P HASE R ETARDATION OF C ONVEX AND C ONCAVE M ODES

The interference pattern is one of the mostly used methods to measure the phase retardation of LC cells. To evaluate the optical properties of GD-LC Lens, we investigated the phase retardation by observing the interference pattern between the ordinary and extraordinary rays passed through the lens cell under crossed polarizers. The rubbing direction of the lens cell is oriented at 45° with respect to the fast axis of the linear polarizer. Two images of GD-LC Lens driven in convex and concave modes are shown in Figure 4-13 (a) and (b) respectively.

In convex mode, as shown in Figure 4-13 (a), GD-LC Lens yielded a phase retardation

approximating to that of ideal lens with 1.6mm effective lens aperture. This result was coherent to that as shown in Figure 4-13, according to the following relation:

_{≅ ∙}

2∆ ∙ (4-8)

where r, and f donate the aperture radius, and focal length respectively. The central grounded electrode with two marginal controlling voltages drives GD-LC Lens in concave mode. The results presented the phase retardation is much closer to that of an effective ideal lens, as Figure 4-13 (b) illustrated.

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(a)

(b)

Figure 4-13 The phase retardation of GD-LC Lens operated in (a) convex, and (b) concave modes. By the different operating arrangement, both convex and concave lenses were achieved.

Table 5 shows the summary of comparison between GD-LC Lens and conventional LC lens with internal electrodes. Through the result, GD-LC Lens dramatically improved 86.0 % operating voltage and can be only driven by less than 5Vrms which has been within the range of normal driver IC.

Table 5 The comparison of focusing time and corresponding operating voltage. The result shows GD-LC Lens significantly improved driving voltage and only can be only driven by less than 5Vrms.

Comparison LC cell Gap Operating Voltage

GD-LC Lens 60 μm (E7) 2.35 Vrms

Conventional LC Lens with

Internal Electrodes 60 μm (E7) ~25Vrms

4.5 Conclusion

GD-LC Lens was proposed exhibiting ultra-low operating voltage for the structure with 60um LC cell gap and 2mm lens aperture. The focal length was voltage and frequency tunable driven by less than 5Vrms. Compare to the conventional result, which usually requires extremely high driving voltages, GD-LC Lens brought the applications utilizing LC lenses to be feasible and practical. Furthermore, both convex and concave effective lenses were achieved by the structure of GD-LC Lens with different operating voltage. Therefore, not only the structure of GD-LC Lens intrinsically solved the issue of high driving voltage in the applications of LC lenses but also the concept of gradient driven would be potentially used for future form of LC lenses.

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Chapter 5.

Over-drive Method for Fast Focusing Liquid Crystal Lens

5.1 Introduction

As mentioned in previous sections, Liquid crystal (LC) is wildly used in many optical applications because of the properties of high sensitivity to electric field and easy control. For lens applications, LC lens owns a very unique property that the focal length is electrically tunable without mechanical moving and shape changing. This advantage made applications for focal length tunable and switchable system slimmer and more compact. However, for lens application, optical power of LC lenses is relatively smaller than that of conventional lenses.

Thus, thickness of the LC layer is usually required few tens or over hundreds of micrometers to generate high optical power rather than that of few micrometers in LCDs. The large thickness results in slow focusing time. For example, as the relation of optical power, P, shows

^2∆ (5-1)

P is dependent on the difference of reflective index between that at border and center, ∆ , cell gap, d, and radius of aperture radius, r. The optical power is difficult to achieve 25m^-1, as the LC lens has 1mm aperture radius and 60um LC cell gap (Merck nematic E7), whose F/# is large as 20. As ∆ is constrained by the index of material and only has a small region of variation due to structures of LC lenses, the cell gap is usually designed much thicker.

According to rising time, of LC devices, larger cell gap results in slow focusing time which is proportional to d from relaxing state to focusing state, as Equation (5-2) shows.

/

/ 1 (5-2)

where is the function of cell gap d, rotational viscosity , elastic constant K, threshold voltage , and applied voltage V. The slow focusing time makes applications unfeasible and unpractical, such as lens head in mobile devices which is requesting very fast auto-focusing or zoom functions, and 3D switchable displays having large numbers of viewing zones. To overcome slow response of LC cells, over-drive is employed for accelerating the response times in LCD industry. By optimized the over driving voltages and switching to target operation, the LC response time can be much reduced.

5.2 Over-drive Method for LC Lens

In this chapter, we propose Over-drive (OD) method for reducing the focusing time of LC lenses. Different from that for LCDs, the OD method investigated focusing behavior of LC lenses to determine the timing for switching. The appropriate operation can significantly reduce the focusing time.

For a test sample, we prepared a homogeneous LC lens structure with external driving electrodes which were patterned upon the glass, as shown in Figure 5-1(a). The electrodes were driven by OD voltage (~65Vrms) which was higher than that for particular focal length (~45Vrms for 5cm). The variation of normalized focusing intensity with time which was used to indicate the variation of focusing performance, as illustrated in Figure 5-1 (b), was measured by GENTEC Beamage Series CCD sensor placed at the focal length of the LC lens illuminated by a laser source (λ = 632.8nm). The initial focusing shows the OD voltage generated an instantaneous but unstable focus. The following focus ability was immediately decay and went to a stable state. This phenomenon was investigated by Fluorescence Confocal Polarizing Microscopy (FCPM) method [67] to analyze the exact director of LC molecules rotated with time, whose details investigations of FCPM for LC lens will be discussed in following section. The analysis showed gradient electric field by the OD voltage

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molecules rapidly and continually rotated to form another phase retardation that generated the focus in the stable state which had similar focusing intensity with the initial focusing, although the two forms of LC directors were totally different.

(a)

(b)

Figure 5-1 (a) General structure of homogeneous LC Lens driven by marginal over-drive voltages, and (b) The curve of focusing response time to normalized intensity on CCD sensor for a particular focal length. The over-drive voltages yielded a initial focusing as shown the first peak.

To perform the process of OD method, the switching operation was examined. In the experiment, the same test sample was driven by the OD voltage (~65Vrms) and then switched to 45Vrms for 5cm focal length while the focusing intensity was almost achieving that of the initial focusing. The variation of intensity is shown by the solid curve in Figure 5-2. From the result, the intensity curve was maintained and further increased to around 90% starting from

the position of the initial focusing instead of intensity decay, although there was tiny variation at the beginning of the switching. This result indicated the initial OD voltage rapidly forced the LC molecules to form a distribution which was approximate to that for optimized focusing.

The switching operation, at this time, can further deform the LC orientations, and finally keep to stable focusing. Comparing with general operation which is shown by the double-line in Figure 5-2, the directly applying ~45Vrms to the electrodes for 5cm exhibited a very slow increasing of focusing intensity. To achieve the same focusing performance, applying OD method can significantly improve the focusing time from around 25.3sec to 7.1sec in this experiment, as we defined the focusing time was when the intensity stably achieved 90%.

As mentioned previously, the director of LC molecules was rapidly rotated by the OD voltage.

In fact, the timing of switching affected the result of focusing behavior. As the OD voltage was switched before the time of the initial focusing, meanwhile the LC directors have not at the orientations for optimized focusing; the maximum focusing intensity was achieved slowly due to the driving voltage was reduced. The similar situation was also occurred if the switching was performed after the initial focusing. The LC directors exceeded the orientations for maintaining the focusing instead rotating back due to the reduced driving voltage. The results are shown in Figure 5-3. The most improved result was obtained when the switching was performed at the time approximated to the initial focusing.

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Figure 5-2 The variation of focusing intensity with time driven by OD method and general driving was measured by CCD sensor.

Figure 5-3 The results of OD method by different switching time. The switching at the time approximated to the initial focusing can obtained the shortest focusing time.

5.3 3-Dimentional Imaging and Analysis for LC Lenses by Fluorescence Confocal Polarizing Microscopy

Fluorescence Confocal Polarizing Microscopy is a confocal microscopy system, whose photo detector would detect the fluorescent light intensity emitted from the dye molecules doped in the LC cell [68]. The transition dipole of the dye molecule is along the direction of the rod-shaped LC molecules, which means the tilt angles of the LC molecules are along the adjacent dye molecules. As shown in Figure 5-4, while the FCPM emits a laser with a linear polarization, P, the probability of the photon absorption is proportional to cos α , where α is the angle between the transition dipole of the dye molecule and the polarization P of the laser. The probability of the fluorescent photon reaches to the detector through almost the same polarization is proportional to cos α if we assume the time of emission is much shorter than that of rotation of LC molecules [69]. The function of the pinhole is to filter out the fluorescence emitted from the dyes out of the region of focus. Therefore, the LC cell can be layer-by-layer scanned. The relation between the incident light intensity and the fluorescent light intensity of each scanning is

To investigate the phenomenon of instantaneously focus, FCPM was utilized to reconstruct the 3D orientation of LC layer. According to the fluorescent intensity of each LC layer, the orientation distribution was derived, and scales of fluorescent intensity, IFCPM, to LC tilt angle, α, is shown as follows:

(5-3)

where is the intensity of laser light source, and is the intensity of detected fluorescence emitted from the excited dye molecule. By the calculation of Equation (5-3), the tilt angle of LC molecules in LC cell can be observed.

Through the focusing response time, optical path of the LC layer was calculated. The

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first peak, the phase retardation cannot yield a superior focusing on the CCD sensor. The curves of initial focusing and stable state also matched the corresponding focal length.

Figure 5-4 The diagram of the fluorescence confocal polarizing microscopy system.

5.3.1 3-D Imaging of LC lenses

To observe the 3-D image of LC orientations, a circle-electrode LC lens with 60um LC layer was prepared. The LC material used in the device were Merck nematic LC (E7) whose extraordinary refractive index ne = 1.7472 and ordinary refractive index no = 1.5271. The alignment of the LC lens was anti-parallel rubbing and placed along the polarization of incident laser, whose wavelength is λ=488nm, as Figure 5-5 shows. The laser light source scanned the LC cell from the 0um layer to the 60um layer with 5-um vertical resolution, and the scanning area was 1.4mm 1.4mm when utilizing 10X objective. The detected intensity of excited fluorescence was used to represent the tilt angle of LC molecules. The scanned result before applying driving voltage is shown in Figure 5-6(a). The fluorescence intensity detected was uniform indicates almost the same director of LC molecules of the layer. The uniform result also shows the phase retardation was not different over the layer. After applying 20Vrms on the electrode, the LC profile started to be deformed and the fluorescence intensity became central symmetrically darker at the border due to the larger tilt angle of LC molecules.

We choose three of the layers to illustrate this, as shown in Figure 5-6(b-d).

Figure 5-5 The setup of measure by FCPM.

Figure 5-6 (a) The scanned result before applying driving voltage, and (b) ~ (d) that of scanned layers after driving.

In Figure 5-6 (b-d), we can notice the distributions of fluorescence intensity far away from bottom layer were more spherical that is because the different structure of top and bottom electrodes. However, the centers of the bright part are not coaxial, as shown in Figure 5-6 (c-d). This result not only indicates defect of fabrication but also the errors of measurement. The errors may be caused by non-horizontal scanning due to the placing of samples and the non-uniform of the doped dyes. According to Equation (5-3), the ratio of the intensity with and without driving voltage is equal to , as following:

0 (5-4)

To calculate tilt angle, α, Equation (5-4) was more useful rather than Equation (5-3) due to limitation of dynamic range of the FCPM. The low dynamic range resulted in the initial

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of the LC lens is known, we can calculate the effective index of each layer by Equation (2-8). By digital image processing, contours were plotted to indicate refractive index and tilt angle, as shown in Figure 5-7. As the contour shows, the index distribution of each layer was around 1.58 to 1.68, and ∆n was only 0.1, which was not achieving the maximum

∆ ≅ 0.22 of E7 material due to the deformation of the LCs at the lens center and not vertical tilt angle of LCs at the border. Actually, in many structures designed for LC lenses, the ∆n is difficult to be achieved. Therefore, the cell gap of LC layer is usually thicker, assuming the high optical power is required. In Figure 5-7, we can find the lens profile is not symmetrical, as mentioned previously. The contours are more broken in the top and bottom layers which were closed to the boundaries, while the layers near the middle layer had more smooth distribution. In order to image the 3-D profile of LC molecules, rod-like LC modal were used to simulate the tilt angles of each layer according the contours, as shown in Figure 5-8. The variance of LC directors was around 20 to 80 degree. The smaller degree at the center indicates the central area of the lens has larger phase retardation which resulted in a convex lens effect.

Figure 5-7 The contours indicating the refractive index and LC tilt angle of each layer.

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Figure 5-8 The 3D diagram of LC profile.

5.3.2 Phase retardation

To verify the optical properties of the scanning results, the scanning layers were used to calculate phase retardation. Typically, f-number (F/#) of LC lenses is relative large compared to conventional lenses. Thus, paraxial approximation can be employed for the propagation of rays in LC lenses to calculate the phase. If we assume light traveling straightly through the LC lens, the phase retardation of the LC lens can be approximated to

² ∙ 2

∙ , , (5-5)

where d is thickness of the LC layer, and is wavelength of the laser. Optic axis of the lens was set to be along z-axis. From Equation (5-5), the integral result of phase retardation, φ, from layer 0-um to 60-um was calculated, as shown in Figure 5-9 (a). Figure 5-9 (a) clearly indicated the absolute phase retardation at the center of the lens was around 371 π, and OPD between the center and border was around +12 π. The phase retardation was also measured by fringing pattern method, as shown in Figure 5-9 (b). Within the scanning area, the relative phase was also around 12 π, but the signs cannot be directly evaluate.

To compare the accuracy of measured results, focal length corresponding to the phase retardations was investigated. Under the 20 Vrms driving voltage, the LC lens had maximum focal intensity at 5cm distance away from the lens when illuminated by a laser source ( = 632.8nm). The focal intensity was measured by GENTEC Beamage Series CCD sensor placed at the focal length in back of the LC lens. The ideal distribution of phase retardation,

, for lenses with focal length, f, can be represented by following equation:

²

2 (5-6)

where r is aperture ray height in cylindrical coordinate system. Equation (5-6) indicates the form of the phase retardation, , is a parabolic curve if no spherical aberration occurs and light focus at a single point. Compare the results by capturing the cross section of profiles in Figure 5-9 (a) and Figure 5-9 (b), the curves, as well as the ideal parabolic curve for 5cm focal length, are shown in Figure 5-10. These tree curves are almost matched indicates the measured results of FCPM method and fringing pattern were coherent, and also matched with the measurement of focal length.

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(a)

(b)

Figure 5-9 The phase retardation calculated by (a) FCPM method and (b) measured by fringing pattern approach.

Figure 5-10 The cross sections of phase retardation for 5 cm focal length by FCPM method and fringing pattern approach, as well as the ideal curve.

5.4 Experiments and Optimization of Over-Driving Voltages

Not only the switching time but also the values of OD voltage affected the result.

Optimization for OD voltages was analyzed by applying different OD voltages to the electrodes. Typically, the focusing time was improved when the OD voltage was increased, as Figure 5-11 shows. The dash line shown in the result indicates directly applying the 45Vrms without OD method which yielded 25sec focusing time, as illustrated in Figure 5-2. As we applied OD method and increased the OD voltage to exceed 45Vrms, the focusing time started to be improved. Of course, the improvement cannot be obtained if the OD voltage is lower than that of general operation. In this experiment, the improvement was saturated as the over-drive voltage was increased to exceed 65Vrms, which means the extremely high

304

67 

Figure 5-11 The result of relation between focusing time and OD voltages. Improvement was saturated as the voltage was larger than 65Vrms.

In this experiment, GD-LC Lens was examined, as structure shown in Figure 5-12(a).

The focusing time was dramatically reduced down to 0.2sec by operating 15Vrms OD voltage.

Figure 5-12 (b) shows the relation of OD voltages to focusing time. This result also showed OD method is necessary for different structures if the fast focusing was required. A comparison of focusing response time, over-drive voltage, and stable driving voltage are listed in Table 6. Two main LC lens structures mentioned above driven by Over-drive Method with 2mm aperture size and 60um LC cell gap were discussed. The LC model was Merck nematic LC (E7). Although the external structure had slower focusing time due to insertion of high K glass, by Over-drive Method, the focusing time was effectively improved from 23.5sec to 7.1 sec. For GD-LC lens, the focusing time was significantly reduced down to 0.2sec with only 15Vrms over-drive voltage.

Table 6 The results and comparisons of two main structures of LC lenses driven by OD method. In the internal structure part, focusing response time was reduced to 0.2sec by 15Vrms over-drive voltage.

External Electrodes GD‐LC Lens

Over‐drive Voltage OD 65Vrms 15Vrms

Stable driving Voltage 45Vrms 5Vrms

Initial focusing time 23.5sec 12.5sec

Initial focusing time 23.5sec 12.5sec