### 2.1 Basics of liquid crystal physics

A thermotropic LC exists between solid and isotropic phases. An anisotropic LC is more ordered than an isotropic liquid and exhibits a certain degree of orientational order. On the other hand, it is less rigid than a crystalline solid and can flow easily. Several kinds of thermotropic LCs have been discovered, including nematic, smectic, and columnar (or discotic) phases [17]. The order of these phases increases accordingly. Nematic phase has no positional order and it acts like a 3D liquid. The LC molecules can move freely in 3D space, as shown in Fig. 1a. Smectic phase has 1D positional order, which resembles stacked 2D liquid layers, as Fig. 1b depicts. On the other hand, columnar phase has 2D positional order and behaves like an array of 1D liquid tubes, as sketched in Fig. 1c. Due to mechanical instability and defects, smectic and columnar phases are less commonly used in practical applications than nematic.

The presently widely used flat panel displays use nematic LCs. If we dope a chiral compound into a nematic host, the LC will form a helical structure depicted in Fig. 1d. This helical structure is called cholesteric liquid crystal (CLC). Therefore, we can regard CLC as a derivative of nematic. Meanwhile, there is another phase existing in between cholesteric phase and isotropic phase, called blue phase [18]. But the operating voltage of blue phase is quite high so that its application is still limited. Thus, we will not discuss it here. Detailed descriptions of LC phases and their physical properties can be found in [17]. The planar optics we present here mainly involves nematic and CLC phases.

A nematic LC usually consists of elongated rod-like molecules. Thus, we can use a unit vector \(\widehat{{\varvec{n}}}\) called the LC director to describe the averaged orientation of local molecules. As shown in Fig. 2a, the LC director \(\widehat{{\varvec{n}}}\) can be spatially variant. If we look at a local region, the molecular orientations are distributed near director \(\widehat{{\varvec{n}}}\). Therefore, we can define a scalar order parameter *S* as

$$ S = \frac{1}{2}\left\langle {3\cos^{2} \theta - 1} \right\rangle , $$

(1)

where ⟨⟩ denotes the statistical average and \(\theta\) is the angle between the molecules and the director \(\widehat{{\varvec{n}}}\). As the temperature increases, LC undergoes phase transition to an isotropic state. Under such condition, the LC directors are randomly distributed so that S = 0. When the temperature is below the melting point, the LC directors are frozen to a crystalline state, and S = 1. In a nematic state, S is usually in the 0.6 to 0.7 range, depending on the temperature. The most generalized description of LC behavior uses a tensor order parameter *Q* defined as

$$ Q_{ji} = \frac{S}{2}\left( {3n_{i} n_{j} - \delta_{ij} } \right), $$

(2)

where \(\delta_{ij}\) denotes Kronecker delta. Because Q contains information of both order parameter and director, it can describe the thermotropic phase transition and LC dynamics. But in most applications at a given temperature, the order parameter is usually a constant, thus a simpler way to describe the LC dynamic behaviors is to only use the director \(\widehat{{\varvec{n}}}\).

To acquire spatial and temporal information of the LC director \(\widehat{{\varvec{n}}}(x,t)\), free energy model is used. The total free energy *F*_{total} consists of three parts: the elastic energy *F*_{el} from director reorientations, the surface free energy *F*_{s} from interaction between LC and surface anchoring, and the electromagnetic field free energy *F*_{field} from dipole coupling between LC and electric/magnetic fields.

According to Frank-Oseen model [17], the elastic free energy density *f*_{el} can be expressed as

$$ f_{el} = \frac{1}{2}K_{1} \left( {\nabla \cdot \widehat{{\varvec{n}}}} \right)^{2} + \frac{1}{2}K_{2} \left( {\widehat{{\varvec{n}}} \cdot \nabla \times \widehat{{\varvec{n}}} + \frac{2\pi }{P}} \right)^{2} + \frac{1}{2}K_{3} \left| {\widehat{{\varvec{n}}} \times \nabla \times \widehat{{\varvec{n}}}} \right|^{2} , $$

(3)

where *K*_{1}, *K*_{2} and *K*_{3} is the splay, twist, and bend elastic constant, respectively, and *P* is the helical pitch of CLC with the director reorienting from 0 to 2π. When there is no chiral dopant, *P* is infinity. From Eq. (3), we can see that in order for *f*_{el} to be zero, all the spatial derivatives of \(\widehat{{\varvec{n}}}\) should be zero, which indicates \(\widehat{{\varvec{n}}}\) should be a constant across the space. When *P* is finite, it is easy to verify that \(\widehat{{\varvec{n}}} = \left( {\cos \frac{2\pi }{P}z,\sin \frac{2\pi }{P}z,0} \right)\) gives zero *f*_{el}, which corresponds to a helical structure with pitch length *P* and the helix is along the *z* axis. Also, because the value of *f*_{el} is not dependent on the choice of coordinates, any rotation of the coordinate system should produce the same result, which means a helical structure with period *P* and arbitrary helical axis orientation leads to *f*_{el} = 0.

When an LC is subject to an external field, the free energy density arising from LC’s electric and magnetic anisotropies can be expressed as:

$$ f_{field} = - \frac{1}{2}\Delta \varepsilon \left( {\widehat{{\varvec{n}}} \cdot {\varvec{E}}} \right)^{2} - \frac{1}{2}\Delta \chi \left( {\widehat{{\varvec{n}}} \cdot {\varvec{H}}} \right)^{2} , $$

(4)

where *E* and *H* are the electric and magnetic fields, \(\Delta \varepsilon\) and \(\Delta \chi\) are dielectric and diamagnetic anisotropies. Depending on the sign of \(\Delta \varepsilon\) and \(\Delta \chi\), the LC directors could be reoriented to be parallel or perpendicular to the electric field or the magnetic field.

The surface free energy density *f*_{s} determines how the LC directors align at the interface. The most general form of *f*_{s} adopts the following [19, 20]:

$$ f_{s} = - \frac{1}{2}\sum\limits_{i,j} {W_{ij} n_{i} n_{j} } , $$

(5)

where *W*_{ij} is the traceless symmetrical anchoring tensor. We can always diagonalize *W*_{ij} in a local coordinate system \(\widehat{{\varvec{\xi}}},\widehat{{\varvec{\eta}}},\widehat{{\varvec{\varepsilon}}}\), as shown in Fig. 2b. Also, note that \(\widehat{{\varvec{n}}}\) is a unit vector, thus *f*_{s} can be rewritten as:

$$ f_{s} = W_{\xi } \left( {\widehat{{\varvec{n}}} \cdot \widehat{{\varvec{\xi}}}} \right)^{2} + W_{\eta } \left( {\widehat{{\varvec{n}}} \cdot \widehat{{\varvec{\eta}}}} \right)^{2} . $$

(6)

Here, we ignore the constant term. When \(\widehat{{\varvec{\xi}}} = \widehat{{\varvec{z}}}\) and \(W_{\eta } = 0\), \(f_{s} = W_{z} \left( {\widehat{{\varvec{n}}} \cdot \widehat{{\varvec{z}}}} \right)^{2}\). If \(W_{z} < 0\), the LC directors tend to align along surface normal \(\widehat{{\varvec{z}}}\). If \(W_{z} > 0\), LC directors tend to align in *x–y* plane, which corresponds to degenerate planar anchoring. When \(W_{\eta } \ne 0\), \(\widehat{{\varvec{\xi}}} = \widehat{{\varvec{z}}}\) and \(W_{z} > 0\), it corresponds to homogeneous anchoring. Because in this case \(\widehat{{\varvec{\eta}}}\) and \(\widehat{{\varvec{\varepsilon}}}\) have to be in *x–y* plane, the LC directors tend to align parallel to \(\widehat{{\varvec{\eta}}}\) if \(W_{\eta } < 0\), or parallel to \(\widehat{{\varvec{\varepsilon}}}\) if \(W_{\eta } > 0\).

The surface free energy can be viewed as a mechanism to align surface LC with treatment of surface by mechanical rubbing, photoalignment or other techniques. The field free energy offers a way to align bulk LC by applying electric/magnetic fields. The final distribution of bulk LC directors is determined by minimizing the following combined free energy:

$$ F_{total} = \int\limits_{v} {\left( {f_{el} + f_{field} } \right)} dv + \int\limits_{s} {f_{s} } ds. $$

(7)

### 2.2 Formation mechanism

In order to manipulate the LC director configuration, from our above analysis, we can either apply an electric field or by patterning the surface molecular alignment. In practical applications, locally addressable electric field can be achieved by the patterned electrodes. Examples include LC lenses [21, 22] and beam steering devices [23,24,25,26]. However, to achieve a large-aperture LC lens with high optical quality, the wavefront modulation should be precise, meaning more and finer electrodes are required to increase the degree of modulation freedom. This in turn causes great difficulty in electrode fabrication and device driving. For LC beam steering devices, the diffraction angle is governed by the electrode density and is usually small. For a simple estimate, a liquid–crystal-on-silicon device with 3-µm pixel pitch would produce a maximum diffraction angle of only 10° at λ = 550 nm. Furthermore, as the diffraction angle increases, the optical efficiency declines rapidly.

On the other hand, locally patterned surface alignment has advantage in precise wavefront modulation. Several patterning methods have been developed, such as micro-rubbing [27, 28], nanoimprinting [29, 30], and photoalignment [31,32,33,34,35,36]. Among them, photoalignment has the best recording precision and considerably easier fabrication process. The photoalignment technique generally refers to inducing optical anisotropy with the exposure of a polarized light. Among several types of photoalignment approaches are two main methods using photochemical reaction and photoisomerization [32]. The method adopting photochemical reaction like photodimerization enables adjusting LC-surface tilt angle and is essential for applications like multi-domain vertical alignment LCDs [33]. The photoisomerization approach involves a type of polarization-sensitive molecules called azo compounds, where the azobenzene group can transit between low-energy trans state and high-energy cis state, as Fig. 3a depicts. After exposure with an elliptically polarized light, the azo compounds repeatedly undergo trans–cis photoisomerization cycles and gradually rotate to the direction parallel to the short axis of elliptically polarized light, where the absorption is minimum. In the early development stage, azo-dye doped LCs are used to record the wavefront of an interfered polarized light [37,38,39]. The movement of azo compounds changes the alignment of surrounding LC materials, causing the LC to form holograms. But the formed holograms usually disappear after the recording beams being removed, due to the reorientation of LC directors. Later, researchers changed the fluidic LC material to LC polymers [34, 40,41,42,43]. The recorded holograms remain stable after exposure, but the required dosage increases dramatically. The formation mechanism of such a volume hologram is also complicated, involving a lot of nonlinear interactions [44, 45]. Afterwards, instead of using a thick film, a thin photoalignment layer is used to align bulk LC. The photoalignment layer can be a photopolymer [34, 35, 46], an azo-polymer [47], an azo-dye doped polyimide [48,49,50], or just an azo-dye [14, 51,52,53,54,55]. The azo-dye alignment layer generally produces a higher optical performance because of its better freedom of molecular movement. Such method is usually called photoalignment polarization holography (PAPH). It has advantages of low exposure dosage, high diffraction efficiency, negligible scattering, and electric-driving potential. Several types of azo-dyes have been used for photoalignment, like methyl red, brilliant yellow, SD-1 and so forth. The chemical structure of a typical azo-dye-based brilliant yellow is shown in Fig. 3b. The LC material used in PAPH could be a fluidic LC or a reactive mesogen. The latter is basically a polymerizable LC with reactive end groups [56]. The chemical structure of a typical reactive mesogen RM257 is depicted in Fig. 3c. Usually a reactive mesogen and additions like photo-initiator are dissolved in an organic compound. The solution is coated on a substrate. After the solvent evaporates, the reactive mesogen is in liquid crystalline state. Under ultraviolet (UV) light illumination, the reactive mesogen molecules form the polymer network with high thermal and chemical stabilities, as sketched in Fig. 3d.

To understand the working principle of PAPH, we shall firstly describe how patterns are formed in the photoalignment layer with exposure of a polarized light. From above description of how azo compounds respond to an elliptically polarized light, it is natural to think that linear polarization is a preferred choice for photoalignment, because of its highest ratio between the parallel and perpendicular electric field components. To produce a patterned linear-polarization pattern, one approach is to interfere two circularly polarized beams. As shown in Fig. 4a, when two beams with left-handed circular polarization (LCP) and right-handed circular polarization (RCP) interfere, the electric field can be calculated as:

$$ \left[ {\begin{array}{*{20}c} 1 \\ i \\ \end{array} } \right]e^{{ - ik_{0} \sin \theta \cdot x}} + \left[ {\begin{array}{*{20}c} 1 \\ { - i} \\ \end{array} } \right]e^{{ik_{0} \sin \theta \cdot x}} = 2\left[ {\begin{array}{*{20}c} {\cos (k_{0} \sin \theta \cdot x)} \\ {\sin (k_{0} \sin \theta \cdot x)} \\ \end{array} } \right], $$

(8)

where *k*_{0} is the wavenumber and *θ* is the incident angle. The resultant electric field has a sinusoidal linearly polarized pattern. Therefore, in the exposure process, azo compounds would rotate to align perpendicular to the electric field, forming a sinusoidal pattern with a phase shift of π to the original electric field. It is worth mentioning that Eq. (8) is based on an approximation of small incident angle and ignoring the field component vertical to the substrate. But it has been shown [57] that for large incident angles and even asymmetric incidence, the resultant pattern still has a high degree of linear polarization. Therefore, such a photoalignment pattern recording method can accommodate a reasonably wide range of exposure scenarios.

When a nematic LC layer is deposited on top of the patterned photoalignment layer, the bottom LC directors would follow the sinusoidal pattern, as Fig. 4b depicts. We note this structure as planar-nematic structure. The director \(\widehat{{\varvec{n}}}\) has following form:

$$ \widehat{{\varvec{n}}} = \left( {\sin (k_{0} \sin \theta \cdot x),\cos (k_{0} \sin \theta \cdot x),0} \right). $$

(9)

Plugging Eq. (9) into Eq. (3), we can calculate the elastic free energy density as

$$ f_{el} = \frac{{2\pi^{2} }}{{\Lambda_{x}^{2} }}\left( {K_{1} \cos^{2} \frac{2\pi }{{\Lambda_{x} }}x + K_{2} \sin^{2} \frac{2\pi }{{\Lambda_{x} }}x} \right) \approx \frac{{2\pi^{2} }}{{\Lambda_{x}^{2} }}K, $$

(10)

where \(\Lambda_{x} = 2\pi /(k_{0} \sin \theta )\) is the pattern pitch in *x* direction, and the sign ≈ represents the single elastic constant approximation \(K_{1} = K_{2} = K_{3} = K\). Although the single elastic constant approximation does not hold true for most LC materials [58, 59], it provides us a simple physical insight for an intuitive understanding. The elastic energy is non-zero, indicating such LC director configuration is in a deformed high-energy state. As mentioned earlier, the most relaxed state for nematic LC is a constant director \(\widehat{{\varvec{n}}}\). Therefore, as the thickness of such LC configuration increases, or \(\Lambda_{x}\) decreases, the above LC intends to align vertically to form the constant director distribution [60, 61]. The critical thickness for deformation depends on the elastic constants of the employed LC material [61]. But it is generally in the same order as \(\Lambda_{x}\). The LC thickness is usually fixed due to the half-wave retardation condition to be discussed later. Thus, \(\Lambda_{x}\) cannot be smaller than a certain value. As a result, such an LC optical element usually exhibits a relatively small diffraction angle (~ 10°).

Lately, it is found that CLC combined with patterned alignment layer can also form high-quality optical components. If we place a CLC on top of a patterned photo-alignment layer, it is natural to assume that the director would follow the bottom pattern, but with a helical twist in z direction, as shown in Fig. 4c. We call this structure as planar-twisting structure, whose pitch in z direction (\(\Lambda_{z}\)) is equal to the natural helical pitch of CLC (\(\Lambda_{CLC}\)). Under such condition, the director \(\widehat{{\varvec{n}}}\) has following form:

$$ \widehat{{\varvec{n}}} = \left( {\sin \left (\frac{2\pi }{{\Lambda_{x} }}x + \frac{2\pi }{{\Lambda_{z} }}z \right),\cos \left (\frac{2\pi }{{\Lambda_{x} }}x + \frac{2\pi }{{\Lambda_{z} }}z \right),0} \right). $$

(11)

From Eq. (3), we find the elastic free energy density as:

$$ f_{el} = \frac{{2\pi^{2} }}{{\Lambda_{x}^{2} }}\left( {K_{1} \cos^{2} \left (\frac{2\pi }{{\Lambda_{x} }}x + \frac{2\pi }{{\Lambda_{z} }}z \right ) + K_{2} \sin^{2} \left (\frac{2\pi }{{\Lambda_{x} }}x + \frac{2\pi }{{\Lambda_{z} }}z \right)} \right) \approx \frac{{2\pi^{2} }}{{\Lambda_{x}^{2} }}K, $$

(12)

which has a similar form to Eq. (10).

Recall that the most relaxed CLC state is a helical structure. To deform from planar structure in Fig. 4c to helical structure is indeed very easy, which is different from the case of planar nematic structure. As depicted in Fig. 4d, if the bulk LC directors no longer maintain the planar form and are tilted with an angle \(\alpha = \arcsin (\Lambda_{G} /\Lambda_{x} )\), then a helical structure is obtained, with the LC directors parallel to Bragg surface. The k-vector of bottom pattern is also matched. The Bragg pitch \(\Lambda_{G}\) equals to the CLC pitch \(\Lambda_{CLC}\) in this case. We note this structure as tilted-helical structure. Both simulations [62] and experiments [62, 63] indicate there exists a thin layer of transitional region where LC director evolves from bottom planar pattern to bulk tilted-helical structure. Remember the helical structure has zero distortion free energy. The free energy density in the transitional layer has a gradient distribution from bottom [whose value is described by Eq. (12)] to top (whose value is zero). Its thickness is dependent on the elastic constants of the employed LC material and anchoring strength of the alignment layer but is typically in the order of 10 nm according to the simulation results. Therefore, the total free energy of tilted-helical structure is significantly lower than that of planar structure, as shown in Fig. 4e. In this sense, the tilted-helical structure is more stable than the planar-nematic structure in Fig. 4b. Such a grating can accommodate a very small \(\Lambda_{x}\) and therefore a very large diffraction angle.

Although the planar-twisting structure is difficult to exist in a single-layer bulk LC, it can be fabricated with multiple spin-coating procedures. The thickness of each spin-coated layer should be small enough so that the planar form is maintained. Another limiting case is when the chiral concentration is low and therefore \(\Lambda_{CLC}\) is very large. This case is similar to the planar-nematic structure, at a semi-stable state with a high distorted free energy.

### 2.3 Device fabrication

From above discussions, PAPH basically includes two parts: (1) patterning photoalignment layer and (2) deposition of LC material. For the patterning part, our previous discussion focuses on the interference of LCP and RCP beams, which can be achieved using several types of interferometers [51, 55, 64, 65]. Figure 5a shows a typical interferometer for recording an off-axis lens pattern. To record an arbitrary pattern, we need to firstly fabricate such a template using methods like diamond turning. Then we can relay the wavefront of that template onto the sample [51].

In addition to interferometers, several other methods can also be employed to produce linearly polarized pattern, such as spatial light modulator (SLM) [66,67,68], direct laser scanning technique [69,70,71], and replication mask [72,73,74,75]. The SLM approach relies on local modulation of phase retardation to control the final linear polarization direction of each pixel, as depicted in Fig. 5b. The device resolution is limited by the SLM and therefore it can only produce pattern with a relatively large period. The direct laser scanning technique uses a laser whose linear polarization direction is controlled by a polarization rotator. Such a polarization rotator is synchronized with the translation stage to write a spatial pattern in a point-to-point manner, as shown in Fig. 5c. Aside from relatively slow writing speed, it may also face the resolution issue because the smallest feature size is limited by the focused laser spot size, which generally has the same order as the laser wavelength. However, to achieve a large diffraction angle, the corresponding pattern feature size should be much smaller than the wavelength. Finally, the patterned polarizer method directly converts the input circularly polarized or unpolarized light to patterned linear polarization with a mask, as sketched in Fig. 5d. The replication mask can be a polarizer mask fabricated by photolithography [72, 73] or interferometer based PAPH [75]. Photolithography can achieve a sub-wavelength feature size but to write a large-scale (~ cm) sample is time consuming. Alternatively, the mask can also be a waveplate mask [74] fabricated through all other methods. It is worth mentioning that the replication mask method has the highest potential for mass production, which is critical for applications like VR and AR.

The LC material deposition can be categorized into spin-coating and cell formation. In spin-coating, a reactive mesogen solution is overcoated on top of the bottom photoalignment layer. After solvent evaporation, a UV light is used to polymerize the reactive mesogen. This process can be repeated several times. Each time, the top surface of previous layer serves to align the subsequent layer. Such a multiple spin-coating routine can be used to form some specific structures that would otherwise be distorted in a single layer approach [76, 77], as discussed above. On the other hand, the cell formation method uses two substrates to form a cell. The photoalignment layer is placed on one or both inner surfaces of the substrates. A fluidic LC is then infiltrated into the cell and form the optical element. If the substrates are coated with indium tin oxide electrodes, the LC directors can be switched by an external voltage. Such an LC optical element is called active device, while the LC polymer based optical element is called passive device. The cured LC polymer is not switchable. To utilize its polarization selectivity, we need to add an active polarization rotator to control the input polarization.

### 2.4 Response under applied voltage

In the planar-nematic structure, if the employed LC material has a positive dielectric anisotropy (Δε > 0), then we can dynamically switch the cell with an electric field along z direction. From Eq. (10), the LC director has the tendency to reorient perpendicularly to lower the free energy. Similar to the Fréedericksz transition in a homogeneous cell, there is a threshold voltage (*V*_{th}^{′}) over which the planar director starts to deform. With some approximations [61], *V*_{th}^{′} can be expressed as

$$ V_{th}^{^{\prime}} = \pi \sqrt {\frac{{K_{1} }}{{\varepsilon_{0} \Delta \varepsilon }}} \sqrt {1 - \left( {\frac{d}{{d_{c} }}} \right)^{2} } = V_{th} \sqrt {1 - \left( {\frac{d}{{d_{c} }}} \right)^{2} } , $$

(13)

where *V*_{th} is the threshold voltage of a homogeneous cell and *d*_{c} is the critical thickness. From Eq. (13), *V*_{th}^{′} is lower than *V*_{th.} This can be explained by the high-energy state of the planar nematic structure. As discussed above, the LC directors have non-zero free energy and tend to align vertically, which is the same as electric field direction. Therefore, the LC directors in a planar twisting structure are easier to deform than those of a homogeneous cell. The response time has the same expression as a homogeneous cell [78]:

$$ \tau_{rise} = \frac{{\tau_{0} }}{{|1 - (V/V_{th} )^{2} |}}, $$

(14)

$$ \tau_{decay} = \tau_{0} = \frac{{\gamma_{1} d^{2} }}{{K_{1} \pi^{2} }}, $$

(15)

where *τ*_{rise} and *τ*_{decay} represents the rise time and free relaxation time, respectively, \(\gamma_{1}\) is the rotational viscosity, *d* is the cell gap, and *V* is the applied voltage. By choosing a low viscosity LC material and thin cell gap, the response time can reach 1–2 ms.

In a tilted helical structure, the LC dynamic response is more complicated [62, 79]. Although the electric field free energy is lower for the LC directors in z direction, the elastic free energy is high when all the LC directors are aligned vertically. According to numerical simulation and experiments, there is also a threshold voltage over which the helical structure begins to rotate toward z direction. But the pattern period in x direction is still fixed and therefore the diffraction angle does not change. At a certain point, the Bragg surface will be aligned vertically, which has the same configuration as a uniform lying helix (ULH) structure. As the voltage further increases, the ULH structure will be deformed and unwinded, until all the LC directors are aligned vertically.