Spatial photo-patterning of nematic liquid crystal pre-tilt

Abstract

A system and method for precise spatial patterning of the liquid crystal (LC) director through a two-step photo-exposure process that combines the effects of polarized and unpolarized light. The system and method allow for the simultaneous control of both the polar (out-of-plane) and azimuthal (in-plane) orientations of the LC director, a unit-vector that delineates the local orientational alignment of LC molecules, the mesogens. Embodiments in accordance with the present disclosure provide control over the polar azimuthal orientations of the LC director, enabling tunability of intricate and diverse LC- based photonic devices.

Description

Attorney Docket No.0184.0296-PCT / C18024 SPATIAL PHOTO-PATTERNING OF NEMATIC LIQUID CRYSTAL PRE-TILT CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 626,562, filed January 30, 2024, and entitled, “Spatial Photo-Patterning of Nematic Liquid Crystal Pre-Tilt.” GOVERNMENT FUNDING

[0002] This invention was made with government support under grant DMR-2104747 awarded by the National Science Foundation. The government has certain rights in the invention. FIELD

[0003] This disclosure relates generally to liquid crystal director alignment. BACKGROUND

[0004] Liquid crystals (LCs) are versatile, adaptive materials with tunable physical properties arising from the order of their rod-like molecules. Nematic LCs, in particular, are ^ characterized by a director – a unit-vector, ^^^^(^^^^) – that delineates the local orientational ^ alignment of the LC molecules, known as mesogens. Historically, ^^^^(^^^^) has been controlled by using electric fields, shear flows, and rubbed polymer coatings. These methods can favor uniaxial alignment, where mesogens are oriented in the same direction over large regions, which can limit the range and complexity of achievable LC configurations. Moreover, existing techniques can require cumbersome multi-step processes, specialized equipment, or external electric fields to maintain the desired LC alignment. This complexity not only increases the cost and time of production, but also limits the adaptability and scalability of LC-based devices.

[0005] Photoalignment enables microscopic spatial patterning of the LC director. The method employs a substrate with a thin film of photosensitive dye that is selectively ^ illuminated with polarized light to tailor the azimuthal (in-plane) orientation of ^^^^(^^^^). Whenirradiated with polarized light, the photosensitive molecules on the substrate, typicallyderivatives of azobenzene, undergo trans-cis isomerization. In time, they progressively orient their long axes (and thus their dipoles) perpendicularly to the local light polarization, setting ^ the alignment or “anchoring” direction of ^^^^(^^^^) in the liquid crystal adjacent to the substrate. Various approaches exist to pattern the alignment. Direct laser writing, plasmonic masks, and digital micromirror devices are static and dynamic options to imprint intricate patterns as anchoring conditions for LCs. These approaches have been used for applications including LC vortices capable of steering light, phase-based flat microlenses, and actuators made of liquid-crystal elastomers.

[0006] Existing photoalignment techniques allow for continuous variation of the in- plane alignment to be imposed. Control over the director’s pretilt (out-of-plane, polar) angle with photoalignment is more complicated. Presently, only discrete LC domains can be tailored to have a varying pretilt alignment by using multi-step processes involving stacked alignment layers, photo-cross linking polymer layers, and patterned electrodes. The ability to impose a continuous spatial variation in the pretilt would significantly expand the application potential of photoaligned LCs, such as by increasing the accessible degrees of freedom for controlling the actuated shapes of liquid crystal elastomers, or for creating phased-based LC optical devices.

[0007] Photoalignment using unpolarized light can control the nematic pretilt angle. Varying the incident angle of unpolarized light with respect to the sample plane of an LC cell, photosensitive azo-dye Brilliant Yellow (BY) has been shown to re-orient to generate an ^ exposure-dependent pretilt in ^^^^(^^^^)over a limited range of angles. The out-of-plane alignmentarises from the azo-molecules undergoing trans-cis-trans isomerization cycles, ultimately acquiring an out-of-plane angle as they align parallel to the light propagation direction. Filling the sample with LC broadens the range of achievable pretilt angles. This technique relies on physically tilting the sample to generate oblique photo-exposures. Thus, finely controlled, spatially varying patterns of the pretilt angle are inaccessible. Navigating complex, multi-step processes has proven to be a significant barrier in achieving continuous, ^ three-dimensional (3D) patterning of ^^^^(^^^^). Accordingly, there is a need for precise spatialpatterning of a LC director.SUMMARY

[0008] The present disclosure is a system and method for the precise spatial patterning of the liquid crystal (LC) director through a two-step photo-exposure process that combines the effects of polarized and unpolarized light, both propagating normal to the substrate. The system and method allow for the simultaneous control of both the polar (out- of-plane) and azimuthal (in-plane) orientations of the LC director, a unit-vector that delineates the local orientational alignment of LC molecules, the LC mesogens. Embodiments in accordance with the present disclosure provide control over the polar azimuthal orientations of the LC director, enabling tunability of intricate and diverse LC- based photonic devices, and do not require altering the sample's orientation relative to the light source, additional masks, or heterogeneous command surfaces. Embodiments in accordance with the present disclosure optimize the LC patterning process for efficiency and accuracy. The resultant LC configurations are stable in the absence of external electric fields. This stability ensures that once the desired pattern is achieved, recalibrations or adjustments are not needed. Embodiments in accordance with the present disclosure provide the ability to achieve a wide range of pretilt angles, from 0° to 90°, meaning that the system and method can be adapted for a variety of applications, from advanced photonics to soft actuation systems. The system and method provide LC director patterning and address challenges and limitations of existing systems and methods. Advanced photonic devices to precisely control LC orientation of, for example, but not limited to, tunable lenses, beam steerers, and modulators in telecommunications, imaging systems, and optical computing can be enabled by embodiments in accordance with the system and method of the present disclosure. Soft actuation systems for, for example, but not limited to, designing soft robotic systems and actuators that leverage the properties of LCs that can be used in, for example, but not limited to, medical devices, wearable technology, and adaptive materials can also be enabled by embodiments in accordance with the system and method of the present disclosure. The system and method of the present disclosure can be used in reconfigurable optics to, for example, but not limited to, pave the way for optics that can be reconfigured on-the-fly in the absence of external electric fields, in the areas of, for example, but not limited to, adaptive vision systems, augmented reality devices, and advanced camera systems. The system and method of the present disclosure can be used in responsive matter integrated into materials that respond to environmental stimuli, such as light or temperature, by changing their opticalproperties, to be used in, for example, but not limited to, smart windows, adaptive camouflage, and environmental sensors.

[0009] Embodiments in accordance with the present disclosure achieve smoothly varying spatial patterning of the polar and azimuthal components of the LC director by a combination of polarized and unpolarized light exposure. By exposing substrates of a filled nematic LC cell to unpolarized light, the nematic director undergoes an out-of-plane re- orientation with a pretilt angle that varies with the energy dose received. The system and method can be accomplished in the absence of masks, stacked alignment layers, electric fields, or altering the sample’s orientation relative to the light source, facilitating a high spatial resolution of the nematic pretilt. The system and method enable precise, control of tilt angles using a basic microscope setup, and employ a generalizable data analysis method.

[0010] Gradient-index (GRIN) lenses with parabolic refractive index profiles can be fabricated. As the pretilt depends on the anchoring conditions at the alignment layer, the lenses remain stable in the absence of external electric fields. The optical characteristics of the lenses confirm that the secondary exposure with unpolarized light preserves the in-plane ^ orientation of ^^^^ . Beyond the application to GRIN lens fabrication, the system and method of the present disclosure could facilitate the development of controllable liquid crystal elastomers, where the encoded pretilt biases actuation to specific deformation modes. The system and method of the present disclosure could enable creating optical instruments such as, for example, but not limited to, re-writable gradient index lenses and phase-based optical devices, photo-switches, spatial light modulators, sensors, polarimeters for minimally invasive clinical diagnosis, photo alignment of LC displays, and alternate or virtual reality glasses. The system and method control the pretilt angle and viewing angle of the display. The main goal is to control the direction of alignment. The GRIN lenses can be designed explicitly with pretilt profiles that function at low voltages, resulting in faster electro-optical modulation. The ability to smoothly vary the pretilt in LCs enables the design and fabrication of optical devices that precisely structure and shape light, improving the resolution in both augmented and virtual reality systems. Furthermore, by varying specific boundary conditions, the system and method provide a platform to study the behavior of topological defects in LCs during phase transitions, paving the way for insights into the properties of materials.

[0011] A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination ofthem installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. One general aspect includes a method for patterning a liquid crystal (LC) director by controlling polar and azimuthal directions of LC mesogens. The method also includes setting an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light, and tuning a polar angle of the LC director over a range of 0-90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo-dye molecules to unpolarized light for a pre-selected amount of exposure time. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0012] One general aspect includes a method for aligning a LC director. The method also includes coating a substrate with a dye, exposing the coated substrate to linearly polarized light, allowing the coated substrate to absorb the linearly polarized light, covering the coated substrate with LC mesogens, and exposing the coated substrate to unpolarized light to set an orientation of the LC mesogens that come in contact with the dye. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0013] One general aspect includes a method for aligning a LC director. The method also includes patterning a LC director by exposing a photoalignment layer to polarized light, and exposing the photoalignment layer and LC mesogens to unpolarized light, forcing the LC mesogens out of plane and controlling pretilt. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0014] One general aspect includes a method for creating a smooth 3D variation of a LC director photopatterning an azimuthal orientation of a nematic LC in contact with a photoalignment layer on a substrate including exposing a coated substrate to a dosage of linearly polarized light, generating an out-of-plane polar alignment of the LC director based on the dosage, and spatially varying the dosage, the spatial variation of the dosage configured to tune a polar orientation of the LC director between 0-90°. Other embodiments of thisaspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0015] One general aspect includes a GRIN lens including a LC director patterned by controlling polar and azimuthal directions of LC mesogens, the patterning may include: setting an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light to create a uniform planar region of pre-selected shape with uniform planar alignment, and tuning a polar angle of the LC director over a range of 0- 90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo-dye molecules of the uniform planar region to unpolarized light illumination over an annular region with fixed outer radius and time-dependent inner radius for a pre-selected amount of exposure time, where the time-dependent inner radius grows so that an exposure time increases with increasing distance from a center of the annular region, forming the GRIN lens, where the GRIN lens has a lens radius based on a maximum of the inner radius, the GRIN lens having a parabolic index of refraction profile. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] The above and / or other aspects and advantages will become more apparent and more readily appreciated from the following detailed description of examples, taken in conjunction with the accompanying drawings, in which:

[0017] FIGs.1A-1 through 1D are associated with controlling nematic pretilt with unpolarized light;

[0018] FIGs.1A-1 and A-2 are schematic diagrams of photoalignment devices that can be used to generate spatially varying pretilt;

[0019] FIG.1B is a photographic representation of an image taken with polarized microscopy of a sample generated using a method in accordance with embodiments of the present disclosure;

[0020] FIG.1C is a graphical representation of the result of rotating the sample to change the relative light intensity transmitted through the LC cell;

[0021] FIG.1D is a graphical representation of the pretilt angle variability depending upon the exposure energy / area;

[0022] FIGs 1E and 1F are graphical representations of the calculated values of the phase retardance on pretilt angle;

[0023] FIG.1G is a schematic diagram of an optical instrumentation in accordance with embodiments of the present disclosure;

[0024] FIGs.1H and 1I are pictorial representations of molecular structures of BY and 5CB;

[0025] FIG.2A is a pictorial representation of a schematic cross-section of an LC cell forming a GRIN lens;

[0026] FIG.2B is a pictorial representation of the two-step photo-patterning process in accordance with embodiments of the present disclosure;

[0027] FIG.2C is a graphical representation of parameters related to creating a lens;

[0028] FIG.2D is a photographic representation of the polarized-optical microscopy of a GRIN lens;

[0029] FIG.2E is a graphical representation of a radial transmitted light intensity profile;

[0030] FIG.2F is a graphical representation of the transmitted light intensity taking into account θ and φ.

[0031] FIG.3A is a schematic diagram illustrating a device to measure the focal length for a microlens;

[0032] FIG.3B is a graphical representation of the focal length variation with radius of the lens;

[0033] FIG.3C is a photographic bright field representation of the progression of the qualities of the image as the lens is rotated;

[0034] FIGs.4A and 4B are schemaic ray tracing diagrams for images formed by a LC GRIN lens; and

[0035] FIG.5 is a flowchart of a method in accordance with embodiments of the present disclosure.DETAILED DESCRIPTION

[0036] In order for the present disclosure to be more readily understood, certain terms are first defined below. Additional definitions for the following terms and other terms may be set forth throughout the specification. If a definition of a term set forth below is inconsistent with a definition in an application or patent that is incorporated by reference, the definition set forth in this application should be used to understand the meaning of the term.

[0037] As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. Thus, for example, a reference to “a method” includes one or more methods, and / or steps of the type described herein and / or which will become apparent to those persons skilled in the art upon reading this disclosure and so forth.

[0038] It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting. Further, unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure pertains. In describing and claiming the methods, systems, and computer readable media, the following terminology, and grammatical variants thereof, will be used in accordance with the definitions set forth below.

[0039] A system and method in accordance with the present disclosure can be implemented using a microscope and standard optics. The photoalignment process involves an initial exposure with polarized light to set the in-plane director alignment, followed by a second exposure with unpolarized light, both propagating normal to the substrate. By adjusting the exposure duration of the unpolarized light, the polar angle of the director can be precisely tuned over a range from 0°-90°. The degree of pretilt depends on the total dose applied, independent of the light flux. Because the exposure time can be varied as a function of position, the approach can create smooth variations of pretilt with arbitrary spatial profiles. A use of the system and method in accordance with embodiments of the present disclosure includes the production of GRIN microlenses with parabolic refractive index profiles. The lenses are formed by gradually varying a radially dependent director profile that is stable in the absence of an applied electric field.

[0040] Referring now to FIGs.1A-1 and 1A-2, to generate alignment boundary conditions, a region of a substrate is exposed to linearly polarized blue light using aphotoalignment apparatus as described herein. When exposed to linearly polarized light, the molecules align orthogonally to the polarization plane, thereby determining the azimuthal ^ orientation of the nematic director, ^^^^ . To achieve spatial variation of the pretilt, LC cells are filled with 4-cyano-4’-pentylbiphenyl (5CB). Once 5CB cools into the nematic phase, the region is re-exposed to unpolarized light 113 by eliminating the polarizer 105 from the optical path. The pretilt is varied by varying the exposure duration across a region with unpolarized light 113. To fabricate lenses, spatially varying the exposure time is controlled by projecting an annular exposed region with a fixed outer radius and a time-varying inner radius as described herein. After the photoalignment process is finished, LC cells are sealed on their ends using UV curable resin.

[0041] Continuing to refer to FIGs.1A-1 and 1A-2, in some configurations, a microscope such as, for example, but not limited to, a Nikon LV 100N Pol or a TI Eclipse TE2000, is used to image photo-patterned regions. The microscope is equipped with a rotating stage and an objective 103 such as, for example, but not limited to, a 20x Nikon Tu Plan ELWD objective. Samples are placed on a heating stage set to, for example, but not limited to, 23.4°C. Köhler illumination is achieved with a condenser such as, for example, but not limited to, a Nikon LW 0.65. A bandpass filter, for example, but not limited to, a 647 nm filter, can be used to restrict the incident wavelength of light. Other wavelength filters can be used. Optical microscopy images are captured using a camera, such as, for example, but not limited to, a Nikon DS-Ri2 with the adjustable aperture diaphragm lever of the condenserlens set to, for example, but not limited to, ≈ 0.1 to minimize stray light. In someconfigurations, a convex (focusing) lens 107 and an aspheric condenser lens 109 are also included.

[0042] Continuing to still further refer to FIGs.1A-1 and 1A-2, embodiments in ^ accordance with the present disclosure provide a smooth, 3D variation in ^^^^(^^^^) that employs a two-step exposure process to photo-pattern the nematic pretilt. In FIG.1A-1, an empty LC cell 101 with BY-coated substrates is initially exposed to polarized light, setting the azimuthal anchoring direction for LC molecules. In FIG.1A-2, after the cell 111 is filled with nematic LC, a region is re-exposed to unpolarized light, forcing the BY (and LC) molecules to re-align out-of-plane.

[0043] Referring now to FIG.1A-1, the two-step process photo-patterns the azimuthal ^ and polar orientation of ^^^^(^^^^). Embodiments in accordance with the present disclosure canexpose azo-dye films on glass substrates to unpolarized light in the presence of nematic LC, which can promote out-of-plane LC re-orientation. LC cells are assembled out of transparent substrates that have been spin-coated with a thin film of the photosensitive dye BY. When the empty cell 101 is first illuminated with linearly polarized light, the BY molecules in the exposed region align themselves to be orthogonal to the light’s polarization direction.

[0044] Referring now to FIG.1A-2, the cell 111 is filled with a liquid crystal in the nematic phase, for example, but not limited to, 4-cyano-4’-pentylbiphenyl (5CB) and 8CB, inthe isotropic phase (^^^^^^^^^^^^ ≈ 41∘^^^^), and cooled to room temperature (^^^^ ≈ 24∘^^^^), where the5CB is nematic. The alignment of the BY molecules by the initial exposure creates a planar anchoring condition where the director is in-plane and perpendicular to the polarization direction. The polarizer 105 is removed, and regions of the sample are illuminated with unpolarized light 113. As the exposure time, ^^^^, increases, the molecules of the alignment layer tilt out of the plane, progressively aligning their absorption axes towards the propagation vector of the incident light while still maintaining the imprint of their initial in- plane alignment. This orientation by the second exposure alters the LC anchoring condition, introducing a pretilt to the director that depends on the duration of the exposure.

[0045] In some configurations, ITO-coated glass substrates are used to achieve uniform spreading of photosensitive dye. The glass is cut and cleaned using an ultrasonic cleaner, followed by sequential rinsing with acetone, ethanol, and isopropyl alcohol. After rinsing, substrates are dried with ^^^^2gas. Photosensitive azo-dye BY is combined with n,n- dimethylformamide (DMF) to achieve a 1 wt.% solution. The BY-DMF solution is spin- coated onto the substrates at 3000 RPM for 45 seconds. After spin-coating, substrates are baked on a hot plate at 95∘^^^^for 15 minutes. Spin-coating and heating steps are performed in a humidity-controlled environment with relative humidity below 35% to optimize the BY thin film for photoalignment. Substrates are subsequently fixed together using epoxy glue to create a liquid crystal cell. After the epoxy cures, spectroscopic reflectometry is used to measure the cells’ thickness ^^^^, obtained from the absolute reflectance spectra fit.

[0046] Referring now to FIG.1B, when viewed under crossed-polarizers, the transmission of white light in the re-exposed regions varies with the exposure time t of unpolarized light (scale bar: 30 μm). Several hexagonally shaped regions are shown after they are subjected to the two-step photoalignment with varying amounts of unpolarized light. Under crossed-polarizers, the resulting transmitted light intensity, ^^^^, through each regionholds quantitative information about the pretilt angle, ^^^^, due to the birefringence of 5CB. Specifically, at room temperature 5CB has a positive uni-axial optical response andbirefringence, where ^^^^^^^^ = 1.524 and ^^^^^^^^ = 1.708 are the refractive indices perpendicular andparallel to the director, respectively. As the duration of the second unpolarized light exposure changes from 20 seconds to 120 seconds, an overall qualitative decrease in the transmitted intensity is shown. For incident light with wavelength ^^^^, ^^^^ is expected to change as a function ^ of the in-plane orientation of ^^^^ with respect to the analyzer, ^^^^, asThe argument of the second term in Equation (1) represents the relative retardance experienced by the polarization component of the light parallel to the in-plane orientation of^ ^^^^ when traveling through a nematic cell of thickness ^^^^, which for the sample in FIG.1B is^^^^ = 2.81 ± 0.04 ^^^^m.

[0047] The first term in Equation (1) describes the dependence of the transmitted ^ intensity on the azimuthal angle ^^^^ of the nematic director ^^^^ with respect to the analyzer. It ismaximum at ^^^^ = ^^^^ / 4. The second term describes the dependence of the transmission on therelative phase retardance,�^^^^^^^^^^^^^^^^(^^^^) − ^^^^^^^^�, of the polarization component of^ light parallel to the in-plane orientation of ^^^^ . The retardance, in turn, depends on ^^^^ through the effective extraordinary index of refraction,With increasing, ^^^^, Equation (2) approaches ^^^^^^^^ and ^^^^^^^^^^^^^^^^ − ^^^^^^^^ decreases. For a fixed ^^^^ / ^^^^,this causes ^^^^^^^^to decrease monotonically as in FIG.1E. The transmitted light intensity is generally not monotonically decreasing as ^^^^ increases from zero to ^^^^ / 2. Specifically, the LC cell acts as a retarder that converts the incident linearly polarized light to elliptically polarized light whose time-average component parallel to the analyzer depends on ^^^^^^^^. As ^^^^^^^^passes through odd multiples of ^^^^, the time-average component parallel to the analyzer and hence the transmission is maximum. As ^^^^^^^^passes through multiples of 2^^^^, the initial linear polarization of the light is recovered, and thetransmission is zero. When ^^^^ = ^^^^ / 4 and ^^^^0 = 1 in Equation (1), as shown in FIG. 1F,^^^^^^^^^^^^2(^^^^^^^^⁄ 2 ) (for indices of refraction ^^^^^^^^ = 1.524 and ^^^^^^^^ = 1.708), when ^^^^ / ^^^^ is sufficientlysmall (e.g., ^^^^ / ^^^^ = 1, ^^^^ > ^^^^^^^^ > 0 for all values of ^^^^), monotonically decreasing to 0 as ^^^^ →^^^^ / 2. For a range of intermediate values of ^^^^ / ^^^^, ^^^^ is in the range 2^^^^ > ^^^^^^^^ > ^^^^ at ^^^^ = 0, and^^^^^^^^^^^^2(^^^^^^^^⁄ 2 ) passes through a single maximum as ^^^^ increases toward ^^^^ / 2. This behavior isillustrated in FIG. 1E for ^^^^ / ^^^^ = 4.34, which corresponds to the cell thickness in themeasurements whose results are shown in FIG. 1C. For still larger values of ^^^^ / ^^^^, ^^^^^^^^ > 2^^^^when ^^^^ = 0, and ^^^^^^^^^^^^2(^^^^^^^^⁄ 2 ) possesses both maxima and zero points as ^^^^ increases toward^^^^ / 2. An additional parameter, ^^^^, in Equation (1) accounts for small errors in the alignment ^ of ^^^^ with respect to the polarizer, occurring when the sample is shifted to expose new regions. ^^^^ is not monotonic for increasing ^^^^ and varies for a fixed ^^^^ in a way that depends on a combination of ^^^^, ^^^^, and ^^^^.

[0048] In some configurations, a narrow band-pass filter can be used to restrict ^^^^ =647 ± 10 nm. In some configurations, to obtain ^^^^0, the mean light intensity transmitted by anunexposed (^^^^ = 0) uniformly planar aligned region at various ^^^^ is measured, and Equation(1) is fit to these data using known values for ^^^^, ^^^^,^^^^^^^^ ,^^^^^^^^, thus resulting in a fixed value for ^^^^0for subsequent fits for ^^^^ > 0.

[0049] Referring now to FIG.1C, ^^^^(^^^^,^^^^) / ^^^^0are shown as a function of ^^^^ across the six exposure times illustrated in FIG.1B. The sample in FIG.1B is rotated with respect topolarized light, with each region exposed to a power flux of ^|^^^^|^ = 1.63W · cm−2. ^^^^ isobtained as a function of the exposure time by a two-parameter fit using Equation (1), where ^^^^ is obtained from an initial fit to the t=0 exposure data and then held fixed for t>0. Rotatingthe sample by θ changes the relative transmitted light intensity I (θ,β) / I0through the LC cell.The power flux of the light source is ^|S|^ = 1.63 W · cm−2and the wavelength λ = 647 nm.Solid lines are results from fits using Equation 1 to obtain β for different exposure times with d = 2.81 μm. Error bars represent the standard deviation of the transmitted light intensity in the re-exposed region for a given θ.

[0050] Referring now to FIG.1D, plotting β obtained with different ^|S|^ shows that the pretilt angle depends on the exposure energy dose and saturates at a maximum value of ≈90◦, independent of ^|S|^. The solid line is an empirical best fit to the data described by^^^^(^^^^) = ^^^^^^^^^^^^^^^^^^^^^^^^([^^^^^|^^^^|^^^^^]^^^^), where ^^^^ and ^^^^ are free parameters, exemplarily ^^^^ = 3.097cm2 / J, and ^^^^ = 0.87, obtained by non-linear least-square minimization. Error barscorrespond to the standard deviation of diagonal elements of the covariance matrix for thebest-fit parameters. ^^^^ increases monotonically with increasing exposure time. The pretilt canbe made to saturate at ^^^^^^^^^^^^^^^^ ≈ 90∘. At different values of ^|^^^^|^, the exposure time tounpolarized light specifies a specific value of ^^^^. Re-scaling the exposure time by ^|^^^^|^ reveals the energy dose received by each re-aligned region. For the range of fluxes shown in FIG. 1D, the variation of ^^^^ depends on the energy dose received. The transmitted intensities in ^ FIG.1C also vary with ^^^^, implying that the azimuthal alignment of ^^^^ is preserved as ^^^^ varies with the exposure to unpolarized light.

[0051] Referring now to FIGs.1E and 1F, a relative phase retardance Φtarises for the ^ polarization component of light that is parallel to the in-plane orientation of ^^^^ . In FIG.1E, ^ the phase lag that the component along ^^^^ accumulates while traveling through a liquid crystal cell depends on pretilt angle β, as shown for several thicknesses d of the cell. In FIG.1F, for sufficiently large d / λ > 1, the normalized intensity exhibits non-monotonic variations as β increases.

[0052] Referring now to FIGs.1G-1I, a photo-patterning instrument includes an LED source 137 feeding into the side port of a bright-field inverted microscope body. Projected images onto the sample plane are generated using an LED-based projector connected to a peripheral optical path. In some configurations, the projector operates using three time- modulated laser diodes. Images generated by the projector first pass through two asphericcondenser lenses 133 with focal lengths ^^^^ = 32^^^^^^^^, separated by 64^^^^^^^^. These lensescollimate the illumination. Two achromatic doublet lenses 131 bring the image into focus at the infinity-corrected plane near the microscope’s side port. A removable linear polarizer 135 is mounted directly before the side port. Once inside the microscope body, the image is reflected by a dichroic mirror 127, picked up by an infinity-corrected tube lens, and collected by a microscope objective 123 that focuses the light onto a substrate. In FIG.1G, a computer connects to an LED projector to generate images focused onto substrates at the sample plane 121. Images are collimated by a set of aspheric condenser lenses 133, expanded by a relay of convex lenses 131. A removable linear polarizer 135 sets the polarization direction of the light before it enters a microscope body containing a dichroic mirror 127 that reflects the image toward an infinity-corrected tube lens 125. Images are then picked up by a microscope objective 123 and focused onto the sample plane 121. In some configurations, an x-y samplestage 129 is included. In FIG.1H, the molecular structure of Brilliant Yellow (BY) is shown. In FIG.1I, the molecular structure of 4-cyano-4’-pentylbiphenyl (5CB) is shown.

[0053] Referring now to FIGs.2A-2F, photographic, pictorial, and graphical representations of the process of crafting gradient-index lenses by varying pretilt profile are shown. In FIG.2A, the illumination method in accordance with the present disclosure enables diverse pretilt patterns within a sample to be created. In FIG.2B, the two-step photo- patterning process for varied pretilt is illustrated. An initial rectangular region is illuminated with linearly polarized light, ensuring uniform LC alignment. In a second exposure, unpolarized light illuminates an annular region with a time-dependent inner radius to vary theexposure time radially. In FIG. 2C, shown are the time-dependent inner radius, Ri(t), leadingto β (r) (inset, dashed curve) and a parabolic neff(r) (inset, solid curve) that create a lens withradius RL= 38 μm. In FIG. 2D, shown is the polarized-optical microscopy of a GRIN lens ofradius RL= 38 μm (dashed line). The lens radius RLis smaller than the outer radius Ro= 70μm of the region exposed to unpolarized light. In the inset of FIG.2D is the predicted polarized optical microscopy (POM) texture for a lens with radius RL, reconstructed from Jones calculus from the director field defined by β (r) (scale bar: 40 μm). In FIG.2E, shown is a radial transmitted light intensity profile, I (θ, r), for θ = π / 4. In FIG.2F, accounting for θand φ, I (θ, r) at different θ collapse onto a single curve defined by Ĩ=I (θ,r) / sin2(2(θ+φ)).

[0054] Referring now to FIG.2A, the two-step illumination method for controlling ^^^^ allows for specific regions to be illuminated with different amounts of unpolarized light. As a result, the illumination method enables diverse pretilt patterns within a sample to be created in the absence of physical masks or altering the sample orientation with respect to the incident light. The illumination method can create a varying pretilt, for example, but not limited to, a varying pretilt by smoothly tuning the exposure time across a region. Micro-scale LC GRIN lenses that have a ^^^^ ranging from 0 to a peak of ≈ 90∘ can be created inwhich ^^^^^^^^^^^^^^^^has a parabolic profile.

[0055] Referring now to FIG.2B, to fabricate a lens, linearly polarized light is applied to create a rectangular region with uniform planar alignment. During the second exposure, an annular region of fixed outer radius ^^^^^^^^and time-dependent inner radius ^^^^^^^^(^^^^) are exposed to unpolarized light. The inner radius grows from zero so that the exposure time (and hence the resulting pretilt ^^^^) increases with increasing distance ^^^^ from the center.Specifically, ^^^^^^^^(^^^^) is chosen to form a GRIN lens with radius ^^^^^^^^and parabolic index of refraction profile,where ^^^^^^^^ ≡ ^^^^^^^^^^^^^^^^ − ^^^^^^^^^^^^^^^^, and ^^^^^^^^^^^^^^^^corresponds to the minimum effective refractive indexachieved at ^^^^^^^^^^^^^^^^. In some configurations, for a GRIN lens, ^^^^^^^^^^^^^^^^and ^^^^^^^^^^^^^^^^are specified to be ^^^^^^^^and ^^^^^^^^, respectively. ^^^^^^^^is less than ^^^^^^^^so that the secondary exposure creates anannular region, ^^^^^^^^ < ^^^^ < ^^^^^^^^, around the lens with ^^^^ ≈ ^^^^^^^^^^^^^^^^. This configuration spatiallyseparates the lens from the boundary and the surrounding area with planar anchoring.

[0056] Referring now to FIG.2C, the parabolic index of refraction profile is obtained through a pretilt profile ^^^^(^^^^)that is found by combining Equations (2) and Equation (3), and solving numerically for ^^^^. To derive the corresponding form forthe measurements^^^^(^^^^) (see FIG. 1D) at varying ^|^^^^|^ are fitted using the empirical form ^^^^(^^^^) =^^^^^^^^^^^^^^^^^^^^^^^^([ ]^^^^), where ^^^^ and ^^^^ are free parameters, exemplarily ^^^^ = 3.097 cm2 / J, and^^^^ = 0.87 by non-linear least-square minimization. Combining these fitting parameters withthe light flux used for lens fabrication, ^|^^^^|^ = 0.51 W / cm2, interpolating between ^^^^(^^^^) and^^^^(^^^^)reveals exposure time as a function of ^^^^. The exposure time is then inverted to obtain and the parabolic profile given by Equation (3) in FIG.2C is generated.illustrates a polarized optical microscopy (POM) image of a GRIN lens with ^^^^^^^^ = 38 ^^^^mfabricated in this way.

[0057] To analyze the index of refraction profile of the lens, the transmitted light intensity of the lens is measured. As ^^^^ increases with ^^^^, the transmitted intensity, described by Equation (1), becomes radially dependent:with increasing ^^^^ and hence also with the distance from the lens center. The obtained POM textures qualitatively match those obtained using Jones matrix calculations for the refractive index (see FIG.2D inset). Jones calculus is used to qualitatively compare experimentally obtained POM textures of GRIN lenses to those expected from the procedure to determine the pretilt profile ^^^^(^^^^)described herein. The numerically derived director field is discretized intovolume elements (voxels) on a 3D grid. The position in the ^^^^ − ^^^^ plane, denoted by ^^^^,contains ^^^^ voxels of thickness ^^^^ stacked along the ^^^^ −direction (^^^^^^^^ = ^^^^). Given that the^ change in ^^^^between consecutive voxels is small, ^^^^ ≪where ^^^^ is the light’s wavelength. A voxel, represented by ^^^^, behaves as a uniaxial birefringent optical element.This is characterized by a 2 × 2 Jones matrix, ℳ^^^^, which depends on the extraordinary neand ordinary norefractive indices of the LC. The light’s propagation through a voxel is ^ subject to an neffthat depends on the polar angle ^^^^^^^^between ^^^^ and the light’s propagationdirection, ^^^^ . With^ 0^^^^0 = ^^^^ , the plane of polarization is defined in the ^^^^ − ^^^^ plane. Thecorresponding Jones matrix is expressed as:

[0059] ℳ^^^^(^^^^) is computed based on the refractive indices of 5CB at a pre-selected temperature and wavelength ^^^^. An operator, ^^^^, integrates the effects of voxels at a given ^^^^.2 × 2 Jones matrices are established for both the polarizer, ^^^^, and the analyzer, ^^^^. Thepropagation of plane waves, ^^^^ , through the sequence of ^^^^,, and ^^^^ provides a vector, . The reconstructed POM texture is derived from the light intensitythrough the voxels, represented as ^^^^^^^^(^^^^) = |^^^^^^^^(^^^^)|2.

[0060] Referring now to FIG.2E, using the target profile of ^^^^^^^^^^^^^^^^(^^^^) from Equation(3), an expected ^^^^(^^^^, ^^^^) for various ^^^^ can be calculated. Measurements for ^^^^(^^^^, ^^^^) for ^^^^ =^^^^ / 4, align well with the predictions given by Equation (6). The solid curve represents the theoretical model (Equation 4), with the shaded area indicating the associated uncertainty. In some configurations, a procedure can be used to estimate the uncertainty in the predicted transmitted intensity through the GRIN lens. For the GRIN lens, the radially dependent intensity profile arising from a spatially varying pretilt is given by Equation (4). The results in FIG.2E are shown for fixed ^^^^ as a function of ^^^^. Uncertainties in ^^^^0, ^^^^, ^^^^, ^^^^, ^^^^, ^^^^^^^^^^^^^^^^,^^^^^^^^,and ^^^^^^^^ can contribute to the uncertainty in the predicted ^^^^(^^^^, ^^^^). With respect to theuncertainty in ^^^^^^^^^^^^^^^^, ^^^^^^^^^^^^^^^^^^^^(^^^^) from Equation (3), this quantity depends on the uncertaintiesassociated with ^^^^^^^^and ^^^^^^^^– both of which are temperature-dependent and correlated – as well as fluctuations in the power flux, ^^^^^|^^^^|^. For 5CB, ^^^^^^^^and ^^^^^^^^, are known, so their uncertainties stem from uncertainties in the temperature. When the measurement temperature is maintained at 23.4°C, ^^^^^^^^^^^^ / ^^^^^^^^and ^^^^^^^^^^^^ / ^^^^^^^^, associated with ambient temperaturefluctuations, are < 1%. ^^^^^|^^^^|^ / ^|^^^^|^ is determined by measuring the fluctuation range of ^|^^^^|^over a 100^^^^ interval and, under some conditions, is ≈ 7.6%. When contributions to ^^^^^^^^^^^^^^^^^^^^(^^^^)are dominated by ^^^^^|^^^^|^, ^^^^^^^^^^^^^^^^^^^^(^^^^)can be approximated asThe uncertainties, ^^^^^^^^0, ^^^^^^^^^^^^^^^^^^^^, ^^^^^^^^, ^^^^^^^^^^^^, ^^^^^^^^^^^^, ^^^^^^^^, ^^^^^^^^^^^^, ^^^^^^^^, ^^^^^^^^ can be propagatedthrough Eq. (4) in quadrature as,^^^^^^^^ =

[0061] The GRIN lens described herein is a converging lens. As the lens radius is larger than ^^^^, optical effects in an LC-based GRIN lens can be approximated by geometric optics. The focal length, ^^^^, can be obtained by considering the maximum angle of refraction experienced by light at the periphery of the lens,where ^^^^^^^^is the index of refraction of the glass substrate . In terms of the lens radius, the focal length is given by, ^^^^^^^^2 =^^^^. (9) ^^^^^^^^2^^^^^^^^^^^^ Equation (9) establishes a correlation between a lens’s focal length and ^^^^^^^^ achieved through photoalignment.

[0062] Referring now to FIG.2F, the expected spatial variation in the pretilt provides ~results for ^^^^(^^^^, ^^^^) at different ^^^^ that fall onto a single curve defined by ^^^^ = ^^^^(^^^^, ^^^^) / ^^^^^^^^^^^^2�2(^^^^ + ^^^^)�, which depends on the LC lens’s refractive properties.

[0063] Referring now to FIG.3A, a "JHU" mask 311 is positioned immediately after the light source 313. The LC lens 303 is mounted on a rotating stage 305, enabling^ adjustment of the orientation of ^^^^ relative to the light’s polarization. Testing the imaging properties of lenses provides an independent measurement of ^^^^^^^^. Measuring ^^^^ of GRINlenses fabricated with radii ranging from ^^^^^^^^ = 24 to 38 ^^^^^^^^ across samples with thicknessesranging between ^^^^ = 2.32 to 4.64 ^^^^^^^^ provides the imaging properties of the GRIN lenses.Lowering the lens 303 brings the mask’s image into focus within the focal plane of the microscope objective 301.

[0064] Referring now to FIG.3B, the linear relation between the 2^^^^ / ^^^^^^^^and ^^^^^2^^^ / ^^^^ is shown. The solid line represents the predicted slope, 1 / (∆n), with ∆n = 0.184 derived fromthe peak value of β at β(r =RL) (based on ^^^^ ∘^^^^^^^^^^^^ = 90 , and ^^^^^^^^ = 1.5).

[0065] Referring now to FIG.3C, a mask containing die-cut letters "JHU" is placed in front of a light source on an upright microscope to serve as an object to image as in FIG.3A. The GRIN lens is set on a rotating stage and brought into focus. Lowering the stage causes the mask’s image to appear in the microscope objective’s focal plane. Since the distance from the mask to the lens is larger than ^^^^, the magnitude of this measured displacement provides adirect measure of ^^^^⁄ ^^^^^^^^ as discussed herein. The lens is rotated by ^^^^ / 2 while keeping the^ image of the mask in focus. The image’s contrast diminishes when the orientation of the ^^^^ changes from being parallel to perpendicular to the illumination’s polarization direction. This property is a stable, equilibrium configuration of the LC, existing in the absence of applied external fields. The birefringent properties of the photoaligned lenses indicate that the out-of- plane tilting of the nematic director is responsible for modulating the refractive index, confirming that the in-plane component maintains its alignment direction during the second ^ photo-exposure step. As ^^^^(^^^^)varies, the lens is a spatially variant waveplate array, which can manipulate the polarization and phase of any incident light beam. Lensing is contingent upon ^ ^ the alignment of ^^^^ with the illumination’s polarization direction (P). When ^^^^ is orthogonal to P, the microlens fails to form an image (scale bar: 20 μm).

[0066] Referring now to FIGs.4A and 4B, measuring the focal length of LC GRIN lenses is addressed. The stage is lowered until the image formed by the lens comes into focus, and the displacement ^^^^ is measured. The apparent position of the lens is affected by the refraction of light at the interface of the glass substrate and the air between the glass and the microscope objective. In addition, because of the thickness of the glass substrate relative to the focal lengths of the lenses, two scenarios may occur with regard to the focused image:the image may form either within the glass, as in FIG.4A, or in the air between the glass and the objective, as in FIG.4B. In both scenarios, the geometric optics dictate that the measured position of the focal point is affected by refraction at the air-glass interface such that, in the small angle approximation, ^^^^ is reduced from the focal length of the GRIN lens in glass, ^^^^,by a factor of 1⁄ ^^^^^^^^ , where ^^^^^^^^ is the index of refraction of the glass; that is, ^^^^ = ^^^^^^^^^^^^. Thedistance ^^^^ is equivalent to the focal length of the GRIN lens in air, and obtained by shifting the LC cell to bring one into focus and then the other. The focal length changes depending on the LC cell gap, d, and the lens’s radius RL. When the focus is within the glass substrate as in FIG.4A, both the apparent position of the lens and the apparent position of the image shift when imaged by a microscope objective in air. When the focal point of the lens is outside of the glass, as in FIG.4B, rays refract at the air-glass interface prior to reaching the focal point. In this case, the measured focal length m is again smaller than the focal length inglass by a factor of 1 / ng.

[0067] Elliptical (non-toric) focal conic domains in smectic LCs form GRIN lenses sensitive to light polarization. The same effect can also be seen in nematic GRIN lenses fabricated using the disclosed method. During exposure with unpolarized light, the in-plane ^ orientation of ^^^^ created during the initial photo-exposure remains fixed, as shown in FIG.1C. ^ Therefore, light incident on the GRIN lens with polarization direction along ^^^^ could experience a refractive index gradient. Light polarized along the direction perpendicular to the in-plane alignment will experience ^^^^^^^^and little to no gradient between the periphery and the center of the lens.

[0068] Referring now to FIG.5, a method 500 for patterning a liquid crystal (LC) director by controlling polar and azimuthal directions of LC mesogens includes, but is not limited to including, setting 502 an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light, and tuning 504 a polar angle of the LC director over a range of 0°-90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo-dye molecules to unpolarized light for a pre-selected amount of exposure time.

[0069] In some configurations, a method for aligning a liquid crystal (LC) director includes coating a substrate with a photosensitive dye, exposing the coated substrate to linearly polarized light, allowing the coated substrate to absorb the linearly polarized light,covering the coated substrate with LC mesogens, and exposing the coated substrate to unpolarized light to set the orientation of the LC mesogens that come in contact with the dye.

[0070] In some configurations, a method for aligning a liquid crystal (LC) director includes patterning a LC director by exposing a photoalignment layer to polarized light, and exposing the photoalignment layer and LC mesogens to unpolarized light, forcing the LC mesogens out of plane and controlling pretilt. The method can further include exposing different regions of a substrate holding the LC mesogens with different dosages of the unpolarized light to obtain spatial control over a 3D orientation of the LC director.

[0071] In some configurations, a method for creating a smooth 3D variation of a liquid crystal (LC) director includes photopatterning an azimuthal orientation of a nematic LC in contact with a photoalignment layer on a substrate by exposing the photoalignment layer to linearly polarized light. The photopatterning also includes exposing a coated substrate to a dosage of unpolarized light, generating an out-of-plane polar alignment of the LC director based on the dosage, and spatially varying the dosage, the spatial variation of the dosage configured to tune a polar orientation of the LC director between 0°-90°.

[0072] While the invention has been described with reference to the exemplary embodiments thereof, those skilled in the art will be able to make various modifications to the described embodiments without departing from the true spirit and scope. The terms and descriptions used herein are set forth by way of illustration only and are not meant as limitations. In particular, although the method has been described by examples, some steps of the method can be performed in a different order than illustrated or simultaneously. Those skilled in the art will recognize that these and other variations are possible within the spirit and scope as defined in the following claims and their equivalents. All patents, patent applications, other publications or documents, and the like cited herein are incorporated by reference in their entirety for all purposes to the same extent as if the individual item were specifically and individually indicated to be so incorporated by reference.

Claims

CLAIMS 1. A method for patterning a liquid crystal (LC) director by controlling polar and azimuthal directions of LC mesogens comprising: setting an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light; and tuning a polar angle of the LC director over a range of 0°-90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo-dye molecules to unpolarized light for a pre-selected amount of exposure time.

2. The method as in claim 1 further comprising: creating a spatial variation in the polar angle including exposing different parts of the photoalignment layer and the LC mesogens to the unpolarized light for different amounts of time, corresponding to a varying dosage of the unpolarized light.

3. The method as in claim 2 further comprising: obtaining the spatial variation of the unpolarized light dosage including: using a LED projector with a spatial light modulator to generate the unpolarized light that passes through one or more lenses before entering a microscope.

4. The method as in claim 1 further comprising: determining an amount of the exposure time for the polar angle by creating a calibration data set, the calibration data set configured to correlate the exposure time to an out-of-plane tilt of the LC director.

5. The method as in claim 1 further comprising: coating the substrate with an azo-dye film.

6. The method as in claim 1 wherein the substrate comprises: a transparent material.

7. The method as in claim 6 wherein the substrate comprises: glass microscope slides coated with indium tin oxide and spin-coated with a film of a photosensitive dye.

8. The method as in claim 7 wherein the photosensitive dye comprises: a dye containing azobenzene.

9. The method as in claim 7 wherein the photosensitive dye comprises: chemical abstracts service (CAS) number 3051-11-4.

10. The method as in claim 1 further comprising: obtaining the linearly polarized light from an optical signal passing through one or more lenses and a linear polarizer before entering a microscope.

11. The method as in claim 10 wherein the optical signal comprises: light from a light emitting diode.

12. The method as in claim 10 wherein the one or more lenses comprises: one or more aspheric lens.

13. The method as in claim 10 wherein the one or more lenses comprises: one or more convex lens.

14. A computer system for patterning a liquid crystal (LC) director by controlling polar and azimuthal directions of LC mesogens comprising: a hardware processor; a non-volatile storage medium storing instructions that when executed by the hardware processor perform operations comprising: setting an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light; and tuning a polar angle of the LC director over a range of 0°-90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo-dye molecules to unpolarized light for a pre-selected amount of exposure time.

15. The computer system of claim 14, wherein the operations comprise: creating a spatial variation in the polar angle including exposing different parts of the photoalignment layer and the LC mesogens to the unpolarized light for different amounts of time, corresponding to a varying dosage of the unpolarized light.

16. The computer system of claim 15, wherein the operations comprise: obtaining the spatial variation of the unpolarized light dosage including: using a LED projector with a spatial light modulator to generate the unpolarized light that passes through one or more lenses before entering a microscope.

17. The computer system of claim 14, wherein the operations comprise: determining an amount of the exposure time for the polar angle by creating a calibration data set, the calibration data set configured to correlate the exposure time to an out-of-plane tilt of the LC director.

18. A gradient-index (GRIN) lens comprising: a liquid crystal (LC) director patterned by controlling polar and azimuthal directions of LC mesogens, the patterning comprising: setting an in-plane alignment of the LC director by exposing a photoalignment layer on a substrate to linearly polarized light to create a uniform planar region of pre- selected shape with uniform planar alignment; and tuning a polar angle of the LC director over a range of 0°-90° by covering the photoalignment layer with the LC mesogens and exposing the LC mesogens and azo- dye molecules of the uniform planar region to unpolarized light illumination over an annular region with fixed outer radius and time-dependent inner radius for a pre- selected amount of exposure time, wherein the time-dependent inner radius grows so that an exposure time increases with increasing distance from a center of the annular region, forming the GRIN lens, wherein the GRIN lens has a lens radius based on a maximum of the time-dependentinner radius, and a user-defined index of refraction profile.

19. The GRIN lens as in claim 18 wherein the exposure time is based on a pretilt profile.

20. The GRIN lens as in claim 18 wherein the patterning further comprises: creating a spatial variation in the polar angle including exposing different parts of the photoalignment layer and the LC mesogens to the unpolarized light illumination for different amounts of time, corresponding to a varying dosage of unpolarized light.

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