反射式單晶矽液晶面板邊際場效應及繞射效應之研究

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國立交通大學

光電工程研究所

博士論文

反射式單晶矽液晶面板邊際場效應及繞射效應之研究

Study on the Fringing-Field and Diffraction Effects of

LCOS Panels

研究生: 范姜冠旭

指導教授: 王淑霞教授

吳詩聰教授

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反射式單晶矽液晶面板邊際場效應及繞射效應之研究

Study on the Fringing-Field and Diffraction Effects of

LCOS Panels

國立交通大學光電工程學系暨研究所

博士論文

A Dissertation

Submitted to Department of Photonics and Institute of Electro-Optical Engineering College of Electrical Engineering and Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Electro-Optical Engineering June 2005

Hsinchu, Taiwan, Republic of China

中華民國九十四年六月

研究生：范姜冠旭

指導教授：王淑霞

吳詩聰

Student：Kuan-Hsu Fan-Chiang

Advisor：Shu-Hsia Chen

Shin-Tson Wu

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反射式單晶矽液晶面板邊際場效應及繞射效應之研究

研究生：范姜冠旭

指導教授：王淑霞教授

吳詩聰教授

國立交通大學光電工程學系暨研究所

摘要

近年來，反射式單晶矽液晶(LCOS)元件一直是顯示產業持續注意的焦點，尤其是在投影顯示器的應用方面，其產品包括前投式投影機、背投式電視以及頭戴式虛擬顯示器等。由於單晶矽的電子漂移率甚高，因此 LCOS 元件可以擁有非常高的解析度，除此之外，其周圍的驅動電路也可以整合在單一的晶片上以減少生產成本。在製造方面，LCOS 液晶面板乃建立在國內兩大產業的基礎上，即半導體與液晶顯示器產業，因此，這對於國內發展 LCOS 不啻為一大利基。由於矽基板的製造是屬於標準的半導體製程，所以具有低價格的潛力與優勢，也因此許多廠商已紛紛投入 LCOS 投影顯示器這個產業。然而，目前 LCOS 顯示技術仍然存在著許多困難有待克服。在面板方面，由於要在微小的面板上做出高解析度的影像，其像素以及像素間距也相對地變得非常小。當相鄰像素的施加電壓不同時，其邊緣電場會被扭曲，使得此處的液晶分子產生不正常的排列，進而影響其光學特性，這就是所謂的邊際場效應。此外，當解析度的要求提高而使得像素電極的大小接近可見光的波長時，其對光波的作用類似於一反射光柵。當光入射在面板上時會產生明顯的繞射效應，而斜向傳播的繞射光可能無法進入光學系統，因而造成嚴重的光損失。在本論文中，我們探討八個常用液晶模態的邊際場效應，並且改變液晶盒參數，研究其對 LCOS 面板光學表現的影響。我們發現混和式扭轉向列型液晶模態的邊際場效應相對較弱，而扭轉向列型和垂直排列型液晶模態則會受到嚴重的邊際場效應影響而降低其影像品質。特別是對於垂直排列型液晶模態，其邊際場效應不但影響其靜態

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顯示的品質，同時也嚴重地拖慢了動態影像的切換速度。在液晶盒結構方面，根據電腦程式的模擬結果，我們也發現像素節距、液晶盒厚度、預傾角和電極斜率都是影響邊際場效應的重要參數。為了要設計一個高對比度且同時不受邊際場效應影響的 LCOS 面板，我們針對擁有完美暗態的垂直排列型液晶盒做分析。根據電腦模擬出的液晶指向矢分佈，我們發現利用圓偏振光的特性，可以有效地保留因邊際場效應所損失的光效率進而提高影像的銳利度；同時，在動態響應上也解決了因緩慢切換過程而產生的影像模糊問題。相關的光學原理可以由著名的 de Varies 理論來解釋。在探討 LCOS 面板的繞射效應方面，由於傳統的瓊斯矩陣法並無法分析光的繞射效應，因此我們將以往應用在光波導計算的光束傳播法延伸到 LCOS 元件的光學計算，而撰寫出有考慮繞射效應的光學模擬程式。利用此程式，我們針對垂直排列型液晶模態以及工研院電子所研發的 FOP(finger-on-plane)模態進行分析。模擬結果顯示繞射效應對高解析度的 LCOS 元件影響甚巨；使用傳統瓊斯矩陣法會造成嚴重的誤差，唯有使用更嚴謹的光束傳播法才能正確的預估其光學行為。此外，我們發現稍微修改 FOP 模態的液晶盒結構能有效地降低在特定波段的繞射效應，若配合使用三片或雙片式的 LCOS 投影光學系統，則可以有效地提升因繞射效應所損失的光效率。

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Study on the Fringing-Field and Diffraction Effects of

LCOS Panels

Student: Kuan-Hsu Fan-Chiang

Advisor: Prof. Shu-Hsia Chen

Prof. Shin-Tson Wu

Department of Photonics and Institute of Electro-Optical Engineering

National Chiao Tung University

Abstract

In recent years, the display industries keep showing great interests in liquid-crystal-on-silicon (LCOS) devices, especially in the application of projection display. The products of LCOS devices include data projectors, rear-projection TV and the head-mounted virtual display. Due to the advantage of intrinsic high electron mobility of crystalline silicon, LCOS devices can be fabricated with very high resolution. In addition, its peripheral driving circuits can be integrated on a single chip, which greatly reduces the cost of manufacturing. Technically, LCOS devices are based on two major domestic industries: the semiconductor and liquid crystal display industries. Since the fabrication of the silicon backplane is based on the standard manufacturing process of semiconductor, LCOS devices have great potential of low price. Therefore, many manufacturers have already invested in this industry.

However, there are still many challenges in the LCOS industry. The two major issues of LCOS panels are the fringing-field and diffraction effects. As the resolution increases, the pixel size and the inter-pixel gap will become very small. When the applied voltages between adjacent pixels are different, the electric fields near the pixel edges will be distorted. Hence, the liquid crystal molecules near this region are aligned abnormally, which, in turns, degrades the optical performance of the device significantly. This is the so-called fringe field effect.

In addition, as the pixel pitch becomes comparable to the wavelength of the visible light, the LCOS panel acts as a reflective grating. Therefore, obvious diffraction effect can be

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observed. The oblique diffracted light may not be able to enter the optical system, and consequently results in serious light loss.

In this dissertation, we investigate the fringing-field effects of eight commonly used liquid crystal modes. We also investigate the influence of the LC cell structure on the optical performance of LCOS devices. It is found that the mixed-mode twist nematic (MTN) has weaker fringing-field effect while the twist nematic mode (TN) and vertically aligned mode (VA) suffer from the effect significantly. The fringing-field effect is particularly severe in VA mode. It not only degrades the static image qualities but also deteriorates the dynamic response of the LCOS panel. The pixel pitch, cell gap, pretilt angle and electrode slope are all found critical to the fringing-field effect.

In order to design a high-contrast-ratio LCOS panel without fringing-field effect, we focus on the analyses of VA mode which possesses an excellent dark state. Based on the simulated results of the LC director profile, we find that, by utilizing the properties of circularly polarized light, the light loss caused by fringing-field effect can be preserved and the sharpness of the image can be enhanced dramatically. Moreover, the dynamic response is also improved and the imaging blurring effect is successfully eliminated. The results can be qualitatively illustrated by the de Vries theory.

With regard to the effect of diffraction, a rigorous simulator is needed to investigate the optical performance of a high-definition LCOS panel. The conventional Jones matrix method is no longer suitable in this condition. We extend the beam propagation method (BPM), which is commonly employed in waveguide calculations, to the optical simulation of LCOS devices. Two promising LC operation modes are analyzed by BPM, i.e. VA and finger-on-plane (FOP) modes. The calculated light efficiencies by Jones matrix method and BPM with respect to the pixel pitch are compared. It is shown that the diffraction effect is critical to the light efficiency. Using Jones matrix method may give rise to significant miscalculation. By using BPM, it is found possible to reduce the diffraction effect for certain waveband by slightly modifying the FOP cell structure, suggesting that the light efficiency can be boosted effectively in a two- or three-panel LCOS projection system.

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致謝 Acknowledgement

時光匆匆，在交大已經待了九個年頭。回想起來，能夠拿到博士學位，需要感謝的人實在太多。其中，家人當然是我最重要的精神支柱。感謝我的父親范姜超沐先生對我的栽培，您始終是我內心中最重要的依靠。感謝我的母親翁明甘女士對我無微不至的照顧，讓我遇到挫折時有一個安全的避風港。感謝我的哥哥范姜冠宇先生陪我一起成長，一起分享心情，讓我的生活更加精彩。在此要特別感謝我的指導教授王淑霞教授。您在課業上和做人處事上的教導讓我一生受用無窮。您對我們的關懷照顧，早已超越了一般師生的關係。對我來說，您更像是一位親人。很難用短短的幾句話表達對老師的感激，只希望自己未來能夠不辜負老師的期望，繳出一張漂亮的成績單。也要特別感謝的是美國的指導教授吳詩聰教授。感謝吳老師讓我有機會能去美國深造。在美國一年的期間中，我在吳老師的指導下學到許多寶貴的知識和經驗，並且讓我的眼界更加寬廣。同時也感謝吳師母在我初到美國時貼心的照顧，使我感覺雖然身在異鄉，卻有在家鄉的溫暖。由衷的感謝您們。在美國的期間，我要感謝林怡欣學姊和吳勇勳學長的照顧。有你們的陪伴讓我感覺很溫馨。感謝朱新羽博士給我許多寶貴的意見，和您討論經常能夠激盪出許多 有趣的新點子。同時我還要感謝許多在美國的朋友：Mr. Tim Wilson, Mr. and Mrs. Kim, Mr.

and Mrs. Matsubara, Mr. and Mrs. Teruaki. Thanks for sharing all the good times with me in Orlando. You guys are the best!!

感謝液晶實驗室的學長姐們: 秋蓮、志勇、阿寬、俊雄、芝珊、揚宜、彥廷、信全、乾煌、梓傑、佳成; 我的同學們: 怡安、惠雯、庭瑞、朝旭、英豪; 以及學弟妹們：舒展、德源、建宏、世郁、品發、美琪、家榮、庭毅、瑞傑。你們讓我的研究生生活多采多姿。尤其要感謝芝珊、庭毅、瑞傑在最後關頭拼畢業的同時也分擔了實驗室許多繁重的工作。很高興我們一起做到了。僅以此論文獻給我最親愛的家人、師長、以及所有關心我和幫助過我的人。由衷的感謝。范姜冠旭新竹交通大學 2005 年 6 月

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List of Figures

1.1 Schematic of pixel apertures in (a) transmissive and (b) reflective light valves. 1.2 Cross-section of a common LCOS structure.

1.3 A Typical three-panel LCOS projection system. M1, M2, M3 and M4 are mirrors. DM1 is the cyan/red dichroic mirror and DM2 is the green/blue dichroic mirror. PBS is the polarizing beam splitter and PSC is the polarization state converter.

1.4 Intensity homogenization optical parts: (a) the fly’s eye lenslet elements; (b) light pipe.

1.5 (a) The polarization state converter; (b) the typical front end of a LCOS optical engine.

1.6 The time-sequential system using a rotating color wheel with red, green and blue segments.

1.7 The time-sequential rotating-prism system.

1.8 Single-panel LCOS system using holographic color filter.

1.9 (a) Sketch of the Brewster angle reflection; (b) an calculated results of normalized reflectance with respect to incident angle when light propagating from air to Ti2O3

(n=2.45).

1.10 Sketch of the fringing fields at the pixel edges when adjacent pixels are operated at different voltages.

1.11 Sketch of the diffraction effect of LCOS devices.

2.1 Structure of chiral nematic liquid crystals. P0 is the natural pitch.

2.2 (a)The dispersion relation of a CLC with ne=1.5578, no=1.4748 and P0=0.25 µm; the

insert denotes the characteristics of eigenmodes at some specific frequencies; (b) Calculated ellipticity with respect to l2/q0 when –1<l2/q0<0; (c) Calculated ellipticity

with respect to l2/q0 when 0< l2/q0<103; (d) Calculated ellipticity with respect to l1/q0

when 0< l1/q0<600.

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2.4 The represented Jones matrix can be derived by subdividing the LC layer into a large number, N, of thin layers.

2.5 Sketch of the mirror image of a left-handed twist LC cell. 2.6 The angle definitions in numerical calculations.

2.7 Sketches of the concepts of the (a) Jones matrix method and (b) the beam propagation method.

3.1 The LC cell structure used for the two-dimensional simulations.

3.2 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of NW 90°-MTN mode with β=20° and d∆n=240 nm.

3.3 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of 63.6°-MTN mode with β=0°, d∆n=203 nm, (d∆n)film=24 nm, and 110° film angle

related to x-axis.

3.4 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of 45°-MTN mode with β=78°, d∆n=195 nm, (d∆n)film=27 nm, and 110° film angle

related to x-axis.

3.5 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of NB 63.6°-TN mode with β=0°, d∆n=508 nm, and d/p=0.6.

3.6 Calculated results of the contrast ratio with respect to d/p ratio for 63.6°-TN mode with d=2.1 µm.

3.7 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of NB 45°-TN mode with β=0°, d∆n=533 nm.

3.8 Simulated reflectance of NB 45°-TN mode with d=2.2 µm.

3.9 Simulation results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profile of

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NB 52°-TN mode with β=0°, d∆n=517 nm.

3.10 Simulated reflectance of NB 52°-TN mode with d=2.2 µm.

3.11 Simulation results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profile of VA mode with β=45°, d∆n=192 nm and (d∆n)c-film=-183 nm.

3.12 The simulated green band reflectance profile of FCH mode, d1∆n1 =184, β=45° (a)

without compensation film, and (b) with compensation film, d2∆n2=52 nm.

3.13 Sketch of the influence of the pixel pitch to the fringing-field effect.

3.14 Calculated contrast ratio of each LC operation mode at dark-bright-dark state with respect to the pixel size.

3.15 Sketch of the effect of cell gap.

3.16 Calculated contrast ratio of 63.6°-TN, 45°-TN and 52°-TN at dark-bright-dark state with respect to cell gap.

3.17 Sketch of the effect of pretilt angle to negative liquid crystal molecules.

3.18 (a) Simulated reflectance profiles of VA mode at dark-bright-dark state with pretilt angle varies from 85° to 88°, and (b) the calculated optical filled factor with respect to the pretilt angle.

3.19 Calculated optical filled factor with respect to evolution time of VA cell after switched from dark-bright-dark state to the all-bright state with pretilt angle varies from 85° to 88°.

3.20 (a) Sketch of the electrode slope; (b) Calculated contrast ratio with respect to the electrode slope ranging from 0.3 to ∞ at dark-bright-dark state for 90°-MTN, 45°-MTN and VA mode.

3.21 The calculated transient states of VA mode when switching from the dark-bright-dark state to the all bright state. The cell gap is 2.3 µm and the pretilt angle is 88°.

4.1 Schematic drawing of the systems used for (a) the LPVA device and (b) the CPVA device. PBS=polarizing beam splitter; P=polarization axis, R is the LC alignment direction at the front surface.

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RLP(x) and RCP(x), at the alternate dark and bright state for the CPVA and the LPVA

systems, respectively. The on-state voltage Von=5 V and the off-state voltage Voff=0.

4.3 Calculated azimuthal angles (f) of the LC directors along the z direction at x=x1, xa,

xb and x2 as denoted in Fig. 4.2.

4.4 Photos captured by CCD through the polarizing microscope of the LC panel operated at the alternate bright and dark states for (a) the LPVA and (b) the CPVA devices. Photos of the LC panel switched from the alternate bright and dark states to the all-bright state for (c) the LPVA device at 198 ms after switching and (d) the CPVA device at 33 ms after switching.

4.5 Measured reflectance with respect to the elapsed time after switching from the alternate bright and dark states to the all-bright state for the CPVA device.

4.6 Sketch of the broadband circular polarizer which comprises a linear polarizer, a 1/2λ-plate and a 1/4λ-plate.

4.7 The off-axis optical system for reflective CPVA device.

4.8 The optical system using hologram film for reflective CPVA device. 4.9 The optical system using Faraday rotator for reflective CPVA device.

4.10 The three-panel optical system for transmissive CPVA high-temperature poly-Si LCD.

5.1 Schematic representation of angle definitions and mirror image of reflective LCOS device.

5.2 Sketch of the acceptance angle θa of the light waves propagating from the LCOS

panel to the projection lens.

5.3 Cell structures used for 2D computer simulations of (a) FOP mode (b) VA mode. 5.4 Computer simulated light efficiencies (θa=10°) with respect to P/λ value of

FOP-LCOS devices at voltage-off state by extended BPM and Jones matrix method. 5.5 Simulated intensity angular spectrum with P=7.7 µm and λ=540 nm for FOP mode. 5.6 (a) Simulated light efficiencies with respect to P/λ value of VA-LCOS devices at

voltage-off state by extended BPM and Jones matrix method, and (b) simulated intensity angular spectrum with P=7.7 µm and λ=540 nm.

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5.7 Computer simulated results of VA mode with off-on-off pixel configuration by extended BPM and Jones matrix method: (a) light efficiencies (θa=10°) of LCOS

devices with respect to P/λ value, (b) intensity angular spectrum with P=7.7 µm and λ=540 nm, and (c) the intensity of zeroth-order diffracted light, I0, calculated with

respect to P/λ value.

5.8 Calculated light efficiency by reflective BPM with respect to wavelength for FOP mode (P=15.5 µm) with di=115 nm and 150 nm.

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List of Symbols

n index of refraction

n director of liquid crystal

g free energy density

K11 elastic constant of the splay deformation

K22 elastic constant of the twist deformation

K33 elastic constant of the bend deformation

P0 natural pitch of the structure of chiral liquid crystals

q0 2π/P0

ε dielectric constant

D electric displacement

E electric field

G total free energy

no ordinary refractive index of liquid crystal

ne extraordinary refractive index of liquid crystal

ω angular frequency of light ρ ellipticity of light

k wavevector

d thickness of liquid crystal cells

θ tilt angle of liquid crystal director

φ azimuthal angle of liguid crystal director

α the angle between the optical axis of compensation film and x-axis

β the angle between the transmission axis of linear polarizer and the entrance liquid crystal director

F optical filled factor

F# f-number

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Chapter 1 Introduction

1.1 Overview of LCOS devices

Information displays can be broadly categorized into direct-view and projection types. Direct-view displays generate images directly on the surfaces that can be viewed by human eyes. On the other hand, projection displays present their images on an auxiliary surface which is separated from the image-generating devices. These images can be either real or virtual depending on different applications.

Generally, there are three main technologies being developed for projection displays: poly-silicon thin-film-transistor (TFT) liquid crystal displays (LCDs) [1], digital light processing (DLP) [2] and liquid-crystal-on-silicon (LCOS) devices [3]. Table 1.1 compares the advantages and disadvantages of these three technologies. Among them, LCOS is especially attractive due to its standard semiconductor manufacturing process. This technology is particularly suitable to its development in Taiwan because of the mature semiconductor industry. When compared to transmissiv-type LCDs, reflective-type LCOS light valves have higher aperture ratio since all of the electronic circuits can be hidden behind the reflective pixels. Figure 1.1 illustrates the schematic of pixel apertures in transmissive and reflective light valves. The black areas represent the light-absorbing material for blocking light to prevent any current leakage. The aperture ratio of reflective light valves can be above 90% while that of the transmissive one is only around 70%. On the aspect of DLP system, it has the advantage of fast response, high light efficiency and compactness which are very suitable for data projectors. However, when playing movies, the so-called color-breaking effect appears as a momentary flash of rainbow-like stripes

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typically trailing the bright objects when looking from one side of the screen to the other [4]. This effect is intrinsically generated from the field-sequential color generating method which degrades the image quality significantly. Another drawback of DLP is the complex manufacturing process of the micro-electro-mechanical systems (MEMS) which becomes one of the main barriers for many investors.

•Low yield

•Fringing field effect •Diffraction effect •Complex manufacturing

•Less color saturation •Color breaking

(rainbow effect) •Low aperture ratio

•Bulky

disadvantage

•Standard semiconductor manufacturing

•High aperture ratio

•High resolutions (5-8 µ m pitch)

•Fast response (µs level) •High light efficeincy •Compact •Mature technology •Mass productive •Simple optical engines Advantages

JVC, Aurora, Sony, Brillian, TMDC, Himax, etc. Taxes Instruments

Sony and Epson, etc.

Companies Reflective Refelective Transmissive Light valves LCOS DLP LCD Technologies •Low yield

•Fringing field effect •Diffraction effect •Complex manufacturing

•Less color saturation •Color breaking

(rainbow effect) •Low aperture ratio

•Bulky

disadvantage

•Standard semiconductor manufacturing

•High aperture ratio

•High resolutions (5-8 µ m pitch)

•Fast response (µs level) •High light efficeincy •Compact •Mature technology •Mass productive •Simple optical engines Advantages

JVC, Aurora, Sony, Brillian, TMDC, Himax, etc. Taxes Instruments

Sony and Epson, etc.

Companies Reflective Refelective Transmissive Light valves LCOS DLP LCD Technologies (a) (b) (a) (b)

Figure 1.2 demonstrates the cross-section of a common LCOS structure. Basically, LCOS comprises a liquid crystal (LC) layer sandwiched between a cover glass slide (top substrate) and a crystalline silicon wafer (bottom substrate). The LC layer functions as an electro-optic modulator which delivers signals of displaying images to incident light. The LC effect will be discussed in Chap. 2. Owing to the high electron mobility of the silicon substrate, the pixel size can be shrunk to as small as 5-8 µm, which results in a very high resolution. Some characteristics of LCOS microdisplay compared to those of TFT-LCDs are listed in Table 1.2. Another benefit of LCOS

Table 1.1 Comparison of LCD, DLP and LCOS for the application of projection display.

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from high electron mobility is the integration capability of driving circuits, which can reduce cost significantly and make the system more compact.

Optical engines play an important role in LCOS projection systems. There is a great diversity of the optical systems for LCOS devices. Basically, the engine is composed of an LCOS panel (or panels), color management optics, projection lens, basic display driving electronics and an illumination system. Optical engines can be roughly classified according to the number of LCOS panels employed in the system. In general, three-panel and single-panel systems are more popular than others, as will be discussed later. Other optical systems are less mature and will not be discussed in this dissertation.

Glass ITO LC Alignment LC Al Reflector Light blocking layer

Source Gate

Silicon Substrate Capacitor Drain

Glass ITO LC Alignment LC Al Reflector Light blocking layer

Source Gate

Silicon Substrate Capacitor Drain

Great integration capability Reflective

~500 cm2_/Vs

High mobility Small chip area Line Width: sub-microns

Small geometry LCOS

Poor integration capability Transmissive

α-Si:~1.5; p-Si:150 (cm2_/Vs)

Low mobility Large panel area

Line Width: several µm

Medium geometry TFT-LCDs

Great integration capability Reflective

~500 cm2_/Vs

High mobility Small chip area Line Width: sub-microns

Small geometry LCOS

Poor integration capability Transmissive

α-Si:~1.5; p-Si:150 (cm2_/Vs)

Low mobility Large panel area

Line Width: several µm

Medium geometry TFT-LCDs

Figure 1.3 shows a typical three-panel LCOS projection system. Usually, an

Table 1.2 Comparison of characteristics of TFT LCD and crystalline silicon based microdisplays. Figure 1.2 Cross-section of a common LCOS structure.

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ultra-high-pressure (UHP) mercury arc lamp is used to generate stable light throughput. This kind of lamps offers a small arc gap, acceptable efficiency and long lifetime [5]. However, some UV and IR radiations are also generated by the UHP lamp. UV light can be harmful to the LC alignments and IR light generates additional heat to the system. Therefore, UV/IR filters must be applied in front of the lamp in order to protect the system from these unwanted UV and IR components.

Integrated lenses including intensity-homogenizing elements are placed in the optical path behind the UV/IR filters. There are two main functions of the integrated lens. The first one is to homogenize the light intensity profile generated by the UHP lamp. The second is to transform the circular cross-section of light beams into a rectangular shape so that the beams can fit in with the size of LCOS light valves. Two approaches are commonly adopted: the fly’s eye lenslet [6] and the light pipe elements [7]. The fly’s eye approach consists of two lenslet elements with different patterns as shown in Fig. 1.4 (a). When light passes through these lenslets, a more uniform light intensity profile can be obtained. The other approach utilizes a hollow or solid rod as shown in Fig. 1.4 (b). Light coming from the lamp is coupled into the rod and is reflected off the internal walls by multiple total internal reflection. As light exits the light pipe, the light intensity will be evened out and produce uniform brightness.

LCOS light valves are polarization-dependent devices. Therefore, a polarizer must be used to polarize the light before it is incident on LCOS panels. A conventional dichroic polarizer theoretically absorbs half of the light which limits the light efficiency to 50%. A polarization state converter (PSC) can significantly recapture the light that is absorbed or dumped by a conventional polarizer. Figure 1.5 (a) illustrates one of such designs [8]. Unpolarized light is decomposed into p-wave and s-wave after it passes through a polarizing beam splitter (PBS). The reflected s-wave is then

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Project

ion

lens

Lamp

Integrated

lens

M1

PS

C

M4

LCOS(G)

DM1

M2

M3

DM

2 PBS(G)

LCOS(R)

LCOS(B)

PBS(B)

_PBS

(R

)

UV/IR fil

te

r

s

B

G

R

Project

ion

lens

Lamp

Integrated

lens

M1

PS

C

M4

LCOS(G)

DM1

M2

M3

DM

2 PBS(G)

LCOS(R)

LCOS(B)

PBS(B)

_PBS

(R

)

UV/IR fil

te

r

s

B

G

R

Figure 1.3 A Typi cal th ree -panel L C OS proje ction system. M1, M2, M3 and M 4 are mi rro rs. DM1 is the cya n/red di ch roi c mirro r an d DM2

is the green/blue dichroi

c

mirror. PBS is the polarizi

ng

beam

split

ter and PSC i

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Lamp

Light pipe

(a)

(b)

Lamp

Light pipe

Lamp

Light pipe

(a)

(b)

p-wave

s-wave

Half- wave plate PBS Mirror Lamp UV/IR filter Integrated Lens PSC

(a)

(b)

p-wave

s-wave

Half- wave plate PBS

Mirror

p-wave

s-wave

Half- wave plate PBS Mirror Lamp UV/IR filter Integrated Lens PSC

(a)

(b)

Figure 1.5 (a) The polarization state converter; (b) the typical front end of a LCOS optical engine Figure 1.4 Intensity-homogenizing optical parts: (a) the fly’s eye lenslet elements; (b) light pipe.

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directed to a half-wave plate which rotates the polarization plane of the transmitted light to the direction parallel with p-wave. The light efficiency is, therefore, boosted effectively. A typical front end of an optical engine is formed comprising a lamp, UV/IR filters, integrated lens and PCS as shown in Fig. 1.5 (b).

A three-panel system needs to separate the red (R), green (G) and blue (B) wavebands from the white light. The most commonly employed element for color separation is a dichroic mirror (DM). The DM is constructed with thin-film stacks which induce interference for certain waveband and leave the others unaffected [9]. Hence, R, G and B wavebands can be separated by using the corresponding DMs as shown in Fig. 1.3. The basic requirements for these DMs are optical efficiency and high-temp durability.

Since LCOS systems are reflective-type devices, the PBS must be applied on the on-axis optical system in order to provide a crossed-polarizer condition. A PBS can pass light of one polarization state and reflect light of the other polarization state orthogonal to the former. When an s-wave is incident on a voltage-on LCOS panel, the reflected light will become p-wave which can pass through the PBS heading to the projection lens and display bright image on the screen. When the voltage is off, the reflected light will remain its polarization state, i.e. s-wave, and be reflected off by the PBS so that the image is dark. The gray level can be acquired by controlling the applied voltage on the LCOS panel. There are still several issues of PBS which will be addressed in the next section.

Another kind of optical engines commonly employed in LCOS projectors is the one-panel systems. Since only one light valve is needed, one-panel systems have the advantages of lower cost and compactness. Various designs have been developed in one-panel systems. There are in general two main approaches: time-sequential and sub-pixelated systems. For the time-sequential approach, the LCOS panel is illuminated by R, G and B lights subsequently. When the field-switching speed is fast enough that human eyes cannot catch up, the colors will be mixed together. Eventually, full color images are

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formed. Figure 1.6 demonstrates a typical time-sequential system using a rotating color wheel with red, green and blue segments [10]. The emerging light after passing the colored segments is incident on the LCOS panel with R, G and B bands subsequently. The basic requirement of this system is the fast response time of LC. The rotating frequency is at least 180 Hz which means the minimum response time of LC is only 5.5 ms. To achieve this requirement, one can either use faster LC material, e.g. ferroelectric LC [11], or shrink the cell gap. Another concern of this system is the low light efficiency due to the absorption of 2/3 of the light after passing the color segments. In order to overcome this drawback, Philips constructed a rotating prism system [12] as shown in Fig. 1.7. In this system, the white light is divided into R, G, B bands by dichroic mirrors. A rotating prism is placed in the optical path of each waveband. The function of these prisms is to produce colored stripes scrolling on the LCOS panel. In this condition, three wavebands are displayed at the same time, implying the light energy is preserved. The main challenge of this system is the reduction of etendue because only 1/3 of the panel area is covered by a certain color.

PBS Lamp Projection lens

R

G

B

_B

_R

_G

White light Mirror S-wave P-wave LCOS panel Color wheel PBS Lamp Projection lens

R

G

B

_B

_R

_G

White light Mirror S-wave P-wave LCOS panel Color wheel

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Scrolling sequence R G B White Light PBS Microdisplay G B R LCOS Panel Lamp Scrolling sequence R G B White Light PBS Microdisplay G B R LCOS Panel Lamp

JVC announced a single-panel system using a holographic color filter (HCF) with sub-pixelated architecture [13] as shown in Fig. 1.8. As white light encounters the HCF from a particular angle, it will be separated into R, G, B lights and directed to the corresponding sub-pixels. After modulated by the LC layer, the light will pass through the HCF again without diffractions. Finally, the light will be collected by the projection lens and displayed on the screen. This system has the advantage of high efficiency because three wavebands can be utilized simultaneously. However, the alignment of the HCF needs to be very precise which becomes a burden for yields.

Overall speaking, LCOS projection displays comprise various designs suitable for different applications. Several key issues of LCOS projection displays will be discussed in the next section.

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Silicon backplane

Spacer glass

_Holographic

color filter White Light

Glass substrate

R G B

B

G

R

LC

Silicon backplane

Spacer glass

_Holographic

color filter White Light

Glass substrate

R G B

B

G

R

LC

1.2 The issues of LCOS projection displays

1.2.1 The issues of optical engines

There are different kinds of considerations for different optical engines. Basically, a lower f-number (F#) is preferred because the light throughput is approximately inversely

proportional to the square of F# [14]. However, the lower F# implies a larger incident angle

which can cause depoloarization when light passes through the PBS.

Typically, a PBS is formed utilizing the physical effect of Brewster angle reflection. Based on Fresnel’s theory, the normalized reflectance of s-wave (Rs) and p-wave (Rp) when

passing an interface between two media can be expressed as: [15]

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2 2 2 2 2 s sin cos sin cos ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − − = θ n θ θ n θ R (1.1) 2 2 2 2 2 2 2 p sin cos sin cos ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − − = θ n θ n θ n θ n R (1.2)

where n is the relative index of refraction of the two media, θ is the angle between the

incident ray and the surface normal. The Brewster angle reflection occurs at the condition of θB=tan−1n. At this condition, the reflected ray becomes linearly polarized with the

electric vector transverse to the plane of incidence. The transmitted light is partially polarized as sketched in Fig. 1.9 (a). A calculated result of normalized reflectance with respect to the incident angle when light propagating from air to Ti2O3 (n=2.45) is shown in

Fig. 1.9 (b). MacNeille and Banning enhanced this effect by increasing the number of interfaces at which reflection occurred using multiple layers of thin inorganic films [16]. Modern designs can reach over 99% reflectance of s-wave and over 95% of p-wave transmission. 0.2 0.4 0.6 0.8 1.0 0 30 60 90

Angle of incident (deg.)

No rm al ize d re flec ta nc e s-wave

p-wave Brewster _{angle 67.8°} Incident ray

p-wave+s-wave

Reflected ray

s-wave Transmitted ray

p-wave+s-wave θ_B Surface normal (a) (b) 0.2 0.4 0.6 0.8 1.0 0 30 60 90

p-wave+s-wave

Reflected ray

p-wave+s-wave θ_B Surface normal 0.2 0.4 0.6 0.8 1.0 0 30 60 90

p-wave+s-wave

Reflected ray

p-wave+s-wave

θ_B Surface normal

(a) (b)

The weaknesses of the MacNeille PBS include sensitivity of transmission of p-wave and reflection of s-wave to the angle of incidence and wavelength [17]. The strong angular dependence of the reflectance can also be seen in Fig. 1.9 (b). For light with a small cone

Figure 1.9 (a) Sketch of the Brewster angle reflection; (b) an calculated results of normalized reflectance with respect to incident angle when light propagating from air to Ti2O3 (n=2.45).

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angle, there can be a significant amount of p-wave in the reflected beam, leading to depolarization of light.

Alternatives for MacNeille PBS have been developed to earn allowance of angle deviations. These technologies include the wire-grid polarizer (WGP) [18], polymer multiple layer [19] and quarter-wave plate compensations [20]. Details of these technologies are beyond the scope of this thesis and will not be mentioned. Recently, WGPs have been employed in some high-end LCOS projection displays not only due to its stability over a great range of incident angle, but also because of its durability from thermal effect. Heat-induced problems could be significant owing to high light flux generated by UHP lamps.

1.2.2 The issues of LCOS panels

The demand of ever-increasingly high resolution of LCOS devices leads to a smaller pixel size. Recall that the pixel size can be as small as 5-8 µm and the inter-pixel gaps are in sub-microns. However, several problems occur as the pixel size and the inter-pixel gaps are shrunk. For instance, the aperture ratio is decreased and the diffraction loss and fringe fields are both increased. How to maintain high brightness, high contrast ratio, high image sharpness and good dynamic response for a high-resolution LCOS projector is a challenging task.

Figure 1.10 sketches the fringing fields at the pixel edges when adjacent pixels operate at different voltages. As the inter-pixel gap is reduced, the lateral components of the fringing fields, generated by the voltage difference between the adjacent pixels, play a critical role in the optical performance. Different liquid crystal modes have different responses to these fringing fields. The influences of the fringing-field effect in different LC modes will be discussed in Chapter 3. Some other parameters of a LC cell, including the pretilt angle, cell thickness, electrode slopes and chiral dopant, are also related to this crucial effect.

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Glass ITO Silicon Backplane Al electrode Von Voff Electric field LC layer Glass ITO Silicon Backplane Al electrode Von Voff Electric field Electric field LC layer

Another important effect generated from tiny pixel pitches is diffraction. The periodic pixel electrodes act as a reflective grating as the pixel pitches become comparable to wavelengths of visible light. Therefore, part of the reflected light is diffracted or scattered out of the projection system as sketched in Fig. 1.11. This intrinsic physical effect can cause significant light loss especially when a high-definition panel is employed. Detailed analyses of diffraction effect of LCOS devices will be introduced in Chap. 5.

1.3 Aims of the research

LCOS devices have great potential for microdisplay applications ranging from data projectors, large-screen rear-projection televisions (RPTVs) to near-to-eye (NTE) virtual projections devices. The major advantage of the single-crystalline silicon is the high electron mobility, which allows very small pixel structures to be fabricated on the silicon substrate. However, as we have mentioned, the fringing-field effect and the diffraction

Figure 1.10 Sketche of the fringing fields at the pixel edges when adjacent pixels are operated at different voltages.

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effect are critical as the pixel pitch becomes too small. Efficiency can be decreased drastically due to diffraction loss. Fringing-field effects often cause poor image sharpness and degrade the dynamic response [21,22].

In this dissertation, we investigate the influences of fringing fields in different boundary conditions and distinct LC cell structures. Varying these parameters can give rise to different optical effects when light passes through the LC layer. Starting from understanding the LC director distributions under the influence of fringing fields, we are able to solve this long-standing problem. On the aspect of diffraction effect in high resolution LCOS devices, we developed an extended beam propagation method to simulate the optical performance of the devices when the pixel pitches become comparable to the visible wavelengths. By using this powerful tool, it is capable of designing a LCOS panel with least diffraction loss. In the followings, the basic theories and algorithm of simulations for LC director orientation and optical performances of LCOS devices are described in Chap. 2. Fringing-field effects in some commonly employed LC modes are investigated in Chap. 3. The influences of changing LC cell structure on fringing-field effect are also addressed in this chapter. In Chap. 4, we present a novel design called the circularly polarized light illuminated vertically aligned (CPVA) LCOS device which firstly eliminates the annoying fringing-field effect in vertically aligned (VA) LC mode. In Chap. 5, the analyses on diffraction effect of LCOS devices are studied by using BPM. A phase-compensated finger-on-plane mode is presented to save the light which is lost due to the strong diffraction in a conventional finger-on-plane LCOS device [22]. Finally, we summarize our work and make a brief conclusion. Suggestions are also given for future study.

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References

[1] Y. Yamamoto, T. Morita, Y. Yamana, F. Funada and K. Awane. Proc. 15th Intl. Research Conf. (Asia Display 95), pp.941 (1995).

[2] J. R. Van Raalte, “A new schlieren light valve for television projection.” Applied Optics,

9, pp.2225 (1970).

[3] P. M. Alt, “Single crystal silicon for high resolution displays.” Intl. Display Res. Conf.

Record, pp. M19-M22 (1997).

[4] L. Arend, J. Lubin, J. Gille and J. Larimer, “Color breakup in sequential scanned LCDs.” Soc. Information Display Tech. Digest, pp. 201 (1994).

[5] E. Schnedler and H. von. Wijngaarde, “Ultrahigh-intensity short-arc ling-life lamp system.” Soc. Information Display Tech. Digest, paper 11.1, pp. 131 (1995).

[6] A. H. J. van den Brandt, and W. A. G. Timmers, “Optical illumination system and projection apparatus comprising such a system.” U.S. patent 5,098,184 (1992).

[7] P. Michaloski and P. Tompkins, “Design and Analysis of illumination systems that use integrating rods or lens arrays. Intl. Optical Design Conference Technical Digest, pp-229 (1994).

[8] Y. Itoh, J. I. Nakamure, K. Yoneno, H. Kamakura and N. Okamaoto, “Ultra-high-efficiency LC projector using a polarized light illuminating system.” Soc.

Information Display Tech. Digest, pp. 993 (1997).

[9] J. Rancourt, Optical Thin-Films User Handbook (SPIE press, 1996). [10] Unaxis Optics Inc., http://optics.unaxis.com.

[11] M. D. Wand, W. N. Thurmes, R. T. Vohra, and K. M. More, “Advances in ferroelectric liquid crystals for microdisplay applications.” Soc. Information Display Tech. Digest, pp. 157 (1996).

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Display Tech. Digest, pp. 1072 (2001).

[13] T. Yamazaki, M. Tokumi, T. Suzuki, S. Nakagaki and S. Shimizu, “The Single-Panel D-ILA Hologram Device for ILATM Projection TV.” Intl. Display Workshop, pp. 1077 (2000).

[14] M. L. Jepsen, M. J. Ammer, M. Bolotski, J. J. Drolet, A. Gupta, Y. Lai, D. Huffman, H. Shi and C. Vieri, “High resolution LCOS microdisplay for single-, double- or triple-panel projection systems.” Displays 23, pp. 109 (2002).

[15] G. R. Fowles, Introduction to Modern Optics 2nd Edition (Holt, Rinehart, and Winston,

Inc., 1975).

[16] S. M. MacNeille, “Beam Splitter.” U.S. patent 2,403,731 (1946).

[17] E. H. Stupp and M. S. Brennesholtz, Projection Displays (John Wiley & Sons, 1999). [18] S. Arnold, E. Gardner, D. Hansen and R. Perkins, “An Improving Polarizing

Beamsplitter LCOS Projection Display Based on Wire-Grid Polarizers.” Soc.

Information Display Tech. Digest, pp. 1282 (2001).

[19] M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant Birefringence Optics in Multilayer Polymer Mirrors.” Science 287, pp.2451 (2000). [20] Y. Ji, J. Gandhi and M. Stefanov, Soc. Information Display Tech. Digest, pp.750

(1999).

[21] S. Zhang, M. Lu, and K. H. Yang, Soc. Information Display Tech. Digest, pp.898 (2000).

[22] W. Y. Chou, C. H. Hsu, S. W. Chang, H. C. Chiang and T. Y. Ho, Jpn. J. Appl. Phys.

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Chapter 2 Theory and Numerical Simulation

2.1 Theories of deformations and optical properties of

liquid crystals

To investigate the distributions of LC molecules under external fields, the continuum theory is often employed. It provides an insight of LC physic and was initiated by Oseen [1] and Zocher [2] back in 1933. Afterward, Frank had reexamined the theory critically and presented the theory of curvature elasticity [3]. With regard to the optical properties of nematic LCs, the optical rotation and birefringence effects are often induced. These phenomena had been studied by many authors, the most clear, rigorous and accessible reference being that of de Vries [4].

2.1.1 Continuum theory of liquid crystals

Many important physical properties of LC can be explained by continuum theory disregarding the details of the structure on the molecular scale. When external fields are applied, the LC molecules are rotated according to the field strength; when the fields are withdrew, the LC molecules return back to their original alignments. This effect is similar to the elastic deformations in solids. Therefore, it can be predicted that the energy generated by the deformations of LC molecules has the same form as that described by Hooke’s law.

LCs are either rod-like or disk-like molecules. In macroscopic point of view, LC molecules tend to align in a common direction, which is called “director” n. The continuum theory describes the distortions of n under some external forces. A good starting point is to explore the minimum energy condition. At first, some assumptions have to be addressed according to the characteristics of nematic:

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2. n and –n are indistinguishable, i.e. there is no permanent dipole in nematic.

3. All properties are identical when the coordinate is rotated around n, which is usually called C¶ symmetry.

4. The reflective symmetry of the plane including n. The properties of the system at two sides of the plane including n are the same.

Despite the influence of the surface energies, the free energy density glc due to the

distortion of n can be expressed as:

l k j i ijkl j i ij a a _, _, _, lc n n n g = + , (2.1)

where aij and aijkl are the constant coefficients depend on the material, and i, j, k, l =x, y ,z

which are the axes of a right-handed Cartesian coordinate system. This equation uses second-order approximation and omits the higher order terms. Based on the assumptions listed above, one can obtain a finally form of free energy density of a nematic as:

[

2

]

33 2 22 2 11 lc ( ) ( ) ( ) 2 1 g = k ∇⋅n +k n⋅∇×n +k n×∇×n , (2.2)

where kii, i=1, 2, 3 are the elastic constants associated with three basic types of LC

deformations, i.e. splay, twist and bend. It is shown in this equation that g=0 when there is no deformation of n. there

When nematic is doped with some chiral material, the LC directors present a helical structure as shown in Fig. 2.1. P0 denotes the natural pitch of the structure which represents

the distance along the helical axis for the directors to rotate 360°. A right-handed twist structure gives a positive P0 and the left-handed twist structure gives a negative P0. In this

case, the forth assumption is not satisfied. The orientations of LC directors are varied along the z axis in the natural balanced state. Therefore, an additional term must be added in Eq. (2.2) to interpret this phenomenon. The modified free energy density is given as:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × ∇ × + + × ∇ ⋅ + ⋅ ∇ = 2 33 2 0 22 2 11 lc ₂ ( ) ( ) ( ) 1 g n n n 2 k n n p k k π . (2.3)

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n n n n x y z 2 0 P n n n n x y z n n n n x y z 2 0 P

The effect of static electric field

The influence of static electric field on nematic LC can be explored by understanding the anisotropy of its dielectric constant. Usually, the dielectric constant ε can be expressed as a tensor: , ε ε ε n n ε δ ε ε // j i a ij ⊥ ⊥ + = + = a ij (2.4)

where δij is Kronecker symbol (δij=1 when i=j; otherwise δij=0; i, j=x, y ,z), ε// is the

dielectric constant measured along n and ε⊥ is the dielectric constant measured

perpendicular to n. When external electric fields are applied on liquid crystal, the electric displacement D can is given as:

) ( ε ε E n E

D= _⊥ + _a ⋅ . (2.5)

Hence, the electric free energy ge can be expressed as:

(

)

[

ε ε

]

. 2 1 2 1 g 2 2 e E n E D E ⋅ + = ⋅ = ⊥ a (2.6)

Since the first term in Eq (2.6) is not related to the director distribution and introduce a constant value for the energy of the system, it can be ignored when deriving the minimum

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energy condition. Therefore, the free energy density with external electric field can be expressed as:

(

)

[

2 2

]

2 33 2 0 22 2 11 e lc tot ε ε 2 1 ) ( ) ( ) ( 2 1 g g g E n E n n 2 n n n _⎥− + ⋅ ⎦ ⎤ ⎢ ⎣ ⎡ × ∇ × + + × ∇ ⋅ + ⋅ ∇ = − = ⊥ a k p k k

π

(2.7) Note that this equation is under the assumption of strong boundary condition. For weak boundary condition, a surface energy term must be included. There are several various approaches [5-9], the details are beyond the scope of this thesis and will not be addressed.

Euler-Lagrange equation

To determine the director distribution, we have to minimize the energy generated by the LC deformations. For simplicity, we suppose that all parameters in Eq. (2.7) depend on z-direction only. Therefore, the total free energy is given as:

(

)

[

]

dz, z ; dz ) ( dn , (z) n g dz ε ε 2 1 -dz ) ( ) ( ) ( 2 1 G 2 / d 2 / d i i tot 2 / d 2 / d 2 2 2 / d 2 / d 2 33 2 0 22 2 11 tot

∫

− − ⊥ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⋅ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × ∇ × + + × ∇ ⋅ + ⋅ ∇ = z k p k k a n E E n n 2 n n n π (2.8) where i=x, y ,z and d denotes the thickness of the LC layer. This is a function related to the integral path. Our purpose is to find a path, ni(z), from a to b which gives the minimum

integral value Gm. In this case, an arbitrary path, ni(z)+αδni(z), from a to b gives the value

of Gtot as:

( )

dz, dz ) ( δn d α dz ) ( dn , (z) αδn (z) n g α G i i i i tot tot

∫

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = b a z z (2.9) where α is a variable ranged from -1 to 1. Gtot(α) is either larger or equal to Gm, which

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0 ) α ( G α tot _α ₀ = ∂ ∂ = (2.10) Hence, 0 dz ] dz δn d dz dn g δn n g [ ) α ( G α i i tot i i tot tot ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ∂ ∂

∫

ab (2.11)

The second term in the integrand can be integrated by parts:

dz δn dz dn g dx d δn dz dn g dz dz δn d dz dn g i i tot i i tot i i tot ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂

∫

b a b a b a (2.12)

The integrated term vanishes because δni(a)= δni(b)=0. Therefore, Eq. (2.11) becomes:

0 dz

δn

]

dz

dn

g

dx

d

n

g

[

dz

]

δn

dz

dn

g

dx

d

δn

n

g

[

)

α

(

G

α

i i tot i tot i i tot i i tot tot

=

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

∂

−

∂

=

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

∂

−

∂

=

∂

∫

b a b a (2.13) Because δni is an arbitrary function, the integrand in Eq. (2.13) must vanish itself:

0 dz dn g dx d n g i tot i tot ₌ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ (2.14)

This result is known as Euler-Lagrange equation which is commonly used for solving the static problems of LC director distribution.

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2.1.2 Optical properties of an ideal helix: de Vries theory

As we mentioned previously, the LC directors present a helical structure when molecules possessing chirality are doped. The same structure is also found in pure cholesteric esters. Therefore, this helical structure is termed cholesteric. The general optical properties of an ideal cholesteric LCs (CLCs) contains many effects occurred in common LC devices [10,11]. The optical behaviors of light propagating in some commonly used LC modes, e.g. twisted nematic (TN), super-twisted nematic (STN) and homogenous, etc., are only the special cases of that in CLCs.

Let us recall the structure shown in Fig. 2.1. In this specific case, n is rotated with a helix axis, i.e. the z-axis. We can describe n in this ideal state as:

constant z q 0 n sin n cos n 0 z y x + = = = = φ φ φ (2.15)

where q0=2π/P0. In this configuration, n is horizontal everywhere and it is usually called the

“planar texture”. To analyze the optical properties of CLCs, the local dielectric constant at any point r must be given at first:

), ( )n ( )n ε ε ( δ ε ) ( ε_ij r = _⊥ _ij + _// + _⊥ _i r _j r (2.16)

where i, j=x, y, z. It is noteworthy that although this equation looks the same as Eq. (2.5), the physical meanings are different. Eq. (2.16) accounts for the optical behaviors of light propagating inside a uniaxial material and Eq. (2.5) accounts for the anisotropy of electric properties with external fields. In Eq. (2.16), the dielectric constants can be related associated with the refraction indices as:

, n ε ε 2 o 0 = ⊥ ε// =ε0n2e (2.17)

where ε0 is the permittivity of vacuum, no is the ordinary refractive index of LC and ne is

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propagates along the helix axis (i.e. the z-axis), the field components are: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ t t iω -y -iω x y x (z)e E (z)e E Re (zt) E (zt) E (2.18)

where t represents time and Re stands for the real part. The electric field vectors must satisfy the well-known wave equation:

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − y x * 2 y x 2 2 E E ) z ( ε c ω E E dz d _(2.19)

where c is the speed of light in vacuum and ε*_{can be deduced from Eqs. (2.15) and (2.16)}

that: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = ⊥ z) cos(2q z) sin(2q z) sin(2q z) cos(2q 2 ε 1 0 0 1 2 ε ε (z) ε 0 0 0 0 a // * _(2.20)

The wave equation can be simplified by using circularly polarized light:

y x y x iE E E iE E E − = + = − + (2.21)

Hence, Eq. (2.19) becomes:

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ₋ ₂ + -0 z 2iq 2 1 z 2iq 2 1 2 2 -2 2 E E k e k e k k c ω E E dz d 0 0 0 (2.22) where 2 ε ε c ω k , 2 ε ε c ω k // 2 2 1 // 2 2 0 ⊥ ⊥ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (2.23)

Solving Eq. (2.22) give the following forms of propagating waves:

)z q i(l )z q i(l 0 0 e E e E − − + + = = b a (2.24)

where a and b are constants. Substitute Eq. (2.24) into Eq. (2.22) gives the relations

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(

)

[

]

(

)

[

l q k

]

0. k , 0 k k q l 2 0 2 0 2 1 2 1 2 0 2 0 = − − + − = − − + b a b a (2.25)

The non-trivial solution of this equation is obtained only when the determinant vanishes: 0 k l 4q ) q l k ( 4 1 2 2 0 2 2 0 2 2 0 + + − − = − (2.26)

This equation gives the dispersion relation of CLCs, i.e. the relation between frequency ω and wavenumber l. Figure 2.2 (a) shows a typical dispersion relation of CLCs with ne=1.5578, no=1.4748 and P0=0.25 µm. It is shown that the curve is separated into two

branches by a band gap. The range of frequency of this band gap can be determined by setting l=0 in Eq. (2.26): 4 1 2 0 2 0 q k k − =± (2.27)

Using Eq. (2.23), the boundaries of the band gap can be given as:

e 0 0 0 n cq (0) ω , n cq (0) ω+ = − = (2.28)

where ω+(0) corresponds to the upper branch and ω-(0) corresponds to the lower branch.

For a incident wave with fixed ω, we can have four eigenmodes. Two propagate along +z direction (transmissive modes) and two propagate along –z direction (reflective modes). Here we focus on the transmissive modes, i. e. ∂ω/∂l>0.

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Fi gu re 2 .2 (a )T he di sp ersi on rel at io n of a C L C wi th ne =1 .5 578 , no =1 .47 48 and P0 =0.25 µm; th e in sert de no tes t he ch aracteristics of eig en m od es at so me sp ecific frequ en cies; (b ) Calcu lat ed ellip ticity with resp ect t o l 2 /q0 whe n –1<l 2 /q0 < 0; (c ) Calc ulated ellip ticity with resp ect t o l2 /q0 whe n 0< l2 /q0 <1 0 3 ; (d) Calcu lated ellip ticity with resp ect t o l1 /q0 whe n 0< l 1 /q0 <600.

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The reflective modes have similar behaviors. The wavenumber of the modes can be derived from Eq. (2.26):

(

)

(

q k

)

4k q k . ) ω ( l , k q 4k k q ) ω ( l 4 1 2 0 2 0 2 0 2 0 2 2 4 1 2 0 2 0 2 0 2 0 2 1 + − + = + + + = (2.29)

The polarization states of the eigenmodes can be deduced using the definition of ellipticity: . ρ b a b a + − = (2.30)

according to Eq. (2.25), we have

2 1 2 0 2 0 2 0 2 0 2 1 k k ) q (l k ) q (l k ₌ − − − + = b a (2.31)

Therefore, Eq. (2.30) becomes

2 1 2 0 2 2 0 0 2 1 2 0 2 0 2 0 2 0 2 1 2 1 2 0 2 0 2 0 2 0 2 1 2 1 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 1 k q l k q 2 k k ) q (l k ) q (l k k ] k ) q (l [ k ) q (l k k k ) q (l k ] k ) q (l [ k ) q (l k k ) q (l k ρ − − − = + − − + − + + − − − + + + − = + − − − − − = − + + + + − = l (2.32)

By using Eq. (2.26), we can rewrite Eq. (2.32) as

2 1 2 0 k s 2lq ρ − ± = (2.33) where 2 2 0 4 1 2 _k _4q _l s = + (2.34)

The sign of s in Eq. (2.33) depends on the branch used for ω(l) as shown in Fig. (2.2). Using Eq. (2.33), we calculate the frequency-dependent ellipticities of the eigenmodes as shown in Fig. 2.2 (b)-(d). They can be roughly divided into three regimes:

1. Elliptical Regime: ω->ω≥0 and 2q0c/∆n>>ω>ω+. There are two elliptical regimes

separated by the band gap of Bragg reflection. The eigenmodes are elliptically polarized with opposite sense of handness. For ω=0, the ellipticities become ±1, which indicates the eigenmodes become circularly polarized in this particular case , i. e. denoted by A1

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and A2 in Fig. 2.2 (a).

2. Bragg Regime: ω+>ω>ω-. In this regime, l2 is imaginary which indicates that the

corresponding eigenmode becomes an evanescent wave. On the other hand, the eigenmode correspond to l1 has its ellipticity close to -1. Thus, the transmitted wave is

almost (but not really) left-handed circularly polarized. Recall that P0=0.25 µm>0

corresponds to a right-handed helix. Hence, it can be concluded that the transmitted wave is almost circularly polarized with opposite rotational sense to that of the helix structure, and the reflective wave will have its rotational sense identical to that of the helix structure. For an extreme case when ne=no, the band gap shrinks to one frequency

ω=cq0/no. The corresponding wavelength λ= noP0 is in agreement with the usual Bragg

reflection condition. For the evanescent wave, the wavenumber can be expressed as:

(

)

[

]

2 1 2 0 2 0 4 1 2 0 2 0 2(ω) iκ i 4k q k q k l = = + − + (2.35)

From Eq. (2.24), we have the field vector written as:

.

e

Be

e

E

,

e

Ae

e

E

z iq κz -i z iq κz -)z q i(l z iq z -κ i z iq z -κ )z q i(l 0 0 0 2 0 0 0 2 − − − − + + + +

=

b a

b

a

φ φ (2.36)

where a and b can be imaginary which are then expressed as real numbers, A and B, multiplied by phasors, exp(iφa) and exp(iφb). Based on Eq. (2.25), we have

b b a a a b φ φ φ φ iBsin Bcos ) iAsin Acos ( k k ) q (iκ k k ) q (iκ 2 1 2 0 2 0 2 1 2 0 2 0 + = + − + = − + = (2.37)

Therefore, we can derive the relationship between A and B, that is

b a a b a a φ φ φ φ φ φ sin k cos A κq 2 sin ] κ k q [ cos k sin A κq 2 cos ] κ k q [ A B 2 1 0 2 2 0 2 0 2 1 0 2 2 0 2 0 + − − = − − − = (2.38)

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elliptically polarized. The angle between the long axis of polarization ellipse and the local director is (φa-φb)/2. From Eq. (2.38), the angle can be derived as:

. ] κ k q [ κq 2 ) tan( ₂ ₂ 0 2 0 0 − − = − _a b φ φ (2.39)

Figure 2.3 shows the calculate B/A and φb-φa with respect to frequency. As shown in

the Fig., the evanescent wave is linearly polarized. The vibration direction of electric field is varied from parallel to perpendicular to the local director n when changing the frequency from ω- to ω+.

3. Maugauin Regime: ω>>2q0c/∆n. In this high frequency regime, the eigenmodes have

their ellipticities either near-zero or near-infinity, which means they are almost linear polarized parallel or perpendicular to the local LC director. As shown in the insert diagram of Fig. 2.2 (a), E1 and E2 indicate the case when ω=500q0c, the ellipticities

become -0.024 and 42.693.

Table 2.1 summarizes the properties of eigenmodes in each regimes.

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Table 2.1 Th e prop ertie s o f eigenmod es in each regi mes.

反射式單晶矽液晶面板邊際場效應及繞射效應之研究

國立交通大學

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