Studies Of Liquid Crystal Response Time

Transcription

1 Uiversity f Cetral Flrida Electric Theses ad Dissertatis Dctral Dissertati (Ope Access) Studies Of Liquid Crystal Respse Time 005 Haiyig Wag Uiversity f Cetral Flrida Fid similar wrks at: Uiversity f Cetral Flrida Libraries Part f the Electrical ad Electrics Cmms STARS Citati Wag, Haiyig, "Studies Of Liquid Crystal Respse Time" (005). Electric Theses ad Dissertatis This Dctral Dissertati (Ope Access) is brught t yu fr free ad pe access by STARS. It has bee accepted fr iclusi i Electric Theses ad Dissertatis by a authrized admiistratr f STARS. Fr mre ifrmati, please ctact lee.dts@ucf.edu.

2 STUDIES OF LIQUID CRYSTAL RESPONSE TIME by HAIYING WANG B.S. Hagzhu Diazi Uiversity, P.R. Chia, 99 M.S. Hagzhu Diazi Uiversity, P.R. Chia, 995 M.S. Uiversity f Cetral Flrida, 004 A dissertati submitted i partial fulfillmet f the requiremets fr the degree f Dctr f Philsphy i the Departmet f Electrical ad Cmputer Egieerig i the Cllege f Egieerig ad Cmputer Sciece at the Uiversity f Cetral Flrida Orlad, Flrida Fall Term 005 Majr Prfessrs: Shi-Ts Wu Thmas X. Wu i

3 ABSTRACT I this dissertati, the respse time issue f the liquid crystal (LC) devices is ivestigated i meetig the challeges fr display ad phtic applicatis. The crrelati betwee the LC directr respse time ad the ptical respse time is derived theretically ad cfirmed experimetally. This thesis begis with a descripti f liquid crystal materials ad their physical prperties, ad the itrduces the simulati methdlgies. These brief but relevat itrductry chapters pave the fudati fr fully uderstadig the dyamic respse f LC devices. After that, three chapters pertaiig t ptical respse time are preseted. A majr ctributi f this thesis is that, based the small agle apprximati, we derive rigrus aalytical slutis fr crrelatig the LC directr respse time t its csequet ptical respse times (bth rise ad decay) f a vertical-aliged ematic LC cell. Pretilt agle effect the LC dyamics is studied, ad it is fud that a mdified rtatial viscsity is eeded i rder t explai the experimetal results. Grayscale switchig is als aalyzed umerically. This wrk successfully fills the gap i the literature f LCD switchig dyamics. A imprtat effect related t respse time, backflw is aalyzed usig a hmgeeus LC cell i a ifrared wavelegth. Due t the relatively high vltage applied i a ptical phased array (OPA), the backflw effect which takes place i the first few millisecds iflueces the LC respse time dramatically. Hwever, the crrespdig Leslie viscsity cefficiets, which are crucial i ivestigatig the dyamic respse f LC devices with backflw, ca hardly be fud i the literature. A ew effective apprach t estimate the Leslie cefficiets f LC ii

4 mixtures based MBBA data is prpsed i this dissertati. Usig this methd, the Leslie cefficiets f the LC material uder study ca be extracted based its rder parameters. The simulati results agree with the experimetal data very well. This methd prvides a useful tl fr aalyzig the dyamic respse icludig backflw, i rder t btai accurate ptical respse time uder a high biased vltage. Cell gap is a imprtat factr affectig the LC respse time. Usually a thier cell gap is chse t achieve faster respse time, sice rmally bth rise ad decay times are kw t be prprtial t d. Hwever, they are valid ly i the V < V < V regi, where V stads fr the threshld vltage f a LC cell. I the large vltage regi wherev th th π th < V < V i, the ptical decay time is idepedet f d. I this thesis, we fid that betwee these tw extremes the respse time is basically liearly prprtial t d. Our aalytical derivati is validated by experimetal results. Therefre, i the whle vltage regi, the physical picture f the ptical respse time as a fucti f the cell gap is cmpleted. This aalysis is useful fr uderstadig the grayscale switchig behavirs f the LC phase mdulatrs. With the help f cell gap effect i phase mdulatr, we ca effectively reduce the respse time by usig a thick cell r duble cells t achieve the iteded phase retardati i a aticipated peratig wavelegth. I cclusi, this dissertati has slved sme imprtat issues related t LC ptical respse time ad supplied valuable tls fr scietists ad egieers t umerically aalyze the LC dyamics. iii

5 T Yujig ad Lipig iv

6 ACKNOWLEDGMENTS I wuld like t thak thse wh ctributed ad supprted my effrts thrughut my graduate studies at the Uiversity f Cetral Flrida. First ad fremst I wish t recrd my gratitude t my advisr, Dr. Shi-Ts Wu fr his fiacial supprt durig these years, withut that, I culd t have ctiued my graduate studies at UCF. Meatime, I wuld like t thak him fr his creative sese,cstat help, mtivati ad valuable guidace i this area, which pes a ew wider wrld t me Liquid Crystal. I wuld like t thak him fr his detailed ad kid istructis i the labratry, fr his patiece ad istructi t imprve my ral ad writte Eglish skills. I am sicerely idebted t my advisr, Dr. Thmas X. Wu fr his elighteig discussi derivati f the crrelati fucti abut the ptical ad directr respse times. I wuld like t thak him fr his patiece ad time t plish my mauscripts. Uder his guidace, I have gtte deeper kwledge i electrmagetic field. I have als imprved my capability t perfrm umerical simulati, which icludes Fiite Elemet, Fiite Differece, ad ther methds. I wuld als like t thak my dissertati cmmittee members, Dr. Aiket Bhattacharya, Dr. Issa Batarseh, ad Dr. Jui J. Liu, fr their help ad ivaluable suggestis i my dissertati. I gratefully ackwledge all the Pstdctral Fellws i the Phtics ad Display grup ad i particular Dr. Xiyu Zhu ad Dr. Yuhua Huag fr their helpful discussis i liquid crystal displays. I appreciate Dr. Yaqig Lu fr his itesive preparati ad psitive attitude. I am idebted t Dr. Hgwe Re ad Dr. Sebastia Gauza fr metrig me i the experimets. I wuld like t give thaks t Dr. W. K. Chi fr his metrship i -D dimmos ad Simulati v

7 sftware as well as useful discussi i techical issues. I am grateful t all the studets i the Phtics ad Display grup ad High-Speed Electric System grups, mst f them have helped me i e way r ather. I particular I wuld like t thak Dr. Claire Y. Fa, Ms. Yi-Hsi Li, ad Ms. Chie-Hui We, fr their help ad ecuragemet. Fially, it is a pleasure t give thaks t my husbad, Lipig Zheg. He als pursued his Ph. D. degree this perid ad we have a lvely yug daughter t be take care f by urselves. He lives cheerfully ad withut cmplaits. I am frever grateful t my parets fr their mral supprt durig this perid. vi

8 TABLE OF CONTENTS LIST OF FIGURES... xi LIST OF TABLES... xvi LIST OF ACRONYMS/ABBREVIATIONS... xvii CHAPTER : INTRODUCTION.... Mtivati.... Thesis Overview... 5 CHAPTER : LIQUID CRYSTAL MATERIALS AND PHYSICAL PROPERTIES LC Mixtures Basic Physical Prperties Birefrigece Wavelegth Effect Temperature Effect Dielectric Aistrpy Mlecular Structure Effect Temperature Effect Frequecy Effect Visc-elastic Prperties Elastic Cstats Rtatial Viscsity Figure f Merit....3 Achrig Eergy... 4 vii

9 .4 Factrs Affectig LC Respse Time... 5 CHAPTER 3: SIMULATION METHODOLOGIES Simulati f Directr Distributi Withut Backflw Effect With Backflw Effect Optical Simulati Matrix Methd Matrix r Exteded Jes Matrix Methd CHAPTER 4: CORRELATIONS BETWEEN LIQUID CRYSTAL DIRECTOR REORIENTAION AND OPTICAL RESPONSE TIME OF A HOMEOTROPIC CELL Itrducti Thery Decay Time Rise Time Results ad Discussi Pretilt Agle Effect Gray Scale Switchig Detailed Crrelatis Decay Time Rise Time Cclusi... 6 CHAPTER 5: BACKFLOW EFFECT ON THE DYNAMIC RESPONSE OF LIQUID CRYSTAL PHASE MODULATOR... 6 viii

10 5. Itrducti Theretical Backgrud Ericks-Leslie Equati Temperature Effect Temperature-depedet Leslie Cefficiets Methd t Estimate Leslie Cefficiets Experimet Results ad Discussi E UCF Cclusi... 8 CHAPTER 6: CELL GAP EFFECT ON THE DYNAMICS OF LIQUID CRYSTAL MODULATORS Itrducti Theretical Backgrud Small Sigal Regi Large Sigle Regi Middle Sigal Regi Experimet Results ad Discussi Middle Sigal Regi Large Sigal Regi Cclusi ix

11 CHAPTER 7: SUMMARY LIST OF REFERENCES LIST OF PUBLICATIONS... 0 x

12 LIST OF FIGURES Figure : Schematics f three thermtrpic LCs: (a) ematic, (b) smectic, ad (c) chlesteric Figure : Wavelegth-depedet birefrigece i VIS [38].... Figure 3: Wavelegth depedet birefrigece i IR [38].... Figure 4: Temperature-depedet refractive idices f 5CB at λ = 450 (triagles), 550 (circles) ad 650 m (squares). Ope dts are fr ad clsed dts are fr [43] e Figure 5: Demstratis f three basic defrmatis f LC... 9 Figure 6: Temperature-depedet birefrigece f UCF-. Blue dts are experimetal data at λ =633 m.... Figure 7: Temperature-depedet visc-elastic cefficiet γ /k f UCF-. Blue dts are experimetal data... Figure 8: Temperature depedet figure-f-merit f UCF- (squares) ad E7 (circles). Slid lies are fittigs t the experimetal data f UCF- usig E =340 mev, β =0.5, ad T c =4 C at λ = 633 m Figure 9: The crdiate system f LC directr... 8 Figure 0: Schematic diagram f a LC pael, which is divided it N layers Figure : The VA cell used fr this study. The LC cell is sadwiched betwee tw crssed plarizers. The ier side f each glass substrate is cated with a thi layer f idium-tixide ad plyimide fr prducig hmetrpic aligmet. The LC has a small pretilt agle xi

13 Figure : The simulated vltage-depedet trasmittace f a VA cell at λ = 550 m with three differet pretilt agles, = 0.0 (dashed lie), (slid lie) ad 5 (dashed dt lie). The parameters used i simulatis are listed i the text Figure 3: (a) Optical decay time (90% 0%) ad (b) rise time (0% 90%) as a fucti f V/V th at fur differet pretilt agles, =,, 3, ad Figure 4: The eight gray levels f the VA cell at λ = 550 m LC: MLC-6608, d = 4.64 µm ad pretilt agle = Figure 5: (a) The calculated LC directr distributi φ(z) as a fucti f rmalized cell gap (z/d). (b) Time-depedet l[ δ / δ ( t) f the VA cell. Dts are calculated data ad slid ] lie is the fittig curve. Frm the slpe f the straight lie, τ is fud t be ~6 ms Figure 6: The crrelati f the ptical decay time T decay (90% 0%) vs. the LC directr rerietati time ( τ ) as a fucti fδ. Circles represet the simulati results usig the Ericks-Leslie equati, while the slid lie is the crrelati btaied frm the small agle apprximati [Eq. (4.4)] Figure 7: The crrelati f ptical rise time T rise (0% 90%) vs. the directr rerietati time ( τ ) as a fuctiδ. Circles represet the simulati results usig the Ericks-Leslie equati, while the slid lie is the crrelati btaied frm the small agle apprximati Figure 8: The crdiate system f LC directr, where θ is the tilt agle f the LC directr, which is the agle betwee the LC directr ad the x-y plae, ad φ is the agle betwee prjecti f the LC directr the x-y plae ad the x-axis xii

14 Figure 0: The Fittig results f fr MBBA (a) usig Eq. (5.8) ad (b) usig Eq. (5.). Ope dts are measured data ad filled dts are the fittig results Figure : The temperature-depedet rtatial viscsity f MBBA. Dts are the experimetal data calculated frm Table 6, dashed lies are fittig results usig Eq. (5.9), ad slid lie is fittig usig Eq. (5.0) Figure : Vltage-depedet trasmissi f a 3.4-µm-thick hmgeeus E7 cell betwee crssed plarizers at three differet temperatures: T =0.3 C (gray lie), 33.3 C (dt-dashed lie) ad 46.9 C (slid lie). λ =.55 µm Figure 3: The temperature depedet γ f E7. Dts are measured data ad slid lies are the fittig curves usig Eq. (5.0) Figure 4: The experimetal ptical dyamic respses f a hmgeeus E7 cell at three differet temperatures: T = 0.3 C (grey lie), 33.3 C (dt-dashed lie) ad 46.9 C (slid lie). d =3.4 µm ad λ =.55 µm Figure 5: The simulated tilt agle distributi f a hmgeeus E7 cell at T = 46.9 C durig its first 30 ms relaxati. d =3.4 µm ad λ =.55 µm. The parameters used fr simulatis are listed i Table Figure 6: The trasiet ptical dyamic respses f a hmgeeus E7 LC cell at three differet temperatures: (a) T = 0.3 C, (b) T =33.3 C ad (c) T =46.9 C. d =3.4 µm ad λ =.55 µm. The slid lies represet the experimetal results, while the dt-dashed es are the simulati results. The parameters used fr simulatis are listed i Table xiii

15 Figure 7: The temperature-depedet Leslie cefficiet 4 fr E7. The slid lie represets the fittig curve usig Eq. (5.4) with E =0.35 ev, while the dts represet the ptimal values f 4 at each temperature Figure 8: The trasiet ptical dyamic respses f a hmgeeus cell usig UCF-0 LC mixture at T =70 C. d =7.8 µm ad λ =.55 µm. The slid lie shws the experimetal results ad the dt-dashed lies are the simulati results. The parameters used fr simulatis are listed i Table Figure 9: Simulated vltage-depedet trasmittace curve f a 5.6-µm hmgeeus E7 cell with 3 pretilt agle betwee paralleled plarizers at λ = 633 m ad 3 C, where i small sigal regi (Part I), small agle apprximati hlds well; while i high sigal regi (Part III), the trasiet ematic effect is satisfied Figure 30: Directr distributi f a 6.6-µm hmgeeus E7 cell betwee paralleled plarizers at λ = 633 m ad 3 C. Slid lie crrespds t its free decay frm iitial biased vltage V i = 4.8 V rms t its first trasmittace miimum, while dtted lie is its free decay frm iitial phase δ i =π t π Figure 3: Simulated trasiet phase chage i the middle cycle f a 0-µm hmgeeus E7 cell with 3 pretilt agle at T = 3 C ad λ =633 m released frm V i =.85 V rms. The gray lie shws the simulated trasiet phase chage, while the red dash lie is the fittig data usig Eq. (6.6). Thus, the trasiet phase chage i the middle cycle fllws the small agle apprximati xiv

16 Figure 3: Measured ad calculated ptical decay times i the middle cycle as a fucti f the cell gap, where the filled ad the empty squares shw the experimetal ad calculati results i Eq. (6.5), respectively... 9 Figure 33: Experimetal ptical decay curve f a 0.7-µm hmgeeus E7 cell betwee crssed plarizers at T =3 C ad λ = 633 m frm the iitial bias vltage V i =4.8 V rms (light blue lie) Figure 34: Optical respse time t p at the last cycle is as a fucti f the cell gap. LC used is E7 at T = 3 C ad λ = 633 m. Beyd d >0 µm, t p is isesitive t d as expected i Sec xv

17 LIST OF TABLES Table : The effects f differet factrs respse time... 5 Table : Simulati results f phase decay time ad ptical decay time at differet vltages f a VA cell. LC: MLC-6608, d = 4.64 µm, pretilt agle = 0.0 ad V th =.9 V rms. Here, t p is the phase decay time, T decay is the ptical decay time, ad τ =.4 ms is the directr s decay time as defied i Eq. (4.5) Table 3: Same as Table except the pretilt agle = * Table 4: Pretilt agle effect the LC directr s decay time τ Table 5: The calculated eight gray level ptical rise time (0% 90%) ad decay time (90% 0%) f the VA cell shw i Figure Table 6: Recmmeded Leslie cefficiets fr MBBA (frm Ref [90]). All i s are i uit f Pa β s. S is calculated frm S = T / T ) with β = 0.88 ad T c = 39. K ( c Table 7: Parameters btaied frm fittig MBBA data with the IO thery (Eqs. (5.) (5.7)). All the parameters are i uits f Pa s Table 8: Sme measured ad calculated LC parameters f E7 at varius temperatures. λ =.55 µm ad T c =60 C Table 9: Sme measured ad calculated LC parameters f UCF-0 at T =70 C ad λ =.55 µm Table 0: Measured ptical respse times f three varius cells d = 7.78, 0.7 ad 6. µm i differet cycles. The ptical respse time i the last cycle is iitially biased at V i =9.5 V rms at t =0 ms xvi

18 LIST OF ACRONYMS/ABBREVIATIONS AM CRT DAP DCC FLC FM IO thery IC LC LCD MTN MCIF MVA OPA OCB PDLC PNLC SLM TFT VA Active-Matrix Cathde Ray Tube Defrmati f Aliged Phases Dyamic Capacitace Cmpesati Ferrelectric Liquid Crystal Figure f Merit Imura ad Oka Thery Itegrated Circuits Liquid Crystal Liquid Crystal Display Mixed-mde Twisted Nematic Mti-Cmpesated Iverse Filterig Multi-dmai Vertical Aligmet Optical Phased Arrays Optical Cmpesati Bed Plymer Dispersed Liquid Crystal Plymer Netwrk Liquid Crystal Spatial Light Mdulatr Thi Film Trasistr Vertical-Aliged xvii

19 CHAPTER : INTRODUCTION. Mtivati Liquid crystal techlgy, which is related t multi-discipliary sciece, has fud its way it every mder husehld i displays ad phtics. I display area, liquid crystal displays (LCDs) ca be fud i hadheld (0.5-4 ), tebk cmputers (0-5 ), desktp mitrs (5- ), LCD-TVs (7-46 +), HDTVs (8 ), ad digital mvies (4 7 ). I mbile cmmuicatis, LCDs are cmmplace i palm pilts, hadheld termials, mbile Iterets, wearable cmputers, wearable TVs, ad car avigatis. I phtics area, liquid crystal (LC) spatial light mdulatr (SLM) has bee used as a phase-ly mdulati fr laser beam steerig, tuable-fcus les, ad ther phtic devices. Liquid crystal devices have bee imprved remarkably ver the last several years. Mrever, ivati has bee takig place every e ad a half years. LCDs have uique features such as flat ad cmpact structure, high image quality, lw cst, lw pwer csumpti tha Cathde Ray Tubes (CRTs), ad they are easily iterfaced with itegrated circuits (IC). Hwever, there are still sme techical issues remai t be vercme. Cmpared t flat scree CRTs ad Plasma mitrs, their vide refresh rate is slwer, ad the viewig agles are arrwer. I this dissertati, we fcus the respse time issue. Fr TV applicatis, LCD is expected t displace CRT very s, as it des i the cmputer mitrs. The rapidly shiftig vides require the LC device t perate at the true vide rate that is less tha 6 ms. Hwever, the respse time f LCDs usig ematic is still t fast

20 eugh fr vide display. Blurred edges i mvig images impede its acceptace i the TV market. Fr trasmissive LCDs, the preset respse time (rise + decay) is apprachig 0 ms, but i sme fast-mvig pictures, the respse time shuld be less tha 5 ms. Fr prjecti displays, the respse time culd be faster tha 6 ms, thaks t the thermal effect f the high pwer lamp. I hadheld displays, the respse time i the Guest-Hst (GH) display is abut 50 ms ad that i the Mixed-mde Twisted Nematic (MTN) mde is ~0 ms, depedig the cell gap. Hwever, fr a clr sequetial micrdisplay, the required frame time shuld be less tha ms. This presets a techical challege t ematic LCDs, especially i the lw temperature regi. I meetig the challeges, the gal f this research wrk is t discver the apprpriate fast respse appraches f LC devices. I pursuit f imprvig the respse time f ematic LCDs, a lt f appraches have bee carried ut. Fr istace, t imprve the material perfrmace ew mlecular structures with lw rtatial viscsity (γ ) ad high birefrigece ( ) [],[] have bee develped. With the adaptive verdrive ad udersht methd cceived i late 980s [3],[4], the itrisic slw iter-gray respse time ca be reduced t less tha e furth that f the cvetial methd [5],[6]. With the itrducti f Dyamic Capacitace Cmpesati (DCC) fr active-matrix (AM) LCDs i 000 [7], LCDs made a jump i image quality. By icrpratig DCC drivig scheme ad faster liquid crystal material, Samsug develped a TFT-LCD with all gray levels less tha 0 ms [8]. Amg all LC mdes, Optically Cmpesated Bed (OCB) mde has the fastest respse ptetial. Hwever, tw majr techlgical issues still eed t be addressed: ) hw t speed up the iitial 'splay t bed' trasiti, ad ) hw pssibly t accmplish a excellet image quality

21 that surpasses thse f cvetial wide viewig LCD mdes. T realize its bed trasiti at a lw vltage ad speed up this trasiti, Uchida grup [9] studied bed-shaped plymer etwrk frmati ad Bs grup [0] develped high pretilt aligmet layer t keep the bed rietati at lw vltage. Nakamura et al. [] prpsed a methd t speed up the 'splay t bed' trasiti by applyig repeated pulse type bias vltage. Ezhv et al. [] develped a 3D system based LC-shutters ad plarizati switchable large paels. Lee grup [3] develped ew drivig methd i 7.0 SVGA pael t speed up the iitial splay t bed trasiti time ad withut sacrificig the perfrmace f OCB pael. T achieve fast respse time, several appraches have bee reprted. Oe apprach is t use dual-frequecy liquid crystal (DFLC) materials [4],[5]. I a DFLC device, electric field is preset durig bth tur- ad tur-ff perids t accelerate the LC rerietati prcesses. As a result, the rise ad decay times are greatly imprved. Hwever, a DFLC material is a mixture ctaiig psitive ad egative ε LC cmpets. Therefre, their ε values at lw ad high frequecy regimes are t large. That meas, the required peratig vltage is relatively high. Mrever, the crssver frequecy shifts ticeably as the peratig temperature icreases. This makes precise grayscale ctrl difficult. Plymer etwrk liquid crystal (PNLC) [6] exhibits a fast respse time. Usually, a PNLC scatters light i the visible regi. Hwever, as the wavelegth icreases, light scatterig dimiishes i the λ=.55 µm regi. Fr a -µm plaar LC cell, the respse time fr achievig a π phase chage is abut ms at rm temperature. Hwever, due t the 0 wt% plymer ccetrati ad small etwrk dmais, the peratig vltage is relatively high, 7 V rms /µm. T reduce vltage, a high birefrigece ad large ε LC material eeds t be develped. 3

22 Liquid crystal respse time is limited by its slw free relaxati. T shrte its decay time, ather rthgal electrde alg the substrate is suggested, which is used t ctrl its decay prcess. I ther wrds, the decay is drive by the electric field. The ccept f this crssed-field effect was first prpsed i 975 [7], hwever, it had t bee actually implemeted i display devices because f the fllwig shrtcmigs: high vltage, lw ctrast, mre cmplicated drivig, mre cmplicated structure, ad -uifrm trasmissi. I the past tw years, Xiag et al. [8]-[] aalytically explred the electr-ptic prperties f VA ad HAN cells with differet electrde cfiguratis (three electrdes [8]-[0] r fur electrdes [],[]) t achieve a fast decay time usig the crssed-field effect. T imprve the mti image quality, Lee et al. [7] ad Pa et al. [3] idepedetly fud that the slw respse f LCD is t a dmiat factr f mti blur. The slw respse f LCD ly ctributes t 30% f the mti blur, while the hld-type rederig mde ctributes t 70%. T reduce the hld-type mti blur, several cmpaies, such as Samsug, Philips, NEC, et al. have sme useful slutis. The first apprach is t itrduce a impulsive behavir t the trasmittace, either by isertig black data betwee every tw frames [4], r by usig blikig backlight t elimiate the flickers [5],[6]. Hwever, these slutis have the itrisic shrtcmigs f lw trasmittace ad ctrast rati. Ather apprach is t duble the frame rate [7]. Hwever, it requires extesive sigal prcessig especially fr mti estimati ad iterplati. Phillips applied MCIF (Mti-cmpesated iverse filterig) a 30 LCD-TV pae i 004 ad btaied a CRT-like display withut mti blurrig [8]. 4

23 . Thesis Overview The research wrk discussed i this dissertati cvers liquid crystal devices with a gal f meetig the respse time challeges fr displays ad ptical cmmuicatis. It ca be divided it three categries: ) Mtivati ad itrducti, ) Varius vel electr-ptical effects that related t ptical respse time i ematic LC mixtures, ad 3) Summary. The dissertati is rgaized as fllws. Chapter itrduces the LC mixtures ad their crrespdig physical prperties. Chapter 3 reviews the umerical simulati methds fr liquid crystal devices. I Chapter 4, we give the detailed derivati f the crrelatis betwee the LC directr rerietati time ad the ptical respse (bth decay ad rise) time f a Vertically Aliged (VA) cell. Pretilt agle effect i respse time is als calibrated by the mdified rtatial viscsity. I Chapter 5, a imprtat effect related t respse time, backflw is aalyzed usig a hmgeeus LC cell at λ =.55 µm, which is cmmly used fr laser cmmuicati. Due t the relatively high vltage applied i a ptical phased array, the backflw takes place i the first few millisecds. We have mdified the expressis f tw Leslie cefficiets ( ad ) i rder t exted their validity t tw high birefrigece LC mixtures, E7 ad UCF- based the Imura ad Oka (IO) thery. With these mdificatis, the simulati results agree quite well with the experimetal data. I Chapter 6, a iterestig apprach t achieve fast respse time was ivestigated. This chapter prvides a cmplete physical picture f ptical respse time as a fucti f cell gap (d) i a wide vltage regime. Nrmally bth rise ad decay times are kw t be prprtial t d 5

24 i the small agle apprximati ( V < V < V ), ad i the high vltage regime where th th V < V < π V i the ptical decay time is idepedet f d. I this chapter, we fud that betwee these tw extremes, there is a regi where the respse time is liearly prprtial t d. Our aalytical derivati is cfirmed by experimetal results. Fially, i Chapter 7, we summarize the key achievemets fr this dissertati. 6

25 CHAPTER : LIQUID CRYSTAL MATERIALS AND PHYSICAL PROPERTIES Althugh liquid crystal ly ccupies a small prti i a LC device, it plays a key rle i determiig the device perfrmaces. Fr example, the LC material ad mlecular aligmet jitly determie the device ctrast rati, perati vltage, respse time, viewig agle, ad peratig temperature. Eve the material stability is critical t the lifetime f the devices. LC material is the fudametal f the whle LCD idustries ad researches. Therefre, this chapter will briefly itrduce the LC classes ad its imprtat physical prperties.. LC Mixtures Liquid crystal is a rgaic mlecule. Liquid crystals pssess physical prperties that are itermediate betwee slid crystals ad istrpic liquids [9]. They have the flw ad cfrmability prperties f liquid, but exhibit crystallie prperties, e.g., ptical, dielectric, ad elastic aistrpies, whe the LC mlecules are aliged by a surfactat. There are three majr LCs f iterest: thermtrpic, lytrpic, ad plymeric. Amg these three, the thermtrpic LCs have bee studied extesively ad their applicatis mature. Thermtrpic LCs ca exist i three phases: ematic, smectic ad chlesteric, as illustrated i Figure. I the ematic phase shw i Figure (a), the rd-like mlecules are, average, lie up parallel t a preferred directi, which is characterized by a uit vectr, called the directr. The directr ca be rerieted by a exteral electric field whe the applied vltage exceeds the Freedericksz trasiti threshld. Sice the liquid crystals are birefrigece media, the fieldiduced directr rerietati imparts a large phase chages a traversig wave. Due t their 7

26 simple mlecular aligmet ad atural gray scale capability, ematic LCs have becme the maistream fr display ad tuable phtics applicatis. I ur research iterest, emphasis is maily the ematic phase. (a) (b) (c) Figure : Schematics f three thermtrpic LCs: (a) ematic, (b) smectic, ad (c) chlesteric. I the smectic phase shw i Figure (b), the mlecules are i layered structures ad arrage well with each ther. There are at least ie distict phases depedig the degree f rderig withi thse plaes. A excitig feature f the chiral smectic-c phase is that it exhibits ferrelectricity. Usig its sptaeus plarizati, the respse time f a bistable ferrelectric liquid crystal (FLC) mdulatr is i the 0-00 µs rage. FLC has bee demstrated a silic backplae fr virtual ad prjecti displays [30]. Due t its fast respse time, the FLC ca perfrm clr sequetial ad pulse-width mdulati t btai gray scales. I the chlesteric phase shw i Figure (c), it is thermdyamically equivalet t ematic phase except fr the chiral-iduced helix i the directrs. There is parallel rderig i the same plae, but helical rtati alg the axis perpedicular t differet plaes. The LC directrs fllw the helical 8

27 rtati f the chiral. The plarizati states f the reflected ad trasmitted waves deped the pitch legth f the chlesteric. Because f its bistable ature (lw pwer csumpti), the chlesteric display is a attractive cadidate fr electric bks [3].. Basic Physical Prperties Liquid crystal exhibits a certai degree f rder i the mlecular arragemet. As a result, there is aistrpy i the mechaical, electrical, magetic, ad ptical prperties. A umber f uique characteristics make LC particularly suitable fr displays. Fr electr-ptic applicatis emplyig a ematic LC, birefrigece, elastic cstats, dielectric aistrpy, ad rtatial viscsity all play imprtat rle affectig the device perfrmace. The fllwig sectis will give a verview the basic physical prperties f LC materials. The birefrigece ( = e.. Birefrigece ) f a uiaxial liquid crystal represets the differece betwee the refractive idices f the extrardiary ray ( ) ad the rdiary ray ( ). High birefrigece LC materials are essetial fr bth display ad ptical cmmuicati applicatis [3]. Fr a chlesteric liquid crystal display (Ch-LCD) [4],[3], the reflecti badwidth ( λ) is liearly prprtial t the LC birefrigece ad pitch legth (p) as: λ = p. I the visible spectral regi, the pitch legth is ~350 m. Thus, if =0.6, the λ =0 m which wuld resemble a rmally white Ch-LCD. Fr a Plymer-Dispersed Liquid Crystals (PDLC) [33],[34] r hlgraphic PDLC [35], high ehaces the light scatterig efficiecy ad thus imprves the display ctrast rati. Fr fiber-ptic switches (λ =.55 µm) usig ptical phased arrays [36], e 9

28 the required phase chage ( δ = πd / λ ) is π. If we wat t retai the cell gap d =4 µm t achieve a fast respse time, the required shuld be ~0.4 at λ =.55 µm, which implies ~0.5 i the visible regi. I rder t describe the refractive idex dispersi f LC materials, tw issues eed t be take it accut: wavelegth effect ad temperature effect.... Wavelegth Effect Vuks prpsed the fllwig equati t crrelate the macrscpic bservables (refractive idices) f a aistrpic LC t its micrscpic prperties (the mlecular plarizability ) [37]: e, 4π = N + 3 e,, (.) where = ( e + )/ 3is the average value f the refractive idices i the ematic phase, N is the umber f mlecules per uit vlume. Wu prpsed a sigle-bad mdel, where the wavelegth-depedet birefrigece f a LC ca be expressed as fllws [3],[38],[39]: * λ λ e ( λ, T ) i ( λ) + GS, (.) * 3 λ λ * GS λ λ 0 ( λ, T ) i ( λ), (.3) * 3 λ λ * λ λ = G. (.4) * λ λ 0

29 I Eq. (.4), the parameter G = gnzs( f // f ) is temperature depedet but wavelegth idepedet, ad * λ is the mea electric trasiti wavelegth. Fr a LC substace, the electric trasiti bads are lcated i the UV regi. Equati (.4) ivlves tw parameters G ad λ *. Thus, by measurig the birefrigece at tw wavelegths, these tw parameters ca be determied ad the etire birefrigece dispersi curve ca be btaied. Frm Eq. (.4), as the wavelegth icreases, decreases gradually ad saturates i the ear ifrared regi. I the IR r millimeter regi where λ>>λ *, Eq. (.4) is * reduced t Gλ, except i the viciities f sme lcal mlecular vibrati bads. I the IR regi, sme harmics f mlecular vibrati bads exist. Figure ad Figure 3 shw the wavelegth-depedet birefrigece curves f tw eutectic mixtures E-7 (cyabiphey mixtures) ad ZLI-3 (cya-pheyl-cyclhexaes mixtures) i the visible ad IR regi, respectively, at rm temperature. The dashed lies represet the theretical results usig Eq. (.4). I Figure 3, psitive dispersi f C N bds (cetered at 4.5 µm) ad C-H bads (~3.4 µm) are bserved i bth materials. As a result, the itrisic absrpti ca be relatively large depedig the detailed mlecular cmpsiti. A LC with a lg alkyl chai wuld ehace the absrpti itesity f the C-H bad. The birefrigece saturates i the ear ifrared regi except i the viciities f sme lcal mlecular vibrati bads. The measured birefrigece results (E-7 ad BL006 at 30GHz are 0.9 ad 0.3, respectively) by Lim [40] are csistet with this aalysis.

30 Figure : Wavelegth-depedet birefrigece i VIS [38]. Figure 3: Wavelegth depedet birefrigece i IR [38]. Wu als prpsed a three-bad mdel, which takes all three electric trasiti bads: e σ σ * ad tw majr π π * it csiderati [4],[4]:, ) ( 0 λ λ λ λ λ λ λ λ λ λ λ λ λ = e e e e g g g (.5)

31 λ λ λ λ λ λ 0 ( λ) = + g 0 + g + g, (.6) λ λ λ λ λ λ where g ~ NZ f are prprtiality cstats, Z 0 is the umber f σ electrs, ad Z = Z is i i i the umber f π electr i the LC mixture. I the ff-resace regi, the three-bad mdel ( ( λ ) 0. 6) agrees well with measured birefrigece results (5CB at 35 GHz are 0.58). Fr LC mlecules with maily saturated bds (such as CH 3, CH, C 6 H ), the σ σ * trasitis makes the primary ctributi t birefrigece. Their resace wavelegth is i deep UV (<80 m) s that their birefrigece is lw. While fr the LC mlecules with usaturated bds (such as pheyl rig), the tw π π * trasitis make dmiat ctributi t LC birefrigece. As cjugati legth icreases, π π * trasiti is mre imprtat. The first π π * trasiti bad (λ ) stays at arud 0 m while the secd π π * trasiti bad (λ ) icreases rapidly tward the visible wavelegth [4] resultig i a higher birefrigece. Wu et al. [4] ad Li et al. [43] exteded the Cauchy equatis t LC mixtures: Be Ce e = Ae + +, (.7) 4 λ λ B C 0 = A + +, (.8) 4 λ λ where A e,, B e,, ad C e, are the Cauchy cefficiets f the LC. This mdel fit the experimetal data well i the ff-resace spectral regi.... Temperature Effect Temperature is ather imprtat factr affectig LC refractive idices. The temperature- 3

32 depedet refractive idex is gvered by the rder parameter S i Eqs.(.) ad (.3), where S is described as: S T β = ( / Tc ) (.9) Eq. (.9) is Haller s apprximati fr the rder parameter. This apprximati hlds reasably well if the temperature is t t clse t the clearig temperature (T c ). The expet β is depedet mlecular structure, t wavelegth. Fr mst LC cmpuds studied, β is arud Li et al. derived the fur-parameter mdel fr describig the temperature effect the LC refractive idices [44]: ( ) 0 T β e ( T ) A BT + ( ), (.0) 3 T ( ) T c c 0 T β 0 ( T ) A BT ( ). (.) 3 Figure 4 plts the measured temperature-depedet refractive idices f 5CB (K 4 N 35.3 I) at three wavelegths: λ =450, 550, ad 650 m, respectively. The filled data pits are fr, while the pe pits are fr e. The slid lies are fittig results usig the fur-parameter mdel. The agreemet is excellet. As the temperature icreases, the refractive idex decrease gradually, this tred is mre pruced as temperature appraches T c. Fr a high T c mixture, its refractive idex is less sesitive t temperature i the viciity f rm temperature. 4

33 .80 Refractive Idices, e ad m 550 m 650 m Temperature, K Figure 4: Temperature-depedet refractive idices f 5CB at λ = 450 (triagles), 550 (circles) ad 650 m (squares). Ope dts are fr ad clsed dts are fr [43]. e By takig temperature derivati f Eqs. (.0) ad (.), Li et al. further derived the fllwig equatis [44]: d e dt d 0 dt ( ) β 0 = B T 3T c ( ) T c ( ) β 0 = B + T 3T c ( ) T c β β,. (.) (.3) This implies that e decreases as the temperature icreases thrughut the etire ematic rage, while 0 may iitially icrease r decrease with temperature, depedig the sig f Eq. (.3). As T appraches T c, d / dt jumps t a large psitive umber. I the itermediate, there exists a trasiti temperature (T 0 ) where d 0 0 / dt 5 =0. Li et al. aalyzed tw critical parameters

34 affectig d 0 / dt, which are high birefrigece ad lw clearig temperature. Based this, they frmulated tw exemplary LC mixtures usig the laterally substituted isthicyaat cmpuds. The birefrigece effect the respse time is thrugh the cell gap depedece. High birefrigece LC materials lead t a thier cell gap which, i tur, reduces the respse time... Dielectric Aistrpy The dielectric aistrpy is defied as ε = ε // ε, where ε // ad ε dete the dielectric permittivity parallel ad perpedicular t the directr, respectively. I tebk PC ad ther applicatis, lw pwer csumpti is critical. Thus, 3.3V ad eve.5v drivig vltage are csidered fr TFT-LCDs. Therefre, high dielectric aistrpy LC-materials are attractive fr lw peratig vltage. Frm the mlecular pit f view, the rigi f the dielectric aistrpy is the aistrpic distributi f the mlecular diples i the liquid crystal phases. Therefre, ematic phases frmed by elgated mlecules carryig lgitudial ad trasverse diples have respectively psitive ad egative dielectric aistrpy, whse magitude icreases with that f the mlecular diples ad with the degree f rderig. Sme sftware i.e. MOPAC [46] ca be used t cmpute the mlecular prperties (such as diple mmet) befre sythesis. Maier ad Meier [47] exteded Osager thery t ematic LC. I their thery, a mlecule is represeted by a aistrpic plarizability with pricipal elemets l ad t i spherical cavity f radius a. Detig the diple mmet with l at a agle θ, the LC dielectric cmpets ε //, ε ad ε ca be expressed as: 6

35 ll ll ) [ ( 3cs θ ) S] ε = NhF{ + ( Fµ / 3kT }, (.4) [ + ( 3cs θ ) S / ] ε = NhF{ + ( Fµ / 3kT ) }, (.5) ε = NhFS {( ) ( Fµ / kt )( 3cs θ )}, l t (.6) where N is the mlecular packig desity, µ is the diple mmet, F is the Osager reacti field ad is average refractive idex. Here h ad F are depedet the average dielectric cstat ε ad average refractive idex : 3 ε h =, (.7) ε + F = ( ε + )( + ) 3( ε + ). (.8) Based Eq. (.6), the dielectric aistrpic f a LC material is iflueced by three factrs: mlecular structure, temperature, ad frequecy. Let us discuss each factr separately.... Mlecular Structure Effect Fr a -plar liear LC cmpud, its diple mmet µ ~0, thus its dielectric aistrpic ε is small. Fr a plar LC cmpud with a plar grup, e.g., cya r flur, its ε ca be psitive r egative depedig the psitis f the plar grups. Whe θ < 55, the diple ctributi t ε is psitive, while whe θ > 55 it is egative, as illustrated i Eq.(.6). The sig f the dielectric aistrpy ε will affect the mlecular aligmet. Psitive ε material teds t alig parallel t the electric field, which egative e rmal t the electric field. Mst disctic shaped mlecules have egative ε. 7

36 I rder t measure either ε // r ε, capacitace methd is cmmly used. It is achieved by keepig the electric field parallel r perpedicular t the directr with ac excitati at lw frequecy (rmally khz). Tw-cell methd r e-cell at tw vltages apprach ca bth be used. T icrease ε, mre tha e plar grup ca be csidered. The fial ε value is determied by the vectr sum f the ttal diple mmets. The magitude f ε ad plar grup type f a LC mixture will affect the threshld vltage, peratig vltage, pwer csumpti, resistivity, ad viscsity.... Temperature Effect The temperature depedece f ε is prprtial t the rder parameter S. Usually as temperature icreases, ε // decreases while ε icreases gradually, resultig i a decreased ε. As T >T c, the istrpic state is reached ad the dielectric aistrpy is vaished....3 Frequecy Effect Frequecy depedet dielectric cstat is ather imprtat factr i aistrpic LC material. At lw frequecy, ε > 0, while at high frequecy, ε culd becme egative. As f icreases, ε // decreases while ε stays cstat, which results i a decreased ε. The dielectric aistrpy ε chages its sig at the crssed frequecy f c. Fr a pure LC cmpud, its f c is usually higher tha 0 MHz, which is t high t be emplyed. Dual-frequecy effect is a useful apprach fr achievig fast respse time. Hwever, the DFLC fte exhibit a small ε (i.e., 8

37 high vltage) ad high viscsity, ad its crssver frequecy is t sesitive t the peratig temperature. These issues limit the widespread applicatis f DFLC materials...3 Visc-elastic Prperties High birefrigece is t the slely csiderati f LC material, we eed t accut the verall perfrmace, which is the Figure f Merit (FM). Thus, ather imprtat parameter, visc-elastic cefficiet, eeds t be itrduced, which is related t viscsity ad elastic cstats characteristics. Each factr will be aalyzed i detail.... Elastic Cstats writte as: The free vlume elastic eergy (Frak eergy) desity f a ematic liquid crystal is f elastic r r r r r = K ( ) ( ) ( ) + K + K33, (.9) where K, K, ad K 33 are the elastic cstats assciated with spray, twist, ad bed defrmati, as shw i Figure 5, respectively. a) Splay (Hmgeeus) b) Bed (hmetrpic) c) Twist (twisted cell) Figure 5: Demstratis f three basic defrmatis f LC. 9

38 Each f them relates t a specific aligmet f the LC cell, i that K fr hmgeeus cell, K fr twisted ematic cell, ad K 33 fr hmetrpic cell. Frm Maier-Saupe thery, the elastic cstats are liearly prprtial t S as: K 33 > ~ S > K K. (.0) A small elastic cstat is favrable t achieve a lw perati vltage, but it leads t a slw respse time because the respse time is prprtial t /K ii. I practical uses, respse time ad viewig agle are usually mre imprtat tha threshld vltage. Thus, multi-dmai vertical aligmet is a favrable apprach fr LCD TVs....3 Rtatial Viscsity Fr almst all LC-related applicatis, fast respse time is critical issue. Therefre, lw viscsity LC materials are essetial fr TFT-LCDs. Rtatial viscsity γ plays a critical rle t LC dyamics. Bth rise time ad decay time are liearly prprtial t γ. Thus, fr mst LC devices, lw rtatial viscsity γ LC material is favrable. The temperature-depedet rtatial viscsity culd be writte as [9]: γ = bs exp( E / k0t ), (.) where E is the activati eergy, T is the Kelvi temperature ad S is the rder parameter shw i Eq. (.9). Frm the mlecular stadpit, the rtatial viscsity depeds the mlecular cstituets, dimesis, mlecular iteractis, ad mmet f iertia. Thus, a liearly cjugated liquid crystal shuld exhibit a relatively lw rtatial viscsity. At the elevated temperature, γ decreases dramatically. Fr every 0 C temperature icrease, viscsity is 0

39 decreased by ~X. Lw viscsity ad high resistivity are essetial fr TFT-LCDs where the LC birefrigece is abut 0.. The cmmly used mlecular structures are fluriated cyclhexaepheyl (CP) tw-rig ad CCP three-rig cmpuds. Usig a lw viscsity LC mixture r peratig a LC mixture at a elevated temperature, a small visc-elastic cefficiet ca be btaied. As a csequece, respse time is reduced....4 Figure f Merit T characterize the verall perfrmace f LC materials, a figure-f-merit (FM) which takes birefrigece ad respse time it accut has bee defied as [9]: FM =, (.) γ K / where K is the splay elastic cstat, is the birefrigece, γ is the rtatial viscsity. All f these parameters are temperature depedet. Bth viscsity ad elastic cstats are als depedet the rder parameter S ad ca be apprximated as: ( ), = (.3) 0 S K ii = as. (.4) Substitutig Eqs. (.)-(.4) it Eq.(.), the temperature depedet FM is derived as: 3β FM = ( a / b)( ) ( T / Tc ) exp( E / k0 0 T ), (.5) where 0 is the birefrigece at S =, E is activati eergy f the liquid crystal ad k 0 is the Bltzma cstat. The value f the β parameter is arud 0.5 ad isesitive t liquid crystal structures.

40 0.45 Birefrigece, UCF Temperature, C Figure 6: Temperature-depedet birefrigece f UCF-. Blue dts are experimetal data at λ =633 m. 5 0 UCF- 5 γ /k Temperature, C Figure 7: Temperature-depedet visc-elastic cefficiet γ /k f UCF-. Blue dts are experimetal data.

41 Figure 6 shws the temperature depedet birefrigece f UCF- at λ =633 m. The ematic rage f UCF- is frm 0.3 C t 4 C. As the temperature icreases, the birefrigece decreases gradually. Figure 7 gives the visc-elastic cefficiet f UCF-. As ca be see frm the figure, γ /k decreases as the temperature icreases. Whe the temperature icreases frm T =3 C t T =70 C, the visc-elastic cefficiet drps frm γ /k = t 4. Sice LC switchig time is prprtial t γ /k, as a result, a fast respse time is achieved by the reduced visc-elastic cefficiet at a elevated temperature. Figure 8: Temperature depedet figure-f-merit f UCF- (squares) ad E7 (circles). Slid lies are fittigs t the experimetal data f UCF- usig E =340 mev, β =0.5, ad T c =4 C at λ = 633 m. Frm abve discussis, it ca be see that elastic cstat, viscsity ad birefrigece are all depedet temperature. Accrdigly, the figure f merit strgly depeds the temperature. The temperature-depedet figure f merit fr UCF- is depicted i Figure 8. At lw-temperature regi, eve thugh birefrigece, dielectric aistrpy, elastic cstats ad viscsity all reduce as temperature icreases, the drppig speed f visc-elastic cefficiet i 3

42 Figure 7 is faster tha that f. Therefre the figure f merit first icreases with icreasig temperature ad reaches a maximum value at ~5 C. As the temperature appraches the clearig temperature T c, birefrigece has very steep drp as shw i Figure 6, which causes a sharp decrease i the figure f merit. The higher figure f merit a LC pssesses, the faster respse time it exhibits. Thus peratig a LC device at a elevated temperature is beeficial fr reducig respse time. Hwever, it brigs cmplicati t the drivig scheme by the additial temperature ctrl system. Als icluded i Figure 8 fr cmparis is a cmmercial E7 mixture. At 70 C, the UCF- mixture has 0X higher FM tha that f E7 at 48 C..3 Achrig Eergy The structure f liquid crystal directrs i clse prximity t a iterface is differet frm that i the bulk, ad this surface structure chages budary cditis ad iflueces the behavir f the LC directr i the bulk. Therefre, the achrig eergy is a imprtat parameter fr a LC cell. It affects t ly the LC aligmet but als the electr-ptic prperties such as threshld vltage ad respse time. The achrig eergy depeds cell thickess. A thier cell exhibits a higher achrig eergy. I ctrast t the strg achrig eergy, weak achrig ca results i: ) Lwer threshld, ) Rise time decreases, ad 3) Fall time icreases 4

43 .4 Factrs Affectig LC Respse Time T summarize the abve discussis, we fid that the LC respse time depeds several factrs, icludig the LC layer thickess, viscsity, temperature ad surface treatmet, as well as the drivig wavefrm. The pretilt agle effect will be umerically aalyzed i Sec i detail. The effects f these factrs the respse time are listed i Table. Table : The effects f differet factrs respse time Factrs T rise T decay Viscsity (γ ) Elastic cstats (K ii ) Dielectric aistrpy ( ε) Thickess (d) Pretilt agle (θ 0 ) Achrig eergy (W) Temperature (T) Vltage (V) T achieve a fast respse time, lw rtatial viscsity (γ ) LC mixtures are preferred [6],[49]. Ather straightfrward apprach is t use a thi cell gap filled with a high birefrigece ( ) ad lw viscsity LC mixture [36],[50]. High birefrigece als ehaces the display brightess ad ctrast rati f plymer-dispersed liquid crystal (PDLC) [33],[34], hlgraphic PDLC [35], chlesteric LCD [4],[3], ad LC gels [5]-[53]. Recetly, may maufacturers have reprted the display devices with reduced cell gaps f belw 4 µm i rder t achieve fast respse time. 5

44 CHAPTER 3: SIMULATION METHODOLOGIES Numerical simulati is a imprtat tl i the desig ad ptimizati f liquid crystal (LC) devices with high quality perfrmace, which icludes fast respse time ad wide viewig agle. I this chapter, we will review recet develpmet f LCD mdelig methds. I geeral, the mdelig f liquid crystal devices ca be divided it tw steps: () fid directr distributi; ad () simulate ptical respse. 3. Simulati f Directr Distributi Fr a hmgeeus LC cell that is strg achred at the surface f z =0 ad z =d with a small pretilt agle (~3 ) t the rmal, whe vltage is applied, the LC mlecules are parallel t the electrdes except a small pretilt agle t avid reverse tilt discliati. Whe the applied vltage exceeds Freedericksz trasiti vltage V th, the LC mlecules will rtate ad be rerieted by the electric field, causig the chage f the permittivity f substrate. The directr distributi uder applied vltage f LC devices shuld be slved first. Whether the backflw effect shuld be csidered depeds the applied vltage. Thus we divide it it tw categries: withut r with backflw effect. 3.. Withut Backflw Effect Uder the circumstace whe backflw effect is t csidered, tw-step apprach is used: slve the vltage distributi usig the fiite elemet methd (FEM) [54], ad the btai the directr distributi usig the fiite differece methd (FDM) [55]. The steady-state LC directr distributi is ctrlled by the balace f the trques 6

45 betwee the elastic free eergy ad electric eergy. The electric eergy uder applied vltage is defied as [56]: f Electric D E) dv = ε V V )dv. (3.) = where V is the vltage distributi, ad ε is the dielectric tesr f the LC cell. The relative dielectric tesr f the LC cell is give by [57]: ε xx ε xy ε xz ε r = ε ε yy ε yz, (3.) yx ε zx ε zy ε zz where ε ε ε xx yy zz = = = + ( + ( e e + ( ) cs θ cs φ, )cs θ si φ, e )si θ, (3.3) (3.4) (3.5) ε yz = ε zy = ( e )siθ csθ siφ, (3.6) ε xy = ε yx = ( e )cs θ siφ csφ, (3.7) ε xz = ε zx = ( e )siθ csθ csφ, 0 (3.8) Here, ad e is the rdiary ad extrardiary refractive idices f the LC medium, respectively, θ is the tilt agle f the LC directr, which is the agle betwee the LC directr ad the x-y plae, ad φ is the azimuthal agle betwee prjecti f the LC directr the x-y plae ad the x-axis as shw i Figure 9. 7

46 Figure 9: The crdiate system f LC directr. Usig rectagular elemets, the vltage distributi ca be expressed as a set f liear equatis [57]: Ne e e V ( x, y, z) = V N ( x, y, z), (3.9) e= e where V is the ukw vltage at the de i f elemet e, N e is the ttal umber f elemets, i i i i ad N (x, y, z e i ) is the shape fucti. T miimize this fuctial, the variati f the electric eergy F the vltage V m f de m is equal t 0. After slvig the fllwig liear algebraic equati [56], the vltage distributi is btaied. Electric A A M A Nde A A A M Nde L L O K A A A Nde Nde M NdeNde V V VN de = 0, (3.0) where 8

47 A m e FElectric =. (3.) V V m I the fllwig, we will discuss the sluti f directr distributi. The mai idea is t use Euler-Lagragia equatis t miimize the free eergy. After applyig the Gibbs free-eergy desity f g t the Ericks-Leslie thery, we have [59]: [ f ] +, i x, y, z di γ = g λ i = (3.) i dt where γ is the rtatial viscsity, i is Cartesia cmpet f the directr, λ is a Lagrage multiplier used t maitai the uit legth f the directr ad [f g i ] is the Euler-Lagragia equatis. Hwever, it is impssible t slve fr bth a update frmula fr the directr ad λ, s we drp the Lagrage multiplier term ad simply rermalize the directr after each iterati. The Euler-Lagragia equatis [f g i ] are expressed as [59]: f f f f = dx i i / dx dy i / dy dz i / dz (3.3) g d g d g d g [ f ], i = x, y, z g i where f g is the Gibbs free eergy desity give by: f g = f f. (3.4) elastic Electric Here f elastic ad f Electric are the elastic free eergy desity ad electric eergy desity, as expressed i Eqs. (.9) ad (3.), respectively. With the kw electric eergy desity f Electric ad elastic eergy desity f elastic, the updated directr cmpets after each time step t ca be derived frm equati Eq. (3.) as = t γ [ f ], i x, y, z. ew ld i i g = i (3.5) 9

48 After this update, the directr must be rermalized t have a uit legth. This rermalizati als depeds the umerical time step t. Therefre, the settig time step will be critical t the calculati accuracy. I ur simulati, we use a flexible time step t, which is less tha the maximum time step give by [59]: x γ tmax =, (3.6) k where x is the miimum meshig size f the x, y, z directi. At each time step, the directr distributi is visualized, which will be used t slve the electr-ptical perfrmace f the simulated LC device. After the LC directr distributi is updated, the vltage distributi ad the electric eergy distributi must be updated t represet the chages f the vltage distributi. This iterati ctiues util settig cvergece criteria is reached. This is a gd apprach i reducig the cmputati time while maitaiig a reasable accuracy. 3.. With Backflw Effect Whe backflw effect is csidered, ather step shuld be added i the directr update usig fiite differece methd. The directr is therefre uder the tilt defrmati, assciated with the viscus trque. I the Cartesia crdiate, the directr is fully described as = ( x, y, z ), the directr distributi i -D case will fllw the rule as: F F vx γ & x = γx + z, (3.7) z z x x, z F F vy γ & y = γy + z, (3.8) z z y y, z 30

49 , 3 3 0, z v z v E F z F y y x x z z z z z z + + = ε ε γ γ & (3.9) where F is the Frak free eergy desity, γ is the rtatial viscsity,,, ad are the differetial f t time t, ad ν x, ν y are flw velcity alg x ad y directi. Ntice the iertial terms have bee eglected i the directr equati. The last terms i the abve equatis are related t the viscus trque. S we have ather tw parameters eed t be slved first: x & y & z & z y x,, z v x ad z v y, which are give by the fllwig tw flw equatis:, ) ) ( ) ( ( = z v z v z v x x z z x y y x x x z x x ρ & & & (3.0). ) ) ( ) ( ( = z v z v z v y y z z y x y x y z z y y ρ & & & (3.) where t 6 are six Leslie cefficiets, ad ρ is the fluid desity. Because the iertia is small, ad ca be mitted, the fllwig equatis are derived:, ) ) ( ) ( ( x z z x x x z z x y y x C z v z v & & = (3.). ) ) ( ) ( ( y z z y y y z z y x y x C z v z v & & = (3.3) 3

50 T get the slutis f v x z ad v y z, we eed t fid the C first. With the,c budary cditi v x z v y = z = 0 at z =0 ad d, ad the directr distributi ( x, y, z ) at time step (i), we ca btai the slutis f C,C. After that, the values f v x z ad v y z, ca be btaied frm Eqs.(3.) ad (3.3). Substitutig them it Eqs. (3.7)-(3.9), the updated directr distributi ( ', ', ') at this time step (i) is btaied. Usig ur i-huse develped x y z prgram, the dyamics f LC cells with backflw effect is ivestigated ad the results will be explred i Chapter Optical Simulati I this secti, tw cmmly emplyed appraches fr calculatig the trasmittace is itrduced: ) the 4 by 4 matrix methd [6]-[64], ) the exteded Jes matrix ( by matrix) [65]-[69]. Berrema s 4 4 Matrix apprach takes it accut the effects f the refracti ad multiple reflectis betwee adjacet plate iterfaces [6]. Hwever, it eeds legthy calculatis ad it is geerally emplyed ly fr the rmal icidece. Over these years may apprximatis t this scheme has bee prpsed. Yeh [65] ad Lie s [67] exteded Jes matrix methds were derived t geeralize the 4 4 Matrix frmulati t the blique icidece. Huag et al. [69] used a imprved 4 4 Matrix fr ptimizig the liquid crystal Fabry-Pert etal fr telecm applicati. 3

51 Matrix Methd Let us csider a uplarized light eterig a LC pael at a blique directi. We chse a crdiate system i which the wave vectr k f the icidet plae wave lies i the x-z plae. Here the +z axis pits frm the bttm glass substrate t the exit plarizer. The wave vectr f the icidet wave i this crdiate system is give by k = xk siθ + y 0 + z csθ, (3.4) i 0 k k k where k 0 is the wave umber i free space, θ k is elevati agle f the icidet plae wave, ad x, y, ad z are the uit vectr i the x, y, z directis, respectively. The whle LCD system is divided it N layers i the z directi. N is rmally a umber less tha 00 such that each layer is csidered as a hmgeeus e. The liquid crystal layer shw i the figure usually csists f layers. I simulatis, each layer is treated as a sigle hmgeus e. Figure 0: Schematic diagram f a LC pael, which is divided it N layers. Fr simplicity, the magetic field itesity H is rmalized as: 33

52 Hˆ µ 0 = ε 0 / H (3.5) expressed as: Assumig E ad Ĥ are exp(-iwt) depedet, the time harmic Maxwell equati ca be E = ik Ĥ, (3.6) Hˆ = ik εe. (3.7) With / y = 0 ad / x = ik x, after expasi f Maxwell s equatis, we ca derive the fllwig frm: E E z H ˆ H x y ˆ = ik0 x where Q is a cuplig matrix give by y Ex E y Q ˆ, (3.8) H x ˆ H y ε ε si zx zy θ k siθ k siθ k 0 ε zz ε zz ε zz Q = ε ε zy ε zx yz si 0 si. ε yx + ε yz ε yy + ε yz + θ k θ (3.9) k ε zz ε zz ε zz ε ε zx zy ε xz ε 0 si xx ε xz ε xy ε xz θ k ε zz ε zz ε zz Fr each sub-layer, the matrix Q ca be trasfrmed t a diagalized matrix: Q q q = T T, q 3 q4 (3.30) 34

53 where q t q 4 are the fur eigevalues, ad T is cmpsed f the crrespdig eigevectrs. q t q 4 are give by q (3.3) (, si / k θ = ), si si cs si / + = k e e zz zz e k zz xz q θ ϕ θ ε ε θ ε ε (3.3) 3 q, q = (3.33) 4 q, q = (3.34) where q ad q crrespd t tw frward eigewaves, ad q 3 ad q 4 crrespd t tw backward eigewaves. With the diagalized Q matrix, we ca further cduct a variable trasfrmati f the tagetial field cmpets as:. 4 3 = U U U U H H E E y x y x T ) ) (3.35) where T is expaded as:. = T T T T T (3.36) Substitutig Eqs. (3.30) ad (3.35) it Eq. (3.8), we have the decupled equatis as: = U U U U q q q q ik U U U U z (3.37) Fr the th sub-layer, the sluti f Eq. (3.37) is give by: 35

54 (3.38),,0 4 3, 4 3 d U U U U U U U U = H where ad are the eigewaves the iput ad utput budaries f the th sublayer, respectively ad d is the thickess f the crrespdig th layer, The matrix H is,0 4 3 U U U U d U U U U, 4 3, ) exp( ) exp( ) exp( ) exp( 4 3 = z z z z d ik d ik d ik d ik H (3.39) where (3.40), 0 q k z = k, 0 q k k z = (3.4) 3, 0 3 q k k z = (3.4) q k k z = (3.43) Substitutig Eq. (3.38) it Eq. (3.35), the electric field the utput budary f the th layer is related t the electric field the iput budary f the same layer by (3.44),,,0 y x y x d y x y x H H E E H H E E = ) ) ) ) P where P is the 4-by-4 matrix f the th sub-layer ad is give by 36

55 P = T HT. (3.45) Sice the tagetial E field cmpets are ctiuus i the iterfaces f adjacet layers, the 4-by-4 matrix f the simulated LC device ca be expressed as: P = P N P N KPP. (3.46) 3.. Matrix r Exteded Jes Matrix Methd If we assume the reflecti iside the LC device is egligible ad ly csider the frward eigewaves, Eq. (3.35) becmes: E E x y = T U U U = S U. (3.47) Fr the eigewave i the th layer, we have U U, d = G U U,0, (3.48) where G exp( ik = z d ) exp( ik z. d ) (3.49) Applyig equati (3.47) t equati (3.46), the exteded Jes matrix f the th layer is J = SGS, (3.50) frm which we ca btai the exteded Jes matrix f the simulated LC device as J = J N J N KJ J. (3.5) Csiderig the trasmissi lss i the air-lcd iterface, the trasmitted electric field is related t the icidet electric field by: 37

56 , = = + y x y x N y x E E E E E E Et N N Ext J J J J J J J K (3.5) where J Ext ad J Et are the matrices csiderig the trasmissi lsses i the air-lcd iterfaces, which are give by, cs cs cs 0 0 cs cs cs + + = p p k k k p p p θ θ θ θ θ θ Et J (3.53), cs cs cs 0 0 cs cs cs + + = p p k p p k p p k p θ θ θ θ θ θ Ext J (3.54) where p is the refracti idex f the plarizer, ad θ p is give by, Re / si si,, + = p p e k p θ θ (3.55) where e,p ad,p are the tw refractive idices f the plarizer. Thus, the verall ptical trasmittace t p is give by. cs cs,,,, y p x N y p N x p E E E E t θ θ + + = + + (3.56) 38

57 CHAPTER 4: CORRELATIONS BETWEEN LIQUID CRYSTAL DIRECTOR REORIENTAION AND OPTICAL RESPONSE TIME OF A HOMEOTROPIC CELL 4. Itrducti Respse time is e f the mst critical issues fr early all liquid crystal (LC) devices ivlvig dyamic switchig. Based the small agle apprximati, Jakema ad Rayes derived the LC directr rerietati times [7]. Sice the, umerus papers dealig with LC respse time have bee published, hwever, the respse time frmula that mst literatures refer t is the LC directr rerietati time rather tha the ptical respse time. Fr amplitude mdulati, e.g., liquid crystal display devices [3], the LC device is usually sadwiched betwee tw plarizers. The measured quatity is trasmittace chage ad the assciated dyamic respse is ptical rise r decay time. O the ther had, fr a phase-ly mdulatr such as ptical phased arrays [36], the measured respse time is phase chage. There is dubt that the ptical respse time fr amplitude mdulati ad phase respse time fr phase mdulati must be related t the LC directr rerietati time. T quatify a display device, the rise ad decay time is usually defied as itesity chage betwee 0% ad 90%. Hwever, the crrelati betwee the directr rerietati time ad the ptical ad phase respse time has t bee carefully studied. Based a simplified mdel, Wu [73] fud that the ptical decay time culd be ~X faster tha the directr rerietati time i a hmgeeus LC cell. It is imprtat t establish the detailed crrelati betwee the LC directr rerietati time ad the ptical ad phase respse time. 39

58 I this chapter, we derived the aalytical crrelati betwee the directr rerietati time ad its csequet ptical rise ad decay times based the small agle apprximati. A vertical-aliged (VA, als kw as hmetrpic, DAP cell) [74] ematic LC cell was used fr these studies due t its simple electr-ptic characteristics ad widespread applicatis i trasmissive direct-view ad reflective prjecti displays [75]-[77]. I that situati, the icmig liearly plarized light with a wave vectr k i parallel t the z axis des t ecuter birefrigece, ad arrives at the secd substrate with a uchaged state f its plarizati, If the aalyzer is crssed with the plarizer, the light is blcked at the utput fr all the wavelegth ad idepedet f d, This is the rmally black state. T validate the derived crrelatis, i Sec. 4. we umerically slved the dyamic Ericks-Leslie equati. Results idicate that the ptical respse time is liearly prprtial t the directr rerietati time ad is weakly depedet the iitial bias vltage. Pretilt agle effect is fud t make a imprtat ctributi t the LC dyamics. Gray scale switchig f the VA cell is als studied ad results are discussed i Sec Thery Whe the backflw ad iertial effects are igred, the dyamics f the LC directr rerietati is described by the fllwig Ericks-Leslie equati [78],[79]: ( K φ cs φ + K 33 si φ) + ( K z φ + ε εe siφ csφ = γ, t 33 K φ )siφ csφ( ) z (4.) where γ is the rtatial viscsity, K ad K 33 represet the splay ad bed elastic cstats, respectively, ε εe is the electric field eergy desity, ε is the LC dielectric aistrpy, ad φ 40

59 is the tilt agle f the LC directrs. I geeral, Eq. (4.) ca ly be slved umerically. Hwever, whe the tilt agle is small ( si φ ~ φ ) ad K 33 ~K (s called small agle apprximati) [7], the Ericks-Leslie equati is reduced t: K φ z 33 + ε E ε φ φ = γ. t (4.) slutis. Uder such circumstaces, bth rise time ad decay time have simple aalytical with 4.. Decay Time Whe the electric field is switched ff, i.e., E=0, Eq. (4.) is further simplified as: φ φ K 33 = γ. (4.3) z t The sluti f Eq. (4.3) ca be expressed as πz t φ( z, t) φm si( ) exp( ), (4.4) d τ γ d τ =, (4.5) K π 33 where φ m is the maximum tilt agle f the LC directrs i the respse f the applied vltage, d is the LC cell gap, z is the psiti f the rieted LC layer uder discussi, ad τ is the LC directr rerietati time ( /e). It shuld be pited ut that i the Ericks-Leslie equati the strg surface achrig ad zer pretilt agle at the surface budaries are assumed. Uder such cditis, the Freedericksz trasiti threshld exists [80]: 4

60 V th = π K33. (4.6) ε ε fllws: The time-depedet phase chage assciated with this agle chage is described as ( t) = d π e dz, λ ( cs φ + 0 e si φ) (4.7) where ad are the refractive idices fr the extrardiary ad rdiary rays, respectively. If a VA cell is iitially biased at a vltage (V b ) which is t t far abve V th, ad the vltage is remved istataeusly at t =0, the trasiet phase chage ca be apprximated frm Eq. (4.7) as [8]: e t δ ( t) δ exp( ), (4.8) τ where δ is the et phase chage frm V=V b t V=0. Frm Eq. (4.8), the phase decay time cstat ( /e) is τ / which is X faster tha the LC rerietati time. T fid ptical respse time, we eed t calculate the itesity chage. The timedepedet rmalized itesity chage I(t) f the VA cell uder crssed plarizers ca be calculated usig the fllwig relatiship: δ ( t) I( t) = si ( ). (4.9) Substitutig Eq. (4.8) it Eq. (4.9), we fid I( t) = si t δ exp( ) τ. (4.0) 4

61 I a display device, tw defiitis f respse time are ecutered: ptical trasmittace chage frm 90% t 0% r frm 00% t 0%. The prcess fr crrelatig the ptical respse time t the directr rerietati time is similar. Let us csider the frmer case first. Based Eq. (4.0), the rmalized trasmittace at t =0 has the fllwig simple expressi: I si δ = ( ). (4.) Let us assume frm t t t the trasmittace decays frm I = 90% t I = 0%. Frm Eq. (4.0), I ad I have the fllwig frms: I 0.9I t δ exp( ) si τ =, = (4.) I = 0.I = si t δ exp( τ ). (4.3) Usig Eqs. (4.), (4.), ad (4.3), we ca easily slve the ptical decay time T decay (90% 0%) as fllws T decay = t t si τ = l si δ 0.9 si( ). (4.4) δ 0.si( ) 43

62 Equati (4.4) crrelates the ptical decay time t the LC directr rerietati time ( τ ). Similarly, the ptical decay time frm 00% t 0% ca be derived easily ad result is shw belw: τ δ / T decay = t = l. (4.5) δ si 0.si( ) Based Eqs. (4.4) ad (4.5), the ptical decay time f a VA cell is liearly prprtial t the directr decay time. The iitial phase retardati (δ ) als plays a imprtat rle, but t t substatially. The detailed umerical results will be shw i Sec Rise Time Rise time is much mre cmplicated t deal with tha relaxati. The rigial small agle apprximati used by Jakema ad Rayes fr rise time is versimplified [7]. They have assumed that the LC directr s tilt agle icreases expetially with time. This apprximati is valid ly i a very shrt time regime. Bliv has csidered the secd rder term ad the fact that the LC directrs will evetually reach a equilibrium stage. Thus, Eq.(4.) is rewritte as [8]: φ 3 φ ξ + φ φ = λ, (4.6) z t where γ λ =, (4.7) ε εe 44

63 33 E K ε ε ξ =. (4.8) I Eqs. (4.7) ad (4.8), the electric field itesity E is expressed as: (4.9) / d, E = V where the bias vltage V shuld satisfy (V-V th )/V th. Uder such a circumstace, the sluti f Eq. (4.6) ca be apprximated as: ). si( ) ( d z t m π φ φ = (4.0) Substitutig Eq. (4.0) t Eq. (4.6), we btai: dt d V V m m m th φ λ φ φ = 3. (4.) Equati (4.) has fllwig sluti:, exp + = r m t τ φ φ φ φ (4.) where is the steady-state value f ( ) = t φ φ m φ crrespdig t the biased vltage, is the iitial directrs fluctuati, ad ( = 0) t = φ φ r τ is the directrs rise time: = = th r V V K d E τ π ε ε γ τ. (4.3) Uder small agle apprximati, the trasiet phase chage is btaied as:. exp ) ( + r t t τ φ φ δ δ (4.4) 45

64 where δ is the ttal phase chage frm the vltage-ff state t the vltage- state. By substitutig Eq. (4.4) it Eq. (4.8), we btai the trasiet trasmittace state: + = r t t I τ φ φ δ exp / si ) (. (4.5) At t, the expetial term i Eq. (4.5) vaishes ad I(t) reaches a plateau: ) ( si ) ( I δ =. (4.6) T slve the ptical rise time, let us assume the trasmittace rises frm I t I as the time icreases frm t t t. Substitutig t ad t t Eq. (4.5), we btai the crrespdig trasmittace at 0% ad 90%:, exp / si 0. + = = r t I I τ φ φ δ (4.7) + = = r t I I τ φ φ δ exp / si.9 0. (4.8) By slvig t ad t frm Eqs. (4.7) ad (4.8), we derive the ptical rise time T rise (0% 90%) as: 46

65 T rise = t t = V V th τ si l si δ / δ 0.si( ). (4.9) δ / δ 0.9 si( ) Equati (4.9) crrelates the ptical rise time (T rise ) t the cmmly used directr rise time ( τ r ) as described i Eq. (4.3). Basically, it is a liear relatiship except fr the additial lgarithm term f the phase depedece. As will be discussed i Sec. 4.3, this phase depedece is relatively mdest. 4.3 Results ad Discussi T validate Eqs. (4.4) ad (4.9), we umerically slve Eq. (4.) usig the fiite elemet methd (FEM) [54]. Oce the LC directr distributi is btaied, we the use the exteded Jes matrix methd [57], [6] t calculate the trasiet phase chage δ(t). Figure shws the system cfigurati f the VA LC uder study. A cmmercial Merck egative ematic MLC-6608 mixture was used i ur cmputer simulatis. The material parameters f MLC 6608 are: =.4748 e =. 5578, the dielectric aistrpy ε = 4., the rtatial viscsity γ = 86 mpas at 0 C, the splay elastic cstat K = N, twist elastic cstat K = 7.0 N, ad bed elastic cstat K33 = 8. 0 N. The buffig 0 iduced pretilt agle is assumed t be frm surface rmal uless therwise metied. 47

66 + Aalyzer Glass LC Glass Plarizer Figure : The VA cell used fr this study. The LC cell is sadwiched betwee tw crssed plarizers. The ier side f each glass substrate is cated with a thi layer f idium-ti-xide ad plyimide fr prducig hmetrpic aligmet. The LC has a small pretilt agle. Fr a thi-film-trasistr liquid crystal display usig trasmissive VA cell, the -state vltage is preferred t be restricted t ~5 V rms fr the iterest f lw pwer csumpti. Therefre, we chse d = 0.7λ, which is slightly larger tha the required half-wave phase retardati i rder t reduce the -state vltage. By usig MLC-6608, the crrespdig cell gap is d =4.64 µm ad the ttal phase retardati is δ =.4 π at λ = 550 m. Based Eq. (4.5), the threshld vltage V is calculated t be.9 V rms. At the first trasmissi peak (i.e., δ =π), th V =.46 V th. We have als studied the respse time betwee gray scales Pretilt Agle Effect Fr a VA cell, pretilt agle () affects the device ctrast rati ad respse time. Here, we defie pretilt agle as the agle f the LC directrs deviated frm cell rmal. If =0, it implies that the LC directrs are aliged perpedicular t the substrate surfaces. Figure plts 48

67 the vltage-depedet trasmittace (r called VT curve) at =0.0,, ad 5. Please te that τ was derived by assumig =0. Hwever, i a real LC device a small pretilt agle is required fr LC directrs t relax back withut creatig dmais. Therefre, we use =0.0 t aimate the results fr =0. As the pretilt agle deviates frm the cell rmal, the threshld behavir is gradually smeared ad the tur- vltage is decreased. We have calculated the quatitative LC directr rerietati time ad ptical respse time at varius vltages fr differet pretilt agles. Hwever, it will be tedius t tabulate all the results here. T fid the tedecy while t lsig geerality, we chse the simulati results with =0.0 ad, as shw i Table ad Table 3, respectively. Figure : The simulated vltage-depedet trasmittace f a VA cell at λ = 550 m with three differet pretilt agles, = 0.0 (dashed lie), (slid lie) ad 5 (dashed dt lie). The parameters used i simulatis are listed i the text. 49

68 Frm Table, the phase decay time data (the t p /τ clum) agree with the LC directr decay time quite well i the lw vltage regime. Here, t cmpare withτ, we simply calculated the phase decay frm t /e (Eq. (4.8)). As the vltage icreases, the t p /τ rati gradually deviates frm uity. At V/V th ~.6, t p /τ icreases by ~4%. At δ π, the phase decay time is ~3% lger tha τ. O the ther had, the ptical decay time (frm 90% t 0%) remais relatively cstat (T decay /τ ~0.65±0.03) i the V th <V<.4V th regime. As the vltage icreases t V/V th ~.5, T decay /τ icreases t Table : Simulati results f phase decay time ad ptical decay time at differet vltages f a VA cell. LC: MLC-6608, d = 4.64 µm, pretilt agle = 0.0 ad V th =.9 V rms. Here, t p is the phase decay time, T decay is the ptical decay time, ad τ =.4 ms is the directr s decay time as defied i Eq. (4.5). Vltage (V rms ) V/V th Phase (π) t p ( /e ) (ms) t p /τ T decay (90 0%) (ms) T decay /τ

69 Table 3: shws the calculated results fr =. A similar tred as that f =0.0 is still bserved except that bth phase ad ptical decay times are smewhat slwer. The theretical directr decay time τ is assumed uchaged. The slwer phase ad ptical decay time is believed t rigiate frm the slightly weaker restrig elastic trque due t the icreased pretilt agle. Table 3: Same as Table except the pretilt agle =. Vltage V/V th Phase t p ( /e ) t p /τ T decay (90 0%) T decay /τ (V rms ) (π) (ms) (ms) I Figure 3(a) ad (b), we plt the calculated ptical decay time (90% 0%) ad rise time (0 90%), respectively, as a fucti f V/V th at =,, 3, ad 5 pretilt agles. The Ericks-Leslie equati was used fr these calculatis. I geeral, at a give V/V th a smaller pretilt agle wuld lead t a faster decay time but slwer rise time. I the viciity f threshld, the rise time is particularly slw, as described i Eq. (4.3). As the vltage icreases, the ptical 5

70 rise time is decreased rapidly. At V ~.V th (peak trasmittace), the rise time is reduced t ~0 ms. 3 Decay Time, ms (a) V b / V th Rise Time, ms (b) V / V th Figure 3: (a) Optical decay time (90% 0%) ad (b) rise time (0% 90%) as a fucti f V/V th at fur differet pretilt agles, =,, 3, ad 5. 5

71 Strictly speakig, the threshld behavir exists ly whe the pretilt agle is zer. Hwever, i mst LC devices a -zer pretilt agle is required i rder t avid dmai frmati durig mlecular rerietati. The free relaxati time τ is derived based the assumptis that =0 ad the applied vltage is t t far abve the threshld. I reality, these assumptis may t be valid. Takig it accut the pretilt agle effect, we mdify the free relax time accrdig t the fllwig equati: * τ =, (4.30) βτ where τ is the free relaxati time whe pretilt agle is zer, ad ca be calculated accrdig t Eq. (4.5). I Eq. (4.3), τ is depedet the pretilt agle. Sice mst f display cells have a pretilt agle, this crrecti factr is ecessary t match thery with experimetal results [84]. We have used the Ericks-Leslie equati t calculate the LC respse time icludig pretilt agle effect but withut usig the small agle apprximati. The β values we fud are listed i Table 4. At a very small pretilt agle ~0.0, τ * ~ τ ; i.e., the crrecti factr β=, as expected. As the pretilt agle icreases, β gradually icreases. At =5, β is fud t be higher tha the ideal value, which is uity, by early 30%. The pretilt agle is depedet the plyimide aligmet layer, rubbig stregth, ad LC material emplyed [85], [86]. Fr a give plyimide aligmet layer, differet LC materials may have a slightly differet pretilt agle depedig the mlecular iteractis. The typical pretilt agle fr a VA cell is ~. Thus, β ~.6 has bee takig it csiderati wheever we calculated the respse time f a VA cell with =. 53

72 * Table 4: Pretilt agle effect the LC directr s decay time τ. * Pretilt agle ( ) τ / τ Gray Scale Switchig The beauty f a ematic LCD is that it has atural gray scales. Each primary clr (red, gree, ad blue) ca display 8-bits gray scales. Thus, a full-clr display with 6 milli clrs ca be btaied. T ivestigate gray scale switchig, we divide the vltage-depedet trasmittace curve it eight equal itesity gray levels, as shw i Figure 4. Level represets the dark state ad level 8 fr the brightest state. The maximum trasmissi shw i Figure 4 is 35% after takig the absrpti f the plarizer ad aalyzer it csiderati. Figure 4: The eight gray levels f the VA cell at λ = 550 m LC: MLC-6608, d = 4.64 µm ad pretilt agle =. 54

73 Table 5 lists the calculated ptical respse time usig the fiite elemet ad fiite differece time dmai methds fr bth decay ad rise prcesses f the eight-level gray scales. The data i the right tp triagle represet the rise time, while the left bttm are the decay time. The rise time frm gray level t is the slwest because the applied vltage is s clse t the threshld vltage. I a VA cell, gray scale represets the switchig frm the darkest state t the secd darkest state. Althugh it is slw, it is frgive because huma eye culd t reslve this chage t well. I the high vltage regime, gray scale switchig is relatively fast. The actual switchig time depeds the cell gap ad visc-elastic cefficiet (γ /K 33 ) f the LC material emplyed. T imprve the switchig speed betwee gray levels, the ver-drive ad udersht methd has bee prpsed ad implemeted i real display devices [3], [4]. The data i the first rw ad the first clum will be further used i ur crrelati. Table 5: The calculated eight gray level ptical rise time (0% 90%) ad decay time (90% 0%) f the VA cell shw i Figure 4. Rise time, ms Decay time, ms

74 4.3.3 Detailed Crrelatis I this secti, we shw the detailed simulati results betwee the directr rerietati time ad ptical respse time. The Ericks-Leslie equati was used fr these calculatis. A gd crrelati betwee the LC directr rerietati time ad ptical respse time is fud Decay Time I the small agle apprximati, e f the imprtat assumptis is si( θ ) ~ θ. Uder such a circumstace, the aalytical frm f LC directr rerietati time cstat τ ca be derived. Hwever, the LC directr rerietati time is difficult t measure directly i a experimet. Fr display applicatis, the ptical respse time is a mre practical term. Hw t crrelate the LC directr rerietati time t the measurable ptical respse time is a imprtat task. Figure 5(a) depicts the directr distributi (φ(z)) as a fucti f rmalized cell gap (z/d) at V ~.37 V th. Althugh the vltage is t t high frm threshld, a large directr defrmati has already ccurred. I the middle layer, the maximum directr tilt agle (φ m ) has reached ~53. Therefre, it is difficult t fresee whether the small agle apprximati still hlds. If it des, the Eq. (4.8) shuld be valid ad l[ / δ ( t) ] δ shuld be a liear fucti f time with slpe equal t /τ. Frm the slpe measuremet, τ ca be extracted. I experimet, the VA cell sadwiched betwee crssed plarizers is biased at a vltage V b. Whe the LC cell relaxes frm V b t 0, the ttal phase chage isδ. Fr a VA cell iteded fr itesity mdulati, δ π. Whe the vltage is released istataeusly at t =0, the time- 56

75 depedet trasmittace is recrded. This time-depedet trasmittace ca be cverted t the trasiet phase decay described by δ (t). Figure 5(b) plts the calculated l[ δ / δ ( t) ] as a fucti f time fr the VA cell. Ideed, a straight lie with slpe f /ms is btaied. Based this slpe, τ = 6. 5 ms is fud. Usig the LC parameters, we fid τ =. 4 ms frm Eq. (4.5) ad τ = 6 ms frm Eq. (4.8) with β =.6 because f the pretilt agle. The agreemet betwee the small agle apprximati ad the Ericks-Leslie equati is amazigly gd i this case. Next, we validate the crrelati betwee the ptical decay time (T decay ) ad the directr s decay time ( τ ), as expressed i Eq. (4.4). If we eglect the lgarithm term, the T decay =0.5τ ; the bserved ptical respse time is X shrter tha that f the LC directr decay time. With the phase-depedet term icluded, the chage is still t t sigificat. Figure 6 plts T decay / τ at differetδ, as described i Eq. (4.4). I Figure 6, circles represet the simulati results usig the Ericks-Leslie equati, while the slid lie represets the small agle apprximati. I the small δ regi, i.e., V is t t far abve V th, the agreemet betwee these tw methds is reasably gd. As icreases, the discrepacy icreases slightly. At the biased vltage δ V b /V th =.46 which crrespds t ~ π, the maximum errr bserved is ~4 %. δ 57

76 Figure 5: (a) The calculated LC directr distributi φ(z) as a fucti f rmalized cell gap (z/d). (b) Time-depedet l[ δ / δ ( t) f the VA cell. Dts are calculated data ad slid lie is ] the fittig curve. Frm the slpe f the straight lie, τ is fud t be ~6 ms. 58

77 Figure 6: The crrelati f the ptical decay time T decay (90% 0%) vs. the LC directr rerietati time ( τ ) as a fucti fδ. Circles represet the simulati results usig the Ericks-Leslie equati, while the slid lie is the crrelati btaied frm the small agle apprximati [Eq. (4.4)] Rise Time Rise time is much mre difficult t slve tha the decay time because it als depeds the applied vltage. Equati (4.9) crrelates the ptical rise time with the LC directr rise time. At a give vltage, the ptical rise time is X shrter tha the LC directr rise time if we eglect the lgarithm term. Eve the phase depedece term is icluded, the results are t affected t greatly. Figure 7 depicts the rati f T rise / τ at differet δ as described by Eq. (4.9). I Figure 7, circles represet the simulati results usig the Ericks-Leslie equati, 59

78 while the slid lie represets the small agle apprximati. A very gd agreemet is btaied except i the ear threshld regi. Figure 7: The crrelati f ptical rise time T rise (0% 90%) vs. the directr rerietati time ( τ ) as a fuctiδ. Circles represet the simulati results usig the Ericks-Leslie equati, while the slid lie is the crrelati btaied frm the small agle apprximati [Eq. (4.9)]. Whe V b gets clse t V th, the pretilt agle effect becmes mre pruced. I ur simulatis, pretilt agle is assumed t be. Due t the smeared ad decreased threshld, the required vltage fr btaiig δ ~ 0.5π is lwer. As a result, the calculated rise time is lger tha τ, as idicated by Eq. (4.3). 60

79 4.4 Cclusi We have derived the crrelatis betwee the LC directr rerietati time ad the ptical respse (bth decay ad rise) time f a vertically aliged cell. Results idicate that the ptical respse time (T rise ad T decay ) is liearly prprtial t the LC directr rerietati time. The iitial bias vltage effect is t t strg. Pretilt agle is fud t make a imprtat impact t the LC dyamics. T crrect fr the pretilt agle effect, a mdified rtatial viscsity eeds t be used. 6

80 CHAPTER 5: BACKFLOW EFFECT ON THE DYNAMIC RESPONSE OF LIQUID CRYSTAL PHASE MODULATOR 5. Itrducti Liquid crystal (LC) ptical phased array (OPA) [36] is a efficiet device fr laser beam steerig. I a OPA, a blazed phase gratig is geerated by ctrllig the spatial vltage patters f the pixilated electrdes. The icmig liearly plarized light iteracts with the phase gratig ad gets deflected. The steerig agle (r the deflecti agle) ca be as large as 5-7 depedig the electrde gap ad birefrigece f the LC emplyed. Fr achievig a pure phase mdulati, ematic LC with hmgeeus aligmet is cmmly used. I a hmgeeus cell, the maximum btaiable phase differece due t the vltage-iduced mlecular rerietati isδ = πd / λ, where d is the cell gap, is the birefrigece, ad λ is the wavelegth. Fr laser beam steerig at λ =.55 µm, the required π phase chage leads t d ~.55 fr a trasmissive OPA. This d requiremet is ~3X higher tha that f a visible wavelegth. Mrever, due t the birefrigece dispersi effect [87] the LC birefrigece drps ~0% i the ear IR regi as cmpared t that at λ =550 m. Fr the iterest f achievig a fast respse time, usig a thi LC cell with a high birefrigece LC is a favrable apprach. Due t the thi cell gap, a relatively high (0 V rms ) vltage is eeded i rder t cmpletely reriet the LC directrs. Uder such a drivig cditi, backflw is ievitable i the dyamic respse [88],[89]. The respse time f a hmgeeus LC cell is prprtial t γ d / π K, where γis the rtatial viscsity ad K is the splay elastic cstat [7]. T achieve a fast respse 6

81 time, high birefrigece ad lw viscsity materials are particularly desirable [],[]. High birefrigece eables a thier cell gap t be used s that the respse time is faster. Lw viscsity is als helpful fr reducig the respse time. A simple methd fr reducig viscsity is t perate the LC device at a elevated temperature. As the temperature icreases by 0 C, the viscsity decreases by ~X. The available data the viscus prperties f LC materials are almst always icmplete. S far, the ly cmplete set f the six Leslie viscsity cefficiets [78] was published by Keppe ad Scheider abut tw decades ag i their cmplicated rtatig magetic field experimets [90]. The LC cmpud studied was 4-methxybezylidee-4 butylailie (MBBA). Hwever, MBBA is t a practical cmpud fr OPA applicatis because f its pr material stability. Presetly, may cmmercial high birefrigece LC mixtures are based cya-bipheyl ad -terpheyl cmpuds. T study the dyamical respse f OPA, it is ecessary t fid a effective methd t estimate the Leslie cefficiets fr high birefrigece LC mixtures. I ur experimets, we used E7 LC mixture (Merck) ad a high birefrigece ad lw viscsity LC mixture, UCF- [],[], as exemplary materials fr validatig ur mdel. I Sec. 5., we derive the dyamic respse f LC icludig backflw at varius temperatures. I Sec. 5.3, we briefly describe the experimetal methds. I Sec. 5.4, we exted the temperaturedepedet Leslie viscsity cefficiets f MBBA t E7 based the Imura ad Oka (IO) thery [9]. We first justify the ad 4 f E7 usig the experimetal data ad the validate the fittig equatis by the experimetal data. Usig these parameters, we simulate the OPA dyamics icludig backflw effect. The agreemet betwee thery ad experimet is excellet. 63

82 5. Theretical Backgrud 5.. Ericks-Leslie Equati I the dyamic Freedericksz trasiti f a hmgeeus ematic cell, the azimuthal agle φ is cstat s that the LC directr ca be slely described by the tilt agle θ(z,t). The mti f the LC directr is cupled with the flw ν i the x directi. The evluti f θ ad ν uder a applied vltage is gvered by the fllwig Ericks-Leslie equati [78],[79], which takes it accut the balace f elastic ad electric-field-iduced trques: θ θ I + γ = t t ( K cs θ + K si θ ) θ θ 33 + ( K33 K) siθ csθ z z v + ε εe siθ csθ + ( si θ 3 cs θ ), z (5.) where i are the Leslie viscsity cefficiets, I is the iertia f the LC, θ is the plar agle f the LC directr (the agle betwee the LC directr ad the x-y plae, as depicted i Figure 8), Kii are the Frak elastic cstats, ε is the dielectric aistrpy, ad v is the flw velcity. z r x φ θ y 64

83 Figure 8: The crdiate system f LC directr, where θ is the tilt agle f the LC directr, which is the agle betwee the LC directr ad the x-y plae, ad φ is the agle betwee prjecti f the LC directr the x-y plae ad the x-axis. I geeral, the iertial effect is much smaller tha the elastic ad viscus trques ad ca be eglected. Neglectig the iertial term, the flw velcity is gvered by: {( ) [ ]. 0 )cs ( )si ( cs si si cs = z v t z θ θ θ θ θ θ θ (5.) Slvig frm Eq. (5.), ad substitutig this it Eq. (5.), we btai the effective rtatial viscsity as [9]: v z / * γ ], )cs ( )si ( cs si [ ) cs si ( * θ θ θ θ θ θ γ γ = (5.3) where γ is the rtatial viscsity, ad s i ' are the Leslie cefficiets. I the lw vltage regime (V < V th ) f a hmgeeus (splay-mde) cell, ca be apprximated by, where [9]: * γ * γ s. (5.4) 0) ( * * 3 3 γ γ θ γ γ = + + = = = s At a higher ffset vltage (V >> V th ), the LC directrs are rerieted perpedicular t the substrates except the budary layers. Thus, the effective rtatial viscsity ( ) fr the bedmde (θ = 90 ) is [9]: * γ B. 90) ( 5 4 * * γ θ γ γ + + = = = B (5.5) 65

84 It is evidet frm Eqs. (5.4) ad (5.5) that i a splay r bed mde, the directr rerietati depeds largely ad ; the 3 values are e t tw rders f magitude smaller ad are eglected. Similar effect is als bserved i the twisted ematic cell [93]. Fr laser beam steerig, we eed a pure phase mdulatr with phase chage δ π. The pure phase mdulati f a TN cell has bee demstrated i the lw vltage regime; belw the ptical Freederisckz trasiti threshld [94]. Hwever, t archive the required π phase chage, the TN cell gap wuld be t large s that the respse time wuld be t slw. Bth hmgeeus ad hmetrpic cells ca be used fr pure phase mdulati. Hwever, frm a mlecular desig stadpit it is easier t btai LC cmpuds with a large ad psitive ε. Thus, here we fcus the LC directrs defrmati f a hmgeeus cell with strg achrig at surface budaries. 5.. Temperature Effect Temperature has a great ifluece the physical prperties f a thermtrpic LC. As temperature icreases, the birefrigece, dielectric aistrpy, viscsity, ad elastic cstat all decrease but at differet rates. The temperature-depedet physical prperties are gvered by the rder parameter S as [95]: = ( ) S, (5.6) S ε = A, T (5.7) K ii = ( K ii ) S, (5.8) γ = bs exp( E' / k0t ), (5.9) 66

85 where ( ) ad (K ii ) are the birefrigece ad elastic cstat, respectively, at S=, i.e., T=0 K, A ad b are prprtiality cstats, E is the activati eergy f mlecular rtati, ad k 0 is the Bltzma cstat. If the temperature is t t clse t T c, the rder parameter S ca be apprximated as [80]: β S = ( T / ). (5.0) T c Equati (5.0) is Haller s apprximati fr the rder parameter where T is the clearig pit f the LC mixture ad β is a material cstat. The rati f T = T / is called r T c reduced temperature. This apprximati hlds quite well if the temperature is t t clse t the clearig temperature. The expet β is depedet mlecular structure, but t c wavelegth. Fr mst LC cmpuds studied, β 0.0. Fr the E7 LC mixture we studied, T c =333 K. Frm the measured temperature-depedet birefrigece data, we btai ( ) =0.304 ad β = 0.. These results are csistet with the published literature values [40] Temperature-depedet Leslie Cefficiets I the Imura ad Oka thery, all six Leslie cefficiets ( - 6 ) are expressed i terms f the rder parameter S as [95]: = A, (5.) S = ( B + C ) S ( B + C ), (5.) S = ( B C ) S ( B C ), (5.3) 3 S = η as A, (5.4) 4 is + 3S 67

86 3 5 = ( a + B ) S + ( A + B ) S, (5.5) 3 6 = ( a B ) S + ( A B ) S, (5.6) where the cefficiets a, A i, B i ad C i (i =, ) are weakly depedet the temperature. Fr differet LC materials, fr simplicity let use assume that these cefficiets (a, A i, B i ad C i ) remai uchaged but the rder parameter wuld vary due t differet β ad T c. Accrdig t the defiitis, γ = 3 ad γ = 6 5. Based the Pardi relatiship [97], γ ca be rewritte as 3 +. I Eq. (5.4), η is is the flw viscsity f the LC i the istrpic state η = exp( E / k T ), (5.7) is η B where E >0 is the activati eergy f mlecular diffusi, ad η is a prprtiality cstat. Equati (5.7) is the well-kw Arrheius law fr istrpic liquids Methd t Estimate Leslie Cefficiets Based the abve discussi, we prpse a methd t estimate the Leslie cefficiets. S far, the ly cmplete set f Leslie cefficiets was published fr MBBA usig the rtatig magetic field experimets [90]. Usig the MBBA data t fit Eqs. (5.) (5.7), we fud E= 0.44 ev. Durig fittigs, we have assumed that all the cefficiets a, A i, B i ad C i (i =, ) are cstats ad used the recmmeded six Leslie cefficiets f MBBA listed i Table 6. Usig the MBBA data t fit the IO thery, we btai a set f cefficiets, listed i Table 7. Frm these fittigs, we fid that the experimetal data fit reasably well with the IO thery 68

87 fr 4, 5, ad 6, but prly fr,, ad 3. Fr, a better fittig is fud if we iclude a third-rder S term i Eq. (5.): = ( A S + A 4 5 S ), (5.8) where A 4 = 0.07 Pa s ad A 5 = Pa s. The fittig result f i Eq. (5.8) is sketched i Figure 9 (a), where pe circles represet the measured results ad filled circles are the fittig results. The agreemet betwee fittig ad experimet is very gd. Als icluded i Figure 9 (b) fr cmparis are the fittig results usig Eq. (5.). Withut the third rder term, the fittig f Eq. (5.) with experimet is quite usatisfactry. Table 6: Recmmeded Leslie cefficiets fr MBBA (frm Ref [90]). All i s are i uit f Pa β s. S is calculated frm S = T / T ) with β = 0.88 ad T c = 39. K. ( c T (C) S Table 7: Parameters btaied frm fittig MBBA data with the IO thery (Eqs. (5.) (5.7)). All the parameters are i uits f Pa s. 69

88 A B C B C η a A A x , Pa s Temperature, C (a) , Pa s (b) Temperature, C Figure 9: The Fittig results f fr MBBA (a) usig Eq. (5.8) ad (b) usig Eq. (5.). Ope dts are measured data ad filled dts are the fittig results. 70

89 The rtatial viscsity is related t the Leslie cefficiets ( 3 ad ) as γ = 3 -. Frm Eqs. (5.) ad (5.3), we btai the fllwig expressi fr γ : γ = C S + C. (5.9) S Hwever, Eq. (5.9) des t fit the MBBA data well, as shw i Figure 0. A better fittig is fud usig the fllwig expressi prpsed by Wu ad Wu [98]: γ = bs exp( E' / k0t ). (5.0) I Eq. (5.0), b is the prprtiality cstat, E is the activati eergy f mlecular rtati, ad k 0 is the Bltzma cstat. Frm Eq. (5.0), the activati eergy E, which is determied by the LC structures, has a dramatic effect the rtatial viscsity γ, Pa s Temperature, C Figure 0: The temperature-depedet rtatial viscsity f MBBA. Dts are the experimetal data calculated frm Table 6, dashed lies are fittig results usig Eq. (5.9), ad slid lie is fittig usig Eq. (5.0). 7

90 Figure 0 plts the MBBA data (dts) ad fittig results usig Eq. (5.9) (dashed lies) ad Eq. (5.0) (slid lie). It is bvius that Eq. (5.0) has a much better fittig tha Eq. (5.9). Frm Table 6, the 3 f MBBA is egative but very clse t zer, which leads t: γ = bs exp( E' / kt ). (5.) Thus, ca be determied thrugh the measuremet f γ. Therefre, i ur simulatis we use Eqs. (5.8) ad (5.) fr ad, ad Eqs. (5.4)-(5.6) fr 4, 5 ad 6, respectively. I the fllwig, we will validate this methd by cmparig the experimetal ad simulati results usig tw LC mixtures, E7 ad UCF Experimet I experimets, we measured the physical prperties such as birefrigece ad viscelastic cefficiet f E7 ad UCF- usig hmgeeus cells. The ier sides f the idium-tixide glass substrates were cated with a thi plyimide layer ad the rubbed i ati-parallel directis. The buffig iduced pretilt agle is ~3 ad the achrig eergy is ~3 0-4 J/m [0]. The cell gap was determied frm the trasmitted iterferece friges thrugh a spectrphtmeter. The cell gap was fud t be d =3.4µm fr E7 ad 7.8 µm fr UCF-. The elastic ad dielectric cstats f E7 ad UCF- were measured usig a cmputer-ctrlled APT III maufactured by Displaytech. The birefrigece ( ) was determied by measurig the vltage-depedet trasmittace f the LC cell at λ =.55µm thrugh the LabVIEW data acquisiti system [99]. Table 8 lists the measured physical prperties f E7 at T =0.3, 33.3, ad 46.9 C. 7

91 Fr the temperature studies, the cell was held i a ve (INSTEC STC-00D) which has 0. C stability. T mimic the peratig cditi f a OPA, the LC cell was iitially biased at V i =0 V rms. Durig the relaxati prcess, the vltage was remved at t=0 ad the ptical respse mitred by the phtdide detectr ad displayed by a digital scillscpe. Table 8: Sme measured ad calculated LC parameters f E7 at varius temperatures. λ =.55 µm ad T c =60 C. T K K 33 ε // ε ( C) (pn) (pn) (mpa s) (mpa s) (mpa s) (mpa s) (mpa s) (mpa s) Results ad Discussi T cmpare with the experimetal results, we umerically slved Eq.(5.) usig the fiite elemet [83] ad fiite differece time dmai methds assumig that the achrig eergy is ifiite. The dyamic respse f tw LC mixtures was studied: E7 ad UCF-. E7 was used i the first geerati OPA. Hwever, its figure f merit [00], defied as FM = K ( ) γ, is / ly ~4 ms/µm at T ~48 C. T ehace FM, we frmulated a ew mixture (UCF-) csistig f isthicyaat-tlae cmpuds. The FM f UCF- reaches ~40 ms/µm at T =70 C. As a result, the OPA wuld have a much faster respse time. Hwever, the backflw effect eeds t be ivestigated. 73

92 5.4. E7 Figure depicts the vltage-depedet trasmittace curves at λ =.55 µm ad three differet temperatures, T = 0.3 C (gray lie), 33.3 C (dt-dashed lie), ad 46.9 C (slid lie). Frm these curves, we fid that the birefrigece f E7 at λ =.55 µm is =0.90, 0.75, ad 0.48 at T =0.3, 33.3, ad 46.9 C, respectively Trasmittace Vltage (V rms ) Figure : Vltage-depedet trasmissi f a 3.4-µm-thick hmgeeus E7 cell betwee crssed plarizers at three differet temperatures: T =0.3 C (gray lie), 33.3 C (dt-dashed lie) ad 46.9 C (slid lie). λ =.55 µm. The temperature-depedet rtatial viscsity f E7 is shw i Figure where the dts are the measured data ad the slid lie represets the fittig curve usig Eq. (5.0). T fit the experimetal data, we use the same β (=0.) fr the rder parameter (S) ad a, {Ai, Bi, Ci} (i=, ), ad A3, A4, ad A5 as MBBA ad leave b ad E as adjustable parameters. Frm the fittigs, we fid b=.3x0-8 Pa s ad E = 0.439eV fr E7. The btaied activati eergy agrees 74

93 very well with that reprted i [8] (E =0.440 ev). The relatively large E is caused by the dimmer frmati f cya-bipheyls i the E7 mixture γ (Pa.s) Reduced Temperature, T R Figure : The temperature depedet γ f E7. Dts are measured data ad slid lies are the fittig curves usig Eq. (5.0). Figure 3 plts the measured ptical dyamic respse f the E7 LC cell at T= 0.3 C (grey lie), 33.3 C (dashed lie), ad 46.9 C (slid lie), respectively. Befre relaxati starts, the LC cell was biased at V =0 V rms. Uder such a circumstace, the LC directrs are b rerieted almst perpedicularly t the substrates thrughut the bulk except the budary layers. These budary layers still preserve sme residual phase retardati. Thus, uder crssed-plarizer cditis, a small trasmittace is bserved at t =0, as shw i Figure 3. This residual trasmittace is affected by the d, threshld vltage, ad surface achrig eergy f the LC cell. As the temperature icreases, bth birefrigece ad threshld vltage decrease. Thus, the iitial trasmittace is suppressed. Mrever, the rtatial viscsity 75

94 decreases as the temperature icreases. Thus, the time required t reach the first trasmittace peak is reduced Trasmittace Time, ms Figure 3: The experimetal ptical dyamic respses f a hmgeeus E7 cell at three differet temperatures: T = 0.3 C (grey lie), 33.3 C (dt-dashed lie) ad 46.9 C (slid lie). d =3.4 µm ad λ =.55 µm. I Figure 3, the backflw effect is clearly bserved i the first few millisecds. At the first stage f relaxati, the trasmittace curve shws a steeper slpe at each temperature due t the backflw effect. Durig the abrupt relaxati, the LC directrs iitially versht with time ad attai a maximum value at a very shrt time befre decayig mtically t zer. Figure 4 shws the simulated tilt agle distributis f the E7 cell at T = 46.9 C durig its first 30 ms relaxati. At t =0, the 0 V rms biased vltage is remved istataeusly. Right befre the electric field is remved, the bulk mlecules are rerieted t be perpedicular t the substrates 76

95 except the budary layers due t their pretilt agle ad strg surface achrig. The tilt agle icreases t beyd 90 with time s that the LC mlecules are drive t the ppsite directi. At t ~0 ms, the tilt agle reaches its maximum value. Afterwards, the tilt agle starts t decay ad the mlecules are drive back t the frward directi. This flw-iduced pheme ccurs rmally whe the stregth f the field is well abve the threshld ad the field trasiti is abrupt. At the ceter f the cell, there exists equal but ppsite trques, s the mlecules are drive t the ppsite directi temprarily. After the iitial backflw, the velcity decreases quickly, chages sig, ad fially appraches zer. Va Dr [88] ad Berrema [89] have umerically aalyzed the directr rerietati idepedetly. Figure 4: The simulated tilt agle distributi f a hmgeeus E7 cell at T = 46.9 C durig its first 30 ms relaxati. d =3.4 µm ad λ =.55 µm. The parameters used fr simulatis are listed i Table 8. 77

96 The trasiet ptical dyamic respse f the E7 cell at three differet temperatures, T = 0.3, 33.3, ad 46.9 C are shw i Figure 5 (a-c), respectively. I these figures, the slid lies represet the experimetal results ad the dashed lies shw the simulati results. The parameters used fr the simulatis are listed i Table 8. The geeral agreemet betwee the simulatis ad experimetal results is quite gd. As already discussed i thery, the directr rerietati depeds largely ad i the frm f Eq. (5.4). I ur simulatis, we ly adjusted the 4 value t best fit the experimetal results at each temperature. The values f 4 i Table 8 have already bee updated fr the best fittigs. Here, we eed t meti that the chice f 4 is sesitive t the elastic cstats, K ad K 33. As the elastic cstat icreases, a larger 4 is required t decrease the frictial trque, thus gettig the ew equilibrium betwee the elastic ad viscus trques Trasmittace (a) Time, ms 78

97 Trasmittace Time, ms (b) Trasmittace Time, ms (c) Figure 5: The trasiet ptical dyamic respses f a hmgeeus E7 LC cell at three differet temperatures: (a) T = 0.3 C, (b) T =33.3 C ad (c) T =46.9 C. d =3.4 µm ad λ =.55 µm. The slid lies represet the experimetal results, while the dt-dashed es are the simulati results. The parameters used fr simulatis are listed i Table 8. 79

98 After havig ptimized the 4 values at all three temperatures, we still use Eq. (5.4) t d the fittig by ly chagig the activati eergy f mlecular diffusi. Figure 6 shws the fittig results. Frm the fittigs, the activati eergy (E) f the istrpic state ca be extracted. Fr E7, we btai E = 0.43 ev, which is cmparable t that f MBBA (E = 0.44 ev) due t similar mlecular cfrmatis (Pa.s) Temperature, C Figure 6: The temperature-depedet Leslie cefficiet 4 fr E7. The slid lie represets the fittig curve usig Eq. (5.4) with E =0.35 ev, while the dts represet the ptimal values f 4 at each temperature UCF- Further t verify the essetial crrectess f this apprach we als tried the high birefrigece LC mixture, desigated as UCF-, that we frmulated. Its ematic rage is frm 80

99 0.3 t 39.7 C. All the parameters used fr simulatis are listed i Table 9. At λ =.55 µm ad T =70 C, UCF- has ~0.39 which is ~X higher tha that f E7 at T ~47 C. Due t the higher birefrigece, the required cell gap t achieve π phase chage is reduced i prprti. By cmparig Table 9 with Table 8, we fid that the (r γ ) f E7 at T ~47 C is abut the same as that f UCF- at T =70 C. Hwever, UCF- has a much higher clearig temperature tha E7, which leads t a higher K. The visc-elastic cefficiet ( γ / K ) f UCF- at T =70 C is ~3X smaller tha that f E7 at T ~47 C. As a result, UCF- shuld have ~0X higher FM tha E7. Table 9: Sme measured ad calculated LC parameters f UCF-0 at T =70 C ad λ =.55 µm. T K K 33 ε // ε ( C) (pn) (pn) (mpa s) (mpa s) (mpa s) (mpa s) (mpa s) (mpa s) Figure 7 plts the ptical dyamic respse f the UCF-0 LC cell (d =7.8µm) at T = 70 C. The simulati results (dashed lie) fit very well with the experimetal results (slid lie). Frm the value f 4, we fid that E = ev, which is ~0% higher tha that f E7. Recall that E is the activati eergy i the istrpic state. Althugh UCF- exhibits a lwer viscsity tha E7 i the ematic phase, its activati eergy i the istrpic state is slightly higher tha that f E7. I the istrpic state, the iter-mlecular assciati is much weaker ad the dimmers f cya-bipheyls are separated. The mmet f iertia f cya-bipheyl is smaller tha that f isthicyaat-tlae, the majr cmpsiti f UCF-. As a result, E7 exhibits slightly smaller activati eergy tha UCF-. 8

100 Trasmittace Time, ms Figure 7: The trasiet ptical dyamic respses f a hmgeeus cell usig UCF-0 LC mixture at T =70 C. d =7.8 µm ad λ =.55 µm. The slid lie shws the experimetal results ad the dt-dashed lies are the simulati results. The parameters used fr simulatis are listed i Table Cclusi Backflw is fud t make a imprtat impact t the respse time f the phase mdulatr usig a hmgeeus LC cell. Fr laser beam steerig at a ear ifrared wavelegth, the required cell gap is arud 5-7 µm, depedig the LC birefrigece. Due t the relatively high vltage applied, the backflw takes place i the first few millisecds. We have mdified the expressis f tw Leslie cefficiets ( ad ) i rder t exted their validity t tw high birefrigece LC mixtures, E7 ad UCF-. With these mdificatis, the simulati results agree quite well with the experimetal data. The trasiet LC directr distributis durig backflw ccurrece are aalyzed. 8

101 CHAPTER 6: CELL GAP EFFECT ON THE DYNAMICS OF LIQUID CRYSTAL MODULATORS 6. Itrducti Liquid crystal spatial light mdulatr (SLM) has bee used as a phase-ly mdulati fr laser beam steerig [36], tuable-fcus les [0]-[06], ad ther phtic devices[07],[08]. T btai a large phase shift while keepig peratig vltage belw 0 V rms, hmgeeus cell (als called parallel-aliged cell) is a favrable chice. Respse time f a hmgeeus cell is a critical issue. T achieve a fast respse time, lw rtatial viscsity (γ ) LC mixtures are preferred [45],[49],[50]. Ather straightfrward apprach is t use a thi cell gap filled with a high birefrigece ( ) ad lw viscsity LC mixture [6],[5]. The LC directrs rise time τ rise ad decay time τ decay are kw t be prprtial t d, where d is the cell gap. Hwever, the theretical derivati f this d depedece is based the small agle apprximati [7]. Thus, these equatis are valid ly i the V < V < V regi, marked as Part I i Figure 8, where V is the applied vltage ad V th is the threshld vltage. Sice the ptical respse time is liearly prprtial t the LC directr rerietati time [08], the crrespdig ptical respse time is als prprtial t d. I the large sigal regime where V π < V < Vi th, marked as Part III i Figure 8, the surface mdes dmiate, where V is the vltage crrespdig t last trasmittace miimum ad V π the iitial vltage at the high vltage regime. As a result, bth rise ad decay times are fast. Mrever, the ptical decay time is idepedet f the cell gap d [0],[]. th i 83

102 Hwever, the ptical respse time i the middle vltage regime has t bee studied yet. I this paper, we fud that betwee Part I ad Part III there is a regi where the respse time is liearly prprtial t d. T validate this experimetal bservati, i Sec. 6., a cmplete derivati based small agle apprximati usig a parallel-aliged cell is give. Experimetal methds are described i Sec I Sec 6.4, the validity f ur aalytical derivati is cfirmed by experimetal results ad the cmplete physical picture f ptical respse time as a fucti f cell gap i the whle vltage regime is give. Figure 8: Simulated vltage-depedet trasmittace curve f a 5.6-µm hmgeeus E7 cell with 3 pretilt agle betwee paralleled plarizers at λ = 633 m ad 3 C, where i small sigal regi (Part I), small agle apprximati hlds well; while i high sigal regi (Part III), the trasiet ematic effect is satisfied. 84