Twisting Structures in Liquid Crystal Polarization Gratings and Lenses

: Recently, diverse twisting structures have been discovered to be a potential approach to design liquid crystal polarization gratings and lenses (LCPGs and LCPLs) with a high diffraction efﬁciency, broad bandwidth, wide view, and large diffraction angle. In this review, we divide these twisting structures into two main types, namely, multi-layer twisting structures with phase compensation and twisting structures forming Bragg diffraction. We found that multi-layer twisting structure LCPGs and LCPLs presented a broader bandwidth and a wider view angle by phase compensation. While for transmissive or reﬂective Bragg LCPGs, a large diffraction angle with high diffraction efﬁciency could be achieved. Based on the theoretical analysis in the review, potential research directions on novel twisting structures were prospected.


Introduction
Since the geometrical phase [1][2][3] was proposed in 1956, a new degree of freedom was provided to modulate the phase, amplitude, polarization and wave front of light waves. Many kinds of geometrical phase devices have been discovered and researched over the years. Liquid crystal (LC) based devices, such as liquid crystal polarization gratings and lenses (LCPGs and LCPLs), were one of the most important types due to their excellent performance for various applications, like non-mechanical beam deflection [4][5][6], virtual/augmented reality (VR/AR) display, near-eye display and waveguide-based display devices [7][8][9][10][11][12][13][14], etc.
From the previous studies [4,5,[15][16][17], LCPGs and LCPLs with typical planar structure can include 100% diffraction into a single order. However, the diffraction efficiency equations are derived using the paraxial approximation, which means the devices are considered "thin" [18][19][20]. Thus, a diffraction efficiency of 100% can only be achieved at a specific wavelength, which meets the half wave condition (∆nd/λ = (2k + 1)/2, where k is an integer). Therefore, if it deviates from the design wavelength, the diffraction efficiency of the device will decrease rapidly. Besides, for LCPGs, a large diffraction angle could be achieved by decreasing its period Λ. However, for a small-period LCPG (usually, Λ < 2 µm [18,21]), which no longer meets the condition of a "thin" grating, its diffraction efficiency falls sharply. The above restrictions would seriously affect practical applications, such as near-eye and full-color displays [10], which need a large diffraction angle.
In order to solve the problems above, several in-depth studies have been developed. Combining the self-organized structure and optical properties of twisted liquid crystals with the traditional PGs and PLs is the main research direction at present. Based on phase compensation, Oh et al. [19,22,23] have proved that the dual-twist structure can broaden the bandwidth of high diffraction efficiency, and Li Tan [15,24,25] has also confirmed that a twisting structure can also provide certain compensation in terms of view angle. Tao Zhan [11,14], Lingshan Li [26] et al. studied wide-band lens. Under certain conditions, a Bragg diffraction could be achieved by a twisting structure. Xiao Xiang et al. [27][28][29][30] introduced transmissive Bragg PGs based on single and dual-layer twisting structures with large diffraction angles. Yishi Weng et al. [7,9,10] studied Bragg gratings and reflective polarized volume gratings (PVGs) with high efficiency and large diffraction angle. Kun Yin [13,14] and Li et al. [12] studied polarized volume lens (PVLs). These reflective devices can be applied in various fields like wearable displays. From those works reported above, we found that the twisting structures, which work in different methods [7,14], are well used by researchers to produce a series of devices with better performance.
In this work, we review recent advances of LCPGs and LCPLs with twisting structure. First in Section 2, we begin with the LC director configuration and optical properties of the twisted liquid crystals. Then we delve into the theoretical model and the differences between various kinds of gratings and lenses. In Section 3, we discuss LCPGs and LCPLs with multi-layer twisting structures, which provide a better performance in terms of broadband and wideview angle. In Section 4, we focus on Bragg diffraction LCPGs and LCPLs with high efficiency, large-angle diffraction generated by twisting structures. Then, several kinds of reflective and transmissive devices are expounded. Figure 1 illustrates the twisting structure of LC. Note that the LC is still in a nematic phase in each plane (x-y plane) perpendicular to the helical axis (z axis), which could be regarded as a uniaxial crystal.

Twisted LC with LCPG Structure
with the traditional PGs and PLs is the main research direction at present. Based o compensation, Oh et al. [19,22,23] have proved that the dual-twist structure can b the bandwidth of high diffraction efficiency, and Li Tan [15,24,25] has also confirm a twisting structure can also provide certain compensation in terms of view ang Zhan [11,14], Lingshan Li [26] et al. studied wide-band lens. Under certain condi Bragg diffraction could be achieved by a twisting structure. Xiao Xiang et al. [27 troduced transmissive Bragg PGs based on single and dual-layer twisting structur large diffraction angles. Yishi Weng et al. [7,9,10] studied Bragg gratings and re polarized volume gratings (PVGs) with high efficiency and large diffraction ang Yin [13,14] and Li et al. [12] studied polarized volume lens (PVLs). These reflective can be applied in various fields like wearable displays. From those works reported we found that the twisting structures, which work in different methods [7,14], a used by researchers to produce a series of devices with better performance.
In this work, we review recent advances of LCPGs and LCPLs with twistin ture. First in Section 2, we begin with the LC director configuration and optical pro of the twisted liquid crystals. Then we delve into the theoretical model and the diff between various kinds of gratings and lenses. In Section 3, we discuss LCPGs and with multi-layer twisting structures, which provide a better performance in te broadband and wideview angle. In Section 4, we focus on Bragg diffraction LCP LCPLs with high efficiency, large-angle diffraction generated by twisting stru Then, several kinds of reflective and transmissive devices are expounded. Figure 1 illustrates the twisting structure of LC. Note that the LC is still in a n phase in each plane (x-y plane) perpendicular to the helical axis (z axis), which c regarded as a uniaxial crystal. Equation (1) represents the LC director field:

Twisted LC with LCPG Structure
where ψ is the azimuth angle at z, φ0 is the azimuth angle of the initial position (z = helical pitch P was defined as the distance in which the orientation of LC molec tated 360 degrees. The direction of the twist is represented by the sign of the helic P, usually the right-handed twisting is defined as positive (P > 0). According to the relationship between the wavelength and the helical pitch, tical property of the LC layer could be divided into four different regions [31]: 1. Mauguin region: In this region (λ << 0.5ΔnP), the polarization ellipses eigenmodes have either near-infinite ellipticity with the long axis parallel or dicular to the local director, which could be approximated as a pair of orth Equation (1) represents the LC director field: where ψ is the azimuth angle at z, ϕ 0 is the azimuth angle of the initial position (z = 0). The helical pitch P was defined as the distance in which the orientation of LC molecules rotated 360 degrees. The direction of the twist is represented by the sign of the helical pitch P, usually the right-handed twisting is defined as positive (P > 0). According to the relationship between the wavelength and the helical pitch, the optical property of the LC layer could be divided into four different regions [31]:

1.
Mauguin region: In this region (λ << 0.5∆nP), the polarization ellipses of the eigenmodes have either near-infinite ellipticity with the long axis parallel or perpendicular to the local director, which could be approximated as a pair of orthogonal linear po-larization states. As the eigenmodes propagate in the medium, the polarization axes will follow the twist of the local director, which is called waveguiding (or adiabatic following). TN-LCDs correspond to this situation [32]; 2.
Short-wavelength circular birefringence region (SW-CBR): In this region (λ < P), the eigenmodes are elliptically polarized with the opposite handedness. As helical pitch decreases, the waveguiding phenomenon no longer holds. When the incident polarized light propagates in the LC layer, its polarization axis still rotates, but, usually, it will not remain parallel (or perpendicular) to the local LC director. Besides, its ellipticity will change along with the propagation distance. In essence, the Mauguin region is a limiting case of this region; 3.
Bragg region: In this region (n o P < λ < n e P), the liquid crystal layer will provide a strong selective circular reflection to the incident light, that is, a circular polarized incident light with the same rotation direction of the LC layer will be reflected, meanwhile the opposite one could propagate through. The twisting LC configuration produces a Bragg structure: its period T (T = 0.5P) could be regarded as a series of the crystal plane, shown in Figure 2. According to Equation (2), which is known as Bragg's law: supposed that the medium within a period is homogeneous and provides an average refractive index n, θ is the angle between the incident light and the "crystal plane". For normal incidence (θ = 90 • ), the central wavelength of the reflection is λ 0 = nP, and the reflectance band is n o P < λ < n e P. For tilted incidence, the central wavelength of the reflection is λ 0 = nP sin θ.

4.
Long-wavelength circular birefringence region (LW-CBR): In this region (λ >> P), both eigenmodes are almost circularly polarized, and would pass through the LC layer. However, due to the difference in the wavenumber (refractive index), their propagation speeds are different, and the principal axes of their resultant, a beam of linearly polarized light, has a net rotation relative to the LC director at the exit plane.
Crystals 2021, 11, x FOR PEER REVIEW 3 of 21 linear polarization states. As the eigenmodes propagate in the medium, the polarization axes will follow the twist of the local director, which is called waveguiding (or adiabatic following). TN-LCDs correspond to this situation [32]; 2. Short-wavelength circular birefringence region (SW-CBR): In this region (λ < P), the eigenmodes are elliptically polarized with the opposite handedness. As helical pitch decreases, the waveguiding phenomenon no longer holds. When the incident polarized light propagates in the LC layer, its polarization axis still rotates, but, usually, it will not remain parallel (or perpendicular) to the local LC director. Besides, its ellipticity will change along with the propagation distance. In essence, the Mauguin region is a limiting case of this region; ( )3 supposed that the medium within a period is homogeneous and provides an average refractive index n , θ is the angle between the incident light and the "crystal plane". For normal incidence (θ = 90°), the central wavelength of the reflection is P n λ = 0 , and the reflectance band is noP < λ < neP. For tilted incidence, the central wavelength of the reflection is 0 sin λ nP θ = . 4. Long-wavelength circular birefringence region (LW-CBR): In this region (λ >> P), both eigenmodes are almost circularly polarized, and would pass through the LC layer. However, due to the difference in the wavenumber (refractive index), their propagation speeds are different, and the principal axes of their resultant, a beam of linearly polarized light, has a net rotation relative to the LC director at the exit plane.
It is necessary for us to know well the optical properties of twisting liquid crystals, which could help us to understand the reasons why polarization gratings in such structures could present different effects. Furthermore, it would provide theoretical guidance and new ideas for the research of a new generation of high-performance LC devices.
Based on the optical characteristics of the twisted liquid crystals mentioned above, combining the periodic structure Λx in the horizontal plane introduced by the typical planar LCPG alignment surface and Λy = P/2 along the vertical direction introduced by twisted LC, a series of slanted and periodical refractive index planes [7] have been set up, as shown by the green dashed line in Figure 3. The slant angle is φ = ± arctan (Λy/Λx), (0° < φ < 90°). It is necessary for us to know well the optical properties of twisting liquid crystals, which could help us to understand the reasons why polarization gratings in such structures could present different effects. Furthermore, it would provide theoretical guidance and new ideas for the research of a new generation of high-performance LC devices.
Based on the optical characteristics of the twisted liquid crystals mentioned above, combining the periodic structure Λ x in the horizontal plane introduced by the typical planar LCPG alignment surface and Λ y = P/2 along the vertical direction introduced by twisted LC, a series of slanted and periodical refractive index planes [7] have been set up, as shown by the red dashed line in Figure 3. The slant angle is ϕ = ± arctan (Λ y /Λ x ), (0 • < ϕ < 90 • ).
in a slanted dimension and, by different given ΛB values, the diffraction behavio grating changes obviously: 1. Bragg diffraction can be established when ΛB meets the condition of the wavelength, according to Equation (2) (where d = ΛB, θ = π/2 -φ) and LC layer enough to generate enough periodical refractive index planes. Then a large-an fraction with high efficiency can be achieved [7,28,30,33]. 2. When the pitch ΛB is much greater than that in the Bragg condition, the optic erties of the twisted LC itself (in short-wave circular birefringence region) is re It has a potential to create a LCPG or LCPL with broadband and wide-vie [22,23,26,34,35], by means of phase compensation.
We will discuss these two situations in detail separately below by reviewin research.

Broadband and Wide-View Angle LCPGs and LCPLs --Based on Phase C pensation of Multi-Layer Twisted LC Structures
In most applications, PB phase optical devices with broad bandwidth, wide-v gle, and high diffraction efficiency, are highly desirable. There are some well-know niques to implement achromatic waveplates [36][37][38][39], which can be employed in P optical devices. Note that broadband polarization transformation could be a through additional degrees of freedom arising from using multiple uniaxial re whose optical axis varies discretely through the compound structure.
The multi-uniaxial-layer design could also be extended to birefringent layer continuously varying optical axis. For example, the self-compensation of retard done using self-aligning multi-twist structures (2013, Komanduri et al. [38]). Du These refractive index surfaces constitute a periodicity Λ B : in a slanted dimension and, by different given Λ B values, the diffraction behavior of the grating changes obviously: 1. Bragg diffraction can be established when Λ B meets the condition of the incident wavelength, according to Equation (2) (where d = Λ B , θ = π/2 − ϕ) and LC layer is thick enough to generate enough periodical refractive index planes. Then a largeangle diffraction with high efficiency can be achieved [7,28,30,33].

2.
When the pitch Λ B is much greater than that in the Bragg condition, the optical properties of the twisted LC itself (in short-wave circular birefringence region) is revealed. It has a potential to create a LCPG or LCPL with broadband and wide-view angle [22,23,26,34,35], by means of phase compensation.
We will discuss these two situations in detail separately below by reviewing recent research.

Broadband and Wide-View Angle LCPGs and LCPLs-Based on Phase Compensation of Multi-Layer Twisted LC Structures
In most applications, PB phase optical devices with broad bandwidth, wide-view angle, and high diffraction efficiency, are highly desirable. There are some well-known techniques to implement achromatic waveplates [36][37][38][39], which can be employed in PB phase optical devices. Note that broadband polarization transformation could be achieved through additional degrees of freedom arising from using multiple uniaxial retarders, whose optical axis varies discretely through the compound structure.
The multi-uniaxial-layer design could also be extended to birefringent layers with a continuously varying optical axis. For example, the self-compensation of retardation is done using self-aligning multi-twist structures (2013, Komanduri et al. [38]). During the layer-by-layer spin coating process, subsequent LC layers are directly aligned with previous ones. This process has greatly simplified fabrication, and reduced transmittance loss and alignment errors. By using chiral-doped LC materials, retardation control for nearly arbitrary bandwidths could be achieved. Besides, we can also control the thickness of each layer by spin-coating to avoid the deterioration of LC alignment and device quality caused by exceeding the critical thickness [40].

Dual-Layer Twisted PGs with Broadband and Wide-View Angle
Based on the principle of phase compensation, Oh et al. [19,22,23]. studied and fabricated a wide-band LCPG (or achromatic PG) by using a double-layer low-twisted structure with opposite twist direction illustrated in Figure 4a,b. The results show that the PG has high diffraction efficiency in a broadband ( Figure 4c) by using extended Jones matrix analysis (EJM) and the finite difference time domain (FDTD) method. Compared with the traditional LCPG (or circular PG), the high diffraction bandwidth is increased by five times (Figure 4d). The bandwidth would increase gradually with the increase of the twist angle Φ for the twist angles less than 70 • . Besides, the bandwidth would decrease sharply with the increase in twist angle for a condition with a twist greater than 70 • . The achromatic behavior can be explained by the counter chromatic dispersions of retardation caused by linear birefringence and induced circular birefringence due to twist [31]. The former birefringence becomes larger in the short-wavelength band while the latter becomes larger in the long-wavelength band. When the thickness of the LC layer reaches the halfwave condition with the 70 • twist angle, the achromatic diffraction is achieved by balancing these two effects for phase compensation. layer by spin-coating to avoid the deterioration of LC alignment and device quality caused by exceeding the critical thickness [40].

Dual-Layer Twisted PGs with Broadband and Wide-View Angle
Based on the principle of phase compensation, Oh et al. [19,22,23]. studied and fabricated a wide-band LCPG (or achromatic PG) by using a double-layer low-twisted structure with opposite twist direction illustrated in Figure 4a-b. The results show that the PG has high diffraction efficiency in a broadband ( Figure 4c) by using extended Jones matrix analysis (EJM) and the finite difference time domain (FDTD) method. Compared with the traditional LCPG (or circular PG), the high diffraction bandwidth is increased by five times (Figure 4d). The bandwidth would increase gradually with the increase of the twist angle Φ for the twist angles less than 70°. Besides, the bandwidth would decrease sharply with the increase in twist angle for a condition with a twist greater than 70°. The achromatic behavior can be explained by the counter chromatic dispersions of retardation caused by linear birefringence and induced circular birefringence due to twist [31]. The former birefringence becomes larger in the short-wavelength band while the latter becomes larger in the long-wavelength band. When the thickness of the LC layer reaches the halfwave condition with the 70° twist angle, the achromatic diffraction is achieved by balancing these two effects for phase compensation. In the case of oblique incidence, the diffraction efficiency of the primary type of PG will be sharply reduced [20]. Tan Li et al. [15,24,25] simulated the diffraction characteristics of oblique incidence by using the extended Jones matrix (EJM) method and optimized a dual-layer PG with wide viewing angle (WVA). An example [15] was given: a WVA PG, Λ = 2 μm, Δnd/λ = 0.61, and a primary PG with the same Λ, Δnd/λ = 0.5. Results showed that, at an incident angle of 20°, the dual-layer PG maintains a high diffraction efficiency of 99.5% at 550 nm, while the primary PG has a 0th order leakage of 10%.
Note that a WVA PG requires a special optimization and the feature of broadband disappeared in their design. That is to say, it is difficult for a dual-layer PG to achieve two good features at the same time. More studies on angular response using numerical models like FDTD [22], and rigorous coupled wave analysis (RCWA) method [41], etc., have been proposed by Oh, Xiang Xiao et al., and will not be detailed in this review.

Three-Layer Multi-Twist PGs and PLs with Ultra-Broadband and Wide-View Angle
As is mentioned above, although the dual-layer twisted PG has wider spectral bandwidth than the traditional one, it still cannot meet the application requirements (especially its angular response [41]). So, a three-layer multi-twist grating structure was proposed by In the case of oblique incidence, the diffraction efficiency of the primary type of PG will be sharply reduced [20]. Tan Li et al. [15,24,25] simulated the diffraction characteristics of oblique incidence by using the extended Jones matrix (EJM) method and optimized a dual-layer PG with wide viewing angle (WVA). An example [15] was given: a WVA PG, Λ = 2 µm, ∆nd/λ = 0.61, and a primary PG with the same Λ, ∆nd/λ = 0.5. Results showed that, at an incident angle of 20 • , the dual-layer PG maintains a high diffraction efficiency of 99.5% at 550 nm, while the primary PG has a 0th order leakage of 10%.
Note that a WVA PG requires a special optimization and the feature of broadband disappeared in their design. That is to say, it is difficult for a dual-layer PG to achieve two good features at the same time. More studies on angular response using numerical models like FDTD [22], and rigorous coupled wave analysis (RCWA) method [41], etc., have been proposed by Oh, Xiang Xiao et al., and will not be detailed in this review.

Three-Layer Multi-Twist PGs and PLs with Ultra-Broadband and Wide-View Angle
As is mentioned above, although the dual-layer twisted PG has wider spectral bandwidth than the traditional one, it still cannot meet the application requirements (especially its angular response [41]). So, a three-layer multi-twist grating structure was proposed Crystals 2021, 11, 243 6 of 16 by Zou and Wu et al. [35], which could achieve high efficiency in both visible light band and large incident angle. The device structure is shown in Figure 5. Note that the twist angle is exceedingly small (3.7 • in 1.32 µm) in the 2nd layer according to simulation. So, they add no chiral dopant to the 2nd layer, and that brings negligible error to the efficiency performance.
Crystals 2021, 11, x FOR PEER REVIEW Zou and Wu et al. [35], which could achieve high efficiency in both visible light b large incident angle. The device structure is shown in Figure 5. Note that the tw is exceedingly small (3.7° in 1.32 μm) in the 2 nd layer according to simulation. So, no chiral dopant to the 2 nd layer, and that brings negligible error to the efficienc mance. The rigorous coupled wave analysis (RCWA) method [42,43] is used to simu three-layer multi-twist grating structure, and several interchangeably optimizat rithms are used to optimize its parameters. Moreover, they reported the fabrica cess and materials in detail. The thickness and the twist angle of each layer are a controlled by Jones matrix method combined with experiments. Compared wit gular response of double-twist PG, the three-layer twist PG (or called Pancha Berry phase deflector, PBD in their paper) has obvious advantages at a large inc gle. The results ( Figure 6) showed that the diffraction efficiency of the three-la PG at 50° in simulation and experiment is 87.2% and 84.1%, respectively, in th much higher than that of dual-twist PG (75.7%) [35]. Based on the idea of broadband PGs above, in 2020, Tao Zhan et al. [11] pr three-layer multi-twist polarization lens (or Pancharatnam-Berry lens, PBL). Co The rigorous coupled wave analysis (RCWA) method [42,43] is used to simulate their three-layer multi-twist grating structure, and several interchangeably optimization algorithms are used to optimize its parameters. Moreover, they reported the fabrication process and materials in detail. The thickness and the twist angle of each layer are accurately controlled by Jones matrix method combined with experiments. Compared with the angular response of double-twist PG, the three-layer twist PG (or called Pancharatnam-Berry phase deflector, PBD in their paper) has obvious advantages at a large incident angle. The results ( Figure 6) showed that the diffraction efficiency of the three-layer twist PG at 50 • in simulation and experiment is 87.2% and 84.1%, respectively, in this work, much higher than that of dual-twist PG (75.7%) [35]. The rigorous coupled wave analysis (RCWA) method [42,43] is used to simulate their three-layer multi-twist grating structure, and several interchangeably optimization algorithms are used to optimize its parameters. Moreover, they reported the fabrication process and materials in detail. The thickness and the twist angle of each layer are accurately controlled by Jones matrix method combined with experiments. Compared with the angular response of double-twist PG, the three-layer twist PG (or called Pancharatnam-Berry phase deflector, PBD in their paper) has obvious advantages at a large incident angle. The results ( Figure 6) showed that the diffraction efficiency of the three-layer twist PG at 50° in simulation and experiment is 87.2% and 84.1%, respectively, in this work much higher than that of dual-twist PG (75.7%).  [35].
Based on the idea of broadband PGs above, in 2020, Tao Zhan et al. [11] proposed a three-layer multi-twist polarization lens (or Pancharatnam-Berry lens, PBL). Combining with a refractive Fresnel lens, the chromatic aberration (CA) of the system was significantly improved. The opposite dispersion between each other illustrated in Figure 7a-c It has a better bandwidth response and can cover most of the light from the display panel (diffraction efficiency > 95%), shown in Figure 7e. Meanwhile, a broadband circular poth Figure 6. Simulated and measured results: spectral (a) and angular response at (b) R (635 nm), (c) G (532 nm), (d) B (457 nm) of the three-layer multi-twist PBD [35]. Adapted with permission from [35] © The Optical Society.
Based on the idea of broadband PGs above, in 2020, Tao Zhan et al. [11] proposed a three-layer multi-twist polarization lens (or Pancharatnam-Berry lens, PBL). Combining with a refractive Fresnel lens, the chromatic aberration (CA) of the system was significantly

High Efficiency and Large-Angle Diffraction Gratings and Lenses-Based on Bragg Diffraction Generated by Twisted LC Structures
A much smaller horizontal period Λ is required to increase the diffraction angle. Oh, Escuti et al. [18] simulated the diffraction efficiency of LCPGs at different diffraction angles (which embodied the horizontal period Λ) and birefringence ∆n using the finite difference time domain (FDTD) method. In their work, parameter Q and ρ were introduced to describe the relationship between the diffraction efficiency and the period of LCPGs.
Results show in Figure 8, illustrated that a LCPG can have near-100% diffraction efficiency only when ρ < 1. That is, the efficiency will be reduced sharply if grating period Λ is comparable to the indecent wavelength λ, especially when a LC material with low ∆n is utilized (Figure 8b). Therefore, in order to achieve a large-angle diffraction with high efficiency, another form of grating diffraction, i.e., Bragg diffraction, needs to be considered. In Section 2, we have mentioned that Bragg diffraction could be achieved with a reasonable design of the horizontal period Λ x , and the pitch of LC (P = 2Λ y ). By adjusting the two pitches (Λ x , Λ y ), both reflective and transmissive gratings can be formed [7], as shown in Figure 9. When the incident angle θ i = 0 • , the 1st diffraction angle θ diff can be defined by the slanted angle ϕ, illustrated in Equation (5) below, and at this point, a larger diffraction angle can be generated. to describe the relationship between the diffraction efficiency and the period of LCPGs.
Results show in Figure 8, illustrated that a LCPG can have near-100% diffraction efficiency only when ρ < 1. That is, the efficiency will be reduced sharply if grating period Λ is comparable to the indecent wavelength λ, especially when a LC material with low Δn is utilized (Figure 8b). Therefore, in order to achieve a large-angle diffraction with high efficiency, another form of grating diffraction, i.e., Bragg diffraction, needs to be considered. In Section 2, we have mentioned that Bragg diffraction could be achieved with a reasonable design of the horizontal period Λx, and the pitch of LC (P = 2Λy). By adjusting the two pitches (Λx, Λy), both reflective and transmissive gratings can be formed [7], as shown in Figure 9. When the incident angle θi = 0°, the 1 st diffraction angle θdiff can be defined by the slanted angle φ, illustrated in Equation (5) below, and at this point, a larger diffraction angle can be generated.
, , Therefore, in order to achieve a large-angle diffraction with high efficiency, another form of grating diffraction, i.e., Bragg diffraction, needs to be considered. In Section 2, we have mentioned that Bragg diffraction could be achieved with a reasonable design of the horizontal period Λx, and the pitch of LC (P = 2Λy). By adjusting the two pitches (Λx, Λy), both reflective and transmissive gratings can be formed [7], as shown in Figure 9. When the incident angle θi = 0°, the 1 st diffraction angle θdiff can be defined by the slanted angle φ, illustrated in Equation (5) below, and at this point, a larger diffraction angle can be generated.
, , In addition, Bragg gratings with twisted LC structures have the characteristic of polarization selection [7,8,31,44]. It will only diffract the circularly polarized incident light which has the same handedness of the LC structure (the 1 st order), while the opposite circularly polarization will be transmitted directly (the 0 th order) [45,46]. For the diffraction beam, the polarization state depends on the method of the Bragg grating: for a reflective Bragg grating, the diffraction beam will keep the same handedness of the incident beam, while for a transmissive Bragg grating, it will be converted to the orthogonal polarization.

Reflective Bragg Volume Polarization Gratings (PVGs) and Lens (PVLs)
Since PVGs have excellent wavelength and polarization selectivity, they can be used in various devices, such as beam steering, optical switches, and wearable displays, etc, which have been reviewed in detail by Kun Yin et al. 2020 [14]. Here we select several representative works to expound. In addition, Bragg gratings with twisted LC structures have the characteristic of polarization selection [7,8,31,44]. It will only diffract the circularly polarized incident light which has the same handedness of the LC structure (the 1st order), while the opposite circularly polarization will be transmitted directly (the 0th order) [45,46]. For the diffraction beam, the polarization state depends on the method of the Bragg grating: for a reflective Bragg grating, the diffraction beam will keep the same handedness of the incident beam, while for a transmissive Bragg grating, it will be converted to the orthogonal polarization.

Reflective Bragg Volume Polarization Gratings (PVGs) and Lens (PVLs)
Since PVGs have excellent wavelength and polarization selectivity, they can be used in various devices, such as beam steering, optical switches, and wearable displays, etc, Crystals 2021, 11, 243 9 of 16 which have been reviewed in detail by Kun Yin et al. 2020 [14]. Here we select several representative works to expound.
According to the polarization selectivity of PVGs, in 2016 Weng et al. [7] designed a reflective PVG for 2D/3D wearable displays and performed a series of theoretical simulations. They used a dual-layer twisted structure with opposite directions in the PVGs, where Λ x = 404.6 nm, Λ y = 233.6 nm, ∆n = 0.2, the slant angle ϕ of the refractive index plane is 30 • , and 4 µm for each layer. As shown in Figure 10, for the left-handed and right-handed circular incident light at 550 nm, they can be reflected in two different directions (±60 • ) at the same time. As a result, 3D near-eye display devices can be obtained by using only one panel. Then in 2017, Lee et al. [8] prepared a high-efficiency PVG for augmented reality (AR) displays. The measured diffraction efficiency can reach 90% @ 650 nm (Λ x = 540 nm, Λ y = 250 nm, d = 3 µm, ∆n = 0.18) and the deflection angle can reach 50 • . Using three different chiral LC materials (1.4 wt%, 1.8 wt% and 2.2 wt%, 3 different concentrations of chiral dopant R5011 mixed in LC monomer RM257), a three-period reflective PVG with was fabricated. The test results showed that the sample can broaden the bandwidth. According to the polarization selectivity of PVGs, in 2016 Weng et al. [7] designed a reflective PVG for 2D/3D wearable displays and performed a series of theoretical simulations. They used a dual-layer twisted structure with opposite directions in the PVGs, where Λx = 404.6 nm, Λy = 233.6 nm, Δn = 0.2, the slant angle φ of the refractive index plane is 30°, and 4 μm for each layer. As shown in Figure 10, for the left-handed and righthanded circular incident light at 550 nm, they can be reflected in two different directions (± 60°) at the same time. As a result, 3D near-eye display devices can be obtained by using only one panel. Then in 2017, Lee et al. [8] prepared a high-efficiency PVG for augmented reality (AR) displays. The measured diffraction efficiency can reach 90% @ 650 nm (Λx = 540 nm, Λy = 250 nm, d = 3 μm, Δn = 0.18) and the deflection angle can reach 50°. Using three different chiral LC materials (1.4 wt%, 1.8 wt% and 2.2 wt%, 3 different concentrations of chiral dopant R5011 mixed in LC monomer RM257), a three-period reflective PVG with was fabricated. The test results showed that the sample can broaden the bandwidth.  In 2018, Weng et al. proposed a PVG-based dual-layer waveguide which could be applied in full-color AR displays [10]. In order to ensure the field of view, two PVGs are placed in different waveguide layers to separate the propagations of red (R) and cyan (GB) spectra light, shown in figure 11a-c. The prepared PVGs showed a higher diffraction efficiency (over 80%) with large diffraction angles at spectra of RGB (630 nm; 532 nm; 457 In 2018, Weng et al. proposed a PVG-based dual-layer waveguide which could be applied in full-color AR displays [10]. In order to ensure the field of view, two PVGs are placed in different waveguide layers to separate the propagations of red (R) and cyan (GB) spectra light, shown in Figure 11a-c. The prepared PVGs showed a higher diffraction efficiency (over 80%) with large diffraction angles at spectra of RGB (630 nm; 532 nm; 457 nm). The prototype, in Figure 11d, showed a clear full-color display with an approximately 35 • diagonal field of view and the transparency was up to 72%.
In 2019, Kun Yin et al. [47] proposed a PVG attached to a flexible material film to make it stretchable and rollable. A 3-µm thick PVG on a 160-µm thick Polydimethylsiloxane (PDMS) substrate was fabricated. With mechanical stretching, the horizontal direction of the PVG was lengthened and the total thickness reduced, as shown in (Figure 12). That is, the Bragg period Λ B values were changed, thus shifting the reflection band and deflection angle. In their work, the central wavelength blue-shifted from 507.5 nm to 474.5 nm, and the deflection angle could be modulated from 35 • to 46.5 • with a tuning range of 11.5 • . This study showed a promising potential for laser beam steering applications and provided a possibility for fabricating flexible devices.  In 2018, Weng et al. proposed a PVG-based dual-layer waveguide which could be applied in full-color AR displays [10]. In order to ensure the field of view, two PVGs are placed in different waveguide layers to separate the propagations of red (R) and cyan (GB) spectra light, shown in figure 11a-c. The prepared PVGs showed a higher diffraction efficiency (over 80%) with large diffraction angles at spectra of RGB (630 nm; 532 nm; 457  nm). The prototype, in Figure 11d, showed a clear full-color display with an approximately 35° diagonal field of view and the transparency was up to 72%. In 2019, Kun Yin et al. [47] proposed a PVG attached to a flexible material film to make it stretchable and rollable. A 3-μm thick PVG on a 160-μm thick Polydimethylsiloxane (PDMS) substrate was fabricated. With mechanical stretching, the horizontal direction of the PVG was lengthened and the total thickness reduced, as shown in Figure 12. That is, the Bragg period ΛB values were changed, thus shifting the reflection band and deflection angle. In their work, the central wavelength blue-shifted from 507.5 nm to 474.5 nm, and the deflection angle could be modulated from 35° to 46.5° with a tuning range of 11.5°. This study showed a promising potential for laser beam steering applications and provided a possibility for fabricating flexible devices. It should be noted that, some latest experiments [14] and simulation [30,33] prove that the LC director configuration in a PVG is not planar (Figure 13a); instead, it tends to present a slanted structure (Figure 13b) due to the trend of the lowest volume free energy [14]. The optical properties of planar and slanted PVGs are very similar in terms of diffraction efficiency, wavelength, and angular response. However, for polarization response, the two structures show different performance. At normal incidence, the diffracted light from the planar PVG significantly deviates from the circular polarization, while the circular polarization state of the slanted PVG remains. Figure 13. Different LC director structures of (a) planar-PVG and (b) slanted-PVG. According to the latest research, the real configuration of LC director in PVG tend to be slanted like (b) [14]. It should be noted that, some latest experiments [14] and simulation [30,33] prove that the LC director configuration in a PVG is not planar (Figure 13a); instead, it tends to present a slanted structure (Figure 13b) due to the trend of the lowest volume free energy [14]. The optical properties of planar and slanted PVGs are very similar in terms of diffraction efficiency, wavelength, and angular response. However, for polarization response, the two structures show different performance. At normal incidence, the diffracted light from the planar PVG significantly deviates from the circular polarization, while the circular polarization state of the slanted PVG remains.  [14]. Reflective polarization volume lenses (or rPVLs) have attracted much more attention in recent years because they can fold the optical path. It could be regarded as a combination of a slanted PVG and a lens while still maintaining the selective characteristics of Bragg diffraction on the polarization and wavelength.
In 2020, Zhan Tao et al. [12] proposed a series of cholesteric LC lenses (CLC lenses) with low f /# = f /D (2, 0.9, 0.45 and 0.33, respectively, diameter D = 2.45 cm) at 550 nm using the optical imprinting technique. The twisting structure of Ch-LC provided a phase change, as shown in Figure 14a,b. These lenses have the same polarization selectivity as Ch-LC. When the circular incident light incident from different directions, the lens presents different effects (positive "+" or negative "-" focal) as shown in Figure 14c,d. The results showed that the diffraction efficiency can reach about 85% and the imaging quality was good for the circular polarization states at normal incidence. CLC lenses could be used as a compact element for expanding the view field in VR/AR displays. Ch-LC. When the circular incident light incident from different directions, the lens pre sents different effects (positive "+" or negative "-" focal) as shown in Figure 14c,d. Th results showed that the diffraction efficiency can reach about 85% and the imaging quality was good for the circular polarization states at normal incidence. CLC lenses could b used as a compact element for expanding the view field in VR/AR displays. Subsequently, Kun Yin et al., who are from the same research group [13], proposed an off-axis reflective polarized volume lens (rPVL), with a low f/# of 0.825 and could obtain an off-axis diffraction angle of 45°, shown in Figure 15a,d. The rPVL has a parabolic phas profile in x-y plane macroscopically, while approximately a PVG with linear phase chang in a tiny area (Figure 15b). Besides, the rPVL presents a slanted helical axis (slant angle γ) Thus, the incident circular polarized light with the same handedness of the rPVL will b reflected and converged with an off-axis angle β (β = 2γ), illustrated in Figure 15c. The thin (about 1 μm thick) rPVLs are designed at Bragg wavelength (λ = 605 nm, 532 nm, and 45 nm) with a measured efficiency of 60%, 60%, and 55% correspondingly. There is still much  (Figure 15b). Besides, the rPVL presents a slanted helical axis (slant angle γ). Thus, the incident circular polarized light with the same handedness of the rPVL will be reflected and converged with an off-axis angle β (β = 2γ), illustrated in Figure 15c. The thin (about 1 µm thick) rPVLs are designed at Bragg wavelength (λ = 605 nm, 532 nm, and 450 nm) with a measured efficiency of 60%, 60%, and 55% correspondingly. There is still much room for improvement. According to their later test, the actual focal spot size is 2.5× larger than the theoretical one, which could be corrected with appropriate optical (AO) systems. The efficiency will also increase by increasing the thickness. to observe the off-axis focusing behavior and diffraction angle [13]. Copyright 2020, WILEY -VCH.

Transmissive Bragg PGs
Philip J. Bos et al. [21,45,48] reported a series of large-angle Bragg PGs using twisting structures in 2015 and 2017, called a dual-twist Pancharatnam phase deflector (DTPPD) in their papers. Here we regard them as transmitted Bragg PGs in the selective circular reflection region. Figure 16a-c is the proposed antisymmetric dual-twist structure, which is similar to the structure proposed by Oh [22] and Li Tan [25]. The single-twist structure only deflects one of the circularly polarized lights while the other is transmitted. Both leftand right-hand circular polarizations can be deflected using an antisymmetric dual-twist structure [45,48]; the research discussed the relationship between retardation and twist angle at specific deflection angles and birefringence to get high efficiency by the FDTD method (Figure 16d-g). Then they optimized Bragg PGs with 20°, 40°, 60°, and 80° deflection angles using birefringence values of Δn = 0.106, 0.179 and 0.3, respectively. A wellperforming dual-twist structure was optimized at the wavelength of 633 nm and 40° deflection angle with the parameters that thickness just met half-wave retardation and twist angle was 75°. Cheng et al. designed and fabricated high-efficiency large-angle transmitted Bragg PGs based on an antisymmetric dual-twist structure. It could deflect ±40° with 90% diffraction efficiency at a 633 nm wavelength.

Transmissive Bragg PGs
Philip J. Bos et al. [21,45,48] reported a series of large-angle Bragg PGs using twisting structures in 2015 and 2017, called a dual-twist Pancharatnam phase deflector (DTPPD) in their papers. Here we regard them as transmitted Bragg PGs in the selective circular reflection region. Figure 16 is the proposed antisymmetric dual-twist structure, which is similar to the structure proposed by Oh [22] and Li Tan [25]. The single-twist structure only deflects one of the circularly polarized lights while the other is transmitted. Both leftand right-hand circular polarizations can be deflected using an antisymmetric dual-twist structure [45,48]; the research discussed the relationship between retardation and twist angle at specific deflection angles and birefringence to get high efficiency by the FDTD method. Then they optimized Bragg PGs with 20 • , 40 • , 60 • , and 80 • deflection angles using birefringence values of ∆n = 0.106, 0.179 and 0.3, respectively. A well-performing dual-twist structure was optimized at the wavelength of 633 nm and 40 • deflection angle with the parameters that thickness just met half-wave retardation and twist angle was 75 • . Cheng et al. designed and fabricated high-efficiency large-angle transmitted Bragg PGs based on an antisymmetric dual-twist structure. It could deflect ±40 • with 90% diffraction efficiency at a 633 nm wavelength. method. Then they optimized Bragg PGs with 20°, 40°, 60°, and 80° deflection angles using birefringence values of Δn = 0.106, 0.179 and 0.3, respectively. A well-performing dualtwist structure was optimized at the wavelength of 633 nm and 40° deflection angle with the parameters that thickness just met half-wave retardation and twist angle was 75°. Cheng et al. designed and fabricated high-efficiency large-angle transmitted Bragg PGs based on an antisymmetric dual-twist structure. It could deflect ±40° with 90% diffraction efficiency at a 633 nm wavelength. Xiao Xiang et al. [28] analyzed the properties of polymer LC Bragg PGs based on twist structure by anisotropic rigorous coupled wave analysis. Without paraxial approximation, they simulated the most important diffraction properties, including the angular, spectral, and polarization responses. They overcame the critical thickness problem by spin-coating multiple sublayers and fabricated a series of single-layer transmitted Bragg PGs with Λ = 335 nm for λ = 450 nm. They varied the chiral material concentration to adjust twist angle ϕ and slant angle θG. The experimental results showed that at the same incident wavelength, the gratings with different twist angle had different peak incident angles and peak power.
Subsequently, they reported a transmitted Bragg PG formed by a double-layer slanted structure [29], shown in Figure 17a,b. Angular bandwidth was significantly increased by arranging two slanted grating layers within the same monolithic film under the premise of ensuring the large deflection angle and broadband. The fabricated sample has excellent properties such as 40° angular bandwidth (Figure 17c), 200 nm spectral bandwidth, 76% for average efficiency, and 96% for peak efficiency (Figure 17d). Xiao Xiang et al. [28] analyzed the properties of polymer LC Bragg PGs based on twist structure by anisotropic rigorous coupled wave analysis. Without paraxial approximation, they simulated the most important diffraction properties, including the angular, spectral, and polarization responses. They overcame the critical thickness problem by spin-coating multiple sublayers and fabricated a series of single-layer transmitted Bragg PGs with Λ = 335 nm for λ = 450 nm. They varied the chiral material concentration to adjust twist angle φ and slant angle θ G . The experimental results showed that at the same incident wavelength, the gratings with different twist angle had different peak incident angles and peak power.
Subsequently, they reported a transmitted Bragg PG formed by a double-layer slanted structure [29], shown in Figure 17a,b. Angular bandwidth was significantly increased by arranging two slanted grating layers within the same monolithic film under the premise of ensuring the large deflection angle and broadband. The fabricated sample has excellent properties such as 40 • angular bandwidth (Figure 17c), 200 nm spectral bandwidth, 76% for average efficiency, and 96% for peak efficiency (Figure 17d). Subsequently, they reported a transmitted Bragg PG formed by a double-layer slanted structure [29], shown in Figure 18a,b. Angular bandwidth was significantly increased by arranging two slanted grating layers within the same monolithic film under the premise of ensuring the large deflection angle and broadband. The fabricated sample has excellent properties such as 40° angular bandwidth (Figure 18c), 200 nm spectral bandwidth, 76% for average efficiency, and 96% for peak efficiency (Figure 18d).

Conclusions and Outlook
In this paper, the research progress and working principle of twisting-based LCPGs and LCPLs in recent years are reviewed. We found that the twisting structures work in

Conclusions and Outlook
In this paper, the research progress and working principle of twisting-based LCPGs and LCPLs in recent years are reviewed. We found that the twisting structures work in different methods in different devices. For a broadband PG, twisted LCs work in the short-wavelength circular birefringence region, presenting the property of retardation compensation. While for transmissive Bragg LCPGs and reflective PVGs, it works in the Bragg region. The slanted and periodical refractive index planes were established by combining the pitch of a twisted LC and the horizontal period on the photo-alignment layer. Based on these theoretical analyses, more devices with excellent characteristics have been achieved. Especially for PVGs and PVLs, they enable promising applications for wearable displays.
Further work in the field of studying novel twisting structures based on combining different method is highly desirable. For instance, the long-wavelength circular birefringence region of twisted LCs has not been used so far, which might present a good performance in infrared (IR) applications. Additionally, combining transmissive Bragg PGs and reflective PVG in one element might gain attention in novel optical systems like waveguides for beam steering and AR display. We believe these studies will gain tremendous applied in the very near future.