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Abstract
The operational bandwidth of a cholesteric liquid crystal deflector based on the Bragg-Berry effect is analyzed using the two-dimensional finite-difference time-domain method. Despite its similarity in structure with a conventional cholesteric mirror under oblique incidence, the bandwidths of selective reflection and selective diffraction are different. The selective reflection wavelength from the cholesteric mirror has a cosine dependence on the Bragg angle, while that of selective diffraction from the cholesteric deflector has a cosine-squared dependence on the slant angle. We also propose equations that approximate the selective diffraction bandwidth of the deflector. The equations can be used to find the helical pitch required to achieve a deflector with a specified deflection angle and operational wavelength, thereby facilitating the development of cholesteric liquid crustal based diffractive optical elements.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Miniature optical elements with the capability to control light are being actively pursued for applications in wearable devices. Various structures in a variety of material platforms, such as meta-surfaces based on metals or dielectrics [1–3], and diffractive and holographic optical elements, both isotropic and anisotropic, based on organic and inorganic materials have been proposed and investigated [4–7]. Liquid crystals (LCs) are attractive materials by which to fabricate diffractive optical elements owing to their high transparency, large birefringence and solution-processability. Patterning of the LC director, or the local orientation direction of the constituent rod-like molecules, gives rise to the Berry-phase effect, in which light acquires a phase in proportion to twice the orientation angle of the director (i.e., optic axis) measured from some reference axis [8–10]. Cholesteric LCs (ChLCs) in which the LC molecules self-assemble into a helical super-structure are additionally capable of reflecting light by Bragg reflection [11]. The helical superstructure confers the material circular polarization selectivity, so that only polarized light having the same circular handedness as the helical structure is reflected, while the opposite polarization is transmitted. Thus, patterning of the LC director in ChLCs, or control in the structural phase of the helical structure (the “helix phase”), enables circularly polarized light to be Bragg reflected and diffracted at the same time [12,13]. After the first proposal that ChLCs can function as holographic optical elements, various beam shaping optical elements have been demonstrated [14–27]. The ChLC diffractive elements have been developed for optical vortex generation based on the geometric Berry phase associated with the circular Bragg reflection [14–18]. The diffractive elements have also been studied for beam steering or wearable displays because the optical patterned ChLC elements provide high diffraction efficiency on large area [20–23]. Recent progress for beam steering is well summarized in a review paper [22].
The selective reflection phenomenon in a conventional cholesteric mirror has been the subject of thorough investigation and is well understood [11,28–31]. For example, the short- and long-band-edge wavelengths, λr,o and λr,e, of selective reflection at normal incidence can be obtained from λr,o = noP and λr,e = neP, where no and ne are the ordinary and extraordinary refractive indices of the ChLC, respectively, and P is the helical pitch. For oblique incidence, the band-edge wavelengths show an approximately cosine dependence on the incidence angle, as demonstrated by numerical calculations of both the reflectance spectra and photonic band-structure. Recently, it was shown that the band-edge wavelengths can also be approximated by a simple expression [31]. On the other hand, the optical properties of ChLC based holographic optical elements have yet to be analyzed in detail. Although there are some studies for broadening the operational wavelength range of patterned ChLCs [26,27], the effect of patterning on the spectral bandwidth of selective reflection has not been clarified explicitly.
In this work, we focus our attention on cholesteric deflectors where the director is modulated linearly in the direction perpendicular to the helical axis, and examine its light transmission and reflection properties using the two-dimensional (2D) finite-difference time-domain (FDTD) method. Considering the similarity in the structure of the cholesteric deflector with a cholesteric mirror under oblique incidence, we compare the optical properties of the two systems and highlight the differences in the spectral position. We also provide equations that can be used as approximations to obtain the band-edge wavelengths of selective diffraction from the cholesteric deflector. The band-edge wavelengths calculated by the proposed method show good agreement with those calculated by the FDTD method, serving as practical tools to calculate the pitch that is necessary to achieve a deflector with a specific deflection angle and bandwidth. Our work thus extends our understanding of ChLC based holographic optical elements, paving the way for practical applications of these novel holograms.
2. Calculation model and method
Figure 1 schematically illustrates the analyzed system. Figure 1(a) shows a conventional cholesteric mirror with uniform alignment, and a light ray incident on the mirror at angle θi from a medium with refractive index ni. P is helical pitch, which is the distance over which the director rotates by 2π, and no and ne are the ordinary and extraordinary refractive indices of the ChLC molecules. The ChLC behaves as a polarization dependent mirror, and partly reflects and transmits light at angles θr (= θi) and θ, respectively. In a conventional cholesteric mirror, the short- and long-band-edge wavelengths of the selective reflection λr,o and λr,e are given by the following equation [31]:
where θo and θe are the propagation angles in the cholesteric mirror for λr,o and λr,e, and ${\bar{n}_o}$ and ${\bar{n}_e}$ are the average refractive indices along each propagation angle, respectively.
Fig. 1. (a) Schematic of a conventional cholesteric mirror and its selective diffraction effect under oblique incidence. (b) Schematic of a cholesteric deflector and its selective diffraction effect at normal incidence.
Figure 1(b) shows a schematic of a cholesteric deflector and its selective diffraction effect. The deflector has the same refractive index and pitch as the mirror, but the director orientation at the boundary changes linearly along a single direction [the x-axis in Fig. 1(b)] over a period of 2Λ (i.e., the director rotates by π radians over a distance of Λ). The deflector has a slant angle, which we denote by α,
The patterned ChLC acts as a reflective deflector where the diffraction angle θd for an incident light with wavelength λ is given by the diffraction grating equation [19]:3. Results and discussion
We first present electric field distributions of light that is reflected or diffracted by the two systems. The modeled mirror has a reflection band spanning from 750 to 850 nm for normal incidence, and the reflection band blue shifts with an increase in incident angle. Figure 2(a) shows the electric field distribution of light with λ = 750 nm incident on the cholesteric mirror for θi = 20°. In the calculation, the cholesteric mirror was rotated in space, and an s-polarized light was used. It is clearly seen that the incident light is reflected obeying the law of reflection with θr = 20°. The black lines in Fig. 2(a) are guides showing mirror reflection (θr = θi) from the sample. Since a linearly polarized light is used as the incident light, we can see both transmitted and reflected light. The reflected light slightly penetrates the boundary between the ChLC and surrounded medium. This is because the selective reflection is caused by multiple reflections in the ChLC as a periodic medium. Figure 2(b) shows the electric field distribution for the condition with θi = 30° and λ = 700 nm. Here, a shorter wavelength than for the case of θi = 20° is used because the reflection band of the cholesteric mirror shifts to shorter wavelength with increasing incident angle. As is evident from the two calculations, we can confirm that the angle of reflection is equal to the angle of incidence.
Fig. 2. Calculated electric field distributions of light on selective reflection from the conventional cholesteric mirror: (a) λ = 750 nm and θi = 20° and (b) λ = 700 nm and θi = 30°. Calculated electric field distributions of light on selective diffraction from the cholesteric deflector: (c) λ = 700 nm and α = 20° and (d) λ = 600 nm and α = 30°.
Figure 2(c) shows the electric field distribution of selective diffraction from the cholesteric deflector for θi = 0°, α = 20° and λ = 700 nm. The incident polarization here is also s- polarized. From Fig. 2(c), we can see that the reflected component of the light is diffracted by the cholesteric deflector. Since the cholesteric deflector also has circular polarization dependence, the FDTD simulation shows transmitted light and diffracted light. Figure 2(d) shows the electric field distribution for the condition with θi = 0°, α = 30° and λ = 600 nm. The FDTD results in Figs. 2(c) and 2(d) show good agreement with the theoretical diffraction angles of 42.8° and 67.5° given by Eq. (4), which are shown in Figs. 2(c) and 2(d) as black lines. Within the rectangular area representing the ChLC, grey lines are drawn to indicate the periodic modulation of the dielectric tensor. While the directions of the grating vectors of the two systems are the same (have the same inclination angles), the light reflection properties – both the reflection angle and spectral bandwidth – are different. For example, the angle between the incident and reflected lights for the cholesteric mirror is 2θi, while the angle between the incident and diffracted light for the cholesteric deflector is not exactly 2α. Furthermore, light with a wavelength of 700 nm is reflected by the mirror with θi = 30°, while the same light is not diffracted by the deflector with α = 30°. Below we give a physical interpretation for this difference and present equations to quantitatively estimate the operational bandwidth of the cholesteric deflector.
We start by explaining why the selective reflection from the cholesteric mirror shifts as a function of cosθ, as described by Eqs. (1) and (2). Figure 3(a) schematically shows the director distribution of a cholesteric mirror and propagating light rays (shown as black arrows). Because of the head-tail equivalence of the director, the ChLC has a periodicity of P/2, represented by the green lines in Fig. 3(a). For oblique propagation at an angle of θ to the helical axis, the geometrical path difference between two rays is Pcosθ, represented by the red line. Moreover, the light is affected by an average refractive index $\bar{n}$ that depends on the molecular tilt observed by the propagation light. As shown in the three-dimensional (3D) view in Fig. 3(a), the effective anisotropy of the local director varies depending on the position, with the extraordinary refractive index varying from ne to n(θ), and the ordinary refractive index remaining constant as no. Since the total optical path difference is given by the product of Pcosθ and $\bar{n}$, the selective reflection from the cholesteric mirror is represented as Eqs. (1) and (2), explaining the blue shift of selective reflection with increasing incident angle.
Fig. 3. (a) Schematic of the director configuration and propagating light in the cholesteric mirror for oblique incidence and the ChLC molecules seen from the light’s point of view. (b) Schematic of the director configuration and propagating light in the cholesteric deflector for normal incidence and the ChLC molecules seen from the light’s point of view. (c) Relationship between ChLC molecules and traveling light before and after reflection in the cholesteric deflector
Figure 3(b) schematically illustrates the director distribution in the cholesteric deflector. Although the helical axis is still in the z-direction, one realizes that now the grating vector is tilted away from the helical axis [green arrow in Fig. 3(b)]. In terms of the pitch, the grating vector has a period of $\frac{P}{2}\,\cos \alpha $, shortened by a factor of cosα due to the slant. The geometrical path difference between two rays is Pcos2α [red line in Fig. 3(b)]. However, in this case, the effective refractive index is not the same between two red lines before and after reflection in the deflector. This is because before reflection the direction of the light ray is parallel to the helix, while after reflection the light ray travels in the deflector with a certain angle to the helix. The top view in Fig. 3(b) shows the director orientation as seen by the light before reflection. Since the propagation light observes no tilt in the director, the effective low and high refractive indices in the cholesteric deflector are no and ne, respectively. On the other hand, the 3D view in Fig. 3(b) shows the director orientation as seen by the light after the light is reflected in the deflector. The effective anisotropy of the local director varies depending on the position, with the extraordinary refractive index varying from ne to n(α). Therefore, the characteristics of wavelengths for the selective diffraction phenomenon – the short- and long-band-edge wavelengths are given by the following equations:
The center wavelength of selective diffraction λd,c can be expressed as
The band-edge wavelengths obtained from Eqs. (1,2) and (5,6) are compared with those calculated by the FDTD method. Figure 4(a) shows the calculated transmission spectra of the cholesteric mirror for θi = 0°, 10°, 20°, and 30°. Here, we used a 45° linear polarized Gaussian pulse as the input, and obtained the transmission spectrum by calculating the Fourier transform of the temporal response. We confirmed that the transmission spectra obtained by the FDTD method showed good agreement with those obtained by Berreman’s 4×4 matrix method [28]. It is seen that the transmittance at the center wavelength is 46% at θi = 0°, and then decrease with increasing angle. Ideally, the reflectance of ChLCs in the selective reflection band is 50% for a linearly polarized light at normal incidence, because the sinusoidal dielectric tensor profile reflects one circular polarization completely while transmitting the other. The slight decrease in reflectance originates from the change in polarization due to Fresnel reflection at the boundary of the ChLC layer. As θi and hence θ increases, the modulation in dielectric tensor experienced by the light becomes no longer perfectly sinusoidal, and starts to reflect light with both circular polarization senses. The loss in circular polarization selectivity leads to the appearance of the so-called total reflection band at large values of θi. In other words, the circular polarization selectivity is an inherent characteristic of chiral LCs because conventional dielectric multilayers do not exhibit circular polarization characteristics. Strictly, the circular polarization selectivity only appears for normal incidence. For oblique incidence, chiral LCs exhibit elliptical polarization characteristics and show eventually polarization independent reflection [28,30,32].
Fig. 4. (a) Calculated transmission spectra of the cholesteric mirror for θi = 0°, 10°, 20°, and 30°. (b) Calculated transmission spectra of the cholesteric deflector for α = 0°, 10°, 20°, and 30°.
Figure 4(b) shows the calculated transmission spectra of the cholesteric deflector for α = 0°, 10°, 20°, and 30°. The deflector for α = 0° is not slanted and therefore is the same as the cholesteric mirror. In contrast to the mirror, the transmittances in the reflection band are similar for all conditions, ∼50%. This is because the helical axis in the cholesteric deflector is normal to the substrates regardless of α, and the incident light always interacts with a sinusoidally modulated refractive index. Moreover, the rotation of the ChLC molecules at the boundary surface cancels out the change in polarization at the boundary. Thus, a near ideal transmittance of 50% is found for the cholesteric deflectors with α = 10°, 20°, and 30°.
To evaluate the band-edge wavelengths from the FDTD calculations, the wavelengths at which 80% of the maximum reflectance is reached were regarded as the band-edges. Figure 5 compares the angular dependence of the band-edge wavelengths of the cholesteric mirror and deflector. The blue and red broken lines represent the values obtained from Eqs. (1) and (2), respectively, while the blue and red solid lines represent the values obtained from Eqs. (5) and (6), respectively. Therefore, the dark and light gray areas correspond to the reflection band of the mirror and the diffraction band of the deflector, respectively. Triangular and circular markers show the values obtained by the FDTD calculations shown in Figs. 4(a) and 4(b). The FDTD results show good agreement with the analytical curves. As evident from Fig. 5, the band-edge wavelengths of selective reflection shift as a function of cosθ, and those of selective diffraction shift as a function of cos2α. The proposed method thus can determine the band-edge wavelengths of selective diffraction from the cholesteric deflector with practical accuracy.
Fig. 5. Angular dependence of the band-edge wavelengths of the cholesteric mirror and deflector. Blue and red markers are obtained by the FDTD method. The blue and red lines are calculated using Eqs. (1), (2), (5), and (6). Dark and light gray areas are the reflection band of the mirror and the diffraction band of the deflector, respectively.
Fig. 6. (a) Wavelength dependence of θd for α = 10°, 20°, 30°, and 40°, in which solid lines are theoretical curves calculated from Eq. (4), and broken lines are the theoretical center wavelengths of diffraction given by Eq. (9). Results for the same α value are represented using the same color. (b) Dependence of θd,c on α which is calculated from Eq. (14).
We also consider a further generalization of Eqs. (5), (6), and (9). If the equation satisfies for all diffractions in the deflector, the diffraction wavelength can be written by
where $n^{\prime}$ is the effective refractive index at the wavelength of interest. Since the traveling light in the deflector is reflected as shown in Fig. 3(c), the light returns back into the incident medium with ni at an angle of 2α. Therefore, Snell’s law at the top boundary from the deflector is given by By substituting Eq. (11) into Eq. (12), we obtainFinally, we discuss the range of α for which reflective diffraction can be observed. Deflection of reflected light can only be observed when the condition θd < 90° is fulfilled, and when the diffraction wavelength of Eq. (4) falls within the reflection bandwidth of the ChLC. This is illustrated in Fig. 6(a), where the solid lines show the wavelength dependence of θd on α calculated from Eq. (4), and the broken lines represent center wavelengths of reflection, λd,c obtained by Eq. (9). One can see that even when θd < 90°, reflective diffraction does not appear at all wavelengths, i.e., the solid and broken curves have an intersection for limited α-values, because the diffraction from the cholesteric deflector is wavelength-selective. To find the limit of α that diffracts light, we denote the diffraction angle at the center wavelength as θd,c (= θd at λd,c). By substituting Eq. (9) into Eq. (4), θd,c is given by
4. Conclusions
The optical properties of a cholesteric deflector based on the Bragg-Berry effect have been analyzed using the 2D-FDTD method. Similar to the selective reflection from a conventional cholesteric mirror, the diffraction from the cholesteric deflector shows both wavelength and polarization selectivity. Analysis of the diffraction wavelength revealed that the operational bandwidth of the cholesteric deflector varies in proportion to the cosine-squared of the slant angle of the grating, which is in contrast to the cholesteric mirror whose reflection bandwidth varies in proportion to the cosine of the incident angle. A physical explanation was given based on the interaction of light with the director in the ChLC, as well as equations that approximate the short- and long-band-edge wavelengths of selective diffraction. The equations serve as practical tools to calculate the ChLC pitch required to realize deflectors with specific diffraction angles and bandwidths, thereby facilitating the development of diffractive deflectors based on the Bragg-Berry effect.
Funding
Japan Society for the Promotion of Science (17H02766, 18H04514, 19H02581); Precursory Research for Embryonic Science and Technology (JPMJPR151D); Osaka University (Innovation Bridge Grant).
Disclosures
The authors declare no conflicts of interest.
References
1. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]
2. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatiall resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015). [CrossRef]
3. G. Zheng, H. Muhlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015). [CrossRef]
4. T. Todorov, L. Nikolova, and N. Tomova, “Polarization holography 2: Polarization holographic gratings in photoanisotropic materials with and without intrinsic birefringence,” Appl. Opt. 23(24), 4588–4591 (1984). [CrossRef]
5. G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light: Sci. Appl. 3(9), e203 (2014). [CrossRef]
6. K. Wakunami, P. Hsieh, R. Oi, T. Senoh, H. Sasaki, Y. Ichihashi, M. Okui, Y. P. Huang, and K. Yamamoto, “Projection-type see-through holographic three-dimensional display,” Nat. Commun. 7(1), 12954 (2016). [CrossRef]
7. V. Cazac, A. Meshalkin, E. Achimova, V. Abashkin, V. Katkovnik, I. Shevkunov, D. Claus, and G. Pedrini, “Surface relief and refractive index gratings patterned in chalcogenide glasses and studied by off-axis digital holography,” Appl. Opt. 57(3), 507–513 (2018). [CrossRef]
8. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: Switchable helical mode generation,” Appl. Phys. Lett. 88(22), 221102 (2006). [CrossRef]
9. C. Oh and M. J. Escuti, “Achromatic Diffraction from Polarization Gratings with High Efficiency,” Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef]
10. N. V. Tabiryan, S. R. Nersisyan, D. M. Steeves, and B. R. Kimball, “The promise of diffractive waveplates,” Opt. Photonics News 3, 40–45 (2010). [CrossRef]
11. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4(3), 219–226 (1951). [CrossRef]
12. J. Kobashi, H. Yoshida, and M. Ozaki, “Planar optics with patterned chiral liquid crystals,” Nat. Photonics 10(6), 389–392 (2016). [CrossRef]
13. M. Rafayelyan, G. Tkachenko, and E. Brasselet, “Reflective Spin-Orbit Geometric Phase from Chiral Anisotropic Optical Media,” Phys. Rev. Lett. 116(25), 253902 (2016). [CrossRef]
14. J. Kobashi, H. Yoshida, and M. Ozaki, “Polychromatic Optical Vortex Generation from Patterned Cholesteric Liquid Crystals,” Phys. Rev. Lett. 116(25), 253903 (2016). [CrossRef]
15. M. Rafayelyan and E. Brasselet, “Bragg-Berry mirrors: reflective broadband q-plates,” Opt. Lett. 41(17), 3972–3975 (2016). [CrossRef]
16. M. Rafayelyan and E. Brasselet, “Spin-to-Orbital Angular Momentum Mapping of Polychromatic Light,” Phys. Rev. Lett. 120(21), 213903 (2018). [CrossRef]
17. P. Chen, L. Ma, W. Duan, J. Chen, S. Ge, Z. Zhu, M. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y. Lu, “Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing,” Adv. Mater. 30(10), 1705865 (2018). [CrossRef]
18. M. G. Nassiri, S. Y. Cho, H. Yoshida, M. Ozaki, and E. Brasselet, “High-order Laguerre-Gauss polychromatic beams from Bragg-Berry flat-optics,” Phys. Rev. A 98(6), 063834 (2018). [CrossRef]
19. J. Kobashi, Y. Mohri, H. Yoshida, and M. Ozaki, “Circularly-Polarized Large-Angle Reflective Deflectors Based on Periodically Patterned Cholesteric Liquid Crystals,” Opt. Data Process. Storage 3(1), 61–66 (2017). [CrossRef]
20. Y. H. Lee, K. Yin, and S. T. Wu, “Reflective polarization volume gratings for high efficiency waveguide-coupling augmented reality displays,” Opt. Express 25(22), 27008–27014 (2017). [CrossRef]
21. K. Yin, Y. Lee, Z. He, and S. T. Wu, “Stretchable, flexible, rollable, and adherable polarization volume grating film,” Opt. Express 27(4), 5814–5823 (2019). [CrossRef]
22. Z. He, F. Gou, R. Chen, K. Yin, T. Zhan, and S. T. Wu, “Liquid Crystal Beam Steering Devices: Principles, Recent Advances, and Future Developments,” Crystals 9(6), 292 (2019). [CrossRef]
23. I. Nys, M. Stebryte, Y. Y. Ussembayev, J. Beeckman, and K. Neyts, “Tilted Chiral Liquid Crystal Gratings for Efficient Large-Angle Diffraction,” Adv. Opt. Mater. 7(22), 1901364 (2019). [CrossRef]
24. J. Kobashi, H. Yoshida, and M. Ozaki, “Circularly-polarized, semitransparent and double-sided holograms based on helical photonic structures,” Sci. Rep. 7(1), 16470 (2017). [CrossRef]
25. Y. Mohri, J. Kobashi, H. Yoshida, and M. Ozaki, “Morpho-Butterfly-Inspired Patterning of Helical Photonic Structures for Circular-Polarization-Sensitive, Wide-Angle Diffuse Reflection,” Adv. Opt. Mater. 5(7), 1601071 (2017). [CrossRef]
26. J. Kobashi, H. Yoshida, and M. Ozaki, “Broadband optical vortex generation from patterned cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 646(1), 116–124 (2017). [CrossRef]
27. M. Rafayelyan, G. Agez, and E. Brasselet, “Ultrabroadband gradient-pitch Bragg-Berry mirrors,” Phys. Rev. A 96(4), 043862 (2017). [CrossRef]
28. D. W. Berreman and T. J. Scheffer, “Bragg Reflection of Light from Single-Domain Cholesteric Liquid-Crystal Films,” Phys. Rev. Lett. 25(9), 577–581 (1970). [CrossRef]
29. R. Dreher, G. Meier, and A. Saupe, “Selective Reflection by Cholesteric Liquid Crystals,” Mol. Cryst. Liq. Cryst. 13(1), 17–26 (1971). [CrossRef]
30. H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental Studies on Reflection Spectra in Monodomain Cholesteric Liquid Crystal Cells: Total Reflection, Subsidiary Oscillation and Its Beat or Swell Structure,” Jpn. J. Appl. Phys. 22(Part 1, No. 7), 1080–1091 (1983). [CrossRef]
31. R. Ozaki, “Simple model for estimating band edge wavelengths of selective reflection from cholesteric liquid crystals for oblique incidence,” Phys. Rev. E 100(1), 012708 (2019). [CrossRef]
32. R. Ozaki and H. Moritake, “Wavelength and bandwidth tunable photonic stopband of ferroelectric liquid crystals,” Opt. Express 20(6), 6191–6196 (2012). [CrossRef]
