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Abstract
Topological defect arrays in liquid crystal is an emerging optical material for smart windows, displays, gratings, and optical vortex generators. Formation of defect arrays is investigated using vertically aligned nematic liquid crystal cells with pad, crossed-strips, and porous electrodes. The location and types of the defects are identified using a polarized optical microscope. The pads and crossed-strips generate alternative radial and hyperbolic defects. Unexpectedly, the holes create dipoles of radial and hyperbolic defects, and the dipoles align in parallel order. The best dense packing of defects is achieved with the 15μm × 15μm unit cell.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Topological defect array in liquid crystal is an emerging optical material for a wide range of applications, from household items to precision optics, including smart windows [1–6], gratings [7,8], and optical vortex generators [9,10]. Liquid crystal is a birefringent material that has long been used to switch the polarization state of light. Displays, lenses, gratings, and phase modulators are based on the tunable birefringence of liquid crystal. In addition to birefringence, defects in liquid crystal also give rise to scattering and diffraction of light [6,11]. Defects are discontinuities or displacements in the liquid crystal. The refractive index and the optical axis of the liquid crystal around the defect have drastic change, leading to strong scattering of light. Defects can be thought of as nanoparticles in liquid. They scatter the light, giving a milky hazy texture [6]. At the defect core, the nematic liquid crystal is strongly compressed and melts into an isotropic state. Optically, the defects in liquid crystal are similar to the apertures in a thin plastic film, and the passing light diffracts [6,8]. Birefringence, scattering, and diffraction are all highly tunable and switchable since liquid crystal is a fluid and the molecules tend to align with an applied electric field. Topological defect arrays are topologically protected, self-retaining, and self-healing, making them stable and durable optical devices.
Formation of defect arrays can occur spontaneously, induced by oscillatory forcing [12] with ionic impurities [13]. The periodic pattern is composed of vortices of electroconvection (Carr-Helfrich mechanism [14,15]). The specific pattern is determined by the applied wave form [12], and the period can range from the micrometer to submicrometer scale. To ensure the precise positioning of defects, surface treatments such as patterned electrodes [6,16], patterned surface alignment [17], micropillars [8], and cavities [18] are employed. Electric force [6,8,16] and magnetic force [19] can generate radial and azimuthal defects (or umbilical escapes), respectively, which is the Freedericksz Transition [20]. In this research, we choose patterned electrodes to fix the defects without inclusions and for the highly controllable switching.
An array is a two-dimensional crystal, and a crystal is the periodic repetition of a unit cell. Topological defects in liquid crystal can be generated by an electrode. The electric field aligns the liquid crystal and fixes the defect in the designed location. Usually, a unit cell contains one pixel of the electrode, and a defect array is generated by a pixel array of electrodes. Nematic liquid crystals (NLC) are rod-like molecules. Their average orientation is denoted by a director, $\vec n$. The wrapping or winding number of the directors around the singularity is invariant under continuous deformation, called the topological charge. The surface and line defects are characterized by the winding number, $s$. The Euler characteristic of the electrode determines the topological charge of the defect [21,22]. However, additional defects appear around the designed electrode because of the topological charge conservation [23]. The total topological charge before and after the voltage application must be the same, but the number and topological charge of each additional defect can have many combinations. The rotational and mirror symmetry decides the defect combination [24]. The combination of defects resulting from rotational symmetry has been studied in our previous work [16,24]. In this research, we focus on how the size affects the defect combination in a unit cell.
Pad, crossed strips, and a hole on the electrode can generate and fix a topological defect in a vertically aligned liquid crystal cell filled with liquid crystal with a negative dielectric anisotropy (Fig. 1). The liquid crystal director reclines and be perpendicular to the electric field when the voltage is applied. Defect formation with pads [16] and crossed strips [25,26] have been studied before. They are the reference samples in this research. The pad electrode generates a converging radial defect with topological charge of +1. The pad array generates alternative hedgehog and hyperbolic defects, of which the topological charges are +1 and -1, respectively. The crossed strip electrode generates a hyperbolic defect [26]. The crossed strip electrodes also generate alternative +1 and -1 defects. A hole on an electrode was supposed to generate a diverging radial director field, and it was assumed that an array of holes would give a +1 -1 defect array. However, the experiment shows an unexpected result. The +1 and -1 defects form a pair like a dipole. On a porous electrode, the dipoles align parallel to each other.
Fig. 1. The liquid crystal cell. (a) Schematic diagram of a test cell. The electrode is pixelized. Each pixel is a unite cell of the defect array. Liquid crystal has negative dielectric anisotropy ($\Delta \epsilon <0$). Indium-Tin-Oxide (ITO) is the transparent electrode. Alignment layer is CTAB. (b) Schematic diagram of a unit cell. (c) Top view of the liquid crystal director field. The pad, crossed strips, and a hole generate the converging radial, hyperbolic, and diverging radial defects, respectively. Topological charge ($s$) is marked under the director field. Top electrode, light blue. Bottom electrode, blue. Electric field lines, dark blue arrows. Ellipsoids, liquid crystal molecules. Short dark lines, directors.
Small unit cells leads to crowded defects. The defects are closely packed in periodic order, supported by the electric field. The +1 and -1 attract each other and then annihilate to minimize the elastic stress. The closer the defects, the stronger the attraction [27]. The resulting texture is an escape lying on the substrate plane. The lower bound for the unit cell size is primarily governed by the array structure and the liquid crystal elasticity. To determine the minimal unit cell capable of holding a stable defect array, we investigated three distinct electrode configurations and pixel sizes ranging from $10~\mu m \times 10~\mu m$ to $25~\mu m \times 25~\mu m$.
In this research, the defect arrays made of pads, crossed strips and porous electrode are compared. Vertically aligned NLC cells equipped with a large array of patterned electrodes generate the defect array. Polarized optical microscope shows the texture and birefringence color of the defect array. Then defects are identified, and the director field of the liquid crystal is derived based on the birefringence color. The translational and rotational symmetry of the defect array is analyzed to determine the unit cell. The porous electrode produces dipole of topological defects. The spontaneous dipole alignment and domain formation is observed and analyzed. Our investigation reveals that the lower limit of unit cell size is $15~\mu m \times 15~\mu m$.
2. Methods
The structure of the liquid crystal cell is shown in Fig. 1(a). Two sheets of Indium-Tin-Oxide (ITO) glasses coated with alignment material are separated by ball spacers, and liquid crystal is filled in the gap. ITO is the transparent electrode. The top and bottom electrodes are shown in Fig. 1(b). The patterns are fabricated with lithography and wet etching. Three typical electrodes are fabricated: 1) a pad on the bottom and fully covered electrode on the top, 2) crossed strips on the top and the bottom, 3) a hole on the bottom electrode and fully covered electrode on the top. The pixel sizes range from 10 $\mu$m $\times$ 10 $\mu$m to 25 $\mu$m $\times$ 25 $\mu$m.
Nematic liquid crystal (NLC) is with negative dielectric anisotropy ($\Delta \epsilon$), provided by Merck Taiwan. Refer to the patent TW I663250 B [28] for the formulation of the applied NLC. The dielectric constants on the direction parallel ($\epsilon _{\parallel }$) and perpendicular ($\epsilon _{\bot }$) to the NLC director are 3.0 and 6.0, respectively, and thus $\Delta \epsilon$, which is $\epsilon _{\parallel }- \epsilon _{\bot }$, is -3.0. The extraordinary ($n_{e}$) and ordinary ($n_{o}$) refractive indices are 1.589 and 1.485, respectively. Birefringence ($\Delta n$) of the NLC is 0.104. The splay ($k_{11}$) and bend ($k_{33}$) elastic constants of the NLC are 11.7 pN and 15.2 pN, respectively. The liquid crystal is in nematic state at temperature between 0 and 75$^\circ$C. The experiments were performed in room temperature between 24 and 26$^\circ$C.
The ITO glass was in an ultrasonic bath to remove the grease and dirt. The cleaning solutions are diluted neutral detergent, Acetone, Methanol, and deionized water. Each bath is for 10 minutes. The cleaned glasses are blow dried by an air gun to prevent water stain, and then baked in 100$^\circ$C for 30 minutes. CATB in deionized water is with concentration of $0.002~wt\%$. The cleaned substrates were dipped in the CTAB solution for 10 minutes, and pulled out of the bath slowly for the formation of a thin layer of CTAB on the surface of the ITO. Dipped ITO glasses were baked at 110$^\circ$C for 40 minutes to harden the CTAB film. The glasses were separated by resin beads and then assembled, and the liquid crystal was injected by capillary effect.
Applying an AC square wave voltage causes the vertically standing NLC to turn in the plane. In our samples, electroconvection is significant when the frequency is smaller than 50 Hz and voltage higher than 20 V. 100 Hz and 10 V of AC voltage was applied to all the test cells to avoid the electroconvection [15]. The NLC on the edges of the electrode initiates translation and bends away from the glass towards the ITO. The pad or the hole were supposed to generated a converging or a diverging director field, respectively. The NLC in crossed strip electrodes exhibits hyperbolic curvatures [26]. In thin NLC cells, the three-dimensional (3D) NLC configuration can be an escape, a point defect, or a vortex line. Among the three, the umbilical escape is the most stable due to having the lowest elastic free energy [20,29,30]. The two ends of an escape are fixed by the surface defects on the top and bottom substrates. The pad electrode generates an umbilical escape, of which the top view is a converging radial defect with topological charge of +1. The crossed strips generate an hyperbolic escape, and the top view is a hyperbolic defect with topological charge of -1. A hole on the electrode was supposed to create a diverging radial escape with topological charge of +1. However, the result was a pair of radial and hyperbolic defects (details in Defects and Unit Cells).
The polarized optical microscope (POM) is used to identify the topological defect. The defects in the NLC cell under an optical microscope appear as dark dots, as shown in the bright field image in Fig. 3. When the NLC cell is placed between crossed polarizer and analyzer under POM, it shows the Schlieren texture. Dark brushes elongate from the defect core, and the topological charge is the number of brushes divided by 4. Four dark brushes indicate that the topological charge of the defect is +1 or -1. The sign of a defect is determined by rotating the NLC cell in the polarizer and analyzer. If the brushes and the NLC cell rotate in the same direction, the topological charge is positive; if they rotate in the opposite direction, the topological charge is negative.
A standard microscope without polarizers and wave plates shows the bright field image. Defects appear as dark textures in the bright field image, making the location of defects clearly identifiable. Point defects appear as dark dots, and when the defect core size is in the micrometer range, the dark spot shows a concentric circular diffraction pattern [31]. Refer to the bright field images in Fig. 2 and Fig. 3 for examples. Vertical escapes show a transparent liquid texture. Escapes lying on the substrate plane look like transparent tubes [23,31]. Disclinations with topological charge of $\pm 1/2$ show dark thin lines [23,31].
Fig. 2. Unit cells with pad electrode (a), crossed-strips electrode (b), and a hole on the electrode (c). The column on the left show the picture of the electrode and the schematic diagram of the unit cell. The bright field, POM, and POM with retardation plate pictures were taken when the applied voltage was 4 V and 10 V. A and P are the linear polarization of the analyzer and polarizer, respectively. $\gamma$, the slow axis of the retardation plate. The retardation is 530 nm. The black lines in the right column shows the director field.
Fig. 3. Defect arrays generated by the pads, the crossed strips, and the porous electrode. The bright field picture, POM picture, POM with retardation plate picture, and director field are from the top to the bottom. A and P are the linear polarization of the analyzer and the polarizer, respectively. $\gamma$, the slow axis of the retardation plate. The retardation is 530 nm. The applied voltage was 10 V. scale bar, 10 $\mu$m. symbols for the director and defects are in the legend. The area of a unit cell is in the green shade.
NLC cell under POM shows birefringent colors. The color depends on the optical path difference (OPD) between the ordinary and extraordinary rays, and the transmittance of the light depends on the angle between the polarization of the incident light and the optical axis of the NLC. The transmittance of light ($I/I_0$) can be calculated by the following equation [32] :
where $I$ and $I_0$ are the intensity of the transmitted and incident light, respectively; $\beta$ is the angle between the optical axis of NLC and the polarization of the incident light (optical axis in parallel to $\vec n$ in our case); $\beta$ is the angle between the projection of the LC optical axis on the plane perpendicular to the light wave vector and the polarization direction of the incident light. and $\lambda$ is the wavelength of the light. OPD between the extraordinary and ordinary light through the cell is where $d$ is the cell gap, $z$ is the cell gap axis. $n_e$ and $n_o$ are the refractive indices on the extraordinary (long) and ordinary (short) axes. $n_e$ depends on the tilt angle of the NLC. When the applied electric field is sufficiently strong and most of the NLC are in plane, $n_e$ can be considered as a constant. Therefore, The brightness of the light, $I/I_0$, is proportional to $\sin ^2 (2 \beta )$. The OPD can be estimated referring to the Michel-Lévy birefringence chart. In the following experiments, The linear polarizer and analyzer are on 0$^{\circ }$ and 90$^{\circ }$, respectively. When the projection of the director on the substrate plane is on 0$^{\circ }$ and 90$^{\circ }$, the director shows dark brushes. Those with projection on 45$^{\circ }$ and 135$^{\circ }$ show the brightest color.An optical retardation wave plate is required to distinguish the directors on 45$^{\circ }$ and 135$^{\circ }$. The full wave plate is a slab of birefringent crystal. The slow axis (marked by $\gamma$) and the fast axis of the wave plate are on the 45$^{\circ }$ and 135$^{\circ }$, respectively. The OPD provided by the wave plate is 530 nm. The applied NLC has refractive indices of 1.589 (slow) and 1.485 (fast) on the long and the short axes, respectively. When the NLC director is on 45$^{\circ }$, the slow axis of the NLC and the slow axis of the wave plate are parallel, which is called the additive position. At this position, the 530 nm is added to the OPD of NLC. On the other hand, when the NLC director is on 135$^{\circ }$, the slow axis of the NLC and the slow axis of the wave plate are orthogonal, called the subtractive position. At this position, 530 nm is subtracted from the OPD of NLC. Therefore, the NLC on 45$^{\circ }$ and 135$^{\circ }$ exhibit different birefringent colors. With the help of the wave plate, the radial and azimuthal director fields can be clearly distinguished. Based on the brightness and colors in the POM images, the director field of the topological defect array can be derived and illustrated.
3. Defects
The POM images of the defects generated with the pad, crossed strips, and a hole on the electrode are shown in Fig. 2. The voltage applied to the NLC cell was 4 V and 10 V, respectively, to show the process of defect generation. The cell gap range is from 3.0 $\mu$m to 3.5 $\mu$m. The maximum OPD should be from 300 nm to 350 nm. The NLC cell between crossed polarizer and analyzer shows the brushes of Schlieren texture under POM. When the applied voltage is 4 V, the NLC on the additive (45$^{\circ }$) and subtractive (135$^{\circ }$) position give the birefringence color of blue and yellow. When the applied voltage is 10 V, the birefrinegnece colors are yellowish green (45$^{\circ }$) and pinkish gray (135$^{\circ }$). The corresponding director fields are next to the POM pictures in Fig. 2.
Pad The pad electrode generates a converging defect with topological charge of +1 as expected. The defects may appear anywhere on the electrode, and can be vortex-like, spiral, or radial. The 4 V POM picture shows a vortex-like director field. With a voltage application of 10 V, the director field forms a spiral and the defect settles at the center of the pad electrode for static balance.
Crossed strips The crossed strips generate a topological defect with charge of -1, called the hyperbolic defect. The transition starts form two $-1/2$ disclinations, and then they merge to form a hyperbolic defect. The crossed strips fix better than the pads. The brushes of the Schlieren texture are straight and symmetric. The hyperbolic defect never leave the center of the cross.
Hole The hole on the electrode render an unexpected pair of +1 and -1 defects. We supposed that the hole should generate a +1 defect of diverging director field. However, the NLC starts with a hyperbolic defect right on the edge of the hole. The hyperbolic defect emerges prior to the anticipated diverging +1 defect. The a pair of +1 and -1 defect is fixed by the hole on the electrode. The pair is a dipole of topological defects [33].
4. Unit cells
Figure 3 shows the $3 \times 3$ unit cells of a defect array. The arrays made of pads, crossed strips, and porous electrode are shown in the left, middle, and right columns in Fig. 3, respectively. The bright field image shows the defect cores. The POM picture of NLC cell sandwiched in crossed polarizer and analyzer shows the Schlieren texture. The POM picture taken with a wave plate distinguishes the NLC molecules on 45$^{\circ }$ and 135$^{\circ }$. The pad cell and the porous cell have cell gap of 3 $\mu$m and 3.3 $\mu$m, respectively. Their OPD is in the range from 0 to 330 nm, showing grey scale brightness under POM. The cell gap of the crossed-strips cell is 5.4 $\mu m$. The OPD of the crossed-strips cell can be as much as 540 nm, so the POM picture shows birefringent color of blue and yellow.
Pads The pad array generates a square array of alternative radial (+1) and hyperbolic (-1) defects. The converging radial +1 defect is imposed by the pad electrode. Then the hyperbolic -1 defect appears between the pad electrodes. Finally the diverging radial +1 defect emerges on the corner of the unit cell. A unit cell contains 1 converging radial defect ($s_{converge}$), 4 half of hyperbolic defects ($s_{hyperbolic}$), and 4 quarter of diverging radial defects ($s_{diverge}$). The total topological charge is
Crossed strips The crossed strips generate a square array of alternative radial (+1) and hyperbolic (-1) defects, too. The hyperbolic defect on the crossing intersection of the strips appears firstly. Between the hyperbolic defects, converging and diverging radial defects appear on the strips. Finally, a hyperbolic defect appears among the strips on the corner of a unit cell. One hyperbolic defect, two half of converging radial defect, two half of diverging radial defect, and 4 quarter of hyperbolic defects ($s_{diverge}$) are in a unit cell. The total topological charge is
Porous electrode The porous electrode generate an array of topological defect dipoles. Every hole on the electrode holds a hyperbolic -1 and a diverging +1 defects. The -1 defect is on the edge of the hole, and the +1 is in the hole. The dipoles are parallel to each other. In the POM picture in Fig. 3, the dipoles are in parallel on the horizontal direction. The dipoles align and form a chain of +1 and -1 defects on the vertical direction. The NLC directors between the dipoles are continuous. When the voltage is just applied, the orientation of the dipole can be pointing to any direction (isotropic). With respect to the increasing applied voltage, the dipoles rotate, align, and settle on the same direction. A pair of diverging radial and hyperbolic defect is in a unit cell. The total topological charge is
5. Arrays
The POM picture of large-scale defect arrays are shown in Fig. 4. The arrays of pads, crossed strips, and holes are the left, middle, and right columns, respectively. The thickness of the pad, the crossed-strips, and the porous cells are $3~\mu m$, $5.4~\mu m$, and $3.3~\mu m$, respectively. The unit cells sizes are from $25~\mu m \times 25~\mu m$ to $10~\mu m \times 10~\mu m$ from the top to the bottom. The NLC cells are placed in crossed polarizer and analyzer.
Fig. 4. Dense Packing of defects created by the pads, the crossed strips, and the porous electrode. The unit cell sizes are $25~\mu m \times 25~\mu m$, $15~\mu m \times 15~\mu m$, and $10~\mu m \times 10~\mu m$ from the top to the bottom. All pictures taken by POM with crossed polarizer (P) and analyzer (A). The thin white lines show the director field. The thick white lines show the escape on the substrate plane. Thick white arrows show the defect dipoles.
Pads The pad array generates defects of alternative +1 and -1 points. Four defects are packed in a unit cell. Two of them are radial +1 defect, and the other two are hyperbolic -1 defects. The detect arrays are regular in $25~\mu m \times 25~\mu m$ unit cells. The $15~\mu m \times 15~\mu m$ array starts to show the escape texture (refer to the 4th row of Fig. 4). The +1 and -1 defects attracts each other and annihilate, resulting in long escapes lying on the substrate plane. The $10~\mu m \times 10~\mu m$ array does not show defect arrays but only domains of NLC in plane. The fringe electric field created by the pads is not strong enough to deform the NLC. The NLC tips and lies on the substrate plane, showing a Schlieren texture on a undulated surface.
Crossed strips The crossed strips generate defects of alternative +1 and -1 points. The cell gap in the crossed-strip cell is $5.4~\mu m$, which is thicker than the other two samples, resulting in yellow, orange, and dark blue colors in its POM picture. Each unit cell contains four defects. The largest unit cells happened to be $25~\mu m \times 15~\mu m$, so the unit cells are rectangular. Despite the small size of the unit cell, the defect arrays are regular. The thin straight strips effectively fix the defects in their designed locations. The array of $15~\mu m \times 15~\mu m$ unit cells is still regular. The +1 and -1 defects slide along the strips and annihilate. In the $10~\mu m \times 10~\mu m$ array, $80 \%$ of the area is covered by point defects, while only $20 \%$ of the area is covered by the escapes.
Porous electrodes The holes on the electrode generate dipoles of +1 and -1 defects. The dipoles align with respect to the increasing voltage, forming domains of parallel dipoles. Each unit cell contains 1 radial and 1 hyperbolic defects. The defect density is the smallest among the three cases. However, the hole is circular, and thus the dipole is free to point to any direction (isotropic). Now the orientation of the dipole is determined by statistical fluctuation and little bumps in the NLC cell. The schematic diagram in Fig. 4 shows the dipoles on the $0^\circ$ and $90^\circ$ directions. The aligned dipole domains appear even when the unit cell is only 10 $\mu m$ x 10 $\mu m$. However, the domains of aligned dipoles are small, resulting in a disordered large-scale texture. To achieve a larger area of aligned dipoles, solutions such as asymmetric holes or surface alignment with a pretilt angle may be effective.
With the current electrode design, the porous electrode creates the densest aligned defects. The unit cell can be as small as $10~\mu m \times 10~\mu m$. The lower bound of the unit cell size is $15~\mu m \times 15~\mu m$ for pads and crossed strips. In Table 1, the three designs are compared and their differences are summarized.
Table 1. Comparison of the defect arrays generated by the pads, crossed strips, and porous electrodes. A check (v) means that the defect array is stable and regular. v* notes that the alignment is in the dipole domain.
6. Conclusion
NLC cells with pad, crossed-strips, and porous electrodes were utilized to generate topological defect arrays. The periodicity and symmetry of the defect arrays were analyzed to find the unit cell. The defects in a unit cell were identified, and the director fields were derived. The optimal packing of ordered defects was achieved with $15 \mu m \times 15\mu m$ unit cells of pads and crossed strips. The porous electrodes generated topological defect dipoles, which aligned spontaneously in domains. The dipoles present a new technology platform for defect array design. Further investigation is required to determine the key parameters for aligning the dipoles develop methods to enlarge the aligned dipole domains. Switchable dipole alignment holds great potential for applications in precision optics and quantum devices.
Funding
Ministry of Science and Technology, Taiwan (MOST 110-2112-M-A49-031); National Science and Technology Council (NSCT 111-2112-M-A49-042-MY3).
Acknowledgments
We would like to express our gratitude to Merck Taiwan for providing us the liquid crystal. We also appreciate that Industrial Technology Research Institute (ITRT) fabricated the patterned electrodes for this study. Their support were crucial to the success of this work.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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