wide-angle beam propagation method for liquid-crystal device calculations

e

Wide-angle beam propagation method forliquid-crystal device calculations

Emmanouil E. Kriezis and Steve J. Elston

A wide-angle beam propagation method suitable for analyzing anisotropic devices involving liquid crys-tals is presented. The mathematical formulation is based on a system of coupled differential equationsinvolving an electric and a magnetic field component. The contribution of all dielectric tensor elementsis included. A numerical implementation based on finite differences is used. Numerical examples arefocused on light-wave propagation within twisted nematic pixels found in microdisplays, with all effectsarising at pixel edges that are incorporated. A comparison between the results obtained and theprediction of finite-difference time-domain simulations is conducted, showing satisfactory agreement.The required computational effort is found to be minimal. © 2000 Optical Society of America

OCIS codes: 160.3710, 260.1180, 350.5500, 000.4430.

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1. Introduction

The mainstream techniques currently used for pre-dicting liquid-crystal ~LC! optics for displays arematrix-type solvers, the most common of them beingthe Berreman method.1 Matrix-type solvers arebased on the stratified medium approximation, as-suming variation of the LC orientation only along thedirection normal to the supporting glass surfaces.This simplification seems to be reasonable when LCdevices are analyzed with sufficiently slow variationalong the transverse directions, but may be underdispute in a significant number of currently evolvingtechnologies, such as microdisplays, zenithal bistablenematic devices, and ferroelectric LC displays, wheretransverse variation can take place on a scale similarto the wavelength of light. An alternative path toaddress the LC optics can be provided when more-rigorous numerical methods are employed for light-wave propagation within anisotropic media, thusaccounting for the spatial variation along the trans-verse directions. For example, the finite-differencetime-domain ~FDTD! method, being a general tool forlectromagnetic simulations, has been introduced

The authors are with the Department of Engineering Science,University of Oxford, Parks Road, Oxford OX1 3PJ, UK. E. E.Kriezis’s e-mail address is [email protected].

Received 15 March 2000; revised manuscript received 20 July2000.

0003-6935y00y05707-08$15.00y0© 2000 Optical Society of America

uccessfully to the prediction of light-wave propaga-ion within LC devices.2–4

A highly successful tool for the numerical model-ing of guided wave optics is the beam propagationmethod ~BPM!. Also the BPM has been extendedo model anisotropic devices, especially in the areaf integrated optics.5–7 In this case, paraxiality as

well as slow refractive-index variation wereassumed—approximations were fully justified bythe guiding nature of the structures analyzed. Anapproach to the theory for a BPM in the generalizedgeometrical-optics approximation has also beenproposed,8 and a finite-element wide-angle BPM,suitable for anisotropic waveguide analysis, re-cently has been published.9

A common limitation of existing BPM’s for aniso-tropic media relates to the fact that these methodsoriginally were developed under the assumption thatguiding devices are to be analyzed. This results insome natural simplifications if various dielectric ten-sor elements are explicitly set to zero or if variousinteractions between dielectric tensor elements andfield components are neglected.5–7,9 In the case ofLC optics the lack of a strong light-guiding mecha-nism ~as opposed to waveguide devices! and the rapid

C reorientation along the axial and the transverseirections suggests the need to avoid such simplifica-ions. A beam propagation scheme for LC wouldave to allow for wide-angle propagation to overcomehe severe limitation, which is otherwise imposed byaraxiality, and must further retain the contributionf all dielectric tensor elements.An initial attempt to apply a BPM in LC optics has

1 November 2000 y Vol. 39, No. 31 y APPLIED OPTICS 5707

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5

been the proposal of a wide-angle scheme for tilted-only LC structures,10 based on the transverse-magnetic field. Because most practical LC devicesoperate by use of a tilted–twisted director profile, ageneralization of the previously proposed scheme isnecessary. Our scope in this paper is to develop awide-angle BPM that is capable of handling tilted–twisted structures, retaining the contribution of allnine dielectric tensor elements. A formulationbased on the coupled equations involving an electricand a magnetic field component is followed.

We undertake an assessment of the proposed BPMby analyzing a microdisplay pixel, and comparisonswith the FDTD method are performed. Both meth-ods lead to almost identical results. Although itmust clearly be understood that the BPM will neverbe able to achieve the generality of the FDTD method,allowing for ultrawide angles of propagation, it can beseen to be an indispensable tool in its own regime ofapplicability. In its regime of validity, it can repro-duce the computational-intensive FDTD simulationswith an extreme improvement in computational effi-ciency. This will allow for the convenient and accu-rate numerical modeling of demanding LC devicesthat fall well beyond the point of valid application forthe classical matrix-type solvers.

2. Mathematical Formulation

A. Maxwell’s Equations

Light-wave propagation is considered within a two-dimensional tilted–twisted LC device, as shown inFig. 1. The concurrent presence of a tilted–twisteddeformation in the director orientation results in adielectric tensor expression where all nine tensor el-

Fig. 1. Schematic representation of a two-dimensional deviceshowing the various conventions used for the measurement of theangles involved. PBC, periodic boundary condition. kinc, inci-dent wave vector.

708 APPLIED OPTICS y Vol. 39, No. 31 y 1 November 2000

ements are nonzero. All field components, as well asthe dielectric tensor elements, are considered to beexplicit functions of the spatial coordinates ~x, z!.The formulation is based on Maxwell’s equations,which are written accordingly:

2]Ey

]z5 2jvm0 Hx, (1a)

2]Ez

]x1

]Ex

]z5 2jvm0 Hy, (1b)

]Ey

]x5 2jvm0 Hz, (1c)

2]Hy

]z5 jvε0~εxxEx 1 εxyEy 1 εxzEz!, (2a)

2]Hz

]x1

]Hx

]z5 jvε0~εyxEx 1 εyyEy 1 εyzEz!, (2b)

]Hy

]x5 jvε0~εzxEx 1 εzyEy 1 εzzEz!. (2c)

Equations ~2a! and ~2c! can be solved with respect toEx and Ez, resulting in

Ex 5 21

jvε0 a Sεzz

]Hy

]z1 εxz

]Hy

]x D 2ca

Ey, (3a)

Ez 51

jvε0 a Sεxx

]Hy

]x1 εzx

]Hy

]z D 2ba

Ey, (3b)

where

a 5 εxxεzz 2 εzxεxz, b 5 εxxεzy 2 εzxεxy,

c 5 εxyεzz 2 εzyεxz.

In complete analogy, Eqs. ~1a! and ~1c! can be solvedwith respect to Hx and Hz, resulting in

Hx 51

jvm0

]Ey

]z, (4a)

Hz 5 21

jvm0

]Ey

]x. (4b)

Substituting Eqs. ~3a!, ~3b!, ~4a!, and ~4b! into Eq.2b! and substituting Eqs. ~3a! and ~3b! into Eq. ~1b!,e can develop a system of two coupled partial dif-

erential equations, involving only the electric andagnetic field components along the y direction:

2Ey

]x2 1]2Ey

]z2 1 k02Sεyy 2 εyz

ba

2 εyx

caDEy

2jvm0

a Sb]Hy

]x2 c

]Hy

]z D 5 0, (5)

t0

h

ad

T

t

s

h

jvε0F ]

]z ScaD 2

]

]x SbaDGEy 1 jvε0Sc

a]Ey

]z2

ba

]Ey

]x D1

εxx

a]2Hy

]x2 1εzz

a]2Hy

]z2 1 2εxz

a]2Hy

]x]z1 F ]

]x Sεxx

a D1

]

]z Sεxz

a DG ]Hy

]x1 F ]

]x Sεzx

a D 1]

]z Sεzz

a DG ]Hy

]z

1 k02Hy 5 0, (6)

where

b 5 εyzεxx 2 εyxεxz, c 5 εyxεzz 2 εyzεzx.

It is important to note the limiting cases, which canbe obtained from Eqs. ~5! and ~6!. In the absence ofwist @i.e., f~x, z! 5 0 and thus εxy 5 εyx 5 εyz 5 εzy 5#, Eq. ~5! is decoupled from Eq. ~6!; therefore, for a

purely tilted structure, TE and TM fields can be con-sidered separately, as was done in our previous pub-lication.10 If the medium is considered to be a

omogeneous anisotropic one ~i.e., ]εuvy]x 5 ]εuvy]z 5 0; u, v 5 x, y, z!, then Eqs. ~5! and ~6! reduce tothose given in Ref. 11, which deals with the finite-element analysis of anisotropic gratings.

Our purpose in Subsection 2.B below is to developa wide-angle BPM for the solution of these coupledequations, without introducing any simplificationsregarding the dielectric tensor elements, as is usuallydone when anisotropic optical waveguides are ad-dressed.7,9 This is necessary because practical LCdevices may involve significant variation of the direc-tor orientation, and therefore of the optical dielectrictensor, along both spatial directions. Moreover, thelack of a strong guiding mechanism ~as opposed towaveguide structures! further suggests that any sim-plification of Eqs. ~5! and ~6! is not fully justifiable.

B. Wide-Angle Beam Propagation Method

A common step toward the development of any BPMis writing the field components as the product of afast varying reference exponential term and a spatialfield envelope:

Ey~x, z! 5 %y~x, z!exp~2jkref z!, (7a)

Hy~x, z! 5 *y~x, z!exp~2jkref z!. (7b)

Equations ~7a! and ~7b! are substituted in Eqs. ~5!nd ~6!, leading to the following system of coupledifferential equations:

F1 00 qG ]2

]z2 F%y

*yG 1 HFR11 R12

R21 R22G 1 F0 0

0 S22]y]xGJ3

]

]z F%y

*yG 1 FA11 A12

A21 A22GF%y

*yG 5 0. (8)

he elements of matrix @R# are space dependent @i.e.,Rij 5 Rij~x, z!; i, j 5 1, 2#, whereas the elements ofmatrix @A# are operators involving the transverse de-rivatives and spatial functions @i.e., Aij 5 Aij ~]y]x,]2y]x2, x, z!; i, j 5 1, 2#. The explicit expressions for

all elements appearing in Eq. ~8! are given in Appen-dix A. It should be noted that Eq. ~8! is derived fromEqs. ~5! and ~6! without any simplifications.

If one omits the first term of Eq. ~8!, which corre-sponds to the second derivatives with respect to the zdirection, a system of coupled parabolic differentialequations is then formed. A propagation scheme isthen feasible by an implicit numerical method suchas the Crank–Nicolson scheme. However, omissionof the second-order derivatives of the spatial enve-lopes with respect to z results in a paraxial descrip-tion of light-wave propagation, with the severelimitation that only components that are propagatingwithin a cone of approximately 67° around the axial~ z! direction can be resolved accurately. It is evidentthat such a restriction significantly limits the appli-cability of the BPM in LC devices. However, a sig-nificant enhancement can be provided if Eq. ~8! istransformed into its wide-angle counterpart. Suchenhancement will extend the useful applicability ofthe BPM to a range of a few tens of degrees withrespect to the z direction.

The wide-angle development is based on the Paderecurrence relation.12 It is obvious from Eq. ~8! thathe following operational relation holds:

]

]z; 2

@A#

F1 00 qG ]

]z1 @R# 1 @S#

. (9)

Identity ~9! suggests the following recurrence formu-la:

]

]zUn11

5 2@A#

F1 00 qG ]

]zUn

1 @R# 1 @S#

, ]

]zU0

5 0. (10)

By setting index n 5 0, we obtain the correspondingparaxial equations. Setting n 5 1 in Eq. ~10! andubstituting for ]y]zu1 5 2@R#21@A# in the denomina-

tor, we obtain the first wide-angle scheme. We de-rived the above expression for ]y]zu1 by dropping thecontribution of matrix @S# to develop a realizable nu-merical scheme. Among all other terms, matrix @S#

as by far the smallest influence here10 as its contri-bution is weak compared with that of the R22 term,and everywhere else it is fully retained. The wide-angle approximation of Eq. ~8! can then be expressedby

$@R# 1 @S#%]

]z F%y

*yG 5 2@A#F%y

*yG , (11)

@R# 5 @R# 2 F1 00 qG@R#21@A#. (12)

It is evident from Eq. ~12! that the numerical imple-mentation of a wide-angle BPM will not present anymore difficulties than the implementation of the cor-responding paraxial equations. In essence oneshould consider replacing matrix @R# in the paraxialcase by matrix @R# in the wide-angle case. The as-


c

S

rc

povt

kcpεsf

E

d

1

5

sociated increase in computational burden is ex-pected to be small, and in Subsection 2.C we discussthe numerical implementation of the method.

C. Numerical Implementation

Equation ~11! is discretized when the implicit Crank–Nicolson finite-difference scheme is applied, with aweight ~controlling! parameter a. The resulting dis-rete form of Eq. ~11! along the axial direction is

accordingly written as

$@Rr11y2# 1 @Sr11y2# 1 adz@Ar11y2#%@Cr11#

5 $@Rr11y2# 1 @Sr11y2# 1 ~a 2 1!dz@Ar11y2#%@Cr#. (13)

uperscript r refers to the axial plane, z 5 rdz, andthe vector @C# is defined as @C# 5 @%y *y#

T. Utilizingthe definition of matrix @R# given in Eq. ~12!, one canewrite Eq. ~13! in the following form, which is moreonvenient for numerical implementation:

$@Rr11y2# 1 @Sr11y2# 1 @Cr11y2~a!#@Ar11y2#%@Cr11#

5 $@Rr11y2# 1 @Sr11y2# 1 @Cr11y2~a 2 1!#@Ar11y2#%@Cr#,

(14)

@C~a!# 5 3adz 21

uRuR22

1uRu

R12

quRu

R21 adz 21

uRuR11

4 . (15)

All matrices present in Eq. ~14! are defined in Ap-endix A and in Eq. ~15!. By further unfolding allperators present in Eq. ~14! involving the trans-erse derivatives, we can form a sparse linear sys-em that involves as unknowns the coupled %y, *y

values at the nodes of plane z 5 ~r 1 1!dz. Asusual, the weight parameter a varies within theinterval @0.5, 1.0#, and the numerical scheme is un-conditionally stable. Increased values of a are

nown to be associated with increased numeri-al dissipation,13 and in most cases optimumerformance of the scheme is obtained for a 5 0.5 11, where ε1 is a small positive quantity. Theparse linear system acquires the following genericorm:

F@H11# @H12#@H21# @H22#

GF@%yr11#

@*yr11#G 5 F@B1#

@B2#G . (16)

ach of the four subblocks @Hij#—i, j 5 1,2—corresponds to a tridiagonal or nearly tridiagonalmatrix, depending on the type of boundary conditionsused. If transparent boundary conditions14 areused, all subblocks are perfectly tridiagonal. How-ever, in the analysis of LC devices it is more naturalto consider periodic boundary conditions, which re-sult in the inclusion of two more nonzero elements ineach subblock. The periodic boundary conditionsare given in the following expressions, and they im-pose on the propagating fields a phase shift that is


identical to the phase shift in the illumination be-tween neighboring periods:

%y~L, z! 5 %y~0, z!exp@2jk0 nglass sin~uinc!#, (17a)

*y~L, z! 5 *y~0, z!exp@2jk0 nglass sin~uinc!#. (17b)

The quantity L represents the period span.The linear system of Eq. ~16! is solved by a precon-

itioned biconjugate gradient method.15

3. Applications

We can verify the applicability of the proposed wide-angle BPM by studying light-wave propagation in atwisted nematic ~TN! microdisplay pixel. Cross-check is provided by the FDTD method, which is ageneral method for solving Maxwell’s equations inboth space and time. Recently, the FDTD methodhas been applied to the optics of LC devices.2–4

However, although it is robust and reliable it is com-putationally extremely time-consuming when com-pared with any spectral-domain method such as theBPM.

The geometry of a small-sized TN pixel, togetherwith the electrode layout is shown in Fig. 2. The LCorientation was calculated by the commercially avail-able software package 2dimMOS from Autronics forvarious applied static voltages and typical E7 mate-rial parameters ~K11 5 12.5 pN, K22 5 7.3 pN, K33 57.9 pN, εi

static 5 14.1, ε'static 5 4.09!. Light-wave

illumination is provided by a monochromatic planewave with free-space wavelength l 5 650 nm at nor-mal or oblique incidence. For simplicity, an extraor-dinary refractive index ~parallel to the molecularorientation! of ne 5 1.6 and an ordinary refractiveindex ~perpendicular to the molecular orientation! ofno 5 1.5 were assumed. These values, together withthe calculated tilt and twist angles @u~x, z! and f~x, z!,respectively# as exported from the 2dimMOS soft-ware, are used for the computation of the relativedielectric tensor through

Fig. 2. Application example geometry representing a 25-mm-wideby 5-mm-thick TN microdisplay pixel. The driving electrodes areshown together with an intergap of 10 mm. The figure corre-sponds to a single period of the model used and is composed of halfof the electrode width associated with the pixel to the left of theintergap, the intergap, and half of the electrode associated with thepixel to the right. ITO, indium tin oxide.

scEir

a

F

tsv

2 2 2 2

The TN pixel is sandwiched between two ideal polar-izing elements oriented along the directions fin andfout, for the lower and upper glass plates, both mea-ured with respect to the positive x direction ~sameonvention as for the twist angle measurement!.xtension to the inclusion of real polarizing elements

s straightforward, as discussed below. For the sur-ounding glass medium, a refractive index of nglass 5

1.5 was assumed.Initially the normalized transmitted optical inten-

sity between crossed polarizers is studied for variousapplied static voltages. The input polarizer is ori-ented along fin 5 0°, whereas the output analyzer islong fout 5 90°, leading to a normally white mode of

operation. Illumination is at normal incidence, andthe applied static voltage has values of 2, 3, 4, and 5V. Figure 3 shows the BPM prediction ~dottedcurve! and the FDTD prediction ~solid curve!, thelatter corresponding to the results presented in Ref.4. It is important to note that the FDTD resultswere calculated on a grid with 1250 3 250 nodes~dx 5 dz 5 20 nm! for the LC region to guaranteeacceptable low numerical error because of the time-domain nature of the method. On the other hand,the BPM calculation was performed on a grid with500 3 100 nodes ~dx 5 dz 5 50 nm!, which was foundto be more than sufficient because the method utilizesthe spatial envelope of the fields involved in the spec-tral domain. Almost identical results were also ob-tained for the case of an even coarser grid consistingof 250 3 50 ~dx 5 dz 5 100 nm! in conjunction with

ε~x, z! 5 Fno 1 Dεr cos u cos f Dεr cDεr cos2 u sin f cos f no

2 1

Dεr sin u cos u cos f Dεr s

Dεr 5 ne2 2 no

2.

Fig. 3. Normalized transmitted near-field optical intensity plot-ted versus the distance along the cell ~ x axis! for different appliedstatic voltages ~2, 3, 4, and 5 V!. The curves shown were calcu-lated by the proposed BPM and the FDTD method. Illuminationis at normal incidence.

the BPM. The difference in memory requirementsand execution speed between the two numericalmethods is tremendous. Nevertheless, it is appar-ent that the general agreement between the twomethods is exceptionally good. The agreement ishighly satisfactory even in the regions of the struc-ture where some discrepancy might be expected. Inparticular, both methods provide the same predictionfor the influence of the disinclination line below theleft electrode, which is formed by the fringing staticfield. The influence of this feature corresponds tothe peak that can be seen on the left part of all curves.Application of the Berreman method would have pro-duced results that significantly deviate from bothpredictions, as has already been demonstrated.4This is expected, as the Berreman method is not suit-able for handling abrupt LC reorientations, such asthe one in the vicinity of this disinclination line.

A second set of results deals with the more de-manding case of oblique incidence, which seriouslytests the effectiveness of the proposed wide-anglescheme. Figure 4~a! presents the normalized trans-mitted optical intensity for an applied voltage of 4 Vwhen the pixel is illuminated at uinc 5 615°, whereas

ig. 4~b! corresponds to uinc 5 630°. The BPM pre-diction is the dotted curve and the FDTD prediction isthe solid curve. Once more, the agreement is foundto be highly satisfactory because the BPM closelyreproduces the FDTD results. We obtained theFDTD results by solving Maxwell’s equations underno approximations or assumptions. Any attempt toapply the classical matrix solvers for LC optics, suchas the Berreman method, will result in highly erro-neous predictions, as this specific problem falls wellbeyond their natural regime of applicability. Itshould be noted that the above angles of incidence aremeasured within the lower glass plate and thereforecorrespond to much steeper angles of incidence in air.This means that one can effectively analyze, with theabove level of accuracy, angles of incidence in excessof 50°, when measured in air.

We provide a further in-depth assessment of theproposed BPM by decomposing the fields at the end ofpropagation into a plane-wave spectrum. This isimportant because it corresponds to the far-field im-age of the device, which is what will be seen by aviewer of a display. Because of the inherent period-icity of the problem, a spectrum consisting of Floquetmodes is calculated. We then obtain the amplitudeof each mode ~plane wave! by applying a fast Fourierransform. Figure 5 presents the correspondingpectra for the fundamental field components in-olved in the development of the proposed BPM ~Ey,

Hy!, at the end of propagation and prior to the en-

u sin f cos f Dεr sin u cos u cos f

r cos2 u sin2 f Dεr sin u cos u sin fcos u sin f no

2 1 Dεr sin2 uG ,

(18)

osDε

in u


flsicrtdwwsdp

t

c

5

forcement of the analyzer, versus their angle of prop-agation with respect to the positive z direction.Spectra are also provided for the same field compo-nents, as calculated by the FDTD method. Figures5~a!–5~d! deal with the case of normal incidence, andFigs. 5~e!–5~h! deal with the oblique incidence case atuinc 5 30°. The director profile used for these calcu-lations is the same as in Fig. 4 ~static voltage of 4 V!.This shows the range of angles over which the struc-ture of the pixel could scatter light. It is also impor-tant to further note the good agreement between theBPM and the FDTD calculations.

To further illustrate the usefulness of the proposedBPM, a direct comparison is made between the com-putational burden and the memory requirements ofthe BPM and the FDTD method. The grid used forthe FDTD method is denser, and this is absolutelynecessary for the results to be accurate because of thecompletely different nature of the method. Further-more, the FDTD grid includes some more nodes for

Fig. 4. Normalized transmitted near-field optical intensity plot-ted versus the distance along the cell ~ x axis! for different angles ofincidence. A static voltage of 4 V was applied to the electrodes.The curves shown were calculated by the proposed BPM and theFDTD method. ~a! uinc 5 615°. ~b! uinc 5 630°. Angles of in-idence are measured within the lower supporting glass plate.


the implementation of the perfectly matched layerboundary condition. All these results are summa-rized in Table 1. Execution times correspond to asingle run, and they were recorded on a Pentium III550-MHz Xeon desktop computer. The improve-ment for the BPM over the FDTD method is nearly 2orders of magnitude for the execution speed and thememory usage. This significant improvement andthe fact that almost identical results are produced byboth methods in the regime of the BPM applicabilityfully establish the proposed BPM as a useful tool forthe analysis of LC optics.

It should further be noted that inclusion of realpolarizing elements in the BPM calculations isstraightforward and does not lead to any further dif-ficulties. Real polarizers can be modeled as homo-geneous lossy anisotropic slabs with a relativedielectric tensor expression similar to the one givenin Eqs. ~18! but with the presence of a negative imag-inary part to account for the losses.

4. Conclusions

A wide-angle BPM for LC device calculations exhib-iting bend–splay–twist deformations in two dimen-sions has been developed. The mathematicalformulation is based on coupled equations involvingthe electric and magnetic field components along they direction. The Pade recurrence relation provideswide-angle enhancement. Numerical results dem-onstrate that the method effectively allows for accu-rate light-wave propagation at least up to 30° withrespect to the z axis. Comparisons have been per-ormed between the BPM and the FDTD method, theatter corresponding to the most general numericalolution available. The methods resulted in almostdentical results, with the BPM requiring a minimalomputational effort. It has been shown that in theange of its applicability the BPM can be used advan-ageously instead of the FDTD method, providing aefinite enhancement over any matrix-type solver,ithout the computational drawbacks associatedith the FDTD method. This means that extensive

imulations for realistic LC devices, such as micro-isplays, are practically feasible on a desktop com-uter.

Appendix A

The elements of matrix @R# are defined through

FR11 R12

R21 R22G

5 322jkref jvm0

ca

jvε0

ca

22jkref

εzz

a1

]

]x Sεzx

a D 1]

]z Sεzz

a D4 . (A1)

The quantities a, b, c, b, and c are defined in Subsec-ion 2.A.

p

Table 1. Comparison between the Execution Time and the Memory
For the composite operator @A#, the following ex-
ressions hold:

FA11 A12

A21 A22G 5 FP11 P12

P21 P22G ]2

]x2 1 FQ11 Q12

Q21 Q22G ]

]x

1 FT11 T12

T21 T22G , (A2)

ng the y direction. The amplitude of each plane-wave component! Hy ~BPM! at uinc 5 0°, ~c! Ey ~FDTD! at uinc 5 0°, ~d! Hy ~FDTD!Ey ~FDTD! at uinc 5 30°, ~h! Hy ~FDTD! at uinc 5 30°.

Requirements for the BPM and the FDTD Methodsa

Method Griddx 5 dz

~nm!Memory~Mbytes!

ExecutionTime

BPM 250 3 50 100 3.8 7 sBPM 500 3 100 50 12.1 72 sFDTD 1250 3 314 20 119 238 min

aRecorded on a Pentium III 550-MHz Xeon desktop computer.

Fig. 5. Plane-wave spectrum representation of the field components alois plotted versus its angle of propagation. ~a! Ey ~BPM! at uinc 5 0°, ~bat uinc 5 0°, ~e! Ey ~BPM! at uinc 5 30°, ~f ! Hy ~BPM! at uinc 5 30°, ~g!


tCL

magnetic simulation of liquid crystal displays,” J. Opt. Soc.

5

where the matrices @P#, @Q#, and @T# are equal to

@P# 5 31 0

0εxx

a4 , (A3)

@Q# 5 3 0 2jvm0

ba

2jvε0

ba

22jkref

εxz

a1

]

]x Sεxx

a D 1]

]z Sεxz

a D4 ,

(A4)

The remaining quantities appearing in Eqs. ~8! aregiven by

q 5εzz

a, (A6)

S22 5 2εxz

a. (A7)

The authors acknowledge the financial support ofhe Engineering and Physical Sciences Researchouncil of the UK and the collaboration of Sharpaboratories of Europe at Oxford.

References1. D. W. Berreman, “Optics in stratified and anisotropic media:

4 3 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 ~1972!.2. B. Witzigmann, P. Regli, and W. Fichtner, “Rigorous electro-

@T# 5 3 k02Sεyy 2 εyz

ba

2 εyx

caD 2 kref

2

krefvε0

ca

1 jvε0F2]

]x SbaD 1

]

]z ScaDG k0

2 2εzz

ak


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