three-dimensional simulations of light propagation in periodic liquid-crystal microstructures

Three-dimensional simulations of light propagationin periodic liquid-crystal microstructures

Emmanouil E. Kriezis, Christopher J. P. Newton, Timothy P. Spiller, andSteve J. Elston

A composite scheme based on the finite-difference time-domain method and a plane-wave expansionmethod is developed and applied to the optics of periodic liquid-crystal microstructures. This is used toinvestigate three-dimensional light-wave propagation in grating-induced bistable nematic devices withdouble periodicity. Detailed models of realistic devices are analyzed with emphasis on two differentunderlying surface-relief grating structures: a smooth bisinusoidal grating and a square-post array.The influence of the grating feature size is quantified. Device performance is examined in conjunctionwith an appropriate compensation layer, and the optimum layer thickness is determined for the differentgrating geometries. © 2002 Optical Society of America

OCIS codes: 160.3710, 260.1180, 350.5500, 000.4430.

1. Introduction

For a long period of time liquid-crystal �LC� opticshave been pursued under the assumption that the LCmaterial can be considered as a stratified mediumwith the direction of stratification coinciding with thecell normal. Matrix-type solvers have been devel-oped, introducing different levels of complexitywithin the stratified medium framework. The sim-plest of all is the Jones method, restricted to forward-only propagation at normal incidence1; this was laterfollowed by the extended Jones method, which stilltraces the forward waves but now allows for obliqueincidence.2,3 Forward and backward waves are con-sidered in the Berreman method,4 which is the mostelaborate of the above matrix methods for optics.Matrix methods for optics can accurately describelight-wave propagation in LC devices when uniformtransverse material orientation is maintained over ascale far exceeding the optical wavelength. For ex-ample, typical pixels found in flat-panel displays withtransverse dimensions of hundreds of micrometersfall well within the stratified medium approximation.

Em. E. Kriezis �[email protected]� and S. J. El-ston are with the Department of Engineering Science, Universityof Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom. C. J. P.Newton and T. P. Spiller are with Hewlett-Packard Laboratories,Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom.

Received 24 January 2002.0003-6935�02�255346-11$15.00�0© 2002 Optical Society of America

5346 APPLIED OPTICS � Vol. 41, No. 25 � 1 September 2002

When the matrix methods are applied to devices withtransverse variation, some approximations are nec-essarily introduced. A decomposition of the directorprofile into stratified columns, together with someaveraging of the optical response associated witheach individual column, is one approach.5 Alterna-tively some type of averaging has to be applied to theoriginal director profile to yield an average opticalresponse. It is obvious that these approximationscan be justified only in the limit of transverse varia-tion of device characteristics and director orientation,which are extremely slow in comparison with theoptical wavelength. Otherwise, the error intro-duced is difficult to estimate a priori and certainlycan be substantial in many practical small-sized LCdevices, which involve significant LC reorientation�LC director gradients� along a transverse direction.

A step forward in the optics of LC devices is toconsider rigorously the LC variation both along thenormal to the cell surfaces and along a single trans-verse direction, leading to a two-dimensional treat-ment of light propagation. This approach hasproven to be successful, and it has been implementedwith the finite-difference time-domain �FDTD�method6–10 and the vector beam propagationmethod.10–12 The above methods were applied tovarious structures with rapid LC reorientation on theoptical wavelength scale, such as reverse tilt disin-clinations in twisted nematic pixel edges,7,9,12 ferro-electric LC domain walls,8 and also zenithal bistablenematic devices with surface-relief monogratings.10

It was demonstrated that both methods were consis-

tent within a small error margin. It was furtherevident that in many cases the more commonly usedmatrix methods led to highly erroneous results be-cause the rapid LC variation caused strong scatteringand diffractive effects that were simply ignored.

Bistable nematic devices fall in a particular class ofnematics with significant technological interest forlow-power applications, primarily because of theirbistable operation. A surface-relief grating with agrating pitch of around a micrometer is the key ele-ment introducing surface bistability. Devices basedon monogratings are usually operating with LC dis-tortion in the zenithal plane �zenithal bistable nemat-ics� and exhibit a defect state and a defect-freestate.13 Azimuthal bistable nematic �ABN� devicescan also be made, if supporting surfaces with suitabledouble-periodicity surface-relief gratings are intro-duced.14,15 This latter class can potentially demon-strate improved optical viewing properties. Arigorous three-dimensional �3-D� optical wave prop-agation study with an appropriate numerical methodis required for devices based on such fine features.In this paper we introduce rigorous 3-D modeling ofoptical wave propagation in LC devices, with ABNsurface-relief gratings as the sample application.To the best of our knowledge this is the first modelingof true LC optical studies in three dimensions.

The proposed numerical scheme uses the FDTDmethod for the region of the device that involves 3-Dvariation in geometry and material properties,whereas it uses plane-wave expansions for the lowerand upper supporting layers, which are uniformalong both transverse directions. This hybrid ap-proach will greatly reduce the computational burdenand will allow for realistic device parameters to beconsidered in the model.

The structure of the paper is as follows. In Sec-tion 2 we provide a mathematical and numerical for-mulation for the optical wave study based on thecombined FDTD and plane-wave expansion method.This is then followed in Section 3 by a numericalsimulation in which we first review how the LC ori-entation profiles are calculated in three dimensionsusing a free-energy minimization technique,16 and wethen proceed with the optical wave simulations forrealistic device models. Detailed results are thenobtained for two representative grating geometries.The first one is a smooth bisinusoidal grating, and thesecond one corresponds to an array of square posts.The influence of a compensation layer in the device isthoroughly investigated in terms of its optical perfor-mance.

2. Mathematical and Numerical Formulation

A. Device Description and Computational Regions

A general layout for the class of bistable nematicdevices to be analyzed is shown in Fig. 1. A trans-missive device is modeled, but the proposed method isalso applicable to reflective devices with a small num-ber of modifications and is described in Subsection2.B. At least one of the supporting surfaces has a

relief-type grating with double periodicity along theaxes x and y. Pitch lengths �Px, Py� of the order of amicrometer are usually expected. The LC materialcontained between the two grating surfaces or be-tween the single grating and a top flat surface, sub-ject to the surface anchoring of the LC, will exhibit aperiodic deformation in the director �n� orientationwith significant variation along all three spatial di-rections:

n� x � Px, y � Py, z� � n� x, y, z�. (1)

The same will also be true for the corresponding op-tical dielectric tensor ε, which is periodic along x, yand varies with respect to all three spatial directions�x, y, z�:

ε� x � Px, y � Py, z� � ε� x, y, z�. (2)

As the optical tensor variation is on the optical wave-length scale, so we can rigorously calculate light-wave propagation inside the LC material and thegrating structure, a true 3-D treatment is necessary,without the introduction of any approximations.

The average LC layer thickness necessary for aswitchable electro-optic response of the device is usu-ally of the order of 2–4 �m for most available nematicmaterials, and the grating depth is usually compara-ble to the grating pitch, that is, around 1 �m or less.Therefore a combined LC and grating thickness of afew micrometers encompasses all the 3-D variation inoptical properties. This is in clear contradistinctionfrom the layers found above and below the combinedLC and grating structure, such as the substrate, su-perstrate, entrance polarizer, analyzer, and possiblysome compensation layers, which are all uniform iso-

Fig. 1. LC display geometry with a periodic surface-relief 3-Dmicrostructure.

1 September 2002 � Vol. 41, No. 25 � APPLIED OPTICS 5347

tropic or anisotropic slabs without any lateral �x, y�variation. A combined thickness of hundreds of mi-crometers is typical for the stack of layers at theentrance face of the device �we refer to this as thepolarizer stack�, and the same is also true for thestack of layers at the exit face �analyzer stack�. It isstraightforward to conclude that such excessive de-vice thickness along the z direction, equivalent tohundreds or thousands of optical wavelengths, willprevent a unified numerical modeling based on amethod that includes subwavelength spatial discreti-zation �such as the FDTD method�. An effective ap-proach is to split the problem space into threecomputational regions, designated as regions I, II,and III in Fig. 1 and then apply a suitable method ineach particular region.

Region II accommodates the LC material togetherwith the grating structure, and it extends up to alimited depth inside the substrate and the super-strate. The interfaces between regions I and II andregions II and III are purely fictitious �they do notcorrespond to any physical interface�, and for reasonsthat will be clear in the following paragraphs, theyhave to be placed inside the substrate and the super-strate, respectively. As region II exhibits a varia-tion of the optical dielectric tensor along all threespatial directions, light-wave propagation inside thisregion will be formulated in terms of Maxwell’s equa-tion in three dimensions, here written in the spectraldomain:

� � H � j�ε� x, y, z�E, (3a)

� � E � �j��0H. (3b)

A general solution of Eqs. �3a� and �3b� can be pro-vided by the FDTD method, without the introductionof any approximations apart from the numerical dis-cretization error. Concise details on the FDTD im-plementation are given in Subsection 2.B, and thereader can also consult the detailed relevant litera-ture.17

Regions I and III are stacks of uniform layers, andtherefore light propagation through them can be ef-fectively modeled in the spectral domain with one ofthe available and well-established methods for strat-ified media optics. For these particular regions, theBerreman method4 and the extended Jones method2,3

are perfectly suited for the task. In this paper weuse the extended Jones method to implement thelight-wave propagation through the polarizer stack�region I� and the analyzer stack �region III�. Cou-pling between the FDTD method and the extendedJones is performed along the fictitious interfacesABCD and EFGH, which are shown in Fig. 1.

The composite scheme to be developed for light-wave propagation studies inside the device depictedin Fig. 1 consists of four steps. In the first step anarbitrarily polarized wave in terms of the sinc and pincpolarization vectors, and propagating along the arbi-trary direction of incidence ��inc, �inc� �as emanatingfrom the semi-infinite lower free space�, is assumed to

hit the air–polarizer entrance face. The propagationof this plane wave is traced through region I by theapplication of the extended Jones method,3 until itreaches the interface plane ABCD. In the secondstep, the already determined expression for the planewave along the above interface is used as the excita-tion of the FDTD method. The main numerical bur-den of the composite scheme resides in this step, withthe FDTD numerical simulation continuing until thefields in region II reach their equilibrium or steady-state response. This will eventually provide us withphasor values for all field components along the in-terfaces ABCD and EFGH. As the problem pos-sesses double spatial periodicity of the opticaldielectric tensor, it follows that the various field com-ponents can be thought of as being composed fromdiscrete plane-wave spectra. The third step in-volves the decomposition of the forward-propagatingfield over the interface EFGH into a plane-wave spec-trum and then the tracing of each individual planewave through the layers of the analyzer stack �regionIII� by use of the extended Jones method. Comple-tion of this step yields the transmitted optical wave,which acquires the form of a discrete spectrum ofplane wave recorded in the semi-infinite upper spaceand traveling toward the z direction. The fourthstep is similar to the third step, and it involves thedecomposition of the backward-propagating field overthe interface ABCD into a plane-wave spectrum.Subsequently, every single plane wave of this latterspectrum is now passed through the layers of theentrance polarizer stack �region I�, which are nowpresented in reverse order, by the extended Jonesmethod. At the end of this step, the reflected opticalwave acquiring the form of a discrete spectrum ofplane waves recorded in the semi-infinite lowerspace, and propagating toward the �z direction, isdetermined.

In summary, because of the multidimensional pe-riodic variation of the LC optical tensor and the grat-ing structure, a single plane-wave illumination willresult in a forward- and a backward-propagatingspectrum of plane waves for the transmitted and re-flected fields, respectively. In Subsections 2.B and2.C we provide the necessary information to imple-ment the above composite scheme and discuss therelevant details.

B. Finite-Difference Time-Domain MethodImplementation Details

The FDTD method is ideally suited to the direct so-lution of Eqs. �3a� and �3b� concurrently in space andin time. Recent publications6–9 successfully intro-duced the FDTD method to a range of two-dimensional LC structures. The generalization tothe 3-D case closely follows the analysis outlined inRefs. 8 and 9. Because of the presence of anisotropy,it is more convenient to time march the dielectricdisplacement D and the magnetic field H instead oftime marching directly E and H. This will essen-tially hide the anisotropy from the time-stepping al-gorithm, and the latter will be identical to the one


used in the isotropic case17 with D instead of E. Fora complete time step, we need to time advance thevector D by discretizing Eq. �3a�, calculate vector Efrom D by inverting the optical tensor ε, and finallytime advance the vector H by discretizing Eq. �3b�,which requires knowledge of E. Time stepping iscontinued in a leapfrog scheme until the sinusoidalsteady-state response has been reached under plane-wave illumination conditions.

The FDTD computational space corresponds to re-gion II and it has to be subjected to two differenttypes of boundary condition: periodic and absorb-ing. Faces ADHE and BCGF are periodic walls forthe x direction, whereas faces ABFE and DCGH areperiodic walls along the y direction. Special careshould be taken when the oblique incidence case isimplemented in the FDTD method with periodicboundary conditions. The presence of double peri-odicity will naturally lead to a spectral phase shift offield components along x and y axes from period toperiod. The time-domain origin of the FDTDmethod imposes an inherent difficulty, as the spectralphase shift is equivalent to time advancing or retard-ing the solution, operations that can be extremelyinvolved or even impossible in an in situ algorithm.This difficulty is circumvented by use of dual-sourceconditions, a technique that is based on illuminationof the computational grid with two spatially identicalbut otherwise 90° out-of-phase plane waves.18

Faces ABCD and EFGH will provide space termina-tion along the z direction by imposing suitable ab-sorbing boundary conditions: In particular, theperfectly matched layer absorbing boundary condi-tions19 were used.

Region II includes, apart from the LC material andthe grating structure, a small number of nodes posi-tioned inside the substrate and the superstrate.This means that the FDTD solution will includeFabry–Perot-type effects �multiple reflections� gener-ated between all enclosed physical interfaces �i.e.,substrate and grating, grating and LC, indium tinoxide contacts�. As the perfectly matched layer ter-minations are in essence equivalent to the case ofinfinitely thick substrate and superstrate, no multi-ple reflections from the air–substrate and thesuperstrate–air interfaces are included. In practice,this is not expected to be restrictive, as the distancebetween these two interfaces will be much greaterthan the coherence length of many light sources20

because of the significant thickness of the substrateand superstrate in any realistic device.

The description so far has targeted devices work-ing in transmission. Devices working in reflectioncan also be analyzed by use of the above schemewith a number of modifications. A reflective devicewith an ideal �nonpenetrable� flat mirror and a sin-gle polarizer can be implemented in the FDTDmodel when the upper perfectly matched layer slabis removed and the tangential electric fields are setto zero. This corresponds to the limit of a perfectlyconducting metal � 3 ��. If finite conductivitiesare used, then more realistic mirrors with losses

can be modeled. In the reflective configuration,the transmitted plane-wave spectrum disappears,and the optical response of the device is simply thereflected plane-wave spectrum. The latter crossesthe layers of the polarizer stack in reverse orderbefore exiting to air, and this is accomplished by theextended Jones method as already outlined above.

C. Coupling of the Finite-Difference Time-DomainMethod to the Extended Jones Method

At the end of the FDTD simulation �referred to asstep two in Subsection 2.A� the forward-propagatingfield has been determined along the face EFGH, andthe backward-propagating field has been compiledalong ABCD. As the problem space is periodic alongthe x and y directions, plane-wave decompositioninto Floquet-type modes for the forward and back-ward fields is made possible.21 For the forwardplane-wave spectrum, decomposed into the s and ppolarization components, this results in

E� x, y, z� � �m 0

�

�n 0

�

�Emnssmn � Emn

ppmn�

� exp��j��m x � �n y � �mnz��, (4a)

�m � ksub sin��sub,inc�cos��sub,inc�

�2m�

Px, (4b)

�n � ksub sin��sub,inc�sin��sub,inc�

�2n�

Py, (4c)

�mn � �ksup2 � �m

2 � �n2�1�2. (4d)

In Eqs. �4a�–�4c� the notation sub refers to the sub-strate, whereas sup refers to the superstrate. Theplane-wave amplitudes Emn

s and Emnp are obtained

from the FDTD field values along the EFGH planewhen we implement the orthogonality relations of theFloquet modes. Every individual propagating planewave appearing in Eq. �4a� �those with indices m, nsatisfying �m

2 �n2 � ksup

2� will then be propagatedthrough the analyzer stack by use of the extendedJones method, completing the third step of Subsec-tion 2.A. This concludes the calculation of the totaltransmitted field at the exit face of the device.

Similar expressions to those of Eqs. �4� hold also forthe reflected field spectrum, whose plane-wave am-plitudes are obtained from the FDTD field valuesalong the ABCD plane. Propagating plane waves ofthis spectrum can be used in the fourth step of Sub-section 2.A to determine �by application of the ex-tended Jones method� the reflected field at theentrance face of the device.

3. Numerical Simulations

A. Director Orientation Profiles

In the context of numerical simulations, we addressdevices with a relief-type grating recorded only on


the lower supporting surface, leaving the upperplate flat. In particular, we investigate two differ-ent grating geometries: a smooth bisinusoidalgrating as shown in Fig. 2�a� and an array of squareposts as in Fig. 2�b�. For the bisinusoidal geome-try the grating boundary is represented by the an-alytic expression

z� x, y� �F4 ��2 � cos�2�x

Px� � cos�2�y

Py�� , (5)

with the factor F controlling the feature size. In thecase of the square-post array, the post height is fixedand equal to F with a cross-sectional boundary de-fined by

�x �P2�

6

� �y �P2�

6

� �P4�

6

; P � Px � Py. (6)

Equations �6� represent in essence a square shapewith slightly rounded corners and provide smoothtransition of the surface normal around the post. Inall the following calculations a common pitch along xand y directions was considered �Px Py 1.4 �m�in conjunction with feature sizes of F 300, 400, and500 nm.

We obtained LC director orientation profiles for theabove 3-D structures using a free-energy minimiza-

tion technique.16 This minimizes the sums of thevolume integrals of the bulk distortion energy

f� �12

K11�� n�2 �12

K22�n � � � n�2

�12

K33�n � �� n��2 (7)

and the surface energy

fs �12

K11

l1�n � ns1�

2�� p1� �12

K11

l2�n � ns2�

2�� p2�,

(8)

where l is the anchoring extrapolation length, n, isthe surface normal, and the � functions constrain thecontribution along the surface profiles �p 0�. Weperformed simulations on a 50-nm cubic grid, assum-ing a nematic LC material with equal elastic con-stants �K11 K22 K33�. The LC layer thicknesswas 2 �m when measured from the grating troughs tothe upper flat boundary. Strong planar alignmentwas used on both surfaces �l1 l2 0.01 �m�. Forthe square-post grating two stable states are foundcorresponding to an ABN structure.14,15 One state isaligned mainly along the line y x with the otheraligned mainly along line y �x. In the bisinusoi-dal grating case many stable states exist, as any az-imuthal direction across the surface has the sameenergy. However, director profiles mainly alignedalong the lines y x and y �x were determined toprovide meaningful comparisons with the corre-sponding bistable states of the square-post array.

Figure 3 illustrates the director orientation profilesobtained for both LC structures under consideration,with a typical feature size of F 400 nm. For clar-ity, cross sections along planes of constant z and con-stant y are shown. These discrete profiles are thenused to obtain the corresponding discrete dielectrictensor profiles.

B. Optical Simulations

Before proceeding to the optical simulations of LCmicrostructures, we assessed the FDTD code for thecase of surface-relief isotropic dielectric gratings.The FDTD prediction was compared with results ob-tained by well-established methods for gratings, suchas the differential method and the coupled-wavemethod, which are available in the literature.22–24

Tests were carried out for various grating depths,pitches, and grating profiles. In particular, the caseof a dielectric grating with triangular profile anddepth ranging from 0.1� to � was analyzed, and theFDTD results were consistent with the differentialmethod results presented in Ref. 22 within a relativeerror margin of 0.47%. A lamellar dielectric gratingwith cylindrical grooves and equal periods along xand y directions was also analyzed, and the FDTDprediction was consistent with the results of an im-proved coupled-wave formulation published in Ref.23 within a relative error of 0.86%. Finally, the dif-

Fig. 2. Grating profiles: �a� bisinusoidal, �b� square post.


fraction efficiencies of a two-dimensional surface-relief grating with a square-wave cross section alongthe x and y axes, supporting several diffraction or-ders, was examined. The maximum deviation of theFDTD prediction from published data by the coupled-wave method23,24 was found to be 1.6%.

Returning to the ABN devices, Fig. 4 shows a de-tailed cross section of all layers necessary for thefunctionality of an actual device. Our optical simu-lations are based on the model of Fig. 4 combinedwith the LC dielectric tensor profiles that were cal-culated above for the bisinusoidal and square-post

grating structures. �For the results presented here,we calculated the profiles using a nematic LC ordi-nary refractive index n0 1.52, together with a bi-refringence of �n 0.225.� We performed opticalsimulations for a free-space wavelength of � 650nm using a 25-nm cubic grid. This necessitates thatwe interpolate the dielectric tensor data, which wereobtained on a 50-nm grid over the 25-nm FDTD gridemployed. Polarizers are assumed to be ideal, lead-ing to perfect extinction of the extraordinary waveand zero attenuation of the ordinary one.

Estimating the phase retardation introduced by

Fig. 3. LC director profiles for gratings with a 400-nm feature size: �a� square-post sliced parallel to the xOy plane, 300 nm from thegrating bottom; �b� square-post sliced along the xOz plane; �c� bisinusoidal sliced parallel to the xOy plane at 300 nm; �d� bisinusoidal slicedalong the xOz plane. The shading in the images highlights different directions; it has no other significance. The director density in �b�and �d� was reduced by a factor of 2 compared with �a� and �c� to improve legibility.


the LC layer illustrates the basic understanding ofthe idealized electro-optic response for these de-vices. In the absence of the grating structure andthe associated distortion of the director orientation

in the vicinity of the grating, one can estimate thatthe device will function approximately as a 3��2uniform plate �more precisely 1.4�� with the opticaxis oriented along � 45° �when LC orientation is

Fig. 4. Detailed device cross section showing the modeled layerswith the corresponding material parameters. ITO, indium tinoxide. Fig. 5. Zero-order and total transmitted �TR� optical power ver-

sus feature size. BS, bisinusoidal; SQ, square post.

Fig. 6. Transmitted optical power versus compensation layer thickness for bisinusoidal grating devices: �a� zero-order power of darkstate, �b� total power of dark state, �c� zero-order power of bright state, �d� total power of bright state.


mainly along the line y x� or � �45° �whenalong the line y �x�, depending on the bistablestate. Therefore it seems reasonable to combinethe device with a compensation layer close to aquarter-wave plate ��4 plate� oriented at 45°. Ar-ranging the entrance polarizer at 0° and the exitanalyzer at 90° will result in the � 45° stateappearing dark, as the total retardation will ap-proach 2�, whereas the � �45° will appear brightowing to a total retardation close to �.

Our optical simulations are now used to thoroughlyinvestigate a more realistic model of the optical re-sponse for the devices of Figs. 2–4. Departure fromthe idealized response outlined above occurs for anumber of reasons. First, the presence of the grat-ing will result in diffraction, and therefore at the exitside of the device light will emerge not only along thedirection of incidence but also along discrete direc-tions as predicted for the various diffracted orders.Second, the grating structure will distort the LC di-rector orientation close to its surface, and thus the

phase retardation introduced will strongly depend onthe grating geometry and the extent and variation ofLC orientation in the distortion zones. This meansthat the thickness of the compensation layer must betailored for each particular grating geometry so wecan observe optimum performance. Moreover, it isexpected that the compensation layer will not act, ingeneral, in the same way on the directly transmitted�zero-order� light and on the diffracted �higher-order�light. For example, a particular compensation layermight completely extinguish the zero-order light inthe dark state but still some light may leak throughbecause of higher orders, thus reducing the contrastratio in particular directions. Our combined FDTD–plane-wave expansion method is capable of address-ing the above points and quantifies such effects, as wedemonstrate below.

In Fig. 5 the optical power carried by the zero-ordermode as well as the total transmitted power are plot-ted versus the feature size F for both grating geom-etries. We determined the results at normal

Fig. 7. Transmitted optical power versus compensation layer thickness for square-post grating devices: �a� zero-order power of darkstate, �b� total power of dark state, �c� zero-order power of bright state, �d� total power of bright state.


incidence for the director profiles oriented mainlyalong � 45° by removing the exit analyzer and thecompensation layer. Director profiles orientedalong � �45° will yield identical curves. It isevident that the square-post structure is morestrongly diffracting than the bisinusoidal one for allfeature sizes, as can be clearly seen by the zero-orderpower variation. This will lead into an enhance-ment of the higher diffracted orders at the expense ofthe zero-order mode. The total transmitted power�zero and higher orders� is similar in both cases andis mainly limited by the reflections at the entranceand exit faces of the device. The optical response ofthe square-post device is thus expected to be moresusceptible to the influence of diffraction.

Figure 6 shows the zero-order and total transmit-ted power versus a compensation layer thickness forthe bisinusoidal grating structures at normal inci-dence. The compensation layer considered is a uni-form anisotropic slab with the optic axis alignedalong 45°, ordinary refractive index n0 1.52, andbirefringence �n 0.1. A slab thickness of 1.625�m is equivalent to a quarter-wave plate. Figures6�a� and 6�b� correspond to the dark state of � 45°,and Figs. 6�c� and 6�d� correspond to the bright statealigned at � �45°. It can be seen that the zero-order light can be completely extinguished for a suit-able compensation layer thickness, and theparticular thickness required strongly depends onthe feature size �Fig. 6�a��. Gratings with increasedfeature size require a thicker compensation layer tocancel out the zero-order light. On the other hand,the total transmitted optical power �Fig. 6�b�� exhib-its a minimum, which nearly coincides with the pointof the zero-order extinction. At this compensationlayer thickness, light is carried only by the higherdiffracted orders, and for the bisinusoidal geometry,this can be in the range of 0.7–1.7% of incident light,depending on the grating feature size �Fig. 6�b��.Figures 6�c� and 6�d� record the optical power associ-ated with the bright state for the same range of com-pensation layer thickness. It is evident that, at thepoint of maximum contrast, which coincides with thepoint of lowest dark-state transmission that can beseen in Figs. 6�a� and 6�b�, the corresponding bright-state transmittance is not the highest possible.Note that the contrast defined from the zero-orderlight is effectively infinite; however, the contrast isreduced to 25:1 if the light carried in the diffractedorders is introduced.

Figure 7 shows the equivalent calculations of Fig. 6for the square-post grating structure, and similarqualitative comments to the above apply. Extinc-tion of the zero-order light for the dark state is ac-complished for a compensation layer thicknessdependent on the feature size. However, a thinnerlayer is needed for the same feature size in compar-ison with the bisinusoidal geometry, as can be seen inFig. 7�a� in comparison with Fig. 6�a�. The totaltransmitted power �including higher-order modes�goes through a global minimum in the vicinity of thezero-order null, but now the remaining light power in

the higher-order modes is in the region of 2.2–5.8%�Fig. 7�b��, still strongly dependent on the featuresize. These much higher values compared with theother geometry are clearly expected, as Fig. 5 hasalready suggested that this particular geometry is farmore diffractive. The optical power associated withthe bright state is shown in Figs. 7�c� and 7�d�.Again the contrast between the switched states of thedevice is seriously reduced by the diffracted light, inthis case to around 15:1.

Light leakage at the optimum compensation layerthickness that is due to the higher diffracted orders isclearly demonstrated in Fig. 8�a� for the 400-nm fea-ture bisinusoidal grating dark state illuminated atnormal incidence. Figures 8�a� and 8�b� show thenormalized transmitted optical intensity, as recordedat the exit face of this device over the extent of a unitperiod Px � Py. The compensation layer thickness isequal to 2.4 �m. The corresponding bright state,which is obtained from the � �45° aligned profile,

Fig. 8. Normalized transmitted intensity recorded at the exit faceof a device based on the bisinusoidal grating with a 400-nm featuresize. The compensation layer was set to its optimum thickness�2.4 �m�: �a� dark state, �b� bright state.


is shown in Fig. 8�b�. Figure 9 provides the sameinformation for the 400-nm feature square-post grat-ing, when combined with the optimum compensationlayer thickness, in this case 2.1 �m. Consistentwith the description of Figs. 6 and 7, it can be seenthat the light leaking in the dark state because of thediffractive effects is particularly strong for thesquare-post geometry and will lead to a deteriorationof the contrast ratio along the directions of propaga-tion for the higher diffracted orders. It should benoted that all transmitted intensity patterns shownin Figs. 8 and 9 do not appear to be symmetric withrespect to planes x 0, y 0, x y, and x �y. Inthe case of Fig. 8, this is attributed to a small mis-alignment of the grating peak with respect to thecenter of the unit period and to quantization errors.However, in the case of the square-post array of Fig.9, asymmetries have a physical origin and they areattributed to the off-diagonal dielectric tensor ele-ments, as the latter are breaking the symmetry with

respect to the above planes because of the directorvariation close to the square-post corners.

4. Conclusion

A composite scheme capable of tracing light-wavepropagation in 3-D LC microstructures, such as thosefound in bistable nematic devices has been success-fully demonstrated. The scheme is based on thecombination of the FDTD method with a plane-waveexpansion and can effectively model devices withdouble periodicity on the micrometer scale, combinedwith various accompanying layers far exceeding theoptical wavelength scale. Azimuthal bistable de-vices based on a square-post array and devices em-ploying a bisinusoidal grating have been analyzed forvarious feature sizes. It was found that the appro-priate compensation layer thickness is sensitive tothe grating geometry and the feature size. Althoughzero-order light extinction is possible for a suitablecompensation layer, light leaks through the higherdiffraction orders degrading the dark state. Ourscheme enables direct quantification of such issuesfor realistic device models. This would not be pos-sible with more conventional stratified media approx-imations.

E. Kriezis acknowledges funding from the Hewlett-Packard Laboratories �Bristol� and the Engineeringand Physical Sciences Council of the UK.

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three-dimensional simulations of light propagation in periodic liquid-crystal microstructures

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