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Active control of total internal reflection and its application in electrophoretic displays Grandmaison, Dmitri 1999

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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 T H E S I S 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 O F T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department o f Physics and Astronomy) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A August 1999 © Dmitri Grandmaison, 1999  ln  presenting  degree  this  at the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  freely available for copying  of  department  this or  publication  of  reference  thesis by  this  for  his thesis  and  study.  scholarly  or for  her  financial  of  The University of British Vancouver, Canada  Date  DE-6 (2/88)  Au-f  W ,  purposes  Columbia  OSS  gain  shall  requirements that  agree  may  representatives.  permission.  Department  I further  the  It not  be is  that  the  permission  granted  allowed  an  advanced  Library shall make  by  understood be  for  for  the that  without  it  extensive  head  of  my  copying  or  my  written  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.  ii  Table of Contents ABSTRACT..  ,  ii  TABLE OF CONTENTS  ii f  LIST OF FIGURES  v  ACKNOWLEDGEMENT  . . . vii  1. INTRODUCTION  1  1.1 Definition o f frustrated total internal reflection  1  1.2 Modeling of F T I R  2  2. FRUSTRATED T O T A L INTERNAL REFLECTION: A REVIEW 2.1 Active control of total internal reflection  4 4  2.2 Overview of total internal reflection  .4  2.3 F T I R as a special case o f TIR. Analytic solution o f F T I R reflection/transmission coefficients in a simple planar case 2.4 Methods of A O T R  •.  6 8  2.4.1  Pressure induced membrane control  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  3. FDTD MODELING OF FTIR .; 3.1 Definition of the problem  9  14  16 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 F T I R at a rough surface  31  4. ELECTROCHEMISTRY OF T H E ELECTROPHORETIC C E L L  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  ,  4.6 Calculation of the efficiency o f the electrophoretic cell 4.7 Comparison of the electrophoretic cell with L C D cell  5. RAY TRACING MODELING OF HIGH INDEX STRUCTURES  45 ;  45 50  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 6.1 Conclusions  :  6.2 Suggestions for future work  .64 64 65  REFERENCES  66  APPENDIX 1. SOURCE CODE OF T H E SIMULATION PROGRAM  68  iv  List of Figures Figure 2-1. Transmission coefficients T and T versus normalised optical separation. ±  t  Figure 2-2. State "on" and state " o f f of pressure-induced cell. Figure 2-3. Corner cube reflector. Figure 2-4. Electrostatically actuated FTIR. Figure 3-1. Position o f various field components. Figure 3-2. Schematic diagram o f model setup. Figure 3-3. Position o f 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 o f the same coefficients. Figure 3-6. £"-field component, TER. case. Figure 3-7. £ - f i e l d component, F T I R 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 T (x) for the glass/acetonitrile interface. 0  Figure 5-1. Maximum reflection of a normal incident ray by a prism. Figure 5-2. Single prismatic layer design of the top cover o f the electrophoretic cell. Figure 5-3. Reflectivity o f a single prismatic layer electrophoretic cell versus incident angle of the incoming light. Figure 5-4. Double-sided prismatic design o f 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 o f  years. It occurs when an electromagnetic wave passes from an optically dense medium into one less dense ( « , >n ) 2  and the incident angle o f the wave 6 is greater than the  critical angle for the two media defined by e„ = sin~'(n  In,).  2  Under the conditions o f 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) =  I (0) xp(-j), e  e  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(n  2  6-n f  2  sm  2  2  where n and n are the refractive indices o f the two media, X is the wavelength o f light x  2  0  in vacuum and I (0) e  is an average intensity o f the incoming light incident at the angle  e.  l  Under conditions of TIR there is no net energy flow between the interface o f two surfaces. However, when a higher index media is inserted in the region o f evanescent wave, some energy may flow into this media. This is called frustrated total internal reflection (FTIR).  1.2  Modeling of FTIR The fraction o f 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 F T I R phenomenon is presented in Chapter 2. F T I R can be controlled by different means. The "frustrating" medium can be displaced from its original location preventing F T I R to happen. It can also be returned back to its original position enabling FTIR. Several methods o f active control o f total internal reflection ( A C T I R ) 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 F T I R is happening between flat surfaces. A more general approach based on the direct solution o f 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 ( F D T D ) 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 o f Maxwell equations in a 3 D space with rectangular periodic boundary conditions.  2  1.3  Display cell design Practical application of A C T I R , 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 . A n electrophoretic method df A C T I R appears to be the 2  most promising way. In this method, TIR occurs at the interface between a retro-reflector and a low refractive index liquid containing charged microparticles. B y 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 o f the particles within the electrophoretic cell and gives an estimate o f the energy fraction that could possibly be modulated by means o f 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 o f actively controlling FTIR; we call it A C T I R .  2.2  Overview of total internal reflection When an electromagnetic wave is incident on an interface between two media,  part o f the energy in that wave may be reflected from that interface.  The percentage o f  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 o f the first media is greater than the refractive index o f the second media and the angle o f incidence exceeds the critical angle  (2.1)  6 =sm {n ln ) l  cr  2  x  total internal reflection occurs. The light in this case is completely reflected back to the first media. The Fresnel formulae  3  «j cos# + « cosf?' 2  £  , ^ « 1  cos 0 - f t ,  2  « cosr? + 2  where  cos-fl'^ fl,cos0'  11  £ " is an amplitude o f 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 n sinf?' = n, sinf?,  (2.6)  2  completely characterize the behavior of the transmitted and reflected waves.  However,  when TIR occurs, a small portion o f 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" point 1  is reached at the distance of  (2.7)  where k is the wave vector, 6>is the incident angle and n is the refractive indices of the i  1  ;  media, z is often referred to as the penetration depth of the evanescent wave.  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 o f 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 F T I R happening at two semi-infinite dielectrics and thin film between them was given by N. Court and F.K. von Willisen in 1964. Using the following notations: 4  «, = refractive index of the first medium n = refractive index of the spacing between two media 2  » = refractive index o f the second medium 3  d = spacing between two media 9 - incident angle 0  X = wavelength of electromagnetic wave in vacuum the transmission coefficients 7J and T were found to be given by L  1  T,  (2.8)  flTj sinh y + ^ and  a  s i n h y + ft 2  L  (2.9)  where  (2.10)  6  ^  {N -\pN -l) 2  47V cos0 (N 2  2  (  2  2  o  2  )  2  O  O  [Jn - s i n 0 + cos0 F fi^SL ° °> 2  2  n  n  4cos<9 Jn* - sin 0  a , =^Jr[{N  2  n  +l)sin 0  2  (2.12)  O  -l][(n N  2  2  o  +ft cosc? )  {]n -sin 0  =  U  sin 0 - l ^ w - sin 0  2  2  2  (2.13)  + \)sm 0 -n } 2  2  O  2  0  o  4 « cos0 jn 2  2  2  iy  - sin #  (  2 M  )  2  0  Here,  n = ~ and N = — i n  (2.15)  n  2  Figure 2-1 represents the transmission coefficients for different polarisation o f light as a function of the gap distance between two dielectric media. Four different curves are shown for n =n =1.5, n =1.0 and n =n =l.6;n x  3  2  x  3  2  =1.0 When un-polarised light  is used the transmission coefficient is the average o f 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. A s we can see from Figure 2-1, the transmission coefficient into the second media is practically zero when the thickness o f the spacing is o f the order o f X. Thus, the effective thickness o f 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 , but for 5  many other F T I R 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 o f electromagnetic waves through dielectric media. The more challenging problem o f 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 o f controlling the conditions under which F T I R occurs.  B y physically inserting a third media into the evanescent field region, most o f the energy of the reflected wave can be absorbed by or scattered into this media. This will cause the reflection coefficient o f the T I R 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 T I R 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. O n the front side o f the device (where the incoming wave arrives) we have a corner cube reflector sheet , which reflects light back toward the incident direction by 6  9  TIR case  Air chamber  FTIR case  Air chamber  Figure 2-2. State "on" and state " o f f o f 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. O n 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 frustrating the TIR.  method of  O n the front side of the device (where the incoming wave arrives)  we again have a corner cube reflector , as shown in Figure 2-4. 6  The pyramids are coated with a transparent conducting film (in our case IndiumTin-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 I T O coating and the conductive rubber.  The purpose o f the  paralyne should be clear: a one micron thick paralyne coating can withstand  an  exceedingly high electric field up to 1 0 V / m , which enables substantial electrostatic field 8  pressure. The rubber slab can be returned to the original position by removing the bias  li  Figure 2-3. Corner cube reflector.  12  TIR case Corner cube reflector  ITO film  =\AAAAAAAAAAAAAAAAAA/  Isolating coating Conductive rubber  DC power supply  FTIR case Corner cube reflector  ITO film  AAAAAAAAAAAAAAAAAAA/  Isolating coating 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 o f 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 =  transparent conductive film, such as ITO, coats the pyramids of the acrylic  1.33). film.  A 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 A C T I R 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 F T I R (Figure 24) and the electrophoretically actuated F T I R 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 F T I R 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. A m o n g 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 ( L C D ) may be the fact that with certain geometries of the cells the angular distribution of the reflected light is wider than that o f for the passive L C D . This provides a wider viewing angle for such a display. The use o f the cheaper materials in comparison with liquid crystals may also be added to the list o f the advantages.  15  3. FDTD modeling of FTIR 3.1  Definition of the problem A s mentioned in Chapter 2 an explicit closed form solution of the problem of  finding the transmission / reflection coefficients for the F T I R phenomenon has been found only for a very specific geometry o f 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 F T I R coefficients for any given geometry o f an F T I R 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 ( F D T D ) method in application for solving Maxwell equations has proved to be a powerful tool in solving time-dependent problems o f scattering and propagation o f electromagnetic waves.  3.2  Yee's algorithm Let's start from the Maxwell equations written in the following form  l 0  :  ^dt L - Ifj.v x E - ^juH  (3-1)  ^dt = sI v x H - ^s E  (3-2)  Assuming that e ,ju,cr p' are isotropic, the following system o f scalar equations is equivalent to the Maxwell curl equations (3-1), (3-2) in a rectangular coordinate system  16  (3-3) dt  f  ^  dt dH  ju dz  .  !  //  -  ^  dx  dy  dx  dy  dz  s  y  ^  ,  dE  ,  )  (3.4)  dH  x  e  *  v  dE j^dH dt  -  dz  1 dE  CT  df  ^  dy  2  dz  dx  (3-7)  -oE ) y  =1 ( ^ . ^ - 0 6 . ) . £ . dx dy  (3-8,  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) = E exp(-c,(y- y f) 0  0  exp(iK • r)E  (3-8)  A three dimensional grid has been introduced in space with an F T I R geometry enclosed in it. The grid spacing o f 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 o f time and space as F(i, j, k, n) = F(/Ax, j&y, kAz, nAt).  (3-10)  17  The equivalent set o f 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 o f Yee's algorithm, which we are using in our program, a space-time offset for the nodes o f the mesh has been employed in the algorithm. Yee positioned the components o f E and H at half time steps, which significantly reduces the amount o f computation necessary for a given level of accuracy (Figure 3-1). This resulted in the following finite difference equations:  H (2i + \,2j,2k,2n +1) = H (2i +1,2 j,2k,2n - 1 ) + x  x  E (2i,2j + 2,2k,2n) - E (2i,2 j,2k,2n) + y  y  —  pAd E (2i,2j,2k,2n) - E (2i,2 j,2k + 2,2n) z  p'H (2t,2j,2k,2n)  (3-12)  x  z  H (2i,2j + \,2k,2n + l) = H (2i,2 j + \,2k,2n - 1 ) + y  At  y  E {2i,2j,2k + 2,2/i) - E (2/,2 j,2k,2n) + y  y  —  pAd E (2i,2j,2k,2n) - E (2i + 2,2 j,2k,2n) z  p'H (2i,2j,2k,2n)  (3-13)  y  z  H (2i,2j,2k + l,2/i +1) = H (2i,2 j,2k + \,2n -1) + z  At  z  E (2/ + 2,2j,2k,2n) - E (2/,2 j,2k,2n) + y  At  y  pAd E (2i,2j,2k,2n) - E (2i,2 j + 2,2k,2n) z  p'H (2i,2j,2k,2n)  (3-14)  z  z  E (2i + \,2j,2k,2n +1) = E (2i +1,2 j,2k,2n -1) + x  At  x  H (2/,2y + 2,2*,2n) - H (2/,2 j,2k,2n) + z  z  sAd H (2i,2j,2k,2n) - H (2/,2 j,2k + 2,2n) y  y  -~aE (2i,2j,2k,2n) £  (3-15)  x  E (2i,2j + \,2k,2n +1) = £ (2/,2j + \,2k,2n - 1 ) + y  At  H (2i,2j,2k x  + 2,2«) - H (2i,2 j,2k,2n) + x  sAd \_H (2i,2j,2k,2n) - H (2i + 2,2 j,2k,2n) Z  -~aE (2i,2j,2k,2n)  z  £  (3-16)  y  E (2i,2j,2k + \,2n + \) = E (2/,2 j,2k + 1 , 2 « - 1 ) + z  At £Ad  Z  H {2i + 2,2j,2k,2n) - H (2i,2 j,2k,2n) + y  y  H (2i,2j,2k,2n) - H {2i + 2,2 j,2k,2n) x  x  •~crE (2i,2j,2k,2n) £  (3-17)  z  18  To optimize calculation speed and simplify the equations the spatial increments A x , Ay and Az were selected such that  Ax = Ay = Az = Ad.  (3-18)  The components o f the electric and magnetic field were defined as following: H (iJ,k,n)  forM),2,4,...;y=l,3,5,...;*=l,3,5 ..;«=l,3,5,...;  (3-19)  H {i,j,k,ri)  for/=l,3,5,...;y==0,2,4,...;*=l,3,5,...;Ai=l,3,5,...;  (3-20)  H (i,j,k,n)  for /=l,3,5,...;y=l,3,5,...; fc=0,2,4,-.; n=l,3,5,...;  (3-21)  E (iJ,k,n)  for /=l,3,5,...;y=0,2,4 -.; *=0,2,4,-.; « = 0 , 2 , 4 , - . ;  (3-22)  E (i,j\k,n)  for /=0,2,4,-;y=l,3,5,...; £=0,2,4,...; «=0,2,4,...;  (3-23)  E,(i,j,k,n)  for /=0,2,4,... y=0,2,4,...; A=l,3,5,...; « = 0 , 2 , 4 , -  (3-24)  v  x  y  t  v  x  y  3.3  ;  Boundary and initial conditions Figure 3-2 shows the setup for the modeling o f FTIR in the simulation program.  A 3-D wave vector K , which was expressed as  K sin 9 cos 0  (3-25)  sin 9 sin <f>  (3-26)  K. = K cos 9  where  9 is the angle between K  (3-27)  and 2-axis,  <f> is the angle made between  K '<  projection onto xy-plane and the *-axis, and K| , the magnitude o f K , is In IX ,where A  is the radiation wavelength shows the direction of propagation o f the incident wave.  20  Second dielectric media  FTIR gap  First dielectric media  0  1  Figure 3-2. Schematic diagram o f 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 o f this significant simplification o f the problem set. Then the x and y components o f K were chosen to satisfy the following equations K -S  = 2mn:  m= ± 0 , 1 , 2 , . . .  (3-28)  K -8'  = 2ln  /= ± 0 , 1 , 2 , . . .  (3-29)  where  8  x  y  is the x and <?'is the y-size o f the considered volume of space. For the  simplicity o f the algorithm the values o f 8 and 8' used were equal. O n one side this restriction imposed on the values o f the components o f wave vector K somehow restricts possible directions for the propagation o f the wave, but on the other leads to a tremendous reduction o f computational time. The different  polarizations o f the incident wave were introduced using the  following approach. E was expressed in a form o f E = ^E,+5E  (3-30)  2  where E , and E  2  are mutually perpendicular vectors o f a unit length, and A,B are  arbitrary coefficients that satisfy  A +B 2  2  =1  If E , and E are defined as 2  Ej = K x k  (3-31)  E =kxE,  (3-32)  2  22  where K and k are unit vectors in the directions o f K and the z-axis, respectively, then E is consistent with the requirement o f E • K = 0 . Substitution o f (3-31) into (3-32) and using (3-30) gives the following form o f 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 o f the magnetic field can be obtained from (3-34)  H = KxE  In order to properly observe the reflected and transmitted waves, it was important to eliminate reflections o f these waves at the ends o f 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 o f the mesh, and returning back, as shown on Figure 3-3. In order to avoid reflection at the boundary o f the reflecting region itself, the value o f 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 i n the determination o f reflection coefficients.  T o eliminate this, the onset o f detection o f  reflected radiation was delayed until after the backward-travelling pulse had been absorbed by the rear absorber. Since the geometry o f the system ensured that there was always a substantial time interval between the arrival o f 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 o f the algorithm (in 3-D case) requires t h a t  A  . .  Ad  1  At<  - = ^ 42c max  (3.  1  C  < max '  where c  1 Ax"-  1 '  11,12  1  Ay'-  h 1  1  3 5 )  2  Az \ l  is the maximum electromagnetic phase velocity within the media being  m a x  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 c  max  = 1, A s —  X and At ~ — . 20  The mesh size and the time increment choice have a large effect on 5  calculation time, so these were made as large as possible without significantly distorting the results. Using the selected values, the mesh consisted of 2 5 x 2 5 x 6 0 0 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 xy and xy . t  r  The following  formulae were applied to calculate the numeric values for the fluxes:  ' ZXE( *)  7 =  5  (3-36)  'r=-HEfc) '  *o yo  (3-37)  =  z  Zr  where the z-component of the Poynting vector is given by  25  (3-38)  S =c s (E H -E H ) 2  z  0  0  x  y  y  x  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 «, -n  2  = 1.5. The FTIR gap medium was air. Incident light was polarised at 4 5 ° ,  V2 which was achieved by setting the parameters A and B in the equation (3-30) to — . Figure 3-5 shows the transmission coefficients r and T for the specified indices B  x  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  F T I R 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 o m p o n e n t , TIR  case.  29  30  fixed moment o f time. Figure 3-6 corresponds to a simple TIR case. Figure 3-7 presents an F T I R case. The goal of presenting these charts is to provide a visual understanding of the phenomenon o f F T I R by showing the relative magnitude o f 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 o f modelling F T I R 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 o f the surface at which -*  y  F T I R 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 o f 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 o f modelling of F T I R 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 = d ;y x  = d, y  where d and x  d  y  are linear dimensions o f the considered area at which F T I R happens. The periodic  31  boundary conditions may work with some degree o f accuracy, however, a possible discontinuity in the considered rough surface may affect the results. A few results were obtained for the modelling of F T I R 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 o f 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 c o m p a n y  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 ( U B C ) 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 3 M , 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 retroreflective 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 3 M , St. Paul, M N functions remarkably well as an electrophoresis 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 o f the cell for the viewer. T o prevent specular glare a diffusive coating may be introduced as the first top layer of the cell cover. Figure 4-2(b) shows " o f f state of the cell. The voltage is applied across the cell, moving the charged particles towards  36  the  prismatic  surface  frustrated/absorbed  of  retro-reflector.  The  incident  ray  in  this  case  is  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 o f 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 o f  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 o f 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 x f r o m 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  +  exp  =n  0+  and  kT  n_ = n _ exp  z_ey/  (4.1)  0  kT  where n and n_ are the respective numbers o f 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 +  respectively),  -z_ey/,  « a n d « _ are the corresponding bulk concentrations of particles and 0 +  counter-ions (i.e.  0  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_en_ =  +  = z+en  e x 0+  P|  z_en _ exp 0  kT  z_ey/  (4.2)  kT  The electrophoretic suspension in the bulk region o f the cell is electrostatically neutral, therefore  zn +  = Z-"o- •  0+  (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  d y/ _  p  dx  £  2  2  (4.4)  where s is an absolute electrical permittivity of the suspension given by  s=k s , D  (4.5)  0  where k  D  is the dielectric constant of the solvent.  Combining the equations (4.2), (4.3) and (4.4) gives  d y/ 2  z  exp  dx'  e +  kT  ¥  -exp  z_ey/  (4.6)  kT  In a special case o f 1-1 electrolyte (i.e. charge of the particles is equal to the charge of the counter-ions)  (4.7)  z =z_ +  the expression (4.6) can be solved for the potential y/ in an explicit form with the following boundary conditions imposed: y/ = y/ at x = 0 ; and dy/1 dx = Oat x = °o  1 3  0  Three base types o f 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 o f  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 N H " ' with the 2  charge of -e. The electrokinetic potential £ has been measured by Aveka, Inc. and has been found to be 26.1 mVin the case o f 10% wt silica/acetonitrile suspension. Unfortunately a very common used Debye-Huckel approximation of equation (4.6)  exp  z ey/ +  kT  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 o f the silica particles of 0.5 microns diameter, with 10% mass suspension in acetonitrile, was found to be n = -2.21 x 10" m 1V 8  2  • 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) mobility  14  £ c a n be obtained from their  .  ^i/r-  -  (4 ,0)  41  where rj  »3A-10~ N• 4  f  s/m is  the fluid (acetonitrile) viscosity and k =37.5 is the  2  D  relative dielectric constant of acetonitrile. The zeta potential was found to be 40 m V , 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 o f y/  0  at the surface to the value o f 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 o f 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 en +  dx  2  0+  exp  -exti  kT  z_ey/  kT  y/ = i// , at x = d = 0.25microns;  (4.12)  d  —  dx  z  +  = 0 , at x = oo,  =270,  z_ =1, £ = 37.5^ , n 0  = 7.0 • W  17  0 +  particles / m (10.0 wt. %) 3  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 o f the equation (4.12) with the following  assumption: the derivative  dx  was assumed to be constant throughout the Stern layer  and the value for it was taken from the solution o f (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 o f the values o f 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  dx  - const at x = 0.  What happens if the electric field applied to the cell changes? Well, the potential y/  0  can be related to the charge density cr by equating the surface charge with the net 0  space charge in the diffuse part o f the double layer (i.e. <r =-^pdx) 0  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 a  and the ionic composition o f the  0  medium (i.e. the values of z , z _ and « +  0 +  ) . If the double layer is compressed, then either  fj must increase, or y/ must decrease, or both. 0  0  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 o f the electrostatic field applied o f 2 - 4 1 0 F / m . Such a high value o f the external 4  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 o f 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. A s the graph shows the concentration of the particles near the plane exceeds the concentration o  0 +  bulk  by several orders o f 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 n  = 3.12 • 10 m' . 18  cp  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 o f particles at the TIR interface. The  \ 45  assumption will be made that the refractive index does not change due to the presence o f 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 o f 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 T (x) 0  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 T (x) 0  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  (4.13) o  and the intensity of the reflected light (i.e. reflectivity of the front surface) in the TIR case will be defined as  47  to co  (x)j uoRdjosqv  48  co  (X)0l 'UOISSjLUSUBJl  OO  (4.14)  R = A$T (x)F(x)dx, 0  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 o f 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 o f 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 R  3  = 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 o f 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 o f a twisted neumatic cell doesn't exceed 33%. The reflectivity o f 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 o f 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 o f them in the "white" and  50  "black" states. Absolute change in the intensity o f 75% comparing to 32% will give a better judgement of white and black colours for the human eye. A s a final remark to this chapter lets mention some other companies that are at the present day carrying the research in the field o f electrophoretic displays. They are Copytele Inc. ( C O P Y ) , Research Frontiers Inc. ( R E F R ) and E-ink, Inc., a spin-off company from MIT. Their research, however, does not involve the use o f electrophoresis to control FTIR. Their focus is on changing the diffuse reflectivity of particle suspensions by motion o f 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™\n  2  /«,)  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. W i l 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. A s 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 o f 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 o f 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 o f the prismatic surface is «, sin(90° -0)  = n sin(90° -29)  (5.1)  2  can be solved for the unknown angle 8 given the refractive index of the prism material = 2.27 and of the outside media n  air  = 1.0. The solution yields the following result:  n  1 i - + ^.) = 34.33° 2  9 = arccos(  v 4«  2  \ I6n  2  2  (5.2)  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:  9  = a r c s i n f ^ ^ - ) = 34.02° 2  cr  (5.3)  zns  n  Notice that the incident angle at the TIR surface only slightly exceeds the critical angle. This results in the fact that i f 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  cn  a)  •a o m  a> o> c ca  CD  to 2> »  O) <D  m o i n CO CD a) a> cn c c  ro ro i • ! •  o  O  cu o  L L  U .  U L  (0  Q)  D) <D  T3 - a  ro ro  III  .t; o  > to"  -C i-X  o  O  8  -f—<  "3  = <D CD O  o o  fi S  OH  Q. CD O -Q  tj ® a) ° > a> £ CD  0  O  a>  <*-H  <U  O  T3  O  a!  J5  ro m  '•+-»  o  u  •*- >»  o >  HP  ^ &  o ro  £  >  O  LO  += LO C 0)  —  l-l  o o  J C CD  £  3 CO  m oo  a?  ~-<D "°  "  CO  .S2 t _  O Q.  e g JD CD CD C CO  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 A C T I R . 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 o f 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 o f 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 6 0 ° . 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, M N ) , are glued together at their flat surfaces (shown by a dashed line). Both of the films are made from polycarbonate material with refractive index o f 1.6 The typical size o f 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 o f the behavior o f the reflectivity o f 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 3 M , 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. T w o absorbers have been placed one inside the electrophoretic liquid, the other above the source o f 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 o f the incoming light that could possibly be modified by means of A C T I R . A set o f angles o f both o f 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 o f 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 o f 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 2 D case of the prismatic surface was  considered in this modeling. Essentially a 2 D 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 F T I R for any given geometry. Several ways of modulating of F T I R 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 A C T I R . 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 o f 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 o f 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 U S Patent Application Serial No. 09/133,214.  3.  M . Born and E. Wolf, "Principles of Optics", Pergamon, Oxford, 1975, 5  4.  I. N. Court and F. K. Willisen, "Frustrated Total Internal Reflection and Application  th  ed.  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. A m . , 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 V e n , "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," I E E E Trans. Anten. Prop., A P - 1 4 , pp. 302307, 1966.  10. J. Stratton, "Electromagnetic Theory", New York, M c G r a w - H i l 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. 288373,1990. 12. A.Taflove, and M.E.Brodwin, "Numerical solution o f steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," I E E E Trans. Microwave Theory Tech., M T T - 2 3 , pp. 623-630, 1975. 13. Overbeck, J.Th.G. "Electrochemistry of a double layer", Coll. Sc., Elseveir, V o l . 1 "Irreversible systems", pp. 115-193, 1952. 14. R. J. Hunter. "Foundation o f 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 & C o , Ltd., 1986. 17. E. J. W. Verway and J. T h . 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* * * * * * * * * * * * * * * * * * * * * *  PROGRAM  JN  PROGRAM  ****************  MAIN  PARAMETER(ITE=1,ITH=1,IXT=19,IYT=19,IZT=4 00, PI=3.1415926)  6  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 REAL REAL REAL REAL REAL  EX(O:IXT,0:IYT,0:IZT,0:ITE) EY(0:IXT,0:IYT,0:IZT,0:ITE) EZ(0:IXT,0:IYT,0:IZT,0:ITE) HX(0:IXT,0:IYT,0:IZT,0:ITH) HY(0:IXT,0:IYT,0:IZT,0:ITH) HZ(0:IXT,0:IYT,0:IZT,0:ITH)  REAL REAL REAL REAL REAL REAL  EXS(0:IXT,0:IYT,0:IZT) EYS ( 0 : IXT, 0 :1YT, 0 :1ZT) EZS(0:IXT,0:IYT,0:IZT) HXS(0:IXT,0:IYT,0:IZT) HYS(0:IXT,0:IYT,0:IZT) 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 KVEC=10.0 WAVELG=2.0*PI/KVEC UL=WAVELG/19.0 T0=0 T1=0 C  TIME LIMIT TERMIN=4 0  FOR  THE  CALCULATIONS  c  DLTT=0.5 DLTX=1.0 DLTY=1.0 DLTZ=1.0 KOF=WAVELG/19.0 c  XSTR=0 YSTR=0 ZSTR=0 XND=IXT YND=IYT ZND=IZT _  c  10=0 J0=0 K0=0 OFFSET=100 c  UNIT=1 ABSORB1=100 ABSORB2=ZND-100 c  KX0=KVEC*SIN(THETA)*COS(PHI) KY0=KVEC*SIN(THETA)*SIN(PHI) KZ0=KVEC*COS(THETA) NORML=SQRT(A**2+B**2) EX0=(A*SIN(PHI)+B*COS(THETA)*COS(PHI))/NORML EY0=(B*COS(THETA)*SIN(PHI)-A*COS(PHI))/NORML EZ0=-B*SIN(THETA)/NORML C  "initialize" CALL  INTLZE(EXS,EYS,EZS,HXS,HYS,HZS)  69  C  "calculate" DO 10 T=l,700 THEN EFLD(EXS,EYS,EZS,HXS,HYS,HZS)  IF(MOD(T,2).EQ.O)  ELSE 10  CALL  CALL ENDIF CONTINUE  C- - - " o u t p u t " -  HFLD(EXS,EYS,EZS,HXS,HYS,HZS)  - - - - - - - - - - - - - - - - - - -  150 FORMAT(IX,21(IX,F5.2)) OPEN(UNIT=22,FILE='at71xy.dat ,STATUS='UNKNOWN') REWIND(UNIT=22) DO 310 J=YSTR,YND,1 WRITE(22,150) (EXS(I,J,71),I=XSTR,XND,1) CONTINUE 1  310  CLOSE(UNIT=22)  311  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) 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)  316  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) 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 REAL REAL REAL REAL REAL  EXS(XSTR:XND,YSTR:YND,ZSTR:ZND) EYS(XSTR:XND,YSTR:YND,ZSTR:ZND) EZS(XSTR:XND,YSTR:YND,ZSTR:ZND) HXS(XSTR:XND,YSTR:YND,ZSTR:ZND) HYS(XSTR:XND,YSTR:YND,ZSTR:ZND) 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  I F ( I . E Q . O ) THEN IMN=XND ELSE IMN=I-1 END I F I F ( I . E Q . X N D ) THEN IPL=0 ELSE IPL=I+1 END I F I F ( J . E Q . 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)* ((HZS(I,JPL,K)-HZS(I,JMN,K))/(2*DLTY)-  - -  6 6  (HYS(I, J,K+1)-HYS(I,J,K-1) )/(2*DLTZ)KOF*EXS(I,J,K)*SGMX)  C  end o f t h e i n t e r n a l  50 100 200 C C  cycle  CONTINUE CONTINUE CONTINUE end ex "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 I F 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 I F END I F IF(I.EQ.O) THEN  73  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  c  6 6 6  (HZS(IPL,J,K)-HZS(IMN,J,K))/(2*DLTX)KOF*EYS(I,J,K)*SGMY)  300  CONTINUE  4 00 500  CONTINUE CONTINUE  C C  SCHEME  EYS(I,J,K)=EYS(I,J,K)+(DLTT/PERMIT)* ((HXS(I,J,K+1)-HXS(I,J,K-1))/(2*DLTZ)-  end  ey  "Ez" DO  900  K=ZSTR+2,ZND-2,1  IF(K.LE.ABSORB1)  THEN  ABSBZ=2.0 ABSFZ=0.005 SLOPEZ=(ABSBZ-ABSFZ)/(ABSORB1-ZSTR) SGMZ=ABSBZ-K*SLOPEZ ELSE IF  (K.GE.ABSORB2)  THEN  ABSBZ=2.0 ABSFZ=0.005 SLOPEZ=(ABSBZ-ABSFZ)/(ZND-ABSORB2)  74  SGMZ=ABSFZ+(K-ABS0RB2)*SLOPEZ ELSE SGMZ=0.0 END I F ENDIF DO  800  I=XSTR,XND,1  DO 7 0 0 J = X S T R + 1 , X N D , 1 IF((K.GE.ZSTR).AND.(K.LE.MEDFND)) PERMIT=N**2 ELSE  THEN  IF((K.GT.MEDFND).AND.(K.LE.MEDSND)) PERMIT=1.0 ELSE  END  END IF  THEN  PERMIT=N**2 IF  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  c  7 00  SCHEME  6  EZS(I,J,K)=EZS(I,J,K)+(DLTT/PERMIT)* ((HYS(IPL,J,K)-HYS(IMN,J,K))/(2*DLTX)-  6 6  (HXS(I,JPL,K)-HXS(I,JMN,K))/(2*DLTY)KOF*EZS(I,J,K)*SGMZ) CONTINUE  75  800 900  CONTINUE CONTINUE  C  end  ez  RETURN END  H-fields: SUBROUTINE INTEGER REAL  Hxs,Hys,Hzs HFLD(EXS,EYS,EZS,HXS,HYS,HZS)  XSTR,XND,YSTR,YND,ZSTR,ZND  SGMX,SGMY,SGMZ,PERMEA,DLTT,DLTX,DLTY,DLTZ,KOF  REAL REAL REAL REAL REAL REAL  EXS(XSTR.XND,YSTR:YND,ZSTR:ZND) EYS(XSTR:XND,YSTR:YND,ZSTR:ZND) EZS(XSTR:XND,YSTR:YND,ZSTR:ZND) HXS(XSTR:XND,YSTR:YND,ZSTR:ZND) HYS(XSTR:XND,YSTR:YND,ZSTR:ZND) HZS(XSTR:XND,YSTR:YND,ZSTR:ZND)  COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C3/ D L T T , D L T X , D L T Y , D L T Z , K O F SGMX=0.0 SGMY=0.0 SGMZ=0.0 PERMEA=1.0  C  -"Hx" DO  200 DO  K=ZSTR+2,ZND-2,1 100 DO  I=XSTR,XND,1 50  J=XSTR+1,XND,1  IF(I.EQ.O)  THEN  IMN=XND ELSE IMN=I-1 END I F IF(I.EQ.XND) IPL=0 ELSE IPL=I+1  THEN  76  END  IF  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 C-  D I S C R E T E SCHEME HXS(I,J,K)=HXS(I,J,K)+(DLTT/PERMEA)* ( -(EZS(I,JPL,K)-EZS(I,JMN,K))/(2*DLTY)+ (EYS(I,J,K+l)-EYS(I,J,K-l))/(2*DLTZ) )  6 6 50  CONTINUE  100 200  CONTINUE CONTINUE  C C  end  ex  "Hy" DO 5 0 0  K=ZSTR+2,ZND-2,1  DO 4 0 0  I=XSTR,XND,1  DO 3 0 0  J=XSTR+1,XND,1  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  77  IF(J.EQ.YND)  THEN  JPL=0 ELSE JPL=J+1 END I F C  DISCRETE  SCHEME  H Y S ( I , J , K ) = H Y S ( I , J , K) + ( D L T T / P E R M E A ) * 6  (  6  (HZS(IPL,J,K)-HZS(IMN,J,K))/(2*DLTX)  300  CONTINUE CONTINUE  C C  )  CONTINUE  4 00 500  -(EXS(I,J,K+1)-EXS(I,J,K-1))/(2*DLTZ)+  -  end  ey  "Hz"DO  900  K=ZSTR+2,ZND-2,1  DO 8 0 0  I=XSTR,XND,1  DO 7 0 0  J=XSTR+1,XND,1  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  c  DISCRETE  SCHEME  HZS(I,J,K)=HZS(I,J,K)+(DLTT/PERMEA)*  78  (- (HYS(IPL,J,K)-HYS(IMN,J,K))/(2*DLTX)+ (HXS(I,JPL,K+1)-HXS(I,JMN,K))/(2*DLTY) )  6 6  700  CONTINUE  800 900  CONTINUE CONTINUE  C  end ez RETURN END  C C****"Initialize 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 REAL REAL REAL REAL REAL  EXI(XSTRrXND,YSTR:YND,ZSTR:ZND) EYI(XSTR:XND,YSTR:YND,ZSTR:ZND) EZI(XSTR:XND,YSTR:YND,ZSTR:ZND) HXI(XSTR:XND,YSTR:YND,ZSTR:ZND) HYI(XSTR:XND,YSTR:YND,ZSTR:ZND) 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  i n 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 E Z I ( 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 100 180 190  CONTINUE CONTINUE CONTINUE  200 280 2 90  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 CONTINUE CONTINUE CONTINUE RETURN END  C******"Calculate SUBROUTINE  Poynting  vector"*************************  PYNT(S2,SI,EXS,EYS,HXS,HYS)  INTEGER YF22,Jl,J2,YF11,J3,J4,XSTR,XND,YSTR,YND REAL S22,S11,S2,S1 REAL REAL REAL REAL  EXS(XSTR:XND,YSTR:YND,ZSTR:ZND) EYS(XSTR:XND,YSTR:YND,ZSTR:ZND) HXS(XSTR:XND,YSTR:YND,ZSTR:ZND) HYS(XSTR:XND,YSTR:YND,ZSTR:ZND)  COMMON/C1234/ XSTR,XND,YSTR,YND,ZSTR,ZND COMMON/C24/ ABSORB1,ABSORB2  80  S2=0. 0 S1=0. 0 S22=0. 0 S11=0.0  DO  6 6  200 100  100  I=XSTR,XND,1  DO 2 0 0 J=XSTR,YND,1 S22=EYS(I,J,ABSORB1)*HXS(I,J,ABSORB1)E X S ( I , J , A B S O R B l ) * H Y S ( I , J , ABSORB1) S11=EYS(I,J,ABSORB2)*HXS(I,J,ABSORB2)EXS(I,J,ABSORB2)*HYS(I,J,ABSORB2)  S2=0.5*S22+S2 S1=0.5*S11+S1 CONTINUE CONTINUE RETURN END  81  


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