Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90&#x00B0; twisted nematic liquid crystals

Cheng-Kai Liu; Wei-Hsuan Chen; Ko-Ting Cheng

doi:10.1364/OE.25.022388

1. Introduction

An optical diode, also known as an optical isolator, has attracted substantial attention because of its ability to manipulate light and its potential applications on information technology. Incident light can transmit optical diode in one direction; however, these lights are blocked in the reverse direction [1]. The reported device also shows unique asymmetrical transmission. Studies have successfully fabricated optical diodes based on photonic crystals [2,3]. Moreover, Faraday rotator and linear polarizers can be combined to demonstrate a known optical diode [4]. Several studies reported that the combination of nematic liquid crystals (LCs) and cholesteric LCs (CLCs) could be used to demonstrate non-reciprocal transmissions [1,5,6]. The use of a dye-doped nematic LC, which serves as a defective layer in a periodical structure, to achieve asymmetrical transmission was also investigated [7]. However, the disadvantage of fabricating an optical diode based on LCs/CLCs is that electro-optical properties of these LCs are strongly sensitive to incident light wavelengths; the advantage is that electro-optical tunability of such devices becomes possible [1,8]. The main feature of these optical devices described above is their asymmetrical transmissions.

In the recent decade, polymer network nematic LCs (PN-NLCs) have been studied worldwide because of their optical properties. PN-NLCs can be switched between transparent and scattering states by applying a specific voltage [8–12]. A polymer network consisting of polymer strands/monomers forms along the direction of LC molecules [8–13]. The structures resulting from the orientation of LCs serve as a mold for the formation of polymer networks during photo-polymerization. Generated polymer networks can provide strong anchoring for LC molecules that are close to the polymer fibril and maintain LC orientation. On the other hand, LCs in bulk generate multi LC domains (mLCds) under the applied external voltage. The effective refractive index mismatch between mLCds mainly results in the scattering of incident light [8,14,15]. Moreover, sizes of mLCds determine the scattering of incident lights with various wavelengths [16–19]. Light scattered by homogeneously aligned PN-NLCs was demonstrated to be strongly dependent on the polarization direction of linearly polarized (LP) incident light [8,15,20,21]. Moreover, homeotropically aligned PN-NLCs and PN 90° twisted nematic LCs (PN-90° TNLCs) were introduced to eliminate the scattering performance, which strongly depended on the direction of the LP incident lights [8,9,15,22]. In comparison with homogeneous PN-NLCs, the direct advantages of the LC device using PN-90° TNLCs are better contrast, larger fabrication tolerance, and larger temperature tolerance [23]. Accordingly, considerable attention has been devoted to scattering performances and applications of PN-90° TNLCs in the past decade [9,23–31]. An outgoing light beam through such PN-90° TNLCs maintains LP light and the polarization direction of the incident LP light beam will be rotated 90° as PN-90° TNLCs meet the criteria of Mauguin’s condition [8]. PN-NLCs and/or PN-90° TNLCs based on the well-known recipe (E7 and RM257), consisting of those adopted in this paper, have been widely studied, and many LC devices have been developed based on such an interesting mixture. In 2011, Yamaguchi et al. has investigated a polarization-dependent scattering PN-90° TNLC cell [30,31]. Briefly, they proposed that the light scattering by their PN-90° TNLC cell is caused by the refractive index mismatch between the LCs and the polymer fibrils. Hence, the scattering of incident LP light with its polarization direction parallel (perpendicular) to the rubbing direction is strong (weak). The result indicates that the light scattering by the PN-90° TNLCs with the application of external voltage is strongly dependent on the polarization direction of the incident LP light. By contrast, we herein propose the mechanism from other points of view to elucidate the polarization-dependent light scattering by PN-90° TNLCs based on the polarization transformation of the incident LP light to elliptically polarized (EP) light inside the LC cell and the refractive index mismatch between the generated mLCds. A fully detailed investigation on the optical mechanism of PN-90° TNLCs showing asymmetrical transmission, will be reported. Briefly, regarding the asymmetrical scattering and reflection, the transmittance as a LP light incident from one direction can pass through the LC cell, while exactly the same light incident from the other direction will be scattered. Most importantly, the optically anisotropic reflection, also known as the back scattering, from the PN-90° TNLCs is analyzed. We have theoretically elucidated the corresponding mechanism for the first step, and several experiments have been demonstrated to verify the proposed mechanism.

In this paper, an investigation and systematic analysis of broadband scattering-type optical isolators/linear polarizers with their properties of electrically switchable and asymmetrical transmission using PN-90° TNLCs have been reported. Briefly, the doped reactive mesogen will be polymerized and aligned along 90° TNLCs. LC alignment can continuously rotate from one substrate (along x-axis) to the other (along y-axis), and ± z-axis indicates the propagation directions of the incident lights. Accordingly, the generated polymer network structure will be consistent with that of the 90° TNLCs. Without the application of external voltage, PN-90° TNLCs function as common 90° TNLCs; hence, the polarization of a broadband incident light with linear polarization parallel (or perpendicular) to the front director of LCs penetrating through PN-90° TNLCs will be rotated 90°, as the PN-90° TNLCs meet the criteria of Mauguin’s condition [8]. However, with suitable external voltage application, incident lights propagating along the ± z-axis with a specifically linear polarization direction can penetrate PN-90° TNLCs from one direction, and its polarization direction will be rotated 90°. Such a polarization rotation of 90° is not the key factor to achieve the reported asymmetrical transmission. On the other hand, the LP light, which comes from the opposite direction, can be scattered. The reason for such light scattering results from the refractive index mismatch of the generated mLCds. We discovered that scattering strength is strongly dependent on polarization states and propagation directions of incident lights. Moreover, without applying any external voltage, we observed that the reflection (back-scattering) intensity, resulting from LC and polymer boundaries, of incident light with specifically linear polarization propagating along the ± z-axis is also strongly dependent on its propagation direction. Here, two experiments were designed to directly show the asymmetrical transmission by the PN-90° TNLCs. Moreover, to modulate a LP light by a single LC cell of PN-90° TNLCs, working as a polarization-dependent scattering shutter, the correct side of the LC cell should be confirmed to ensure that the incident polarized light could be scattered [23–29].

2. Methods

2.1 Material and cell preparations

The materials adopted herein were nematic LCs, E7 (~95 wt%, Merck) and reactive mesogen, RM257 (~5 wt%, Merck). The mixture was mixed homogeneously. No photo-initiator was used herein to achieve smooth polymer network structure by slow cross-linking rate. The extraordinary/ordinary refractive indices (n_e/n_o) of E7 and RM257 were 1.746/1.521 and 1.687/1.508, respectively. Notably, the difference of n_e (0.059) between E7 and RM257 was larger than that of n_o (0.013) between E7 and RM257. Two indium tin oxide-coated glass substrates were treated with planar alignment layers and then rubbed along two orthogonal directions (x- and y-axes). An empty cell, with a 12-µm-thick cell gap was fabricated by these two substrates. The empty cell was filled with the homogeneous mixture by capillary action and treated with UV illumination (central wavelength was ~365 nm, intensity was ~10 mW/cm²) for 60 min to gradually generate smooth polymer networks. LC cell surface facing UV light was called the S₁, and the other surface as the S₂. Finally, an LC cell with PN-90° TNLCs structures was completed.

2.2 Fabrications

To generate smooth polymer networks, unpolarized UV light (λ = 365 nm, 10 mW/cm²) was used as light source and illuminated onto the S₁ of the LC cell. During photo-polymerization, RM257 fibrils tended to grow along the direction consistent with LC director orientation [8,9]. Thus, polymer RM257 (p-RM257) network structures were generated following the 90° TNLC structures mold of the original LC orientation. The PN-90° TNLCs were formed; the schematic is shown in Fig. 1(a). Concentration of p-RM257 networks close to S₁ was higher than that close to S₂ because of molecular diffusion [8,9]. However, to elucidate the anisotropic properties of PN-90° TNLCs, we assumed that p-RM257 networks concentration was homogeneously distributed through the whole LC bulk. Moreover, the electro-optical properties of the formed PN-90° TNLCs without the application of an external field were intrinsically consistent with those of the original 90° TNLCs.

Fig. 1 Schematics of (a) PN-90° TNLCs structures under UV illumination (365 nm) and (b) structures of PN-90° TNLCs with an application of external voltage after the photo-polymerization processes are completed. S₁ and S₂ represent command and reference surfaces, respectively.

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3. Mechanism of optically anisotropic properties

To elucidate the mechanism of the optically anisotropic properties of PN-90° TNLCs (Fig. 1), four LP light beams (L1_in, L2_in, L3_in, and L4_in) with different polarization states and propagating directions were introduced as shown in Fig. 1(b). Figure 1(b) presents the structures of PN-90° TNLCs, possessing optical isolator properties by applying a suitable external voltage. As shown in Figs. 1(a) and 1(b), the LC and PVA boundary position that was aligned with x-axis (y-axis) was defined as z = 0 (z = d). The incident lights, L2_in from S₁ and L3_in from S₂, were scattered because they encountered different refractive indices between various mLCds. Refractive indices of mLCds, encountered by L1_in, L2_in, L3_in, and L4_in as they travel into the PN-90° TNLCs, must be investigated to clarify the mechanism of PN-90° TNLCs. Before discussing the refractive indices of various mLCds, changes in linear polarization state of LP lights penetrating through PN-90° TNLCs should be elucidated first. The polarization state of LP light in the bulk of TNLCs satisfying Mauguin’s condition is elliptical polarization [32].

Figure 2 shows the schematic diagrams of the changes of the dimensionless parameter of major (DP_oma) and minor (DP_omi) axis of EP “L1_in or L2_in or L3_in or L4_in” when they were travelling in the LC bulk along ± z-axis in 11 different positions. In Fig. 2(a), L1_in and L2_in travel along the + z-axis from z = 0, and in Fig. 2(b), L3_in and L4_in travel along the −z-axis from z = d. The red, white, green, and blue arrows represent the polarization directions of L1_in, L2_in, L3_in, and L4_in, respectively. The orange–yellow rods, shown in Figs. 2(a) and 2(b), represent the orientation of p-RM257/LC molecules on the xy-plane in 11 different positions at the z-axis, marked as 0, $\frac{d}{10}$ , $\frac{2 d}{10}$ … $\frac{8 d}{10}$ , $\frac{9 d}{10}$ , and d. LCs and p-RM257 fibrils rotate continuously from x-axis to y-axis. Notations of 0, d, and $\frac{5 d}{10}$ represent the positions of S₁, S₂, and middle bulk, respectively. The 90° TNLCs can be divided into homogeneous LC layers with identical thicknesses, and the LC director of each homogeneous layer can be uniformly oriented as an external voltage is applied. Regarding the electro-optical properties, 90° TNLCs can be viewed as the sum of these layers [32]. Moreover, for simplification, we assumed that all major (minor) axes of EP “L1_in or L4_in” and “L2_in or L3_in” are parallel (perpendicular) and perpendicular (parallel) to the fast axis of LC molecules (p-RM257 fibrils) in PN-90° TNLC, respectively, as shown in Figs. 2(a) and 2(b). Moreover, the DP_oma of EP “L1_in or L2_in” (“L3_in or L4_in”) gradually decreases and increases from positions 0 to $\frac{5 d}{10}$ and $\frac{5 d}{10}$ to d (d to $\frac{5 d}{10}$ and $\frac{5 d}{10}$ to 0), respectively. To simplify the analysis, we assume that the changes of the DP_oma of the EP in the bulk can be normalized to 1 and are described as Eqs. (1) and (2),

D P_{o} m a = \cos (\frac{π}{2 d} h), 0 \leq h < \frac{5 d}{10} .

D P_{o} m a = \cos (\frac{π}{2} - \frac{π}{2 d} h), \frac{5 d}{10} \leq h \leq d .

where h (ranging from 0 to d) represents position in the bulk of LC cell. As h = 0 and h = d, Eqs. (1) and (2) show that DP_oma is 1. This result is reasonable because “L1_in or L2_in” and “L3_in or L4_in” are LP at positions 0 and d, respectively. Moreover, DP_omi of EP “L1_in or L2_in” (“L3_in or L4_in”), which gradually increases and decreases from positions 0 to

\frac{5 d}{10}

and

\frac{5 d}{10}

to d (d to

\frac{5 d}{10}

and

\frac{5 d}{10}

to 0), respectively, can also be normalized to 1 and is described as Eqs. (3) and (4),

Fig. 2 Schematics polarization states variations of incident light in the bulks of LC cell. The orange rods represent the orientation of p-RM257/LC molecules on the xy-plane at 11 positions, marked as 0, $\frac{d}{10}$ , $\frac{2 d}{10}$ … $\frac{8 d}{10}$ , $\frac{9 d}{10}$ , and d. (a) L1_in and L2_in travel along the + z-axis from z = 0, and (b) L3_in and L4_in travel along the −z-axis from z = d.

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D P_{o} m i = \sin (\frac{π}{2 d} h), 0 \leq h < \frac{5 d}{10} .

D P_{o} m i = \sin (\frac{π}{2} - \frac{π}{2 d} h), \frac{5 d}{10} \leq h \leq d .

As h = 0 and h = d, Eqs. (3) and (4) show that DP_omi is 1. This result is reasonable because “L1_in or L2_in” and “L3_in or L4_in” are LP at positions 0 and d, respectively. Normalized DP_oma and DP_omi could be viewed as normalized light intensity with polarization along the major and minor axes of EP light. The maximum and minimum values of normalized DP_oma (DP_omi) were defined as 1 (0.707) and 0.707 (0), respectively. At position (h) of $\frac{5 d}{10}$ , DP_oma and DP_omi values remained the same (0.707) because the linear polarization of the incident LP light was transferred to the circular polarization for these two cases.

Figure 3 shows the calculation variations of DP_oma from the perspective view of “L1_in or L4_in” and DP_omi from the perspective view of “L2_in or L3_in” as a function of the positions in LC cell based on Eqs. (1)–(4). Notably, the two curves shown in Fig. 3 were symmetric with respect to the middle of the LC cell, also known as $h = \frac{5 d}{10}$ . Clearly, except at $h = \frac{5 d}{10}$ , all values shown as red diamonds were larger than those shown as blue squares in Fig. 3. The minimum value of the red curve was equal to the maximum value of the blue one at $h = \frac{5 d}{10}$ . The value of red diamonds was considerably larger than that of blue squares as h was close to 0 and d. Considering the above-mentioned assumption on Figs. 2(a) and 2(b), we found that the DP_oma of “L1 or L4” and DP_omi of “L2 or L3” encountered similar refractive indices in different mLCds, which is the ordinary refractive index (n_o) of the used LCs with/without application of external voltage. The assumption is reasonable because the LC director of each homogeneous layer can be uniformly oriented as an external voltage is applied. We also assumed that all major (minor) axes of EP L1_in and L4_in (L2_in and L3_in) are parallel to the fast axis of LC molecule/p-RM257 fibrils, so that they are nearly parallel to the fast axis of LC director of each homogeneous layer when a suitable voltage is applied. Therefore, DP_oma of “L1 and L4” and DP_omi of “L2 and L3” encounter n_o of LC in different mLCds through the whole bulk. Moreover, we also assumed that n_o of used LC polymer was identical with that of used LCs to simplify the discussion. Therefore, incident light could pass through the LC cell without any light scattering [8]. However, the DP_omi of “L1 and L4” and the DP_oma of “L2 and L3” encountered different refractive indices at different mLCds with a suitable application of external voltage. Accordingly, incident light will be scattered [8]. In the next paragraph, we will discuss the scattering performance (SP) of incident lights, including L1_in, L2_in, L3_in, and L4_in, in detail to elucidate the electrically switchable optical isolator of PN-90° TNLCs.

Fig. 3 Variations of DP_oma of “L1_in or L4_in” and DP_omi of “L2_in or L3_in” as a function of position.

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SP of homogeneous PNLC can be described according to Eq. (5), in which the reflection from glass and air boundary was ignored [32–34],

SP = 1 - exp (- C \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} d) = \int_{0}^{d} C \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} e x p (- C \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} h) d h .

where h, C, Δn_eff, and λ_o are position, mLCds average domain size in the bulk, effective refractive index difference, and incident light wavelength, respectively. The h ranged from 0 to d, where d was the cell gap. Notably, Δn_eff values depended on the selection of LCs, whereas effective birefringence of LCs was dependent on applied voltage amplitude [8,24]. The maximum Δn_eff was the birefringence of LCs, also known as (n_e−n_o). Moreover, referring to Eq. (5), C and Δn were considered as constants. The wavelength parameter was

λ_{0}^{2}

, but not

λ_{0}^{4}

, because the incident light was scattered via Mie scattering [34,35]. We mentioned that light scattering was mainly caused by DP_omi of “L1_in or L4_in) and DP_oma of “L2_in or L3_in”. Therefore, based on Eq. (5), SP of “L1_in or L2_in or L3_in or L4_in” can be described by Eqs. (6) and (7) as follows:

\begin{array}{l} S P_{L 1 o r L 4} = \int_{0}^{\frac{5 d}{10}} [\sin (\frac{π}{2 d} h)] C_{D P_{0} m i} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} e x p (- C_{D P_{0} m i} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} h) d h + \\ \int_{\frac{5 d}{10}}^{d} [\sin (\frac{π}{2} - \frac{π}{2 d} h)] C_{D P_{0} m i} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} e x p (- C_{D P_{0} m i} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} h) d h . \end{array}

\begin{array}{l} S P_{L 2 o r L 3} = \int_{0}^{\frac{5 d}{10}} [\cos (\frac{π}{2 d} h)] C_{D P_{0} m a} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} e x p (- C_{D P_{0} m a} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} h) d h + \\ \int_{\frac{5 d}{10}}^{d} [\cos (\frac{π}{2} - \frac{π}{2 d} h)] C_{D P_{0} m a} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} e x p (- C_{D P_{0} m a} \frac{Δ n_{e f f}^{2}}{λ_{0}^{2}} h) d h . \end{array}

SP_{L1 or L4} and SP_{L2 or L3} represent SP of L1_in or L4_in and L2_in or L3_in, respectively, through PN-90° TNLCs.

C_{D P_{0} m a}

and

C_{D P_{0} m i}

represent the averaged domain sizes that DP_oma of “L1 or L4” and DP_omi of “L2 or L3” encountered, respectively. Values of

C_{D P_{0} m a}

and

C_{D P m i}

were considered as constants. According to Fig. 3, all values of red diamonds in each position were larger than those of blue squares, except for the value at

\frac{5 d}{10}

. Therefore, the integral value of Eq. (7) must be larger than that of Eq. (6). Based on the calculation results, we can say that under a suitable application of external voltage, incident light of LP L4_in (L1_in) with polarization direction along the x (y)-axis (Fig. 2) can penetrate through PN-90° TNLCs from S₁; simultaneously, light will be scattered from S_r. Results directly show the evidence of asymmetrical transmission by the reported PN-90° TNLCs. An experiment was designed to further investigate and confirm the statements shown in Eqs. (6) and (7).

4. Results and discussion

4.1 Anisotropic scattering of PN-90° TNLCs

An experiment was designed to demonstrate the properties of asymmetrical transmission by the reported PN-90° TNLCs. The experiment setup, as shown in Fig. 4, was for the measurements of real scattering strengths of L1_in, L2_in, L3_in, and L4_in after passing through PN-90° TNLCs with and without the application of external voltages. Rubbing directions of PVA alignment layers onto S₁ and S₂ were along the x- and y-axes, respectively, to form PN-90° TNLCs. An unpolarized He–Ne laser (λ = 632.8 nm) was used as the probe beam and was incident from S₁ of PN-90° TNLCs. The unpolarized light can be divided into two orthogonally linear polarized lights with equal intensity [(L1_in or L2_in) and (L3_in or L4_in)]. Moreover, an analyzer was placed behind PN-90° TNLCs. A photo-detector was set behind the analyzer to investigate transmittance variations during various external voltage applications. The distance between the photo-detector and LC cell, and the collection angle of the scattering were about 10 cm and 2.3°, respectively. Without applied voltages, the PN-90° TNLC cell acted similar to a common 90° TNLC cell. Referring to Fig. 1(b), degree of linear polarization (DoLP), a value between 0 and 1, was significantly discussed and defined as,

D o L P_{L 1 o r L 2} = \frac{L 1_{o u t} - L 2_{o u t}}{L 1_{o u t} + L 2_{o u t}} .

D o L P_{L 3 o r L 4} = \frac{L 4_{o u t} - L 3_{o u t}}{L 4_{o u t} + L 3_{o u t}} .

DoLP_{L1 or L2} and DoLP_{L3 or L4} were used to examine DoLP of L1_out and L4_out, respectively, after passing through PN-90° TNLCs. Figures 5(a) and 5(b) show the results as He–Ne laser passed through S₁ and S₂ sides, respectively. The red circle (triangle) curves shown in Figs. 5(a) and 5(b) represent transmittance versus voltage (T–V) curves of L1_out (L2_out) and L3_out (L4_out) under applications of various applied voltages, respectively. Regarding the measurements shown in Figs. 5(a) and 5(b), the reflections caused by the glass substrates are considered. Moreover, the red circle and triangle curves in Figs. 5(a) and 5(b) show the T–V curves measured as transmission axis of the analyzer was parallel to x-axis and y-axis, respectively. With applied external voltage of 25 V, L1_in and L4_in penetrated through PN-90° TNLCs, whereas L2_in and L3_in were scattered by PN-90° TNLCs. These experimental results correspond to the analysis described above. The measured transmittances of L1_out, L4_out, L2_out, and L3_out were approximately 72.33%, 77.33%, 2.95%, and 2.63%, respectively. Moreover, both the calculated DoLP values of L1_out and L4_out, as shown in Figs. 5(a) and 5(b), exceeded 0.93 (maximum of 1) as 25 V was applied. Clearly, the obtained results successfully demonstrated the properties of asymmetrical transmission by the reported PN-90° TNLCs. Accordingly, a specific LP light [L2_in and L3_in shown in Fig. 1(b)], travelling from only one direction, can be blocked (scattered) using a single LC cell of PN-90° TNLCs with the application of external voltage. If the applied voltage is turned off, lights with any polarization can pass through the LC cell from both directions. In addition, DoLPs of L1_out and L4_out that were close to 1 strongly indicated that such PN-90° TNLC devices can be applied as scattering-type linear polarizers. Notably, the transmittances of L1_out and L4_out continuously decreased as the applied voltage is higher than 25 V. The orientation of LCs, close to p-RM257 fibrils, rotated as the applied voltage was higher than 25 V; this rotation caused the transmittance to change. Therefore, extra LC domains were generated, which increased light scattering, to decrease the transmittances of L1_out and L4_out. Additionally, the contrast ratios of the polarized lights, L2 and L3, are calculated to be about 12.4 and 13.5, respectively. These two obtained contrast ratios can be tuned to be equal by fabricating the mLCds with the same size in the PN-90° TNLC. The method to approach fixed size mLCds will be discussed in the conclusion section.

Fig. 4 Experiment setup for demonstrating the polarization selective light scattering properties of the reported LC device achieved by PN-90° TNLCs.

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Fig. 5 Variations of transmittance and degree of polarization as functions of applied voltage. The unpolarized lights were incident from (a) command (S₁) and (b) reference (S₂) surfaces. Analyzer_trans. represents the direction of the transmission axis of the analyzer.

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4.2 Anisotropic reflection of PN-90° TNLCs

The transmittance shown as red circle (L1_out) [red triangles (L3_out)] curve was slightly higher than that of the red triangle (L2_out) [red circle (L4_out)] curve, as shown in Fig. 5(a) [5(b)], without the application of external voltage. The difference in their transmittances are mainly caused by the reflections (back-scatterings) resulting from p-RM257 and LCs boundaries [33]. Another cause for the difference could be the imperfection of the generated polymer networks. Figure 6 shows the schematic structures of the proposed PN-90° TNLCs without the application of external voltage. Reflections can be calculated by the following relationships with the condition that all lights were normally incident onto LC cell [34,35],

Fig. 6 Schematic diagram of PN-90° TNLCs structures without any application of external voltage.

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R_{n o} = {(\frac{n_{n o \cdot L C} - n_{n o \cdot p R M 257}}{n_{n o \cdot L C} + n_{n o \cdot p R M 257}})}^{2} = 0.0000184.

R_{n e} = {(\frac{n_{n e \cdot L C} - n_{n e \cdot p R M 257}}{n_{n e \cdot L C} + n_{n e \cdot p R M 257}})}^{2} = 0.000295.

Equations (10) and (11) represent reflections caused by one boundary of p-RM257 and LCs with respect to n_o and n_e, respectively. The n_no-LC, n_no-pRM257, n_ne-LC, and n_ne-pRM257 represent n_o of LC, n_o of p-RM257, n_e of LCs, and n_e of p-RM257, respectively. Under ideal conditions, p-RM257 grows uniformly from one substrate to the other through the bulk of LC cell. Therefore, the DP_oma and DP_omi of “L1 or L4” always encounter R_no and R_ne of both LC and p-RM257, respectively; the DP_omi and DP_oma of “L2 or L3” always encounter R_ne and R_no of both LC and p-RM257, respectively. Moreover, because the polarization state of light traveling in the bulk of PN-90° TNLCs was EP, the reflection calculated from Eqs. (10) and (11) should multiply DP_oma and/or DP_omi to calibrate the light reflections. According to Eqs. (1)–(4), overall reflections of L1 and L2 (R_L1 and R_L2) with LP parallel to y- and x-axes can be described by Eqs. (12) and (13), respectively [34,35].

\begin{array}{l} R_{L 1} \approx (1 - \underset{}{\underset{m - b o u n d a r i e s}{\underset{︸}{{[1 - \cos (\frac{π}{2 d} 0) R_{n o}] \times \dots \times [1 - \cos (\frac{π}{2} - \frac{π}{2 d} d) R_{n o}]}}}}) + \\ (\underset{m - b o u n d a r i e s}{\underset{︸}{1 - \underset{}{\underset{}{{[1 - \sin (\frac{π}{2 d} 0) R_{n e}] \times \dots \times [1 - \sin (\frac{π}{2} - \frac{π}{2 d} d) R_{n e}]}}}}}) . \end{array}

\begin{array}{l} R_{L 2} \approx (\underset{m - b o u n d a r i e s}{\underset{︸}{1 - \underset{}{\underset{}{{[1 - \sin (\frac{π}{2 d} 0) R_{n o}] \times \dots \times [1 - \sin (\frac{π}{2} - \frac{π}{2 d} d) R_{n o}]}}}}}) \\ (\underset{m - b o u n d a r i e s}{\underset{︸}{1 - \underset{}{\underset{}{{[1 - c o s (\frac{π}{2 d} 0) R_{n e}] \times \dots \times [1 - c o s (\frac{π}{2} - \frac{π}{2 d} d) R_{n e}]}}}}}) . \end{array}

Light reflections of R_L1 and R_L2 are functions of the position with respect to the top and bottom substrates. m represents the quantity of the boundary between p-RM257 and LCs. Furthermore, Fig. 7 shows the difference between L1 [R_L1, Eq. (12)] and L2 [R_L2, Eq. (13)] light reflections. The blue triangles and orange circles represent the calculated reflections R_L2 and R_L1, respectively. R_L2 was always higher than R_L1 if the value of m was selected as 6, 10, 30, and 90. Such results gave direct evidence that transmittances (without any application of external voltage) of the red circle (L1_out) [red triangle (L3_out)] curve should be slightly higher than that of the red triangle (L2_out) [red circle (L4_out)] curve in Fig. 5(a) [5(b)]. Theoretically, with proper selection of adopted materials, undesired light reflection can be fully eliminated if the n_no-LC equals the n_no-pRM257, and the n_ne-LC equals the n_ne-pRM257.

Fig. 7 Variations of overall reflection of L1 and L2, also known as R_L1 and R_L2, as a function of m. m represents the quantity of the boundary between p-RM257 and LCs.

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4.3 Asymmetrical polarization-dependent transmission

Furthermore, we experimentally demonstrated that a sole LC cell of PN-90° TNLCs shows asymmetrical polarization-dependent transmission, indicating that a specific LP light, incident from only one side, can pass through the LC cell. The experimental setups and results are shown in Fig. 8. The black and blue arrows represent the rubbing directions of these two substrates, S₁ and S₂, respectively. S₁ (S₂) is close to the camera (light source). The red arrow depicts the linear polarization direction of the light source, which was polarized by a polarizer with the transmission axis parallel to y-axis. Figures 8(b) and 8(d) show the configurations for demonstrating the experimental results presented in Figs. 8(a) and 8(c), respectively. Moreover, the experimental setup shown in Fig. 8(b) [8(d)] was used to demonstrate the real situation of L1_in/out or L4_in/out [L2_in/out or L3_in/out] shown in Fig. 1(b). The rubbing directions of S₁ (S₂) substrates depicted in Figs. 8(b) and 8(d) were perpendicular (parallel) and parallel (perpendicular) to the x-axis, respectively. As a suitable voltage was applied, Fig. 8(a) showed that the backlight [refer to the L1_in and L4_in in Fig. 1(b)] penetrated through the PN-90° TNLCs, and the background image can be observed even if a slight reflection was present. These results are consistent with those shown in Figs. 5(a) and 5(b) (red circle curves). On the other hand, after the PN-90° TNLC cell was rotated 180° along the x-axis, Fig. 8(c) shows that the backlight [refer to the L2_in and L3_in in Fig. 1(b)] did not penetrate through PN-90° TNLCs and was fully scattered, consistent with the results shown in Figs. 5(a) and 5(b) (red triangle curves). Moreover, because the spectrum of backlight covered almost all visible RGB wavelength, the experiment results indicated that the PN-90° TNLC devices can be used for all visible wavelengths.

Fig. 8 Observation of the cases of L1_in/out and L4_in/out [Fig. 2(a)] through PN-90° TNLCs (a) with the application of a suitable external voltage according to the experimental setup shown in (b); (c) observation of the cases of L2_in/out and L3_in/out [Fig. 2(b)] with the application of a suitable external voltage according to the experimental setup shown in (d). Red, black, and blue arrows represent the linear polarization directions of the light source (known as the transmission axis of the polarizer), rubbing directions of the substrate S₁, and rubbing directions of the substrate S₂, respectively (see Visualization 1).

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Furthermore, a video (Visualization 1) for directly exhibiting the asymmetrical polarization-dependent transmission based on the setup, shown in Fig. 8, is given. The initial experimental setup, revealed in the start of the video (Visualization 1), is identical with that shown in Fig. 8(d) without any applied voltage. The LCEO logo can be observed clearly in the beginning, and is blocked by the LC cell at the scattering state when an AC voltage is applied. Such an observation corresponds to that depicted in Fig. 8(c). Thereafter, the LC cell with an applied voltage is rotated 180° along the x-axis, and then the LCEO logo reappears again, consistent with the observation and the experimental setup shown in Figs. 8(a) and 8(b), respectively. One can see the slight light scattering from the LC cell because of the slight mismatch of refractive indices of LC and p-RM257. Finally, when the applied AC voltage is switched off, the LC cell reverts to its common PN-90° TNLC so that one can see the LCEO logo clearly through the LC cell. In summary, the video (Visualization 1) clearly demonstrates the asymmetrical polarization-dependent transmission using a sole LC cell of PN-90° TNLCs. Moreover, the red and blue curves in Fig. 9 show the measured spectra of the PN-90° TNLC device with and without the application of a suitable external voltage using the setup consistent with that shown in Fig. 8(d), respectively. Clearly, the spectra of transmittance and scattering of the LC device cover the wavelengths range of all visible lights (between 450 nm and 650 nm).

Fig. 9 The red and blue curves represent the measured spectra of the PN-90° TNLC device with and without the application of a suitable external voltage using the setup consistent with that shown in Fig. 8(d), respectively.

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Figure 10 depicts the schematic diagrams of the asymmetrical polarization-dependent transmission using a single LC cell of PN-90° TNLCs. Referring to Fig. 10(a), we found that lights with linear polarizations parallel to y-axis traveling along the + z-axis (−z-axis) can (cannot) pass through the PN-90° TNLCs with an applied AC voltage. On the other hand, lights with linear polarizations parallel to x-axis traveling along the + z-axis (−z-axis) cannot (can) pass through the PN-90° TNLCs with an applied AC voltage [Fig. 10(b)]. Notably, the linear polarization direction of the passed lights will be rotated 90°. Considering a special application, this LC device can be utilized to a unidirectional light shutter for LP ambient light when a suitable voltage is applied. One can see through it from only one side, but not from the other side.

Fig. 10 Schematic diagrams of the asymmetrical polarization-dependent transmission using a single LC cell of PN-90° TNLCs with an applied AC voltage for the cases of incident lights with linear polarization directions parallel to (a) y-axis and (b) x-axis traveling along ± z-axis.

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Moreover, response times of PN-90° TNLCs were also measured. The initial voltage off state of PN-90° TNLC was in transmissive state. After the external voltage was applied, mLCds were formed and strongly scattered the incident lights. The rise and fall times were defined as the periods required to change the transmission of PN-90° TNLCs from 90% to 10% and from 10% to 90% of its maximum transmission, respectively. Accordingly, the measured rise and fall times were measured to be approximately 1.25 and 10.35 ms, respectively. Notably, the rise time was considerably shorter than the fall time. This result was reasonable because the orientation of LCs was aligned by the torque produced by the applied external field (rise time), whereas it spontaneously relaxed to its initial state when the applied voltage was turned off (fall time). Regarding the LC devices, response time was short enough for real information/display technology applications [23]. Moreover, such LC devices based on the reported PN-90° TNLCs can be extended to apply to the optical information/communication, requiring asymmetrical transmission.

5. Conclusion

In this paper, we theoretically and experimentally demonstrated the use of PN-90° TNLCs as a broadband and optically asymmetrical scattering and transmission optical device for a specific LP light. The conclusion is based on the homogeneous formation of polymer network through the whole LC cell of PN-90° TNLCs. The key process for generating such structures is the absence of the photo-initiator, which does accelerate the rate of photo-polymerization process. Such electrically switchable PN-90° TNLCs with the asymmetrical polarization-dependent transmission and scattering, achieved by a single LC layer, are almost independent of wavelengths in visible light. Comparing with other LC optical devices showing asymmetric transmission [1,5–7,36,37], we found that their performances are usually dependent on the wavelengths of incident light and need at least two or three LC layers and optical structures. By applying an appropriate voltage onto the LC cell, incident light with a specifically linear polarization (L1_in and L4_in) can penetrate the PN-90° TNLCs from one direction, and its polarization direction will be rotated 90°. However, such a polarization rotation is absolutely not the cause for the reported asymmetrical transmission. On the other hand, referring to Fig. 1(b), the key point is that the LP lights, (L2_in or L3_in) and (L1_in or L4_in), having two different polarization directions with respect to the long axis of the polymerized reactive mesogen (p-RM257) encounter mismatch and match-refractive indices of LC and reactive mesogen, respectively. Regarding our theoretical analysis, we assumed that the sizes of all mLCds in the bulk of LC cell from S₁ to S₂ are the same. However, in reality, the size of mLCds close to S₂ should be larger than that of the others [1]. Referring to Refs [38, 39], the fixed size of mLCds in the bulk of LC cell can be achieved by the off-resonant method. The performance of PN-TNLCs, such as reduction of threshold and operation voltages, further increase of DoLP, decrease of response time, and effect of UV curing light sources polarization state, can be improved further [40,41]. Furthermore, based on the reported results depicted in Figs. 1(b) and 10, modulating a LP light by a single LC cell of PN-90° TNLCs, working as a polarization-dependent scattering shutter, the correct side of the LC cell should be confirmed to ensure that the incident LP light could be scattered. A simplified model is proposed to theoretically analyze the PN-90° TNLCs. A more precise model is under development, and researches on electrically switchable unidirectional light-pass glasses is currently ongoing. Moreover, because a sole cell of PN-90° TNLCs can be used to effectively block unwanted broadband lights with specific linear polarization direction, this device possesses a significant potential for laser/information technology application [1].

Funding

Ministry of Science and Technology of Taiwan (MOST 103-2112-M-008-018-MY3; MOST 106-2112-M-008-002-MY3).

Acknowledgments

We would like to thank Professor Rumiko Yamaguchi of the Graduate School of Engineering and Resource Science, Akita University in Japan, for the helpful discussion. We are also grateful to Mr. Wen-Fa Cheng in our department for making the spectra measurements. Importantly, we sincerely thank the reviewers for their valuable comments and significant suggestions.

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Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals

Abstract

1. Introduction

2. Methods

2.1 Material and cell preparations

2.2 Fabrications

3. Mechanism of optically anisotropic properties

4. Results and discussion

4.1 Anisotropic scattering of PN-90° TNLCs

4.2 Anisotropic reflection of PN-90° TNLCs

4.3 Asymmetrical polarization-dependent transmission

5. Conclusion

Funding

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (10)

Equations (13)

Optics Express