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Active control of total internal reflection and its application in electrophoretic displays 1999
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Title | Active control of total internal reflection and its application in electrophoretic displays |
Creator |
Grandmaison, Dmitri |
Date Created | 2009-06-25T23:58:45Z |
Date Issued | 2009-06-25T23:58:45Z |
Date | 1999 |
Description | The main objective of this thesis is to analyze and quantify methods for controlling the phenomenon known as Frustrated Total Internal Reflection (FTIR). Several methods of numerical and analytical study of FTIR are presented. A direct solution of the Maxwell equations is shown to be a powerful tool in understanding FTIR. An analytic solution is derived for a specific planar case of FTIR. Results that were obtained in the form of transmission coefficients are compared with the numerical modeling, with favorable results. The numerical modeling results were obtained with a new computer code based on the FDTD algorithm that finds the solution of Maxwell equations in 3D space with periodic boundary conditions. Several devices based on the active control of total internal reflection (ACTIR) have been proposed, with the most promising being an electrophoretic display cell, in which the frustration of TIR is done by electrostatically charged particles suspended in liquid medium. An analysis is presented from the physical and chemical points of view. An optimization of the electrophoretic display cell surface parameters has been performed by means of ray tracing modeling, and these results and meir implications to the future cell design are presented. Finally a comparative analysis of presented electrophoretic display technology is presented, in the context of alternative existing technologies. |
Extent | 4691386 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-06-25T23:58:45Z |
DOI | 10.14288/1.0085105 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/9670 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0085105/source |
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Active Control of Total Internal Reflection and its Application in Electrophoretic Displays by DMITRI G R A N D M A I SON B . S c , Moscow Institute of Physics and Technology, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R O F S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A August 1999 © Dmitri Grandmaison, 1999 ln presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date Au-f W , OSS DE-6 (2/88) Abstract The main objective of this thesis is to analyze and quantify methods for controlling the phenomenon known as Frustrated Total Internal Reflection (FTIR). Several methods of numerical and analytical study of FTIR are presented. A direct solution of the Maxwell equations is shown to be a powerful tool in understanding FTIR. An analytic solution is derived for a specific planar case of FTIR. Results that were obtained in the form of transmission coefficients are compared with the numerical modeling, with favorable results. The numerical modeling results were obtained with a new computer code based on the FDTD algorithm that finds the solution of Maxwell equations in 3 D space with periodic boundary conditions. Several devices based on the active control of total internal reflection (ACTIR) have been proposed, with the most promising being an electrophoretic display cell, in which the frustration of TIR is done by electrostatically charged particles suspended in liquid medium. An analysis is presented from the physical and chemical points of view. An optimization of the electrophoretic display cell surface parameters has been performed by means of ray tracing modeling, and these results and meir implications to the future cell design are presented. Finally a comparative analysis of presented electrophoretic display technology is presented, in the context of alternative existing technologies. i i Table of Contents ABSTRACT.. , ii TABLE OF CONTENTS ii f LIST OF FIGURES v ACKNOWLEDGEMENT . . . vii 1. INTRODUCTION 1 1.1 Definition of frustrated total internal reflection 1 1.2 Modeling of FTIR 2 2. FRUSTRATED TOTAL INTERNAL REFLECTION: A REVIEW 4 2.1 Active control of total internal reflection 4 2.2 Overview of total internal reflection .4 2.3 FTIR as a special case of TIR. Analytic solution of FTIR reflection/transmission coefficients in a simple planar case •. 6 2.4 Methods of A O T R 8 2.4.1 Pressure induced membrane control 9 2.4.2 Electrostatic control 11 2.4.3 Electrophoretic control 14 2.5 Practical use of A C T I R in display technology 14 3. FDTD MODELING OF FTIR .; 16 3.1 Definition of the problem 16 3.2 Yee's algorithm 16 3.3 Boundary and initial conditions 20 3.4 Computational stability and computation time 25 iii 3.5 Determination of reflection and transmission coefficients 25 3.6 Results of the modelling ; 26 3.7 FTIR at a rough surface 31 4. ELECTROCHEMISTRY OF THE ELECTROPHORETIC CELL 33 4 1 Introduction 33 4.2 Coaxial light-pipe switch 33 4.3 Electrophoretic cell 33 4.4 Planar electrophoretic switch 37 4.5 Chemistry of the electrophoretic suspension , 45 4.6 Calculation of the efficiency of the electrophoretic cell ; 45 4.7 Comparison of the electrophoretic cell with L C D cell 50 5. RAY TRACING MODELING OF HIGH INDEX STRUCTURES 52 5.1 Outline of the problem 52 5.2 Choice of the materials and approximations used in the modeling 52 5.3 Calculation of the prism angle for the maximum deflection 53 5.4 Modelling of single layer prismatic structure 54 5.5 Modelling of double prismatic surface .59 5.6 2D modelling versus 3D .63 6. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK .64 6.1 Conclusions : 64 6.2 Suggestions for future work 65 REFERENCES 66 APPENDIX 1. SOURCE CODE OF THE SIMULATION PROGRAM 68 iv List of Figures Figure 2-1. Transmission coefficients T± and Tt versus normalised optical separation. Figure 2-2. State "on" and state "of f of pressure-induced cell. Figure 2-3. Corner cube reflector. Figure 2-4. Electrostatically actuated FTIR. Figure 3-1. Position of various field components. Figure 3-2. Schematic diagram of model setup. Figure 3-3. Position of non-reflecting absorbers. Figure 3-4. Position of virtual planes for calculation the transmitted and reflected energies. Figure 3-5. Transmission coefficients obtained as a result of numeric modelling and corresponding analytically obtained values of the same coefficients. Figure 3-6. £"-field component, TER. case. Figure 3-7. £- f ie ld component, FTIR case. Figure 4-1. Coaxial electrophoretic switch. Figure 4-2. Design of the electrophoretic cell. Figure 4-3. Planar electrophoretic switch. Figure 4-4. Surface potential y/ versus the distance x from the charged surface. Figure 4-5. Concentration of the particles n+ versus the distance x from the charged plane. Figure 4-6. Normalised absorption function. Figure 4-7. Transmission function T0 (x) for the glass/acetonitrile interface. Figure 5-1. Maximum reflection of a normal incident ray by a prism. Figure 5-2. Single prismatic layer design of the top cover of the electrophoretic cell. Figure 5-3. Reflectivity of a single prismatic layer electrophoretic cell versus incident angle of the incoming light. Figure 5-4. Double-sided prismatic design of the top surface electrophoretic cell. Figure 5-5. Reflectivity of the electrophoretic cell in the "white" state (double prism design). Acknowledgements First I would like to thank Dr. Lome Whitehead for his support and advice in this project. This work would never be a success without his invaluable assistance. I would like to express my thanks to all past and present members of the Structured Surface Physics Laboratory who were working in the laboratory during my M. Sc. work. I also thank the administration of Physics Department, the secretaries, faculty members and stuff, who has been helpful and supportive at every stage of this project. I would like to express my appreciation to Dr. Andrzej Kotlicki for his assistance and valuable remarks during the preparation of this thesis. Many thanks to Peter Kan for his help with different lab matters and assistance with ray tracing modeling. I would particularly like to thank Robin Coope and Michele Mossman for their help and useful discussions about display technology. I also thank Dr. S.P Rao from 3 M , USA for his help with the measurements and useful remarks. I am grateful to my parents for continuing support during my academic career. vii 1. Introduction 1.1 Definition of frustrated total internal reflection Total internal reflection (TIR) is a phenomenon well known over hundreds of years. It occurs when an electromagnetic wave passes from an optically dense medium into one less dense ( « , >n2) and the incident angle of the wave 6 is greater than the critical angle for the two media defined by e„ = sin~'(n 2 In,). Under the conditions of TIR the incident wave is completely reflected back to the first, optically dense media, A small fraction of energy passes through the physical interface between the two media and forms a so-called evanescent wave. The evanescent wave transports energy parallel to the interface and the intensity in this wave can be expressed as ' I(0,z) = Ie(0)exp(-j), where z is the distance from the interface, 9 is the angle with respect to the interface normal, and / is the penetration depth given by I _ K *n(n2 sm2 6-n2f2 where nx and n2 are the refractive indices of the two media, X0 is the wavelength o f light in vacuum and Ie(0) is an average intensity of the incoming light incident at the angle e. l Under conditions of TIR there is no net energy flow between the interface of two surfaces. However, when a higher index media is inserted in the region of evanescent wave, some energy may flow into this media. This is called frustrated total internal reflection (FTIR). 1.2 Modeling of FTIR The fraction of light transmitted into the "frustrating" media is a function of the refractive indices of a) the original medium b) the layer of lower refraction medium c) the frustrating medium and the incident angle 0 of the incoming wave. A general expression for the transmission coefficients for the FTIR phenomenon is presented in Chapter 2. FTIR can be controlled by different means. The "frustrating" medium can be displaced from its original location preventing FTIR to happen. It can also be returned back to its original position enabling FTIR. Several methods of active control of total internal reflection (ACTIR) are presented in Chapter 2. The analytic solution for the transmission coefficients presented in Chapter 2 can only be found in an explicit compact form for the case when FTIR is happening between flat surfaces. A more general approach based on the direct solution of Maxwell equations has been employed, and is presented in Chapter 3. The values of the transmission coefficients analytically obtained in Chapter 2 have been numerically verified by finding the actual intensities of the electric and magnetic field. A Finite-Difference Time-Domain (FDTD) algorithm has been employed to solve Maxwell equations. A computer code based on F D T D algorithm has been developed to solve the system of Maxwell equations in a 3D space with rectangular periodic boundary conditions. 2 1.3 Display cell design Practical application of ACTIR , an image display cell, has been designed and is described in this thesis. Several ways of controlling the frustration of TIR have been considered for such purpose 2 . A n electrophoretic method df ACT IR appears to be the most promising way. In this method, TIR occurs at the interface between a retro-reflector and a low refractive index liquid containing charged microparticles. By applying an electrostatic potential in the fluid, the charged microparticles can be drawn to the TIR surface, frustrating TIR, and can be drawn back stopping the frustration. Chapter 4 gives a self-consistent solution for the equation describing the distribution of the particles within the electrophoretic cell and gives an estimate of the energy fraction that could possibly be modulated by means of electrophoresis. Based on the literature search it is believed that it is one of the first solutions of such a problem. Chapter 5 describes other possible designs for the top surface of the electrophoretic cell, as alternatives to standard corner-cube retro-reflectors. A ray tracing analysis has been carried out to optimize the surface parameters of proposed designs; the results are summarized. Chapter 6 presents conclusions and suggestions for future work. 3 2. Frustrated Total Internal Reflection: a Review 2.1 Active control of total internal reflection FTIR can be controlled by different means. It can be controlled mechanically by physically inserting frustrating material in the evanescent wave zone. It can be controlled electrostatically by controlling the position of the frustrating material by an electric field. It is also be possible to control FTIR by changing the refractive index ratio of the two media. Several geometrical structures are suggested in this thesis to establish that these are methods of actively controlling FTIR; we call it ACTIR. 2.2 Overview of total internal reflection When an electromagnetic wave is incident on an interface between two media, part of the energy in that wave may be reflected from that interface. The percentage of reflection depends on the refractive index of the two media involved, as well as the angle of the incoming wave with the interface normal. If, however, the refractive index of the first media is greater than the refractive index of the second media and the angle of incidence exceeds the critical angle 6cr=sml{n2lnx) (2.1) total internal reflection occurs. The light in this case is completely reflected back to the first media. The Fresnel formulae 3 «j cos# + « 2 cosf?' £ , ^ « 2 cos 0 - f t , cos- f l '^ 1 « 2 cosr? + fl,cos0' 11 where £" is an amplitude of the reflected wave, E" is an amplitude of the transmitted wave, 6 is the incident angle of the incoming wave and 0' is given by the Snells law n2 sinf?' = n, sinf?, (2.6) completely characterize the behavior of the transmitted and reflected waves. However, when TIR occurs, a small portion of light penetrates into the second media forming the so-called evanescent wave. The Fresnel relations do not describe this phenomenon. We need a full treatment of Maxwell's equations to fully account for the energy transmitted in the evanescent wave. The amplitude of the evanescent wave drops down exponentially from the surface boundary while the wave proceeds parallel to the surface. The e"1 point is reached at the distance of where k is the wave vector, 6>is the incident angle and n ; is the refractive indices of the i1 media, z is often referred to as the penetration depth of the evanescent wave. (2.7) 5 2.3 FTIR as a special case of TIR. Analytic solution of FTIR reflection/transmission coefficients in a simple planar case When a higher refractive index medium is present in the region of evanescent wave, the phenomenon of frustrated total internal reflection (FTIR) occurs. The energy of the evanescent wave is either absorbed by this third medium or scattered further into the second medium. In the most general form the solution of the problem of FTIR happening at two semi-infinite dielectrics and thin film between them was given by N. Court and F.K. von Willisen 4 in 1964. Using the following notations: «, = refractive index of the first medium n2 = refractive index of the spacing between two media » 3 = refractive index of the second medium d = spacing between two media 90 - incident angle X = wavelength of electromagnetic wave in vacuum the transmission coefficients 7J and TL were found to be given by T, 1 (2.8) flTj sinh y + ^ and aL s inh 2 y + ft (2.9) where (2.10) 6 ^ {N2-\pN2-l) ( 2 U ) 47V2 cos0o(N2 s in 2 0O - l ^ w 2 - s in 2 0O [Jn2 - s in 2 0n + cos0n F fi^SL ° °> (2.12) 4cos<9 Jn* - sin 0O a , =^Jr[{N2 + l ) s i n 2 0 o -l][(n2N2 + \)sm2 0O -n2} (2.13) n = {]n2 -sin20o +ft 2 cosc? 0 ) 2 ( 2 M ) 4 « 2 cos0iyjn2 - s in 2 #0 Here, n = ~ and N = — ni n2 (2.15) Figure 2-1 represents the transmission coefficients for different polarisation of light as a function of the gap distance between two dielectric media. Four different curves are shown for nx =n3 =1.5, n2 =1.0 and nx =n3 =l.6;n2 =1.0 When un-polarised light is used the transmission coefficient is the average of that for parallel and perpendicular polarisations: T= " (2.16) The transmission coefficients into the second media die off exponentially with the thin film thickness, d. As we can see from Figure 2-1, the transmission coefficient into the second media is practically zero when the thickness of the spacing is of the order of X. Thus, the effective thickness of the evanescent wave region can be considered to be roughly half the wavelength of the incident light. 7 UOjSSJLUSUBJl The analytical solution for the system described above has been found 5 , but for many other FTIR geometries, the solution for the transmission coefficients can not be found in an explicit, analytical form. In Chapter 3 the analytical solution shown here will be reproduced by numerical modelling of propagation of electromagnetic waves through dielectric media. The more challenging problem of FTIR happening between plane and rough surface will then be considered numerically. 2.4 Methods of ACTIR. 2.4.1 Pressure induced membrane control There are several ways of controlling the conditions under which FTIR occurs. By physically inserting a third media into the evanescent field region, most of the energy of the reflected wave can be absorbed by or scattered into this media. This will cause the reflection coefficient of the TIR surface to decrease. The frustrating medium can be introduced into the evanescent wave region in different ways. One way to introduce an energy absorbing medium into the evanescent wave region is to mechanically displace a rubber membrane by applying pneumatic pressure on the back side of it. When the rubber membrane gets pushed close enough to the back side of the TIR surface, the energy of the incoming wave gets absorbed from the evanescent wave region. Let's consider a device that can demonstrate this mechanism, as shown on Figure 2-2. On the front side of the device (where the incoming wave arrives) we have a corner cube reflector sheet 6 , which reflects light back toward the incident direction by 9 TIR case Air chamber FTIR case Air chamber Figure 2-2. State "on" and state "of f of pressure-induced cell. means of three total internal reflections off prism facets. Specifically, the polycarbonate sheet surface has repeated three-sided pyramids with a ninety-degree angle between the three walls at the top of the pyramid, as depicted on Figure 2-3. On the back side of the device we have attached a cell filled with air. Inside this cell, there is a rubber membrane rigidly attached to the walls of the cell. The enclosed volume between the prismatic sheet and the membrane is at atmospheric pressure. The other side is connected to a pressure actuator. When positive pressure is applied to this side of the cell, the membrane bends up and is introduced into the evanescent wave region. The membrane is easily withdrawn from this region by releasing the pressure in the bottom half of the cell, whereupon the elastic restoring force of the rubber restores the membrane to its original position. 2.4.2 Electrostatic control The device depicted in Figure 2-4 demonstrates an alternative method of frustrating the TIR. On the front side of the device (where the incoming wave arrives) we again have a corner cube reflector 6 , as shown in Figure 2-4. The pyramids are coated with a transparent conducting film (in our case Indium- Tin-Oxide (ITO)). The cell on the back side of the device has a matching shape conductive rubber slab in it, coated on the top with a thin film insulating material, such as paralyne. The rubber slab can be inserted into the evanescent wave region by applying a bias (200V) between the ITO coating and the conductive rubber. The purpose of the paralyne should be clear: a one micron thick paralyne coating can withstand an exceedingly high electric field up to 10 8 V/m, which enables substantial electrostatic field pressure. The rubber slab can be returned to the original position by removing the bias l i Figure 2-3. Corner cube reflector. 12 TIR case ITO film Corner cube reflector Isolating coating = \ A A A A A A A A A A A A A A A A A A / Conductive rubber DC power supply FTIR case ITO film Corner cube reflector Isolating coating A A A A A A A A A A A A A A A A A A A / Conductive rubber DC power supply Figure 2-4. Electrostatically actuated FTIR. 13 between the conducting rubber slab and the ITO, whereupon the elastic restoring force of the rubber causes it to move back. 2.4.3 Electrophoretic control The third proposed device uses charged particles suspended in a liquid contained in the cell on the back side of the corner reflectors. Preferably the liquid should have a low refractive index and have a high electrical resistance. We identified two good candidates: Fluorinert Electronic Liquid (n = 1.27) or acetonitrile (n = 1.33). A transparent conductive film, such as ITO, coats the pyramids of the acrylic film. The general idea is to apply a bias across the solution, and thereby move the charged particles, such as silica particles 7 or polystyrene particles 8 into the evanescent wave region. This device requires careful analysis and is described in detail in Chapter 4. 2.5 Practical use ofACTIR in display technology. The key idea of ACT IR is that the frustration of the TIR has to be controlled in both ways: in the first case it has to occur and in the second case, it has to be cancelled. The rubber membrane method (Figure 2-2), the electrostatically actuated FTIR (Figure 2- 4) and the electrophoretically actuated FTIR device are the three clear possibilities for such a control. The presented devices may also been referred as A C T I R controllable cells. These cells have two possible states: a reflective state when TIR occurs and therefore the cell appears to be white ("on" state) and the FTIR state where the incident light is absorbed or scattered by the medium introduced in the evanescent zone region. In the second state ("off state) the cells appear to be black. 14 A n assembly of these cells in a matrix arrangement with an individual control of each cell may be considered as a flat panel display device. Among the advantages of such a device could be the fact that it is passive, i.e. doesn't need to emit light, and therefore it requires less power to operate. The other advantage over the conventional liquid crystal displays (LCD) may be the fact that with certain geometries of the cells the angular distribution of the reflected light is wider than that of for the passive L C D . This provides a wider viewing angle for such a display. The use of the cheaper materials in comparison with liquid crystals may also be added to the list of the advantages. 15 3. FDTD modeling of FTIR 3.1 Definition of the problem As mentioned in Chapter 2 an explicit closed form solution of the problem of finding the transmission / reflection coefficients for the FTIR phenomenon has been found only for a very specific geometry of the problem. The geometries considered in our case are much more complicated, therefore it would be nice to have a tool which may quantify the FTIR coefficients for any given geometry of an FTIR event. The most general approach is to start from the Maxwell equations and solve them using numerical method. This has been achieved by writing a F O R T R A N solver employing Yee's 9 algorithm. Originally introduced by K. Yee, the Finite-Difference Time-Domain (FDTD) method in application for solving Maxwell equations has proved to be a powerful tool in solving time-dependent problems of scattering and propagation of electromagnetic waves. 3.2 Yee's algorithm Let's start from the Maxwell equations written in the following form l 0 : ^ L - I v x E - ^ H (3-1) dt fj. ju ^ = I v x H - ^ E (3-2) dt s s Assuming that e ,ju,cr p' are isotropic, the following system of scalar equations is equivalent to the Maxwell curl equations (3-1), (3-2) in a rectangular coordinate system 16 (3-3) dt ju dz dy f ^ . ! ^ - ^ - , * , ) (3.4) dt // dx dz dH 1 dE dEv CT dy dx df s dy dz dEyj^dHx dH2 dt e dz dx -oEy) (3-7) ^ = 1 ( ^ . ^ - 0 6 . ) (3-8, . £ . dx dy In order to determine the reflectance and transmittance coefficients the incident plane wave can be represented, at t = 0 , as an incident pulse - a wave train truncated by an gaussian envelope that is substantially larger than the wavelength. Mathematically it can be expressed as following (for the E component) t(x, y, z,0) = E0 exp(-c,(y- y0f) exp(iK • r)E (3-8) A three dimensional grid has been introduced in space with an FTIR geometry enclosed in it. The grid spacing of the cell has been chosen to be cubic to simplify the equations. A grid point of the space has been denoted as (i,j,k) = (iAx,jAy,kAz) (3-9) and any function of time and space as F(i, j, k, n) = F(/Ax, j&y, kAz, nAt). (3-10) 17 The equivalent set of finite difference equations for (3-3) - (3-8) has been constructed as following. The time and space derivatives are approximated using a second order centred finite difference scheme. The key feature of Yee's algorithm, which we are using in our program, a space-time offset for the nodes of the mesh has been employed in the algorithm. Yee positioned the components of E and H at half time steps, which significantly reduces the amount of computation necessary for a given level of accuracy (Figure 3-1). This resulted in the following finite difference equations: Hx(2i + \,2j,2k,2n +1) = Hx(2i +1,2 j,2k,2n -1) + pAd Ey (2i,2j + 2,2k,2n) - Ey (2i,2 j,2k,2n) + Ez (2i,2j,2k,2n) - Ez (2i,2 j,2k + 2,2n) — p'Hx(2t,2j,2k,2n) Hy (2i,2j + \,2k,2n + l) = Hy (2i,2 j + \,2k,2n -1) + At pAd Ey {2i,2j,2k + 2,2/i) - Ey (2/,2 j,2k,2n) + Ez (2i,2j,2k,2n) - Ez (2i + 2,2 j,2k,2n) — p'Hy(2i,2j,2k,2n) Hz (2i,2j,2k + l,2/i +1) = Hz (2i,2 j,2k + \,2n -1) + At pAd Ey (2/ + 2,2j,2k,2n) - Ey (2/,2 j,2k,2n) + Ez (2i,2j,2k,2n) - Ez (2i,2 j + 2,2k,2n) At p'Hz(2i,2j,2k,2n) (3-12) (3-13) (3-14) Ex(2i + \,2j,2k,2n +1) = Ex(2i +1,2 j,2k,2n -1) + At sAd Hz (2/,2y + 2,2*,2n) - Hz (2/,2 j,2k,2n) + Hy (2i,2j,2k,2n) - Hy (2/,2 j,2k + 2,2n) -~aEx(2i,2j,2k,2n) £ (3-15) Ey (2i,2j + \,2k,2n +1) = £ (2/,2j + \,2k,2n -1) + At Hx(2i,2j,2k + 2,2«) - Hx(2i,2 j,2k,2n) + sAd \_HZ (2i,2j,2k,2n) - Hz (2i + 2,2 j,2k,2n) -~aEy(2i,2j,2k,2n) £ (3-16) Ez (2i,2j,2k + \,2n + \) = EZ (2/,2 j,2k +1 ,2« -1) + At £Ad Hy{2i + 2,2j,2k,2n) - H y(2i,2 j,2k,2n) + Hx(2i,2j,2k,2n) - H x{2i + 2,2 j,2k,2n) •~crEz(2i,2j,2k,2n) £ (3-17) 18 To optimize calculation speed and simplify the equations the spatial increments Ax , Ay and Az were selected such that Ax = Ay = Az = Ad. The components of the electric and magnetic field were defined as following: Hx(iJ,k,n) f o rM) ,2 ,4 , . . . ; y= l , 3 , 5 , . . . ; *= l , 3 , 5 v . . ; «= l , 3 , 5 , . . . ; Hy{i,j,k,ri) for/=l,3,5,...;y==0,2,4,...;*=l,3,5,...;Ai=l,3,5,...; Ht(i,j,k,n) for /=l,3,5,...;y=l,3,5,...; fc=0,2,4,-.; n=l,3,5,...; Ex(iJ,k,n) for /=l,3,5,...;y=0,2,4v-.; *=0,2,4,-.; «=0,2 ,4 , - . ; Ey(i,j\k,n) for /=0,2,4,-;y=l,3,5,...; £=0,2,4,...; «=0,2,4,.. .; E,(i,j,k,n) for /=0,2,4,...;y=0,2,4,...; A=l,3,5,...; « = 0 , 2 , 4 , - 3.3 Boundary and initial conditions (3-18) (3-19) (3-20) (3-21) (3-22) (3-23) (3-24) Figure 3-2 shows the setup for the modeling of FTIR in the simulation program. A 3-D wave vector K , which was expressed as K. = K sin 9 cos 0 sin 9 sin <f> K cos 9 (3-25) (3-26) (3-27) where 9 is the angle between K and 2-axis, <f> is the angle made between K '< projection onto xy-plane and the *-axis, and K| , the magnitude of K , is In IX ,where A is the radiation wavelength shows the direction of propagation of the incident wave. 20 Second dielectric media First dielectric media FTIR gap 0 1 Figure 3-2. Schematic diagram of model setup. 21 The system represented on Figure 3-2 has a translational symmetry in the x and_y directions. This means that it is possible to consider only a finite volume of space in x and y directions. The periodic boundary condition has been employed in the algorithm to make use of this significant simplification of the problem set. Then the x and y components of K were chosen to satisfy the following equations Kx-S = 2mn: m= ± 0 , 1 , 2 , . . . (3-28) Ky-8' = 2ln / = ± 0 , 1 , 2 , . . . (3-29) where 8 is the x and <?'is the y-size of the considered volume of space. For the simplicity of the algorithm the values of 8 and 8' used were equal. On one side this restriction imposed on the values of the components of wave vector K somehow restricts possible directions for the propagation of the wave, but on the other leads to a tremendous reduction of computational time. The different polarizations of the incident wave were introduced using the following approach. E was expressed in a form of E = ^ E , + 5 E 2 (3-30) where E , and E 2 are mutually perpendicular vectors of a unit length, and A,B are arbitrary coefficients that satisfy A2+B2 =1 If E , and E 2 are defined as E j = K x k (3-31) E 2 = k x E , (3-32) 22 where K and k are unit vectors in the directions of K and the z-axis, respectively, then E is consistent with the requirement of E • K = 0 . Substitution of (3-31) into (3-32) and using (3-30) gives the following form of E E = (A sin <j) + B cos 9 cos <f>) i + {-A cos <f> + B cos 0 sin <f>) j - (B sin 6>)k (3-33) where i , j , k are, respectively, the unit vectors in the x, y, and z directions. The polarization of the magnetic field can be obtained from H = K x E (3-34) In order to properly observe the reflected and transmitted waves, it was important to eliminate reflections of these waves at the ends of the mesh. It has been achieved by setting up, at both ends, absorption regions in which the electrical conductivity, o , was large enough that a travelling wave would be substantially absorbed in travelling through the region, reflecting off the end of the mesh, and returning back, as shown on Figure 3-3. In order to avoid reflection at the boundary of the reflecting region itself, the value of a was ramped smoothly from 0 at this boundary to a maximum value of 2 siemens/meter at the end of the mesh. The incident pulse we was modelled as a forward-travelling wave. However, when the corresponding initial conditions were set, it was found that this produced a small backward-travelling pulse as well, which could create small errors in the determination of reflection coefficients. To eliminate this, the onset of detection of reflected radiation was delayed until after the backward-travelling pulse had been absorbed by the rear absorber. Since the geometry of the system ensured that there was always a substantial time interval between the arrival of the backward pulse and the reflection radiation, it was easy to employ this method. 23 Figure 3-3. Position of non-reflecting absorbers. 24 3.4 Computational stability and computation time The computational stability of the algorithm (in 3-D case) requires that 1 1 , 1 2 A . . 1 Ad At< - = ^ ( 3 . 3 5 ) 1 1 1 2 1 42c max C < 1 h - max ' - ' - 1 Ax" Ay' Azl\ where c m a x is the maximum electromagnetic phase velocity within the media being modeled. Also, the space mesh has to be such that the electromagnetic field doesn't X change significantly over one space increment. In our model, we set cmax = 1, A s — X and At ~ — . The mesh size and the time increment choice have a large effect on 20 5 calculation time, so these were made as large as possible without significantly distorting the results. Using the selected values, the mesh consisted of 25x25x600 nodes, and we processed 10,000 time increments. This would take about 12 hours of C P U time on a Silicon Graphics ® workstation with a MIPS R4600 processor. 3.5 Determination of reflection and transmission coefficients In order to calculate the reflected and transmitted flux densities we evaluated the z-component of the Poynting vector at two imaginary surfaces xyt and xyr. The following formulae were applied to calculate the numeric values for the fluxes: 7 ' =ZXE( 5*) (3-36) ' r=-HEfc) (3-37) ' *o yo z=Zr where the z-component of the Poynting vector is given by 25 Sz=c20s0(ExHy-EyHx) (3-38) and the minus sign at the reflected flux indicates that it is in negative z-direction. Figure 3-4 shows the position of the imaginary surfaces at which the transmitted and reflected flux densities were calculated. Notice that minus sign in (3-37) refers to the energy transmitted in the negative z direction. 3.6 Results of the modelling In order to find the FTIR transmission coefficients two dielectric media were introduced in the considered volume of space, as depicted on Figure 3-2. The thickness of the FTER gap was varying with every run. The incident angle for the results presented below was chosen to be 45 degrees, both of the dielectric media had the refractive index of «, -n2 = 1.5. The FTIR gap medium was air. Incident light was polarised at 45° , V2 which was achieved by setting the parameters A and B in the equation (3-30) to — . Figure 3-5 shows the transmission coefficients rB and Tx for the specified indices of refraction obtained as a result of analytic solution to the same problem as well as values of the transmission coefficient obtained as a result of numerical modelling. The 45° polarisation of the incident light was produced by equal contributions from parallel and perpendicular polarisation T - — . It can be seen that the numerical modelling agrees well with the analytic solution, which suggests that this will be a useful tool for analysing the more challenging problem of FTIR at a rough interface Figures 3-6 and 3-7 have been created with the help of "Mathematica" ™ software. They show the values of the x-component of electric field in the xy-plane at a 26 y A Plane wave origin V i r t u a l P l a n e s FTIR gap Figure 3-4. Position of virtual planes for calculation the transmitted and reflected energies. 27 (X)l 'UOjSSIUJSUBJl F i g u r e 3-6. £-f i e l d c omp o n e n t , TIR case. 29 3 0 fixed moment of time. Figure 3-6 corresponds to a simple TIR case. Figure 3-7 presents an FTIR case. The goal of presenting these charts is to provide a visual understanding of the phenomenon of FTIR by showing the relative magnitude of the electric component of the electromagnetic field in x direction as a contour plot inyz-plane. 3.7 FTIR at a rough surface The described above method of modelling FTIR has a great advantage over the analytic approach to the same problem. With the help of F D - T D modelling it is possible to find the transmission coefficients for almost any arbitrary shaped frustrating surface. It is obvious that this can not be achieved by the analytic approach. Certain changes for the algorithm may have to be introduced in order to do that. The mesh size in x and y directions has to be significantly increased form the previous values of 7 V = TV = 25. This will increase the considered area of the surface at which -* y FTIR is to be modelled and thus will lead to more statistically accurate results. Each mesh node (/, j, k) has to be assigned a unique value of its refractive index, depending on its location in space. This will require specific modifications of the source code presented in Appendix 1. O f course the problem of modelling of FTIR on a rough surface is a more complicated one in comparison with a flat surface case. The computational time increases rapidly ~ A, where A is the area of the FTIR surface. Also the question of imposing proper boundary conditions appears at x = 0;y = 0 and x = dx;y = dy, where dxand dy are linear dimensions of the considered area at which FTIR happens. The periodic 31 boundary conditions may work with some degree of accuracy, however, a possible discontinuity in the considered rough surface may affect the results. A few results were obtained for the modelling of FTIR happening at a rough surface in a form of the transmission coefficients, however, we believe that they were not accurate enough to be quoted in this thesis because of the small size of the considered area. The modelling at a wider surface requires much more computational time and was left for the future research work. 32 4. Electrochemistry of the electrophoretic cell 4.1 Introduction At present, the most promising and interesting case of active control of total internal reflection is the so called electrophoretic control of total internal reflection, in which the TIR is frustrated by moving the charged scattering or absorbing microparticles into the evanescent wave region, by means of an applied electrostatic field. 4.2 Coaxial light-pipe switch Early research in the area of electrophoretic control of FTIR was conducted in Ford company 7 Researchers there proposed a device named a coaxial light-pipe switch. The switch consisted of an inner electrode, which was a glass rod coated with a transparent conductor and an outer metal electrode with the electrophoretic suspension filling the region in between the electrodes as shown in Figure 4-1. This switch was intended to control the intensity of the light in a single solid dielectric light guide without introducing significant reflection losses at the switch-pipe interface, unlike the other technologies. 4.3 Electrophoretic cell The device that is related in nature but can be used in reflective display technology was invented at the SSP laboratory (UBC) in 1998. Figure 4-2 depicts an image display device cell in which a sheet of high refractive index material is positioned with the flat viewing surface outward and the retro-reflective prism-bearing surface inward. The high refractive index material, for example, may be zinc sulfide (ZnS, n«2.4), titanium 33 Retro-reflective prismatic sheet Electrophoretic suspension A A A A Figure 4-2. Design of the electrophoretic cell. 35 dioxide (TiCh, n«2.5), niobium pentoxide (NbOs, n«2.3) or zirconium oxide (ZrO, n»2.1). Various shapes of the retro-reflective sheeting may be considered. A good example would be the retro-reflective sheeting manufactured by 3M, known as "Diamond Grade" ™ . The thickness of the prismatic surface may be 5 to 10 micron with the size of the prisms being about 2 micron. A transparent conductive coating, such as indium-tin oxide (ITO) covers the inner surface of the retro-reflective sheeting. The outer surface of the prismatic retro- reflective sheeting may be covered with antireflective coating to minimise the reflection from the very top surface. The inner surface of the sheeting, the bottom metal electrode and the side covers of the cell form a reservoir containing electrophoretic suspension. It has been found that a low refractive index, low viscosity, electrically insulating liquid such as Fluorinert™ Electronic Liquid FC-72 (n»1.25) or FC-75 (n»1.27) heat transfer media available from 3M, St. Paul, M N functions remarkably well as an electro- phoresis medium in the display cell. The reason for use of such materials as a high refractive index zinc sulphide and FC-72 is to create a greater index mismatch at the TIR surface so that retro-reflection happens at a wider incident angle 9. Also Fluorinerts™ are excellent dielectrics: p « l 3 M o h m - m , so use of them prevents the electrical current from flowing through the cell and therefore prevents deleterious electrochemical reactions at the electrodes. Figure 4-2(a) shows the "on" state of the cell. The incident ray is retro-reflected from the prismatic surface of the retro-reflector providing a white appearance of the outside surface of the cell for the viewer. To prevent specular glare a diffusive coating may be introduced as the first top layer of the cell cover. Figure 4-2(b) shows "of f state of the cell. The voltage is applied across the cell, moving the charged particles towards 36 the prismatic surface of retro-reflector. The incident ray in this case is frustrated/absorbed by the particles, providing a dark appearance of the outer surface of the cell to the viewer. 4.4 Planar electrophoretic switch For demonstration purposes a simpler device rather than the electrophoretic cell described above has been build and tested in SSP laboratory. Figure 4-3 represents a schematic view of the planar electrophoretic switch. Charged particles are suspended in a low refractive index fluid held between a metallic outer electrode ( z = d ) and an ITO coated glass inner electrode ( z = 0 ). Light, which is incident at the prism is refracted at the air-glass interface at an angle 6, and hits the interface between the liquid and the inner electrode. For interface incident angles greater than the critical angle, an evanescent field extends from the inner electrode into the liquid. When a potential difference between the inner and outer electrodes is applied, particles can be drawn to or from the inner electrode, thereby modulating the reflectivity of the liquid-electrode interface. 4.5 Chemistry of the electrophoretic suspension Let's consider a simplified model describing the behavior of the electrophoretic suspension in the vicinity of the boundary at which TIR occurs. The following approximations will be made: 1. The outside surface is assumed to be flat, of infinite extent and having a uniform charge per unit area. 37 Figure 4-3. Planar electrophoretic switch. 38 2. The ions in the diffuse part of the double layer (i.e. counter-ion cloud surrounding the particle) are assumed to be point charges distributed according to the Boltzmann distribution. 3. The solvent is assumed to influence the double layer only through its dielectric constant, which is assumed to have the same value throughout the diffuse part. In a real test the following electrophoretic substance was used: 10% wt mixture of silica particles of spherical shape of an average size of 0.5 micron suspended in acetonitrile. The counter-ions used were NH2 with a negative effective charge of about one electron, as quoted by the supplier, Geltech, Inc. The effective charge of the particles was positive with the magnitude of the order of hundreds of electrons 1 . Let the electrostatic potential be i//0 at the surface and y/at a distance xfrom the surface in the electrophoretic suspension. Taking the surface to be negatively charged and the particles in the suspension to be positively charged and applying the Boltzmann distribution, n+ =n0+ exp kT and n_ = n0_ exp z_ey/ kT (4.1) where n+ and n_ are the respective numbers of particles and counter-ions per unit volume where the potential isi//- (i.e. where the electric potential energy i s z + e r / a n d -z_ey/, respectively), « 0 + a n d « 0 _ are the corresponding bulk concentrations of particles and counter-ions (i.e. concentrations when the applied field is zero) and z + , z _ are the effective charges of a particle and a counter-ion respectively. The net volume charge density p at the points where the potential is y/ is given by 39 p = z+en+ = z+en0+ e x P | z_en_ = kT z_en0_ exp z_ey/ kT (4.2) The electrophoretic suspension in the bulk region of the cell is electrostatically neutral, therefore z+n0+ = Z-"o- • (4.3) The charge density p is related to the potential y/by Poisson's equation, which takes the following form for a flat double layer d2y/ _ p dx2 £ where s is an absolute electrical permittivity of the suspension given by s = kD s0 , where kD is the dielectric constant of the solvent. Combining the equations (4.2), (4.3) and (4.4) gives d2y/ (4.4) (4.5) dx' exp z + e ¥ kT - e x p z_ey/ kT (4.6) In a special case of 1-1 electrolyte (i.e. charge of the particles is equal to the charge of the counter-ions) z+=z_ (4.7) the expression (4.6) can be solved for the potential y/ in an explicit form with the following boundary conditions imposed: y/ = y/0 at x = 0; and dy/1 dx = Oat x = °o 1 3 Three base types of suspensions have been used in our experiment for frustrating TIR. The first choice was polystyrene latex spheres mixed with water and the second was silica particles suspended in acetonitrile. The third option was to use paint dye particles 40 mixed in the Fluorinert liquid. The typical size of the particles was chosen to be 0.5 micron. The values that are presented here refer to the second choice of the electrophoretic substance. In our particular case the effective charge of the particles was varying from 400e to lOOe depending on the size of the particles. The counter-ions used were NH 2 " ' with the charge of -e. The electrokinetic potential £ has been measured by Aveka, Inc. and has been found to be 26.1 mVin the case of 10% wt silica/acetonitrile suspension. Unfortunately a very common used Debye-Huckel approximation of equation (4.6) exp kT z+ey/ can not be used in this case because kT I e = 25 m V at 25° C and -z^ew — « 1 (4.9) kT V ' is not satisfied for the charged particles because of their high effective charge value. The effective charge of the particles has been determined using mobility measurements that were conducted at Aveka Inc. The electrophoretic mobility of the silica particles of 0.5 microns diameter, with 10% mass suspension in acetonitrile, was found to be n = -2.21 x 10"8 m21V • s. Using the assumption that the concentration of the counter-ions is low in the suspension, i.e. electrostatic screening length is long the electrokinetic potential of the particles (zeta potential) £ c a n be obtained from their mobility 1 4 . ^i/r- (4-,0) 41 where rjf »3A-10~4N• s/m2is the fluid (acetonitrile) viscosity and kD=37.5 is the relative dielectric constant of acetonitrile. The zeta potential was found to be 40 mV, which is fairly consistent with a value of 26.1 mV measured by Aveka, Inc. In a long screening range the effective charge of the particles Q 1 4 was obtained from their zeta potential — , (4.11) where D is the diameter of the particle, and was found to be approximately equal to 270e. The model described above didn't take into account the finite size of the particles. In 1924 Stern 1 5 has proposed a model in which the double layer near a charged interface is divided into two parts separated by a plane located at about an particle radius from the surface. This seems obvious since the centre of the particle can only approach the surface to within its radius. This model assumes that the potential changes linearly from the value of y/ 0 at the surface to the value of y/d at the Stem layer and then decays from y/d to zero in the diffuse part of the double layer. It is common to assume that the value of electrokinetic (zeta) potential £ is identical to the Stern potential y/d for the colloid we are dealing with because of it's relatively low magnitude of 26.1 mV. The thickness d of the Stern layer can be estimated from the size of the particle J = a / 2 = 0.25 micron. In order to model the distribution of the particles and counter-ions the equation has to be solved numerically for the unknown potential y/. The corresponding boundary value second order partial differential equation is as follows: 42 d > _ z+en0+ dx2 exp kT - e x t i z_ey/ kT y/ = i//d, at x = d = 0.25microns; (4.12) — = 0 , at x = oo, dx z+ =270, z_ =1, £ = 37.5^ 0, n 0 + = 7.0 • W17 particles / m3 (10.0 wt. %) A very common Debye-Huckel approximation (4.8) can not be applied to the first exponential term in the equation (4.12). Thus (4.12) has to be solved numerically. Figure 4-4 shows the solution of the equation (4.12) with the following assumption: the derivative was assumed to be constant throughout the Stern layer dx and the value for it was taken from the solution of (4.12) at x = d. After the extrapolation to x = 0 this yields the surface potential y/0 to be equal to 60.6 mV, which lies within a typical range of the values of surface potentials from 50 mVXo 80 mV. One may think that the second order equation (4.12) does not have the sufficient set of boundary conditions to be solved. However the asymptotic condition ^ - = 0 , when dx x = co uniquely defines the solution. In fact it is an equivalent to the boundary condition of the form - const at x = 0. dx What happens if the electric field applied to the cell changes? Well, the potential y/0 can be related to the charge density cr0 by equating the surface charge with the net space charge in the diffuse part of the double layer (i.e. <r0 =-^pdx) and applying the 0 Poisson-Boltzmann distribution. It turns out that at the low potentials 1 6 the surface 43 potential y/ 0 depends both on charge density a0 and the ionic composition of the medium (i.e. the values of z + , z _ and « 0 + ) . If the double layer is compressed, then either fj 0 must increase, or y/0 must decrease, or both. The cell shown at Figure 4-3 has been tested with the value of typical voltage applied of 50-100V over the gap between electrodes of 2.5 mm, which results in the value of the electrostatic field applied o f 2 - 4 1 0 4 F / m . Such a high value of the external applied field certainly affects the double layer model that has been described above. The most probable outcome of this is a creation of a closely packed layer of particles at the charged surface. Lets find the distribution of the particles corresponding to the potential from Figure 4-4. It is given by the equation (4.1). Figure 4-5 represents these results. As the graph shows the concentration of the particles near the plane exceeds the bulk concentration o 0 + by several orders of magnitude. This happens because of the particles being treated as point charges. The fraction of the volume occupied by closely packed 71 particles is —-=. Based on this number the concentration of the particles forming a closed-packed layer can be found. It turns out to be ncp = 3.12 • 10 1 8 m'3. Thus the results from Figure 4-5, while being non-physical, suggest that the particles near the charged surface form a close packed structure. 4.6 Calculation of the efficiency, of the electrophoretic cell 1 For practical reasons it would be helpful to find the percentage of the energy that could be frustrated by the closely packed layer of particles at the TIR interface. The \ 45 assumption will be made that the refractive index does not change due to the presence of the particles in the vicinity of the interface. The depth of evanescent wave varies with the incident angle. This means that percentage of frustration will depend not only on the concentration of the particles at the charged interface but the incident angle of the incoming plane wave as well. Other assumptions that the particles are fully absorptive or scattering and there is no multi-scattering happening will be made. The intensity of the evanescent wave at the prism-liquid interface can be estimated through the transmission coefficients 7j| and T± analytically found in Chapter 2. Lets construct a function, which will serve to estimate the absorption happening at a certain distance. In order to do it lets assume that the absorption at a certain distance x from the plane is proportional to the cross-sectional fraction of the area occupied by the particles. For the closely packed layer it will look as shown in Figure 4-6. The change in the reflectivity due to the presence of the particles may be estimated using the following technique. Lets find the transmission coefficient function T0(x) for TIR happening at the glass-acetonitrile surface. Assuming that the incident light is curricular polarised and the incident angle is, for example, 60 degrees, the transmission coefficient function T0 (x) defined as an average of 7J (x) and T± (x) has found to be as shown on the Figure 4-7. The intensity of the reflected light in the TIR case will be defined as and the intensity of the reflected light (i.e. reflectivity of the front surface) in the TIR case will be defined as (4.13) o 47 to co (x)j uoRdjosqv 48 co (X)0l 'UOISSjLUSUBJl OO R = A$T0(x)F(x)dx, (4.14) o where A is a normalising constant. Using the data from Figure 4-6 and Figure 4-7 the reflectivity R has been found to be 0.69. Notice that the 69% change in the reflectivity happens three times when a retro-reflector is used. Thus the described model predicts the reflectivity of 0.33 i.e. 33% for the electrophoretic cell. This may seem not to be a promising result because the estimated contrast ratio between the reflective and the absorptive case is only 3:1. However this result greatly depends on the shape of the suggested absorption function F(x). For the smaller particles the reflection coefficient R gets smaller reaching it's asymptotic value of = 0.642 and R3 = 0.25 in this case. 2V6 4.7 Comparison of the electrophoretic cell with LCD cell Though maximum estimated contrast ratio for the display cell 4:1 is not large the main advantage of the electrophoretic cell is a high reflectivity in the "white" state. Compare an electrophoretic cell with a twisted neumatic liquid crystal display cell. In the "white" state the reflectivity of a twisted neumatic cell doesn't exceed 33%. The reflectivity of the electrophoretic display cell may be as high as 100% provided the concentration of the particles in the solution is low. In the "black" state the reflectivity of the twisted neumatic cell is 1-0.5%, the reflectivity of the electrophoretic cell can be as low as 25%. From this numbers the contrast ratios for the two types of the displays are 40:1 and 4:1. However the human visual perception system is much more sensitive to the absolute change in the reflection rather than to the ratios of them in the "white" and 50 "black" states. Absolute change in the intensity of 75% comparing to 32% will give a better judgement of white and black colours for the human eye. As a final remark to this chapter lets mention some other companies that are at the present day carrying the research in the field of electrophoretic displays. They are Copytele Inc. (COPY), Research Frontiers Inc. (REFR) and E-ink, Inc., a spin-off company from MIT. Their research, however, does not involve the use of electrophoresis to control FTIR. Their focus is on changing the diffuse reflectivity of particle suspensions by motion of absorbing electrophoretic particles, which requires a much larger size scale of motion of particles and does not provide much optical control. 51 5. Ray tracing modeling of high index structures 5.1 Outline of the problem The electrophoretic cell design described in Chapter 4 incorporates a prismatic structured surface placed in a contact with a liquid electrophoretic substance. It is well known that TIR occurs only if the incident angle of the incoming wave exceeds the critical angle defined by Oar =S™\n2 /«,) A set of questions arises regarding what happens when the incident ray falls on the prismatic surface at an angle different from the normal angle. Wil l it be totally internally reflected from the boundary with the electrophoretic suspension? What portion of the energy of the incident plane wave can be modulated by means of active total internal reflection, as a function of the incident angle? What are the optimum angles for the top prismatic structure to provide TIR for a wide set of incident angles? Is there any other better design of the prismatic surface rather than described in Chapter 4? In this chapter a ray tracing modeling is presented with a goal to answer these questions. 5.2 Choice of the materials and approximations used in the modeling A relatively wide range of the materials and substances may be used for the creation of the electrophoretic cell. As mentioned in the previous chapter, the high index material may be, for example, zinc sulfide (ZnS, n«2.27), titanium dioxide (TiC>2, n«2.5), niobium pentoxide (NbOs, n«2.3) or zirconium oxide (ZrO, n«2.1) and the liquid for the 52 electrophoretic suspension may be Fluorinert™ Electronic Liquid FC-72 (n«1.25), FC-75 (n«1.27), acetonitrile (n«1.33), even water (n«1.33). For practical reasons it is preferable to have a greater index mismatch between the high index material and the liquid used for the electrophoretic suspension. The critical angle in this case will decrease, enabling TIR over a wider range of incident angles. It is also favorable to have the specific gravity of the liquid used for the electrophoretic suspension to be close to the specific gravity of the suspended particles. This choice will help to avoid sedimentation of the solution over the time due to the gravitational force. A set of assumptions for the modeling purposes will be made: 1. The high index material will be considered to be zinc sulfide (ZnS, n«2.27) 2. The liquid used for the electrophoretic suspension will be Fluorinert™ Electronic Liquid FC-75 (n«1.27) 3. We will assume that there is no refractive index change in the liquid due to the presence of the particles. 5.3 Calculation of the prism angle for the maximum reflection For several reasons it is preferable to have TIR happening at a flat surface. The uniformity of the electric field will be preserved is this case, which leads to the formation of a uniform particle layer and uniform luminosity of the surface. The problem of liquid turbulence will be minimal. The diffraction effects will disappear in this case. A n interesting observation is that there is an angle of the maximum reflection for a particular prismatic surface, for which the refracted ray travels parallel to the adjacent prism facet, 53 when the incoming wave is at the normal to the prismatic surface. This idea is shown in Figure 5-1. Let's analyze this particular example to determine 9. Snells law written at the point where the incident ray hits the facet of the prismatic surface is «, sin(90° -0) = n2 sin(90° -29) (5.1) can be solved for the unknown angle 8 given the refractive index of the prism material = 2.27 and of the outside media nair = 1.0. The solution yields the following result: 9 = arccos( v 4 « 2 \ n2 1 i - + ̂ .) = 34.33° (5.2) I6n2 2 The value of the incident angle of the deflected ray at the interface prism/liquid will be the same as 9. Compare this angle to the critical angle at this boundary: 9cr = a r c s i n f ^ ^ 2 - ) = 34.02° (5.3) nzns Notice that the incident angle at the TIR surface only slightly exceeds the critical angle. This results in the fact that if the angle of incident light deviates only slightly from surface normal, it will fail to undergo TIR over approximately half of the area of the prismatic surface. 5.4 Modeling of a single layer prismatic structure A n important result that can be found by means of ray tracing modeling is a dependency between the percentage of the rays that are totally internally reflected from the prism/liquid interface and the incident angle of the incoming light cp on the top prismatic structure. Figure 5-3 shows the results obtained after ray tracing of the structure shown on Figure 5-2. 54 Figure 5-1. Maximum reflection of a normal incident ray by a prism. Figure 5-2. Single prismatic layer design of the top cover of the electrophoretic cell. 56 Q) CD to 2> » a? ~-<D "° .t; o J C CD > to" " LO £ o += LO C 0) — -f—< = <D CD O o ro fi S Q. CD O -Q tj ® a) °> a> £ CD £ 0 •*- >» o J 5 > .S2 '•+-» t _ O Q. eg JD CD CD C CO cn D) O) a) <D <D •a T3 -a o m o m in CO a> CD a) o> a> cn c c c ca ro ro ! • i • cu o O o (0 ro ro L L U . U L III m oo CO 3 CO l-l > o o O - C i-X O 8 HP "3 ^ & o o o OH O u a> <*-H <U O T3 O a! ro m o o o o o o o o o o o o t - o o c o h - c o m ' t c o c M T - 57 The maximum percentage of the rays that do undergo TIR at the high index prismatic structure/liquid interface, as predicted from the considerations presented in the preceding paragraph, does not exceed 85%. This is the maximum fraction of the light that can be modulated by means ACTIR. The percentage of the reflected light may be increased by a significant factor by decreasing the critical angle at the liquid/high index material interface. Unfortunately there is a limited number of materials that can be used for this design in practice. The list of high index materials has been presented in the previous chapter. There exist other substances with a higher refractive index. However they are absorptive at the visible light bandwidth. Lowering the refractive index of the liquid also seems to be problematic. The Fluorinerts™ seem to be the best available liquids for this purpose. A significant fact is that the angle between the facets of the top prismatic film must provide a wide viewing angle as well as a high reflectivity when the incoming light is incident at the normal to the cell. Also it is preferable that the intensity of the reflected light smoothly varies depending on the incident angle. This will provide a graduate transition of the appearance of the brightness of the display cell with the change of the viewing angle. The chart on the Figure 5-3 presents the results of the ray tracing for the three values of the angle between facets of the prisms, 50°,55° and 60° . Reflectivity when the incident light is close to the normal to the surface is 85 % for the 60° prism. It is significantly lower for the smaller prism angle, however there is no sharp transition in the reflectivity for 50° angle prism. Making the angle at the prism smaller will decrease the reflectivity at the normal, making larger will lead to the appearance of sharp changes in 58 reflectivity with the change of the viewing angle. Thus the range of 50° - 60° degrees is optimum for the proposed design of the top cover of the electrophoretic cell. 5.5 Modeling of double prismatic surface The drawback of the structure described above (single prismatic layer) is that it is really hard to manufacture. ZnS is a brittle material, it also appears to be opaque in a thick slab, so making the top prismatic surface thicker will increase rigidity of the structure but may result in the poor transmission properties of the device. A different way of deflecting incident rays by the top surface of the electrophoretic cell has been proposed. The structure presented on Figure 5-4 shows a double-sided prismatic film design for the top cover of the electrophoretic cell. Two prismatic films, which are flat from one side and structured on the other (such as B E F or O L F manufactured by 3 M company, MN) , are glued together at their flat surfaces (shown by a dashed line). Both of the films are made from polycarbonate material with refractive index of 1.6 The typical size of a prism on the top film is about 20 micron, of the bottom film about 5 micron. The bottom film is coated with a high refractive index transparent material such as zinc sulfide (n = 2.27). This film has a typical thickness of about 2 microns (4 micron from the tops of the prisms). The high index coating has a boundary with a liquid substance containing electrophoretic dye. The same questions of the behavior of the reflectivity of this device depending on the incident angle appear as with a single layer prismatic structure. A ray tracing modeling has been carried out to optimize the two parameters: the angle between the prism facets at the top prismatic structure and the angle between the facets at the bottom prismatic structure. 59 Figure 5-4. Double-sided prismatic design of the top surface electrophoretic cell. 60 A proprietary ray tracing software "Hamster" ™ developed in 3M, M N has been used for the modeling. A beam of parallel rays was incident on the structure depicted on Figure 5-4 at an angle <p, where q> is measured from the normal direction of the sheet as a whole. The reflection coefficients of the surfaces were assumes to be negligibly small, which is technically achievable by coating the surfaces with an antireflective film. Two absorbers have been placed one inside the electrophoretic liquid, the other above the source of the light rays. Those rays that has been reflected back could have been reflected only by means of TIR, the other part of the rays hitting the absorber inside the liquid was never totally internally reflected. The modeling gave for a specific incident angle q> the percentage of the incoming light that could possibly be modified by means of ACTIR. A set o f angles of both of the prismatic surfaces has been tested in order to provide a wide range of incident angles at which a significant portion of light is totally internally reflected. In other words the angles at the prisms have been chosen to maximize the portion of light that is totally internally reflected when the incident angle is close to normal. The described range of angles is an equivalent to a so-called viewing angle of the display cell. Figure 5-5 shows the ray tracing results for three sets of selected top and bottom angles between prism facets. The best combination of the angles has been found to be around 60° top and 60° bottom prism angle. The achieved reflectivity for the normal incident light is comparable (86% in maximum) with the one obtained for a single layer prismatic structure. 61 5.6 2D modeling versus 3D For the simplicity of the ray-tracing model a 2D case of the prismatic surface was considered in this modeling. Essentially a 2D model is good enough to understand the basic behavior of the rays in the prismatic structure. The presented two top surface designs have a significant difference from the design considered in the previous chapter. A retro-reflector, like a corner cube retro-reflector, has been used in the previous design and TIR was happening three times at the facets of the retro-reflector. Here TIR occurs only once. The design of the top cover presented in this chapter can be extended to a similar 3D design, where the grooves of the prismatic surface cover the surface in the perpendicular directions, forming pyramids. O f course the obtained 2D results would differ from an exact 3D modeling but they would probably be a good estimate for the expected values of reflectivity. 63 6. Conclusions and Suggestions for Future Work 6.1 Conclusions A quantitative description of the phenomenon of FTIR has been presented in this work. A n analytic solution known for the case when FIR is happening between flat surfaces has been compared with a numeric solution of the electromagnetic field. Such F D - T D modelling of the Maxwell equations has proved to be a useful tool in the understanding the phenomenon of FTIR. A direct solution of Maxwell equations made it possible to obtain the transmission coefficients of FTIR for any given geometry. Several ways of modulating of FTIR has been considered in the thesis. The presented technologies have been found very promising in the sense of being important in the industry, so they are currently in the process of being patented. A detailed description of an image display cell based on electrophoresis has been presented. The electrophoretic control of TIR has been found to be one of the most promising technologies in the field of ACTIR. A quantitative description of the static behaviour of the electrophoretic solution in the cell has been found. Though the model was developed with some assumptions it is believed to be one of the first solutions of such a problem. A ray tracing modelling of a top prismatic surface for the top cover of the electrophoretic cell has been carried out. A n optimisation of the angle of the prismatic surfaces has been done, the best results for the deflection of the rays have been obtained for an angle close to 60 degrees. A n alternative double-sided prismatic design of the top prismatic surface of the cell has been also proposed. It has been proved to give 64 approximately the same results for the reflectivity of the cell with the advantage of being easier to manufacture. 6.2 Suggestions for future work The most exciting further research would be to develop a dynamic model of distribution of the particles in the electrophoretic cell, i.e. find the dependency of the concentration of the particles at the certain distance from the surface of the voltage applied to the cell. A different way of describing the distribution of the particles may be considered based on the D L V O theory 1 7 With a help of the improved capabilities of computer hardware the modelling of FTIR modulated by means of electrophoresis can be carried out with the help of developed F D - T D solver of Maxwell equations. 65 References 1. N. J. Harrick, "Internal Reflection Spectroscopy", Wiley, New York, 1967, pp. 27-30. 2. L. A . Whitehead, R. Coope, D. Grandmaison, M . Mossmann US Patent Application Serial No. 09/133,214. 3. M . Born and E. Wolf, "Principles of Optics", Pergamon, Oxford, 1975, 5 t h ed. 4. I. N. Court and F. K. Willisen, "Frustrated Total Internal Reflection and Application of its Principle to Laser Cavity Design", Appl. Opt. 3, pp. 719,1964. 5. J. H. Harrold, "Matrix Algebra for Ideal Lens Problem", J. Opt. Soc. Am. , 44, pp. 254,1954. 6. J. C. Nelson, M . E . Gardner, R. H. Appeldorn, T. L. Hoopman, U.S. Patent 4938563, 3 Jul, 1990 7. J. T. Remillard, J. M . Ginder, W. H. Weber, "Evanescent-wave Scattering by Electrophoretic Microparticles: a Mechanism for Optical Switching", Appl. Opt. 34, n. 19, pp. 3777, 1995. 8. G. A . Schumacher and T. G. M. van de Ven, "Evanescent Wave Scattering Studies on Latex-Glass Ineteractions", Langmuir 7, pp. 2028-2033, 1991. 9. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Anten. Prop., AP-14, pp. 302- 307, 1966. 10. J. Stratton, "Electromagnetic Theory", New York, McGraw-Hil l , 1941, p. 23. 11. A.Taflove, K.R.Umashankar, "The finite-difference time-domain method for numerical modelling of electromagnetic wave interactions with arbitrary structures," in Progress in Electromagnetic Research 2: Finite Element and Finite Difference Methods in Electromagnetic Scattering, M.A.Morgan, Elsevier, New York, pp. 288- 373,1990. 12. A.Taflove, and M.E.Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech., MTT-23 , pp. 623-630, 1975. 13. Overbeck, J.Th.G. "Electrochemistry of a double layer", Coll . Sc., Elseveir, Vol . 1 "Irreversible systems", pp. 115-193, 1952. 14. R. J. Hunter. "Foundation of Colloid Science", Clarendon Press, Oxford, 1987. 66 15. O. Stern, Z. Elekrochem. 30, pp. 508,1924. 16. D.J.Shaw "Introduction to Colloid and Surface Chemistry", Butterworth & Co, Ltd., 1986. 17. E. J. W. Verway and J. Th. G. Overbeck, "Theory of the Stability of Lyophobic Colloids", Elseveir, 1948. 67 Appendix 1 The source code of the F T - T D solver program is presented as following. It can be compiled and run using F O R T R A N 7 7 or F O R T R A N 9 0 compiler. Q * * * * * * * * * * * * * * * * * * * * * * J N PROGRAM **************** PROGRAM MAIN PARAMETER(ITE=1,ITH=1,IXT=19,IYT=19,IZT=4 00, 6 PI=3.1415926) INTEGER T,TO,TI,XSTR,XND,YSTR,YND,ZSTR,ZND,TERMIN, 6 10,J0,K0,OFFSET REAL THETA,PHI,WAVELG,UL,KVEC,KXO,KYO,KZO,UNIT, 6 A,B,C,CPX,ABSORBl,ABSORB2, 6 EX0,EY0,EZ0, 6 DLTT,DLTX,DLTY,DLTZ,KOF REAL EX(O:IXT,0:IYT,0:IZT,0:ITE) REAL EY(0:IXT,0:IYT,0:IZT,0:ITE) REAL EZ(0:IXT,0:IYT,0:IZT,0:ITE) REAL HX(0:IXT,0:IYT,0:IZT,0:ITH) REAL HY(0:IXT,0:IYT,0:IZT,0:ITH) REAL HZ(0:IXT,0:IYT,0:IZT,0:ITH) REAL EXS(0:IXT,0:IYT,0:IZT) REAL EYS ( 0 : IXT, 0 :1YT, 0 :1ZT) REAL EZS(0:IXT,0:IYT,0:IZT) REAL HXS(0:IXT,0:IYT,0:IZT) REAL HYS(0:IXT,0:IYT,0:IZT) REAL HZS(0:IXT,0:IYT,0:IZT) COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C12/ 10,JO,K0,OFFSET C O M M O N / G I / C,UL,KXO,KYO,KZO,EX0,EY0,EZ0,KVEC COMMON/024/ ABSORB1,ABSORB2 COMMON/C3/ DLTT,DLTX,DLTY,DLTZ,KOF c THETA=0.0 PHI=PI*0.5 C=1.2 A=l. 0 B = 1 . 0 K V E C = 1 0 . 0 W A V E L G = 2 . 0 * P I / K V E C U L = W A V E L G / 1 9 . 0 T 0 = 0 T 1 = 0 C T I M E L I M I T F O R T H E C A L C U L A T I O N S T E R M I N = 4 0 c D L T T = 0 . 5 D L T X = 1 . 0 D L T Y = 1 . 0 D L T Z = 1 . 0 K O F = W A V E L G / 1 9 . 0 c X S T R = 0 Y S T R = 0 Z S T R = 0 X N D = I X T Y N D = I Y T Z N D = I Z T c _ 1 0 = 0 J 0 = 0 K 0 = 0 O F F S E T = 1 0 0 c U N I T = 1 A B S O R B 1 = 1 0 0 A B S O R B 2 = Z N D - 1 0 0 c K X 0 = K V E C * S I N ( T H E T A ) * C O S ( P H I ) K Y 0 = K V E C * S I N ( T H E T A ) * S I N ( P H I ) K Z 0 = K V E C * C O S ( T H E T A ) N O R M L = S Q R T ( A * * 2 + B * * 2 ) E X 0 = ( A * S I N ( P H I ) + B * C O S ( T H E T A ) * C O S ( P H I ) ) / N O R M L E Y 0 = ( B * C O S ( T H E T A ) * S I N ( P H I ) - A * C O S ( P H I ) ) / N O R M L E Z 0 = - B * S I N ( T H E T A ) / N O R M L C " i n i t i a l i z e " C A L L I N T L Z E ( E X S , E Y S , E Z S , H X S , H Y S , H Z S ) 69 C " c a l c u l a t e " DO 10 T=l,700 IF ( M O D ( T,2) . E Q . O ) THEN CALL EFLD(EXS,EYS,EZS,HXS,HYS,HZS) ELSE CALL HFLD(EXS,EYS,EZS,HXS,HYS,HZS) ENDIF 10 CONTINUE C- - -"output"- - - - - - - - - - - - - - - - - - - - 150 FORMAT(IX,21(IX,F5.2)) OPEN(UNIT=22,FILE='at71xy.dat1,STATUS='UNKNOWN') REWIND(UNIT=22) DO 310 J=YSTR,YND,1 WRITE(22,150) (EXS(I,J,71),I=XSTR,XND,1) 310 CONTINUE CLOSE(UNIT=22) OPEN(UNIT=2 4,FILE='at 61xy.dat',STATUS='UNKNOWN') REWIND(UNIT=24) DO 311 J=YSTR,YND,1 WRITE(24,150) (EXS(I,J,61),I=XSTR,XND,1) 311 CONTINUE CLOSE(UNIT=24) 153 FORMAT(IX,601(IX,F5.2)) OPEN(UNIT=2 7,FILE='et ime.dat',STATUS='UNKNOWN') REWIND(UNIT=27) DO 314 K=ZSTR,ZND,1 WRITE(27,153) (EXS(I,15,K),I=XSTR,XND,1) 314 CONTINUE CLOSE(UNIT=27) OPEN(UNIT=28,FILE='htime.dat',STATUS='UNKNOWN') REWIND(UNIT=28) DO 316 K=ZSTR,ZND,1 WRITE(28,153) (HXS(I,15,K),I=XSTR,XND,1) 316 CONTINUE CLOSE(UNIT=28) STOP END 70 c*****» E - f i e l d s : Exs,Eys,Ezs SUBROUTINE EFLD(EXS,EYS,EZS,HXS,HYS,HZS) INTEGER XSTR,XND,YSTR,YND,ZSTR,ZND REAL SGMX,SGMY,SGMZ,PERMIT,DLTT,DLTX,DLTY,DLTZ, 6 ABSORB1,ABSORB2, 6 ABSBX,ABSBY,ABSBZ,ABSFX,ABSFY,ABSFZ, 6 SLOPEX,SLOPEY,SLOPEZ, 6 MEDFND,MEDSND,N,KOF REAL EXS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL EYS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL EZS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HXS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HYS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HZS (XSTR:XND, YSTR: YND, ZSTR-.ZND) COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C24/ ABSORBl,ABSORB2 COMMON/C3/ DLTT,DLTX,DLTY,DLTZ,KOF SGMX=0.0 SGMY=0.0 SGMZ=0.0 PERMIT=1.0 N=1.5 MEDFND=200 MEDSND=200 C "Ex" DO 200 K=ZSTR+2,ZND-2, 1 IF(K.LE.ABSORBl) THEN ABSBX=2.0 ABSFX=0.005 SLOPEX=(ABSBX-ABSFX)/(ABSORB1-ZSTR) SGMX=ABSBX-K*SLOPEX ELSE IF (K.GE.ABSORB2) THEN ABSBX=2.0 ABSFX=0.005 , SLOPEX=(ABSBX-ABS FX)/(ZND-ABSORB2) 71 SGMX=ABSFX+(K-ABS0RB2)*SLOPEX ELSE SGMX=0.0 ENDIF ENDIF DO 100 I=XSTR,XND,1 DO 50 J=YSTR+1,YND,1 b e g i n n i n g o f t h e i n t e r n a l c y c l e - - IF((K.GE.ZSTR).AND.(K.LT.MEDFND)) THEN PERMIT=N**2 ELSE IF((K.GE.MEDFND).AND.(K.LT.MEDSND)) THEN PERMIT=1.0 ELSE PERMIT=N**2 END I F END I F IF(I.EQ.O) THEN IMN=XND ELSE IMN=I-1 END I F IF(I.EQ.XND) THEN IPL=0 ELSE IPL=I+1 END I F IF(J.EQ.O) THEN JMN=YND ELSE JMN=J-1 END I F IF(J.EQ.YND) THEN JPL=0 ELSE JPL=J+1 END I F DISCRETE SCHEME EXS(I,J,K)=EXS(I,J,K)+(DLTT/PERMIT)* ( ( H Z S ( I , J P L , K ) - H Z S ( I , J M N , K ) ) / ( 2 * D L T Y ) - 6 (HYS(I, J,K+1)-HYS(I,J,K-1) )/(2*DLTZ)- 6 KOF*EXS(I,J,K)*SGMX) C end of the i n t e r n a l c y c l e 50 CONTINUE 100 CONTINUE 200 CONTINUE C end ex C "Ey" DO 500 K=ZSTR+2,ZND-2,1 IF(K.LE.ABSORBl) THEN ABSBY=2.0 ABSFY=0.005 SLOPEY=(ABSBY-ABSFY)/(ABSORB1-ZSTR) SGMY=ABSBY-K*SLOPEY ELSE IF (K.GE.ABSORB2) THEN ABSBY=2.0 ABSFY=0.005 SLOPEY=(ABSBY-ABSFY)/(ZND-ABSORB2) SGMY=ABSFY+(K-ABSORB2)*SLOPEY ELSE SGMY=0.0 END IF ENDIF DO 400 I=XSTR,XND, 1 DO 300 J=XSTR+1,XND,1 IF((K.GE.ZSTR).AND.(K.LE.MEDFND)) THEN PERMIT=N**2 ELSE IF((K.GT.MEDFND).AND.(K.LE.MEDSND)) THEN PERMIT=1.0 ELSE PERMIT=N**2 END IF END IF IF(I.EQ.O) THEN 73 I M N = X N D E L S E I M N = I - 1 E N D I F I F ( I . E Q . X N D ) T H E N I P L = 0 E L S E I P L = I + 1 E N D I F I F ( J . E Q . O ) T H E N J M N = Y N D E L S E J M N = J - 1 E N D I F I F ( J . E Q . Y N D ) T H E N J P L = 0 E L S E J P L = J + 1 E N D I F c D I S C R E T E S C H E M E E Y S ( I , J , K ) = E Y S ( I , J , K ) + ( D L T T / P E R M I T ) * 6 ( ( H X S ( I , J , K + 1 ) - H X S ( I , J , K - 1 ) ) / ( 2 * D L T Z ) - 6 ( H Z S ( I P L , J , K ) - H Z S ( I M N , J , K ) ) / ( 2 * D L T X ) - 6 K O F * E Y S ( I , J , K ) * S G M Y ) 3 0 0 C O N T I N U E 4 0 0 C O N T I N U E 5 0 0 C O N T I N U E C e n d e y C " E z " DO 9 0 0 K = Z S T R + 2 , Z N D - 2 , 1 I F ( K . L E . A B S O R B 1 ) T H E N A B S B Z = 2 . 0 A B S F Z = 0 . 0 0 5 S L O P E Z = ( A B S B Z - A B S F Z ) / ( A B S O R B 1 - Z S T R ) S G M Z = A B S B Z - K * S L O P E Z E L S E I F ( K . G E . A B S O R B 2 ) T H E N A B S B Z = 2 . 0 A B S F Z = 0 . 0 0 5 S L O P E Z = ( A B S B Z - A B S F Z ) / ( Z N D - A B S O R B 2 ) 74 S G M Z = A B S F Z + ( K - A B S 0 R B 2 ) * S L O P E Z E L S E S G M Z = 0 . 0 E N D I F E N D I F DO 8 0 0 I = X S T R , X N D , 1 DO 7 0 0 J = X S T R + 1 , X N D , 1 I F ( ( K . G E . Z S T R ) . A N D . ( K . L E . M E D F N D ) ) T H E N P E R M I T = N * * 2 E L S E I F ( ( K . G T . M E D F N D ) . A N D . ( K . L E . M E D S N D ) ) T H E N P E R M I T = 1 . 0 E L S E P E R M I T = N * * 2 E N D I F E N D I F I F ( I . E Q . O ) T H E N I M N = X N D E L S E I M N = I - 1 E N D I F I F ( I . E Q . X N D ) T H E N I P L = 0 E L S E I P L = I + 1 E N D I F I F ( J . E Q . O ) T H E N J M N = Y N D E L S E J M N = J - 1 E N D I F I F ( J . E Q . Y N D ) T H E N J P L = 0 E L S E J P L = J + 1 E N D I F c D I S C R E T E S C H E M E E Z S ( I , J , K ) = E Z S ( I , J , K ) + ( D L T T / P E R M I T ) * 6 ( ( H Y S ( I P L , J , K ) - H Y S ( I M N , J , K ) ) / ( 2 * D L T X ) - 6 ( H X S ( I , J P L , K ) - H X S ( I , J M N , K ) ) / ( 2 * D L T Y ) - 6 K O F * E Z S ( I , J , K ) * S G M Z ) 7 0 0 C O N T I N U E 75 8 0 0 C O N T I N U E 9 0 0 C O N T I N U E C end ez R E T U R N E N D H - f i e l d s : H x s , H y s , H z s S U B R O U T I N E H F L D ( E X S , E Y S , E Z S , H X S , H Y S , H Z S ) I N T E G E R X S T R , X N D , Y S T R , Y N D , Z S T R , Z N D R E A L S G M X , S G M Y , S G M Z , P E R M E A , D L T T , D L T X , D L T Y , D L T Z , K O F R E A L E X S ( X S T R . X N D , Y S T R : Y N D , Z S T R : Z N D ) R E A L E Y S ( X S T R : X N D , Y S T R : Y N D , Z S T R : Z N D ) R E A L E Z S ( X S T R : X N D , Y S T R : Y N D , Z S T R : Z N D ) R E A L H X S ( X S T R : X N D , Y S T R : Y N D , Z S T R : Z N D ) R E A L H Y S ( X S T R : X N D , Y S T R : Y N D , Z S T R : Z N D ) R E A L H Z S ( X S T R : X N D , Y S T R : Y N D , Z S T R : Z N D ) C O M M O N / C 1 2 3 4 / X S T R , X N D , Y S T R , Y N D , Z S T R , Z N D C O M M O N / C 3 / D L T T , D L T X , D L T Y , D L T Z , K O F S G M X = 0 . 0 S G M Y = 0 . 0 S G M Z = 0 . 0 P E R M E A = 1 . 0 DO 5 0 J = X S T R + 1 , X N D , 1 I F ( I . E Q . O ) T H E N I M N = X N D E L S E I M N = I - 1 E N D I F I F ( I . E Q . X N D ) T H E N I P L = 0 E L S E I P L = I + 1 C - " H x " DO 2 0 0 K = Z S T R + 2 , Z N D - 2 , 1 DO 1 0 0 I = X S T R , X N D , 1 76 E N D I F I F ( J . E Q . O ) T H E N J M N = Y N D E L S E J M N = J - 1 E N D I F I F ( J . E Q . Y N D ) T H E N J P L = 0 E L S E J P L = J + 1 E N D I F D I S C R E T E S C H E M E H X S ( I , J , K ) = H X S ( I , J , K ) + ( D L T T / P E R M E A ) * ( - ( E Z S ( I , J P L , K ) - E Z S ( I , J M N , K ) ) / ( 2 * D L T Y ) + ( E Y S ( I , J , K + l ) - E Y S ( I , J , K - l ) ) / ( 2 * D L T Z ) ) 5 0 C O N T I N U E 1 0 0 C O N T I N U E 2 0 0 C O N T I N U E C e n d e x C " H y " DO 5 0 0 K = Z S T R + 2 , Z N D - 2 , 1 DO 4 0 0 I = X S T R , X N D , 1 DO 3 0 0 J = X S T R + 1 , X N D , 1 I F ( I . E Q . O ) T H E N I M N = X N D E L S E I M N = I - 1 E N D I F I F ( I . E Q . X N D ) T H E N I P L = 0 E L S E I P L = I + 1 E N D I F I F ( J . E Q . O ) T H E N J M N = Y N D E L S E J M N = J - 1 E N D I F C- 6 6 77 I F ( J . E Q . Y N D ) T H E N J P L = 0 E L S E J P L = J + 1 E N D I F C D I S C R E T E S C H E M E H Y S ( I , J , K ) = H Y S ( I , J , K ) + ( D L T T / P E R M E A ) * 6 ( - ( E X S ( I , J , K + 1 ) - E X S ( I , J , K - 1 ) ) / ( 2 * D L T Z ) + 6 ( H Z S ( I P L , J , K ) - H Z S ( I M N , J , K ) ) / ( 2 * D L T X ) ) 3 0 0 C O N T I N U E 4 0 0 C O N T I N U E 5 0 0 C O N T I N U E C - e n d e y C " H z " - DO 9 0 0 K = Z S T R + 2 , Z N D - 2 , 1 DO 8 0 0 I = X S T R , X N D , 1 DO 7 0 0 J = X S T R + 1 , X N D , 1 I F ( I . E Q . O ) T H E N I M N = X N D E L S E I M N = I - 1 E N D I F I F ( I . E Q . X N D ) T H E N I P L = 0 E L S E I P L = I + 1 E N D I F I F ( J . E Q . O ) T H E N J M N = Y N D E L S E J M N = J - 1 E N D I F I F ( J . E Q . Y N D ) T H E N J P L = 0 E L S E J P L = J + 1 E N D I F c D I S C R E T E S C H E M E H Z S ( I , J , K ) = H Z S ( I , J , K ) + ( D L T T / P E R M E A ) * 78 6 6 (- (HYS(IPL,J,K)-HYS(IMN,J,K))/(2*DLTX)+ (HXS(I,JPL,K+1)-HXS(I,JMN,K))/(2*DLTY) ) 700 CONTINUE 800 CONTINUE 900 CONTINUE C end ez RETURN END C C * * * * " I n i t i a l i z e Subroutine''*********************** SUBROUTINE INTLZE(EXI,EYI,EZI,HXI,HYI,HZI) INTEGER XSTR,XND,YSTR,YND,ZSTR,ZND,OFFSET,10,JO,K0 REAL C,UL,KXO,KYO,KZO,EX0,EY0,EZ0,KVEC,ABSORBl,ABSORB2 REAL PHASE REAL EXI(XSTRrXND,YSTR:YND,ZSTR:ZND) REAL EYI(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL EZI(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HXI(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HYI(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HZI(XSTR:XND,YSTR:YND,ZSTR:ZND) COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C12/ 10,JO,KO,OFFSET COMMON/C24/ ABSORBl,ABSORB2 COMMON/C1/ C,UL,KXO,KYO,KZO,EXO,EYO,EZO,KVEC C "statement f u n c t i o n " E0(M)=EXP(-C* ( (M-17 0 ) / 2 0 . 0 ) * * 2 ) PHASE(LX,LY,LZ)=ABS(KXO*LY*UL+KYO*LX*UL+KZO*LZ*UL) C C in c i d e n t wave DO 190 K=ZSTR,ZND,1 DO 180 J=YSTR,YND,1 79 DO 100 I=XSTR,XND,1 IF((K.LE.ABSORB1).OR.(K.GE.ABSORB2)) THEN EXI ( I , J,K)=0.0 EYI ( I , J,K)=0.0 EZI(I , J,K)=0.0 ELSE EXI(I,J,K)=SIN(PHASE(I,J,K))*EX0*E0(K) EYI ( I , J,K) SIN(PHASE(I,J,K))*EY0*E0(K) EZI ( I , J,K) SIN(PHASE(I,J,K))*EZ0*E0(K) ENDIF 100 180 CONTINUE CONTINUE 190 CONTINUE DO 290 K=ZSTR,ZND,1 DO 280 J=YSTR,YND,1 DO 200 I=XSTR,XND,1 HXI ( I , J,K)=0.0 HYI ( I , J,K)=0.0 HZI(I,J,K)=0.0 200 CONTINUE 280 CONTINUE 2 90 CONTINUE RETURN END C******"Calculate Poynting v e c t o r " * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE PYNT(S2,SI,EXS,EYS,HXS,HYS) INTEGER YF22,Jl,J2,YF11,J3,J4,XSTR,XND,YSTR,YND REAL S22,S11,S2,S1 REAL EXS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL EYS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HXS(XSTR:XND,YSTR:YND,ZSTR:ZND) REAL HYS(XSTR:XND,YSTR:YND,ZSTR:ZND) COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C24/ ABSORB1,ABSORB2 80 S 2 = 0 . 0 S 1 = 0 . 0 S 2 2 = 0 . 0 S 1 1 = 0 . 0 DO 1 0 0 I = X S T R , X N D , 1 DO 2 0 0 J = X S T R , Y N D , 1 S 2 2 = E Y S ( I , J , A B S O R B 1 ) * H X S ( I , J , A B S O R B 1 ) - 6 E X S ( I , J , A B S O R B l ) * H Y S ( I , J , A B S O R B 1 ) S 1 1 = E Y S ( I , J , A B S O R B 2 ) * H X S ( I , J , A B S O R B 2 ) - 6 E X S ( I , J , A B S O R B 2 ) * H Y S ( I , J , A B S O R B 2 ) S 2 = 0 . 5 * S 2 2 + S 2 S 1 = 0 . 5 * S 1 1 + S 1 2 0 0 C O N T I N U E 1 0 0 C O N T I N U E R E T U R N E N D 81
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