1. Introduction

Liquid crystals (LCs) exhibit attractive features of large birefringence and low driving voltage; consequently, they are applicable not only to displays but also to electrically-controllable optical devices such as spatial light modulators [1,2], lenses [3], and gratings [4,5] One example of a practical application of LC optical devices is the optical pick-up system using LC aberration correction devices [6] for compact disk and digital versatile disk players. Furthermore, the mechanically moving part becomes unnecessary by introducing the LC variable-focus lens; thereby, the high resistance to a mechanical shock of the optical system is expected.

The focusing function in general LC lenses is obtained by spatially distributing the retardation Δn_effd, were Δn_eff is the effective birefringence in the LC layer and d is the cell gap (LC layer thickness). In earlier LC lenses the combination of a plano-concave lens and a flat glass plate were employed to distribute d; however, the thick LC layer causes the long response time and large light scattering. Other way to distribute the retardation is the utilization of the nonuniform electric field generated by hole-patterned electrodes [7]. To obtain the lens function, the voltage application is constantly necessary in this type of lens.

In this article, we propose an LC lens consisting with a spatially distributed pretilt angle, in the in-plane direction, by utilizing a microrubbing process [8]. This LC lens exhibits a focusing function even without a voltage application and its focal length is electrically controllable. So far, the pretilt angle is demonstrated to be reduced by rubbing polyimide films for a homeotropic orientation using a tiny metal ball [8]. The pretilt angle susceptibly depends on a lot of rubbing conditions such as the pushing load of the ball, the number of rubbings, the moving speed of the ball, and so on; thus, accurate control of the pretilt angle have not been attained in the past trial. We investigate a new controlling parameter of the density of rubbings. It has been confirmed that the retardation can be controlled by the area ratio of homeotropic and homogeneous orientations by a photoalignment investigation [9,10], this approach is applicable to electrically controllable LC optical devices [10]. The variation of the rubbing density is equivalent to such the variation of the area ratio of different orientation domains. The microrubbing exhibits the high thermal stability of the LC molecular alignment and the strong surface anchoring compared with the conventional photoalignment.

The photolithographic method is one effective way for forming microscale patterns of LC molecular orientations and it generally requires a photoresponsive alignment film [11]. On the other hand, some polyimide films for vertical-alignment-mode LC displays can available for the pretilt controlling by microrubbing and the LC molecular orientation is stable with the time elapsed, like a conventional rubbing process. Furthermore, an arbitrary azimuthal angle of the orientation direction can easily be obtained, while the arbitrary azimuthal angle is not easy to be obtained by photolithographic methods, because some times of irradiations are necessary.

Furthermore, the microrubbing enables the microscale distribution of the rubbing density unlike the conventional rubbing. Since the spatial resolution of the surface treatment by microrubbing depends on the size of the stylus tip, even a nanoscale modification can be achieved using an atomic force microscope (AFM) [12]. The feasibility of this precise surface treatment is a feature of microscale and nanoscale rubbing against the photolithographic method, of which the resolution is restricted by the wavelength of light.

So far, various variable-focus lenses have been proposed as liquid lenses with an elastic film [13], electrowetting lenses [14], and dielectric lenses [15]. Unlike these lenses, the proposed LC lens utilizes the electrical change of the refractive index that gradually varies in the LC layer. Since there is not clear boundary of the refractive index change, no Fresnel loss arises. Furthermore, it is advantage that the proposed method does not limit the lens size: not only micrometer scale lens but also millimeter scale lens.

In the experiment, the rubbing density is accurately controlled by introducing the microrubbing technique using a tiny diamond stylus. First, the cylindrical lens with a rectangular rubbing area is demonstrated, and the spatial distribution profile of the rubbing density for good focusing properties is discussed. Next, the rectangular lens with a square rubbing area is tried; the fundamental focusing properties are investigated.

2. Pretilt control by rubbing density

In the cell fabrication, polyimide films were spin-coated on pre-cleaned glass substrates with indium-tin-oxide layers: SE150 for homogeneous alignments and SE7511L for homeotropic one are used (provided by Nissan Chemical Industry, Ltd., Japan). The both polyimide films were baked at 200°C for 2 h. The microrubbing was carried out only on the SE7511L-coated substrate and the SE150-coated one was uniformly processed by a conventional rubbing machine with a rubbing cloth. Each empty cell was constructed by combining the SE7511L- and SE150-coated substrates, inserting glass rod spacers. The edge of the filled cell was sealed using an epoxy resign, following a nematic LC (4-cyano-4’-pentyl biphenyl: 5CB) was injected in to the empty cell.

In the microrubbing, the polyimide film was rubbed using a tiny diamond stylus (the curvature of the radius of 2.5 μm) as shown in Fig. 1(a) . The spring was fixed and the movement of the XYZ-stage was controlled by a personal computer, where the moving speed was 0.2 mm/s. The pushing load was controlled by monitoring the spring deformation using a laser displacement sensor. The pushing load during the rubbing is 0.2 mN that causes the rubbed line thickness of 1.5-2.0 μm, according to the preliminary experimental result of AFM observations. The rubbed line thickness is reduced with decreasing the load. However, the submicron-scale rubbed line was not obtained even under the extremely lower load because the stylus tip jumps and is intermittently apart from the polyimide surface during the scanning. The rubbing density was controlled by the pitch of scans p(x) defined as:

 

Fig. 1 (a). Experimental setup for microrubbing. (b) Rubbing pattern of an LC cylindrical lens, where rectangles denote LC molecules. The tilt angle of an LC director on a polyimide film for homeotropic alignment varies depending on the magnitude of rubbing density.

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where p_e and p_c is the pitches at the edge and center of the lens, respectively. r is half the lens size and α is the shape parameter, which is highly related to the retardation distribution profile of the lens. The retardation rises with an increase in the rubbing density, which can be defined as 1/p(x). The rubbing pattern for obtaining an LC cylindrical lens is shown in Fig. 1(b). Note that the microrubbing was made only on the surface of the 7511L-coated substrate and that of the SE150-coated substrate was processed by the conventional uniform rubbing. The rubbing density is low at the edge of the lens, leading almost the hybrid orientation, while, since the rubbing density is high at the center, the LC molecules tend to orient homogeneously.

In our previous work, it was revealed that the pretilt angle decreases with increasing the frictional work of rubbing in a side-chain-type polyimide film for homeotropic alignment [16]. Furthermore, it was pointed out the possibility that the polar anchoring strength increases with increasing the orientation order of the main-chain of rubbed polyimide [8]. Therefore, it is likely that the polar anchoring in the closely rubbed area is stronger than that in the roughly rubbed one when the scan pitch spatially varies. Although the polar anchoring is one important factor that affects an LC molecular orientation, the effect of the spatial elastic relaxation in an LC bulk is also nonnegligible factor. In the proposed LC lens, the surface tilt angle in the rubbed area microscopically slightly modulated in the in-plane direction, depending on the polar anchoring. However, the bulk LC molecular orientation, which is significant for optical properties of LC lenses, depends not only on the anchoring nature but also on the elastic one of LC (the bulk orientation tends to be uniform). Since the influence of the spatial elastic relaxation is not easy to be theoretically investigated, we experimentally examined the relationship between the pretilt angle (averaged in the in-plane direction) and the ratio of rubbed and entire areas [defined as t/p, where t is the rubbed line thickness (≈1 μm) and p is the scan pitch], as shown in Fig. 2 . The pretilt angle was determined by comparing experimental and computer-simulated retardation data [16]. From Fig. 2, it is revealed that the pretilt angle is constantly almost 0° when the scan pitch is narrower than the rubbing line thickness (t/p>1). This means that the each rubbed line is overwrapped and the anchoring strength is very strong. On the other hand, the pretilt angle drastically increases with increasing the scan pitch. Particularly, it almost saturates at the pitch of 4 μm (t/p = 0.25). Therefore, we concluded that the appropriate variation range of the scan pitch is 1-4 μm.

 

Fig. 2 Relationship between the pretilt angle and area ratio t/p, where t denotes rubbed line thickness and p is the scan pitch. The spacer diameter of examined LC cells was 20 μm.

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3. Liquid crystal cylindrical lens

Figures 3(a) and 3(b) show the polarizing microscope images of the fabricated LC cylindrical lens at 0 V and 1.34 V (1 kHz, a square wave), respectively, where α = 2, r = 100 μm, p_e = 4 μm, and p_c = 1 μm. The cell thickness estimated by a retardation measurement was ~46 μm (the spacer diameter used was 40 μm). The down-pointing arrows indicate the position of interference fringes. To obtain clear fringe patterns produced by the interference between ordinary and extraordinary rays, a He-Ne laser (λ = 632.8 nm) was used. The fringe pattern corresponding to the spatial retardation distribution is generated even without a voltage application [Fig. 3(a)]. As the voltage increases, the fringe pattern moves toward the center of the lens and the number of fringes is reduced as shown in Fig. 3(b). Finally, the fringe pattern fully disappears under a sufficiently high voltage application. This moving behavior of the fringes indicates that the retardation at the center is larger than that at the edge; hence, this LC cylindrical lens functions as a variable-focus convex lens. From Fig. 3, it is also seen that the pinstriped pattern near the edge of the lens owing to the wide p_e of 4 μm. Since the rubbed line thickness was estimated as 1.5-2.0 μm, one can optically recognize the difference in the retardation between the rubbed and unrubbed areas increases at the edge of the lens.

 

Fig. 3 Interference fringe patterns observed by a polarizing microscope (crossed polarizers) at (a) 0 V and (b) 1.34 V. The arrows denote the position of the fringes. The cell gap of the LC cylindrical lens is 47 μm.

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The necessary number of scans for α = 2, r = 100 μm, p_e = 4 μm, and p_c = 1 μm can be calculated by Eq. (1) to be 122. Since the processing time per scan was around 5 s, the rubbing process for the cylindrical lens (the rubbed area was 200 × 500 μm²) was not so time-consuming (≈10 min). Therefore, the reduction of the orientation ability by a long-time atmospheric exposure of the rubbed surface was negligible. The long processing time becomes a problem to fabricate arrayed devices; however, the use of an array of stylus will overcome that problem

The spatial distribution profile of the retardation can be obtained by the interference fringe patterns as shown in Fig. 4 : the retardation value at the center of each dark and bright fringes is plotted and the solid lines denote least-square-fitted parabolic curves. The retardation distribution profile highly reflects the distribution profile of the rubbing density determined by α. For instance, it seen that the retardation varies linearly in the x direction for α = 1; consequently, the retardation distribution deviate from the fitted parabolic curve. On the other hand, the retardation distribution profile for α = 2 almost shows a good accordance with the fitted parabolic curve, indicating less aberration. The aberration is most reduced when α = 2, independently of the applied voltage; namely, the focal length is varied while maintaining the aberration small.

 

Fig. 4 Retardation distribution profiles for (a) α = 1, (b) α = 2, and (c) α = 4. The cell gaps derived from the retardation at the edge of the lens is (a) 51 μm, (b) 47 μm, and (c) 47 μm (Δn = 0.174).

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It is further found that the retardation difference between the center and edge decreases with increasing the applied voltage, indicating the increase in the focal length. The maximum retardation difference ΔR_max corresponding to the shortest focal length is theoretically achieved assuming that the LC molecular orientations at the center and edge are homogeneous and hybrid, respectively. Since the retardation in the unrubbed area (the hybrid orientation) is R = 4.1 μm, the cell gap is evaluated to be d = 47 μm for the birefringence of Δn = 0.174 (5CB); hence, ΔR_max is estimated to be 4.1 μm, but the experimentally obtained value is lower (3.2 μm) at 0.58 V for α = 2 (0.72 V for α = 4). Although the voltage of 0.58 V is lower than the threshold voltage of the Fredericks transition (~0.8 V for 5CB), the LC director at the center of the lens somewhat tilts up even though almost the homogeneous orientation, leading to the decrease in the retardation. One possible reason is the elastic in-plane interaction to the surrounding hybrid domain.

4. Liquid crystal rectangular lens

Based on the experimental result of the cylindrical lens, we propose an LC lens with a square area produced by crossing two unidirectionally rubbed substrates as shown in Fig. 5 . The polyimide for the homeotropic alignment (SE7511L) was coated on the both substrates and then the microrubbing was made. The rubbed substrates were combined with the rubbing direction crossed. This lens geometry will be convenient for the fast fabrication of a lens array. The total rubbing density of the both substrates increases toward the center of the lens area. The LC molecular orientation states (denoted by cylinders) at different positions are also shown in Fig. 5. The retardation in the orientation D is the highest while that in A is the lowest, those in the orientations B and C are, furthermore, the same. Namely, the retardation spatially varies depending on the sum of the rubbing densities of the two substrates. Note that the sufficient cell gap is necessary so that the ordinary and extraordinary rays hardly interfere; thus, the focusing function can be obtained only for the extraordinary rays.

 

Fig. 5 Schematic diagram of the proposed LC lens with a two-dimensional focusing ability. The wide lines crossed orthogonally denote the rubbing trajectory. Four LC molecular orientations at the different positions are also depicted.

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Figures 6(a) and 6(b) show the polarizing microscope images at 0 V and 0.69 V, respectively, where d = 40 μm (the diameter of the spacer) and the rubbed square area is 200 × 200 μm². It is seen that the longitudinally and laterally symmetric fringe patterns are generated at the square area and the number of fringes decreases with increasing the applied voltage. The shape of the fringe patterns becomes circular toward the center of the lens, while in the outer area it takes a diamond shape leading to the deterioration of the focusing properties. Thereby, the available area for a lens function is smaller than the rubbed square area.

 

Fig. 6 Polarizing microscope images of the proposed LC lens (a) at 0 V and (b) at 0.69 V. The cell gap is 40 μm, and the square lens area is 200 × 200 μm². The rubbing parameters are the same as that of the cylindrical lens: α = 2, r = 100 μm, p_e = 4 μm, and p_c = 1 μm.

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The LC molecular orientation pattern exhibits high stability for the time elapsed, as well as the conventional rubbing process; that is, the fabricated pattern was maintained even after a two-years-storage at room temperature. To investigate the endurance of the elastic stress, low frequency voltage (10 Hz, 2 V, and a rectangular wave) was applied across the LC cell. The cell exhibited a visible dynamic switching behavior of transmitted light (observed by a polarizing microscope) for 24 h (exceed 80000 cycles). As a result, any deterioration of the orientation pattern was not confirmed after the examination.

The focal length f as a function of the applied voltage is shown in Fig. 7 , where f was defined as the distance between the LC layer and focal point. It is revealed that the focal length dramatically rises when the applied voltage exceeds the threshold voltage of the twisted nematic 5CB cell (~0.8 V). Other interesting find is that the focal length is slightly depressed in the low voltage region below 0.8 V. Since the tilt angle of the LC director at least on one of the two substrates is inclined against the substrate plane in the outer area (orientations A, B, and C), the LC molecular begins to move below 0.8 V, resulting in the slight fall of the focal length due to the increase of the retardation difference between the center and edge of the square lens area. Furthermore, in the high voltage region one problem arises; that is, Mauguin’s condition tends to be unsatisfied, leading to a worse focusing function. The clear focusing spot was recognized in the voltage range up to 1.4 V.

 

Fig. 7 Focal length as a function of the applied voltage, where the focal length was measured as the distance from the LC layer. The spot profile at 0 V is also depicted.

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The inset in Fig. 7 shows the focusing spot profile at 0 V. The full width of half maximum (FWHM) is evaluated from the image to be 3.5 μm. FWHM can be theoretically calculated as λf/nD, where λ is the wavelength, n is the refractive index of the substrate, and D is the lens diameter. The effective lens diameter (so that the fringe pattern is circular) is much smaller than the rubbed width of 200 μm as shown in Fig. 4. Assuming the reasonable diameter as D = 70 μm, we get FWHM = 3.6 μm [λ = 632.8 nm, f = 0.6 mm, and n = 1.5 (the focal point is present in the glass layer)], which is close to the experimental result. This estimation supports that the good focusing function is achieved within the effective diameter of 70 μm (35% of 200 μm). It is expected that the bulk LC molecular orientation depends both on structural and material parameters such as the rubbed area, cell gap, dielectric constant, elastic one, and so on. Therefore, the effective lens diameter possibly increases by optimizing those parameters, resulting in the improvement of the fill factor, which is significant in the case of arrayed devices.

The imaging behavior of the fabricated LC lens was examined using an optical microscope. Figure 8(a) shows the experimental setup, where the LC lens and object plate was set on the sample stage. The wavelength of the incident light was 515 nm (FWHM = 10 nm). Since the lens size is smaller than the size of the object (inversed “A”), the object was located far from the lens (8 mm), where the distance was maintained using glass plates. Figure 8(b) shows an object pattern, which was prepared by a photomask. Figures 8(c)-8(g) show produced images at different applied voltages, where the height of the sample stage was adjusted so that the clearest image could be observed. No image was observed when the voltage was higher than 1.4 V. In the case of 0.4 V, two images under crossed and parallel polarizers were taken as shown in Figs. 8(d) and 8(e), respectively, where the incident polarization direction is parallel to the rubbing direction of the incident-light-side substrate. Furthermore, no image was obtained when the incident polarization direction is perpendicular to the rubbing direction of the incident-light-side substrate regardless of the presence of the analyzer. Therefore, the proposed LC lens functions only for the incident extraordinary light and the linearly-polarized state is almost maintained after passing through the LC cell.

 

Fig. 8 (a) Experimental setup for examining imaging properties of an LC lens. (b) Object pattern and produced images at (c) 0 V, (b) 0.4 V (crossed polarizers), (c) 0.4 V (parallel polarizers), (d) 0.6 V, and (d) 1.0 V. Wavelength of incident light was 515 nm (FWHM = 10 nm).

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It is seen that the size of the image “A” varies with applied voltage. Here, note that the distance between the lens and object (8 mm) is longer than the focal length (<2 mm), the magnification is less than unity (0.17 at 0 V). The magnification temporally decreases to 0.12 (0.6 V) with increasing the applied voltage, and then it increases again. This voltage dependence of the magnification is qualitatively consistent with that of the inversed focal length (1/f). Furthermore, it is revealed that the sharpness of image is different depending on the applied voltage. The clearest image was obtained at 0.6 V and the image was much blurred at 0 V. The sharpness of image is nothing to do with the magnification and is, however, highly related to aberration in the LC lens. It is considered that the aberration is most reduced at 0.6 V.

5. Conclusion

A technique to obtain the spatially distributed pretilt angle by changing the density of the microscale rubbing was proposed and it was applied to LC variable-focus lenses. The almost parabolic retardation profile was obtained when the distribution profile of the scan pitch is also parabolic, but the detailed optimization of the shape parameter of the scan pitch is required for a perfect aberration elimination. The unique variable-focus lens with a square rubbing area was proposed, and the controllable focal length range was measured to be 0.5-1.8 mm. The longer focal length can hardly be obtained because the twisted orientation disappears with raising the voltage. From the discussion of the polarization microscope image and the focusing spot width, the effective lens diameter was estimated to be 35% of the width of the rubbed area. The improvement of the effective lens diameter by optimizing the cell parameters (such as the rubbed area width, cell gap, and so on) is needed.

In this study, the proposed method was adopted to refractive optical devices. On the other hand, it is expected to be extended to the diffractive ones such as Fresnel zone plate, because only the binary control of the LC molecular orientation is required. Furthermore, interesting electrically-tunable optical devices combining refractive and diffractive function may be created. Such the hybrid device may be demonstrated for the purpose of compensating for chromatic aberration.