CHAPTER 1. INTRODUCTION
1.5 M OTIVATION AND O BJECTIVE OF T HIS T HESIS
Figure 1-10 The structure of LC zoom lens and conventional zoom lens.
1.5 Motivation and Objective of This Thesis
To summarize the discussion in section 1.1 to 1.3, we know that LC lenses have the potential to perform the imaging functions for mobile devices in limited spaces. The most unique characteristic of LC lenses is the electrically tunable focal length without the mechanical movement, which can fix the total thickness of the lens system and reduce the integration cost.
Furthermore, AF and optical zoom can be performed simultaneously by the simple structure, as shown in Figure 1-10. As Table 4 summarized, the most competitive advantage of LC lenses are the small dimension and self-tunable focusing. Without complicate integrations, AF and optical zoom can be performed. However, the issues as mentioned, inferior optical performance, slow focusing time, and high driving voltage, are the practical problems which governs LC lens can be commercialized or not.
Therefore, in this thesis, we want to overcome these issues. In the part of optical performance, we try to increase the range to obtain superior focusing. On the other hand, the
focusing time and driving voltage actually indicate the same work to improve the efficiency of driving. In typically, AF should be completed in one second for most mobile phones. This illustrates the focusing time should be reduced down to 200ms, if we perform several times of focusing in one AF. The improvement of focusing time can be achieved by increasing the driving efficiency. That means the same performance should be generated by lower driving voltage. In mobile devices, the driving voltage less than 15V or 10V could be an acceptable range.
Table 4 The comparison of three AF solution for mobile devices.
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Chapter 2.
Theory and Principle of LC Lenses
The basic mechanical theories of LC molecular and optical properties of LC lenses are introduced in this chapter. In the mechanical description, the orientations of LC molecules are usually calculated by the total free energy in a LC cell. This energy is induced by the elastic properties of LC material and effects of electric field on LC, as well as anchoring force on the boundary. In order to search for the director configuration, the total free energy is minimized to obtain the certain state. Response time is also introduced in this chapter. In the part of optical properties, the basic principle of LC lenses is introduced by propagations of plane wavefronts.
2.1 Introduction to Liquid Crystals
Liquid Crystal (LC) is a material which the state can be varied between crystal solid and isotropic liquid. It is believed that LC was discovered in 1888 by Friedrich Reinitzer, an Austrian botanist. He found the phases of LC were changed from a thick and turbid shape to a pure liquid when increased the temperatures. Following, Lehmann, a Germany physicist, further proved the state of crystalline liquid of LC utilizing polarizing microscope, and started the research of LC.
Typically, the phases of LC can be classified into crystal solid, smectic, nematic, and isotropic liquid under different range of temperature, as shown in Figure 2-1. At low temperature, the LC is in crystal solid state and the molecules have very high order in positions and orientations. When the temperature is increased, the state becomes to smectic phase. In this state, not only the orientations but also the positions of LCs become more random, and form a layered structure. Generally, LC in smectic phase higher viscosity and slower response time to applied electric field compared to nematic phase. When the temperature further increased, LC
is transformed to nematic phase, which is the most common state. In this phase, the LC molecules have partial orientation and aligned in one-dimension space. Nematic phase has the advantage of low viscosity and faster response time. Therefore, it is wildly employed in many applications, such as LCD and other kind of displays. In this thesis, we also focus on this advantage, and utilized it for our research. At high temperature, the LC material becomes isotropic phase, which the orientations and positions of molecules are totally random. In this state, LC has no birefringence.
Figure 2-1 The phases of LCs under different range of temperature 2.2 Energies in LC Cells
Direction of LCs is uniform in the LC cell when there is no external energy or confinement. If there is applied energy or other limited conditions, the spatial directions of LCs will be re-orientated, and this change was named the deformation. To investigate and calculate the change of directors, continuum theory is well-used in calculating the equilibrium of energies in LC cells. Generally, these energies are considered including free energy density on uniform state ( ), elastic energy ( ), electric or magnetic energy ( ), anchoring effect ( ), and so on, as shown as Equation (2-1).
(2-1)
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Elastic Energy
In the nematic phase, there are three possible forms of deformation as shown in Figure 2-2.
These forms were called splay, twist, and bend representing each specific variation of LC directors, , which indicates the directions of LC molecules, as shown in Figure 2-3. The general equation used to describe the energy stored by the three forms of deformations is
1
2 ∙ 1
2 ∙ 1
2 (2-2)
Each term in the left side indicates the deformation energy of the three forms. This equation is usually referred to as the Ossen-frank energy, and , , and represent the splay twist, and bend elastic constant of LC material respectively.
(a) (b) (c)
Figure 2-2 The three possible forms of deformation, (a) splay, (b) twist, and (c) bend, of nematic LCs.
Figure 2-3 LC directors, , which indicates the directions of LC molecules in Cartesian coordinate
Electric Field
Electric field is considered to re-orientate LC directors because it induces polarization. While the permittivity of LC molecule in the direction parallel to , ∥, is different from that of perpendicular direction, , the polarization can be changed by applying different field. By decomposing the field into two perpendicular directions along and perpendicular to , as shown in Figure 2-4. The induced polarization can be analyzed by
∥ ∙ (2-3)
Thus, the electric energy of LC is derived as following:
1
2 ∙ 1
2 ∥ ∙ 1
2 (2-4)
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Figure 2-4 The field was decomposed into two perpendicular directions along and perpendicular to .
Anchoring Effect
In reality, to fabricate LC cells, the boundary forces of the substrates affect the equilibrium state. This force is usually achieved by coating an alignment layer on the substrates, and the directions of the layers can determine the initially configuration of LC directors. By arranging the directions of upper and lower substrates, stabilized LC profiles can be generated in specific types. Three of the common alignment methods are shown in Figure 2-5, the perpendicular, anti-parallel, and parallel alignments. Each of them is well-used for producing twist-nematic cell (TN), electrically controlled birefringence (ECB), and pi-cells, respectively.
In the applications of LC lenses, ECB mode is wildly used for simplifying the control and structures. The force describing the anchoring effect can be expressed by Rapini-Papoular approach as following:
(2-5)
where
1
2 (2-6)
1
2 (2-7)
, and is the polar and azimuthal angles, as shown in Figure 2-3, which determine the
anchoring force of two components, , and respectively. and are the constant which are determined by interaction between the alignment layer and the LC molecules.
and are the angles at which the interaction has minimum energy.
(a) (b) (c)
Figure 2-5 Three of the common alignment methods, (a)perpendicular, (b)anti-parallel, and (c)parallel alignments for LC cells.
2.3 Optical Principle of LC Lenses
LC is a material exhibiting birefringence. This property causes the incident light with different polarizations meet different refractive index. For analyzing the distribution of index, the light can be decomposed into two eigen-polarizations, one of which is parallel to the optical axis of the material and the other one is perpendicular to the plane of optical axis. The parallel one which is named ordinary light sees all the same index no matter how the included angle between the incident light and the optical axis, as shown in Figure 2-6. On the other hand, the polarization which is perpendicular to the plane of optical axis, named extraordinary light, sees the refractive index which is direction dependent. Although, the index would be varied by the included angles between the incident light and the optical axis, the effective index can be calculated by Equation (2-8).
(2-8)
where and are the extraordinary and ordinary refractive indies respectively. is
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Figure 2-6 The ordinary and extraordinary lights travelling in the LC material and extraordinary lights see the different indies which are dependent to the included angles between the incident light and the optical axis of LC molecules.
LC lenses, as shown in Figure 2-7, employ the gradient electric field to control the LC layers. The distribution of the field is controlled by the applied voltages and generates phase retardation which is approximate to that of lenses to converge or diverge the incident extraordinary light. The schematic is shown in Figure 2-8. r, d, and f respectively represent the radius of lens aperture, thickness of LC layer, and the effective focal length. To derive the relation of these parameters, optical path length (OPL) of wavefront is used to explain. As shown in Figure 2-8, the LC layer of an LC lens can be considered as a GRIN Lens [61], thus the refractive index is a function of position . To simulate the focusing of the lens, we assume the rays of a plane wave passing the lens and arriving at point E have the same OPL.
Therefore, the optical path difference (OPD) between ABE and CDE should be zero.
∙ ∙ (2-9)
where S , and S are the OPL of ray AB and CD respectively, and n is the refractive index in air. Generally, the lens power of LC lenses is relatively small, thus the refraction in the LC cell can be ignored. Therefore, Equation (2-9) is approximate as
S ∙
⇒ ∙ ∙
⇒ ∆ ∙ ≅ ∙
2
⇒ ∆ ∙ ≅ ∙
2
⇒ ≅ ∙
2∆ ∙ (2-10)
where ∆ indicate the difference of refractive index between border and center. By substituting r, d, and f for the length , , and DE, Equation (2-10) can be rewrite as
≅ ∙
2∆ ∙ (2-11)
In Equation (2-11), we have assumed the straight propagating of rays in LC layer, and the effective index, and at the border and center of the lens. To obtain a unique focal length, f, from the whole lens aperture, the distribution of ∆ is
∆ ≅ ∙
2 ∙ ∙ (2-12)
which is a parabolic form of variable, .
Figure 2-7 The wavefront focusing of LC lens.
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Figure 2-8 The diagram of LC lens considered as a GRIN lens for analyzing the focusing.
Chapter 3.
Multi-electrode Driven LC Lens
To have a tunable lens, the LC lenses were expected to produce superior focusing at each focal length, which can yield similar image quality for focusing objects at different position.
Therefore, a well control for the electric field is required. In this chapter, we proposed Multi-electrode Driven LC Lens to have such high control freedom yielding superior focusing for each focal length by operating different sets of driving voltages.
3.1 Introduction
As mentioned in section 1.4.1.3, to achieve LC lenses exhibiting highly focusing ability, the spot size of PSF should be minimized for each focal length. However, the conventional structures with low control freedom, as introduced, are hard to significantly yield desired electrical field to control the LC orientation for wide range of focusing. Figure 3-1 shows a focusing example of the homogeneous LC lens with internal electrodes. When the higher driving voltage was applied, the higher focusing light intensity can be detected at focal length, 4cm, and the full width at half maximum (FWHM) of the spot size is 65um. When the driving voltage was decreased, however, the maximum light intensity was detected at focal length, 10cm, with a different beam profile from that of 4cm, whose FWHM increased to 121um. If we employed these results for AF, the image performances are different for objects at different distance. To finely control the phase retardation of LC layer, Multi-electrically Driven Liquid Crystal Lens (MeDLC Lens) was propose. MeDLC Lens utilized a large number of electrodes to significantly control the electrical field on LC layer. By optimizing the sets of driving voltages for different focal length, a wide range of superior focusing was yielded.
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Figure 3-1 The focusing profile of the homogeneous LC lens with different focal length, 4cm and 10cm. The focusing profiles are different.
3.2 Multi-electrically Driven Liquid Crystal Lens (MeDLC Lens) 3.2.1 MeDLC Lens
The main concept of MeD-LC Lens is to utilize a large number of electrodes to finely control LC orientation, as the structure shown in Figure 3-2. To generate a smooth electrical field in LC layer, multi-electrode were constructed above a high K material (i.e. the glass). The number of electrodes and width of electrodes and slits, WE and WS, were designed for optimized the phase retardation by different aperture size, LC cell gap, and thickness of glass for different applications. By arrangement of driving voltages for each electrode, the phase retardation over the LC layer was optimized for particular focal length. Typically, the arrangement of driving voltages for MeD-LC Lens was correlated to the LC orientation. For a convex lens application, the driving voltages were higher at the marginal electrodes and decreased to the lowest voltage in the central region.
Figure 3-2 MeD-LC Lens with large number of electrodes. MeD-LC Lens utilized the arrangements of driving voltages to finely control the LC orientation, and yielded a wide range of superior focusing
3.2.2 Optimization
To optimize the number of electrodes, errors of phase retardation generated by MeD-LC Lens was quantified. The difference of reflective index of LC lenses is shown following:
∆
2 ∙ ∙ (3-1)
where Δnl donates the average difference of reflective index between border and center of the lens, r, f(r), and d represent the aperture radius, focal length, and LC cell gap respectively. For an ideal case, a unique focal length for all lens apertures, the distribution of Δnl is a parabolic curve and was used for analysis of phase retardation. As the phase retardation of MeD-LC Lens could be approximated to that of ideal one, which indicated MeD-LC Lens generates phase retardation with lower optical aberration and approaches to the lens with approximate unique focal length. So the optimization of MeD-LC Lens was focused on the design of the size of electrodes to match the ideal parabolic curve. To simplify the optimization, the ratio of WE to WS (WE/WS) was fixed to 1 for 1.5mm lens aperture. The LC cell gap and thickness of the glass were designed to 60um and 700um. LC material was chosen as Merck nematic LC (E7), which the intrinsic difference of reflective index between ordinary and extraordinary, Δn,
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approximate 3.5cm. As the number of electrodes increased, the curve of Δnl generated by MeD-LC Lens approached to that of ideal one. In conventional LC lens design, the large marginal electrodes usually yielded flat electrical field in the border, and cannot generate lenses with full aperture size for lens array. On the other hand, MeD-LC Lens utilized the finer electrodes to yield the phase retardation which was much close to ideal parabolic curve and with full designed aperture size (1.5mm).
A merit function defined as following:
∑ ∆ ∆ (3-2)
was used to quantify the errors between ideal distribution ofΔnl and that generated by MeD-LC Lens, where Δni and ΔnMeD-LC represent Δnl of ideal lens and MeD-LC Lens respectively. Through the result, as illustrated in Figure 3-4, the larger number of electrodes yielded smaller E-value, which means the curves was more approximate to the ideal parabolic one under electrical field of finely control. As the number of electrodes exceeded 9, E started to be saturate. In our study, 9 electrodes for MeD-LC Lens with 1.5mm aperture size would be a balance design.
Figure 3-3 The profile of Δnl generated by MeD-LC Lens and an ideal lens for around 3.5cm focal length. The ideal lens represented by a parabolic curve is compared with that of MeD-LC, presenting a deviation from the ideal case.
Figure 3-4 The quantified errors, E, representing the deviation of the results of MeD-LC 1.5
1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3
5 6 7 8 9 10 11 12 13 14 15 16 The Number of Electrodes
E(%)
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3.3 Experimental Results
Cylindrical MeD-LC Lens with 9-electrode and WE/WS =1 for 1.5mm aperture size was investigated. For convex lens applications, the driving voltages of each electrode, as shown in Figure 3-5, were optimized for different focal length according to the focusing profile. The result showed a symmetrical arrangement. Each driving voltage was adjusted for particular focal length. In the convex MeD-LC Lens, marginal driving voltages are higher than central voltages to form convex-like phase retardation. As the total energy increased, the focal length was shortened. Figure 3-6 (a) and (b) illustrates the experimental setup and the comparison performed by the structures with external, internal electrodes mentioned above, and MeD-LC Lens. All of the structures with 60um LC cell gap, 1.5 mm aperture size and the same substrate were illuminated by 632.8nm Hi-Ne LASER and investigated the optimized focusing profile for focal length of 4cm, 7cm, and 10cm. The focusing profile was measured by GENTEC Beamage Series CCD sensor placed in front of the LC lenses by a distance of focal length. Corresponding cross-section (bottom right) and FWHM (bottom left) of the focusing profile are shown as well. For two of the conventional LC lenses, although the external structure generated acceptable focusing at focal length from 7cm to 10cm, the 4cm showed a broken result in beam profile. On the other hand, internal structure yielded adequate results from 4cm to 7cm, but the poor focusing at 10cm which the FWHM was much increased also indicated the limitation of low freedom of control. Comparing to the results, MeD-LC Lens yielded a wide range of superior focusing at all of 4cm, 7cm, and 10cm focal length that FWHM of focusing profile was maintained narrower.
Figure 3-5 The optimized driving voltages for 9-electrode MeD-LC Lens in convex applications. The symmetrical arrangement of each driving voltage was adjusted for particular focal length. In convex MeD-LC Lens, marginal voltages are higher than central voltages to form convex-like phase retardation.
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(a)
(b)
Figure 3-6(a) Cylindrical MeDLC Lens measured by CCD sensor at a distance of corresponding focal length (EF), and (b) the focusing profile were compared with two conventional LC lens. As the results shows, two conventional LC lenses only yielded a particular range of focusing. On the other hand, the multi-electrode of MeDLC Lens served the LC layer optimized phase retardation for each focal length and a wide range of focusing.
In our experiments, MeDLC Lens provided a smooth variance in focal length and maintained superior focusing profile. By the high freedom of multi-electrically driven, electrical field was significantly modified for different focal length from 3.5cm to infinity with 1.5mm aperture size and 60um LC cell gap. In the comparison, conventional LC lens with low control freedom only yielded focusing in a particular and narrow range. This phenomenon limits the applications for tunable lenses. MeDLC Lens provided stable performance in focusing for a wide range. This benefit can increase the feasibility for applications and achieve a highly tunable lens.
3.4 Discussion - Grating Structure of MeD-LC Lens
The electrodes of MeD-LC Lens, as shown in and the results are shown in Figure 3-2,
The electrodes of MeD-LC Lens, as shown in and the results are shown in Figure 3-2,