3. TN-LC gratings fabricated by one-step holographic photoalignment

#### 3.1. Holographic photoalignment

2.2. Experiment and results

358 Holographic Materials and Optical Systems

By applying the feature of P6CB described above, we propose a one-step photoalignment method as shown in Figure 3. In this one-step photoalignment method, orthogonal alignment direction between the two P6CB substrates is applied by linearly polarized UV beam irradiation to an empty glass cell whose inner walls are coated with P6CB. The one-step photoalignment method is realized by leveraging the phenomenon that the exposure dose between the two P6CB films is different due to the light absorption in the front P6CB film as shown in Figure 3. Therefore, a TN-aligned LC cell can be fabricated by injecting low-molecular-mass LCs in the empty glass cell. The experimental procedure and results of the demonstration

P6CB substrates were prepared by spin coating, a solution of 1.5 wt% P6CB in methylene chloride on cleaned glass substrates. The spin coating in the first step is carried out for 3.0 s at 500 rpm, and then the second step is carried out for 40.0 s at 1500 rpm; these steps are continuous. The thickness of the resultant P6CB films on the glass substrates was 0.3 μm. An empty glass cell was fabricated by interposing 12 μm-thick spacers between two P6CB substrates, and then these were adhered using an epoxy-based adhesive. The empty glass cell was exposed to the linearly polarized UV beam as shown in Figure 3. A 325 nm wavelength He-Cd laser, which operates in TEM00 mode and emits a linear polarization, was used as the light source. The cross-sectional area of the beam was set to 0.04 cm<sup>2</sup> using two planoconvex lenses

Figure 3. Schematic illustration of fabrication procedure of TN-LC cell by one-step photoalignment method.

exposure dose varied from 90 to 525 mJ/cm2 in 72.5 mJ/cm2 steps by changing the exposure time from 1.8 to 10.5 s. After laser irradiation, the empty glass cell was annealed at 150°C for 15 min. After cooling to room temperature, the empty glass cell was filled with the nematic LC 4 pentyl-4′-cyanobiphenyl (5CB, Merck Japan K-15) through capillary action. The transmitted light from the resultant LC cell was observed by crossed Nichols method when a white light was used as the light source. In addition, the polarization state of the transmitted light was measured by the rotation-analyzer method using a Glan-Thompson prism as the analyzer. A 633 nm wavelength linearly polarized He-Ne laser, which was incident normal to the plane of

. In this experiment, the

with different focal lengths. The beam intensity was set to 50 mW/cm2

the P6CB substrates, was used as the probe beam.

experiment of the one-step photoalignment are described below.

LC gratings containing the TN alignment can be fabricated by extended to an interference exposure from the single beam one-step photoalignment described above. By exposure of the empty glass cell to UV interference beam in which polarization states are periodically modulated, LC gratings are fabricated accurately and efficiently. In the present study, three types of LC gratings, which hereafter referred to as the "continuous," "binary," and "planar TN," are fabricated. These LC gratings are fabricated using the common two-beam interference optical system shown in Figure 5. The light source was He-Cd laser which was the same as that used in the previous experiment of the one-step photoalignment. The cross-sectional area of the beam was expanded to 0.04 cm<sup>2</sup> . The crossing angle was 0.3°, and the resultant modulation period of the electric field was 60 μm. The polarization states of the interfering beams were adjusted using the half-/quarter-wave plates. The intensity ratio of the interfering beams can be controlled by adjusting the polarization azimuth, which was an incident on the polarization beam splitter, using the half-wave plate. The one-step holographic photoalignment for the fabrication of the three types of LC gratings is described below individually. The empty glass cells used in the present experiment were the same as that described in Section 2.2. Moreover, the annealing process after the UV exposure and the injection process of nematic LCs were also the same.

Figure 5. Optical system for two-beam interference exposure. NDF, M, PBS, and H/QWP represent the neutral-density filter, mirror, polarization beam splitter, and half-/quarter-wave plates.

#### 3.1.1. Continuous LC gratings

Figure 6 shows the fabrication procedure for the continuous LC grating with 0° planar or with 90° TN alignment. In the continuous LC grating, LC directors are constantly rotated along the grating vector. These were fabricated by one-step circular polarization interference exposure. When two orthogonally circular polarized beams interfere with each other, the polarization azimuth of the linear polarization in the interference field is continuously modulated; however, the polarization ellipticity and the intensity are not modulated. Therefore, the periodic alignment process, in which alignment directions are continuously rotated along the grating vector, can be simultaneously applied to two P6CB alignment films in the empty glass cell. Moreover, the same alignment pattern and the orthogonal pattern with each other (i.e., the period of the pattern is shifted by a half period with each other) can be applied into the empty glass cell by adjusting the exposure dose as shown in Figure 6(b). In this experiment, the beam powers of both interfering two beams were set to 2 mW. The exposure doses to fabricate the continuous LC grating with planar and with TN alignment were set to 400 mJ/cm2 and 200 mJ/ cm<sup>2</sup> , and the exposure times were set to 8.0 s and 4.0 s, respectively. The self-diffraction from the front P6CB substrate during the photoalignment process does not occur because the optical anisotropy of P6CB is induced after annealing. The continuous LC gratings with planar or with TN alignment were fabricated by injecting the nematic LC after annealing process as shown in Figure 6(c).

Figure 6. Schematic of the fabrication procedure for the continuous LC grating. (a) One-step exposure of an empty glass cell to a UV interference beam obtained by interfering reverse rotated circularly polarized beams with each other, (b) periodic alignment patterns after annealing in the P6CB films, and (c) director distributions after injecting with a nematic LC (5CB).

#### 3.1.2. Binary LC gratings

system shown in Figure 5. The light source was He-Cd laser which was the same as that used in the previous experiment of the one-step photoalignment. The cross-sectional area of the

period of the electric field was 60 μm. The polarization states of the interfering beams were adjusted using the half-/quarter-wave plates. The intensity ratio of the interfering beams can be controlled by adjusting the polarization azimuth, which was an incident on the polarization beam splitter, using the half-wave plate. The one-step holographic photoalignment for the fabrication of the three types of LC gratings is described below individually. The empty glass cells used in the present experiment were the same as that described in Section 2.2. Moreover, the annealing process after the UV exposure and the injection process of nematic LCs were also

Figure 6 shows the fabrication procedure for the continuous LC grating with 0° planar or with 90° TN alignment. In the continuous LC grating, LC directors are constantly rotated along the grating vector. These were fabricated by one-step circular polarization interference exposure. When two orthogonally circular polarized beams interfere with each other, the polarization azimuth of the linear polarization in the interference field is continuously modulated; however, the polarization ellipticity and the intensity are not modulated. Therefore, the periodic alignment process, in which alignment directions are continuously rotated along the grating vector, can be simultaneously applied to two P6CB alignment films in the empty glass cell. Moreover, the same alignment pattern and the orthogonal pattern with each other (i.e., the period of the pattern is shifted by a half period with each other) can be applied into the empty glass cell by adjusting the exposure dose as shown in Figure 6(b). In this experiment, the beam powers of both interfering two beams were set to 2 mW. The exposure doses to fabricate the

Figure 5. Optical system for two-beam interference exposure. NDF, M, PBS, and H/QWP represent the neutral-density

. The crossing angle was 0.3°, and the resultant modulation

beam was expanded to 0.04 cm<sup>2</sup>

360 Holographic Materials and Optical Systems

3.1.1. Continuous LC gratings

filter, mirror, polarization beam splitter, and half-/quarter-wave plates.

the same.

Figure 7 shows the fabrication procedure for the binary LC grating with 0° planar or with 90° TN alignment. In the binary LC grating, LC directors are changed periodically and discretely by 90° in the grating vector. These were fabricated by one-step exposure of the empty glass cells to the UV interference field obtained by the interference of linearly polarized beams which were inclined +45° and 45° from the x-axis. The interference field periodically modulates the polarization states only because the +45° and 45° linear polarizations are orthogonality relation. The polarization ellipticity in the interference field is continuously modulated in the range from 0.0 to 1.0, and the polarization azimuth is discretely changed by 90° at the boundary point where the polarization ellipticity is 1.0; the polarization state changes between S3 via S1 in Poincaré sphere. Considering that the alignment direction of P6CB in the alignment films is predominantly in the longitudinal direction of the irradiated elliptical polarization, the alignment direction in the alignment films is periodically and discretely changed by 90° after annealing as shown in Figure 7(b). However, the regions in the alignment films which were irradiated elliptical or circular polarization involve the decline in the anchoring energy. In this experiment, the exposure doses to fabricate the continuous grating with planar and with TN alignment were set to 600 mJ/cm2 and 200 mJ/cm2 . Other experimental conditions and fabrication procedure were identical to those for the previously described continuous LC gratings.

Figure 7. Schematic of the fabrication procedure for the binary LC grating. (a) One-step exposure of an empty glass cell to a UV interference beam obtained by interfering orthogonal linearly polarized beams with each other, (b) periodic alignment patterns after annealing in the P6CB films, and (c) director distributions after injecting with a nematic LC (5CB).

#### 3.1.3. Planar-TN-LC grating

Figure 8 shows the fabrication procedure for the planar-TN-LC grating. In the planar-TN-LC grating, 0° planar and 90° TN alignment are periodically intermixed in the grating vector. This was fabricated by one-step exposure of the empty glass cell to the UV interference field obtained by the interference of linearly polarized beams which were inclined 90° from the xaxis. The interference field periodically modulates the intensity only. When the beam powers of interfering two beams are the same, the visibility of the interference field is maximum. This indicates that there are regions where the intensity in the interference field is nearly zero; the photoalignment is not applied in these regions. Therefore, the beam power ratio of the interfering linearly polarized beams was intentionally shifted from 1:1 to reduce the visibility from 1.0. Moreover, the planar-TN-LC grating can be fabricated by adjusting to the appropriate beam power ratio because the exposure dose varies along the x-axis position. The maximum intensity Imax and minimum intensity Imin in the interference field can be calculated as

Figure 8. Schematic of the fabrication procedure for the planar-TN-LC grating. (a) One-step exposure of an empty glass cell to a UV interference beam obtained by interfering parallel linearly polarized beams with each other, (b) periodic alignment patterns after annealing in the P6CB films, and (c) director distributions after injecting with a nematic LC (5CB).

One-Step Holographic Photoalignment for Twisted Nematic Liquid Crystal Gratings http://dx.doi.org/10.5772/67293 363

$$\begin{cases} I\_{\text{max}} = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \\ I\_{\text{min}} = I\_1 + I\_2 - 2\sqrt{I\_1 I\_2} \end{cases} \tag{1}$$

where I<sup>1</sup> and I<sup>2</sup> represent the beam intensities of the interfering beams. As calculation conditions, Imax = 600 mJ/cm<sup>2</sup> and Imin = 200 mJ/cm2 were set. From these conditions, I<sup>1</sup> and I<sup>2</sup> were determined to be 2.5 mW/cm<sup>2</sup> and 37.5 mW/cm2 , and the derived exposure time is 10 s. However, in the result of the preliminary experiment based on this condition, planar areas were larger than the TN areas in the resultant planar-TN-LC grating (i.e., an overall ratio of the planar and the TN areas was not 1:1). Therefore, the exposure time was experimentally determined in 8.0 s to achieve an overall 1:1 ratio of the planar and the TN areas. Other experimental conditions and fabrication procedure were identical to those for the previously described continuous LC gratings.

#### 3.2. Observation results of director distributions by polarized light microscopy

3.1.3. Planar-TN-LC grating

362 Holographic Materials and Optical Systems

Figure 8 shows the fabrication procedure for the planar-TN-LC grating. In the planar-TN-LC grating, 0° planar and 90° TN alignment are periodically intermixed in the grating vector. This was fabricated by one-step exposure of the empty glass cell to the UV interference field obtained by the interference of linearly polarized beams which were inclined 90° from the xaxis. The interference field periodically modulates the intensity only. When the beam powers of interfering two beams are the same, the visibility of the interference field is maximum. This indicates that there are regions where the intensity in the interference field is nearly zero; the photoalignment is not applied in these regions. Therefore, the beam power ratio of the interfering linearly polarized beams was intentionally shifted from 1:1 to reduce the visibility from 1.0. Moreover, the planar-TN-LC grating can be fabricated by adjusting to the appropriate beam power ratio because the exposure dose varies along the x-axis position. The maximum

Figure 7. Schematic of the fabrication procedure for the binary LC grating. (a) One-step exposure of an empty glass cell to a UV interference beam obtained by interfering orthogonal linearly polarized beams with each other, (b) periodic alignment patterns after annealing in the P6CB films, and (c) director distributions after injecting with a nematic LC (5CB).

intensity Imax and minimum intensity Imin in the interference field can be calculated as

Figure 8. Schematic of the fabrication procedure for the planar-TN-LC grating. (a) One-step exposure of an empty glass cell to a UV interference beam obtained by interfering parallel linearly polarized beams with each other, (b) periodic alignment patterns after annealing in the P6CB films, and (c) director distributions after injecting with a nematic LC (5CB). Figure 9(a) and (b) shows the photographs of the continuous LC grating with planar and with TN alignment, respectively. The transmittance in the photographs continuously varies along the grating vector. The full dark fields, where the LC directors incline by 0° or 90° with respect to the transmission axes of the polarizer and the analyzer, exist in the planar alignment of the continuous LC grating as shown in Figure 9(a). Note that the transmittance of the bright fields, where the LC directors incline by 45° with respect to the transmission axes, depends on the retardation of the LC grating. This is because the polarization state of output electric field changes between �S3 in Poincaré sphere depending on the retardation when the linear polarization inclined by 45° with respect to an optical axis is incident. In the TN alignment shown in Figure 9(b), when the LC directors of the alignment film interface incline by 0° or 90° with transmission axes of the polarizer and the analyzer, the transmittance of white light is maximized. Full dark fields do not exist. Figure 9(c) and (d) shows the photographs of the binary LC grating with planar and with TN alignment, respectively. For the reasons described above, the entire region can be seen as full dark fields in the planar alignment as shown in Figure 9(c), and the entire region can be seen as bright fields in the TN alignment as shown in Figure 9(d). Moreover, vertical lines extending perpendicular to the grating vector direction are observed. These lines represent the "transition regions" between the binary regions, which will be detailed later. Figure 9(e) shows the photographs of the planar-TN-LC grating. The full dark fields and the bright fields are seen to alternate, and the distribution of full dark fields and bright fields is inverted by interchanging the crossed Nicol and parallel Nicol polarizers, as shown in Figure 9(e-1) and (e-2). These results indicated that the director distributions of the resultant LC gratings were nearly the same as the initially designed director distributions shown in Figures 6(c), 7(c), and 8(c).

#### 3.3. Numerical analysis of director distributions and diffraction properties

It is important to consider quantitatively the observed director distributions of the resultant LC gratings based on a physical theory. The director distributions of the LC gratings are not determined uniquely and strictly by only patterns of alignment films. Therefore, in some case, the slightly different director distributions from the ideal distributions were obtained as described in Section 3.2. In this section, the estimation method of the static director distributions based on the elastic continuum theory of a nematic LCs is described below.

Figure 9. Photographs of the resultant LC gratings observed by polarized light microscopy. (a) Continuous LC gratings with planar alignment and (b) TN alignment. (c) Binary LC gratings with planar alignment and (d) TN alignment. (e) Planar-TN-LC gratings under (e-1) crossed Nicol polarizers and (e-2) parallel Nicol polarizers. The part of the figure is reproduced by the kind permission of The Optical Society of America from Applied Optics 54, 6010-6018 (2015).

In LC gratings, LCs are anchored by alignment films, and then spatial nonuniformities in LC directors are induced. Since LCs have elastic properties, restoring forces and elastic free energies are increased by these elastic deformations. Moreover, when the spatial nonuniformities (i.e., total elastic free energies) are minimized, the periodic director distributions of the LC gratings are stable. Therefore, the detailed director distributions can be estimated by calculating the condition which the total free energy is minimized. The total elastic free energy of the LC grating is obtained by spatially integrating the elastic free energy per unit volume shown in the following equation:

$$f = \frac{1}{2}\mathcal{K}\_1(\nabla \cdot \mathbf{n})^2 + \frac{1}{2}\mathcal{K}\_2[\mathbf{n}\cdot(\nabla \times \mathbf{n})]^2 + \frac{1}{2}\mathcal{K}\_3[\mathbf{n}\times(\nabla \times \mathbf{n})]^2,\tag{2}$$

where K1, K2, and K<sup>3</sup> represent the elastic constants the splay, twist, and bend of nematic LCs, respectively. n represents the director of the nematic LCs in the xyz coordinate frame. The xaxis is parallel to the grating vector, and the z-axis is parallel to the thickness direction which follows to the definition shown in Section 3.1. Based on the premise that the director is not tilted in the z-axis, the director n is defined as n = (cosφ, sinφ, 0), where φ represents the angle between the x-axis and the director n. By substituting the director n into Eq. (2), we obtain

$$f = \frac{1}{2}K\_1 \left(\frac{\partial \phi}{\partial x}\right)^2 \sin^2 \phi + \frac{1}{2}K\_2 \left(\frac{\partial \phi}{\partial z}\right)^2 + \frac{1}{2}K\_3 \left(\frac{\partial \phi}{\partial x}\right)^2 \cos^2 \phi,\tag{3}$$

where K<sup>1</sup> = 6.4 pN, K<sup>2</sup> = 3.0 pN, and K<sup>3</sup> = 10.0 pN [12]. The total elastic free energy F of one period is obtained by spatially integrating Eq. (3) as given by

described in Section 3.2. In this section, the estimation method of the static director distribu-

In LC gratings, LCs are anchored by alignment films, and then spatial nonuniformities in LC directors are induced. Since LCs have elastic properties, restoring forces and elastic free energies are increased by these elastic deformations. Moreover, when the spatial nonuniformities (i.e., total elastic free energies) are minimized, the periodic director distributions of the LC gratings are stable. Therefore, the detailed director distributions can be estimated by calculating the condition which the total free energy is minimized. The total elastic free energy of the LC grating is obtained by spatially integrating the elastic free energy per unit volume

reproduced by the kind permission of The Optical Society of America from Applied Optics 54, 6010-6018 (2015).

Figure 9. Photographs of the resultant LC gratings observed by polarized light microscopy. (a) Continuous LC gratings with planar alignment and (b) TN alignment. (c) Binary LC gratings with planar alignment and (d) TN alignment. (e) Planar-TN-LC gratings under (e-1) crossed Nicol polarizers and (e-2) parallel Nicol polarizers. The part of the figure is

<sup>K</sup>2½ � <sup>n</sup> � ð Þ <sup>∇</sup> · <sup>n</sup> <sup>2</sup> <sup>þ</sup>

where K1, K2, and K<sup>3</sup> represent the elastic constants the splay, twist, and bend of nematic LCs, respectively. n represents the director of the nematic LCs in the xyz coordinate frame. The xaxis is parallel to the grating vector, and the z-axis is parallel to the thickness direction which follows to the definition shown in Section 3.1. Based on the premise that the director is not tilted in the z-axis, the director n is defined as n = (cosφ, sinφ, 0), where φ represents the angle between the x-axis and the director n. By substituting the director n into Eq. (2), we obtain

1 2

þ 1 2 K3 ∂φ ∂x <sup>2</sup>

<sup>K</sup>3j j <sup>n</sup> · ð Þ <sup>∇</sup> · <sup>n</sup> <sup>2</sup>

cos <sup>2</sup>

; (2)

φ; (3)

shown in the following equation:

364 Holographic Materials and Optical Systems

<sup>f</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>f</sup> <sup>¼</sup> <sup>1</sup> 2 K1 ∂φ ∂x <sup>2</sup>

<sup>K</sup>1ð Þ <sup>∇</sup> � <sup>n</sup> <sup>2</sup> <sup>þ</sup>

1 2

sin <sup>2</sup> φ þ 1 2 K2 ∂φ ∂z <sup>2</sup>

tions based on the elastic continuum theory of a nematic LCs is described below.

$$F = \int\_0^d \int\_0^\Lambda f dx dz,\tag{4}$$

where d and Λ represent the cell gap and the grating pitch of LC gratings, respectively. As described above, the director distribution is determined by calculating φ when the total elastic free energy is minimized. Specifically, the director distribution can be calculated as ∂F = 0. We adopted the finite element method to solve this variational problem of the functional. In this numerical calculation, we applied periodic boundary conditions for speeding up the calculation. Moreover, we set fixed boundary conditions in the alignment substrates at z = 0 and z = d by assuming strong anchoring conditions of P6CB alignment films shown in Table 1.


Table 1. Boundary conditions of the three types of LC gratings φ(x) at z = 0 and z = d.

Figure 10 shows a cross-sectional view of the director distributions in the LC gratings over one period as calculated by the elastic continuum theory of nematic LCs. The ideal director distributions shown in Section 3.1 were obtained. However, especially in the binary LC gratings [Figure 10(c) and (d)], the LC directors do not change abruptly at the boundary of two regions (i.e., previously described "transition regions"). These results indicate that LC directors in the LC grating are not induced steep elastic deformations in order to prevent increasing locally elastic free energies.

To consider quantitatively the effects on the diffraction properties due to transition regions, the numerical solutions of the diffraction properties are calculated on the basis of Jones calculus using the obtained director distribution shown in Figure 10. To calculate the phase distributions of the nematic LC phases, we consider the birefringence plate of n layers, and the Jones matrix W of the LC gratings can be written as

$$\mathbf{W}(\mathbf{x}) = \mathbf{W}\_{\text{out}}(\mathbf{x}) \cdot \left(\prod\_{m=1}^{n} \mathbf{W}\_{m}(\mathbf{x})\right) \cdot \mathbf{W}\_{\text{in}}(\mathbf{x}),\tag{5}$$

where Win and Wout are Jones matrices for the P6CB alignment films at the input and output sides, respectively, and W<sup>m</sup> represents the LC phase of the mth layer. These are given by

$$\mathbf{W}\_{\rm in/out}(\mathbf{x}) = \mathbf{R} \begin{bmatrix} -\phi(\mathbf{x}) \end{bmatrix} \cdot \begin{bmatrix} \exp(-i\pi\Delta n\_{\rm P}d\_{\rm P}/\lambda) & 0\\ 0 & \exp(i\pi\Delta n\_{\rm P}d\_{\rm P}/\lambda) \end{bmatrix} \cdot \mathbf{R} \begin{bmatrix} \phi(\mathbf{x}) \end{bmatrix},\tag{6}$$

$$\mathbf{W}\_{\mathfrak{m}}(\mathbf{x}) = \mathbf{R}\left[-\phi(\mathbf{x})\right] \cdot \begin{bmatrix} \exp(-i\pi\Delta n d\_{\mathfrak{m}}/\lambda) & 0\\ 0 & \exp(i\pi\Delta n d\_{\mathfrak{m}}/\lambda) \end{bmatrix} \cdot \mathbf{R}\left[\phi(\mathbf{x})\right],\tag{7}$$

where R represents the coordinate rotation matrix. Δn and Δn<sup>P</sup> are the optical anisotropy of the nematic LCs and the P6CB alignment films, respectively. λ is the wavelength of the probe beam. dm and d<sup>P</sup> represent the thicknesses of the nematic LC layers (d/n = dm) and P6CB alignment films, respectively. The parameters used in the calculation are Δn = 0.18, Δn<sup>P</sup> = 0.2, d<sup>P</sup> = 300 nm, and λ = 633 nm. The electric field transmitted through the LC gratings is obtained by multiplying the incident electric field vector by the Jones matrices W. The diffraction properties are obtained by Fourier transform of the spatial distributions of the output electric field vector. The resultant numerical solutions of the diffraction properties will be described later in conjunction with the measurement results.

Figure 10. Cross-sectional view of the director distributions calculated from the elastic continuum theory. (a) Continuous LC gratings with planar alignment and (b) TN alignment. (c) Binary LC gratings with planar alignment and (d) TN alignment. (e) Planar-TN-LC gratings. Fixed boundary conditions shown in Table 1 are applied in gray regions. The part of the figure is reproduced by the kind permission of The Optical Society of America from Applied Optics 54, 6010-6018 (2015).

#### 3.4. Analytical solutions by Jones calculus

It is important to obtain mathematically the analytical solutions of diffraction properties to give exact theoretical solutions. In this section, the analytical solutions of the resultant LC gratings are derived using Jones calculus. However, the transition regions are not considered in this analysis, and the ideal director distributions shown in Section 3.1 are analyzed. Comparisons of the analytical solutions and the numerical solutions will be described in Section 3.5. The variables which are defined in the previous section are taken over.

#### 3.4.1. Continuous LC gratings

<sup>W</sup>in=outð Þ¼ <sup>x</sup> <sup>R</sup> �φð Þ<sup>x</sup> � expð Þ �iπΔnPdP=<sup>λ</sup> <sup>0</sup>

<sup>W</sup>mð Þ¼ <sup>x</sup> <sup>R</sup> �φð Þ<sup>x</sup> � expð Þ �iπΔndm=<sup>λ</sup> <sup>0</sup>

later in conjunction with the measurement results.

366 Holographic Materials and Optical Systems

3.4. Analytical solutions by Jones calculus

(2015).

which are defined in the previous section are taken over.

0 expð Þ iπΔnPdP=λ

0 expð Þ iπΔndm=λ

� <sup>R</sup> <sup>φ</sup>ð Þ<sup>x</sup> ; (6)

� <sup>R</sup> <sup>φ</sup>ð Þ<sup>x</sup> ; (7)

where R represents the coordinate rotation matrix. Δn and Δn<sup>P</sup> are the optical anisotropy of the nematic LCs and the P6CB alignment films, respectively. λ is the wavelength of the probe beam. dm and d<sup>P</sup> represent the thicknesses of the nematic LC layers (d/n = dm) and P6CB alignment films, respectively. The parameters used in the calculation are Δn = 0.18, Δn<sup>P</sup> = 0.2, d<sup>P</sup> = 300 nm, and λ = 633 nm. The electric field transmitted through the LC gratings is obtained by multiplying the incident electric field vector by the Jones matrices W. The diffraction properties are obtained by Fourier transform of the spatial distributions of the output electric field vector. The resultant numerical solutions of the diffraction properties will be described

It is important to obtain mathematically the analytical solutions of diffraction properties to give exact theoretical solutions. In this section, the analytical solutions of the resultant LC gratings are derived using Jones calculus. However, the transition regions are not considered in this analysis, and the ideal director distributions shown in Section 3.1 are analyzed. Comparisons of the analytical solutions and the numerical solutions will be described in Section 3.5. The variables

Figure 10. Cross-sectional view of the director distributions calculated from the elastic continuum theory. (a) Continuous LC gratings with planar alignment and (b) TN alignment. (c) Binary LC gratings with planar alignment and (d) TN alignment. (e) Planar-TN-LC gratings. Fixed boundary conditions shown in Table 1 are applied in gray regions. The part of the figure is reproduced by the kind permission of The Optical Society of America from Applied Optics 54, 6010-6018 In the resultant continuous LC gratings, the alignment direction of the 0° planar alignment or 90° TN alignment is rotated continuously along the grating vector [Figure 6(a)]. Therefore, rotation matrix which depends on the position ξ (=2πx/Λ) in the grating vector is introduced into the Jones matrix of the continuous LC gratings as shown in the following:

$$\mathbf{W}\_{\text{plnar}/\text{TN}}^{\text{continuous}} = \mathbf{R}\left(-\frac{\xi}{2}\right) \cdot \mathbf{W}\_{\text{Planar}/\text{TN}} \cdot \mathbf{R}\left(\frac{\xi}{2}\right),\tag{8}$$

where WPlanar and WTN represent Jones matrices of the 0° planar and 90° TN alignment, respectively, as shown in the following:

$$\mathbf{W}\_{\text{planar}} = \begin{bmatrix} \exp(-i\frac{\mathbf{r}}{2}) & \mathbf{0} \\ \mathbf{0} & \exp(i\frac{\mathbf{r}}{2}) \end{bmatrix},\tag{9}$$

$$\mathbf{W}\_{\text{TN}} = \mathbf{R}(-\boldsymbol{\Phi}) \cdot \begin{bmatrix} \cos X - i \frac{\text{T} \sin X}{2\text{X}} & \mathbf{0} \frac{\sin X}{X} \\ -\mathbf{Q} \frac{\sin X}{X} & \cos X + i \frac{\text{T} \sin X}{2\text{X}} \end{bmatrix},\tag{10}$$

where Γ (=2πΔnd/λ) stands for the phase retardation, Φ (=π/2) represents the twisted angle in the TN alignment, and X is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Φ</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>Γ</sup>=<sup>2</sup> <sup>2</sup> q . By substituting the Eq. (9) or (10) into Eq. (8) and separating perturbation terms which depend on the position ξ, we obtain the Jones matrices of the continuous LC gratings with the planar or with TN alignment which contribute to �1st-order diffraction, as follows:

$$\mathbf{W}\_{\text{planar}}^{\text{continuous}} = \frac{i}{2} \sin \frac{\Gamma}{2} \begin{bmatrix} 1 & \pm i \\ \pm i & -1 \end{bmatrix} \exp(\pm i \xi),\tag{11}$$

and

$$\mathbf{W}\_{\text{TN}}^{\text{continuous}} = i \frac{\Gamma \sin X}{4X} \begin{bmatrix} \mp i & -1 \\ -1 & \pm i \end{bmatrix} \exp(\pm i \xi) \tag{12}$$

The electric field vector of the �1st-order diffracted beam is derived by multiplying the electric field vector of the incident beam shown in the following:

$$\mathbf{E}\_{\rm in} = \begin{bmatrix} (\cos \Psi) \exp(i\delta) \\ \sin \Psi \end{bmatrix},\tag{13}$$

where Ψ and δ represent the amplitude ratio angle and the phase difference. Therefore, the electric field vector of the �1st-order diffracted beams are given by

$$\mathbf{E}\_{\text{out}} \text{ (planar)} = \mathbf{W}\_{\text{planar}}^{\text{continuous}} \cdot \mathbf{E}\_{\text{in}} = \frac{i}{2} \sin \left( \frac{\Gamma}{2} \right) \exp(\pm i\xi) \cdot \left\{ \cos \Psi \exp(i\delta) \pm i \sin \Psi \right\} \begin{bmatrix} 1\\ \pm i \end{bmatrix}, \quad (14)$$

and

$$\mathbf{E}\_{\text{out}\ (\text{TN})} = \mathbf{W}\_{\text{TN}}^{\text{continuous}} \cdot \mathbf{E}\_{\text{in}} = i \frac{\Gamma \sin X}{4X} \exp(\pm i\xi) \cdot \left\{ \mp i \cos \Psi \exp(i\delta) - \sin \Psi \right\} \begin{bmatrix} 1 \\ \mp i \end{bmatrix}. \tag{15}$$

These analytical solutions indicate that the polarization states of the �1st-order diffracted beams from both the planar and the TN alignment are always circular polarization, and this property does not depend on the polarization states of the incident beams. Either of +firstor �first-order diffraction beam only diffracts when the circular polarization is incident (i.e., δ = π/2, and Ψ = π/4); both +first- and �first-order diffraction beams diffract when the linear polarization is incident (i.e., δ = 0). In the case of the entering the circular polarization, the diffraction efficiency is twice in comparison with the diffraction efficiency when the linear polarization is incident; these are obtained by squaring the electric field of the �1st-order diffracted beams shown in Eqs. (14) and (15).

#### 3.4.2. Binary LC gratings and planar-TN grating

Both the binary and the planar-TN-LC gratings are classified into the diffraction gratings in which the amount of the phase shift is rectangularly modulated along the grating vector. The diffraction properties of thin anisotropic diffraction gratings, in which two different anisotropic regions are periodically arranged, can also be analyzed by Jones calculus [12]. Diffraction properties of the binary LC gratings and the planar-TN-LC grating are given by the sum of the emitted electric fields from the two anisotropic regions. Therefore, the Jones matrix W<sup>N</sup> of the Nth-order diffracted beam is given by Fourier transform when the Jones matrices of the two anisotropic regions are defined as W<sup>A</sup> and W<sup>B</sup> and can be written as

$$\mathbf{W}\_{N} = \frac{1}{\Lambda} \left[ \int\_{-\Lambda/2}^{\zeta\Lambda} \mathbf{W}\_{\mathbf{A}} \exp\left(-i\frac{2\pi \mathbf{x}N}{\Lambda}\right) d\mathbf{x} + \int\_{\zeta\Lambda}^{\Lambda/2} \mathbf{W}\_{\mathbf{B}} \exp\left(-i\frac{2\pi \mathbf{x}N}{\Lambda}\right) d\mathbf{x} \right],\tag{16}$$

where ζ is the boundary position of the two anisotropic regions and defined as (� 1/2 ≤ ζ ≤ 1/2). Note that the Jones matrices W<sup>A</sup> and W<sup>B</sup> are not dependent on the x position in the integration interval. The Jones matrix W is given by

$$\mathbf{W}\_{N} = \exp\left[-i\pi N\left(\zeta - \frac{1}{2}\right)\right] \frac{\sin\left[\pi N(\zeta + \frac{1}{2})\right]}{\pi N} (\mathbf{W}\_{\text{A}} - \mathbf{W}\_{\text{B}}).\tag{17}$$

Eq. (17) can also be applied to any grating when W<sup>A</sup> and W<sup>B</sup> in an anisotropic diffraction grating are known. Moreover, when the area ratio of the two anisotropic regions of W<sup>A</sup> and W<sup>B</sup> is 1:1 (i. e., ζ = 0), even-order diffracted beams are not diffracted as shown in Eq. (17). This is because the even-order diffracted beams are negated by the anti-phase emitted electric field.

To derive the Jones matrices of the binary LC grating with 0° planar and with 90° TN alignments, W<sup>A</sup> are defined by the right-hand side of the Eqs. (9) and (10) described in the previous section, respectively. On the other hand, the alignment directions of W<sup>B</sup> are inclined by 90° with the alignment directions of WA, and W<sup>B</sup> are given by

One-Step Holographic Photoalignment for Twisted Nematic Liquid Crystal Gratings http://dx.doi.org/10.5772/67293 369

$$\mathbf{W}\_{\rm B} = \mathbf{R}\left(-\frac{\pi}{2}\right) \cdot \mathbf{W}\_{\rm A} \cdot \mathbf{R}\left(\frac{\pi}{2}\right). \tag{18}$$

The Jones matrices of the �1st-order diffraction from the binary LC grating with 0° planar and with 90° TN alignments are given by substituting the Jones matrices described in Eqs. (9) or (10) and (18) into Eq. (17) and can be written as

$$\mathbf{W}\_{\text{planar}}^{\text{binary}} = \frac{2}{\pi} \sin \frac{\Gamma}{2} \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix},\tag{19}$$

and

<sup>E</sup>out TN ð Þ <sup>¼</sup> <sup>W</sup>continuous

368 Holographic Materials and Optical Systems

TN � Ein ¼ i

diffracted beams shown in Eqs. (14) and (15).

3.4.2. Binary LC gratings and planar-TN grating

<sup>W</sup><sup>N</sup> <sup>¼</sup> <sup>1</sup> Λ

interval. The Jones matrix W is given by

ðζΛ �Λ=2 Γ sin X

These analytical solutions indicate that the polarization states of the �1st-order diffracted beams from both the planar and the TN alignment are always circular polarization, and this property does not depend on the polarization states of the incident beams. Either of +firstor �first-order diffraction beam only diffracts when the circular polarization is incident (i.e., δ = π/2, and Ψ = π/4); both +first- and �first-order diffraction beams diffract when the linear polarization is incident (i.e., δ = 0). In the case of the entering the circular polarization, the diffraction efficiency is twice in comparison with the diffraction efficiency when the linear polarization is incident; these are obtained by squaring the electric field of the �1st-order

Both the binary and the planar-TN-LC gratings are classified into the diffraction gratings in which the amount of the phase shift is rectangularly modulated along the grating vector. The diffraction properties of thin anisotropic diffraction gratings, in which two different anisotropic regions are periodically arranged, can also be analyzed by Jones calculus [12]. Diffraction properties of the binary LC gratings and the planar-TN-LC grating are given by the sum of the emitted electric fields from the two anisotropic regions. Therefore, the Jones matrix W<sup>N</sup> of the Nth-order diffracted beam is given by Fourier transform when the Jones matrices of the

two anisotropic regions are defined as W<sup>A</sup> and W<sup>B</sup> and can be written as

2πxN Λ � �

2 � � � � sin <sup>π</sup><sup>N</sup> <sup>ζ</sup> <sup>þ</sup> <sup>1</sup>

dx þ

where ζ is the boundary position of the two anisotropic regions and defined as (� 1/2 ≤ ζ ≤ 1/2). Note that the Jones matrices W<sup>A</sup> and W<sup>B</sup> are not dependent on the x position in the integration

Eq. (17) can also be applied to any grating when W<sup>A</sup> and W<sup>B</sup> in an anisotropic diffraction grating are known. Moreover, when the area ratio of the two anisotropic regions of W<sup>A</sup> and W<sup>B</sup> is 1:1 (i. e., ζ = 0), even-order diffracted beams are not diffracted as shown in Eq. (17). This is because the even-order diffracted beams are negated by the anti-phase emitted electric field. To derive the Jones matrices of the binary LC grating with 0° planar and with 90° TN alignments, W<sup>A</sup> are defined by the right-hand side of the Eqs. (9) and (10) described in the previous section, respectively. On the other hand, the alignment directions of W<sup>B</sup> are inclined by 90°

" #

ð<sup>Λ</sup>=<sup>2</sup> ζΛ

<sup>2</sup> ½ � ð Þ

WBexp �i

2πxN Λ � �

<sup>π</sup><sup>N</sup> ð Þ <sup>W</sup><sup>A</sup> � <sup>W</sup><sup>B</sup> : (17)

dx

; (16)

WAexp �i

<sup>W</sup><sup>N</sup> <sup>¼</sup> exp �iπ<sup>N</sup> <sup>ζ</sup> � <sup>1</sup>

with the alignment directions of WA, and W<sup>B</sup> are given by

<sup>4</sup><sup>X</sup> expð Þ� �i<sup>ξ</sup> <sup>∓</sup><sup>i</sup> cos <sup>Ψ</sup>expð Þ� <sup>i</sup><sup>δ</sup> sin <sup>Ψ</sup> � � <sup>1</sup>

∓i � �

: (15)

$$\mathbf{W}\_{\text{TN}}^{\text{binary}} = \frac{\Gamma \sin X}{2\pi X} \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}. \tag{20}$$

The Jones vectors for �1st-order diffraction is derived by multiplying the electric field vector of the incident beam shown in Eq. (13) and can be written as

$$\mathbf{E}\_{\rm out} \left( \text{planar} \right) = \mathbf{W}\_{\rm planar}^{\rm binary} \cdot \mathbf{E}\_{\rm in} = \frac{2}{\pi} \sin \frac{\Gamma}{2} \begin{bmatrix} (\cos \Psi) \exp(i\delta) \\ -\sin \Psi \end{bmatrix},\tag{21}$$

and

$$\mathbf{E}\_{\text{out (TN)}} = \mathbf{W}\_{\text{TN}}^{\text{binary}} \cdot \mathbf{E}\_{\text{in}} = \frac{\Gamma \sin X}{2\pi X} \begin{bmatrix} \sin \Psi\\ (\cos \Psi) \text{exp}(i\delta) \end{bmatrix},\tag{22}$$

The sign of the y component in Jones vector for the planar alignment shown in Eq. (21) is different than the Jones vector of the probe beam. On the other hand, the Jones vector for the TN alignment shown in Eq. (22) is the same as the probe beam Jones vector where the x and y components are interchanged. These results indicate that the binary gratings have diffraction properties which convert the polarization azimuth and the rotational direction of the polarized probe beam.

In the planar-TN-LC grating, W<sup>A</sup> and W<sup>B</sup> in the Eq. (17) are given by Eqs. (9) and (10), respectively. In the same way, the Jones vector for the �1st-order diffraction from the planar-TN-LC grating is given by

$$\mathbf{E}\_{\text{out}\{\text{planar-TN}\}} = \mathbf{W}^{\text{planar-TN}} \cdot \mathbf{E}\_{\text{in}\text{-}} = \frac{\dot{\mathbf{i}}}{\pi} \cdot \begin{bmatrix} \frac{\pi \sin X}{2X} - \exp\left(-\frac{\pi}{2}\right) & -\cos X - i\frac{\pi \sin X}{2X} \\\\ \cos X - i\frac{\pi \sin X}{2X} & \frac{\pi \sin X}{2X} - \exp\left(\frac{\pi}{2}\right) \end{bmatrix} \cdot \begin{bmatrix} (\cos \Psi) \exp(i\delta) \\\\ \sin \Psi \end{bmatrix} . \tag{23}$$

In the planar-TN-LC gratings, analytical solutions of the diffraction properties cannot be derived unconditionally because the diffraction properties depend on the phase retardation Γ, as shown in Eq. (23). The diffraction properties can be analyzed by fitting to the experimental results using the phase difference (especially the cell gap, d) as a fitting parameter.

#### 3.5. Measurement results of diffraction properties

The first-order diffracted beams from the resultant LC gratings were probed experimentally with He-Ne laser beam with a wavelength of 633 nm. The polarization states of the probe beam were adjusted to a linear or a circular polarization using a half-wave plate and a quarterwave plate. The polarization azimuth of the probe beam was defined as the inclination angle of the polarization director with respect to the grating vector. The probe beam was incident normal to the plane of the substrate including the grating vector. The first-order diffracted beams were separated from other orders using a pinhole. The diffracted beam intensities were measured using an optical power meter. The polarization states of diffracted beams were measured by a rotating analyzer method using a Glan-Thompson prism as the analyzer.

Figure 11(a) shows the measurement and the calculation results of polarization states of the first-order diffracted beams from the continuous LC gratings. The calculated results are obtained from the numerical solutions described in Section 3.3 and the analytical solutions described in Section 3.4. The calculated diffraction efficiencies were fitted to the experimental results using the cell gaps as the fitting parameter, and the optimum cell gaps in the planar and TN alignment obtained by fitting were 13.2 μm and 13.1 μm, respectively. The polarization states of the first-order diffracted beams from both the planar and the TN alignments are always circular polarization, and this property does not depend on the polarization states of the incident beams. Either of +first- or first-order diffraction beam only diffracts when the circular polarization is incident; both +first- and first-order diffraction beams diffract when the linear polarization is incident. The diffraction efficiency when the circular polarization is incident is twice in comparison with the diffraction efficiency when the linear polarization is incident.

Figure 11(b) shows the measurement and the calculation results of polarization states of the 1st-order diffracted beams from the binary LC gratings. The polarization ellipticity of the diffracted beam is conserved from the probe beam, although the rotation direction is inverted from the probe beam. The polarization azimuth of the diffracted beam varies over a range from 0° to 90° depending on the polarization azimuth of the probe beam. Furthermore, the polarization azimuth of the diffracted beams is 90° offset from the planar and TN alignment when the probe beam is linearly polarized. The diffraction properties in the positive and negative diffraction orders were the same. The polarization states of the diffracted beam do not depend on the phase difference. The calculated diffraction efficiencies were fitted to the experimental results using the cell gaps as the fitting parameter, and the optimum cell gaps in both the planar and TN alignments obtained by fitting were 13.4 μm. There were slight differences in some results between the experimental results and the calculation results. These discrepancies can be mainly attributed to the transition regions at the boundary of the two alignment regions. In the calculated results, there were only slightly differences between the analytical and numerical solutions because the transition regions were considered in the numerical solutions. Specifically, with circular polarization of the probe beam, the polarization ellipticity of the diffracted beams determined with the analytical solution was 1.0; the numerical solution yielded 0.97. From that above, the cause of the slight deviation between the experimental results and the theoretical expectations is that the transition regions in the resultant binary LC gratings were slightly larger than the estimated director distributions.

3.5. Measurement results of diffraction properties

370 Holographic Materials and Optical Systems

The first-order diffracted beams from the resultant LC gratings were probed experimentally with He-Ne laser beam with a wavelength of 633 nm. The polarization states of the probe beam were adjusted to a linear or a circular polarization using a half-wave plate and a quarterwave plate. The polarization azimuth of the probe beam was defined as the inclination angle of the polarization director with respect to the grating vector. The probe beam was incident normal to the plane of the substrate including the grating vector. The first-order diffracted beams were separated from other orders using a pinhole. The diffracted beam intensities were measured using an optical power meter. The polarization states of diffracted beams were measured by a rotating analyzer method using a Glan-Thompson prism as the analyzer.

Figure 11(a) shows the measurement and the calculation results of polarization states of the first-order diffracted beams from the continuous LC gratings. The calculated results are obtained from the numerical solutions described in Section 3.3 and the analytical solutions described in Section 3.4. The calculated diffraction efficiencies were fitted to the experimental results using the cell gaps as the fitting parameter, and the optimum cell gaps in the planar and TN alignment obtained by fitting were 13.2 μm and 13.1 μm, respectively. The polarization states of the first-order diffracted beams from both the planar and the TN alignments are always circular polarization, and this property does not depend on the polarization states of the incident beams. Either of +first- or first-order diffraction beam only diffracts when the circular polarization is incident; both +first- and first-order diffraction beams diffract when the linear polarization is incident. The diffraction efficiency when the circular polarization is incident is twice in

comparison with the diffraction efficiency when the linear polarization is incident.

gratings were slightly larger than the estimated director distributions.

Figure 11(b) shows the measurement and the calculation results of polarization states of the 1st-order diffracted beams from the binary LC gratings. The polarization ellipticity of the diffracted beam is conserved from the probe beam, although the rotation direction is inverted from the probe beam. The polarization azimuth of the diffracted beam varies over a range from 0° to 90° depending on the polarization azimuth of the probe beam. Furthermore, the polarization azimuth of the diffracted beams is 90° offset from the planar and TN alignment when the probe beam is linearly polarized. The diffraction properties in the positive and negative diffraction orders were the same. The polarization states of the diffracted beam do not depend on the phase difference. The calculated diffraction efficiencies were fitted to the experimental results using the cell gaps as the fitting parameter, and the optimum cell gaps in both the planar and TN alignments obtained by fitting were 13.4 μm. There were slight differences in some results between the experimental results and the calculation results. These discrepancies can be mainly attributed to the transition regions at the boundary of the two alignment regions. In the calculated results, there were only slightly differences between the analytical and numerical solutions because the transition regions were considered in the numerical solutions. Specifically, with circular polarization of the probe beam, the polarization ellipticity of the diffracted beams determined with the analytical solution was 1.0; the numerical solution yielded 0.97. From that above, the cause of the slight deviation between the experimental results and the theoretical expectations is that the transition regions in the resultant binary LC

Figure 11. Polar plots of the 1st-order diffracted beam for the resultant LC gratings. Open circles, red solid curves, and blue broken lines represent the experimental data, theoretical solution, and numerical solution, respectively. The numerical values below the polar plots represent the experimental results of diffraction efficiency. (a) Continuous LC gratings and (b) binary LC gratings with planar alignment and with TN alignment. (c) Planar-TN-LC gratings. The part of the figure is reproduced by the kind permission of The Optical Society of America from Applied Optics 54, 6010-6018 (2015).

Figure 11(c) shows the measured and the calculated results of the planar-TN-LC grating. There is good agreement between the measured results and the analytical and numerical solutions. When the probe beam was right- and left-hand circularly polarized, the elliptically polarized beams, with polarization azimuths offset by 90°, were diffracted depending on the rotation direction of the probe beams. The polarization azimuth of both probe beam and the diffracted beam is inversely related. The polarization ellipticity of the diffracted beams periodically varies over the range of 0.0–0.2 depending on the polarization azimuth of the probe beam. When the probe beams were 0° and 90° linearly polarized, the polarization ellipticity of diffracted beams was 0.0. On the other hand, when the probe beams were 45° linearly polarized, the ellipticity was 0.2. It was found from the theoretical analysis that the polarization ellipticity of the diffracted beam varies over the 0.0–0.9 range, depending on the phase difference when the probe beam was 45° linearly polarized. In addition, when the probe beams are 0° and 90° linearly polarized, the diffracted beams are always linearly polarized and do not depend on the phase difference. The calculated diffraction properties were fitted to the experimental results using the cell gap as the fitting parameter, and the optimum cell gap in the planar-TN-LC grating obtained by fitting was 14.8 μm. The diffraction properties in the positive and negative diffraction orders were the same.
