioned above. These results demonstrate that a subwavelength LC grating with a suitable F can realize a polarization-independent phase shift by changing the alignment from a planar to a homeotropic states. The response times for the applied voltage have not been investigated yet. However, we can estimate that the switching-on time and the switching-off time are a few seconds and a few tens of seconds, respectively [20,43]. For practical use, the response times should be improved by using highly birefringent LCs, polymer-stabilized LCs, multi-layered structures, multi-electrode structures, highly birefringent metamaterials, etc [19,21,43,44].
Fig. 7. Phase shifts induced by the applied voltage: (a) the measured data and (b) the calculated data. The red (blue) open circles represent the measured phase shift for ${\boldsymbol E}\parallel {\boldsymbol K}$ (${\boldsymbol E} \bot {\boldsymbol K}$). The red (blue) lines represent the calculated phase shift for ${\boldsymbol E}\parallel {\boldsymbol K}$ (${\boldsymbol E} \bot {\boldsymbol K}$).
4. Conclusions
A zero-birefringence material was designed for the THz spectral range from a subwavelength binary grating of anisotropic media. The subwavelength grating was fabricated using a nematic LC and photoregulated alignment films. The complex transmittance of the LC grating was measured via THz time-domain spectroscopy and calculated based on the effective medium theory and the Fresnel equations. The absolute value of the effective birefringence of the subwavelength LC grating was 0.03, while the birefringence of the used LC was 0.16. This demonstrated that the phase difference for the two eigen linearly polarized waves can become zero by controlling the duty ratio in the grating. The phase shift induced by an applied voltage was also measured. There was minimal dependence of the phase shift on the polarization state. Accordingly, the subwavelength LC grating was viable for application as a polarization-independent phase shifter. We believe that these subwavelength LC structures can be used to realize a variety of useful THz devices because of their responsivity to applied fields.
Funding
Japan Society for the Promotion of Science (JP18K04259).
Disclosures
The authors declare no conflicts of interest.
References
1. K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11(20), 2549–2554 (2003). [CrossRef]
2. J. F. Federici, B. Schulkin, F. Huang, D. Gary, R. Barat, F. Oliveira, and D. Zimdars, “THz imaging and sensing for security applications - Explosives, weapons and drugs,” Semicond. Sci. Technol. 20(7), S266–S280 (2005). [CrossRef]
3. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, 2009).
4. J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010). [CrossRef]
5. H.-J. Song and T. Nagatsuma, “Present and future of terahertz communications,” IEEE Trans. Terahertz Sci. Technol. 1(1), 256–263 (2011). [CrossRef]
6. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011). [CrossRef]
7. I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Physical Communication 12, 16–32 (2014). [CrossRef]
8. T. Nose, S. Sato, K. Mizuno, J. Bae, and T. Nozokido, “Refractive index of nematic liquid crystals in the submillimeter wave region,” Appl. Opt. 36(25), 6383–6387 (1997). [CrossRef]
9. I. C. Khoo, “Nonlinear optics of liquid crystalline materials,” Phys. Rep. 471(5-6), 221–267 (2009). [CrossRef]
10. R. Dąbrowski, P. Kula, and J. Herman, “High birefringence liquid crystals,” Crystals 3(3), 443–482 (2013). [CrossRef]
11. C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497–4499 (2003). [CrossRef]
12. T.-R. Tsai, C.-Y. Chen, R.-P. Pan, and C.-L. Pan, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wireless Compon. Lett. 14(2), 77–79 (2004). [CrossRef]
13. C.-J. Lin, Y.-T. Li, C.-F. Hsieh, R.-P. Pan, and C.-L. Pan, “Manipulating terahertz wave by a magnetically tunable liquid crystal phase grating,” Opt. Express 16(5), 2995–3001 (2008). [CrossRef]
14. R. Wilk, N. Vieweg, O. Kopschinski, and M. Koch, “Liquid crystal based electrically switchable Bragg structure for THz waves,” Opt. Express 17(9), 7377–7382 (2009). [CrossRef]
15. B. Scherger, M. Reuter, M. Scheller, K. Altmann, N. Vieweg, R. Dabrowski, J. A. Deibel, and M. Koch, “Discrete terahertz beam steering with an electrically controlled liquid crystal device,” J. Infrared, Millimeter, Terahertz Waves 33(11), 1117–1122 (2012). [CrossRef]
16. Y. Wu, X. Ruan, C.-H. Chen, Y. J. Shin, Y. Lee, J. Niu, J. Liu, Y. Chen, K.-L. Yang, X. Zhang, J.-H. Ahn, and H. Yang, “Graphene/liquid crystal based terahertz phase shifters,” Opt. Express 21(18), 21395–21402 (2013). [CrossRef]
17. T. Sasaki, K. Noda, N. Kawatsuki, and H. Ono, “Universal polarization terahertz phase controllers using randomly aligned liquid crystal cells with graphene electrodes,” Opt. Lett. 40(7), 1544–1547 (2015). [CrossRef]
18. C.-T. Wang, C.-L. Wu, H.-W. Zhang, T.-H. Lin, and C.-K. Lee, “Polarization-independent 2 pi phase modulation for terahertz using chiral nematic liquid crystals,” Opt. Mater. Express 6(7), 2283–2290 (2016). [CrossRef]
19. X. Lia, N. Tanb, M. Pivnenkoa, J. Sibikb, J. A. Zeitlerb, and D. Chu, “High-birefringence nematic liquid crystal for broadband THz applications,” Liq. Cryst. 43(7), 955–962 (2016). [CrossRef]
20. T. Sasaki, H. Okuyama, M. Sakamoto, K. Noda, H. Okamoto, N. Kawatsuki, and H. Ono, “Twisted nematic liquid crystal cells with rubbed poly(3,4-ethylenedioxythiophene)/poly(styrenesulfonate) films for active polarization control of terahertz waves,” J. Appl. Phys. 121(14), 143106 (2017). [CrossRef]
21. C.-T. Wang, P.-S. Fang, J.-T. Guo, T.-H. Lin, and C.-K. Lee, “Sub-second switching speed polarization-independent 2 pi terahertz phase shifter,” IEEE Photonics J. 9(6), 1–7 (2017). [CrossRef]
22. J.-P. Yu, S. Chen, F. Fan, J.-R. Cheng, S.-T. Xu, X.-H. Wang, and S.-J. Chang, “Tunable terahertz wave-plate based on dual-frequency liquid crystal controlled by alternating electric field,” Opt. Express 26(2), 663–673 (2018). [CrossRef]
23. T. Sasaki, H. Okuyama, M. Sakamoto, K. Noda, N. Kawatsuki, and H. Ono, “Optical control of polarized terahertz waves using dye-doped nematic liquid crystals,” AIP Adv. 8(11), 115326 (2018). [CrossRef]
24. R. Ito, M. Honma, and T. Nose, “Electrically tunable hydrogen-bonded liquid crystal phase control device,” Appl. Sci. 8(12), 2478 (2018). [CrossRef]
25. T. Sasaki, H. Kushida, M. Sakamoto, K. Noda, H. Okamoto, N. Kawatsuki, and H. Ono, “Liquid crystal cells with subwavelength metallic gratings for transmissive terahertz elements with electrical tunability,” Opt. Commun. 431, 63–67 (2019). [CrossRef]
26. A. K. Sahoo, C.-S. Yang, C.-L. Yen, H.-C. Lin, Y.-J. Wang, Y.-H. Lin, O. Wada, and C.-L. Pan, “Twisted nematic liquid-crystal-based terahertz phase shifter using pristine PEDOT: PSS transparent conducting electrodes,” Appl. Sci. 9(4), 761 (2019). [CrossRef]
27. O. Paul, C. Imhof, B. Lägel, S. Wolff, J. Heinrich, S. Höfling, A. Forchel, R. Zengerle, R. Beigang, and M. Rahm, “Polarization-independent active metamaterial for high-frequency terahertz modulation,” Opt. Express 17(2), 819–827 (2009). [CrossRef]
28. T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley, 2007).
29. S. R. Nersisyan, N. V. Tabriyan, D. M. Steeves, and B. R. Kimball, “Optical axis gratings in liquid crystals and their use for polarization insensitive optical switching,” J. Nonlinear Opt. Phys. Mater. 18(01), 1–47 (2009). [CrossRef]
30. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20(1), 364–376 (2012). [CrossRef]
31. T. Sasaki, K. Noda, H. Ono, and N. Kawatsuki, “Liquid crystal diffraction gratings using photocrosslinkable liquid crystalline polymer films as alignment layers,” in Liquid Crystalline Polymers: Processing and Applications, V. K. Thakur and M. R. Kessler, eds. (Springer, 2015).
32. A. Emoto, M. Nishi, M. Okada, S. Manabe, S. Matsui, N. Kawatsuki, and H. Ono, “From birefringence in intrinsic birefringent media possessing a subwavelength structure,” Appl. Opt. 49(23), 4355–4361 (2010). [CrossRef]
33. H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Highly controllable form birefringence in subwavelength-period grating structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 29(9), 2386–2391 (2012). [CrossRef]
34. H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31(1), 11–19 (2014). [CrossRef]
35. S. Ge, P. Chen, Z. Shen, W. Sun, X. Wang, W. Hu, Y. Zhang, and Y. Lu, “Terahertz vortex beam generator based on a photopatterned large birefringence liquid crystal,” Opt. Express 25(11), 12349–12356 (2017). [CrossRef]
36. Z. Shen, S. Zhou, S. Ge, W. Duan, L. Ma, Y. Lu, and W. Hu, “Liquid crystal tunable terahertz lens with spin-selected focusing property,” Opt. Express 27(6), 8800–8807 (2019). [CrossRef]
37. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2(3), 466–475 (1956).
38. R.-P. Pan, C.-F. Hsieh, C.-L. Pan, and C.-Y. Chen, “Temperature-dependent optical constants and birefringence of nematic liquid crystal 5CB in the terahertz frequency range,” J. Appl. Phys. 103(9), 093523 (2008). [CrossRef]
39. N. Vieweg, M. K. Shakfa, B. Scherger, M. Kikulics, and M. Koch, “THz properties of nematic liquid Crystals,” J. Infrared, Millimeter, Terahertz Waves 31(11), 1312–1320 (2010). [CrossRef]
40. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
41. A. Maka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier–Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016). [CrossRef]
42. T. Unuma and S. Maeda, “Phase shift of terahertz Bloch oscillations induced by interminiband mixing in a biased semiconductor superlattice,” Appl. Phys. Express 12(4), 041003 (2019). [CrossRef]
43. D. C. Zografopoulos, A. Ferraro, and R. Beccherelli, “Liquid-crystal high-frequency microwave technology: materials and characterization,” Adv. Mater. Technol. 4(2), 1800447 (2018). [CrossRef]
44. T. Sasaki, Y. Nishie, M. Kambayashi, M. Sakamoto, K. Noda, H. Okamoto, N. Kawatsuki, and H. Ono, “Active terahertz polarization converter using a liquid crystal-embedded metal mesh,” IEEE Photonics J. 11(6), 1–7 (2019). [CrossRef]
