Recent Advances in Photoalignment Liquid Crystal Polarization Gratings and Their Applications

crystals

Review Recent Advances in Photoalignment Liquid Crystal Polarization Gratings and Their Applications

Tiegang Lin, Jin Xie, Yingjie Zhou, Yaqin Zhou, Yide Yuan, Fan Fan * and Shuangchun Wen

Key Laboratory for Micro/Nano Optoelectronic Devices of Ministry of Education & Hunan Provincial Key Laboratory of Low-Dimensional Structural Physics and Devices, School of Physics and Electronics, Hunan University, Changsha 410082, China; [email protected] (T.L.); [email protected] (J.X.); [email protected] (Y.Z.); [email protected] (Y.Z.); [email protected] (Y.Y.); [email protected] (S.W.) * Correspondence: [email protected]

Abstract: Liquid crystal (LC) circular polarization gratings (PGs), also known as Pancharatnam– Berry (PB) phase deﬂectors, are diffractive waveplates with linearly changed optical anisotropy axes. Due to the high diffraction efﬁciency, polarization selectivity character, and simple fabrication process, photoalignment LC PGs have been widely studied and developed especially in polarization management and beam split. In this review paper, we analyze the physical principles, show the exposure methods and fabrication process, and present relevant promising applications in photonics and imaging optics.

Keywords: liquid crystals; polarization gratings; photonics; displays

1. Introduction Citation: Lin, T.; Xie, J.; Zhou, Y.; Zhou, Y.; Yuan, Y.; Fan, F.; Wen, S. Unlike the conventional diffraction gratings that periodically operate the amplitude or Recent Advances in Photoalignment dynamical phase of light, polarization gratings (PGs) periodically modulate the polarization Liquid Crystal Polarization Gratings states or PB phases of light by spatially changing the anisotropy parameters across the plane and Their Applications. Crystals 2021, of the elements in a periodic way [1]. Various anisotropic materials have been proposed and 11, 900. https://doi.org/10.3390/ demonstrated to fabricate PG, such as liquid crystals (LCs) [2], plasmonics [3], dielectric cryst11080900 metasurfaces [4,5], or space-variant polarizers [6], etc. Among them, LC PGs, as diffractive optical elements with an anisotropic periodic index profile using liquid crystal materials, Academic Editor: Shin-Tson Wu play a prominent part due to their easy fabrication process, unique optical properties, and potential applications [7]. Many types of LC PGs have been reported [8,9]. As one of Received: 3 June 2021 the most popular types of LC PGs, LC circular PGs have an anisotropic profile consisting Accepted: 29 July 2021 of a spiraling, constant magnitude, and linear birefringence but are constant along the Published: 31 July 2021 thickness [10,11]. In this review paper, we only discuss and review the most studied LC PGs: LC circular PGs, because LC circular PGs have received great attention owing to its Publisher’s Note: MDPI stays neutral 100% first-order diffraction efficiency and the polarization-selectivity characteristic [11]. with regard to jurisdictional claims in In order to fabricate PGs with high quality and good performance, the LC anisotropic published maps and institutional affil- axes should be precisely aligned through exposing the photoalignment materials under iations. a desired polarized field. Hence, polarization holographic recording has been regarded as a crucial step for LC PG fabrication. To date, various approaches have been applied for polarization hologram generation. Generally, they can be divided into three groups: interference [12,13], imprint [14–17], and digital approaches [18–20]. Based on the pho- Copyright: © 2021 by the authors. toalignment technology, the most widely used PGs are fabricated into reactive mesogen Licensee MDPI, Basel, Switzerland. polymer films [11] or electrically responsive LC cell [21]. In recent years, various PG- This article is an open access article based applications have emerged due to the obvious advantages of PG, such as beam distributed under the terms and steering [22], integrated photonics devices [23], image processing [24], near eye display conditions of the Creative Commons systems [13,25], etc. Attribution (CC BY) license (https:// In this paper, some recent advances in LC PG and their applications in optical systems creativecommons.org/licenses/by/ 4.0/). have been reviewed. In Section2, we analyze the basic operation principle of LC PG. In

Crystals 2021, 11, 900. https://doi.org/10.3390/cryst11080900 https://www.mdpi.com/journal/crystals Crystals 2021, 11, 900 2 of 22

Section3, the photoalignment technology and some polarization holography setup for alignment is introduced. The exposure methods for multi-domain PG are also reviewed in this part. The basic fabrication process of LC PG-based devices is described. In Section4, we review some PG-based integrated photonic devices for structured light generation and manipulation, beam steering and edge detection systems for information processing, and some applications in imaging optics and displays.

2. Operation Principles A large period LC PG in Raman–Nath regime is based on the PB phase of patterned LC wave plate [13]. The theoretical properties of PB phase optical elements can be explored by using the Jones Calculus. Based on the Jones Calculus method, when a wave plate with phase retardation δ and local direction of slow axis ϕ is illuminated by a left/right circular T polarization light formulated as Ein = [1 ± j] , the Jones vector of the output beam can be expressed as:

1 1 E = cos(δ/2) − jsin(δ/2)exp(±j2ϕ) . (1) out ±j ∓j

Here, the phase retardation δ = 2π∆nd/λ, where ∆n is the birefringence of LC, d is the thickness of LC layer, and λ is the wavelength of incident light. As for a half wave plate T (HWP) with δ = π, the corresponding output beam is Eout = −jexp(±j2ϕ) 1 ∓j , manifesting that the handedness of circular polarization is reversed, and at the same time, an additional PB phase shift with exp(±j2ϕ) is generated [26,27]. Its response to left-handed circular polarization (LCP) light and right-handed circular polarization (RCP) light is symmetric [7]. Here, Jones Calculus is the adequate way to describe the completely polarized light or the coherent superposition of polarized light because it operates the amplitudes or phases of light. The most appropriate analysis method is to use Mueller Calculus in terms of unpolarized or partially polarized light, because Stokes vectors and Mueller matrices operate on intensities [28]. The molecular orientation of LC PG is depicted in Figure1. The slow axis rotates along x-axis with an angle ϕ = πx/P and the period P is the length where the slow axis rotates by π. To investigate the diffraction behavior of such a grating in the paraxial domain, the far-ﬁeld electric ﬁeld of the m diffraction order after the Fourier transformation can be calculated as: 1 Z P Dm = T(x)Einexp(−j2πmx/P)dx, (2) P 0 where P is the period of PG and the local Jones matrix can be given by:

πx exp(−jδ/2) 0 πx T(x) = R − R , (3) P 0 exp(jδ/2) P

where R is the rotation matrix. We can simplify Equation (2) as: Dm = ΓmEin, where the 1 R P grating transfer matrix is: Γm = P 0 T(x)exp(−j2πmx/P)dx. The transfer matrix of a PG has the non-zero solutions for 0 and ±1 orders, which are expressed as:

Γ0 = cos(δ/2) I, (4)

1 −j ∓1 Γ± = sin(δ/2) , (5) 1 2 ∓1 j 2 2 The diffraction efﬁciency of each order can be formulated as: ηm = |Dm| /|Ein| , 2 1 0 2 0 which is η0 = cos (δ/2), η±1 = 2 (1 ∓ S3)sin (δ/2), respectively, where S3 = S3/S0 is the normalized Stokes parameter related to the ellipticity of input light. Here, the optical properties of the PG are summarized, as seen in Figure1a. Firstly, only 0 and ±1 orders exist and the total efﬁciency is 100%. The +1 and −1 orders are left- and right-handed circular Crystals 2021, 11, 900 3 of 22

Crystals 2021, 11, 900 3 of 22 the normalized Stokes parameter related to the ellipticity of input light. Here, the optical properties of the PG are summarized, as seen in Figure 1a. Firstly, only 0 and ±1 orders exist and the total efficiency is 100%. The +1 and −1 orders are left- and right-handed circularpolarization, polarization, respectively. respectively. The The diffraction diffraction angle angle of of± 1±1orders orders is isθ 𝜃=±𝑎𝑟𝑐𝑠𝑖𝑛(𝜆/= ±arcsin(λ/P) at 𝑃)normal at normal incidence, incidence, where whereλ is the𝜆 is wavelength the wavelength of incident of incident light. light. Secondly, Secondly,0 order 0 order preserves preservesthe polarization the polarization state and state the efﬁciencyand the effici is determinedency is determined by the phaseby the retardation.phase retardation. In order to Indepict order theto depict LC PG the structure LC PG structure in detail, in the detail, front the view front and view top and view top of view PG of structure PG struc- can be tureseen can in be Figure seen1 inb,c. Figure 1b,c.

Figure 1. Illustration of (a) LC PG structure and the diffraction property. (b) Front-view and (c) side- Figure 1. Illustration of (a) LC PG structure and the diffraction property. (b) Front-view and (c) side-viewview of the of the PG. PG.

InIn addition, addition, one one important important issue issue in LC in LCPG PGis that is that 100% 100% first-order first-order diffraction diffraction effi- effi- ciencyciency can can only only be be achieved achieved at ata single a single wavelength wavelength at normal at normal incidence incidence under under half wave half wave retardationretardation condition. condition. To solve To solve this thisissue, issue, the twisted the twisted structures structures in polymer in polymer films are films de- are signeddesigned for realizing for realizing high highefficiency efficiency in a broa inder a broader spectral spectral and angular and bandwidth, angular bandwidth, extend- ex- ingtending LC PGs LC into PGs more into practical more practical applications applications [29,30]. Here, [29,30 the]. Here, diffraction the diffraction properties properties ana- lyzedanalyzed above above were done were using done the using paraxial the paraxial approximation approximation of waves ofpropagation waves propagation at small at diffractionsmall diffraction angles relative angles to relative the propagation to the propagation axis [31]. If axis the [period31]. If of the PGs period gets smaller of PGs gets (Psmaller 2 µm ()P corresponding< 2 µm) corresponding to a larger diffraction to a larger angle, diffraction the grating angle, performs the grating in the performs Bragg in regime.the Bragg A highly regime. twisted A highly structures twisted [32,33] structures or slanted [32 ,structures33] or slanted [34] is structures usually introduced [34] is usually inintroduced transmissive in transmissiveBragg PG to keep Bragg high PG effici to keepency high at normal efficiency incidence at normal while incidence its response while its toresponse LCP and to RCP LCP is andasymmetric RCP is asymmetric[35]. Recently [35, reflective-type]. Recently, reflective-type Bragg PG is also Bragg proposed PG is also usingproposed cholesteric using LC cholesteric [36–39]. LCThese [36 –PGs39]. with These different PGs with structures different are structures analyzed areand analyzed re- viewedand reviewed by Zhang by et Zhang al. [40] et and al. [the40] coupled-wa and the coupled-waveve approach of approach Kogelnik of [41,42] Kogelnik or rigor- [41,42] or ousrigorous coupled-wave coupled-wave analysis analysis method method[35,43–45] [35 can,43 be–45 used] can to be analyze used tothe analyze LC PG thein Bragg LC PG in regimeBragg [46]. regime [46].

3.3. Device Device Fabrication Fabrication 3.1.3.1. LC LC Alignment Alignment LCLC alignment alignment technology technology is isessential essential for for planar planar patterned patterned LC LCoptical optical elements. elements. Now- Nowa- adays,days, severalseveral techniquestechniques areare proposedproposed for for LC LC alignment alignment such such as as micro-nano micro-nano rubbing rubbing [47 ,48], [47,48],direct-laser direct-laser writing writing [49,50], [49,50], and photoalignment and photoalignment [51–54 ].[51–54]. Rubbing Rubbing technique technique is considered is consideredto be a mature to be techniquea mature technique for most offor LC most displays, of LC displays, but micro-nano but micro-nano rubbing rubbing for LC for PGs are LClow-resolution PGs are low-resolution and the generation and the generation of electrostatic of electrostatic charges will charges bring will damages bring damages to alignment tolayer. alignment Direct-laser layer. Direct-laser writing can writing also produce can also LC produce PG surface LC PG alignment, surface alignment, but the multi-step but thealignment multi-step procedure alignment is procedure time-consuming is time-consuming and complicated. and complicated. LC photoalignment LC photoalign- has been mentstudied has sincebeen studied 1988 [55 since] providing 1988 [55] an providing easy way an to easy control way theto control direction the ofdirection LC molecules. of LCThe molecules. photoalignment The photoalignment method is divided method into is divided two techniques: into two techniques: surface photoalignment surface pho- and toalignmentbulk alignment. and bulk The surfacealignment. photoalignment The surface photoalignment are most widely are used most to widely create LCused PGs to with periods larger than 2 µm. Bulk photoalignment of an LC polymer exhibits the obvious advantages especially in generating the symmetric and slanted PGs with periods smaller than 1um with large diffraction angle [34]. Crystals 2021, 11, 900 4 of 22

create LC PGs with periods larger than 2 µm. Bulk photoalignment of an LC polymer Crystals 2021, 11, 900 exhibits the obvious advantages especially in generating the symmetric and slanted 4PGs of 22 with periods smaller than 1um with large diffraction angle [34].

3.1.1.3.1.1. Surface Surface Photoalignment Photoalignment PhotoalignmentPhotoalignment materials materials are are sensitive sensitive to to the the polarization polarization of of exposed exposed light light and and the the effecteffect of of LC LC photoalignment isis aa directdirect consequence consequence of of the the photo-induced photo-induced optical optical anisotropy anisot- ropyand dichroicand dichroic absorption absorption in the in thin the amorphousthin amorphous ﬁlms, films, which which is induced is induced after exposureafter expo- to sureultraviolet to ultraviolet (UV) or (UV) blue or polarized blue polarized light. Currently,light. Currently, the sulfuric the sulfur azo dyeic azo SD1 dye [56 SD1] is widely[56] is widelyused as used photoalignment as photoalignment materials ma ofterials LCPGs of LC because PGs because of their of extraordinary their extraordinary strong anchor-strong anchoringing force andforce high and photo- high photo- and thermal- and thermal- stability. stability. The mechanism The mechanism of SD1 of is photochemicalSD1 is photo- chemicalreversible reversiblecis-trans isomerizationcis-trans isomerization under the under action the of action UV-visible of UV-visible light [51 ].light The [51]. chemical The chemicalstructure structure of SD1 is of shown SD1 is in shown Figure in2a. Figure 2a.

FigureFigure 2. 2. (a(a) )Chemical Chemical structure structure of of SD1 SD1 and and (b (b) )the the illustration illustration of of photoalignment photoalignment exposure. exposure.

InIn the the experiment, experiment, SD1 SD1 is dissolved is dissolved in the in solvent the solvent dimethylformamide dimethylformamide (DMF) (DMF) in 0.5% in 0.5–1% by weight. The solution was spin coated on the substrate at speed of 800 rpm for −1% by weight. The solution was spin coated on the substrate at speed of 800rpm for 10s 10 s and then 1500 rpm for 30 s. Next, the substrate is baked for 5 min at 100 ◦C to form a and then 1500rpm for 30s. Next, the substrate is baked for 5 minutes at 100°C to form a thin photoalignment layer. As shown in Figure2b, the prepared homogeneous alignment thin photoalignment layer. As shown in Figure 2b, the prepared homogeneous alignment layer on the substrate is exposed to the polarized light, the azobenzene molecules will align layer on the substrate is exposed to the polarized light, the azobenzene molecules will perpendicularly to the long axis of the polarization ellipse [51]. align perpendicularly to the long axis of the polarization ellipse [51]. Based on the surface photoalignment technology, the most widely used PGs are Based on the surface photoalignment technology, the most widely used PGs are fab- fabricated into reactive mesogen polymer films [11] or electrically responsive LC cell [21], ricated into reactive mesogen polymer films [11] or electrically responsive LC cell [21], whose structures can be seen in Figure3. To fabricate a polymeric (passive type) PG whose structures can be seen in Figure 3. To fabricate a polymeric (passive type) PG thin thin film, first a photoalignment layer is spin coated on the clean substrate and then film, first a photoalignment layer is spin coated on the clean substrate and then exposed exposed under a polarized field. Next, the prepared reactive mesogen mixture is coated under a polarized field. Next, the prepared reactive mesogen mixture is coated on the on the surface and the UV light is applied for polymerization to form a stable thin film. surfaceThe thickness and the ofUV the light film is applied layer can for be polymeri preciselyzation controlled to form by a stable the coating thin film. speed The and thick- the nessconcentration of the film oflayer solution. can be To precisely fabricate cont an electricallyrolled by the switchable coating speed (active and type) the LCconcentra- PG, two tionprepared of solution. ITO substrates To fabricate coated an electrically with photoalignment switchable layer (active are type) assembled LC PG, together two prepared to form ITOan emptysubstrates cell. Bycoated doing with so, the photoalignment cell gap can be layer determined are assembled by the dielectric together spacers to form and an the emptyphase retardationcell. By doing can so, be the tuned cell by gap the can applied be determined voltage and by the the electro-optical dielectric spacers properties and the can phasebe seen retardation in [57]. Next, can webe tuned photo by align the the applie celld with voltage PG patternand the and electro-optical after that, we properties fill LC to canthe be cell seen at a in temperature [57]. Next, we above photo the align clear the point cell of with the LCPG and pattern then and the after LC cools that, to we the fill room LC Crystals 2021, 11, 900 5 of 22 totemperature. the cell at a Intemperature addition, the above transmissive the clear casepoint can of bethe extended LC and then to the the reflective LC cools mode to the by roomreplacing temperature. the bottom In glassaddition, substrate the transmissive with a reflective case substrate, can be extended as mentioned to the in reflective [58,59]. mode by replacing the bottom glass substrate with a reflective substrate, as mentioned in [58,59].

Figure 3. The structures of (a) polymer-ﬁlm-based and (b) electrically responsive LC-cell-based polarization gratings. (PAL, Figure 3. The structures of (a) polymer-film-based and (b) electrically responsive LC-cell-based polarization gratings. Photoalignment layer; RM, reactive mesogen; LC, liquid crystal). (PAL, Photoalignment layer; RM, reactive mesogen; LC, liquid crystal).

3.1.2. Bulk Photoalignment In contrast to surface photoalignment technology, bulk photoalignment can be also used as an alternative method to fabricate LC PG. This approach uses a photo-crosslinkable LC polymer material for the creation of LC PG. Photo-crosslinkable LC polymethacrylate consists of mesogenic biphenyl side groups and photoreactive cinnamic ester groups [60]. This polymer material is dissolved in solvent 1,1,1-trichloroethane and then spin coated on a substrate to form a homogeneous thin film. After that, the polymer film is exposed under a space-variant polarized light field at room temperature resulted in the photocycloaddition of cinnamate moieties. Next, film anneal is processed at the glass transition temperature above to enhance the birefringence and eventually cause the bulk alignment [61]. A transmissive-type LC PG based on bulk alignment can be seen in Figure 4. It uses all-in-one material that ensures an independence of the molecular orientation from any patterned surface of the alignment layer to simplify the fabrication procedure. This method creates symmetric and slanted LC PG with higher spatial resolution and higher efficiency [34].

Figure 4. The illustration of an LC PG based on bulk alignment. (PLCP, Photo-crosslinkable liquid crystal polymer).

3.2. Polarization Holography 3.2.1. Interference Methods Polarization interference method can be used to generate the polarization holograms for LC alignment with high-resolution. As seen in Figure 5a, spatial-varying linear polarization are formed by the interference pattern of two plane waves 𝑬 and 𝑬 with equal intensity and orthogonal circular polarization. When 𝑬 and 𝑬 are left-handed circular polarization (LCP) and right-handed circular polarization (RCP), the electric vectors can be written as:

𝒌⋅𝒓 𝑬 =𝑒 𝑐𝑜𝑠 𝛼1𝑒 𝑠𝑖𝑛 𝛼 𝑒 , (6) and

𝒌⋅𝒓 𝑬 =𝑒 𝑐𝑜𝑠 𝛼 1−𝑒 𝑠𝑖𝑛 𝛼 𝑒 , (7)

Crystals 2021, 11, 900 5 of 22

Figure 3. The structures of (a) polymer-film-based and (b) electrically responsive LC-cell-based polarization gratings. (PAL, Photoalignment layer; RM, reactive mesogen; LC, liquid crystal).

3.1.2. Bulk Photoalignment Crystals 2021, 11, 900 5 of 22 In contrast to surface photoalignment technology, bulk photoalignment can be also used as an alternative method to fabricate LC PG. This approach uses a photo-crosslinka- ble3.1.2. LC Bulkpolymer Photoalignment material for the creation of LC PG. Photo-crosslinkable LC polymethacrylate Inconsists contrast toof surface mesogeni photoalignmentc biphenyl technology, side groups bulk photoalignment and photoreactive can be also cinnamic ester groupsused as [60]. an alternative This polymer method tomaterial fabricate is LC dissolved PG. This approach in solvent uses a1,1,1-trichloroethane photo-crosslinkable and then spinLC polymercoated on material a substrate for the creation to form of a LC homogene PG. Photo-crosslinkableous thin film. LC After polymethacrylate that, the polymer film consists of mesogenic biphenyl side groups and photoreactive cinnamic ester groups [60]. is exposed under a space-variant polarized light field at room temperature resulted in the This polymer material is dissolved in solvent 1,1,1-trichloroethane and then spin coated on photocycloadditiona substrate to form a homogeneous of cinnamate thin moieties. film. After Next, that, the film polymer anneal film is isprocessed exposed under at the glass tran- sitiona space-variant temperature polarized above light fieldto enhance at room temperature the birefringence resulted in theand photocycloaddition eventually cause the bulk alignmentof cinnamate [61]. moieties. A transmissive-type Next, film anneal LC is processed PG base atd theon glassbulk transition alignment temperature can be seen in Figure 4. aboveIt uses to all-in-one enhance the material birefringence that and ensures eventually an independence cause the bulk alignmentof the molecular [61]. A orientation transmissive-type LC PG based on bulk alignment can be seen in Figure4. It uses all-in-one frommaterial any thatpatterned ensures ansurface independence of the alignment of the molecular layer orientation to simplify from the any fabrication patterned procedure. Thissurface method of the alignmentcreates symmetric layer to simplify and theslanted fabrication LC procedure.PG with Thishigher method spatial creates resolution and highersymmetric efficiency and slanted [34]. LC PG with higher spatial resolution and higher efficiency [34].

Figure 4. The illustration of an LC PG based on bulk alignment. (PLCP, Photo-crosslinkable liquid Figurecrystal 4. polymer). The illustration of an LC PG based on bulk alignment. (PLCP, Photo-crosslinkable liquid crystal3.2. Polarization polymer). Holography 3.2.1. Interference Methods 3.2. PolarizationPolarization Holography interference method can be used to generate the polarization holograms 3.2.1.for LC Interference alignment with Methods high-resolution. As seen in Figure5a, spatial-varying linear polarization are formed by the interference pattern of two plane waves E1 and E2 with equal intensityPolarization and orthogonal interference circular method polarization. can Whenbe usedE1 and to generateE2 are left-handed the polarization circular holograms forpolarization LC alignment (LCP) andwith right-handed high-resolution. circularpolarization As seen in (RCP), Figure the 5a, electric spatial-varying vectors can be linear polar- written as: ization are formed by the interference pattern ofT two plane waves 𝑬 and 𝑬 with equal h j π j π i jωt−jk ·r E = e 2 cosα 1 e 2 sinα e 1 , (6) intensity and orthogonal1 circular polarization. When 𝑬 and 𝑬 are left-handed circular polarizationand (LCP) and right-handed circular polarization (RCP), the electric vectors can h π π iT −j −j jωt−jk2·r be written as: E2 = e 2 cosα 1 −e 2 sinα e , (7)

2π T 2π T𝒌⋅𝒓 where k1 = λ [ −sinα 0 cos𝑬α=] 𝑒and 𝑐𝑜𝑠k2 = λ [ sinα 𝛼1𝑒 0𝑠𝑖𝑛 cosα 𝛼 ]𝑒 are the wave, vectors (6) of E1 and E2, and θ is half of the angle between these two waves. The electric ﬁled of the andresulting interference pattern at x-y plane can be written as:

T 𝒌jωt ⋅𝒓 E = E1 + E𝑬2== [ −𝑒cosα𝑐𝑜𝑠sinβ cosβ − 𝛼jsin 1−𝑒 αcos𝑠𝑖𝑛β ] 𝛼 2𝑒e ., (8) (7)

The z component can be neglected because θ is usually very small. The eventu- T ally electric ﬁeld in Jones vector can be formulated as E = 2 −sinβ cosβ , where 2πx β = λ sinα. The interference pattern can be seen in Figure5b. The electric ﬁeld intensity is constant and the linear polarization state rotates periodically along the x-axis in counter clockwise direction. Crystals 2021, 11, 900 6 of 22

where 𝒌 = [−𝑠𝑖𝑛𝛼 0] 𝑐𝑜𝑠𝛼 and 𝒌 = [𝑠𝑖𝑛 𝛼 0 𝑐𝑜𝑠𝛼] are the wave vectors of 𝑬 and 𝑬, and 𝜃 is half of the angle between these two waves. The electric filed of the resulting interference pattern at x-y plane can be written as:

𝑬 = 𝑬 +𝑬 =[− 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛𝑗 𝛽𝑐𝑜𝑠− 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠𝛽] 2𝑒 . (8)

The z component can be neglected because θ is usually very small. The eventually electric field in Jones vector can be formulated as 𝑬 =[− 2 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠𝛽] , where 𝛽= 𝑠𝑖𝑛 𝛼. The interference pattern can be seen in Figure 5b. The electric field intensity is Crystals 2021,11, 900 6 of 22 constant and the linear polarization state rotates periodically along the x-axis in counter clockwise direction.

FigureFigure 5. Interference 5. Interference of two of beams two beams with orthogonal with orthogonal circular polarization circular polarization conﬁgurations. configurations. (a) Schematic (a diagrams) Sche- of two-beammatic arrangement. diagrams of (b two-beam) Polarization arrangement. distribution of (b the) Polarization interference ﬁeld. distribution of the interference field.

The periodical polarization holograms mentioned above can be recorded by polarization- The periodical polarization holograms mentioned above can be recorded by polari- sensitive materials and furtherly transferred to LC to form a one-dimensional PG. Several zation-sensitive materialstypes of dual-beam and furtherly interferometer transferred based to on LC the to optical form patha one-dimensional differences can be PG. utilized to Several types of dual-beamgenerate the interferometer high-resolution polarization based on hologram.the optical Figure path6a differences is the conventional can be polariza- utilized to generatetion the interferometer high-resolution [32,62, 63polarization], which is established hologram. for exposingFigure 6a the is PG the with conven- relatively short tional polarizationperiods. interferometer Polarizing [32,62,63], beam splitter which (PBS) is isestablished utilized to splitfor exposing light into reflectedthe PG withs-polarized relatively short periods.and transmitted Polarizingp-polarized beam splitter components. (PBS) For is splitting utilized these to split two components light into withre- equal intensity, an HWP is inserted prior to the PBS to change the linear polarization direction. flected s-polarized and transmitted p-polarized components. For splitting these two com- Subsequently, these two beams are deflected by mirrors and normally incident on the quar- ponents with equalter intensity, wave plates an (QWPs). HWP Here,is inse therted slow prior axis ofto QWP the inPBS two to arms change is oriented the linear at ±45 ◦ with polarization direction.respect Subsequently, to the input polarization these two direction, beams are converting deflecteds-polarization by mirrors and andp-polarization normally incident oninto the LCPquarter and RCP,wave respectively. plates (QWPs). Finally, theseHere, two the orthogonal slow axis circular of QWP polarization in two beams arms is oriented atwith ±45 a certain° with crossing respect angle to the are input superimposed polarization together direction, to get the converting desired polarization s- polarization and phologram-polarization on the into substrate. LCP and To expose RCP, theresp PGectively. with relatively Finally, large these period, two theorthog- optical setup shown in Figure6a is not feasible because of its long optical path in space. Figure6b–d onal circular polarizationare the Mach–Zehnder beams with a [ 30certain], Michelson crossing [64], angle and Sagnac are superimposed interferometers together [65], which can to get the desired polarizationbe established hologram to expose theon the photoalignment substrate. To layer expose on the the substrates PG with with relatively large periods, large period, the opticalcorrespondingly. setup shown These in two Figure coherent 6a is beams not feasible can illuminate because the of photoalignment its long optical layer on path in space. Figurethe substrates6b–d are withthe aMach–Ze small crossinghnder angle [30], by Michelson slightly rotating [64], the and mirror Sagnac or PBS inter- to generate ferometers [65], whichthe spatial-varying can be established linear polarization to expose periodically.the photoalignment In the Mach–Zehnder layer on the interferometer, substrates with large periods,the PBS closed correspondingly. to the input laser These is to two split coherent the laser beams into two can arms illuminate while the the other PBS near the substrates is to combine the two arms together. After combing the two arms photoalignment layertogether, on the only substrates one QWP with oriented a sm atall45 crossing◦ with respect angle to bys-polarization slightly rotating or p-polarization the mirror or PBS to isgenerate arranged tothe generate spatial-varying two orthogonal linear circular polarization polarizations. periodically. Unlike the Mach–ZehnderIn the Mach–Zehnder interferometer,interferometer, Michelsonthe PBS closed counterpart to the fold input the two laser arms is togetherto split sothe that laser only into one PBS is two arms while theutilized other to PBS split thenear input the laser substrates and recombine is to combine the two arms the together. two arms After together. splitting by PBS, the polarization of s-polarized beam in one arm is converted into p-polarization as it travels twice of a QWP oriented at 45◦ with respect to s-polarization. Meanwhile, the polarization of p-polarized beam in the other arm is flipped into s-polarization. After combining by PBS, these two beams with orthogonal linear polarizations are transferred into orthogonal circular polarizations by QWP. However, in these dual-path configurations discussed Crystals 2021, 11, 900 7 of 22

After combing the two arms together, only one QWP oriented at 45° with respect to s- polarization or p-polarization is arranged to generate two orthogonal circular polarizations. Unlike the Mach–Zehnder interferometer, Michelson counterpart fold the two arms together so that only one PBS is utilized to split the input laser and recombine the two arms together. After splitting by PBS, the polarization of s-polarized beam in one arm is converted into p-polarization as it travels twice of a QWP oriented at 45° with respect to to s-polarization. Meanwhile, the polarization of p-polarized beam in the other arm is Crystals 2021flipped, 11, 900 into s-polarization. After combining by PBS, these two beams with orthogonal lin- 7 of 22 ear polarizations are transferred into orthogonal circular polarizations by QWP. However, in these dual-path configurations discussed above, a stable external environment must be guaranteed to obtainabove, the a desired stable external pattern environment with high must accuracy. be guaranteed To decrease to obtain the the impact desired of pattern environmental perturbationwith high accuracy. and vibration, To decrease a modified the impact Sagnac of environmental interferometer perturbation with andcom- vibration, mon path can be designeda modiﬁed for Sagnac exposure, interferometer as shown with in commonFigure 6d. path The can two be designed beams with for exposure, or- as thogonal linear polarizationshown in Figure are superimposed6d. The two beams together with orthogonal and then linear QWPs polarization manipulate are superimposed them into orthogonal circulartogether polarizations. and then QWPs Such manipulate setup themhas reduced into orthogonal sensitivity circular to polarizations. vibration, Such setup has reduced sensitivity to vibration, which improves the stability due to the common which improves thepath stability nature. due to the common path nature.

Figure 6. a b Figure( ) Conventional6. (a) Conventional two-arms two-arms polarization polarization interferometry interf forerometry exposing PGfor withexposing short period.PG with ( )short Mach–Zehnder, pe- (c) Michelson and (d) Sagnac interferometry for exposing PG with long periods. (M, Mirror; S-P, S-polarization; P-P, riod. (b) Mach–Zehnder, (c) Michelson and (d) Sagnac interferometry for exposing PG with long P-polarization; S, Substrates; LP, Linear polarization). periods. (M, Mirror; S-P, S-polarization; P-P, P-polarization; S, Substrates; LP, Linear polarization). Furtherly, two-dimensional (2D) PGs has been demonstrated. In References [66–68], Furtherly, two-dimensionalthe authors assemble (2D) the PGs patterned has been substrates demonstrated. of the LC cell In with References orthogonal [66–68], grating vectors the authors assemblefor thethe top patterned and bottom substrates substrates, of respectively.the LC cell with Reo Amano orthogonal use a self-organizedgrating vec- micro- tors for the top andpixelated bottom LC substrates, structure to resp designectively. the 2D Reo PGs [Amano69]. To directly use a generateself-organized the 2D polarization mi- patterns, the multi-beam interference exposure systems are introduced [70–74]. The po- cro-pixelated LC structure to design the 2D PGs [69]. To directly generate the 2D polari- larization patterns and the corresponding diffraction results raised interest for their rich zation patterns, thediversity. multi-beam The corresponding interference lightexposure intensity systems distribution are introduced and polarization [70–74]. hologram The can be polarization patternsanalyzed and the by thecorresponding same interference diffraction theory as results dual-beam raised interference. interest for Here, their several rich require- diversity. The correspondingments should light be met intensity to get the distribution feasible 2D PG and pattern. polarization Firstly, the intensityhologram of interferencecan be analyzed by thefield same has interference no non-zero value theory in any as location,dual-beam otherwise interference. the photoalignment Here, several materials re- cannot quirements shouldbe be completely met to get aligned. the feasible Secondly, 2D because PG pattern. the abrupt Firstly, change the of intensity the polarization of inter- orientation will cause the random alignment of the molecules in the discontinuous area, the polariza- ference field has no non-zero value in any location, otherwise the photoalignment mate- tion should have 2D chiral superstructure arrangement and vary continuously so that the rials cannot be completelyinduced 2D aligned. PG can obtainSecondly the, high because first-order the abrupt efficiency change and the of polarization-selectivity the polarization orientation willdiffraction cause the property. random In [alignment71], the authors of the demonstrated molecules ain 2D the PG discontinuous pattern via four-beam area, the polarizationpolarization should interferometryhave 2D chiral of superstructure circularly polarized arrang light.ement Soon after and this, vary a 2Dcontin- polarization structure was designed through multi-beam interference by introducing a weak linear beam into the three circular polarized beams configuration [72]. Multi-domain periodical LC structure can be applied in many display fields including holographic polarizers [75], CMOS polarization image sensor [76], and LC displays [77,78].

Crystals 2021, 11, 900 8 of 22

uously so that the induced 2D PG can obtain the high first-order efficiency and the polarization-selectivity diffraction property. In [71], the authors demonstrated a 2D PG pattern via four-beam polarization interferometry of circularly polarized light. Soon after this, a 2D polarization structure was designed through multi-beam interference by introducing a weak linear beam into the three circular polarized beams configuration [72]. Multi-domain periodical LC structure can be applied in many display fields including holographic polarizers [75], CMOS polarization image sensor [76], and LC displays [77,78]. To photo-align the pixelated PG structure where each pixel governs a PG domain Crystals 2021with, 11, independent 900 grating vector, many methods were proposed such as multi-time rub-8 of 22 bing, multi-step interference, building the nanostructure on the surface using the imprint method, or digital method as presented in Section 3.2.3. Among them, multi-step interference method is a relativelyTo photo-align simple the pixelated way to PGphoto-align structure where the eachmulti-domain pixel governs PG a PG mother domain pho- with inde- tomask with large areapendent owing grating to vector,the re-orienta many methodstion property were proposed of phot suchoalignment as multi-time layer. rubbing, Fig- multi- ure 7 depicts an examplestep interference, using the building two-step the nanostructure interference on to the generate surface using the two-domain the imprint method, PG or digital method as presented in Section 3.2.3. Among them, multi-step interference method alignment [79]. Theis whole a relatively area simple is exposed way to photo-alignto the interference the multi-domain of two PGcircularly mother photomaskpolarized with light with orthogonallarge handedness. area owing to theAfter re-orientation that, a second property exposure of photoalignment step through layer. Figurethe same7 depicts interference methodan but example with using two the orthogonal two-step interference circularly to polarized generate the light two-domain in reversed PG alignment hand- [79]. edness is processedThe under whole a area mask is exposed with tometall the interferenceic strip to of cover two circularly the whole polarized substrate. light with The orthogonal handedness. After that, a second exposure step through the same interference method effective exposure zone of the second step generates the PG structure with reversed grat- but with two orthogonal circularly polarized light in reversed handedness is processed ing vector but the blockedunder a mask area with keeps metallic the alignment strip to cover direction the whole substrate.of the primary The effective exposure, exposure as zone shown in Figure 7b.of the second step generates the PG structure with reversed grating vector but the blocked area keeps the alignment direction of the primary exposure, as shown in Figure7b.

Figure 7. Illustration of (a) two-domain patterned PG alignment structure in a two-step interference exposure method and (b)Figure the corresponding 7. Illustration LC director of (a) distribution two-domain of the patterned photo-aligned PG PG.alignment structure in a two-step interference exposure method and (b) the corresponding LC director distribution of the photo-aligned 3.2.2. Imprint Methods PG. The holographic interference method can be commonly used to generate the polarization hologram to expose the PB optical elements. If the mother mask already exists, the 3.2.2. Imprint Methodsimprint method can be used to fabricate PG with the advantages of low cost, convenient The holographicprocess, interference and small occupiedmethod areacan [80be]. commonly used to generate the polarization hologram to exposeA mask the PG PB with optical half waveelements. thickness If the can mother be used mask as the photoalready mask exists, and thethe basic operation proposal was ﬁrst demonstrated by Rolic research Ltd. [81] and it has been imprint method caneven be widelyused to employed fabricate nowadays PG with because the advantages of its characters of low of convenience cost, convenient and easy of process, and small fabricationoccupied [area14]. The[80]. exposure system is illustrated in Figure8a. The calculation for a A mask PG withx-directional half wave linearly thickness polarized can light be passingused as through the photo a uniform mask HWP and can the be basic given by operation proposalJones was Matrixfirst demonstrated as follows: by Rolic research Ltd [81] and it has been even widely employed cosϕ − sinnowadaysϕ exp( because−jπ/2) of its0 charac terscos ϕof conveniencesinϕ 1 and easycos2 ϕof fabrica- E = = −j , (9) tion [14]. Thesin ϕexposurecosϕ system 0is illustrated exp(jπ in/2 )Figure−sin 8a.ϕ Thecos calculationϕ 0 for asin2 x-directionalϕ

where ϕ is the angle between the slow axis and x-axis. As seen in Figure8a, if ϕ is the spatial-varying angle to be formulated as ϕ = πx/P where P is the period of the mother

Crystals 2021, 11, 900 9 of 22

linearly polarized light passing through a uniform HWP can be given by Jones Matrix as follows: 𝑐𝑜𝑠 𝜑 − 𝑠𝑖𝑛𝜑 𝑒𝑥𝑝( − 𝑗𝜋/2) 0 𝑐𝑜𝑠 𝜑 𝑠𝑖𝑛𝜑 1 𝑐𝑜𝑠 2𝜑 𝑬 = =−𝑗 , (9) 𝑠𝑖𝑛 𝜑 𝑐𝑜𝑠𝜑 0𝑒𝑥𝑝(𝑗𝜋/2) −𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑 0 𝑠𝑖𝑛 2𝜑 where 𝜑 is the angle between the slow axis and x-axis. As seen in Figure 8a, if 𝜑 is the spatial-varying angle to be formulated as 𝜑=𝜋𝑥/𝑃 where 𝑃 is the period of the mother PG photomask, the output polarization will be spatial-varying linear polarization and the direction angle is 2𝜑. Based on this imprint method, the PG can be generated from transferring the mother mask with a half-period 𝑃′ =𝑃/2. Thus, such easy configuration can be used to expose the short-period PG from a large-period one and even shorter one via multi-stack way [82,83]. Similarly, phase-integrated type LC devices such as fork grating, spatial-separation lens, and bifocal vortex generators can also be fabricated through this type of exposure configuration [15,23,84]. Besides the patterned HWP, the PG-patterned QWP can be employed as the mother mask to transfer the structure [15,85]. If the illuminated source is circular polarization, the calculation for a circularly polarized light passing through the uniform QWP can be given as: 𝑐𝑜𝑠 𝜑 − 𝑠𝑖𝑛𝜑 10 𝑐𝑜𝑠 𝜑 𝑠𝑖𝑛𝜑 1 𝑐𝑜𝑠( 𝜑 ± 𝜋/4) 𝑬 = =𝑒𝑥𝑝(±𝑗𝜑) . 𝑠𝑖𝑛 𝜑 𝑐𝑜𝑠𝜑 0𝑒𝑥𝑝(𝑗𝜋/2) −𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑 ±𝑗 𝑠𝑖𝑛( 𝜑 ± 𝜋/4) (10) Here, the output light is linear polarization with angle shift of 𝜋/4 and −𝜋/4 from the slow axis of the QWP, besides, a phase shift of 𝑒𝑥𝑝( 𝑗𝜋/4) and 𝑒𝑥𝑝( − 𝑗𝜋/4) also occurs with regard to LCP and RCP. Therefore, as illustrated in Figure 8b, the spatial-changing linear polarization distribution with the same period can be generated to expose the photo alignment layer via transferring from a mother PG with quarter wave thickness. The twisted nematic mode is one of the most commonly used modes in the LC displays [86]. Due to the polarization-guide effect, the twisted nematic cells can be also used as polarization converters. The obtained twisted nematic cell with a linear polarization conversion can be used as a mask to implement the photoalignment [16]. The experi- Crystals 2021, 11, 900 mental setup for LC PG photoalignment is shown in Figure 8c. The top alignment9 of 22 of the twisted nematic cell is uniform, while the alignment on the bottom substrate is designed to be the same alignment structure as PG. Therefore, when the input polarization is par- allelPG to photomask, the top alignment, the output the polarization output polari will bezation spatial-varying will be spatially linear changed polarization following and the bottomthe direction alignment angle structure, is 2ϕ. Based which on this can imprintbe used method, to align thea PG. PG The can photoalignment be generated from method 0 = usingtransferring twisted the nematic mother maskcell has with the a half-period advantagesP of Psimple/2. Thus, configuration, such easy conﬁguration convenient use, can be used to expose the short-period PG from a large-period one and even shorter one via low cost and high efficiency. Moreover, the twisted nematic LC is achromatic for twist multi-stack way [82,83]. Similarly, phase-integrated type LC devices such as fork grating, anglespatial-separation 𝛷≪2𝜋𝛥𝑛𝑑/𝜆 lens, (Mauguin and bifocal condition), vortex generators which canmakes also the be fabricatedapproach through broadband-avail- this abletype [16,87,88]. of exposure conﬁguration [15,23,84].

Figure 8. The schematic illustration of transferring the mother photomask of (a) patterned QWP, (b) patterned HWP, (c) twisted nematic cell, and (d) a wedge cell with QWP to the new device for exposure. (Photoalignment layer, PAL; LP, Linear polarization; PHWP, Patterned HWP; PWQP, Patterned QWP; OP, Output polarization).

Besides the patterned HWP, the PG-patterned QWP can be employed as the mother mask to transfer the structure [15,85]. If the illuminated source is circular polarization, the calculation for a circularly polarized light passing through the uniform QWP can be given as: cosϕ −sinϕ 1 0 cosϕ sinϕ 1 cos(ϕ ± π/4) E = = exp(±jϕ) . (10) sinϕ cosϕ 0 exp(jπ/2) −sinϕ cosϕ ±j sin(ϕ ± π/4)

Here, the output light is linear polarization with angle shift of π/4 and −π/4 from the slow axis of the QWP, besides, a phase shift of exp(jπ/4) and exp(−jπ/4) also occurs with regard to LCP and RCP. Therefore, as illustrated in Figure8b, the spatial-changing linear polarization distribution with the same period can be generated to expose the photo alignment layer via transferring from a mother PG with quarter wave thickness. The twisted nematic mode is one of the most commonly used modes in the LC displays [86]. Due to the polarization-guide effect, the twisted nematic cells can be also used as polarization converters. The obtained twisted nematic cell with a linear polarization conversion can be used as a mask to implement the photoalignment [16]. The experimental setup for LC PG photoalignment is shown in Figure8c. The top alignment of the twisted nematic cell is uniform, while the alignment on the bottom substrate is designed to be the same alignment structure as PG. Therefore, when the input polarization is parallel to the top alignment, the output polarization will be spatially changed following the bottom alignment structure, which can be used to align a PG. The photoalignment method using twisted nematic cell has the advantages of simple conﬁguration, convenient use, low cost and high efﬁciency. Moreover, the twisted nematic LC is achromatic for twist angle Φ 2π∆nd/λ (Mauguin condition), which makes the approach broadband-available [16,87,88]. Crystals 2021, 11, 900 10 of 22

These LC PG photoalignment methods are based on transferring the grating structure on the mother mask to the new substrate. Nevertheless, the mother alignment structure is essential and the spatial resolution of LC PGs depends on the mother mask. Here, a non-interferometric method based on the phase retardation variation is used to form the polarization rotation for photoalignment [17]. The optical system for generating the continuous PG structure can be seen in Figure8d. The input is linearly polarized light with 45 degrees to x-axis and therefore the polarization state can be formulated into ◦ ◦ T Ein = [ cos45 sin45 ] . The light propagates along the z-axis. The wedge cell ﬁlling with LC function as the spatially varying retardation layer and the slow axis of the wedge cell is along the x-axis. The local phase retardation of the wedge cell can be formulated as δ = 2πxtanθ∆n/λ, where ∆n is the birefringence of the LC materials and λ is the wavelength of the illuminated light. Based on the Jones Matrix, the light after passing through the wedge cell can be calculated as: √ e−jδ/2 0 2 e−jδ/2 J = E = . (11) 1 0 ejδ/2 in 2 ejδ/2 The slow axis of the following QWP is at −45 degree with respect to x-axis. The output light after passing through the QWP for photoalignment can be calculated as:

cos − 45◦ sin − 45◦ e−jπ/4 0 cos − 45◦ −sin − 45◦ sin(δ/2 + π/4) E = J = e−jπ/4 . (12) out −sin − 45◦ cos − 45◦ 0 ejπ/4 sin − 45◦ cos − 45◦ 1 cos(δ/2 + π/4)

From Equation (12), we find that the output light is linear polarization and the polarization direction angle is β = δ/2 + π/4 = πxtanθ∆n/λ + π/4 with respect to x-axis, meaning that the output linear polarization can be controlled by the phase retardation of the wedge cell and the output light is linearly changing, which is the same as the PG alignment structure. The principle of generating such a polarization conversion can be also understood in the following way. The x and y components of the input electric field have the equal amplitude in the uniaxial retardation and get a δ phase difference. After passing through the QWP, they become circular polarization with opposite handedness with a δ phase difference and the superposed light is linear polarization whose direction can be determined by δ. Based on the output polarization distribution, we can calculate out that the period of the PG alignment structure is: P = λ/tanθ∆n. Based on the same working principle, the cascaded configuration including a polarizer, a birefringent prism and a QWP with designed orientation can be also set for PG photoalignment [62,63,89], which can continuously modulate the polarization state of exposure beams. These methods can be implemented in a single-step exposure setup and with a single-beam system. Compared with the two-beam interference, these methods have the advantages of being more robust, having a lower cost, occupying a smaller space, and offering a simpler design.

3.2.3. Digital Methods Besides interference and imprint methods, digital methods offer a larger freedom of design in the PG. As shown in Figure9a, arbitrary polarization pattern such as PG structure can be digitally generated using a spatial light modulator (SLM) [18,90]. The principle is the same as the method using a wedge cell in Section 3.2.2. The phase retardation of light in each pixel can be precisely controlled by the pixel level of SLM. The polarization hologram is then generated after passing through the QWP. Secondly, because each pixel of digital micro-mirror device (DMD) can be switched between on and off state independently, DMD based lithography system can form the spatial-changing periodical polarization structure or even arbitrary polarization distribution through multistep process [19,91,92]. As shown in Figure9b, the system consists of a light emission source, a dynamic pattern module, an image focusing module, and a monitor module. The collimated light projected to the on-state micro-mirrors is reﬂected to form a desired pattern. After passing through a motorized rotatable polarizer, linearly polarized light in a designed direction is obtained. Crystals 2021, 11, 900 11 of 22

[19,91,92]. As shown in Figure 9b, the system consists of a light emission source, a dynamic Crystals 2021, 11, 900 pattern module, an image focusing module, and a monitor module. The collimated11 light of 22 projected to the on-state micro-mirrors is reflected to form a desired pattern. After passing through a motorized rotatable polarizer, linearly polarized light in a designed direction is obtained. Subsequently, it illuminates the substrates after focusing by an objective lens. Subsequently, it illuminates the substrates after focusing by an objective lens. Here, a Here, a charge-coupled device (CCD) is utilized to monitor the focusing process. The charge-coupled device (CCD) is utilized to monitor the focusing process. The drawback of drawback of digital method is that patterns are quasi-continuous and a trade-off between digital method is that patterns are quasi-continuous and a trade-off between the effective the effective area and the spatial resolution exists. Figure 9c shows the direct-writing de- area and the spatial resolution exists. Figure9c shows the direct-writing device for complex vice for complex LC patterns [20,93]. The exposure position can be precisely controlled by LC patterns [20,93]. The exposure position can be precisely controlled by a two-dimensional a two-dimensional translation stage. A desired arbitrary polarization pattern from contin- translation stage. A desired arbitrary polarization pattern from continuous scanning the uous scanning the focused spot can be directly written to the substrate. focused spot can be directly written to the substrate.

Figure 9. Schematic illustrationillustration ofof opticaloptical setups setups for for LC LC alignment alignmen tot to generate generate the the polarization polarization pattern pattern digitally digitally based based on on (a) SLM,(a) SLM, (b) ( DMD,b) DMD, and and (c) ( direct-writingc) direct-writing device. device. (P, (P, Polarizer; Polarizer; OL, OL, Objective Objective Lens; Lens; CL, CL, CollimatedCollimated Light;Light; S, Substrates; RP, RP, Rotatable polarizer; TS, Translation stage).stage).

4. Applications Applications 4.1. Multi-Functional Multi-Functional Photonic Devices 4.1.1. Orbital Orbital Angular Angular Momentum Generator and Controller Vortex beambeam carryingcarrying orbital orbital angular angular momentum momentum (OAM) (OAM) has has attracted attracted great great attention atten- tiondue due to their to their ability ability to carry to carry information, informatio detectn, detect shape, shape, and and manipulate manipulate particles particles [94 –[94–98]. 98].OAM OAM of photons of photons can can be expressed be expressed as a as helical a helical phase phase front: front:exp (𝑒𝑥𝑝(jlϕ), where 𝑗𝑙𝜑), wherel and lϕ andare 𝜑the are topological the topological charge charge and azimuthal and azimuthal angle, angle, respectively respectively [99]. [99]. Taking Taking advantage advantage of the of thestrong strong optical optical spin spin to orbitto orbit interaction interaction on on the the traditional traditional LC LC q-plate, q-plate, oneone cancan realizerealize a direct transformation of the optical angular momentum and thereby convert the OAM state of light to a new mode with adding an additional topological charge of ∆l = ±2q state of light to a new mode with adding an additional topological charge of 𝛥𝑙 = ±2𝑞 where the sign depends on the spin state [100–104]. In terms of the spin-split effect of where the sign depends on the spin state [100–104]. In terms of the spin-split effect of PG, PG, it can spatially separate OAMs with different modes into either +1 or −1 diffrac- it can spatially separate OAMs with different modes into either +1 or −1 diffraction tion orders and their propagation directions are determined by spin states [15,105]. As orders and their propagation directions are determined by spin states [15,105]. As seen in seen in Figure 10a, a high-efﬁcient OAM generator and controller based on fork polar- Figure 10a, a high-efficient OAM generator and controller based on fork polarization grat- ization grating can be considered as an integration of traditional one-dimensional PG ing can be considered as an integration of traditional one-dimensional PG and q-plate, and q-plate, whose azimuthal angle of LC directors in the x-y plane can be formulated as whose azimuthal angle of LC directors in the x-y plane can be formulated as 𝛷(𝑥, 𝑦) = Φ(x, y) = qϕ(x, y) + πx/P + Φ , where Φ is a constant. Figure 10b shows the schematic 𝑞𝜑(𝑥, 𝑦) + 𝜋𝑥/𝑃 +𝛷 , where 𝛷 0 is a constant.0 Figure 10b shows the schematic diagram diagram and diffraction property of a liquid crystal fork polarization grating with q = 1 and diffraction property of a liquid crystal fork polarization grating with 𝑞=1 under under plane wave illumination. The +1 and −1 diffraction order is LCP with OAM charge plane wave illumination. The +1 and −1 diffraction order is LCP with OAM charge 𝑙= l = 2 and RCP with OAM charge l = −2, correspondingly. Similarly, PG can be also used 𝑙=2 𝑙=−2 to generate and RCP and with manipulate OAM charge other structured, correspondingly. light such as Airy Similarly, beam [106 PG], can vortex-focusing be also used usedand defocusing to generate light and [manipulate107]. other structured light such as Airy beam [106], vortex- focusing and defocusing light [107].

Crystals 2021, 11, 900 12 of 22

CrystalsCrystals 20212021,, 1111,, 900900 1212 ofof 2222

Figure 10. (a) Schematic illustration of slow axis of fork grating, which is a sum of q-plate and PG and (b) the diffraction propertyFigure 10.10. of (( aana)) SchematicSchematicLC fork grating illustrationillustration with 𝑞=1 ofof slowslow. The axis gray of forkfork level grating,grating, from black which to iswhiteis aa sumsum indicates ofof q-plateq-plate the andandphase PGPG changes andand ((bb) )from thethe diffractiondiffraction 0 to 𝜋. propertyproperty ofof anan LCLC forkfork gratinggrating withwith q𝑞=1= 1.. The The gray gray level level from from black black to to white white indicates indicates the the phase phase changes changes from from 0 0 to toπ .𝜋. 4.1.2. Spin Hall Meta-Lens 4.1.2.In Spin order Hall to realizeMeta-Lens the multidimensional spin-dependent splitting in both transverse and longitudinalIn order to realize direction thethe multidimensionalinmultidimensional free space simultaneously spin-dependentspin-dependent by a splittingsplittingsingle device, inin bothboth a transversephotonicstransverse spinand Halllongitudinal meta-lens direction direction is designed in in freefree and space spaceproposed simultaneously simultaneously by integrating by by a PB asingle singlelens device,and device, PG ainto aphotonics photon- a con- ventionalicsspin spin Hall Hall dynamicmeta-lens meta-lens phase is designed is lens, designed as and seen proposed and in proposedFigure by 11integrating by[23,108]. integrating PBThe lens local PB and lensslow PG and axisinto PG ofa con- intothe meta-lensaventional conventional candynamic be dynamicwritten phase as phase lens,𝜃=𝜋𝑟 as lens, seen/2𝜆𝑓 as in seen+𝜋𝑥/𝑃 Figure in Figure where11 [23,108]. 11the[ 23first ,The108 item]. localThe represents slow local axis slow the of pro- axis the 2 filesofmeta-lens the of lens meta-lens can with be focalwritten can belength as written 𝜃=𝜋𝑟 ±𝑓 as to/2𝜆𝑓 focusθ = +𝜋𝑥/𝑃 πoner / spin2λ f where1 componen+ πx the/P firstwheret and item to the represents defocus first item the the repre-other pro- spinsentsfiles component,of the lens profiles with andfocal of lensthe length second with ±𝑓 focal item to length focusrepresen one± fts1 spin tothe focus linearcomponen one profiles spint and componentof toPG defocus with period and the toother de-P ,focus spinwhich component, the generate other spin theand transverse component,the second spin-dependent item and represen the secondts splitting.the item linear represents After profiles combining of the PG linear with with profilesperiod the dy- ofP namicPG, which with phase generate period lensP the, whichwith transverse focal generate lengthspin-dependent the transverse𝑓 , the totalsplitting. spin-dependent phases After can combining splitting.be written with After as the com-𝜙= dy- ±2(𝜋𝑟biningnamic /2𝜆𝑓 withphase the+𝜋𝑥/𝑃)+𝜋𝑟 lens dynamic with phasefocal/𝜆𝑓 , lenslengthwhere with the𝑓 focal, signthe length correspondstotal fphases2, the total tocan LCP phases be and written can RCP, be respec- writtenas 𝜙= 2 2 tively.as±2(𝜋𝑟φ = It/2𝜆𝑓 demonstrates±2 π+𝜋𝑥/𝑃)+𝜋𝑟r /2λ f1 that+ π thex/𝜆𝑓/ Pmeta-lens, where+ πr isthe/ λa fcombination2sign, where corresponds the of sign an toultrathin corresponds LCP and convex/concave RCP, to LCP respec- and lens,RCP,tively. beam respectively. It demonstrates splitter, and It demonstrates that dynamic the meta-lens phase that lens. theis a Th meta-lenscombinationerefore, isthe aof focal combinationan ultrathin length of convex/concave of the an meta-lens ultrathin canconvex/concavelens, be beam calculated splitter, lens,as: and 𝑓± beam =𝑓dynamic 𝑓 splitter,/(𝑓 phase and±𝑓 lens.) dynamic. Such Th erefore,integration phase the lens. approachfocal Therefore, length for of themeta-lens the focal meta-lens length pro- videsofcan the be one meta-lenscalculated more degree can as: be𝑓± of calculated=𝑓 freedom 𝑓/(𝑓 as: and ±𝑓f± implements=). Suchf1 × fintegration2/( thef1 ± functionsf2) .approach Such with integration for spin-dependent meta-lens approach pro- splitting,forvides meta-lens one imaging, more provides degree and spatiallyoneof freedom more separated degree and implements of focusing. freedom andInthe comparison implementsfunctions with with the spin-dependent functionsintensity type with bifocalspin-dependentsplitting, lens imaging, HOE splitting,[109], and spin spatially imaging, Hall meta-lensseparated and spatially based focusing. separatedon PB In phase comparison focusing. integration Inwith comparison can intensity be consid- withtype eredintensitybifocal as alens special type HOE bifocal kind [109], of lens holographicspin HOE Hall [109 meta-lens optical], spin Hall elementbased meta-lens on (HOE). PB phase based HOE integration onis based PB phase on can polarization integration be consid- holography,canered be as considereda special which kind asis formed aof special holographic by kind recording of optical holographic the element polarization optical (HOE). states element HOE of is an (HOE).based electric on HOE polarization field is based based ononholography, photoinduced polarization which holography, anisotropy is formed [110]. whichby recording The is formednew the PB pola byphase recordingrization optical states theelement polarization of an is electric designed states field by based ofadd- an ingelectricon photoinducedPB lens field and based grating, anisotropy on photoinduced exhibiting [110]. Thea anisotropyhybrid new PB function. phase [110]. optical TheIn addition, new element PB phasePB is phase designed optical optical elementby add-ele- mentsising designed PB have lens the byand addingfeature grating, PBof exhibiting lenspolarization-selecti and grating,a hybrid exhibiting function.vity and ahighIn hybrid addition, efficiency. function. PB Itphase can In addition, beoptical used ele- PBin polarization-multiplexedphasements opticalhave the elements feature haveof multi- polarization-selecti theplane feature off-axis of polarization-selectivity imagingvity and and high displays efficiency. and [111]. highIt canThe efficiency. bepresented used in It can be used in polarization-multiplexed multi-plane off-axis imaging and displays [111]. methodpolarization-multiplexed circumvents the limitations multi-plane of off-axisbulky optical imaging devices and displays in traditional [111]. integratedThe presented op- The presented method circumvents the limitations of bulky optical devices in traditional ticsmethod and it circumvents drastically increasesthe limitations the impact of bulky on multifunctional optical devices opticalin traditional elements. integrated op- integratedtics and it drastically optics and itincreases drastically the increases impact on the multifunctional impact on multifunctional optical elements. optical elements.

Figure 11. Illustration of designed spin Hall meta-lens, which is designed by integrating PB lens and PG with a dynamic phase lens.

Crystals 2021, 11, 900 13 of 22

Figure 11. Illustration of designed spin Hall meta-lens, which is designed by integrating PB lens Crystals 2021, 11, 900 and PG with a dynamic phase lens. 13 of 22

4.2. Information Processing and Displays 4.2.1.4.2. Information Beam Steering Processing System and Displays 4.2.1.Changing Beam Steering and tracking System the optical direction is highly required in free-space optical communications,Changing and laser tracking weapons, the remote optical directionsensing and is highlyfiber-optics required [22]. in Thanks free-space to the optical grat- ingcommunications, diffraction, LC laser PGs weapons, can non-mechanically remote sensing steer and the fiber-optics beam to [ 22a certain]. Thanks angle to the based grating on thediffraction, handedness LC PGsof input can light non-mechanically and the applied steer voltage the beam [112–115]. to a certain As shown angle in based Figure on 12, the a singlehandedness beam steering of input stage light consists and the of applied a polarization voltage [switch112–115 (PS)]. As (e.g., shown a 90 in° TN Figure cell 12in , combinationa single beam with steering a QWP stage [116]) consists and a of switchable a polarization PG. The switch polarization (PS) (e.g., of a 90input◦ TN light cell changesin combination from linear with polarization a QWP [116 to]) LCP and aor switchable RCP after PG.passing The through polarization a PS. of Afterwards, input light LCPchanges or RCP from light linear deflects polarization an angle toand LCP then or diffracts RCP after to passing+1 or −1 through order a by PS. the Afterwards, LC PG. If IfLCP the orLC RCP PG lightis switched deflects on an angleby applying and then a vo diffractsltage, tothe+ diffraction1 or −1 order effect by thedisappears LC PG.If and the thereforeLC PG is switchedlight transmits on by applyingto 0 order. a voltage, Here, it the can diffraction be seen th effectat this disappears system enables and therefore beam steeringlight transmits in three to different0 order. Here,directions it can by be indi seenvidually that this switching system enables the LC beam PG and steering modulating in three PS.different To obtain directions multiple by steering individually angles, switching the obvious the method LC PG andis to modulatingstack multiple PS. stages To obtain con- sistingmultiple of steeringPS and PG. angles, In this the system, obvious the method total number is to stack of multiplesteering angles stages consisting𝑀 are determined of PS and byPG. the In number this system, of stages the total 𝑁. numberIf the LC of PG steering in each angles stageM hasare a determineddifferent grating by the period, number the of totalstages steeringN. If the angles LC PG can in be each expressed stage has as: a different𝑀=3^𝑁 grating. However, period, this the non-mechanical total steering anglesbeam steeringcan be expressed system cannot as: M =steer3N. However,light continuously. this non-mechanical To realize beam the continuous steering system and cannothigh- throughputsteer light continuously. beam steering, To realizeone typical the continuous approach andis to high-throughput use Risley gratings beam comprised steering, one of twotypical independently approach is torotating, use Risley inline gratings PGs [117]. comprised In conclusion, of two independently thanks to the rotating, single-order inline diffraction,PGs [117]. polarization In conclusion, selectivity thanks to behavior the single-order and wide diffraction, acceptance polarization angle of PGs, selectivity optical systemsbehavior based andwide on PGs acceptance can accomplish angle of the PGs, wide- opticalangle systems beam basedsteering on in PGs a high can accomplishefficiency, compact,the wide-angle and light-weight beam steering way. in a high efficiency, compact, and light-weight way.

Figure 12. Beam steering system based on a switchable PG and a polarization switch. (PS, Polarization Figure 12. Beam steering system based on a switchable PG and a polarization switch. (PS, Polari- switch). zation switch). 4.2.2. Edge Detection 4.2.2. Edge Detection Edge detection can significantly improve the system efficiency and accuracy in image processing,Edge detection computing can significantly and machine improve vision because the system it can efficiency obviously and reduce accuracy the in amount image processing,of stored data computing but reserve and the machine key image vision information because it at can the obviously same time. reduce Many the approaches amount ofcan stored realize data the but edge reserve detection, the key mainly image divided information into digital at the same [118] andtime. analog Many [approaches24,119–124] canprocessing realize the methods. edge detection, Among them, mainly an divided optical analoginto digital method [118] based and onanalog the PG [24,119–124] shows the processingadvantages methods. of ultra-fast Among processing them, speedsan optical and analog low power method consumption. based on the The PG fundamental shows the advantagesoptical setup of ultra-fast is proposed processing and demonstrated speeds and inlow Figure power 13 consumption.a. A 4f optical The imaging fundamental system opticalconsisting setup of twois proposed lenses with and identical demonstrated focus length in Figure is installed. 13a. A 4f Here, optical the distanceimaging betweensystem consistingthe two lenses of two is lenses 2f and with the PGidentical is inserted focus inlength the middle is installed. of them. Here, The the objectdistance is placedbetween at theone two focal lenses length is in2f frontand the of the PG first is inserted lens and in the the CCD middle is placed of them. at one The focal object length is placed afterthe at onesecond focal lens. length Thus, in front the first of the lens first conducts lens and the the Fourier CCD transformationis placed at one of focal the objectlength at after its back the secondfocal plane, lens. whichThus, isthe also first the lens location conducts of the the LC Fourier PG. Then, transformation the second lensof the fulfills object another at its Fourier transformation so that the corresponding diffraction pattern can be collected and observed in a CCD. In addition, a PG can implement the optical spatial differentiator while the two orthogonally directional polarizers filter out the edge information. To better understand the edge-detection concept of this work, consider a square-shaped object being Crystals 2021, 11, 900 14 of 22

back focal plane, which is also the location of the LC PG. Then, the second lens fulfills another Fourier transformation so that the corresponding diffraction pattern can be collected and observed in a CCD. In addition, a PG can implement the optical spatial differentiator while the two orthogonally directional polarizers filter out the edge information. To better understand the edge-detection concept of this work, consider a square-shaped object being introduced in our system, as shown in Figure 13b. It is illuminated by a linearly polarized light along x direction and the Jones vector of electric field can be formulated as [10]. The PG will split the RCP and LCP components to different directions and the output electric field at image plane can be written as: 1 1 𝑬 (𝑥, 𝑦) =𝐸 (𝑥 − 𝛥, 𝑦) +𝐸 (𝑥 + 𝛥, 𝑦) −𝑗 𝑗 , (13) where 𝛥=𝜆𝑓/𝑃, 𝜆 denotes the light wavelength, and 𝑓 is the focal length. From Equa- tion (13), it can be found that the output light is composed of LCP and RCP with deviation distances ±𝛥. The final electric field in the image plane after passing through the whole system can be formulated as: 0 𝑬 (𝑥, 𝑦) = (𝐸 [(𝑥 + 𝛥), 𝑦] −𝐸 [(𝑥 − 𝛥), 𝑦]) _ 𝑗 . (14) Crystals 2021, 11, 900 If the shift Δ is much smaller than the image size, the amplitude of edge can be written14 of 22 as: (,) 𝑬 (𝑥, 𝑦) ≃ 2𝛥 . (15) introduced in our system, as shown_ in Figure 13b. It is illuminated by a linearly polarized T Here,light alongbecause x directionthe period and of thethe Jonesutilized vector PG ofis large, electric the field shift can Δ be is formulated much smaller as than1 0 the . imageThe PG size will and split 𝑬_ the RCP(𝑥, and 𝑦) is LCP approximately components proportional to differentdirections to the first-order and the spatial output differentiationelectric field at of image the input plane light. can As be seen written in Figure as: 13b, LCP and RCP images are projected at the image plane with a tiny shift. The overlapped region interferes LCP and RCP again and becomes original linear polarization. Thus,1 the second polarizer1 can eliminate the Eout(x, y) = Ein(x − ∆, y) + Ein(x + ∆, y) , (13) light in the overlapped region but the edge information−j in the non-overlappingj part still exists. where ∆ = λ f /P, λ denotes the light wavelength, and f is the focal length. From Equa- In addition, the resolution of edge detection can be adjusted by applying different tion (13), it can be found that the output light is composed of LCP and RCP with deviation grating period 𝑃 [24]. More recently, changing the angles between the fringe directions distances ±∆. The final electric field in the image plane after passing through the whole of two PGs can realize the adjustable resolution at any real-time without varying the sys- system can be formulated as: tem construction, which provides a more feasible and effective method [122]. Further- more, the edge detection can be extended to two dimensions based on0 a flat PB lens with E (x, y) = (E [(x + ∆), y] − E [(x − ∆), y]) . (14) large focus length [123]out_edge or a customizedin PB element comprisingin a symmetricj phase gradient along the radical direction [124].

Figure 13. (a) Experimental setup for edge detection based on a PG. (b,c) Input and output images of a square object in real space. The red and blue areas represent the edge information through the analyzer. (L1, Lens1; L2, Lens2, P1, Polarizer1, P2, Polarizer2).

If the shift ∆ is much smaller than the image size, the amplitude of edge can be written as: dE (x, y) E (x, y) ' 2∆ in . (15) out_edge dx Here, because the period of the utilized PG is large, the shift ∆ is much smaller than

the image size and Eout_edge(x, y) is approximately proportional to the ﬁrst-order spatial differentiation of the input light. As seen in Figure 13b, LCP and RCP images are projected at the image plane with a tiny shift. The overlapped region interferes LCP and RCP again and becomes original linear polarization. Thus, the second polarizer can eliminate the light in the overlapped region but the edge information in the non-overlapping part still exists. In addition, the resolution of edge detection can be adjusted by applying different grating period P [24]. More recently, changing the angles between the fringe directions of two PGs can realize the adjustable resolution at any real-time without varying the system construction, which provides a more feasible and effective method [122]. Furthermore, the edge detection can be extended to two dimensions based on a ﬂat PB lens with large focus length [123] or a customized PB element comprising a symmetric phase gradient along the radical direction [124]. Crystals 2021, 11, 900 15 of 22

Figure 13. (a) Experimental setup for edge detection based on a PG. (b) and (c) Input and output images of a square object in real space. The red and blue areas represent the edge information Crystals 2021, 11, 900 through the analyzer. (L1, Lens1; L2, Lens2, P1, Polarizer1, P2, Polarizer2.) 15 of 22

4.2.3. High Transmittance Holographic Polarizer 4.2.3.Polarizers High Transmittance play an important Holographic role Polarizerin current display and photonic. Nowadays, the widespreadPolarizers polarizers play an are important absorption-type role in current and reflection-type display and photonic. caused Nowadays,that the transmit- the tancewidespread ratio of polarizersunpolarized are light absorption-type is less than and 50%. reflection-type To improve causedthe light that efficiency, the transmit- a high transmittancetance ratio of polarizer unpolarized is developed light is less based than on 50%. the Topolarization improve the conversion light efficiency, [39,75,125,126]. a high Here,transmittance a holographic polarizer polarizer is developed consisted based of onstructured the polarization PG and conversion patterned [39 QWP,75,125 is, 126shown]. withHere, ideal a holographic 100% efficiency. polarizer As illustrated consisted of in structured Figure 14a, PG the and basic patterned optical QWP function is shown layer is a withstructured ideal 100% LC PG efficiency. film periodically As illustrated patterned in Figure with 14a, four the basicdomains: optical A, function B, C, and layer D while, is a thestructured grating LCvector PG filmin each periodically domain patternedpoints to withthe common four domains: center A, of B, the C, and four D domains. while, the As wegrating analyzed vector earlier, in each when domain the points non-polarized to the common input center light oftransmits the four domains.through the As wePG an- with halfalyzed wave earlier, retardation, when the it non-polarizedwill be spatially input separated light transmits into LCP through and theRCP PG beams, with half which wave cor- retardation, it will be spatially separated into LCP and RCP beams, which correspond to respond to the +1st and −1st diffraction orders, respectively. As shown in Figure 14b, the the +1st and −1st diffraction orders, respectively. As shown in Figure 14b, the propagation propagation directions of light after passing through the PG film in each domain are de- directions of light after passing through the PG film in each domain are determined by terminedtheir local by grating their local vectors. grating Benefited vectors. from Bene thefited complex from arrangement the complex of LCarrangement patterns, LCP of LC patterns,and RCP LCP beams and are RCP spatially beams gathered are spatially and distinguished gathered and after distinguished a short distance after away a short from distancethe PG away layer. from This the distance PG layer. is determined This distance by both is determined the size of the by microscale both the size domain of the and mi- croscalethe period domain of the and LC structures.the period Theof the broader LC st distributionructures. The of LCPbroader and distribution RCP light at theof LCP plane and RCPof this light distance at the canplane be of seen this in distance Figure 14 canc. be seen in Figure 14c.

Figure 14. Schematic illustration of (a) LC directors of the complex-ordered PG for a holographic polarizer and (b) the Figure 14. Schematic illustration of (a) LC directors of the complex-ordered PG for a holographic polarizer and (b) the near-field light propagation within a concrete distance where red domain represents the beam distribution of LCP in each near-field light propagation within a concrete distance where red domain represents the beam distribution of LCP in each domain while black domain represents those of RCP. (c) The patterned QWP with two-domain for polarization conversion. domain while black domain represents those of RCP. (c) The patterned QWP with two-domain for polarization conversion. In order to develop the structured PG into a holographic polarizer, a patterned QWP with two domains is assembled in this given distance. The function of patterned QWP is to convertIn order the circularto develop polarization the structured to linear PG polarization. into a holographic Here, onepolarizer, domain a coverspatterned all theQWP withLCP two beams domains while theis assembled other domain in this covers given all thedistance. RCP beams. The function It should of be patterned noted that QWP the is toslow convert axis the of domain circular 1 polarization is perpendicular to linear to that polarization. of domain 2, Here, which one isset domain for obtaining covers all the the LCPhomogeneous beams while linear the other polarization domain output. covers Based all the on RCP the beams. analysis It above, should the be holographicnoted that the slowthin-film axis of polarizer domain with1 is perpendicular high transmittance to that can of be domain successfully 2, which designed is set for via obtaining fitting the the homogeneousstructured PG linear film and polarizati patternedon QWP. output. Because Based the on transmittance the analysis of above, conventional the holographic absorp- thin-filmtion and reflectionpolarizer typewith polarizer high transmittance is less than 50%, can such be successfully demonstrated designed holographic via polarizer fitting the structuredshows great PG potential film and to doublepatterned the QWP. optical Because efficiency the of polarizingtransmittance optical of systemsconventional like the ab- sorptionbacklights and of reflection LC displays. type Moreover, polarizer the is output less than light 50%, intensity such with demonstrated regard to holographic holographic polarizer remains constant regardless of the direction angle of input polarization. That is to say, besides the property of high transmittance, the holographic polarizer can also present

Crystals 2021, 11, 900 16 of 22

polarizer shows great potential to double the optical efficiency of polarizing optical systems like the backlights of LC displays. Moreover, the output light intensity with regard Crystals 2021, 11, 900 to holographic polarizer remains constant regardless of the direction angle of input polar-16 of 22 ization. That is to say, besides the property of high transmittance, the holographic polarizer can also present the unique character of polarization and intensity maintenance. The discoveredthe unique character character is of of polarization great significance and intensity for many maintenance. photonics The applications discovered including character photonicis of great fibers significance and photonics for many waveguides, photonics which applications can be further including employed photonic in optical fibers com- and municationphotonics waveguides, systems. which can be further employed in optical communication systems. 4.2.4. Enlarging the Eyebox for Near-Eye Display 4.2.4. Enlarging the Eyebox for Near-Eye Display Currently, mismatch of converging and accommodation distance, called vergence- accommodationCurrently, mismatchconflicts (VAC), of converging remains andin mo accommodationst of the current distance, virtual reality called (VR) vergence- and augmentaccommodation reality (AR) conflicts near-eye (VAC), displays remains (NEDs) in most [127]. of This the currentmajor problem virtual increases reality (VR) fusion and timeaugment of binocular reality (AR)imagery near-eye and decreases displays (NEDs) fusion accuracy, [127]. This which major drawn problem the increases researchers fusion to alleviatetime of binocularthis issue. imagery Several and proposals decreases were fusion proposed accuracy, to whichalleviate drawn this theissue researchers including to multi-focusalleviate this display issue. Several[128], light proposals field display were proposed [129], holographic to alleviate thisdisplay issue [130], including and Max- multi- wellianfocus display display [ [25,131].128], light The field working display mechanism [129], holographic of alleviating display VAC [130 in ],Maxwellian-view and Maxwellian displaysdisplay [is25 that,131 ].the The images working can mechanismbe directly ofprojected alleviating to the VAC retina in Maxwellian-view by converging the displays light raysis that to the the eye images pupil. can Therefore, be directly it projectedcan present to the always retina by in-focus converging images the regardless light rays of to thethe focus eye pupil. depth Therefore, of human it eye. can Maxwellian-view present the always NEDs in-focus have images the obvious regardless advantages of the focus in- depth of human eye. Maxwellian-view NEDs have the obvious advantages including cluding simpler optical system and higher optical throughput. Nevertheless, the eyebox simpler optical system and higher optical throughput. Nevertheless, the eyebox size is size is relatively narrow, preventing its broader development. Here, in order to overcome relatively narrow, preventing its broader development. Here, in order to overcome the the small eyebox limitation, a Maxwellian NED architecture with an expanded eyebox small eyebox limitation, a Maxwellian NED architecture with an expanded eyebox based based on PG is demonstrated. As shown in Figure 15, a laser scanning projector is applied on PG is demonstrated. As shown in Figure 15, a laser scanning projector is applied as as the image source and a collimation lens to collimate light. The resulting polarized light the image source and a collimation lens to collimate light. The resulting polarized light rays after passing through the polarization controller (PC) are converged to the eye pupil rays after passing through the polarization controller (PC) are converged to the eye pupil of human by the eyepiece lens. When we insert a PG between the polarization controller of human by the eyepiece lens. When we insert a PG between the polarization controller and eyepiece lens, the efficiency of each order can be controlled by the input polarization and eyepiece lens, the efficiency of each order can be controlled by the input polarization and the phase retardation of PG. Thus, the collimation light is split into three orders with and the phase retardation of PG. Thus, the collimation light is split into three orders with differentdifferent diffraction diffraction angles andand eventuallyeventually be be focused focused on on different different spot spot positions positions to enlargeto en- dx= tanθ θ largethe eyebox. the eyebox. In addition, In addition, the distancethe distance of adjacent of adjacent focus focus spot spot is d is= xtanθ, where, whereθ is the isdiffraction the diffraction angle angle and xand is thex isdistance the distance between between the PGthe andPG and the focusthe focus plane plane of eyepiece of eye- piecelens. lens. In order In order to avoidto avoid the the double double image image effect effect in in the the real real scene,scene, thisthis space is is usually usually designeddesigned to to be be a a little little larger than thethe eyeeye pupilpupilwidth. width. Further, Further, the the full-color full-color display display can can be berealized realized based based on on a a double double LC LC layer layer withwith twistedtwisted structuresstructures [[25,29]25,29] and the the eyebox eyebox size size cancan be be extended extended in in two-dimensions two-dimensions through through a a customized customized Dammann Dammann grating grating [131,132]. [131,132]. In In conclusion,conclusion, our our approach providesprovides aa simple simple way way to to enlarge enlarge the the eyebox eyebox for Maxwellian-viewfor Maxwellian- viewNEDs NEDs while while adding adding the negligible the negligible weight weight and volume.and volume.

Figure 15. Schematic illustration of the optical setup in Maxwellian-view near-eye displays with an expanded eyebox. (PC, polarization controller; CL, collimation lens; EL, eyepiece lens).

5. Conclusions In conclusion, we have reviewed some advances in the physical principles, fabrication methods of LC PGs. The relevant novel applications for photonics and displays are also introduced. Based on the photoalignment technology, the existing exposure methods Crystals 2021, 11, 900 17 of 22

including interference, patterned wave plate imprint, and digital approaches are crucial for LC alignment with high stability and good performance. Based on LC PG, many applications were demonstrated and developed. Firstly, the PG can be integrated to other PB phase elements or dynamic phase elements to generate and manipulate the structured light in micro-nano photonics area. It can be also applied to the meta-lens to accomplish spin-dependent splitting, spatially separated multi-focusing, and imaging, simultaneously. These findings provide extra degrees of freedom to manage polarization photonics and show more possibilities for exploiting multi-functional devices. Moreover, it can be used as a beam steering device or designed as a high-efficient holographic polarizer. It can also implement the edge detection to reduce the information amount and enlarge the eyebox in the near-eye displays. In the future, more theoretical and experimental improvement in terms of LC PGs should be concerned. On the one hand, theoretical analysis of LC PGs with large diffraction angle, customized spectral and angular response are highly desirable for satisfying requirements of practical optical system. One the other hand, experimental demonstrations of LC PG based on photoalignment technology should be focused. Various generation methods of polarization hologram should be explored for exposing large area and high- resolution LC PG in a high-stable and low-cost way. In addition, benefiting from the optical properties of LC PG, a polarization-dependent off-axis imaging in a free space near-eye displays can be implemented by introducing a compact combiner with PG and lens phase profiles [111,133]. Additionally, eye-tracking [116], multifocal display [134], or extending the field of view [135,136] in the waveguide displays can be performed using LC PG as a prominent beam coupler. Hence, an increasing attention should be paid on applying LC PGs into more optical systems to address some difficulties. The widespread applications of LC PGs especially in imaging and wearable displays are foreseeable.

Author Contributions: T.L., J.X. and Y.Z. (Yingjie Zhou) wrote the manuscript. T.L., J.X., Y.Z. (Yaqin Zhou) and Y.Y. revised the manuscript. F.F. and S.W. supervised the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: National Natural Science Foundation of China (Grant No. 61605046 and 11574079), Natural Science Foundation of Hunan Province (Grant No. 2018JJ3069) and Hunan Provincial Innovation Foundation for Postgraduate (CX20190318). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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