Abstract

Liquid crystal thin films obtained by spincoating and photopolymerization are widely used nowadays for geometric phase optical components. The liquid crystal-air interface during photopolymerization plays a crucial role as it should not disturb the photoalignment induced by the other interface. When photopolymerizing the liquid crystal layer in vacuum, processing parameters need to be optimized carefully to avoid alignment issues at the air interface. In this work we demonstrate that domains are formed due to a tilt angle which is induced at the air interface under certain conditions. A measurement method is developed to quantify the tilt angle at the interface. The method consists of optical analysis of the domains using crossed-polarizer transmission microscopy with an oblique sample holder. By careful analysis of the obtained tilt angles for different parameters, it is assumed that a non-negligible homeotropic anchoring strength at the air interface is responsible for the domain formation. The results are explained with a theoretical model that assumes a finite anchoring strength at both the photoaligned interface and the air interface.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Liquid crystal polymers with well controlled orientation are attractive due to their extraordinary optical, electrical and mechanical properties [13]. The general approach to obtain ordered polymer networks is through the photo-induced polymerisation of liquid crystal (LC) monomers. Several phases with variety of molecular order, can be frozen into a polymer network by a photo-polymerization procedure, such as nematic LC phases, including patterned nematic phases, and chiral nematic alignments. Moreover smectic phases, which possess not only orientational order but also positional order by aligning the molecules within layers, can be frozen into a polymer layer thereby producing various highly ordered supramolecular materials [4]. Photo-polymerisation proceeds quickly and has the advantage that phase separation and phase conversion during the polymerization process are kinetically suppressed. Free-radical polymerization is used frequently and it is initiated by a photoinitiator which has the required absorption wavelength. During the process, the liquid crystal’s orientation is fixed, including the anisotropic optical properties of the film. Typically, polymerization is carried out in a nitrogen environment, which generally requires the process to be carried out in a glove box. This is because water and oxygen in the air can inhibit the polymerization reaction and affect the quality of the polymer film [2]. A cheaper and less cumbersome alternative is to photopolymerize the film in vacuum.

Polyimide (PI) or polyamide (PA) rubbing technology has been conventionally used to align liquid crystals. The main disadvantage of this method is the mechanical contact with the aligned substrate, which can lead to surface charging and dust. The LC alignment produced by rubbing is not completely homogeneous, especially at the microscopic level. Photoaligment is another alignment technique in which linearly polarized UV or blue light is used to illuminate a photo-responsive material to set the alignment direction [5]. It is a high-quality non-contact method which offers the possibility of micro-patterning the liquid crystal alignment. Micro-patterning can be achieved by controlling the polarization direction of the UV light on the microscale by UV laser interference [6], direct writing [7], spatial light modulators [8] or plasmonic photopatterning [9]. Photoalingment and photopatterning technologies have various applications including functional displays such as LC e-paper devices [10]; flat optical elements such as a thin Pancharatnam lens [11,12]; augmented or virtual reality systems [13,14] and optical communications [8]. Brilliant Yellow (BY) is widely used as photoalignment layer. However, it has been reported that BY is sensitive to environmental humidity [15,16]. Once the humidity exceeds 45%, the photoaligment ability of BY decreases dramatically. Because BY is a water-dissolvable ionic dye and is susceptible to hydrogen bonding, it is essential to pay attention to the moisture in the air during the BY film preparation and usage.

In this work, we deposit polymerized LC layers with different processing parameters on substrates with uniform planar alignment. UV polymerization of the layers occurs in a vacuum box. For certain parameters domains are formed that indicate a non-negligible tilt angle in the layer. We investigate the orientation of the LC at the LC-air interface using a polarization microscope and we interpret these results by combining them with optical simulations using Comsol. With this result, we explore the anchoring strength of the top and bottom interfaces of the LC bulk. By looking at the interface alignment for different operation parameters during polymerization we find a correlation between anchoring strength and processing parameters. Finally, we discuss the results and point out the importance of these findings for functional elements based on in-plane photoalignment and polymerization technology.

2. Sample preparation

2.1 Alignment layer

The process of making the polymerized LC samples is schematically shown in Fig. 1. The ITO coated glass substrates are put into an ultrasonic cleaning machine, successively immersed in soap, acetone, isopropanol, and deionized water for 15 minutes. The ITO has a sheet resistance of about 100 $\Omega$/sq. Afterwards, the substrates are baked at 100 $^{\circ }$C for 4 hours and treated in an ozone plasma. Two different methods are used to align the polymerizable liquid crystals: photoalignment and rubbing. Brilliant yellow is the photoalignment material dissolved in N, N-Dimethylformamide (DMF from Merck) with a concentration of 0.2wt%. The mixture is spincoated on glass substrates (3000rpm, 30s) and baked for 10 min. We tried 90$^{\circ }$C, 100$^{\circ }$C and 110$^{\circ }$C and verified that the baking temperature of the BY layer does not result in any appreciable difference in the sample’s texture. Subsequently, the sample is illuminated with a continuous wave UV laser emitting at $\lambda$ = 355 nm (Coherent, Genesis CX SLM, 100 mW) to achieve uniform planar orientation of the liquid crystal. The required UV laser dose for photoalignment is 17 J/cm$^{2}$. The rubbing alignment material is nylon which is dissolved in 2,2,2-trichloroethanol with 1wt% concentration. The mixture is spincoated onto the glass substrates and rubbed by a rubbing machine after baking at 180$^{\circ }$C for 4 hours.

 figure: Fig. 1.

Fig. 1. Processes of making a polymerized LC sample using different alignment materials: BY and nylon.

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2.2 LC polymer layer

The mixture of reactive mesogens (RMs) contains reactive mesogen RM257 (Haihang Industry), photoinitiator Irgacure 819 (BASF) and inhibitor tert-Butylhydroquinone (Sigma-Aldrich). The mixture is dissolved in toluene with concentrations of 5wt%, 10wt% and 20wt%. After both photo-alignment and rubbing steps, the RM mixture is spincoated onto the alignment layers with 3000 rpm, 30 s. Then, the samples are encapsulated in a chamber with quartz cover (transparent to UV) which is sucked vacuum, heated up to a certain temperature and illuminated with a mercury UV lamp. The illumination dose for polymerization step is less then 8 J/cm$^{2}$.

2.3 Single-layer samples

Three different concentrations of reactive mesogen solution are used to achieve a range of thicknesses, while the rotation speed for spin coating is kept the same. The phase sequence of RM257 is solid to nematic LC at 87$^{\circ }$C and nematic LC to isotropic at 118$^\circ$C [17]. It is necessary to heat the spincoated film up to a certain temperature, first of all because the solvent needs to be evaporated. Second, it is desired that the film is in the nematic state during photopolymerization. The temperature during photopolymerization could be one of the factors affecting the quality of the films, so samples with different photopolymerization temperature are processed, ranging from 60$^\circ$C to 80$^\circ$C, with 5$^\circ$C intervals.

It is vital for the further analysis to know the thickness of the LC layer accurately. For that reason we use a Veeco Dektak profilometer to measure the thickness of the RM polymer layer for each sample. To measure the thickness of a sample we remove the polymer partially from one edge of the glass substrate to create an height difference between polymer and glass surface. A typical profile is shown in Fig. 2. The graph shows that the polymer layer is thoroughly cleaned over the position 0 to 400 $\mu$m. Between 400 $\mu$m and 800 $\mu$m is a transition region. The polymer is not eliminated and the surface flatness is destroyed. After 800 $\mu$m, the surface of the polymer is intact.

 figure: Fig. 2.

Fig. 2. Surface profile of LC polymer (LCP) film.

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Table 1 summarizes the parameters of the different single layer samples that are used in the further analysis. The first 15 samples were not subjected to a pre-baking process, which appears to be vital for avoiding the formation of domains. For nylon samples we use the same processing method including identical concentrations (5%, 10% and 20% ) and curing temperatures (from 60$^\circ$C to 80$^\circ$C) to cast polymer on top. To ensure that the second UV irradiation does not influence the photoalignment of the BY, we examined the images before and after polymerization and found no observable differences.

Tables Icon

Table 1. Information for each BY sample including concentration of RM mixture in toluene, curing temperature and measured thickness of polymer film. Samples 1 - 15 did not go through the pre-baking step. From microscope images, there are no domains found from Samples 16 - 19.

2.4 Multi-layer samples

To obtain samples with larger thickness, the procedure for a single layer is repeated several times to get polymer films with a thickness of up to 2 $\mu$m as shown in Table 2.

Tables Icon

Table 2. Information for each mutilayer BY sample including concentration of RM mixture in toluene and thicknesses.

3. Optical analysis

3.1 Microscope analysis

From observation using a polarization microscope, it is found that the samples with BY alignment layer are prone to defects, while the samples with nylon layer are not. Figure 3 shows pictures of a multi-layer sample with a BY alignment layer, using a polarization microscope. A multi-layer sample is taken as an example because it offers a higher contrast due to the higher retardation, but the single layer samples show similar behavior. When the imposed alignment direction is parallel to the polarizer, the sample looks dark which proves that the alignment is working and that there is no twist. Pictures of the sample at the same position at three tilting angles of the sample: 10$^{\circ }$, 0$^{\circ }$ and −10$^{\circ }$ are taken and oriented 45$^{\circ }$ between the imposed alignment direction and polarizer. This sample shows two types of domain boundaries when observed at 0$^{\circ }$ angle: the bright lines grow along the imposed alignment direction, while the dark lines are deviating from this direction. These lines divide the sample into two types of domains. At zero tilt angle , the domains exhibit the same intensity. At non zero tilt angle, it is clear that two types of domains exist. One type of domain (for example, the small domain in the dashed circle) becomes brighter at 10$^{\circ }$ tilt angle while it becomes darker for a negative tilt angle. Only two discrete brightnesses appear in the entire field of view in the tilt observation. This implies that the azimuthal angle does not deviate from the imposed alignment direction, but there is a tilt angle with opposite sign for the two types of domains. This is very similar to the weak anchoring effect described by Ravi K. Komanduri [18] who mentioned if the layer is thicker, the molecules at the surface will tend to be homeotropic. This implies that at the polymer-air interface there is a tendency for homeotropic alignment. Depending on the anchoring energy at the bottom, the elastic constants and the layer thickness, also the bottom LC alignment may deviate from the planar orientation.

 figure: Fig. 3.

Fig. 3. Microscope images of polymerized LC on BY. (a) Optical scheme of sample orientation. (b) Three pictures are taken with different tilt angle; left picture with a tilt of 10$^\circ$ (right view clockwise rotation); picture in the middle without tilt; right picture with a tilt of −10$^\circ$ (right view anticlockwise rotation).

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3.2 Model for liquid crystal orientation

The LC orientation in the layer is determined by the interplay between elastic forces in the bulk and anchoring at the two interfaces. The elastic energy density can be expressed by the Oseen-Frank’s formula, which in the case of zero twist and only tilt, is reduced to

$$f = \frac{1}{2} K_1 \cos^2\theta \left( \frac{d \theta}{d z} \right)^2 + \frac{1}{2} K_3 \sin^2\theta \left( \frac{d \theta}{d z} \right)^2$$

In this equation, $K_1$ and $K_3$ are the elastic constants for splay and bend, respectively. In the one elastic constant approximation $K_1 = K_3 = K$, this simplifies to

$$f = \frac{1}{2} K \left( \frac{d \theta}{d z} \right)^2$$

The surface anchoring energy is commonly expressed by the Rapini-Papoular expression in which the anchoring energy varies with a sine square with respect to the easy direction [19]. For top and bottom interfaces, this becomes:

$$f_\textrm{b} = W_\textrm{b} \sin^2\left(\theta_\textrm{b0} - \theta(0)\right)$$
$$f_\textrm{t} = W_\textrm{t} \sin^2\left(\theta_\textrm{t0} - \theta(d)\right)$$

$\theta _\textrm {b0}$ and $\theta _\textrm {t0}$ are the preferential orientation directions at the bottom and top interface. $d$ is the thickness of the polymer layer. Using the Euler-Lagrange formalism, the minimization of the energy can be transformed into a differential equation and two boundary conditions.

$$\frac{d^{2}\theta(z)}{dz^{2}}=0$$
$$\frac{W_\textrm{b}}{2K}\sin(2(\theta_\textrm{b0}-\theta(0))-\frac{d\theta(z)}{dz} \bigg|_{z=0} = 0$$
$$\frac{W_\textrm{t}}{2K}\sin(2(\theta_\textrm{t0}-\theta(d))+\frac{d\theta(z)}{dz} \bigg|_{z=d} = 0$$

The solution of this equation is a linearly varying tilt angle as a function of $z$ with a certain value for the angle $\theta$ at bottom and top: $\theta _b = \theta (0)$ and $\theta _t = \theta (d)$. In other words, the values $\theta _b$ and $\theta _t$ fully determine the orientation of the LC in the layer, because the angle $\theta$ varies linearly between these two values. In the remainder of the analysis, we assume that the bottom surface anchoring strength is weak to medium in the case of BY with a planar easy direction $\theta _\textrm {b0} = 0^\circ$, while for the nylon-LC interface we assume strong anchoring with planar aligment with a small pretilt angle in the order of 2$^\circ$. At the top interface (LC-air) we assume that a weak homeotropic anchoring is induced.

Figure 4 shows the relationship between surface inclination, bulk thickness and bottom anchoring energy. The $W'_\textrm {t}$ and $W'_\textrm {b}$ in the left figure are the dimensionless top and bottom anchoring energies, respectively, and the expressions are as follows:

$$W'_\textrm{b}=\frac{W_\textrm{b}d}{2K}$$
$$W'_\textrm{t}=\frac{W_\textrm{t}d}{2K}$$

 figure: Fig. 4.

Fig. 4. Top surface tilt angle $\theta _\textrm {t}$ indicating the liquid crystal orientation as a function of sample thickness and the anchoring energy $W_\textrm {b}$: (a) LC bulk cross-sectional schematic. (b) Relation between top surface tilt angle and dimensionless top anchoring energy. (c) Top surface tilt angle as a function of thickness $d$ when top anchoring energy is $10 \times 10^{-6}$ J/m$^2$. (d) Top surface tilt angle as a function of fixed bottom pretilt angle when normalized top anchoring energy is 0.2, 0.4 and 0.6.

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From Fig. 4(b) we can see that if $\theta _\textrm {b}=0^\circ$, $\theta _\textrm {t}$ equals $\theta _\textrm {b}$ when the top anchoring energy $W'_\textrm {t}$ is small. The angle increases rapidly after reaching a certain critical energy. If $\theta _\textrm {b0}=2^\circ$, $\theta _\textrm {t}$ increases more gradually but remains small until $W'_\textrm {t}=0.4$. When $W'_\textrm {t}$ reaches the value of about 0.7, the curve coincides with the purple curve for $\theta _\textrm {b}=0^\circ$ and $W'_\textrm {b}=10$. If there is a weak preferential top surface angle $\theta _\textrm {t0}=10^\circ$, $\theta _\textrm {t}$ increases more gradually and converges to a smaller $\theta _\textrm {t}$.

Assuming a value of $W_t = 1 \times 10^{-5}$ J/m$^2$, it is possible to estimate the surface angle $\theta _\textrm {t}$ with thickness $d$ in case the bottom interface has weak planar anchoring as indicated in Fig. 4(c). In the case of weak $W_\textrm {t}$, the surface tilt angle can be kept to 0 within a specific thickness range. After the critical thickness is exceeded, the inclination angle will increase steadily and for very large thickness reach 90$^\circ$ (not shown in the graph). The smaller $W_\textrm {b}$, the smaller this critical thickness. In the case of multilayer samples, the next layer is spincoated on top of the previous layer. It is assumed that the top surface of the previous layer induces relatively strong alignment at the bottom of the next layer. The top surface angle of the previous layer thus accumulates to the next layer. The tilt angle at the top surface increases as the pretilt angle of the bottom surface increases, as shown in Fig. 4(d): when the top anchoring energy is relatively small, the relationship is approximately linear. As the normalized top anchoring increases to 0.4, the top surface tilt angle first increases rapidly from 0, and then increases more slowly. After a certain level of top anchoring, the top surface tilt angle does not increase from zero, e.g. $W'_\textrm {t}$= 0.6 and the angle increases from 30$^\circ$.

3.3 Quantifying the Nylon+RM257 tilt angle

To ensure the top surface tilt angle of Nylon+RM257 samples, we implemented a pretilt measurement setup based on the the crystal rotation measurement [20]. T10 nylon samples have been fabricated with varying thickness of the RM layer. The top surface tilt angle that is extracted from these measurements is shown in Fig. 5. In this graph, also simulations with different anchoring energies are included. For the simulations, we chose infinite bottom anchoring $W_{b}$ for nylon-RM interface and a 2$^\circ$ pretilt. For the RM-Air interface $W_{t}$, different values are chosen as indicated in the graph. The figure shows that the measured points are more or less corresponding to a top anchoring energy value between 5e-5J/m$^{2}$ and 1e-4J/m$^{2}$.

 figure: Fig. 5.

Fig. 5. Top surface tilt angle measurements for 10 Nylon+RM257 samples based on the crystal rotation method are shown as green dots. Top surface tilt angle as a function of fixed bottom pretilt angle 2.8$^\circ$ when top anchoring energy is 2e-5J/m$^2$, 5e-5J/m$^2$, 1e-4J/m$^{2}$ and 2e-4J/m$^{2}$ are shown as lines.

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3.4 Quantifying the BY+RM257 tilt angle

Quite a number of methods exist to accurately measure the pretilt angle of liquid crystal layers, but they require uniform samples and a measurement area of several square millimeter [21]. The crystal rotation method is widely used for planar aligned samples, while spectroscopic measurements can be used for layers with homeotropic alignment [22]. Detecting the pretilt angle of the liquid crystal layers in this work is difficult because the size of these domains is on the micrometer scale, much smaller than the typical light beam diameter of a photospectrometer. Our method consists of taking microscope pictures and analyzing the brightness ratio between the two types of domains. Instead of using the light from the incandescent bulb in the microscope, red LEDs are used which offer the benefit of having a much narrower wavelength range. The LED is switched on and we wait several minutes before starting the measurement because there is an initial drop of power. Afterwards, The power remains stable with a maximum variation of 0.12%. Moreover, the current to the LEDs is fixed to avoid fluctuations of the intensity as a function of time. We then analyse the brightness of the two domains and compare it with calculated values to obtain the pretilt angle. For this analysis, an accurate thickness measurement is required as explained before. To take microscope pictures at an inclination angle, an inclined sample holder ($20^\circ$ with respect to the horizontal plane) is placed onto the sample stage of the polarization optical microscope (POM). The sample stage is rotated and every 5$^\circ$ an image is recorded. The illumination configuration is illustrated in Fig. 6(a). Figures 6(b) and (c) show two ways of placing the sample onto the holder. The imposed alignment direction is either orthogonal to the tilt axis (b), or it is parallel to the tilt axis (c). An example photograph is shown in Fig. 6(d). The different figures are then analysed numerically by generating the histogram of the brightnesses as shown in Fig. 6(e) for one certain figure. The histogram has two main peaks, one corresponding to the darker region and the other to the brighter region. From the values of the horizontal coordinates corresponding to the two peaks, the brightness of the two regions is obtained separately. As shown in Fig. 7, in the orthogonal method, for domains one and two, the transmission peak appears at the same angle, but the value of the transmission is different. One domain has a larger retardation because the LC is on average more parallel to the plane of polarization, as schematically illustrated in Fig. 7(a). In the parallel method, both domains have the same peak transmission values, but at different rotation angles as schematically illustrated in Fig. 7(b). With the tilt angle of the sample, the projection of the LC director on the plane of polarization has different angles for domain one and domain two, which explains the angular shift of the transmission graph. It is clear that both the parallel and the orthogonal measurement can be used to estimate the average LC tilt angle in the layer.

 figure: Fig. 6.

Fig. 6. Diagram of the polarization microscope used to characterize the LC director for oblique illumination: (a) Schematic diagram of POM with an oblique frame set on a sample stage; (b) schematic diagram of an oblique frame with an angle of $20^\circ$. The yellow arrows indicate the alignment direction. In this arrangement, the alignment direction is parallel to the tilt axis; (c) the alignment is orthogonal to the tilt axis; (d) Microscope images of LCP on nylon and BY substrates, respectively; (e) Histogram of microscope image of a BY sample.

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 figure: Fig. 7.

Fig. 7. Illustration of orthogonal and parallel configuration: (a) The cross section of LC film in orthogonal measurement indicating the LC orientation. (b) Cross section of LC film in parallel configuration. (c) Transmission curve for rotating the sample over $180^\circ$ using orthogonal method. (d) Transmission curve for rotating the sample over $180^\circ$ in parallel configuration.

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3.5 Quantifying the domain size

Liquid crystal which is infiltrated in a cavity with weak surface anchoring tends to organize in a Schlieren texture with larger domain size compared to conditions with strong anchoring [23]. Analysis of the domain size in the samples can thus compliment the measurement of the tilt angle as it is related to the anchoring energy. The domain size is calculated using image processing tools in Matlab.

3.6 Optical simulation of sample under an angle

From Eq. (5), we know that the tilt angle in the bulk varies linearly along the thickness. Assuming that the anchoring strength at the BY surface is larger than the anchoring strength at the air interface, we can use the approximation that the pretilt angle at the BY interface $\theta _b$ is zero. In the simulations, the angle $\theta$ starts from $0^\circ$ and it linearly varies towards the top surface tilt angle at the air interface $\theta _t$, while the azimuthal angle $\phi$ remains zero:

$$\theta(y) = \frac{(\theta _\textrm{t}-\theta _\textrm{b})y}{d} \hspace{0.5cm} 0\leq y\leq d$$

Optical transmission simulations are carried out for a liquid crystal layer based on the finite-element method implemented in the commercial software COMSOL Multiphysics, wave optics module. In the simulation, the plane wave is incident from the glass substrate, enters the LC layer and exits towards the air. The refractive index of the glass is set to 1.5 and the liquid crystal layer is expressed by a birefringent medium with ordinary and extraordinary index of refraction of respectively $n_o = 1.508$, $n_e = 1.687$ [24], surrounded by periodic conditions and two perfectly matched layers (PMLs).

3.6.1 Orthogonal configuration

As shown in Fig. 7(c), when the sample is 20$^\circ$ tilted with respect to the horizontal plane and the alignment direction is perpendicular to the tilt axis, there is a difference in the maximum transmission because one domain exhibits a larger retardation than the other one. The transmission under crossed polarizers for the two domains ($\theta _{t}$ from −20$^\circ$ to 20$^\circ$) is simulated as a function of thickness $d$ for values ranging from 0.1 $\mu$m to 0.9 $\mu$m. The resulting transmission is plotted as a function of the surface tilt angle $\theta _\textrm {t}$ and the thickness of the sample in Fig. 8(a). The ripples that appear as a function of thickness are due to Fabry-Perot resonances arising from the reflection at the glass/LC and LC/air interface. To further compare with the experimental data, we are interested in the transmission ratio of the two domains which is shown in Fig. 8(b).

 figure: Fig. 8.

Fig. 8. Results from simulations with the assumption of $\theta _b=0^\circ$ and the sample is tilted at $20^\circ$: (a) 3D transmittance plot with the tilting angle $\theta _t$ and the sample thickness $d$ as axes in orthogonal method. For example the red dots are indicating the transmission of domain 1 and domain 2 from a 0.3$\mu$m thick sample with −10$^\circ$ top surface tilt, respectively. (b) 3D normalized transmittance ratio plot with the tilting angle $\alpha$ and the sample thickness $d$ as axes in orthogonal method. The red dot indicating the transmittance ratio between domain 2 and domain 1 corresponding to the red dots in (a). (c) The relation between the distance of two curves and the tilting angle $\theta _t$ in parallel method which applies to all the thicknesses.

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3.6.2 Parallel configuration

As shown in Fig. 7(b), when the sample is tilted concerning the horizontal plane and the tilting axis is parallel to the LC alignment direction, there is an angular shift in the position of the minimum and maximum for the two domains. The transmission is simulated as a function of angle $\phi$ with varying LC layer thickness $d$ and different surface tilting angles $\theta _\textrm {t}$. The simulations produce a series of curves similar to the ones extracted from experiments. The angular shift between the two curves is approximately linearly dependent on the angle $\theta _\textrm {t}$ and independent of the thickness $d$ as shown in Fig. 8(c).

4. Experimental results and discussion

Consistently through the samples and processing conditions it is found that the polymerizable LC mixture appears in different morphologies on nylon and BY after polymerization. The environment humidity is well controlled in the lab. The LC layer exhibits uniform alignment on the nylon substrate without any observable domains. On the BY substrates, two different types of domains can be observed in samples 1 to 15. From Fig. 2 it is clear that the surface of the samples is flat. These samples are still intact after six months of storage at room temperature and do not exhibit observable degradation, indicating a stable polymerized layer. A first important explanation for the difference is that the rubbed nylon substrate has a pretilt angle, making the LC tilt in the same direction, suppressing the domain with the other tilt direction. Another explanation is that nylon has a stronger anchoring strength than BY.

After spincoating the solution of RM257 on the substrate, the substrate is directly put into the vacuum box at a certain temperature to start the polymerization process. Although most of the solvent has evaporated during the spin-coating process, it is possible that a small amount of solvent is present at photo-polymerization step. The presence of the solvent could alter the properties of the RM257 layer, resulting in altered elastic constants of an altered anchoring energy at the interfaces during the polymerization step. After further experiments it is found that adding a baking step before polymerization makes the domains disappear (sample 16 to 19). A duration of the pre-baking time of 30 s seems to be sufficient to avoid any domains.

By comparing data obtained from experiments on samples with domains with simulated transmission curves, the surface tilt angle for samples 1 to 15 is extracted. The results are plotted as a function of the measured sample thickness in Fig. 9(a). In principle, the orthogonal and parallel configuration should yield the same value for the surface tilt angle. For larger sample thickness, the values for the surface tilt angle correspond well, while some deviations can be observed for smaller sample thickness. This is probably caused by the fact that samples with small thickness have small retardation and hence low transmission under crossed polarizers, which increases noise on the transmission values. From the graph, a clear correlation between thickness and surface tilt angle can be observed: the larger the thickness of the sample, the larger the surface tilt angle $\theta _t$. The surface tilt angle increases from an average of about 6$^\circ$ for small thicknesses up to about 9$^\circ$ for larger thicknesses. The increase of $\theta _t$ as a function of thickness is in agreement with the theoretical predictions of the weak anchoring model of Eqs. (5) to (7), although it is not possible to obtain a quantitative fit of the results with the simple theoretical model. For small thickness, the model predicts a zero top surface tilt angle, which is indeed the case for sample M1 in Fig. 10.

 figure: Fig. 9.

Fig. 9. Results from comparison of optical measurement and simulation: (a) Top surface tilt angle $\theta _t$ as a function of thickness with both orthogonal and parallel method; (b) Relation between sample’s feature size and thickness.

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 figure: Fig. 10.

Fig. 10. Measured $\theta _t$ values for samples M1-M4 (circles) and simulated $\theta _t$ values for different values of the BY/LC anchoring energies $W_b$ and LC/air anchoring energies $W_t$.

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The size of the domains decreases with increasing thickness as can be observed from Fig. 9(b). As mentioned before, it is expected that the domain size decreases with increasing anchoring strength. In this case, only the thickness of the sample changes while other parameters are kept the same, so we expect the anchoring strength to be the same for all sample in Fig. 9(b). When we look at the dimensionless anchoring strength of Eq. (8) we do see that an increase in thickness increases the dimensionless anchoring strength $W_t$. The boundaries of these domains also exhibit a trend, for example, the brighter boundaries grow in the imposed alignment direction and are slightly longer, and conversely the darker boundaries deviate from the direction and are slightly shorter. This indicates that the brighter boundaries require less elastic energy to form.

Often it is required to achieve a certain layer thickness to meet the requirement on the retardation of the film. Spincoating multilayer samples with several stacked layers is a straightforward way to achieve sufficient thickness. However, the individual layers need to be thin enough such that the surface tilt angle is sufficiently small. For samples M2 to M4 it was decided to start with a first layer which is fairly thick. This results in a considerable surface tilt angle at the air interface. Further spincoating of layers on top of this layer then results in a further increase of the surface tilt angle. This method allows to extract the top surface tilt angle which can further indicate the value for the anchoring energy.

As sample M1 is very thin (only 0.05 $\mu$m), the anchoring strength of BY is strong enough to keep the top surface tilt angle in plane. For sample M2, M3 and M4 fairly large first layers thickness is chosen, such that the surface tilt angle is not negligible. Further spincoating of consequent layers then results in a further increase of the top surface tilt angle $\theta _t$ as shown by the circles in Fig. 10. The data of samples M1 to M4 is now used to get an estimation of the top and bottom anchoring energies. For the simulated curves shown in Fig. 10, we assume weak anchoring $W_b$ at the BY interface, strong anchoring with a pretilt equal to $\theta _t$ of the previous interface and the same values for the top anchoring $W_t$ irrespective of whether the layer is deposited onto a BY interface or a polymerized LC interface. Also, we assume a value of $K = 10$ pN. The simulated curves in Fig. 10 show what $\theta _t$ would be if a sample with a certain thickness is fabricated. Two curves are shown for $W_b = 2 \times 10^{-5}$ J/m$^2$ and $W_t = 9\times 10^{-6}$ or $1 \times 10^{-5}$ J/m$^2$. This demonstrates that a small change in $W_t$ can lead to strongly different values for $\theta _t$. An optimization procedure is also run to find the values that give the best match between simulations and measured data. The optimization procedure gave a good match ($W_t = 3.8 \times 10^{-6}$ J/m$^2$, $W_b = 4.8 \times 10^{-6}$ J/m$^2$) for the data in Fig. 10. Note that the surface tilt angle does not decrease a lot when moving from sample M2 to M4, which indicates that the previous layer indeed offers rather strong alignment for the next layer. It must however be stressed that these values still do not provide a good match for the results of the single layer samples, presented in Fig. 9. We do not observe a strong increase of $\theta _t$ in single layer samples when the thickness is increased as expected from the theoretical model (see Fig. 4(c)). A possible explanation is that our model for the anchoring energy is too simple. Better agreement might be obtained by adding higher order terms in the anchoring energy of Eqs. (3) and (4) and/or assuming a $\theta _\textrm {t0}$ which is not equal to 90$^\circ$.

Another aspect worth noting is that for the Nylon+RM257 samples the top surface tilt is larger than for the BY+RM257 samples. As a result, the homeotropic surface anchoring strength appears to be much stronger for Nylon+RM257 compared to BY+RM257. The exact reason at this moment is not yet entirely clear and needs further research to verify. One mayor difference is that in the BY samples the LCs have different pretilt directions (with a positive or negative tilt angle). At the boundary between different domains, this distortion causes a high elastic energy. It is possible that in the BY samples the surface tilt is lowered compared to a uniform sample to reduce the amount of elastic energy. In the nylon sample, there is a uniform pretilt angle and the LCs require less elastic energy to deviate from the plane. Another possibility is that spincoating the RM mixture onto the BY and the nylon layer may cause some molecules to be dissolved and end up inside the LC layer. This may cause a change in the behaviour of the LC, such as the anchoring energy at the air interface. Finally, although many processing conditions, such as humidity and temperature, are accurately controlled, we do see some variation in batches of samples that are processed on a different day.

Since homeotropic alignment strength at the air interface affects the homogeneous alignment of uniformly aligned LC, it will also play a role in in-plane patterned alignments for geometric optical components, such as optical axis gratings and lenses. There, even small values of homeotropic anchoring energy at the LC/air interface may induce strong out-of-plane tilt of the LC. The work of Man-Chun Tseng et al. demonstrates that the polar anchoring strength can be greater if a thin RM polymer layer is applied on top of photo-aligned Azo Dye [25]. This is however part of our future research efforts and is out of the scope of the current work.

5. Conclusion

A novel method for measuring the surface tilt angle is worked out for samples with micrometer scale domains. We demonstrate that the value of the LC surface tilt angle is linked to the surface anchoring energies at the alignment-LC and the LC-air interfaces. The anchoring energy of both nylon and BY alignment materials is analyzed for different processing conditions of polymerized LC layers. Strong influence of processing conditions is observed, i.e. samples exhibited domains or not. It is concluded that baking before photo-polymerization decreases the homeotropic anchoring energy at the LC/air interface drastically, thereby avoiding the formation of a non-negligible surface tilt and associated domains. The temperature during polymerization does not seem to be crucial. The results in this work provide a number of vital guidelines for the production of geometric phase optical components using photoaligned polymerized LC layers.

Funding

Fonds Wetenschappelijk Onderzoek (1S88220N).

Acknowledgment

Migle Stebryte acknowledges support and funding as an SB-PhD Fellow of the research foundation–Flanders (FWO, Grant number 1S88220N).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article. [CrossRef]  

2. K. Gao, H.-H. Cheng, A. K. Bhowmik, and P. J. Bos, “Thin-film Pancharatnam lens with low f-number and high quality,” Opt. Express 23(20), 26086 (2015). [CrossRef]  

3. D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014). [CrossRef]  

4. T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012). [CrossRef]  

5. V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008). [CrossRef]  

6. T. Zhan, J. Xiong, Y.-H. Lee, R. Chen, and S.-T. Wu, “Fabrication of Pancharatnam-Berry phase optical elements with highly stable polarization holography,” Opt. Express 27(3), 2632–2642 (2019). [CrossRef]  

7. X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017). [CrossRef]  

8. B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021). [CrossRef]  

9. Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016). [CrossRef]  

10. V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020). [CrossRef]  

11. T. Zhan, J. Xiong, Y.-H. Lee, and S.-T. Wu, “Polarization-independent Pancharatnam-Berry phase lens system,” Opt. Express 26(26), 35026 (2018). [CrossRef]  

12. K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021). [CrossRef]  

13. J. Xiong and S.-T. Wu, “Planar liquid crystal polarization optics for augmented reality and virtual reality: from fundamentals to applications,” eLight 1(1), 3 (2021). [CrossRef]  

14. S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019). [CrossRef]  

15. X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016). [CrossRef]  

16. O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

17. J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry. [CrossRef]  

18. R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007). [CrossRef]  

19. A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969). [CrossRef]  

20. G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976). [CrossRef]  

21. K.-H. Chen, W.-Y. Chang, and J.-H. Chen, “Measurement of the pretilt angle and the cell gap of nematic liquid crystal cells by heterodyne interferometry,” Opt. Express 17(16), 14143–14149 (2009). [CrossRef]  

22. P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009). [CrossRef]  

23. G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999). [CrossRef]  

24. C.-K. Liu, W.-H. Chen, and K.-T. Cheng, “Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals,” Opt. Express 25(19), 22388–22403 (2017). [CrossRef]  

25. M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015). [CrossRef]  

References

  • View by:

  1. Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
    [Crossref]
  2. K. Gao, H.-H. Cheng, A. K. Bhowmik, and P. J. Bos, “Thin-film Pancharatnam lens with low f-number and high quality,” Opt. Express 23(20), 26086 (2015).
    [Crossref]
  3. D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014).
    [Crossref]
  4. T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
    [Crossref]
  5. V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
    [Crossref]
  6. T. Zhan, J. Xiong, Y.-H. Lee, R. Chen, and S.-T. Wu, “Fabrication of Pancharatnam-Berry phase optical elements with highly stable polarization holography,” Opt. Express 27(3), 2632–2642 (2019).
    [Crossref]
  7. X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
    [Crossref]
  8. B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
    [Crossref]
  9. Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
    [Crossref]
  10. V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
    [Crossref]
  11. T. Zhan, J. Xiong, Y.-H. Lee, and S.-T. Wu, “Polarization-independent Pancharatnam-Berry phase lens system,” Opt. Express 26(26), 35026 (2018).
    [Crossref]
  12. K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
    [Crossref]
  13. J. Xiong and S.-T. Wu, “Planar liquid crystal polarization optics for augmented reality and virtual reality: from fundamentals to applications,” eLight 1(1), 3 (2021).
    [Crossref]
  14. S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
    [Crossref]
  15. X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
    [Crossref]
  16. O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).
  17. J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
    [Crossref]
  18. R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007).
    [Crossref]
  19. A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969).
    [Crossref]
  20. G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
    [Crossref]
  21. K.-H. Chen, W.-Y. Chang, and J.-H. Chen, “Measurement of the pretilt angle and the cell gap of nematic liquid crystal cells by heterodyne interferometry,” Opt. Express 17(16), 14143–14149 (2009).
    [Crossref]
  22. P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
    [Crossref]
  23. G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
    [Crossref]
  24. C.-K. Liu, W.-H. Chen, and K.-T. Cheng, “Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals,” Opt. Express 25(19), 22388–22403 (2017).
    [Crossref]
  25. M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
    [Crossref]

2021 (3)

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

J. Xiong and S.-T. Wu, “Planar liquid crystal polarization optics for augmented reality and virtual reality: from fundamentals to applications,” eLight 1(1), 3 (2021).
[Crossref]

2020 (1)

V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
[Crossref]

2019 (3)

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

T. Zhan, J. Xiong, Y.-H. Lee, R. Chen, and S.-T. Wu, “Fabrication of Pancharatnam-Berry phase optical elements with highly stable polarization holography,” Opt. Express 27(3), 2632–2642 (2019).
[Crossref]

2018 (1)

2017 (2)

X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
[Crossref]

C.-K. Liu, W.-H. Chen, and K.-T. Cheng, “Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals,” Opt. Express 25(19), 22388–22403 (2017).
[Crossref]

2016 (2)

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

2015 (2)

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

K. Gao, H.-H. Cheng, A. K. Bhowmik, and P. J. Bos, “Thin-film Pancharatnam lens with low f-number and high quality,” Opt. Express 23(20), 26086 (2015).
[Crossref]

2014 (2)

D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014).
[Crossref]

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

2012 (1)

T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
[Crossref]

2009 (2)

K.-H. Chen, W.-Y. Chang, and J.-H. Chen, “Measurement of the pretilt angle and the cell gap of nematic liquid crystal cells by heterodyne interferometry,” Opt. Express 17(16), 14143–14149 (2009).
[Crossref]

P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
[Crossref]

2008 (1)

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

2007 (2)

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007).
[Crossref]

1999 (1)

G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
[Crossref]

1976 (1)

G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
[Crossref]

1969 (1)

A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969).
[Crossref]

Abbott, N. L.

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Aida, T.

T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
[Crossref]

Baur, G.

G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
[Crossref]

Beeckman, J.

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
[Crossref]

Berreman, D. W.

G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
[Crossref]

Berteloot, B.

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

Bhowmik, A. K.

Bos, P. J.

Broer, D. J.

D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014).
[Crossref]

Bryan-Brown, G. P.

G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
[Crossref]

Bukusoglu, E.

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Cai, S.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Chang, W.-Y.

Chen, J.-H.

Chen, K.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Chen, K.-H.

Chen, R.

Chen, S.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Chen, W.-H.

Cheng, H.-H.

Cheng, K.-T.

Chigrinov, V.

V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
[Crossref]

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

Chigrinov, V. G.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

de Pablo, J. J.

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Escuti, M.

R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007).
[Crossref]

Escuti, M. J.

X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
[Crossref]

Gao, K.

Guo, Y.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Hasebe, H.

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

He, Q.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Hegde, G.

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

Hou, Y.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Jang, C.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Jiang, M.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Kamal, T.

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Kim, J.

X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
[Crossref]

Komanduri, R.

R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007).
[Crossref]

Kwok, H.

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

Kwok, H.-S.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

Lavrentovich, O.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Lee, B.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Lee, C.-K.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Lee, C.-Y.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

Lee, G.-Y.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Lee, H.-S.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Lee, J.-H.

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Lee, Y.-H.

Liu, C.-K.

Liu, D.

D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014).
[Crossref]

Meijer, E. W.

T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
[Crossref]

Miller, D. S.

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Minori, A.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Moon, S.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Nam, S.-W.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Neyts, K.

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
[Crossref]

Nys, I.

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

Papoular, M.

A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969).
[Crossref]

Park, S.-Y.

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Peng, C.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Rapini, A.

A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969).
[Crossref]

Roth, S. V.

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Sage, I. C.

G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
[Crossref]

Seo, W.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Srivastava, A. K.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

Stupp, S. I.

T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
[Crossref]

Sun, J.

V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
[Crossref]

Sun, K.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Sung, G.

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Takada, H.

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

Takatsu, H.

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

Tolley, M. T.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Tseng, M.-C.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

Vanbrabant, P. J.

P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
[Crossref]

Wang, X.

V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
[Crossref]

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Wang, Y.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Wang, Z.

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Wei, Q.-H.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Wittwer, V.

G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
[Crossref]

Wood, E. L.

G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
[Crossref]

Wu, S.-T.

Xiang, X.

X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
[Crossref]

Xiong, J.

Xue, X.

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

Yaroshchuk, O.

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

Yaroshchuk, O. V.

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

Ye, H.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Yuan, D.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Zhan, T.

Zhang, P.

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Zhou, G.

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Adv. Mater. (1)

Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q.-H. Wei, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals,” Adv. Mater. 28(12), 2353–2358 (2016).
[Crossref]

Advanced Optical Materials (1)

K. Chen, Y. Hou, S. Chen, D. Yuan, H. Ye, and G. Zhou, “Design, fabrication, and applications of liquid crystal microlenses,” Advanced Optical Materials 9, 2100370 (2021).
[Crossref]

Colloq. Phys. (1)

A. Rapini and M. Papoular, “Distorsion D’une Lamelle NÉmatique Sous Champ Magnétique Conditions D’ancrage aux Parois,” Colloq. Phys. 30(C4), C4-54–C4-56 (1969).
[Crossref]

Crystals (2)

V. Chigrinov, J. Sun, and X. Wang, “Photoaligning and photopatterning: new LC technology,” Crystals 10(4), 323 (2020).
[Crossref]

X. Xiang, J. Kim, and M. J. Escuti, “Far-field and fresnel liquid crystal geometric phase holograms via direct-write photo-alignment,” Crystals 7(12), 383 (2017).
[Crossref]

eLight (1)

J. Xiong and S.-T. Wu, “Planar liquid crystal polarization optics for augmented reality and virtual reality: from fundamentals to applications,” eLight 1(1), 3 (2021).
[Crossref]

IDW ’07 - Proceedings of the 14th International Display Workshops (1)

O. Yaroshchuk, G. Hegde, V. Chigrinov, H. Kwok, H. Hasebe, and H. Takatsu, “Photoalignment properties of brilliant yellow dye,” IDW ’07 - Proceedings of the 14th International Display Workshops 3, 1665–1668 (2007).

J. Mol. Liq. (1)

B. Berteloot, I. Nys, X. Xue, J. Beeckman, and K. Neyts, “Rotationally invariant ring-shaped liquid crystal structures between two substrates with different photoalignment,” J. Mol. Liq. 337, 116238 (2021).
[Crossref]

J. Soc. Inf. Disp. (1)

V. G. Chigrinov, H.-S. Kwok, H. Hasebe, H. Takatsu, and H. Takada, “Liquid-crystal photoaligning by azo dyes (Invited Paper),” J. Soc. Inf. Disp. 16(9), 897 (2008).
[Crossref]

Langmuir (1)

D. Liu and D. J. Broer, “Liquid crystal polymer networks: preparation, properties, and applications of films with patterned molecular alignment,” Langmuir 30(45), 13499–13509 (2014).
[Crossref]

Liq. Cryst. (1)

P. J. Vanbrabant, J. Beeckman, and K. Neyts, “Tunable quasi-homeotropic liquid crystal pretilt angle based on competing alignment layers,” Liq. Cryst. 36(12), 1373–1377 (2009).
[Crossref]

Nat. Mater. (1)

X. Wang, D. S. Miller, E. Bukusoglu, J. J. de Pablo, and N. L. Abbott, “Topological defects in liquid crystals as templates for molecular self-assembly,” Nat. Mater. 15(1), 106–112 (2016).
[Crossref]

Nature (1)

G. P. Bryan-Brown, E. L. Wood, and I. C. Sage, “Weak surface anchoring of liquid crystals,” Nature 399(6734), 338–340 (1999).
[Crossref]

Opt. Express (5)

Phys. Lett. A (1)

G. Baur, V. Wittwer, and D. W. Berreman, “Determination of the tilt angles at surfaces of substrates in liquid crystal cells,” Phys. Lett. A 56(2), 142–144 (1976).
[Crossref]

Phys. Rev. E (1)

R. Komanduri and M. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007).
[Crossref]

RSC Adv. (1)

J.-H. Lee, T. Kamal, S. V. Roth, P. Zhang, and S.-Y. Park, “Structures and alignment of anisotropic liquid crystal particles in a liquid crystal cell,” RSC Adv. 4(76), 40617–40625 (2014). Publisher: The Royal Society of Chemistry.
[Crossref]

Sci. Adv. (1)

Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai, “Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation,” Sci. Adv. 5(10), eaax5746 (2019). Publisher: American Association for the Advancement of Science Section: Research Article.
[Crossref]

Sci. Rep. (1)

S. Moon, C.-K. Lee, S.-W. Nam, C. Jang, G.-Y. Lee, W. Seo, G. Sung, H.-S. Lee, and B. Lee, “Augmented reality near-eye display using Pancharatnam-Berry phase lenses,” Sci. Rep. 9(1), 6616 (2019).
[Crossref]

Science (1)

T. Aida, E. W. Meijer, and S. I. Stupp, “Functional supramolecular polymers,” Science 335(6070), 813–817 (2012).
[Crossref]

SID Symp. Dig. Tech. Pap. (1)

M.-C. Tseng, C.-Y. Lee, O. V. Yaroshchuk, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “67.3: Photo-stable Azo dye photoalignment polymer surface for in plane switching liquid crystal displays,” SID Symp. Dig. Tech. Pap. 46(1), 997–1000 (2015).
[Crossref]

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)

Fig. 1.
Fig. 1. Processes of making a polymerized LC sample using different alignment materials: BY and nylon.
Fig. 2.
Fig. 2. Surface profile of LC polymer (LCP) film.
Fig. 3.
Fig. 3. Microscope images of polymerized LC on BY. (a) Optical scheme of sample orientation. (b) Three pictures are taken with different tilt angle; left picture with a tilt of 10$^\circ$ (right view clockwise rotation); picture in the middle without tilt; right picture with a tilt of −10$^\circ$ (right view anticlockwise rotation).
Fig. 4.
Fig. 4. Top surface tilt angle $\theta _\textrm {t}$ indicating the liquid crystal orientation as a function of sample thickness and the anchoring energy $W_\textrm {b}$: (a) LC bulk cross-sectional schematic. (b) Relation between top surface tilt angle and dimensionless top anchoring energy. (c) Top surface tilt angle as a function of thickness $d$ when top anchoring energy is $10 \times 10^{-6}$ J/m$^2$. (d) Top surface tilt angle as a function of fixed bottom pretilt angle when normalized top anchoring energy is 0.2, 0.4 and 0.6.
Fig. 5.
Fig. 5. Top surface tilt angle measurements for 10 Nylon+RM257 samples based on the crystal rotation method are shown as green dots. Top surface tilt angle as a function of fixed bottom pretilt angle 2.8$^\circ$ when top anchoring energy is 2e-5J/m$^2$, 5e-5J/m$^2$, 1e-4J/m$^{2}$ and 2e-4J/m$^{2}$ are shown as lines.
Fig. 6.
Fig. 6. Diagram of the polarization microscope used to characterize the LC director for oblique illumination: (a) Schematic diagram of POM with an oblique frame set on a sample stage; (b) schematic diagram of an oblique frame with an angle of $20^\circ$. The yellow arrows indicate the alignment direction. In this arrangement, the alignment direction is parallel to the tilt axis; (c) the alignment is orthogonal to the tilt axis; (d) Microscope images of LCP on nylon and BY substrates, respectively; (e) Histogram of microscope image of a BY sample.
Fig. 7.
Fig. 7. Illustration of orthogonal and parallel configuration: (a) The cross section of LC film in orthogonal measurement indicating the LC orientation. (b) Cross section of LC film in parallel configuration. (c) Transmission curve for rotating the sample over $180^\circ$ using orthogonal method. (d) Transmission curve for rotating the sample over $180^\circ$ in parallel configuration.
Fig. 8.
Fig. 8. Results from simulations with the assumption of $\theta _b=0^\circ$ and the sample is tilted at $20^\circ$: (a) 3D transmittance plot with the tilting angle $\theta _t$ and the sample thickness $d$ as axes in orthogonal method. For example the red dots are indicating the transmission of domain 1 and domain 2 from a 0.3$\mu$m thick sample with −10$^\circ$ top surface tilt, respectively. (b) 3D normalized transmittance ratio plot with the tilting angle $\alpha$ and the sample thickness $d$ as axes in orthogonal method. The red dot indicating the transmittance ratio between domain 2 and domain 1 corresponding to the red dots in (a). (c) The relation between the distance of two curves and the tilting angle $\theta _t$ in parallel method which applies to all the thicknesses.
Fig. 9.
Fig. 9. Results from comparison of optical measurement and simulation: (a) Top surface tilt angle $\theta _t$ as a function of thickness with both orthogonal and parallel method; (b) Relation between sample’s feature size and thickness.
Fig. 10.
Fig. 10. Measured $\theta _t$ values for samples M1-M4 (circles) and simulated $\theta _t$ values for different values of the BY/LC anchoring energies $W_b$ and LC/air anchoring energies $W_t$.

Tables (2)

Tables Icon

Table 1. Information for each BY sample including concentration of RM mixture in toluene, curing temperature and measured thickness of polymer film. Samples 1 - 15 did not go through the pre-baking step. From microscope images, there are no domains found from Samples 16 - 19.

Tables Icon

Table 2. Information for each mutilayer BY sample including concentration of RM mixture in toluene and thicknesses.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

f=12K1cos2θ(dθdz)2+12K3sin2θ(dθdz)2
f=12K(dθdz)2
fb=Wbsin2(θb0θ(0))
ft=Wtsin2(θt0θ(d))
d2θ(z)dz2=0
Wb2Ksin(2(θb0θ(0))dθ(z)dz|z=0=0
Wt2Ksin(2(θt0θ(d))+dθ(z)dz|z=d=0
Wb=Wbd2K
Wt=Wtd2K
θ(y)=(θtθb)yd0yd