1. Introduction

Liquid crystals (LCs) are widely used for optical devices due to the large electro-optic modulation resulting from the high optical anisotropy, the electrically tuning capability at low voltages, and the design flexibility on various substrates [1–6]. Among a number of the LC devices, the LC grating elements play an important role in many optical systems such as 3-dimentional (3D) displays, optical data storage, and optical communication systems by means of the change in the phase or the polarization state of light [2,7–11]. Although the LC grating devices with simple binary patterns have been commonly used [12–15], more sophisticated gratings including step-wise discrete gratings and continuous gratings are needed for specific applications [6,16–22]. The step-wise LC gratings fabricated using a micro-rubbing method [11,21] or a micro-electrode patterning method [22] suffer from the discontinuity of the phase retardation due to the uniform alignment of the LC molecules within each step-wise pattern in alternating domains. On the other hand, a rather continuous version of the LC grating obtained through photoalignment of the LC using holographic methods [6,16–20], shown in Fig. 1(a) , is capable of eliminating higher-orders of diffraction except for ±1st orders whose polarization states are circular. This type always involves the disclination in the LC alignment from one domain to the other and thus possesses the mirror symmetry. The resultant diffraction properties were found to depend on the thickness of the LC cell. Therefore, it is very important to understand the diffraction properties of the continuous LC gratings in term of the symmetry argument from the fundamental as well as practical viewpoints.

 

Fig. 1 Schematic representation of the distribution of the LC director in (a) a continuous grating structure with a wall (indicated in red) separating two domains and (b) a fully continuous grating structure with no wall in alternating semi-circular (quadrant) forms and the LC director in y-z plane. The green bars represent the LC molecules.

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In this paper, we demonstrated a fully continuous LC grating device with the rotational symmetry of the LC alignment which exhibits the switching effect between the diffraction orders independent of the thickness of the LC cell. A simple and low-cost imprinting technique [23] was employed to produce the alternating semi-circular alignment of the LC as shown in Fig. 1(b). It was found that the LC molecules were well aligned along the direction of the microgrooves defined by the imprinting process due to the Berreman effect [24]. Our LC grating showed the perfect continuity of the phase retardation and the switchable diffraction with the diffraction efficiency of 44% at ±1st orders. Due to the rotational symmetry, two symmetric grating configurations were produced depending on the input polarization direction with respect to the grating patterns. The experimental results were found to agree well with the theoretical predictions made in the transfer matrix formalism.

2. Transfer matrix formalism for diffraction

Consider the distribution of the LC director on the bottom substrate in one period of a continuous LC grating shown in Fig. 1(b). In the hybrid LC geometry used in this study, a schematic diagram of the LC director in the y-z plane was shown in the right of Fig. 1(b). Here, the LC molecules are aligned in alternating semi-circular (quadrant) forms and the director of LC molecules is then written as

Here, Λ means the period of the LC alignment patterns. In this configuration, the LC director (or the phase retardation) changes in an alternating semi-circular (quadrant) form along the x-axis.

The electric field D_m of the output beam through our diffraction grating device at the diffracted order m can be expressed as

Here, Δn is the birefringence of the LC, d is the thickness of the LC layer, and λ is the wavelength of the input beam. The two domains in Fig. 1(a), represented by Eq. (1) and Eq. (2), were used in the previous works using the photoalignment method [6,16–18,20]. Note that the disclination in the LC alignment between two adjacent domains exists in Fig. 1(a) while the continuous symmetry with no wall is shown in Fig. 1(b).

Let us calculate the electric fields of the output beam at several low orders of diffraction from Eqs. (3)-(5) in the hybrid alignment geometry which has the rotational symmetry with respect to the substrate normal. From the fact that under no applied voltage in the hybrid geometry, the phase retardation along the easy axis is λ/2 regardless of the thickness of LC layer, we have cos(Δndπ/λ) = 0, and sin(Δndπ/λ) = 1 in Eqs. (4) and (5). As a result, the electric fields at the diffraction orders of 0th, ±1st, and ±2nd are calculated as

It should be noted that the diffracted intensities at higher orders than ±2nd are negligibly small. Using Eqs. (6)-(8), the diffraction efficiencies at 0th, ±1st, and ±2nd orders are calculated as 40.5%, 50%, and 9%, respectively. In the presence of an applied voltage, the magnitude of the phase retardation itself changes, but the switching phenomenon among the diffraction orders is independent of the LC cell thickness.

3. LC cell with alternating semi-circular alignment

Figure 2(a) depicts the schematic diagram of our LC grating cell with the fully continuous, alternating semi-circular alignment of the LC. The fabrication process of the bottom substrate with alternating quadrant patterns by imprinting was shown in Fig. 2(b). A photo-curable polymer (NOA65; Norland) was first prepared on an indium-tin-oxide (ITO) coated glass substrate by spin coating at the rate of 2000 rpm for 30 s. The substrate was then imprinted with a master mold which was fabricated with poly(dimethylsiloxane) (PDMS; Dow Corning). The master mold had alternating quadrant patterns with a period of 60 μm. The imprinted substrate together with the master mold was exposed to ultraviolet (UV) light at the intensity of 100 mW/cm² for 10 min. The master mold was finally peeled-off. The top substrate in Fig. 2(a) was made of an ITO coated glass substrate on which the vertical alignment layer of polyimide (AL00010; JSR) was prepared by spin coating at the rate of 3000 rpm for 30 s. The alignment layer was baked at 180 °C for 90 min. The thickness of the LC cell was maintained using glass spacers of 9.5 μm thick and a nematic LC (ZLI-2293, Δn = 0.13, Δε = 10; Merck) was injected into the cell by capillary action at room temperature.

 

Fig. 2 (a) The schematic diagram of our LC grating cell and (b) the fabrication process of the bottom substrate with sinusoidal patterns by imprinting (d₁ = d₂ = 3 μm and h = 1 μm).

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Figures 3(a) and 3(b) show the optical microscopic image of the alternating semi-circular patterns on the bottom substrate and the image of the microgrooves observed with a scanning electron microscopy (SEM; XL30FEG, Phillips), respectively. The height of each microgroove was about 1 μm and the separation between two adjacent microgrooves was 3 μm.

 

Fig. 3 (a) The optical microscopic image of alternating semi-circular patterns on the bottom substrate. (b) The SEM image of the microgrooves in the semi-circular patterns on the bottom substrate.

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Figures 4(a) , 4(b), and 4(c) show optical microscopic textures together with the LC distortions in our continuous LC grating device under parallel polarizers in the presence of the applied voltage of (a) 0, (b) 25, and (c) 100 V, respectively. Under no applied voltage, the LC molecules were aligned along the microgrooves defined by imprinting on the bottom substrate and vertically on the top substrate as shown in Fig. 4(a). Depending on the angle between the easy axis (or optic axis) and the polarizer, the transmitted light intensity is continuously modulated within each domain as seen in Fig. 4(a). As the applied voltage increases, the LC molecules in the bulk are reoriented along the electric field direction and the grating effect from the alternating semi-circular alignment on the bottom substrate becomes to diminish as shown in Fig. 4(b). At 100 V, the grating patterns almost disappear as shown in Fig. 4(c). It is noted that due to the relatively large feature size (3 μm) of the microgrooves, the static background patterns were seen in Fig. 4 and the LC molecules would be less aligned.

 

Fig. 4 Optical microscopic textures together with the LC distortions in our continuous LC grating device under parallel polarizers in the presence of the applied voltage of (a) 0, (b) 25, and (c) 100 V.

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4. Diffraction properties

Let us examine the diffraction properties of our fully continuous LC grating device as a function of the applied voltage. Measurements were carried out using a He-Ne laser with the wavelength λ = 632.8 nm as a light source in conjunction with a charge-coupled device and a photodetector. The input beam was linearly polarized and the input polarization direction with respect to the grating patterns was set to be 90þ or 45þ. Figure 5(a) shows the experimental geometry for measuring the diffraction patterns in the input polarization direction of 90þ. The diffracted patterns under no analyzer, a crossed analyzer, and a parallel analyzer are shown in Figs. 5(b), 5(c), and 5(d), respectively. As clearly seen in Fig. 5(b), the diffraction orders of 0th and ±1st are dominant under no analyzer and the orders higher than ±2nd are negligible. Under a crossed analyzer, ±1st (odd) orders of diffraction are essentially observed as shown in Fig. 5(c) while under a parallel polarizer, the 0th and the ±2nd (even) orders are appeared as in Fig. 5(d). This indicates that 0th and ±2nd orders maintain the input polarization state while ±1st orders change it by 90þ.

 

Fig. 5 (a) Experimental geometry for measuring the diffracted patterns in the input polarization direction of 90þ. The diffraction patterns under (b) no analyzer, (c) a crossed analyzer, and (d) a parallel analyzer.

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Figure 6(a) shows the experimental geometry for measuring the diffraction patterns in the input polarization direction of to 45þ with respect to the grating patterns. The diffraction patterns under no analyzer, a crossed analyzer, and a parallel analyzer are shown in Figs. 6(b), 6(c), and 6(d), respectively. It should be noted that under no analyzer, the diffraction orders observed are exactly the same as those for the case in the input polarization direction of 90þ. Moreover, from Figs. 5(c), 5(d), 6(c), and 6(d), it was found that the rotation of the input polarization with respect to the grating patterns by 45þ corresponds to the interchange between the crossed configuration and the parallel configuration. In other words, for the case of the input polarization direction of 45þ, the polarization states of ±1st orders remain unchanged while those of 0th and ±2nd orders are rotated by 90þ. Such diffraction properties can be well described from Eqs. (6)-(8) where the geometric symmetry of the LC alignment in our LC grating is reflected. The non-vanishing diffracted intensity at 0th order in Fig. 5(c) or Fig. 6(d) may be attributed to the static background microgroove patterns and the LC alignment quality that originate solely from the microgroove pattern features but not from the intrinsic symmetry of the LC grating structure.

 

Fig. 6 Experimental geometry for measuring the diffracted patterns in the input polarization direction of 45þ. The diffraction patterns under (b) no analyzer, (c) a crossed analyzer, and (d) a parallel analyzer.

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We now describe the switching effect between the diffraction orders as a function of the applied voltage. Figures 7(a) and 7(b) show the diffracted patterns at the applied voltages of 0 and 100 V, respectively. Under no applied voltage, there exists ±1st orders of diffraction while at a high applied voltage of 100V, only 0th order appears. It is noted that the diffraction orders observed in the transverse-periodic structure [17] are appeared either positive or negative orders depending on the input polarization. In contrast, in our LC grating structure with high symmetry, both positive and negative orders of diffraction are always produced irrespective of the input polarization. The measured diffraction efficiency at ±1st orders was found to be about 44% which is somewhat less than the theoretical value (50%) calculated from Eq. (7). In fact, the anchoring energy on the homeotropic surface (the top substrate) plays an important role on the diffraction efficiency. However, in our case, the anchoring energy of the homogeneous, bottom substrate influences the diffraction properties more strongly since the anchoring strength achieved by the grooves of about 3 μm is relatively weak compared to the homeotropic case. In addition, the incidental defects involved in the LC alignment contribute to the difference. The voltage-dependent switching effect between the diffraction orders of 0th and ±1st is shown in Fig. 7(c). At about 15 V, the three diffracted intensities are nearly identical. On further increasing the applied voltage above 15 V, the diffracted intensity at ±1st order decreases and eventually vanishes while those at 0th orders increase and become saturated. Note that the driving voltage, being somewhat high, can be simply reduced with decreasing the voltage drop across the imprinted alignment layer, meaning that the thinner the imprinted alignment layer is, the lower the driving voltage is. The switchable features of our LC grating device will be useful for separating an original image into two different types of diffractive images.

 

Fig. 7 Diffraction patterns under a parallel analyzer in the input polarization direction of 45þat (a) 0 V and (b) 100 V. (c) The normalized intensities at 0th and ±1st orders as a function of the applied voltage. The insets in (c) show the diffraction patterns at 8 V and 20 V.

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5. Concluding remarks

We demonstrated a new type of the LC grating device possessing the continuously graded phase retardation within an individual grating pattern. The fully continuous phase retardation was achieved using the semi-circular alignment of the LC on a micro-grooved substrate fabricated through an imprinting process. Our continuous LC grating with rotation symmetry exhibited the switching effect between the diffraction orders of 0th and ±1st. The diffraction efficiency at ±1st orders was found to be about of 44%. The continuous grating architecture with high symmetry presented here will be applicable for devising a variety of optical elements including a tunable beam splitter and a switchable diffractive component.