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Article

Customized Design of Aperiodic Liquid Crystal Grating for Generation of Multiple Optical Patterns

1
SCNU-TUE Joint Lab of Device Integrated Responsive Materials (DIRM), National Center for International Research on Green Optoelectronics, South China Normal University, Guangzhou 510006, China
2
Institute of Electronic Paper Displays and Guangdong Provincial Key Laboratory of Optical Information Materials and Technology, South China Academy of Advanced Optoelectronics, South China Normal University, Guangzhou 510006, China
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Crystals 2023, 13(2), 300; https://doi.org/10.3390/cryst13020300
Submission received: 28 December 2022 / Revised: 31 January 2023 / Accepted: 7 February 2023 / Published: 11 February 2023
(This article belongs to the Special Issue Liquid Crystals and Their Advanced Applications)

Abstract

:
Liquid crystal (LC) grating has drawn extensive attention due to its merits, including tunability, easy preparation process, cost effectiveness, and high diffraction efficiency. However, limited efforts have been made to explore LC grating for the generation of customized diffraction patterns, despite the fact that the patterning size of LC molecular director orientation has been tremendously reduced even to subwavelength scale via high-precision photoalignment techniques. Here, we propose a versatile method based on the binary particle swarm optimization algorithm to efficiently design aperiodic LC gratings for customized optical patterns generation, including needle beam, multi-foci, and hollow beam. The optical performance of the LC gratings was simulated using both the vectorial Rayleigh–Sommerfeld method (with discrete zero or π phase, in this article) and the actual director orientations. These results matched each other well, which proves the effectiveness of the optimization method based on discrete phase. This study paves the way for designing LC gratings with engineered functionalities, which may find applications in optical tweezers, laser delivery systems, and laser scanning imaging.

1. Introduction

Diffractive grating is one of the most fundamental components of various optical systems, ranging from optical rulers [1] and spectrometers [2,3] to two/three-dimensional (2D/3D) displays [4,5] and LiDAR [6]. Generally, the diffractive gratings can be predominantly categorized into wire-grid grating [1,7,8,9], surface relief grating [4,10,11], holographic grating [12,13,14,15], or liquid crystal (LC) grating [16,17,18,19]. Especially, LC gratings, consisting of LC molecules with a patterned or self-assembled director configuration, have attracted continuous attention in both industry and scientific research, owing to their high diffraction efficiency, tunability, simple preparation process, and polarization sensitivity [17,18,19,20,21,22,23,24]. Historically, the functionalities of conventional LC gratings were very limited due to the lack of fabrication techniques that could pattern the alignment surfaces into microscopic or nanoscopic domains. The fabrication process used to be arduous and challenging when the alignment involved a large number of alignment angles [25]. Thanks to the rapid advances in nanofabrication techniques, the last decade has witnessed tremendous progress in controlling alignment surface accuracy [25,26,27,28,29]. Nowadays, patterning alignment surfaces with micro/nanoscale features has become technically feasible [25,26,27,28], thus opening new avenues for designing LC gratings with designable functionalities and multi-functional devices [15,30].
So far, several approaches have been developed to realize high-resolution alignment surfaces with designable director distributions, including nanogroves [31], digital micromirror device (DMD) photolithography [26,32], photoalignment based on direct laser writing [33], and plasmonic metamasks [27,28]. These methods have been demonstrated to allow the fabrication of arbitrary surface patterns with resolution down to 1 μm [26] and even smaller than the wavelength of blue light [28]. They are therefore highly promising options for designing LC devices with engineered director orientations. On the one hand, LC gratings with designable directors, which rely on LC molecules and photoalignment agents, help to abandon the fabrication of nanoscatters [1,4,5,34], making them cost-effective and efficient. On the other hand, LC gratings based on tuning the in-plane azimuthal angles are flat and compact. The LC thickness can be smaller than 2 μm in the visible spectrum, as it merely has to satisfy the half-wave conditions. Although LC gratings can be equivalent to metagratings based on metallic and dielectric materials [4,5,30,35], the efforts devoted to studying LC gratings with designable functionalities are still very limited. Hence, the optimization and customized method for designing LC gratings is urgently needed, especially when various computational approaches [36,37,38] have recently been reported for metadevices [39,40] with engineered functionalities.
In this study, we propose a versatile method for efficiently designing aperiodic LC gratings to realize various optical patterns, including optical needle, multi-foci, and hollow beam. The director orientation (the in-plane azimuthal angle of the LC molecules) of the LC grating was optimized and designed using the binary particle swarm optimization (BPSO) algorithm [36]. The vectorial Rayleigh–Sommerfeld (VRS) method was employed to simulate the optical performance of the LC gratings. We also simulated the optical performance of the devices, based on the actual LC director distribution of the designed LC gratings, which was realized by minimizing the Q-tensor free energy. The results from these two methods match each other, which proves the effectiveness of the design method based on discrete phases. This study paves the way for designing LC gratings with engineered functionalities.

2. Theory and Methodology

2.1. Design Principle of LC Gratings Based on PB Phase

Figure 1a illustrates the principle of adopting aperiodic LC gratings for diverse optical pattern generation. The core idea is to optimize the Pancharatnam–Berry (PB) phase configuration and then obtain the rigid director orientations of the LC molecules. According to the optical birefringence of LC materials, the LC devices should first satisfy the half-wave condition, which means the thickness of the LC layer should be properly chosen so as to make the accumulated dynamic phase retardation along the z-direction meet the half-wave condition. Under this condition, the transmission of LC devices can be represented by the Jones matrix J = ( cos 2 α sin 2 α sin 2 α cos 2 α ) , assuming that the in-plane azimuthal angle of the LC director is α. For simplicity, the LC directors in this study are assumed to orient at π/4 and −π/4 (Figure 1b), corresponding to phase retardations of π and 0, respectively. As a proof of concept, this study aims to optimize aperiodic LC gratings to realize multiple optical patterns, including needle beam, multiple foci, and hollow beam (Figure 1c). When the linearly x-polarized light impinges on the LC grating, the emitted electric field E is given as:
( E outx E outy ) = J ± 45 ° ( E inx E iny ) = ( cos ( ± 2 × 45 ° ) sin ( ± 2 × 45 ° ) sin ( ± 2 × 45 ° ) cos ( ± 2 × 45 ° ) ) ( 1 0 ) = ( 0 ± 1 )
where Einx(x, y, 0) and Einy(x, y, 0) denote the incident electric field components along the x and y directions, respectively. This implies that the transmitted light becomes linearly y-polarized, as E outx ( x , y , 0 ) = 0 and E outy ( x , y , 0 ) = ± 1 . We propose to simulate the propagation property of the diffracted light along the optical axis using the VRS method [41], where the electric field components of the diffracted light can be written as
E ± 45 ° x ( u , v , z ) = 0 E ± 45 ° y ( u , v , z ) = 1 2 π R 2 z exp ( i k n R ) R 2 ( i k n 1 R ) d x d y E ± 45 ° z ( u , v , z ) = 1 2 π R 2 ( y v ) × [ exp ( i k n R ) R 2 ( i k n 1 R ) ] d x d y
where R2 = z2 + (xu)2 + (y − v)2, k is the wave number, n is the refractive index of the medium space, and (u, v, z) represent the observation planes. The total field intensity of the diffraction pattern in the observation plane can then be calculated as follows: I = | E x | 2 + | E y | 2 + | E z | 2 , according to Equation (2).

2.2. The Binary Particle Swarm Optimization Algorithm

The building blocks of the LC grating are 0.5 μm wide rectangular strips, each with either π/4 or −π/4 director orientation, corresponding to a PB phase retardation of π or 0, respectively. The aim of the customized design is to find the optimal phase profile for the objective LC grating, based on the unit building blocks. The BPSO algorithm, which is proven to be suitable for generating 0 or π phase change with fast convergence [36,42], is proposed to optimize the orientation configuration of the target light field. The flow chart of BPSO for LC grating design is depicted in Figure 2a. First, a large number of 0 and 1 arrays, corresponding to the phase distribution of 0 and π, are generated by the random function to initialize the population, where each phase distribution group represents a random LC grating. After that, the light intensity distribution generated by each group of phase profiles (namely the LC grating) is calculated using the VRS method. According to the optical performance of each random LC grating, the local optimum phase distribution is selected. A comparison between the local optimum and the global optimum is conducted to update the new global optimum. Meanwhile, the position and speed of 0 and π phase distribution are updated in the iteration. The iteration ends after the focal spot corresponding to the globally optimal phase distribution meets the target termination condition. Otherwise, the iteration goes back to step 2. Figure 2b shows an example of the LC grating consisting of rectangular blue and yellow regions, where the smallest width is 0.5 μm.
In the process of optimization, the LC grating director orientation field is considered to update the BPSO algorithm’s velocity and position during the iterative process. It can be expressed as [36]:
v ( k , : ) = ω v ( k , : ) + c 1 r a n d [ p b e s t ( k , : ) x ( k , : ) ] + c 2 r a n d [ g b e s t ( k , : ) x ( k , : ) ]
where ω is a dynamic weight, c1 and c2 are the learning factors, rand is an isolated random function to generate director orientation arrangement, x(k,:) and v(k,:) are the particle position and velocity respectively, and pbest(k,:) and gbest(k,:) are the local best and global best light field fitness, respectively. After abstracting with the sigmoid ambiguity function vs(k,:) = 1/(1 + e−v(k,:)), we can map the velocity change from v(k,:) to vs(k,:) with a value between 0 and 1.
Similarly, we can establish a rectangle function to decide the particle position x(k,t) by changing the mapping to 0 or 1 using function x ( k , t ) = { 1 , v s ( k , t ) > r a n d 0 , v s ( k , t ) r a n d , which corresponds to the phase of 0 or π in the LC grating. Hence, different kinds of light fields can be realized and optimized by setting different objective functions and convergence conditions, such as needle beam, multi-foci, or hollow beam.

3. Results and Discussion

3.1. LC Grating for Needle Beam

We start by introducing how to design LC gratings for needle beam generation. We first construct a rectangle function as the objective function to define the needle beam, which is given as follows:
I n e e d l e ( z ) = { 1 , ( z 1 < f < z 2 ) 0 , ( e l s e )
where z1 and z2 are the front and rear endpoints of the target focus, respectively, and f is the focus position. We propose controlling the shape of the focal spot using a set of convergence conditions for the needle beam, as follows:
{ 185 μ m < f < 210 μ m 2 × | z h z f | 8 λ F W H M f < λ
where zh is the position with half-maximum intensity, zf is the center of the optical needle beam, and FHHMf is the full width at half maximum (FWHM) along the x-axis. Equations (4) and (5) helps to define a needle beam with subwavelength size along the x-axis and length larger than 8 λ.
For all the studies in this article, the working wavelength of incidence with linear x-polarization is assumed to be λ = 632.8 nm. The length of the LC grating along the y-axis is set to a = 200 μm, and the size of the unit pixel for building the grating is 0.5 μm. Based on the MATLAB platform, the BPSO algorithm, together with the objective function in Equation (4) and the terminate conditions in Equation (5), are coded to optimize the LC grating for forming a needle beam. Table 1 shows one of the optimal LC grating geometrical parameters for needle beam generation, which consists of 86 strips with designed phase distribution. Each group of positions denotes the two margins (inner and outer) of each strip with an identical phase (0 or π). Figure 3a depicts the simulated intensity distribution of the needle beam in the xz-plane based on the VRS method. The polarization of the optical pattern extending along the y-direction is linearly y-polarized. The FWHM of the needle along the optical axis is 8 λ (Figure 3b), and the FWHM along the x-axis is 0.97 λ (Figure 3c). These results reveal that the designed grating successfully realizes the objectives given by Equations (4) and (5).
However, it should be noted that the simulation results (Figure 3a–c) assume that the LC grating is composed strictly of strips with either zero or π phase modulation. This may be different from the realistic director distribution or PB phase of the LC grating, due to the director transition between neighboring strips with discrete PB phases, namely jumping director orientations. Hence, we also simulated the LC director distribution of the designed LC grating by minimizing the Q-tensor free energy [43]. We assumed that E7 (k11 = 11.1 pN, k22 = 7.4 pN, k33 = 17.1 pN, ne = 1.7396, and no = 1.522) was adopted to produce the designed LC grating. The calculated light field distribution of the LC grating (Figure 3d) shows a needle beam at the designed position, which matches well with the result shown in Figure 3a. We also calculated the polarized optical microscope (POM) image (Figure 3e) of the LC grating, based on the simulated spatial LC director distribution. These results indicate that the optimized LC grating is capable of generating a needle beam. In order to obtain the directors of LC molecules in the steady state, the phase retardations of all the strips were set to zero or π (directors orient at + or −45 degrees) according to the optimized phase profile of the LC grating at the beginning. However, in real cases, the director in the middle region of the LC layer along the z-axis does not exactly distribute according to the ideal orientations (+ or −45 degrees), due to the influence of the orientations of adjacent stripes. Therefore, there is twist deformation due to the difference in the azimuthal angle of the LC director in the z-direction, when the steady state of the LC director configuration is realized after the lowest Q-tensor free energy is reached. This phenomenon can be observed in Figure 3f, where the black curve plots the director angles of LC molecules at the top of the LC layer as a function along the x-axis, and the dashed blue curve denotes the director angles in the middle of the LC layer. Here, the phase retardation at the adjacent regions with different preset orientation angles is not strictly 0 or π in the steady state.

3.2. LC Grating for Linear Multi-Foci

In the design of multi-foci, we assume that these foci are located linearly along the optical axis. In order to achieve relatively pure and uniform focal spots without prominent sidelobes, the variance of the peak and valley of each focus is adopted to construct the objective function, which is expressed below:
S t r i f o c i = i = 1 3 1 3 ( I h - a v g I i ) 2 + j = 1 2 1 2 ( I l a v g I j ) 2
where Ih-avg and Il-avg are the average intensities of the peak and valley of the foci along the optical axis, respectively, and Ii and Ij are the average intensities of the foci. The convergence condition is set as follows:
{ 100 μ m < f i < 135 μ m   ( i = 1 , 2 , 3 ) F W H M i < λ   ( i = 1 , 2 , 3 ) I i 0.9 & I i 0.1
where fj is the position of each focus along the z-axis, and FWHMi is the FWHM of each focus along the x-axis.
Table 2 shows one of the optimal LC grating geometrical parameters for multi-foci generation, which consists of 109 strips with designed phase distribution. Figure 4a depicts the simulated intensity distribution of the multi-foci in the xz-plane, which is linearly y-polarized. The simulation results clearly show three foci at focal lengths of f1 = 110.7 μm, f2 = 115.7 μm, and f3 = 124.2 μm (Figure 4b). The FWHM of the multi-foci is 0.67 λ along the x-axis (Figure 4c). These results reveal that the designed grating successfully realizes the objective given by Equations (6) and (7). We also simulated the LC director distribution of the designed LC grating and calculated the light field distribution of the LC grating (Figure 4d). Our results match those shown in Figure 4a, except for the slight difference in the purity of the foci and the sidelobes. This may be caused by the director transition between adjacent strips with different PB phases. The POM image (Figure 4e) of the LC grating was also simulated based on the simulated spatial LC director distribution. These results indicate that the optimized LC grating is capable of generating multi-foci. We also plotted the director angles of the LC molecules at the top and in the middle of the LC layer as a function of the x-axis (Figure 4f), resulting in a trend similar to that in Figure 3f. It can be inferred that the intensity difference between Figure 4a,d arises from the director transition, which is depicted by Figure 4f.

3.3. LC Grating for Hollow Beam

Hollow beam, also known as dark hollow beam, is a kind of beam with low or null intensity in the center, surrounded by a strong-intensity region [12,44]. Different from a conventional 2D hollow beam [44], the hollow beam based on an aperiodic LC grating is a 1D beam with a hollow channel, where a dark channel is formed in the direction perpendicular to the optical axis. Hence, the hollow channel beam should contain two adjacent peaks along the optical axis. The valley between these two peaks should also be the valley between the two lateral peaks along the x-axis. Similar to the design of multi-foci, the variance of the peaks along the z-axis and those along the x-axis are adopted to construct the objective function:
S hollow - pipe = i = 1 2 1 2 ( I z - a v g I i ) 2 + j = 1 2 1 2 ( I x a v g I j ) 2
where Iz-avg and Ix-avg are the average intensities of the two peaks along the z-axis and x-axis, respectively. We can set the convergence conditions for the hollow beam as follows:
200 μ m < f i < 230 μ m ( i = 1 , 2 ) I z min 10 2 I z a v g 0.9 & I x a v g 0.9 5 λ z 2 l e f t z 1 r i g h t 15 λ λ x 2 l e f t x 1 r i g h t 2 λ
where Izmin is the intensity of the hollow beam in the center; fj is between two focus positions on the z-axis; and z2left, z1right and x2left, x1right are the positions with half-maximum intensity on the z-axis and x-axis, respectively.
Table 3 shows one of the optimal LC grating geometrical parameters for hollow beam generation, which consists of 93 strips with designed phase distribution. Figure 5a depicts the simulated light field intensity distribution of the hollow beam in the xz-plane, which is linearly y-polarized. Figure 5b plots the light field intensity distributions along the white dashed lines (p1 and p2) in Figure 5a, which reveal that a null field is encircled by a high-intensity wall. The light field intensity distributions in the yz-plane at x = 0 (Figure 5c), along the z-axis at x = 0 (Figure 5d), in the xy-plane at z = 215 μm (Figure 5e), and along the x-axis at z = 215 μm (Figure 5f) are also plotted. These simulation results indicate that the designed grating successfully realizes the objective given by Equations (8) and (9). The size of the null field inside the hollow is estimated to be 10 λ by 1.8 λ. As in the studies of needle beam and multi-foci, we also simulated the LC director distribution of the designed LC grating and calculated the light field distribution of the LC grating (Figure 5g), which matches the result shown in Figure 5a. The POM image (Figure 5h) of the LC grating was also simulated based on the simulated spatial LC director distribution. In Figure 5i, we plotted the director angles of the LC molecules at the top and in the middle of the LC layer as a function along the x-axis, resulting in a similar trend to those seen in Figure 3f and Figure 4f. These results indicate that the optimized LC grating is capable of generating a hollow beam. A one-dimensional pipe or channel with low energy (zero energy) in the center is formed perpendicular to the propagation direction, which may inspire applications in dark-field massive-particle manipulation [45]. It should also be mentioned that this method can be extended to design LC gratings with other functionalities, similar to those of metagratings [4,5,35,39]. However, the LC gratings with designable functionalities have many advantages including an easy and cost-effective fabrication process, as it is no longer necessary to fabricate nanoscatters [1,34] relying on sophisticated and time-consuming lithography processes.
In this study, the minimum orientation region of our device is assumed to be 0.5 μm, with an LC layer thickness of approximately 1.45 μm (to satisfy the half-wave condition). In principle, a transition status exists in the nematic LC molecules between two adjacent regions with different director orientations, as shown in Figure 3f, Figure 4f, and Figure 5i. The smaller the strips with similar orientation, the greater the transition effect. Theoretically, the line width could be smaller if the LC layer thickness is further reduced. However, it is rather challenging to reduce the LC thickness to less than 1 μm in the experiment, due to the availability of the nanospacers with high unity to create such a cell gap [25,26,27]. Hence, the line widths with a strictly identical phase cannot be made infinitely small. Indeed, there is a compromise between LC layer thickness and line width. The reported line width that can be reached using plasmonic metamasks is in the order of one working wavelength [27,28], which may enable the creation of various flat optics based on LC materials apart from the aperiodic gratings with designable functionalities in this study.

4. Conclusions

In summary, a versatile method based on BPSO for efficiently designing aperiodic LC gratings to produce customized optical patterns is proposed and theoretically demonstrated in this study. Three kinds of typical optical patterns, optical needle, multi-foci, and hollow beam, are successfully designed using LC gratings. We simulated the optical performance of these LC gratings using both the VRS method (with discrete zero or π phase, in this article) and the actual director orientations. We found that these results matched each other well, thus proving the effectiveness of the optimization method based on discrete zero or π phase. This study paves the way for designing LC gratings with engineered functionalities, which may find applications in optical tweezers, laser delivery systems, and laser scanning imaging.

Author Contributions

Conceptualization, H.Y. and G.Z.; methodology, J.W., B.T. and D.Y.; data curation, C.L. and Z.G.,; writing—review and editing, H.Y., C.L. and Z.G.; All authors have read and agreed to the published version of the manuscript.

Funding

This work was partially supported by the National Key R&D Program of China (2021YFB3600602), the Science and Technology Program of Guangzhou (202102020604), the Special Program on Key Fields for Colleges and Universities of Guangdong Province (2021ZDZX1048), the Natural Science Foundation of Guangdong Province (2021A1515010623), the Guangdong Provincial Key Laboratory of Optical Information Materials and Technology (2017B030301007), and the 111 project.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

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

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The principle of adopting LC gratings for diverse optical pattern generation. (a) Schematic illustration showing the generation of multiple patterns when x-polarized light is impinging on the LC grating. (b) The director orientation of the LC molecules corresponding to phase retardation of π and 0. (c) Multiple optical patterns including optical needle, multiple foci, and hollow beam.
Figure 1. The principle of adopting LC gratings for diverse optical pattern generation. (a) Schematic illustration showing the generation of multiple patterns when x-polarized light is impinging on the LC grating. (b) The director orientation of the LC molecules corresponding to phase retardation of π and 0. (c) Multiple optical patterns including optical needle, multiple foci, and hollow beam.
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Figure 2. (a) The flow chart of BPSO for LC grating design. (b) Phase distribution of the LC grating.
Figure 2. (a) The flow chart of BPSO for LC grating design. (b) Phase distribution of the LC grating.
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Figure 3. Optimized LC grating for needle beam generation. (a) Intensity distribution along the xz-plane. The white-circled dot represents that the polarization direction of the light is along the y-axis. (b) Intensity distribution along the optical axis. (c) Transverse intensity distribution along the focal plane. (d) The simulated light field distribution based on the simulated director distribution of E7. (e) The simulated POM image of the LC grating. Scale bar: 50 μm. (f) The LC director angles of the grating as a function of the x-axis.
Figure 3. Optimized LC grating for needle beam generation. (a) Intensity distribution along the xz-plane. The white-circled dot represents that the polarization direction of the light is along the y-axis. (b) Intensity distribution along the optical axis. (c) Transverse intensity distribution along the focal plane. (d) The simulated light field distribution based on the simulated director distribution of E7. (e) The simulated POM image of the LC grating. Scale bar: 50 μm. (f) The LC director angles of the grating as a function of the x-axis.
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Figure 4. Optimized LC grating for linear multi-foci generation. (a) Intensity distribution in the xz-plane. The white-circled dot represents that the polarization direction of the light is along the y-axis. (b) Intensity distribution along the optical axis. (c) Intensity distribution along the three focal planes (ffocus1 = 110.7 μm, ffocus2 = 115.7 μm, ffocus3 = 124.2 μm). (d) The simulated light field distribution based on the simulated director distribution of E7. (e) The simulated POM image of the LC grating. Scale bar: 50 μm. (f) The LC director angles of the grating as a function of the x-axis.
Figure 4. Optimized LC grating for linear multi-foci generation. (a) Intensity distribution in the xz-plane. The white-circled dot represents that the polarization direction of the light is along the y-axis. (b) Intensity distribution along the optical axis. (c) Intensity distribution along the three focal planes (ffocus1 = 110.7 μm, ffocus2 = 115.7 μm, ffocus3 = 124.2 μm). (d) The simulated light field distribution based on the simulated director distribution of E7. (e) The simulated POM image of the LC grating. Scale bar: 50 μm. (f) The LC director angles of the grating as a function of the x-axis.
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Figure 5. Optimized LC grating for hollow beam generation. (a) The simulated light field intensity distribution in the xz-plane. The blue arrows denote that the hollow beam is linearly y-polarized. (b) The light field intensity distributions along the white dashed lines p1 and p2 in Figure 5a. (c) Intensity distribution in the yz-plane at x = 0. (d) Intensity along the z-axis at x = 0. (e) Intensity distribution in the xy-plane at z = 215 μm. (f) Intensity along the x-axis at z = 215 μm. (g) The simulated light field distribution based on the simulated director distribution of E7. (h) The simulated POM image of the LC grating. Scale bar: 50 μm. (i) The LC director angles of the grating as a function of the x-axis.
Figure 5. Optimized LC grating for hollow beam generation. (a) The simulated light field intensity distribution in the xz-plane. The blue arrows denote that the hollow beam is linearly y-polarized. (b) The light field intensity distributions along the white dashed lines p1 and p2 in Figure 5a. (c) Intensity distribution in the yz-plane at x = 0. (d) Intensity along the z-axis at x = 0. (e) Intensity distribution in the xy-plane at z = 215 μm. (f) Intensity along the x-axis at z = 215 μm. (g) The simulated light field distribution based on the simulated director distribution of E7. (h) The simulated POM image of the LC grating. Scale bar: 50 μm. (i) The LC director angles of the grating as a function of the x-axis.
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Table 1. The Optimal LC Grating Geometrical Parameters for Needle Beam Generation *.
Table 1. The Optimal LC Grating Geometrical Parameters for Needle Beam Generation *.
NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)
InnerOuterInnerOuterInnerOuterInnerOuterInnerOuter
1 0219 283037 47.54955 64.56973 85.586
2 22.520 303138 495056 697074 8687
3 2.54.521 313439 505157 7070.575 8788
4 4.56.522 343540 5151.558 70.57276 8888.5
5 6.51223 353741 51.552.559 7272.577 88.589
6 1212.524 3737.542 52.55360 72.57378 8990
7 12.51425 37.53843 5353.561 7373.579 9090.5
8 1414.526 3838.544 53.55562 73.575.580 90.592
9 14.51527 38.53945 555763 75.576.581 9294.5
10 1515.528 3939.546 5757.564 76.579.582 94.596
11 15.51929 39.54047 57.55965 79.58083 9697
12 1921.530 4040.548 5959.566 808184 9797.5
13 21.52231 40.541.549 59.56167 8181.585 97.599.5
14 2222.532 41.542.550 6161.568 81.58286 99.5100
15 22.52533 42.543.551 61.56369 8283
16 2525.534 43.544.552 6363.570 8383.5
17 25.52635 44.54753 63.56471 83.584.5
18 262836 4747.554 6464.572 84.585.5
* Odd number denotes phase retardation of zero, and even is π.
Table 2. The Optimal LC Grating Geometrical Parameters for Linear Multi-foci Generation *.
Table 2. The Optimal LC Grating Geometrical Parameters for Linear Multi-foci Generation *.
NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)
InnerOuterInnerOuterInnerOuterInnerOuterInnerOuter
1 01.523 2020.545 37.538.567 58.55989 7878.5
2 1.52.524 20.521.546 38.539.568 5959.590 78.580.5
3 2.53.525 21.52247 39.54069 59.56191 80.582
4 3.55.526 222448 4040.570 6161.592 8283.5
5 5.5627 2424.549 40.54171 61.56293 83.584.5
6 66.528 24.525.550 4141.572 6262.594 84.586
7 6.5729 25.52651 41.542.573 62.56395 8687
8 7830 2626.552 42.543.574 6367.596 8788.5
9 88.531 26.52853 43.54575 67.56897 88.589
10 8.59.532 282954 4546.576 6868.598 8992.5
11 9.51033 2930.555 46.54877 68.56999 92.593
12 101134 30.53156 484978 6970100 9394.5
13 111235 313257 4949.579 7070.5101 94.595.5
14 1212.536 3232.558 49.550.580 70.572102 95.596
15 12.513.537 32.53359 50.55281 7272.5103 9696.5
16 13.516.538 3333.560 5253.582 72.574.5104 96.597
17 16.51739 33.53461 53.55583 74.575105 9797.5
18 1717.540 3434.562 5555.584 7575.5106 97.598
19 17.51841 34.53563 55.55685 75.576.5107 9898.5
20 1818.542 3536.564 565786 76.577108 98.599
21 18.51943 36.53765 5757.587 7777.5109 99100
22 192044 3737.566 57.558.588 77.578
* Odd number denotes phase retardation of zero, and even is π.
Table 3. The Optimal LC Grating Geometrical Parameters for Hollow Beam Generation *.
Table 3. The Optimal LC Grating Geometrical Parameters for Hollow Beam Generation *.
NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)NumberPosition (μm)
InnerOuterInnerOuterInnerOuterInnerOuterInnerOuter
1 0220 2222.539 4041.558 5959.577 8585.5
2 2321 22.525.540 41.54259 59.560.578 85.586
3 3422 25.52741 424360 60.561.579 8686.5
4 44.523 2727.542 4343.561 61.56280 86.588
5 4.55.524 27.52843 43.54562 626381 8889.5
6 5.5625 283044 4545.563 6364.582 89.590
7 6826 3030.545 45.546.564 64.565.583 9091
8 88.527 30.53146 46.547.565 65.56784 9191.5
9 8.59.528 3131.547 47.549.566 676885 91.592
10 9.51029 31.53248 49.55067 6870.586 9292.5
11 101130 323349 5050.568 70.57187 92.593
12 1111.531 3334.550 50.55169 7172.588 9393.5
13 11.513.532 34.53551 515270 72.573.589 93.595
14 13.515.533 3535.552 5253.571 73.57590 9595.5
15 15.51634 35.536.553 53.555.572 757691 95.597.5
16 1616.535 36.538.554 55.55673 7676.592 97.599
17 16.52036 38.53955 5657.574 76.577.593 99100
18 202137 3939.556 57.558.575 77.584
19 212238 39.54057 58.55976 8485
* Odd number denotes phase retardation of zero, and even is π.
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Liu, C.; Guo, Z.; Wu, J.; Yuan, D.; Zhou, G.; Tang, B.; Ye, H. Customized Design of Aperiodic Liquid Crystal Grating for Generation of Multiple Optical Patterns. Crystals 2023, 13, 300. https://doi.org/10.3390/cryst13020300

AMA Style

Liu C, Guo Z, Wu J, Yuan D, Zhou G, Tang B, Ye H. Customized Design of Aperiodic Liquid Crystal Grating for Generation of Multiple Optical Patterns. Crystals. 2023; 13(2):300. https://doi.org/10.3390/cryst13020300

Chicago/Turabian Style

Liu, Chen, Zhenghao Guo, Jun Wu, Dong Yuan, Guofu Zhou, Biao Tang, and Huapeng Ye. 2023. "Customized Design of Aperiodic Liquid Crystal Grating for Generation of Multiple Optical Patterns" Crystals 13, no. 2: 300. https://doi.org/10.3390/cryst13020300

APA Style

Liu, C., Guo, Z., Wu, J., Yuan, D., Zhou, G., Tang, B., & Ye, H. (2023). Customized Design of Aperiodic Liquid Crystal Grating for Generation of Multiple Optical Patterns. Crystals, 13(2), 300. https://doi.org/10.3390/cryst13020300

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